Dynamic Response of Double Elastic Cantilever Beam...

18
Research Article Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable Uniformly Distributed Load Wei He 1 and Yanjing Wei 2,3 College of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao , China Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou , China University of Chinese Academy of Sciences, Beijing , China Correspondence should be addressed to Wei He; [email protected] Received 3 January 2019; Revised 14 February 2019; Accepted 18 February 2019; Published 13 March 2019 Academic Editor: Saeed Eſtekhar Azam Copyright © 2019 Wei He and Yanjing Wei. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Based on the double-layer elastic foundation beam theory, the rails and the wall plates of the electromagnetic launching device are modeled as a double-layer elastic cantilever foundation beam. Aſter establishing the kinetic differential equation and setting the boundary condition of the cantilever beam, the displacement solution of double-layer elastic cantilever foundation beam under no load condition is obtained. Applying the Heaviside Function, the deflection equation of the upper and lower beam, the expression of the bending moment, and the stress are obtained. In the case of given motion parameters and structural parameters, the analytical solutions of the rail and the wall plate are calculated. e ANSYS numerical analysis is carried out under the same condition and the results of both solutions are in good agreement. e results can provide theoretical basis for the design of strength and stiffness of the electromagnetic launch device. 1. Introduction Since the beginning of the last century, the electromagnetic railguns have seen considerable progress in development. As the electromagnetic railgun has unusual advantages, many countries attach great importance to it. Recently, many experts and researchers have used different theoretical mod- els to study the stiffness, strength, and vibration problem of the electromagnetic launcher elements [1, 2]. Based on classic material mechanical theory in which the electromagnetic launching device is modeled as the elastic foundation beam [3–5], Tzeng investigated the strength and the deformation of launching device shell, deducing the control equation solution, and simulated the strain and stress field in the track under the magnetic field pressure through ANSYS soſtware. Johnson et al. also simplified the electromagnetic launching device as elastic foundation beam [1], using material mechan- ical method for calculation, and preliminarily analyzed the passing characteristics of stress wave for electromagnetic track under electromagnetic pressure [6–8]; Jin et al. studied dynamic response of the rail for the electromagnetic railguns with the rail being modeled as an Euler–Bernoulli beam and used the Matlab soſtware to carry out the numeri- cal simulation [9], and Che et al. established a dynamic model of rail of single-layer beam based on the principle of Bernoulli–Euler, having discussed the influence of different constraints and pretightening force on vibration and stiffness in railgun [10]. Cao et al. used the Winkler beam to analyze the dynamic deformation of the rails [11], and Young-Hyun Lee et al. used the Timoshenko beam model to calculate the dynamic response of electromagnetic launcher with a C- shaped armature [12]. Tian et al. simplified the composite rail as an elastic foundation beam and analyzed its mechanical characteristics [13]. Hassanabadi, Attari, and Nikkhoo et al. made several investigations on the dynamics response of thin/thick subjected to moving mass by semi-analytical approaches [14–17]. However, only simplifying the electromagnetic launcher rail as a single-layer elastic foundation to analyze bending moment, stress, and vibration is far from the actual condition. Our group discussed the double-layer elastic foundation beam model in the condition of electromagnetic track beam Hindawi Mathematical Problems in Engineering Volume 2019, Article ID 2657271, 17 pages https://doi.org/10.1155/2019/2657271

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Research ArticleDynamic Response of Double Elastic Cantilever Beam Attributedto Variable Uniformly Distributed Load

Wei He 1 and YanjingWei 23

1College of Civil Engineering and Mechanics Yanshan University Qinhuangdao 066004 China2Northwest Institute of Eco-Environment and Resources Chinese Academy of Sciences Lanzhou 730000 China3University of Chinese Academy of Sciences Beijing 100049 China

Correspondence should be addressed to Wei He heweiysueducn

Received 3 January 2019 Revised 14 February 2019 Accepted 18 February 2019 Published 13 March 2019

Academic Editor Saeed Eftekhar Azam

Copyright copy 2019 Wei He and Yanjing Wei This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Based on the double-layer elastic foundation beam theory the rails and the wall plates of the electromagnetic launching device aremodeled as a double-layer elastic cantilever foundation beam After establishing the kinetic differential equation and setting theboundary condition of the cantilever beam the displacement solution of double-layer elastic cantilever foundation beam under noload condition is obtained Applying theHeaviside Function the deflection equation of the upper and lower beam the expression ofthe bending moment and the stress are obtained In the case of given motion parameters and structural parameters the analyticalsolutions of the rail and the wall plate are calculated The ANSYS numerical analysis is carried out under the same condition andthe results of both solutions are in good agreement The results can provide theoretical basis for the design of strength and stiffnessof the electromagnetic launch device

1 Introduction

Since the beginning of the last century the electromagneticrailguns have seen considerable progress in development Asthe electromagnetic railgun has unusual advantages manycountries attach great importance to it Recently manyexperts and researchers have used different theoretical mod-els to study the stiffness strength and vibration problem ofthe electromagnetic launcher elements [1 2] Based on classicmaterial mechanical theory in which the electromagneticlaunching device is modeled as the elastic foundation beam[3ndash5] Tzeng investigated the strength and the deformationof launching device shell deducing the control equationsolution and simulated the strain and stress field in the trackunder the magnetic field pressure through ANSYS softwareJohnson et al also simplified the electromagnetic launchingdevice as elastic foundation beam [1] usingmaterial mechan-ical method for calculation and preliminarily analyzed thepassing characteristics of stress wave for electromagnetictrack under electromagnetic pressure [6ndash8] Jin et al studieddynamic response of the rail for the electromagnetic railguns

with the rail being modeled as an EulerndashBernoulli beamand used the Matlab software to carry out the numeri-cal simulation [9] and Che et al established a dynamicmodel of rail of single-layer beam based on the principle ofBernoullindashEuler having discussed the influence of differentconstraints and pretightening force on vibration and stiffnessin railgun [10] Cao et al used the Winkler beam to analyzethe dynamic deformation of the rails [11] and Young-HyunLee et al used the Timoshenko beam model to calculatethe dynamic response of electromagnetic launcher with a C-shaped armature [12] Tian et al simplified the composite railas an elastic foundation beam and analyzed its mechanicalcharacteristics [13] Hassanabadi Attari and Nikkhoo etal made several investigations on the dynamics responseof thinthick subjected to moving mass by semi-analyticalapproaches [14ndash17]

However only simplifying the electromagnetic launcherrail as a single-layer elastic foundation to analyze bendingmoment stress and vibration is far from the actual conditionOur group discussed the double-layer elastic foundationbeam model in the condition of electromagnetic track beam

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2657271 17 pageshttpsdoiorg10115520192657271

2 Mathematical Problems in Engineering

Panel

Rail

Inputcurrent

Outputcurrent

Magneticthrust

ArmatureX

(a) Model of railgun

Rail Insulation(Elastic support)

Wall plate

Cover plate(Elastic support)

qq

(b) Section of railgun

Figure 1 Schematic of railgun

Figure 2 Exterior of rectangular electromagnetic railgun

as simply supported [18ndash21] despite the factual model thatis close to cantilever beam Hence simplifying electromag-netic launching device under force as a double-layer elasticcantilever beam will meet the requirement of actual workingcondition well

This paper takes rectangular-caliber electromagneticlauncher as research object and the cover plates and insu-lation which provide support to the rails are modeled aselastics supports to the rails and wall plates (shown in Figures1 and 2) The insulation between the guide rails and the wallplates is equivalent to a layer of elastic foundation and thewall plates are usually connected with the cover plates bybolts Cover plates and bolts in work state of deformationare equivalent to another layer elastic foundation for the wallplate Calculating the deformation and stress state of therails and the wall plates should be carried out in launchingprocess [18ndash21] During launching process when electriccurrent is guided into rails electric armature is subjected topowerful electromagnetic force and due to import of thecurrent two guide rails produce mutual repulsion 119902 The

repulsion 119902 is seen as uniformly distributed load so due tothemovement of armature the whole device can be simplifiedas a mechanics scenario in which the model of double-layerelastic cantilever beam is exposed to variable and uniformlydistributed electromagnetic force 119902

2 Mechanical Analysis

21 Mechanics Model and Differential Equations The launchrails and wall plates are simplified as a double-layer cantileverbeam system on Winkler elastic foundation as shown inFigure 3 where the rail is viewed as the upper beam andthe wall plate is viewed as the lower beam 119897 representsthe length of the beam and 119897

1marks the location of the

moving armature 119902 is the electromagnetic force applied tothe rails 119907 is the velocity of the armature 119864

11198681and 119864

21198682are

the bending stiffness of upper and lower beam respectivelyElastic constant between the upper and lower beam is 1198881 andelastic constant between the lower beam and the foundationis 1198882 119907 is the velocity of armature

Mathematical Problems in Engineering 3

l1l

q

c1

c2

E1 I1

E2 I2

v

Figure 3 Mechanical model of double-layer cantilever beam

The dynamic equations of upper and lower beam are

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 119891 (119909 119905)

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + 11988821199082 = minus1198881 (1199082 minus 1199081)

(1)

where 1199081and 119908

2represent the deflection of the upper and

lower beam respectively 1198981 = 12058811198781 and 1198982 = 12058821198782are the mass per unit length of upper and lower beamrespectively 1205881 and 1205882 are the mass density of upper andlower beam respectively 1198781 and 1198782 are the cross-sectionalarea of upper and lower beam respectively 119905 is the timewith armaturersquosmove f (x t) is the uniformly distributed loadelectromagnetic force per unit length applied to the rail

22 Homogeneous Solution Based on interactions betweenupper and lower beam in the case of no load the dynamicequations of upper and lower beam are

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 0 (2)

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + (1198881 + 1198882) 1199082 minus 11988811199081 = 0 (3)

From (2) we obtain

1199082 = 119864111986811198881 120597411990811205971199094 + 11989811198881 120597

211990811205971199052 + 1199081 (4)

Applying (4) to (3) and simplifying the solution we obtain

1198641119868111986421198682 120597811990811205971199098 + (119864211986821198981 + 119864111986811198982) 1205976119908112059711990941205971199052

+ [119888111986421198682 + (c1 + 1198882) 11986411198681] 120597411990811205971199094 + 11989811198982 120597411990811205971199054

+ [11988811198982 + (1198881 + 1198882)1198981] 120597211990811205971199052 + 119888111988821199081 = 0(5)

According to the displacement forms of the beam we assumethe homogeneous solution of 1199081 and 1199082 respectively as1199081 = infinsum

119894=1

119883119894 (119860 119894 sin119875119894119905 + 119861119894 cos119875119894119905) = infinsum119894=1

119883119894119884119894 (6)

1199082 = infinsum119894=1

[119864111986811198881 1205974119883119894120597119909 + (1 minus 1198981119875

21198941198881 )119883119894]119884119894 =

infinsum119894=1

119885119894119884119894 (7)

with 119875119894 as circular frequency Yi = Ai sin Pit + Bi cos Pit and119885119894 = (119864111986811198881)(1205974119883119894120597119909) + (1 minus 11989811198752119894 1198881)119883119894Applying (6) to (5) we obtain

120572119894 12059781198831198941205971199098 + 120573119894 12059741198831198941205971199094 + 120574119894119883119894 = 0 (8)

where 120572119894 = 1198641119868111986421198682 120573119894 = 119888111986421198682+(1198881+1198882)11986411198681minus1198752119894 (119864211986821198981+119864111986811198982) and 120574119894 = 119898111989821198754119894 + 11988811198882 minus 1198752119894 [11988811198982 + (1198881 + 1198882)1198981]The solution of (8) can be expressed as

119883119894 = 1198621 cos119872119894119909 + 1198622 sin119872119894119909 + 1198623ch119872119894119909+ 1198624sh119872119894119909 + 1198625 cos119873119894119909 + 1198626 sin119873119894119909+ 1198627ch119873119894119909 + 1198628sh119873119894119909

(9)

where C1simC8 is undetermined coefficients and the coeffi-cients

119872119894 = (minus120573119894 + (1205732119894 minus 4120572119894120574119894)12

2120572119894 )14

and 119873119894 = (minus120573119894 minus (1205732119894 minus 4120572119894120574119894)12

2120572119894 )14

(10)

Applying (9) to (7) we can obtain from 1199082119885119894 = 1198621119869119894 cos119872119894119909 + 1198622119869119894 sin119872119894119909 + 1198623119869119894ch119872119894119909

+ 1198624119869119894sh119872119894119909 + 1198625119870119894 cos119873119894119909 + 1198626119870119894 sin119873119894119909+ 1198627119870119894ch119873119894119909 + 1198628119870119894sh119873119894119909

(11)

4 Mathematical Problems in Engineering

with 119869119894 = (119864111986811198721198944 + 1198881 minus 11989811198751198942)1198881 and119870119894 = (119864111986811198731198944 + 1198881 minus11989811198751198942)1198881Accordingly we obtain the bendingmoment of upper and

lower beam as follows

119872119906119901119901119890119903 = 11986411198681 120597211990811205972119909 = 11986411198681infinsum119894=1

12059721198831198941205971199092 119884119894 = [infinsum119894=1

11986411198681sdot (minus11986211198722119894 cos119872119894119909 minus 11986221198722119894 sin119872119894119909+ 11986231198722119894 ch119872119894119909 + 11986241198722119894 sh119872119894119909 minus 11986251198732119894 cos119873119894119909minus 11986261198732119894 sin119873119894119909 + 11986271198732119894 ch119873119894119909 + 11986281198732119894 sh119873119894119909)]sdot (119860 119894 sin119875119894119905 + 119861119894 cos119875119894119905)

(12)

119872119897119900119908119890119903 = 11986421198682 120597211990821205972119909 = 11986421198682infinsum119894=1

12059721198851198941205971199092 119884119894 = [infinsum119894=1

11986421198682sdot (minus11986211198691198941198722119894 cos119872119894119909 minus 11986221198691198941198722119894 sin119872119894119909+ 11986231198691198941198722119894 ch119872119894119909 + 11986241198691198941198722119894 sh119872119894119909minus 11986251198701198941198732119894 cos119873119894119909 minus 11986261198701198941198732119894 sin119873119894119909+ 11986271198701198941198732119894 ch119873119894119909 + 11986281198701198941198732119894 sh119873119894119909)] (119860 119894 sin119875119894119905+ 119861119894 cos119875119894119905)

(13)

The rotation angles of upper and lower beam are

1205791 = 11986411198681 1205971199081120597119909 = 11986411198681infinsum119894=1

120597119883119894120597119909 119884119894 = [infinsum119894=1

11986411198681sdot (minus1198621119872119894 sin119872119894119909 + 1198622119872119894 cos119872119894119909 + 1198623119872119894sh119872119894119909+ 1198624119872119894ch119872119894119909 minus 1198625119873119894 sin119873119894119909 + 1198626119873119894 cos119873119894119909+ 1198627119873119894sh119873119894119909 + 1198628119873119894ch119873119894119909)]

(14)

1205792 = 11986421198682 1205971199082120597119909 = 11986421198682infinsum119894=1

120597119885119894120597119909 119884119894 = [infinsum119894=1

11986421198682sdot (minus1198621119869119894119872119894 sin119872119894119909 + 1198622119869119894119872119894 cos119872119894119909+ 1198623119869119894119872119894sh119872119894119909 + 1198624119869119894119872119894ch119872119894119909minus 1198625119870119894119873119894 sin119873119894119909 + 1198626119870119894119873119894 cos119873119894119909+ 1198627119870119894119873119894sh119873119894119909 + 1198628119870119894119873119894ch119873119894119909)]

(15)

The shear forces of upper and lower beam are

1198761 = 11986411198681 120597311990811205971199093 = 11986411198681infinsum119894=1

12059731198831198941205971199093 119884119894= infinsum119894=1

(11986211198723119894 sin119872119894119909 minus 11986221198723119894 cos119872119894119909+ 11986231198723119894 sh119872119894119909 + 11986241198723119894 ch119872119894119909 + 11986251198733119894 sin119873119894119909minus 11986261198733119894 cos119873119894119909 + 11986271198733119894 sh119873119894119909 + 11986281198733119894 ch119873119894119909)

(16)

1198762 = 11986421198682 120597311990821205971199093 = 11986421198682infinsum119894=1

12059731198851198941205971199093 119884119894 =infinsum119894=1

11986421198682 (11986211198691198941198723119894sdot sin119872119894119909 minus 11986221198691198941198723119894 cos119872119894119909 + 11986231198691198941198723119894 sh119872119894119909+ 11986241198691198941198723119894 ch119872119894119909 + 11986251198701198941198733119894 sin119873119894119909 minus 11986261198701198941198733119894sdot cos119873119894119909 + 11986271198701198941198733119894 sh119873119894119909 + 11986281198701198941198733119894 ch119873119894119909)

(17)

where the constants 1198621sim1198628 can be determined from bound-ary conditions of the rail According to the double-layerbeam model where one end of the rail is constrained andthe another is under condition of freedom we can obtain theboundary conditions of both ends as follows

11990811003816100381610038161003816119909=0 = 012057911003816100381610038161003816119909=0 = 011990821003816100381610038161003816119909=0 = 012057921003816100381610038161003816119909=0 = 011987211003816100381610038161003816119909=119897 = 011987611003816100381610038161003816119909=119897 = 011987221003816100381610038161003816119909=119897 = 011987621003816100381610038161003816119909=119897 = 0

(18)

Substituting (6) (7) (14) and (15) into (18) from the leftboundary conditions we can obtain

1198621 + 1198623 + 1198625 + 1198627 = 01198622119872119894 + 1198624119872119894 + 1198626119873119894 + 1198628119873119894 = 01198621119869119894 + 1198623119869119894 + 1198625119870119894 + 1198627119870119894 = 0

1198622119869119894119872119894 + 1198624119869119894119872119894 + 1198626119870119894119873119894 + 1198628119870119894119873119894 = 0(19)

Substituting (12) (13) (16) and (17) into (18) from the rightboundary conditions we can obtain

minus 11986211198722119894 cos119872119894119897 minus 11986221198722119894 sin119872119894119897 + 11986231198722119894 ch119872119894119897+ 11986241198722119894 sh119872119894119897 minus 11986251198732119894 cos119873119894119897 minus 11986261198732119894 sin119873119894119897+ 11986271198732119894 ch119873119894119897 + 11986281198732119894 sh119873119894119897 = 0

Mathematical Problems in Engineering 5

11986211198723119894 sin119872119894119897 minus 11986221198723119894 cos119872119894119897 + 11986231198723119894 sh119872119894119897minus 11986241198723119894 ch119872119894119897 + 11986251198733119894 sin119873119894119897 minus 11986261198733119894 cos119873119894119897+ 11986271198733119894 sh119873119894119897 + 11986281198733119894 ch119873119894119897 = 0

minus 11986211198691198941198722119894 cos119872119894119897 minus 11986221198691198941198722119894 sin119872119894119897 + 11986231198691198941198722119894 ch119872119894119897+ 11986241198691198941198722119894 sh119872119894119897 minus 11986251198701198941198732119894 cos119873119894119897minus 11986261198701198941198732119894 sin119873119894119897 + 11986271198701198941198732119894 ch119873119894119897+ 11986281198701198941198732119894 sh119873119894119897 = 0

11986211198691198941198723119894 sin119872119894119897 minus 11986221198691198941198723119894 cos119872119894119897 + 11986231198691198941198723119894 sh119872119894119897minus 11986241198691198941198691198941198723119894 ch119872119894119897 + 11986251198701198941198733119894 sin119873119894119897minus 11986261198701198941198733119894 cos119873119894119897 + 11986271198701198941198733119894 sh119873119894119897+ 11986281198701198941198733119894 ch119873119894119897 = 0

(20)

From (19) we obtain 1198621 = minus11986231198622 = minus11986241198625 = minus11986271198626 = minus1198628

(21)

From (20) we obtainminus1198625 cos119873119894119897 minus 1198626 sin119873119894119897 + 1198627ch119873119894119897 + 1198628sh119873119894119897 = 0minus1198621 cos119872119894119897 minus 1198622 sin119872119894119897 + 1198623ch119872119894119897 + 1198624sh119872119894119897 = 01198625 sin119873119894119897 minus 1198626 cos119873119894119897 + 1198627sh119873119894119897 + 1198628ch119873119894119897 = 01198621 sin119872119894119897 minus 1198622 cos119872119894119897 + 1198623sh119872119894119897 + 1198624ch119872119894119897 = 0

(22)

Applying (21) into (22) we obtain

1198627 (cos119873119894119897 + ch119873119894119897) + 1198628 (sin119873119894119897 + sh119873119894119897) = 01198623 (cos119872119894119897 + ch119872119894119897) + 1198624 (sin119872119894119897 + sh119872119894119897) = 01198627 (minus sin119873119894119897 + sh119873119894119897) + C8 (cos119873119894119897 + ch119873119894119897) = 01198623 (minus sin119872119894119897 + sh119872119894119897) + C4 (cos119872119894119897 + ch119872119894119897) = 0

(23)

From (23) we obtain1198627 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198628 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198623 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198624 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 0

(24)

From (24) we obtainch119873119894119897 cos119873119894119897 + 1 = 0ch119872119894119897 cos119872119894119897 + 1 = 0 (25)

Assume C3=1 C7=1 with1198621 = 1198625 = minus11198624 = minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 1198622 = sh119872119894119897 minus sin119872119894119897

ch119872119894119897 + cos119872119894119897 1198626 = sh119873119894119897 minus sin119873119894119897

ch119873119894119897 + cos119873119894119897 1198628 = minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897

(26)

Accordingly the mode function for upper and lower beamcan be obtained as follows119883119894 = minus cos119872119894119909 + ch119872119894119909

minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 (sh119872119894119909 minus sin119872119894119909)minus cos119873119894119909 + ch119873119894119909minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897 (sh119873119894119909 minus sin119873119894119909)

(27)

119885119894 = minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 119869119894 (sh119872119894119909 minus sin119872119894119909)minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897119870119894 (sh119873119894119909 minus sin119873119894119909)

(28)

Then 1199081 and 1199082 can be expressed as

1199081 = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] (119860 119894sdot sin119875119894119905 + 119861119894 sin119875119894119905)

(29)

1199082 = infinsum119894=1

[minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)] (119860 119894 sin119875119894119905 + 119861119894 sin119875119894119905)

(30)

with 120589 = (sh119872119894119897minussin119872119894119897)(ch119872119894119897+cos119872119894119897) and 120585 = (sh119873119894119897minussin119873119894119897)(ch119873119894119897 + cos119873119894119897)23 Particular Solution of Upper Beam Assume f (x t) as theforce imposed on upper beam then substituting it into (2)and we obtain

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 119891 (119909 119905) (31)

6 Mathematical Problems in Engineering

Make 1199081 = suminfin119894=1119883119894(119909)119884119894(119905) and then we obtain

infinsum119894=1

[11986411198681119883119894(4)119884119894 + 119898111988410158401015840119894 119883119894 + 1198881 (119883119894119884119894 minus 1119869119883119894119884119894)]= 119891 (119909 119905)

(32)

Make (32) multiplied by 119883119895(119909) and integrate it in thewhole beam considering (27) and the orthogonality of modefunction of upper beam and then obtaining

Y10158401015840119894 + 1198752119894 119884119894 = 119876119894 (119905) (33)

where 119876119894(119905) = int1198970 119891(119909 119905)119883119894(119909)119889119909 named i-order generalizedloads The general solution of (33) is

Y119894 (119905) = 1119875119894 int119905

0119876119894 (120591) sin119875119894 (119905 minus 120591) d120591 (34)

Solving (34) then we can obtain the dynamic response ofupper beam under load as follows

1199081 (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minuscos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] int119905

0119891 (119909 119905)

sdot sin119875119894 (119905 minus 120591) 119889120591 119889119909(35)

24 Particular Solution of Lower Beam For lower beamthe load imposed on it is the pressure from upper beamAccording to (3) we obtain

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + (1198881 + 1198882) 1199082 minus 11988811199081 = 119891 (119909 119905) (36)

where 11988811199081 determined by (35) is the load imposed on lowerbeam Make 1199082 = suminfin119894=1 119885119894(119909)Y119894(119905) and we obtain

infinsum119894=1

[11986421198682119885119894(4)119884119894 + 119898211988410158401015840119894 119885119894 + (1198881 + 1198882) 119885119894119884119894] = 11988811199081 (37)

Make (37) multiplied by 119885119895(119909) and integrate it in thewhole beam considering (28) and the orthogonality of modefunction of lower beam and then obtaining

11988410158401015840 + (1198752119894 + 11988811198691198941198982)119884119894 = 119876119894 (119905) (38)

where 119876119894(119905) = (1198981199051198692119894 1198982) int1198970 11988811199081119885119894119889119909 is defined as gener-alized loads imposed on the lower beam Then make 119899119894 =(1198752119894 + 11988811198691198941198982)12 and the general solution of (38) is

119884119894 (119905) = 1119899119894 int119905

0119876119894 (120591) sin 119899119894 (119905 minus 120591) d120591 (39)

Substituting 119876119894(119905) (28) and (39) into 1199082 = suminfin119894=1 119885119894(119909)119884119894(119905)we obtain the dynamic response of lower beam under load asfollows

1199082= infinsum119894=1

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894

sdot 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)

minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]sdot int11990501199081 (119909 120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(40)

where 1199081 is the dynamic response of upper beam under loadobtained from (35)

25 Expression of Uniformly Distributed Load According tothe loading status of upper and lower beam the functionfor variable uniformly distributed load can be expressed asfollows

119901119902(119909119905) = minus119902119867 (V119905 minus 119909) (0 le 119905 le 119897V) (41)

where 119902 is the distributed electromagnetic force between therails and119867(119909) is the Heaviside Function3 Displacement of Double-Layer Beam underVariable Uniformly Distributed Load

31 Derivation of Dynamic Deflection Curve Equations ofUpper Beam Based on (31) we can obtain the dynamic

Mathematical Problems in Engineering 7

equation of upper beam attributed to uniformly distributedload as follows 11986411198681 120597411990811205971199094 + 1198981 120597

211990811205971199052 + 1198881 (1199081 minus 1199082) = 119901119902(119909119905) (42)

Substituting 119901119902(119909119905) into (42) we obtain

1199081119902(119909119905) (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)]

sdot int1199050119901119902(119909120591) sin119875119894 (119905 minus 120591) 119889120591 119889119909

(43)

Considering the property of Heaviside Function and sub-stituting 119901119902(119909119905) = minus119902119867(V119905 minus 119909) into (43) we can obtainedthe dynamic response of upper beam under uniformly dis-tributed load q through the integral as follows

1199081119902(119909119905) (119909 119905)= infinsum119894=1

minus119902[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 )+ 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 )

minus 1198722119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 )+ 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 )minus 120585 [1198732119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(44)

32 Bending Moment and Stress of Upper Beam Having gotthe dynamic deflection curve equations and then consideringthe expression of1198721 and 1205901 we can directly obtain from (44)

1198721 = 11986411198681 119889211990811199021198891199092 =infinsum119894=1

minus11986411198681119902 [1198721198942 cos119872119894119909 +1198721198942ch119872119894119909 minus 1205891198721198942 (sh119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942ch119873119894119909 minus 1205851198731198942 (sh119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(45)

8 Mathematical Problems in Engineering

1205901 = 1198641ℎ12 119889211990811199021198891199092 =

infinsum119894=1

minus1198641ℎ11199022 [1198721198942 cos119872119894119909 +1198721198942119888ℎ119872119894119909 minus 1205891198721198942 (119904ℎ119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942119888ℎ119873119894119909 minus 1205851198731198942 (119904ℎ119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ] (46)

33 Derivation of Dynamic Deflection Curve Equations ofLower Beam According to the determined 1199081119902(xt) similarlywe obtain

1199082119902(119909119905) = infinsum119894=0

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894+ 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]

sdot int11990501199081119902(119909120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(47)

Substituting (44) into (47) and simplifying the solutionwe can obtain

1199082119902(119909119905) = minus 119902119888111989811990511989911989411989821198692119894 (1198831 + 1198832) (1198671 + 1198672) (1198791 + 1198792) (48)

where the expression of11986711198672 1198791 and 1198792 can be reviewedin the appendix

34 Bending Moment and Stress of Lower Beam

1198722= minus11990211988811198981199051198641119868111989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792) (49)

1205902= minus11990211988811198981199051198641ℎ2211989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792)

(50)

where119883110158401015840 = 1198722119894 119869119894(cosh119872119894119909+ cos119872119894119909)minus 1205891198722119894 119869119894(sinh119872119894119909+sin119872119894119909) and 119883210158401015840 = 1198732119894 119870119894(cosh119873119894119909 + cos119873119894119909) minus1205851198732119894 119870119894(sinh119873119894119909 + sin119873119894119909)

4 Instance Analysis

41 Parameters The dynamic model for electromagneticlaunching rail due to uniformly distributed load is shownin Figure 4 The length of the beam is l=3000mm Thesection sizes of upper beam are 1198671=45mm and ℎ1=15mmrespectively The section sizes of lower beam are 1198672=75mmand ℎ2=30mm respectively The material of upper beamis copper of which the elastic modulus is 1198641=110GPa andthe mass density is 1205881=8290kgm3 the material of lowerbeam is copper of which the elastic modulus is 1198642 =283GPaand the mass density is 1205882 =980kgm3 The elastic constantbetween upper and lower beam is 1198881=3MPa the elasticconstant between lower beamand foundation is 1198882=6MPaTheuniformly distributed force is119902=3000NmThevelocity of thearmature is assumed as1000ms

42 Consequence and Analysis In the case of given motionparameters and structure parameters we have calculated thedisplacement bending moment and stress of double-layerelastic cantilever foundation beam and obtained analyticalsolutions and numerical results got from ANSYS which areshown in Figures 5ndash12

Mathematical Problems in Engineering 9

RailInsulation

qq

H 1H2

h2 h1 h2h1b

Cover plate

Figure 4 The cross-section of electromagnetic rail launcher

0 05 1 15 2 25 3x (m)

minus14minus12minus10minus8minus6minus4minus2

024

W1

(mm

)

l1=05m

Matlab ResultANSYS Result

Figure 5 Comparison of analytical solution with numerical solution for deflection of upper beam

Figures 5ndash8 show the dynamic response of the upperbeam attributed to moving load Analytical solution andnumerical solution for deflection of upper beam are depictedin Figure 5 in which peak values of displacement of analyticaland numerical solution are similar and the values occur atclose positions However the analytical solution begins tofluctuate near x=05m after which the numerical solutionremains stable Figure 6 shows the changing displacementof upper beam at x = 15m in which the analytical solutionand the numerical solution are in good agreement Figures7 and 8 depict the bending moment and stress of upperbeam at different positions due to the motion of load Dueto the linear relationship between the bending moment andthe stress both of them are identical in trend Similar to thechange of displacement solution the numerical solution andthe analytical solution are close before x=05 and after thatthe analytical solution fluctuates

Meanwhile dynamic response of the lower beam is inves-tigated The displacement results of analytical and numericalsolution of lower beam differentiate obviously as shown inFigure 9 Figure 10 shows the displacement of lower beamchanging with time at x = 15m in which similar to upperbeam the analytical solution and the numerical solutionare in good agreement in trend In Figures 11 and 12 both

analytical and numerical solution trend of bending momentand stress of lower beam follow the same patterns of upperbeam in Figures 7 and 8

It is apparent that the peak values of dynamic responseincluding displacement bending moment and stress ofupper beam are bigger than that of lower beam becauseof upper beam dissipating majority of the vibration energydue to the moving load There are some analytical solu-tion fluctuations compared with numerical solution in Fig-ures 5 7 8 9 11 and 12 which may be aroused byneglecting the layer damping In summary the analyticaland numerical solutions of the dynamic response of theupper and lower beams in the above mentioned figuresare similar in the overall trend despite some acceptabledeviation

In order to investigate the properties of analytical andnumerical solutions more specifically we compare the peakvalues of displacement bending moment and stress (Figures5ndash12) of the upper and lower beams of dynamic response asshown in Table 1

It is clear that the relative error of the peak values betweenthe analytical solution and the numerical solution obtainedby ANSYS is within a reasonable range and fluctuates around5

10 Mathematical Problems in Engineering

0 05 1 15 2 25 3t (s)

minus3minus2minus1

01234

v=1000msx=15m Position

Matlab ResultANSYS Result

times10-3

W1

(mm

)

Figure 6 Upper beam x = 15m comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus700minus600minus500minus400minus300minus200minus100

0100200300

M1

(Nlowast

m)

Matlab ResultANSYS Result

l1=05m

Figure 7 Comparison of analytical solution with numerical solution for bending moment distribution of upper beam

0 05 1 15 2 25 3x (m)

minus400

minus300

minus200

minus100

0

100

200

Matlab ResultANSYS Result

l1=05m

1(MPa

)

Figure 8 Comparison of analytical solution with numerical solution of stress distribution on upper beam

0 05 1 15 2 25 3x (m)

minus2minus15minus1

minus050

051

152

W2

(mm

)

Matlab ResultANSYS Result

Figure 9 Comparison of analytical solution with numerical solution for deflection of lower beam

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Page 2: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

2 Mathematical Problems in Engineering

Panel

Rail

Inputcurrent

Outputcurrent

Magneticthrust

ArmatureX

(a) Model of railgun

Rail Insulation(Elastic support)

Wall plate

Cover plate(Elastic support)

qq

(b) Section of railgun

Figure 1 Schematic of railgun

Figure 2 Exterior of rectangular electromagnetic railgun

as simply supported [18ndash21] despite the factual model thatis close to cantilever beam Hence simplifying electromag-netic launching device under force as a double-layer elasticcantilever beam will meet the requirement of actual workingcondition well

This paper takes rectangular-caliber electromagneticlauncher as research object and the cover plates and insu-lation which provide support to the rails are modeled aselastics supports to the rails and wall plates (shown in Figures1 and 2) The insulation between the guide rails and the wallplates is equivalent to a layer of elastic foundation and thewall plates are usually connected with the cover plates bybolts Cover plates and bolts in work state of deformationare equivalent to another layer elastic foundation for the wallplate Calculating the deformation and stress state of therails and the wall plates should be carried out in launchingprocess [18ndash21] During launching process when electriccurrent is guided into rails electric armature is subjected topowerful electromagnetic force and due to import of thecurrent two guide rails produce mutual repulsion 119902 The

repulsion 119902 is seen as uniformly distributed load so due tothemovement of armature the whole device can be simplifiedas a mechanics scenario in which the model of double-layerelastic cantilever beam is exposed to variable and uniformlydistributed electromagnetic force 119902

2 Mechanical Analysis

21 Mechanics Model and Differential Equations The launchrails and wall plates are simplified as a double-layer cantileverbeam system on Winkler elastic foundation as shown inFigure 3 where the rail is viewed as the upper beam andthe wall plate is viewed as the lower beam 119897 representsthe length of the beam and 119897

1marks the location of the

moving armature 119902 is the electromagnetic force applied tothe rails 119907 is the velocity of the armature 119864

11198681and 119864

21198682are

the bending stiffness of upper and lower beam respectivelyElastic constant between the upper and lower beam is 1198881 andelastic constant between the lower beam and the foundationis 1198882 119907 is the velocity of armature

Mathematical Problems in Engineering 3

l1l

q

c1

c2

E1 I1

E2 I2

v

Figure 3 Mechanical model of double-layer cantilever beam

The dynamic equations of upper and lower beam are

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 119891 (119909 119905)

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + 11988821199082 = minus1198881 (1199082 minus 1199081)

(1)

where 1199081and 119908

2represent the deflection of the upper and

lower beam respectively 1198981 = 12058811198781 and 1198982 = 12058821198782are the mass per unit length of upper and lower beamrespectively 1205881 and 1205882 are the mass density of upper andlower beam respectively 1198781 and 1198782 are the cross-sectionalarea of upper and lower beam respectively 119905 is the timewith armaturersquosmove f (x t) is the uniformly distributed loadelectromagnetic force per unit length applied to the rail

22 Homogeneous Solution Based on interactions betweenupper and lower beam in the case of no load the dynamicequations of upper and lower beam are

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 0 (2)

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + (1198881 + 1198882) 1199082 minus 11988811199081 = 0 (3)

From (2) we obtain

1199082 = 119864111986811198881 120597411990811205971199094 + 11989811198881 120597

211990811205971199052 + 1199081 (4)

Applying (4) to (3) and simplifying the solution we obtain

1198641119868111986421198682 120597811990811205971199098 + (119864211986821198981 + 119864111986811198982) 1205976119908112059711990941205971199052

+ [119888111986421198682 + (c1 + 1198882) 11986411198681] 120597411990811205971199094 + 11989811198982 120597411990811205971199054

+ [11988811198982 + (1198881 + 1198882)1198981] 120597211990811205971199052 + 119888111988821199081 = 0(5)

According to the displacement forms of the beam we assumethe homogeneous solution of 1199081 and 1199082 respectively as1199081 = infinsum

119894=1

119883119894 (119860 119894 sin119875119894119905 + 119861119894 cos119875119894119905) = infinsum119894=1

119883119894119884119894 (6)

1199082 = infinsum119894=1

[119864111986811198881 1205974119883119894120597119909 + (1 minus 1198981119875

21198941198881 )119883119894]119884119894 =

infinsum119894=1

119885119894119884119894 (7)

with 119875119894 as circular frequency Yi = Ai sin Pit + Bi cos Pit and119885119894 = (119864111986811198881)(1205974119883119894120597119909) + (1 minus 11989811198752119894 1198881)119883119894Applying (6) to (5) we obtain

120572119894 12059781198831198941205971199098 + 120573119894 12059741198831198941205971199094 + 120574119894119883119894 = 0 (8)

where 120572119894 = 1198641119868111986421198682 120573119894 = 119888111986421198682+(1198881+1198882)11986411198681minus1198752119894 (119864211986821198981+119864111986811198982) and 120574119894 = 119898111989821198754119894 + 11988811198882 minus 1198752119894 [11988811198982 + (1198881 + 1198882)1198981]The solution of (8) can be expressed as

119883119894 = 1198621 cos119872119894119909 + 1198622 sin119872119894119909 + 1198623ch119872119894119909+ 1198624sh119872119894119909 + 1198625 cos119873119894119909 + 1198626 sin119873119894119909+ 1198627ch119873119894119909 + 1198628sh119873119894119909

(9)

where C1simC8 is undetermined coefficients and the coeffi-cients

119872119894 = (minus120573119894 + (1205732119894 minus 4120572119894120574119894)12

2120572119894 )14

and 119873119894 = (minus120573119894 minus (1205732119894 minus 4120572119894120574119894)12

2120572119894 )14

(10)

Applying (9) to (7) we can obtain from 1199082119885119894 = 1198621119869119894 cos119872119894119909 + 1198622119869119894 sin119872119894119909 + 1198623119869119894ch119872119894119909

+ 1198624119869119894sh119872119894119909 + 1198625119870119894 cos119873119894119909 + 1198626119870119894 sin119873119894119909+ 1198627119870119894ch119873119894119909 + 1198628119870119894sh119873119894119909

(11)

4 Mathematical Problems in Engineering

with 119869119894 = (119864111986811198721198944 + 1198881 minus 11989811198751198942)1198881 and119870119894 = (119864111986811198731198944 + 1198881 minus11989811198751198942)1198881Accordingly we obtain the bendingmoment of upper and

lower beam as follows

119872119906119901119901119890119903 = 11986411198681 120597211990811205972119909 = 11986411198681infinsum119894=1

12059721198831198941205971199092 119884119894 = [infinsum119894=1

11986411198681sdot (minus11986211198722119894 cos119872119894119909 minus 11986221198722119894 sin119872119894119909+ 11986231198722119894 ch119872119894119909 + 11986241198722119894 sh119872119894119909 minus 11986251198732119894 cos119873119894119909minus 11986261198732119894 sin119873119894119909 + 11986271198732119894 ch119873119894119909 + 11986281198732119894 sh119873119894119909)]sdot (119860 119894 sin119875119894119905 + 119861119894 cos119875119894119905)

(12)

119872119897119900119908119890119903 = 11986421198682 120597211990821205972119909 = 11986421198682infinsum119894=1

12059721198851198941205971199092 119884119894 = [infinsum119894=1

11986421198682sdot (minus11986211198691198941198722119894 cos119872119894119909 minus 11986221198691198941198722119894 sin119872119894119909+ 11986231198691198941198722119894 ch119872119894119909 + 11986241198691198941198722119894 sh119872119894119909minus 11986251198701198941198732119894 cos119873119894119909 minus 11986261198701198941198732119894 sin119873119894119909+ 11986271198701198941198732119894 ch119873119894119909 + 11986281198701198941198732119894 sh119873119894119909)] (119860 119894 sin119875119894119905+ 119861119894 cos119875119894119905)

(13)

The rotation angles of upper and lower beam are

1205791 = 11986411198681 1205971199081120597119909 = 11986411198681infinsum119894=1

120597119883119894120597119909 119884119894 = [infinsum119894=1

11986411198681sdot (minus1198621119872119894 sin119872119894119909 + 1198622119872119894 cos119872119894119909 + 1198623119872119894sh119872119894119909+ 1198624119872119894ch119872119894119909 minus 1198625119873119894 sin119873119894119909 + 1198626119873119894 cos119873119894119909+ 1198627119873119894sh119873119894119909 + 1198628119873119894ch119873119894119909)]

(14)

1205792 = 11986421198682 1205971199082120597119909 = 11986421198682infinsum119894=1

120597119885119894120597119909 119884119894 = [infinsum119894=1

11986421198682sdot (minus1198621119869119894119872119894 sin119872119894119909 + 1198622119869119894119872119894 cos119872119894119909+ 1198623119869119894119872119894sh119872119894119909 + 1198624119869119894119872119894ch119872119894119909minus 1198625119870119894119873119894 sin119873119894119909 + 1198626119870119894119873119894 cos119873119894119909+ 1198627119870119894119873119894sh119873119894119909 + 1198628119870119894119873119894ch119873119894119909)]

(15)

The shear forces of upper and lower beam are

1198761 = 11986411198681 120597311990811205971199093 = 11986411198681infinsum119894=1

12059731198831198941205971199093 119884119894= infinsum119894=1

(11986211198723119894 sin119872119894119909 minus 11986221198723119894 cos119872119894119909+ 11986231198723119894 sh119872119894119909 + 11986241198723119894 ch119872119894119909 + 11986251198733119894 sin119873119894119909minus 11986261198733119894 cos119873119894119909 + 11986271198733119894 sh119873119894119909 + 11986281198733119894 ch119873119894119909)

(16)

1198762 = 11986421198682 120597311990821205971199093 = 11986421198682infinsum119894=1

12059731198851198941205971199093 119884119894 =infinsum119894=1

11986421198682 (11986211198691198941198723119894sdot sin119872119894119909 minus 11986221198691198941198723119894 cos119872119894119909 + 11986231198691198941198723119894 sh119872119894119909+ 11986241198691198941198723119894 ch119872119894119909 + 11986251198701198941198733119894 sin119873119894119909 minus 11986261198701198941198733119894sdot cos119873119894119909 + 11986271198701198941198733119894 sh119873119894119909 + 11986281198701198941198733119894 ch119873119894119909)

(17)

where the constants 1198621sim1198628 can be determined from bound-ary conditions of the rail According to the double-layerbeam model where one end of the rail is constrained andthe another is under condition of freedom we can obtain theboundary conditions of both ends as follows

11990811003816100381610038161003816119909=0 = 012057911003816100381610038161003816119909=0 = 011990821003816100381610038161003816119909=0 = 012057921003816100381610038161003816119909=0 = 011987211003816100381610038161003816119909=119897 = 011987611003816100381610038161003816119909=119897 = 011987221003816100381610038161003816119909=119897 = 011987621003816100381610038161003816119909=119897 = 0

(18)

Substituting (6) (7) (14) and (15) into (18) from the leftboundary conditions we can obtain

1198621 + 1198623 + 1198625 + 1198627 = 01198622119872119894 + 1198624119872119894 + 1198626119873119894 + 1198628119873119894 = 01198621119869119894 + 1198623119869119894 + 1198625119870119894 + 1198627119870119894 = 0

1198622119869119894119872119894 + 1198624119869119894119872119894 + 1198626119870119894119873119894 + 1198628119870119894119873119894 = 0(19)

Substituting (12) (13) (16) and (17) into (18) from the rightboundary conditions we can obtain

minus 11986211198722119894 cos119872119894119897 minus 11986221198722119894 sin119872119894119897 + 11986231198722119894 ch119872119894119897+ 11986241198722119894 sh119872119894119897 minus 11986251198732119894 cos119873119894119897 minus 11986261198732119894 sin119873119894119897+ 11986271198732119894 ch119873119894119897 + 11986281198732119894 sh119873119894119897 = 0

Mathematical Problems in Engineering 5

11986211198723119894 sin119872119894119897 minus 11986221198723119894 cos119872119894119897 + 11986231198723119894 sh119872119894119897minus 11986241198723119894 ch119872119894119897 + 11986251198733119894 sin119873119894119897 minus 11986261198733119894 cos119873119894119897+ 11986271198733119894 sh119873119894119897 + 11986281198733119894 ch119873119894119897 = 0

minus 11986211198691198941198722119894 cos119872119894119897 minus 11986221198691198941198722119894 sin119872119894119897 + 11986231198691198941198722119894 ch119872119894119897+ 11986241198691198941198722119894 sh119872119894119897 minus 11986251198701198941198732119894 cos119873119894119897minus 11986261198701198941198732119894 sin119873119894119897 + 11986271198701198941198732119894 ch119873119894119897+ 11986281198701198941198732119894 sh119873119894119897 = 0

11986211198691198941198723119894 sin119872119894119897 minus 11986221198691198941198723119894 cos119872119894119897 + 11986231198691198941198723119894 sh119872119894119897minus 11986241198691198941198691198941198723119894 ch119872119894119897 + 11986251198701198941198733119894 sin119873119894119897minus 11986261198701198941198733119894 cos119873119894119897 + 11986271198701198941198733119894 sh119873119894119897+ 11986281198701198941198733119894 ch119873119894119897 = 0

(20)

From (19) we obtain 1198621 = minus11986231198622 = minus11986241198625 = minus11986271198626 = minus1198628

(21)

From (20) we obtainminus1198625 cos119873119894119897 minus 1198626 sin119873119894119897 + 1198627ch119873119894119897 + 1198628sh119873119894119897 = 0minus1198621 cos119872119894119897 minus 1198622 sin119872119894119897 + 1198623ch119872119894119897 + 1198624sh119872119894119897 = 01198625 sin119873119894119897 minus 1198626 cos119873119894119897 + 1198627sh119873119894119897 + 1198628ch119873119894119897 = 01198621 sin119872119894119897 minus 1198622 cos119872119894119897 + 1198623sh119872119894119897 + 1198624ch119872119894119897 = 0

(22)

Applying (21) into (22) we obtain

1198627 (cos119873119894119897 + ch119873119894119897) + 1198628 (sin119873119894119897 + sh119873119894119897) = 01198623 (cos119872119894119897 + ch119872119894119897) + 1198624 (sin119872119894119897 + sh119872119894119897) = 01198627 (minus sin119873119894119897 + sh119873119894119897) + C8 (cos119873119894119897 + ch119873119894119897) = 01198623 (minus sin119872119894119897 + sh119872119894119897) + C4 (cos119872119894119897 + ch119872119894119897) = 0

(23)

From (23) we obtain1198627 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198628 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198623 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198624 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 0

(24)

From (24) we obtainch119873119894119897 cos119873119894119897 + 1 = 0ch119872119894119897 cos119872119894119897 + 1 = 0 (25)

Assume C3=1 C7=1 with1198621 = 1198625 = minus11198624 = minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 1198622 = sh119872119894119897 minus sin119872119894119897

ch119872119894119897 + cos119872119894119897 1198626 = sh119873119894119897 minus sin119873119894119897

ch119873119894119897 + cos119873119894119897 1198628 = minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897

(26)

Accordingly the mode function for upper and lower beamcan be obtained as follows119883119894 = minus cos119872119894119909 + ch119872119894119909

minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 (sh119872119894119909 minus sin119872119894119909)minus cos119873119894119909 + ch119873119894119909minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897 (sh119873119894119909 minus sin119873119894119909)

(27)

119885119894 = minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 119869119894 (sh119872119894119909 minus sin119872119894119909)minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897119870119894 (sh119873119894119909 minus sin119873119894119909)

(28)

Then 1199081 and 1199082 can be expressed as

1199081 = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] (119860 119894sdot sin119875119894119905 + 119861119894 sin119875119894119905)

(29)

1199082 = infinsum119894=1

[minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)] (119860 119894 sin119875119894119905 + 119861119894 sin119875119894119905)

(30)

with 120589 = (sh119872119894119897minussin119872119894119897)(ch119872119894119897+cos119872119894119897) and 120585 = (sh119873119894119897minussin119873119894119897)(ch119873119894119897 + cos119873119894119897)23 Particular Solution of Upper Beam Assume f (x t) as theforce imposed on upper beam then substituting it into (2)and we obtain

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 119891 (119909 119905) (31)

6 Mathematical Problems in Engineering

Make 1199081 = suminfin119894=1119883119894(119909)119884119894(119905) and then we obtain

infinsum119894=1

[11986411198681119883119894(4)119884119894 + 119898111988410158401015840119894 119883119894 + 1198881 (119883119894119884119894 minus 1119869119883119894119884119894)]= 119891 (119909 119905)

(32)

Make (32) multiplied by 119883119895(119909) and integrate it in thewhole beam considering (27) and the orthogonality of modefunction of upper beam and then obtaining

Y10158401015840119894 + 1198752119894 119884119894 = 119876119894 (119905) (33)

where 119876119894(119905) = int1198970 119891(119909 119905)119883119894(119909)119889119909 named i-order generalizedloads The general solution of (33) is

Y119894 (119905) = 1119875119894 int119905

0119876119894 (120591) sin119875119894 (119905 minus 120591) d120591 (34)

Solving (34) then we can obtain the dynamic response ofupper beam under load as follows

1199081 (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minuscos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] int119905

0119891 (119909 119905)

sdot sin119875119894 (119905 minus 120591) 119889120591 119889119909(35)

24 Particular Solution of Lower Beam For lower beamthe load imposed on it is the pressure from upper beamAccording to (3) we obtain

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + (1198881 + 1198882) 1199082 minus 11988811199081 = 119891 (119909 119905) (36)

where 11988811199081 determined by (35) is the load imposed on lowerbeam Make 1199082 = suminfin119894=1 119885119894(119909)Y119894(119905) and we obtain

infinsum119894=1

[11986421198682119885119894(4)119884119894 + 119898211988410158401015840119894 119885119894 + (1198881 + 1198882) 119885119894119884119894] = 11988811199081 (37)

Make (37) multiplied by 119885119895(119909) and integrate it in thewhole beam considering (28) and the orthogonality of modefunction of lower beam and then obtaining

11988410158401015840 + (1198752119894 + 11988811198691198941198982)119884119894 = 119876119894 (119905) (38)

where 119876119894(119905) = (1198981199051198692119894 1198982) int1198970 11988811199081119885119894119889119909 is defined as gener-alized loads imposed on the lower beam Then make 119899119894 =(1198752119894 + 11988811198691198941198982)12 and the general solution of (38) is

119884119894 (119905) = 1119899119894 int119905

0119876119894 (120591) sin 119899119894 (119905 minus 120591) d120591 (39)

Substituting 119876119894(119905) (28) and (39) into 1199082 = suminfin119894=1 119885119894(119909)119884119894(119905)we obtain the dynamic response of lower beam under load asfollows

1199082= infinsum119894=1

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894

sdot 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)

minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]sdot int11990501199081 (119909 120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(40)

where 1199081 is the dynamic response of upper beam under loadobtained from (35)

25 Expression of Uniformly Distributed Load According tothe loading status of upper and lower beam the functionfor variable uniformly distributed load can be expressed asfollows

119901119902(119909119905) = minus119902119867 (V119905 minus 119909) (0 le 119905 le 119897V) (41)

where 119902 is the distributed electromagnetic force between therails and119867(119909) is the Heaviside Function3 Displacement of Double-Layer Beam underVariable Uniformly Distributed Load

31 Derivation of Dynamic Deflection Curve Equations ofUpper Beam Based on (31) we can obtain the dynamic

Mathematical Problems in Engineering 7

equation of upper beam attributed to uniformly distributedload as follows 11986411198681 120597411990811205971199094 + 1198981 120597

211990811205971199052 + 1198881 (1199081 minus 1199082) = 119901119902(119909119905) (42)

Substituting 119901119902(119909119905) into (42) we obtain

1199081119902(119909119905) (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)]

sdot int1199050119901119902(119909120591) sin119875119894 (119905 minus 120591) 119889120591 119889119909

(43)

Considering the property of Heaviside Function and sub-stituting 119901119902(119909119905) = minus119902119867(V119905 minus 119909) into (43) we can obtainedthe dynamic response of upper beam under uniformly dis-tributed load q through the integral as follows

1199081119902(119909119905) (119909 119905)= infinsum119894=1

minus119902[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 )+ 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 )

minus 1198722119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 )+ 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 )minus 120585 [1198732119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(44)

32 Bending Moment and Stress of Upper Beam Having gotthe dynamic deflection curve equations and then consideringthe expression of1198721 and 1205901 we can directly obtain from (44)

1198721 = 11986411198681 119889211990811199021198891199092 =infinsum119894=1

minus11986411198681119902 [1198721198942 cos119872119894119909 +1198721198942ch119872119894119909 minus 1205891198721198942 (sh119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942ch119873119894119909 minus 1205851198731198942 (sh119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(45)

8 Mathematical Problems in Engineering

1205901 = 1198641ℎ12 119889211990811199021198891199092 =

infinsum119894=1

minus1198641ℎ11199022 [1198721198942 cos119872119894119909 +1198721198942119888ℎ119872119894119909 minus 1205891198721198942 (119904ℎ119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942119888ℎ119873119894119909 minus 1205851198731198942 (119904ℎ119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ] (46)

33 Derivation of Dynamic Deflection Curve Equations ofLower Beam According to the determined 1199081119902(xt) similarlywe obtain

1199082119902(119909119905) = infinsum119894=0

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894+ 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]

sdot int11990501199081119902(119909120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(47)

Substituting (44) into (47) and simplifying the solutionwe can obtain

1199082119902(119909119905) = minus 119902119888111989811990511989911989411989821198692119894 (1198831 + 1198832) (1198671 + 1198672) (1198791 + 1198792) (48)

where the expression of11986711198672 1198791 and 1198792 can be reviewedin the appendix

34 Bending Moment and Stress of Lower Beam

1198722= minus11990211988811198981199051198641119868111989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792) (49)

1205902= minus11990211988811198981199051198641ℎ2211989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792)

(50)

where119883110158401015840 = 1198722119894 119869119894(cosh119872119894119909+ cos119872119894119909)minus 1205891198722119894 119869119894(sinh119872119894119909+sin119872119894119909) and 119883210158401015840 = 1198732119894 119870119894(cosh119873119894119909 + cos119873119894119909) minus1205851198732119894 119870119894(sinh119873119894119909 + sin119873119894119909)

4 Instance Analysis

41 Parameters The dynamic model for electromagneticlaunching rail due to uniformly distributed load is shownin Figure 4 The length of the beam is l=3000mm Thesection sizes of upper beam are 1198671=45mm and ℎ1=15mmrespectively The section sizes of lower beam are 1198672=75mmand ℎ2=30mm respectively The material of upper beamis copper of which the elastic modulus is 1198641=110GPa andthe mass density is 1205881=8290kgm3 the material of lowerbeam is copper of which the elastic modulus is 1198642 =283GPaand the mass density is 1205882 =980kgm3 The elastic constantbetween upper and lower beam is 1198881=3MPa the elasticconstant between lower beamand foundation is 1198882=6MPaTheuniformly distributed force is119902=3000NmThevelocity of thearmature is assumed as1000ms

42 Consequence and Analysis In the case of given motionparameters and structure parameters we have calculated thedisplacement bending moment and stress of double-layerelastic cantilever foundation beam and obtained analyticalsolutions and numerical results got from ANSYS which areshown in Figures 5ndash12

Mathematical Problems in Engineering 9

RailInsulation

qq

H 1H2

h2 h1 h2h1b

Cover plate

Figure 4 The cross-section of electromagnetic rail launcher

0 05 1 15 2 25 3x (m)

minus14minus12minus10minus8minus6minus4minus2

024

W1

(mm

)

l1=05m

Matlab ResultANSYS Result

Figure 5 Comparison of analytical solution with numerical solution for deflection of upper beam

Figures 5ndash8 show the dynamic response of the upperbeam attributed to moving load Analytical solution andnumerical solution for deflection of upper beam are depictedin Figure 5 in which peak values of displacement of analyticaland numerical solution are similar and the values occur atclose positions However the analytical solution begins tofluctuate near x=05m after which the numerical solutionremains stable Figure 6 shows the changing displacementof upper beam at x = 15m in which the analytical solutionand the numerical solution are in good agreement Figures7 and 8 depict the bending moment and stress of upperbeam at different positions due to the motion of load Dueto the linear relationship between the bending moment andthe stress both of them are identical in trend Similar to thechange of displacement solution the numerical solution andthe analytical solution are close before x=05 and after thatthe analytical solution fluctuates

Meanwhile dynamic response of the lower beam is inves-tigated The displacement results of analytical and numericalsolution of lower beam differentiate obviously as shown inFigure 9 Figure 10 shows the displacement of lower beamchanging with time at x = 15m in which similar to upperbeam the analytical solution and the numerical solutionare in good agreement in trend In Figures 11 and 12 both

analytical and numerical solution trend of bending momentand stress of lower beam follow the same patterns of upperbeam in Figures 7 and 8

It is apparent that the peak values of dynamic responseincluding displacement bending moment and stress ofupper beam are bigger than that of lower beam becauseof upper beam dissipating majority of the vibration energydue to the moving load There are some analytical solu-tion fluctuations compared with numerical solution in Fig-ures 5 7 8 9 11 and 12 which may be aroused byneglecting the layer damping In summary the analyticaland numerical solutions of the dynamic response of theupper and lower beams in the above mentioned figuresare similar in the overall trend despite some acceptabledeviation

In order to investigate the properties of analytical andnumerical solutions more specifically we compare the peakvalues of displacement bending moment and stress (Figures5ndash12) of the upper and lower beams of dynamic response asshown in Table 1

It is clear that the relative error of the peak values betweenthe analytical solution and the numerical solution obtainedby ANSYS is within a reasonable range and fluctuates around5

10 Mathematical Problems in Engineering

0 05 1 15 2 25 3t (s)

minus3minus2minus1

01234

v=1000msx=15m Position

Matlab ResultANSYS Result

times10-3

W1

(mm

)

Figure 6 Upper beam x = 15m comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus700minus600minus500minus400minus300minus200minus100

0100200300

M1

(Nlowast

m)

Matlab ResultANSYS Result

l1=05m

Figure 7 Comparison of analytical solution with numerical solution for bending moment distribution of upper beam

0 05 1 15 2 25 3x (m)

minus400

minus300

minus200

minus100

0

100

200

Matlab ResultANSYS Result

l1=05m

1(MPa

)

Figure 8 Comparison of analytical solution with numerical solution of stress distribution on upper beam

0 05 1 15 2 25 3x (m)

minus2minus15minus1

minus050

051

152

W2

(mm

)

Matlab ResultANSYS Result

Figure 9 Comparison of analytical solution with numerical solution for deflection of lower beam

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Page 3: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

Mathematical Problems in Engineering 3

l1l

q

c1

c2

E1 I1

E2 I2

v

Figure 3 Mechanical model of double-layer cantilever beam

The dynamic equations of upper and lower beam are

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 119891 (119909 119905)

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + 11988821199082 = minus1198881 (1199082 minus 1199081)

(1)

where 1199081and 119908

2represent the deflection of the upper and

lower beam respectively 1198981 = 12058811198781 and 1198982 = 12058821198782are the mass per unit length of upper and lower beamrespectively 1205881 and 1205882 are the mass density of upper andlower beam respectively 1198781 and 1198782 are the cross-sectionalarea of upper and lower beam respectively 119905 is the timewith armaturersquosmove f (x t) is the uniformly distributed loadelectromagnetic force per unit length applied to the rail

22 Homogeneous Solution Based on interactions betweenupper and lower beam in the case of no load the dynamicequations of upper and lower beam are

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 0 (2)

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + (1198881 + 1198882) 1199082 minus 11988811199081 = 0 (3)

From (2) we obtain

1199082 = 119864111986811198881 120597411990811205971199094 + 11989811198881 120597

211990811205971199052 + 1199081 (4)

Applying (4) to (3) and simplifying the solution we obtain

1198641119868111986421198682 120597811990811205971199098 + (119864211986821198981 + 119864111986811198982) 1205976119908112059711990941205971199052

+ [119888111986421198682 + (c1 + 1198882) 11986411198681] 120597411990811205971199094 + 11989811198982 120597411990811205971199054

+ [11988811198982 + (1198881 + 1198882)1198981] 120597211990811205971199052 + 119888111988821199081 = 0(5)

According to the displacement forms of the beam we assumethe homogeneous solution of 1199081 and 1199082 respectively as1199081 = infinsum

119894=1

119883119894 (119860 119894 sin119875119894119905 + 119861119894 cos119875119894119905) = infinsum119894=1

119883119894119884119894 (6)

1199082 = infinsum119894=1

[119864111986811198881 1205974119883119894120597119909 + (1 minus 1198981119875

21198941198881 )119883119894]119884119894 =

infinsum119894=1

119885119894119884119894 (7)

with 119875119894 as circular frequency Yi = Ai sin Pit + Bi cos Pit and119885119894 = (119864111986811198881)(1205974119883119894120597119909) + (1 minus 11989811198752119894 1198881)119883119894Applying (6) to (5) we obtain

120572119894 12059781198831198941205971199098 + 120573119894 12059741198831198941205971199094 + 120574119894119883119894 = 0 (8)

where 120572119894 = 1198641119868111986421198682 120573119894 = 119888111986421198682+(1198881+1198882)11986411198681minus1198752119894 (119864211986821198981+119864111986811198982) and 120574119894 = 119898111989821198754119894 + 11988811198882 minus 1198752119894 [11988811198982 + (1198881 + 1198882)1198981]The solution of (8) can be expressed as

119883119894 = 1198621 cos119872119894119909 + 1198622 sin119872119894119909 + 1198623ch119872119894119909+ 1198624sh119872119894119909 + 1198625 cos119873119894119909 + 1198626 sin119873119894119909+ 1198627ch119873119894119909 + 1198628sh119873119894119909

(9)

where C1simC8 is undetermined coefficients and the coeffi-cients

119872119894 = (minus120573119894 + (1205732119894 minus 4120572119894120574119894)12

2120572119894 )14

and 119873119894 = (minus120573119894 minus (1205732119894 minus 4120572119894120574119894)12

2120572119894 )14

(10)

Applying (9) to (7) we can obtain from 1199082119885119894 = 1198621119869119894 cos119872119894119909 + 1198622119869119894 sin119872119894119909 + 1198623119869119894ch119872119894119909

+ 1198624119869119894sh119872119894119909 + 1198625119870119894 cos119873119894119909 + 1198626119870119894 sin119873119894119909+ 1198627119870119894ch119873119894119909 + 1198628119870119894sh119873119894119909

(11)

4 Mathematical Problems in Engineering

with 119869119894 = (119864111986811198721198944 + 1198881 minus 11989811198751198942)1198881 and119870119894 = (119864111986811198731198944 + 1198881 minus11989811198751198942)1198881Accordingly we obtain the bendingmoment of upper and

lower beam as follows

119872119906119901119901119890119903 = 11986411198681 120597211990811205972119909 = 11986411198681infinsum119894=1

12059721198831198941205971199092 119884119894 = [infinsum119894=1

11986411198681sdot (minus11986211198722119894 cos119872119894119909 minus 11986221198722119894 sin119872119894119909+ 11986231198722119894 ch119872119894119909 + 11986241198722119894 sh119872119894119909 minus 11986251198732119894 cos119873119894119909minus 11986261198732119894 sin119873119894119909 + 11986271198732119894 ch119873119894119909 + 11986281198732119894 sh119873119894119909)]sdot (119860 119894 sin119875119894119905 + 119861119894 cos119875119894119905)

(12)

119872119897119900119908119890119903 = 11986421198682 120597211990821205972119909 = 11986421198682infinsum119894=1

12059721198851198941205971199092 119884119894 = [infinsum119894=1

11986421198682sdot (minus11986211198691198941198722119894 cos119872119894119909 minus 11986221198691198941198722119894 sin119872119894119909+ 11986231198691198941198722119894 ch119872119894119909 + 11986241198691198941198722119894 sh119872119894119909minus 11986251198701198941198732119894 cos119873119894119909 minus 11986261198701198941198732119894 sin119873119894119909+ 11986271198701198941198732119894 ch119873119894119909 + 11986281198701198941198732119894 sh119873119894119909)] (119860 119894 sin119875119894119905+ 119861119894 cos119875119894119905)

(13)

The rotation angles of upper and lower beam are

1205791 = 11986411198681 1205971199081120597119909 = 11986411198681infinsum119894=1

120597119883119894120597119909 119884119894 = [infinsum119894=1

11986411198681sdot (minus1198621119872119894 sin119872119894119909 + 1198622119872119894 cos119872119894119909 + 1198623119872119894sh119872119894119909+ 1198624119872119894ch119872119894119909 minus 1198625119873119894 sin119873119894119909 + 1198626119873119894 cos119873119894119909+ 1198627119873119894sh119873119894119909 + 1198628119873119894ch119873119894119909)]

(14)

1205792 = 11986421198682 1205971199082120597119909 = 11986421198682infinsum119894=1

120597119885119894120597119909 119884119894 = [infinsum119894=1

11986421198682sdot (minus1198621119869119894119872119894 sin119872119894119909 + 1198622119869119894119872119894 cos119872119894119909+ 1198623119869119894119872119894sh119872119894119909 + 1198624119869119894119872119894ch119872119894119909minus 1198625119870119894119873119894 sin119873119894119909 + 1198626119870119894119873119894 cos119873119894119909+ 1198627119870119894119873119894sh119873119894119909 + 1198628119870119894119873119894ch119873119894119909)]

(15)

The shear forces of upper and lower beam are

1198761 = 11986411198681 120597311990811205971199093 = 11986411198681infinsum119894=1

12059731198831198941205971199093 119884119894= infinsum119894=1

(11986211198723119894 sin119872119894119909 minus 11986221198723119894 cos119872119894119909+ 11986231198723119894 sh119872119894119909 + 11986241198723119894 ch119872119894119909 + 11986251198733119894 sin119873119894119909minus 11986261198733119894 cos119873119894119909 + 11986271198733119894 sh119873119894119909 + 11986281198733119894 ch119873119894119909)

(16)

1198762 = 11986421198682 120597311990821205971199093 = 11986421198682infinsum119894=1

12059731198851198941205971199093 119884119894 =infinsum119894=1

11986421198682 (11986211198691198941198723119894sdot sin119872119894119909 minus 11986221198691198941198723119894 cos119872119894119909 + 11986231198691198941198723119894 sh119872119894119909+ 11986241198691198941198723119894 ch119872119894119909 + 11986251198701198941198733119894 sin119873119894119909 minus 11986261198701198941198733119894sdot cos119873119894119909 + 11986271198701198941198733119894 sh119873119894119909 + 11986281198701198941198733119894 ch119873119894119909)

(17)

where the constants 1198621sim1198628 can be determined from bound-ary conditions of the rail According to the double-layerbeam model where one end of the rail is constrained andthe another is under condition of freedom we can obtain theboundary conditions of both ends as follows

11990811003816100381610038161003816119909=0 = 012057911003816100381610038161003816119909=0 = 011990821003816100381610038161003816119909=0 = 012057921003816100381610038161003816119909=0 = 011987211003816100381610038161003816119909=119897 = 011987611003816100381610038161003816119909=119897 = 011987221003816100381610038161003816119909=119897 = 011987621003816100381610038161003816119909=119897 = 0

(18)

Substituting (6) (7) (14) and (15) into (18) from the leftboundary conditions we can obtain

1198621 + 1198623 + 1198625 + 1198627 = 01198622119872119894 + 1198624119872119894 + 1198626119873119894 + 1198628119873119894 = 01198621119869119894 + 1198623119869119894 + 1198625119870119894 + 1198627119870119894 = 0

1198622119869119894119872119894 + 1198624119869119894119872119894 + 1198626119870119894119873119894 + 1198628119870119894119873119894 = 0(19)

Substituting (12) (13) (16) and (17) into (18) from the rightboundary conditions we can obtain

minus 11986211198722119894 cos119872119894119897 minus 11986221198722119894 sin119872119894119897 + 11986231198722119894 ch119872119894119897+ 11986241198722119894 sh119872119894119897 minus 11986251198732119894 cos119873119894119897 minus 11986261198732119894 sin119873119894119897+ 11986271198732119894 ch119873119894119897 + 11986281198732119894 sh119873119894119897 = 0

Mathematical Problems in Engineering 5

11986211198723119894 sin119872119894119897 minus 11986221198723119894 cos119872119894119897 + 11986231198723119894 sh119872119894119897minus 11986241198723119894 ch119872119894119897 + 11986251198733119894 sin119873119894119897 minus 11986261198733119894 cos119873119894119897+ 11986271198733119894 sh119873119894119897 + 11986281198733119894 ch119873119894119897 = 0

minus 11986211198691198941198722119894 cos119872119894119897 minus 11986221198691198941198722119894 sin119872119894119897 + 11986231198691198941198722119894 ch119872119894119897+ 11986241198691198941198722119894 sh119872119894119897 minus 11986251198701198941198732119894 cos119873119894119897minus 11986261198701198941198732119894 sin119873119894119897 + 11986271198701198941198732119894 ch119873119894119897+ 11986281198701198941198732119894 sh119873119894119897 = 0

11986211198691198941198723119894 sin119872119894119897 minus 11986221198691198941198723119894 cos119872119894119897 + 11986231198691198941198723119894 sh119872119894119897minus 11986241198691198941198691198941198723119894 ch119872119894119897 + 11986251198701198941198733119894 sin119873119894119897minus 11986261198701198941198733119894 cos119873119894119897 + 11986271198701198941198733119894 sh119873119894119897+ 11986281198701198941198733119894 ch119873119894119897 = 0

(20)

From (19) we obtain 1198621 = minus11986231198622 = minus11986241198625 = minus11986271198626 = minus1198628

(21)

From (20) we obtainminus1198625 cos119873119894119897 minus 1198626 sin119873119894119897 + 1198627ch119873119894119897 + 1198628sh119873119894119897 = 0minus1198621 cos119872119894119897 minus 1198622 sin119872119894119897 + 1198623ch119872119894119897 + 1198624sh119872119894119897 = 01198625 sin119873119894119897 minus 1198626 cos119873119894119897 + 1198627sh119873119894119897 + 1198628ch119873119894119897 = 01198621 sin119872119894119897 minus 1198622 cos119872119894119897 + 1198623sh119872119894119897 + 1198624ch119872119894119897 = 0

(22)

Applying (21) into (22) we obtain

1198627 (cos119873119894119897 + ch119873119894119897) + 1198628 (sin119873119894119897 + sh119873119894119897) = 01198623 (cos119872119894119897 + ch119872119894119897) + 1198624 (sin119872119894119897 + sh119872119894119897) = 01198627 (minus sin119873119894119897 + sh119873119894119897) + C8 (cos119873119894119897 + ch119873119894119897) = 01198623 (minus sin119872119894119897 + sh119872119894119897) + C4 (cos119872119894119897 + ch119872119894119897) = 0

(23)

From (23) we obtain1198627 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198628 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198623 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198624 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 0

(24)

From (24) we obtainch119873119894119897 cos119873119894119897 + 1 = 0ch119872119894119897 cos119872119894119897 + 1 = 0 (25)

Assume C3=1 C7=1 with1198621 = 1198625 = minus11198624 = minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 1198622 = sh119872119894119897 minus sin119872119894119897

ch119872119894119897 + cos119872119894119897 1198626 = sh119873119894119897 minus sin119873119894119897

ch119873119894119897 + cos119873119894119897 1198628 = minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897

(26)

Accordingly the mode function for upper and lower beamcan be obtained as follows119883119894 = minus cos119872119894119909 + ch119872119894119909

minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 (sh119872119894119909 minus sin119872119894119909)minus cos119873119894119909 + ch119873119894119909minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897 (sh119873119894119909 minus sin119873119894119909)

(27)

119885119894 = minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 119869119894 (sh119872119894119909 minus sin119872119894119909)minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897119870119894 (sh119873119894119909 minus sin119873119894119909)

(28)

Then 1199081 and 1199082 can be expressed as

1199081 = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] (119860 119894sdot sin119875119894119905 + 119861119894 sin119875119894119905)

(29)

1199082 = infinsum119894=1

[minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)] (119860 119894 sin119875119894119905 + 119861119894 sin119875119894119905)

(30)

with 120589 = (sh119872119894119897minussin119872119894119897)(ch119872119894119897+cos119872119894119897) and 120585 = (sh119873119894119897minussin119873119894119897)(ch119873119894119897 + cos119873119894119897)23 Particular Solution of Upper Beam Assume f (x t) as theforce imposed on upper beam then substituting it into (2)and we obtain

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 119891 (119909 119905) (31)

6 Mathematical Problems in Engineering

Make 1199081 = suminfin119894=1119883119894(119909)119884119894(119905) and then we obtain

infinsum119894=1

[11986411198681119883119894(4)119884119894 + 119898111988410158401015840119894 119883119894 + 1198881 (119883119894119884119894 minus 1119869119883119894119884119894)]= 119891 (119909 119905)

(32)

Make (32) multiplied by 119883119895(119909) and integrate it in thewhole beam considering (27) and the orthogonality of modefunction of upper beam and then obtaining

Y10158401015840119894 + 1198752119894 119884119894 = 119876119894 (119905) (33)

where 119876119894(119905) = int1198970 119891(119909 119905)119883119894(119909)119889119909 named i-order generalizedloads The general solution of (33) is

Y119894 (119905) = 1119875119894 int119905

0119876119894 (120591) sin119875119894 (119905 minus 120591) d120591 (34)

Solving (34) then we can obtain the dynamic response ofupper beam under load as follows

1199081 (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minuscos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] int119905

0119891 (119909 119905)

sdot sin119875119894 (119905 minus 120591) 119889120591 119889119909(35)

24 Particular Solution of Lower Beam For lower beamthe load imposed on it is the pressure from upper beamAccording to (3) we obtain

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + (1198881 + 1198882) 1199082 minus 11988811199081 = 119891 (119909 119905) (36)

where 11988811199081 determined by (35) is the load imposed on lowerbeam Make 1199082 = suminfin119894=1 119885119894(119909)Y119894(119905) and we obtain

infinsum119894=1

[11986421198682119885119894(4)119884119894 + 119898211988410158401015840119894 119885119894 + (1198881 + 1198882) 119885119894119884119894] = 11988811199081 (37)

Make (37) multiplied by 119885119895(119909) and integrate it in thewhole beam considering (28) and the orthogonality of modefunction of lower beam and then obtaining

11988410158401015840 + (1198752119894 + 11988811198691198941198982)119884119894 = 119876119894 (119905) (38)

where 119876119894(119905) = (1198981199051198692119894 1198982) int1198970 11988811199081119885119894119889119909 is defined as gener-alized loads imposed on the lower beam Then make 119899119894 =(1198752119894 + 11988811198691198941198982)12 and the general solution of (38) is

119884119894 (119905) = 1119899119894 int119905

0119876119894 (120591) sin 119899119894 (119905 minus 120591) d120591 (39)

Substituting 119876119894(119905) (28) and (39) into 1199082 = suminfin119894=1 119885119894(119909)119884119894(119905)we obtain the dynamic response of lower beam under load asfollows

1199082= infinsum119894=1

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894

sdot 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)

minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]sdot int11990501199081 (119909 120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(40)

where 1199081 is the dynamic response of upper beam under loadobtained from (35)

25 Expression of Uniformly Distributed Load According tothe loading status of upper and lower beam the functionfor variable uniformly distributed load can be expressed asfollows

119901119902(119909119905) = minus119902119867 (V119905 minus 119909) (0 le 119905 le 119897V) (41)

where 119902 is the distributed electromagnetic force between therails and119867(119909) is the Heaviside Function3 Displacement of Double-Layer Beam underVariable Uniformly Distributed Load

31 Derivation of Dynamic Deflection Curve Equations ofUpper Beam Based on (31) we can obtain the dynamic

Mathematical Problems in Engineering 7

equation of upper beam attributed to uniformly distributedload as follows 11986411198681 120597411990811205971199094 + 1198981 120597

211990811205971199052 + 1198881 (1199081 minus 1199082) = 119901119902(119909119905) (42)

Substituting 119901119902(119909119905) into (42) we obtain

1199081119902(119909119905) (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)]

sdot int1199050119901119902(119909120591) sin119875119894 (119905 minus 120591) 119889120591 119889119909

(43)

Considering the property of Heaviside Function and sub-stituting 119901119902(119909119905) = minus119902119867(V119905 minus 119909) into (43) we can obtainedthe dynamic response of upper beam under uniformly dis-tributed load q through the integral as follows

1199081119902(119909119905) (119909 119905)= infinsum119894=1

minus119902[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 )+ 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 )

minus 1198722119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 )+ 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 )minus 120585 [1198732119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(44)

32 Bending Moment and Stress of Upper Beam Having gotthe dynamic deflection curve equations and then consideringthe expression of1198721 and 1205901 we can directly obtain from (44)

1198721 = 11986411198681 119889211990811199021198891199092 =infinsum119894=1

minus11986411198681119902 [1198721198942 cos119872119894119909 +1198721198942ch119872119894119909 minus 1205891198721198942 (sh119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942ch119873119894119909 minus 1205851198731198942 (sh119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(45)

8 Mathematical Problems in Engineering

1205901 = 1198641ℎ12 119889211990811199021198891199092 =

infinsum119894=1

minus1198641ℎ11199022 [1198721198942 cos119872119894119909 +1198721198942119888ℎ119872119894119909 minus 1205891198721198942 (119904ℎ119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942119888ℎ119873119894119909 minus 1205851198731198942 (119904ℎ119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ] (46)

33 Derivation of Dynamic Deflection Curve Equations ofLower Beam According to the determined 1199081119902(xt) similarlywe obtain

1199082119902(119909119905) = infinsum119894=0

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894+ 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]

sdot int11990501199081119902(119909120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(47)

Substituting (44) into (47) and simplifying the solutionwe can obtain

1199082119902(119909119905) = minus 119902119888111989811990511989911989411989821198692119894 (1198831 + 1198832) (1198671 + 1198672) (1198791 + 1198792) (48)

where the expression of11986711198672 1198791 and 1198792 can be reviewedin the appendix

34 Bending Moment and Stress of Lower Beam

1198722= minus11990211988811198981199051198641119868111989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792) (49)

1205902= minus11990211988811198981199051198641ℎ2211989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792)

(50)

where119883110158401015840 = 1198722119894 119869119894(cosh119872119894119909+ cos119872119894119909)minus 1205891198722119894 119869119894(sinh119872119894119909+sin119872119894119909) and 119883210158401015840 = 1198732119894 119870119894(cosh119873119894119909 + cos119873119894119909) minus1205851198732119894 119870119894(sinh119873119894119909 + sin119873119894119909)

4 Instance Analysis

41 Parameters The dynamic model for electromagneticlaunching rail due to uniformly distributed load is shownin Figure 4 The length of the beam is l=3000mm Thesection sizes of upper beam are 1198671=45mm and ℎ1=15mmrespectively The section sizes of lower beam are 1198672=75mmand ℎ2=30mm respectively The material of upper beamis copper of which the elastic modulus is 1198641=110GPa andthe mass density is 1205881=8290kgm3 the material of lowerbeam is copper of which the elastic modulus is 1198642 =283GPaand the mass density is 1205882 =980kgm3 The elastic constantbetween upper and lower beam is 1198881=3MPa the elasticconstant between lower beamand foundation is 1198882=6MPaTheuniformly distributed force is119902=3000NmThevelocity of thearmature is assumed as1000ms

42 Consequence and Analysis In the case of given motionparameters and structure parameters we have calculated thedisplacement bending moment and stress of double-layerelastic cantilever foundation beam and obtained analyticalsolutions and numerical results got from ANSYS which areshown in Figures 5ndash12

Mathematical Problems in Engineering 9

RailInsulation

qq

H 1H2

h2 h1 h2h1b

Cover plate

Figure 4 The cross-section of electromagnetic rail launcher

0 05 1 15 2 25 3x (m)

minus14minus12minus10minus8minus6minus4minus2

024

W1

(mm

)

l1=05m

Matlab ResultANSYS Result

Figure 5 Comparison of analytical solution with numerical solution for deflection of upper beam

Figures 5ndash8 show the dynamic response of the upperbeam attributed to moving load Analytical solution andnumerical solution for deflection of upper beam are depictedin Figure 5 in which peak values of displacement of analyticaland numerical solution are similar and the values occur atclose positions However the analytical solution begins tofluctuate near x=05m after which the numerical solutionremains stable Figure 6 shows the changing displacementof upper beam at x = 15m in which the analytical solutionand the numerical solution are in good agreement Figures7 and 8 depict the bending moment and stress of upperbeam at different positions due to the motion of load Dueto the linear relationship between the bending moment andthe stress both of them are identical in trend Similar to thechange of displacement solution the numerical solution andthe analytical solution are close before x=05 and after thatthe analytical solution fluctuates

Meanwhile dynamic response of the lower beam is inves-tigated The displacement results of analytical and numericalsolution of lower beam differentiate obviously as shown inFigure 9 Figure 10 shows the displacement of lower beamchanging with time at x = 15m in which similar to upperbeam the analytical solution and the numerical solutionare in good agreement in trend In Figures 11 and 12 both

analytical and numerical solution trend of bending momentand stress of lower beam follow the same patterns of upperbeam in Figures 7 and 8

It is apparent that the peak values of dynamic responseincluding displacement bending moment and stress ofupper beam are bigger than that of lower beam becauseof upper beam dissipating majority of the vibration energydue to the moving load There are some analytical solu-tion fluctuations compared with numerical solution in Fig-ures 5 7 8 9 11 and 12 which may be aroused byneglecting the layer damping In summary the analyticaland numerical solutions of the dynamic response of theupper and lower beams in the above mentioned figuresare similar in the overall trend despite some acceptabledeviation

In order to investigate the properties of analytical andnumerical solutions more specifically we compare the peakvalues of displacement bending moment and stress (Figures5ndash12) of the upper and lower beams of dynamic response asshown in Table 1

It is clear that the relative error of the peak values betweenthe analytical solution and the numerical solution obtainedby ANSYS is within a reasonable range and fluctuates around5

10 Mathematical Problems in Engineering

0 05 1 15 2 25 3t (s)

minus3minus2minus1

01234

v=1000msx=15m Position

Matlab ResultANSYS Result

times10-3

W1

(mm

)

Figure 6 Upper beam x = 15m comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus700minus600minus500minus400minus300minus200minus100

0100200300

M1

(Nlowast

m)

Matlab ResultANSYS Result

l1=05m

Figure 7 Comparison of analytical solution with numerical solution for bending moment distribution of upper beam

0 05 1 15 2 25 3x (m)

minus400

minus300

minus200

minus100

0

100

200

Matlab ResultANSYS Result

l1=05m

1(MPa

)

Figure 8 Comparison of analytical solution with numerical solution of stress distribution on upper beam

0 05 1 15 2 25 3x (m)

minus2minus15minus1

minus050

051

152

W2

(mm

)

Matlab ResultANSYS Result

Figure 9 Comparison of analytical solution with numerical solution for deflection of lower beam

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Page 4: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

4 Mathematical Problems in Engineering

with 119869119894 = (119864111986811198721198944 + 1198881 minus 11989811198751198942)1198881 and119870119894 = (119864111986811198731198944 + 1198881 minus11989811198751198942)1198881Accordingly we obtain the bendingmoment of upper and

lower beam as follows

119872119906119901119901119890119903 = 11986411198681 120597211990811205972119909 = 11986411198681infinsum119894=1

12059721198831198941205971199092 119884119894 = [infinsum119894=1

11986411198681sdot (minus11986211198722119894 cos119872119894119909 minus 11986221198722119894 sin119872119894119909+ 11986231198722119894 ch119872119894119909 + 11986241198722119894 sh119872119894119909 minus 11986251198732119894 cos119873119894119909minus 11986261198732119894 sin119873119894119909 + 11986271198732119894 ch119873119894119909 + 11986281198732119894 sh119873119894119909)]sdot (119860 119894 sin119875119894119905 + 119861119894 cos119875119894119905)

(12)

119872119897119900119908119890119903 = 11986421198682 120597211990821205972119909 = 11986421198682infinsum119894=1

12059721198851198941205971199092 119884119894 = [infinsum119894=1

11986421198682sdot (minus11986211198691198941198722119894 cos119872119894119909 minus 11986221198691198941198722119894 sin119872119894119909+ 11986231198691198941198722119894 ch119872119894119909 + 11986241198691198941198722119894 sh119872119894119909minus 11986251198701198941198732119894 cos119873119894119909 minus 11986261198701198941198732119894 sin119873119894119909+ 11986271198701198941198732119894 ch119873119894119909 + 11986281198701198941198732119894 sh119873119894119909)] (119860 119894 sin119875119894119905+ 119861119894 cos119875119894119905)

(13)

The rotation angles of upper and lower beam are

1205791 = 11986411198681 1205971199081120597119909 = 11986411198681infinsum119894=1

120597119883119894120597119909 119884119894 = [infinsum119894=1

11986411198681sdot (minus1198621119872119894 sin119872119894119909 + 1198622119872119894 cos119872119894119909 + 1198623119872119894sh119872119894119909+ 1198624119872119894ch119872119894119909 minus 1198625119873119894 sin119873119894119909 + 1198626119873119894 cos119873119894119909+ 1198627119873119894sh119873119894119909 + 1198628119873119894ch119873119894119909)]

(14)

1205792 = 11986421198682 1205971199082120597119909 = 11986421198682infinsum119894=1

120597119885119894120597119909 119884119894 = [infinsum119894=1

11986421198682sdot (minus1198621119869119894119872119894 sin119872119894119909 + 1198622119869119894119872119894 cos119872119894119909+ 1198623119869119894119872119894sh119872119894119909 + 1198624119869119894119872119894ch119872119894119909minus 1198625119870119894119873119894 sin119873119894119909 + 1198626119870119894119873119894 cos119873119894119909+ 1198627119870119894119873119894sh119873119894119909 + 1198628119870119894119873119894ch119873119894119909)]

(15)

The shear forces of upper and lower beam are

1198761 = 11986411198681 120597311990811205971199093 = 11986411198681infinsum119894=1

12059731198831198941205971199093 119884119894= infinsum119894=1

(11986211198723119894 sin119872119894119909 minus 11986221198723119894 cos119872119894119909+ 11986231198723119894 sh119872119894119909 + 11986241198723119894 ch119872119894119909 + 11986251198733119894 sin119873119894119909minus 11986261198733119894 cos119873119894119909 + 11986271198733119894 sh119873119894119909 + 11986281198733119894 ch119873119894119909)

(16)

1198762 = 11986421198682 120597311990821205971199093 = 11986421198682infinsum119894=1

12059731198851198941205971199093 119884119894 =infinsum119894=1

11986421198682 (11986211198691198941198723119894sdot sin119872119894119909 minus 11986221198691198941198723119894 cos119872119894119909 + 11986231198691198941198723119894 sh119872119894119909+ 11986241198691198941198723119894 ch119872119894119909 + 11986251198701198941198733119894 sin119873119894119909 minus 11986261198701198941198733119894sdot cos119873119894119909 + 11986271198701198941198733119894 sh119873119894119909 + 11986281198701198941198733119894 ch119873119894119909)

(17)

where the constants 1198621sim1198628 can be determined from bound-ary conditions of the rail According to the double-layerbeam model where one end of the rail is constrained andthe another is under condition of freedom we can obtain theboundary conditions of both ends as follows

11990811003816100381610038161003816119909=0 = 012057911003816100381610038161003816119909=0 = 011990821003816100381610038161003816119909=0 = 012057921003816100381610038161003816119909=0 = 011987211003816100381610038161003816119909=119897 = 011987611003816100381610038161003816119909=119897 = 011987221003816100381610038161003816119909=119897 = 011987621003816100381610038161003816119909=119897 = 0

(18)

Substituting (6) (7) (14) and (15) into (18) from the leftboundary conditions we can obtain

1198621 + 1198623 + 1198625 + 1198627 = 01198622119872119894 + 1198624119872119894 + 1198626119873119894 + 1198628119873119894 = 01198621119869119894 + 1198623119869119894 + 1198625119870119894 + 1198627119870119894 = 0

1198622119869119894119872119894 + 1198624119869119894119872119894 + 1198626119870119894119873119894 + 1198628119870119894119873119894 = 0(19)

Substituting (12) (13) (16) and (17) into (18) from the rightboundary conditions we can obtain

minus 11986211198722119894 cos119872119894119897 minus 11986221198722119894 sin119872119894119897 + 11986231198722119894 ch119872119894119897+ 11986241198722119894 sh119872119894119897 minus 11986251198732119894 cos119873119894119897 minus 11986261198732119894 sin119873119894119897+ 11986271198732119894 ch119873119894119897 + 11986281198732119894 sh119873119894119897 = 0

Mathematical Problems in Engineering 5

11986211198723119894 sin119872119894119897 minus 11986221198723119894 cos119872119894119897 + 11986231198723119894 sh119872119894119897minus 11986241198723119894 ch119872119894119897 + 11986251198733119894 sin119873119894119897 minus 11986261198733119894 cos119873119894119897+ 11986271198733119894 sh119873119894119897 + 11986281198733119894 ch119873119894119897 = 0

minus 11986211198691198941198722119894 cos119872119894119897 minus 11986221198691198941198722119894 sin119872119894119897 + 11986231198691198941198722119894 ch119872119894119897+ 11986241198691198941198722119894 sh119872119894119897 minus 11986251198701198941198732119894 cos119873119894119897minus 11986261198701198941198732119894 sin119873119894119897 + 11986271198701198941198732119894 ch119873119894119897+ 11986281198701198941198732119894 sh119873119894119897 = 0

11986211198691198941198723119894 sin119872119894119897 minus 11986221198691198941198723119894 cos119872119894119897 + 11986231198691198941198723119894 sh119872119894119897minus 11986241198691198941198691198941198723119894 ch119872119894119897 + 11986251198701198941198733119894 sin119873119894119897minus 11986261198701198941198733119894 cos119873119894119897 + 11986271198701198941198733119894 sh119873119894119897+ 11986281198701198941198733119894 ch119873119894119897 = 0

(20)

From (19) we obtain 1198621 = minus11986231198622 = minus11986241198625 = minus11986271198626 = minus1198628

(21)

From (20) we obtainminus1198625 cos119873119894119897 minus 1198626 sin119873119894119897 + 1198627ch119873119894119897 + 1198628sh119873119894119897 = 0minus1198621 cos119872119894119897 minus 1198622 sin119872119894119897 + 1198623ch119872119894119897 + 1198624sh119872119894119897 = 01198625 sin119873119894119897 minus 1198626 cos119873119894119897 + 1198627sh119873119894119897 + 1198628ch119873119894119897 = 01198621 sin119872119894119897 minus 1198622 cos119872119894119897 + 1198623sh119872119894119897 + 1198624ch119872119894119897 = 0

(22)

Applying (21) into (22) we obtain

1198627 (cos119873119894119897 + ch119873119894119897) + 1198628 (sin119873119894119897 + sh119873119894119897) = 01198623 (cos119872119894119897 + ch119872119894119897) + 1198624 (sin119872119894119897 + sh119872119894119897) = 01198627 (minus sin119873119894119897 + sh119873119894119897) + C8 (cos119873119894119897 + ch119873119894119897) = 01198623 (minus sin119872119894119897 + sh119872119894119897) + C4 (cos119872119894119897 + ch119872119894119897) = 0

(23)

From (23) we obtain1198627 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198628 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198623 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198624 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 0

(24)

From (24) we obtainch119873119894119897 cos119873119894119897 + 1 = 0ch119872119894119897 cos119872119894119897 + 1 = 0 (25)

Assume C3=1 C7=1 with1198621 = 1198625 = minus11198624 = minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 1198622 = sh119872119894119897 minus sin119872119894119897

ch119872119894119897 + cos119872119894119897 1198626 = sh119873119894119897 minus sin119873119894119897

ch119873119894119897 + cos119873119894119897 1198628 = minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897

(26)

Accordingly the mode function for upper and lower beamcan be obtained as follows119883119894 = minus cos119872119894119909 + ch119872119894119909

minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 (sh119872119894119909 minus sin119872119894119909)minus cos119873119894119909 + ch119873119894119909minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897 (sh119873119894119909 minus sin119873119894119909)

(27)

119885119894 = minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 119869119894 (sh119872119894119909 minus sin119872119894119909)minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897119870119894 (sh119873119894119909 minus sin119873119894119909)

(28)

Then 1199081 and 1199082 can be expressed as

1199081 = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] (119860 119894sdot sin119875119894119905 + 119861119894 sin119875119894119905)

(29)

1199082 = infinsum119894=1

[minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)] (119860 119894 sin119875119894119905 + 119861119894 sin119875119894119905)

(30)

with 120589 = (sh119872119894119897minussin119872119894119897)(ch119872119894119897+cos119872119894119897) and 120585 = (sh119873119894119897minussin119873119894119897)(ch119873119894119897 + cos119873119894119897)23 Particular Solution of Upper Beam Assume f (x t) as theforce imposed on upper beam then substituting it into (2)and we obtain

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 119891 (119909 119905) (31)

6 Mathematical Problems in Engineering

Make 1199081 = suminfin119894=1119883119894(119909)119884119894(119905) and then we obtain

infinsum119894=1

[11986411198681119883119894(4)119884119894 + 119898111988410158401015840119894 119883119894 + 1198881 (119883119894119884119894 minus 1119869119883119894119884119894)]= 119891 (119909 119905)

(32)

Make (32) multiplied by 119883119895(119909) and integrate it in thewhole beam considering (27) and the orthogonality of modefunction of upper beam and then obtaining

Y10158401015840119894 + 1198752119894 119884119894 = 119876119894 (119905) (33)

where 119876119894(119905) = int1198970 119891(119909 119905)119883119894(119909)119889119909 named i-order generalizedloads The general solution of (33) is

Y119894 (119905) = 1119875119894 int119905

0119876119894 (120591) sin119875119894 (119905 minus 120591) d120591 (34)

Solving (34) then we can obtain the dynamic response ofupper beam under load as follows

1199081 (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minuscos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] int119905

0119891 (119909 119905)

sdot sin119875119894 (119905 minus 120591) 119889120591 119889119909(35)

24 Particular Solution of Lower Beam For lower beamthe load imposed on it is the pressure from upper beamAccording to (3) we obtain

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + (1198881 + 1198882) 1199082 minus 11988811199081 = 119891 (119909 119905) (36)

where 11988811199081 determined by (35) is the load imposed on lowerbeam Make 1199082 = suminfin119894=1 119885119894(119909)Y119894(119905) and we obtain

infinsum119894=1

[11986421198682119885119894(4)119884119894 + 119898211988410158401015840119894 119885119894 + (1198881 + 1198882) 119885119894119884119894] = 11988811199081 (37)

Make (37) multiplied by 119885119895(119909) and integrate it in thewhole beam considering (28) and the orthogonality of modefunction of lower beam and then obtaining

11988410158401015840 + (1198752119894 + 11988811198691198941198982)119884119894 = 119876119894 (119905) (38)

where 119876119894(119905) = (1198981199051198692119894 1198982) int1198970 11988811199081119885119894119889119909 is defined as gener-alized loads imposed on the lower beam Then make 119899119894 =(1198752119894 + 11988811198691198941198982)12 and the general solution of (38) is

119884119894 (119905) = 1119899119894 int119905

0119876119894 (120591) sin 119899119894 (119905 minus 120591) d120591 (39)

Substituting 119876119894(119905) (28) and (39) into 1199082 = suminfin119894=1 119885119894(119909)119884119894(119905)we obtain the dynamic response of lower beam under load asfollows

1199082= infinsum119894=1

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894

sdot 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)

minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]sdot int11990501199081 (119909 120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(40)

where 1199081 is the dynamic response of upper beam under loadobtained from (35)

25 Expression of Uniformly Distributed Load According tothe loading status of upper and lower beam the functionfor variable uniformly distributed load can be expressed asfollows

119901119902(119909119905) = minus119902119867 (V119905 minus 119909) (0 le 119905 le 119897V) (41)

where 119902 is the distributed electromagnetic force between therails and119867(119909) is the Heaviside Function3 Displacement of Double-Layer Beam underVariable Uniformly Distributed Load

31 Derivation of Dynamic Deflection Curve Equations ofUpper Beam Based on (31) we can obtain the dynamic

Mathematical Problems in Engineering 7

equation of upper beam attributed to uniformly distributedload as follows 11986411198681 120597411990811205971199094 + 1198981 120597

211990811205971199052 + 1198881 (1199081 minus 1199082) = 119901119902(119909119905) (42)

Substituting 119901119902(119909119905) into (42) we obtain

1199081119902(119909119905) (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)]

sdot int1199050119901119902(119909120591) sin119875119894 (119905 minus 120591) 119889120591 119889119909

(43)

Considering the property of Heaviside Function and sub-stituting 119901119902(119909119905) = minus119902119867(V119905 minus 119909) into (43) we can obtainedthe dynamic response of upper beam under uniformly dis-tributed load q through the integral as follows

1199081119902(119909119905) (119909 119905)= infinsum119894=1

minus119902[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 )+ 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 )

minus 1198722119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 )+ 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 )minus 120585 [1198732119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(44)

32 Bending Moment and Stress of Upper Beam Having gotthe dynamic deflection curve equations and then consideringthe expression of1198721 and 1205901 we can directly obtain from (44)

1198721 = 11986411198681 119889211990811199021198891199092 =infinsum119894=1

minus11986411198681119902 [1198721198942 cos119872119894119909 +1198721198942ch119872119894119909 minus 1205891198721198942 (sh119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942ch119873119894119909 minus 1205851198731198942 (sh119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(45)

8 Mathematical Problems in Engineering

1205901 = 1198641ℎ12 119889211990811199021198891199092 =

infinsum119894=1

minus1198641ℎ11199022 [1198721198942 cos119872119894119909 +1198721198942119888ℎ119872119894119909 minus 1205891198721198942 (119904ℎ119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942119888ℎ119873119894119909 minus 1205851198731198942 (119904ℎ119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ] (46)

33 Derivation of Dynamic Deflection Curve Equations ofLower Beam According to the determined 1199081119902(xt) similarlywe obtain

1199082119902(119909119905) = infinsum119894=0

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894+ 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]

sdot int11990501199081119902(119909120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(47)

Substituting (44) into (47) and simplifying the solutionwe can obtain

1199082119902(119909119905) = minus 119902119888111989811990511989911989411989821198692119894 (1198831 + 1198832) (1198671 + 1198672) (1198791 + 1198792) (48)

where the expression of11986711198672 1198791 and 1198792 can be reviewedin the appendix

34 Bending Moment and Stress of Lower Beam

1198722= minus11990211988811198981199051198641119868111989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792) (49)

1205902= minus11990211988811198981199051198641ℎ2211989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792)

(50)

where119883110158401015840 = 1198722119894 119869119894(cosh119872119894119909+ cos119872119894119909)minus 1205891198722119894 119869119894(sinh119872119894119909+sin119872119894119909) and 119883210158401015840 = 1198732119894 119870119894(cosh119873119894119909 + cos119873119894119909) minus1205851198732119894 119870119894(sinh119873119894119909 + sin119873119894119909)

4 Instance Analysis

41 Parameters The dynamic model for electromagneticlaunching rail due to uniformly distributed load is shownin Figure 4 The length of the beam is l=3000mm Thesection sizes of upper beam are 1198671=45mm and ℎ1=15mmrespectively The section sizes of lower beam are 1198672=75mmand ℎ2=30mm respectively The material of upper beamis copper of which the elastic modulus is 1198641=110GPa andthe mass density is 1205881=8290kgm3 the material of lowerbeam is copper of which the elastic modulus is 1198642 =283GPaand the mass density is 1205882 =980kgm3 The elastic constantbetween upper and lower beam is 1198881=3MPa the elasticconstant between lower beamand foundation is 1198882=6MPaTheuniformly distributed force is119902=3000NmThevelocity of thearmature is assumed as1000ms

42 Consequence and Analysis In the case of given motionparameters and structure parameters we have calculated thedisplacement bending moment and stress of double-layerelastic cantilever foundation beam and obtained analyticalsolutions and numerical results got from ANSYS which areshown in Figures 5ndash12

Mathematical Problems in Engineering 9

RailInsulation

qq

H 1H2

h2 h1 h2h1b

Cover plate

Figure 4 The cross-section of electromagnetic rail launcher

0 05 1 15 2 25 3x (m)

minus14minus12minus10minus8minus6minus4minus2

024

W1

(mm

)

l1=05m

Matlab ResultANSYS Result

Figure 5 Comparison of analytical solution with numerical solution for deflection of upper beam

Figures 5ndash8 show the dynamic response of the upperbeam attributed to moving load Analytical solution andnumerical solution for deflection of upper beam are depictedin Figure 5 in which peak values of displacement of analyticaland numerical solution are similar and the values occur atclose positions However the analytical solution begins tofluctuate near x=05m after which the numerical solutionremains stable Figure 6 shows the changing displacementof upper beam at x = 15m in which the analytical solutionand the numerical solution are in good agreement Figures7 and 8 depict the bending moment and stress of upperbeam at different positions due to the motion of load Dueto the linear relationship between the bending moment andthe stress both of them are identical in trend Similar to thechange of displacement solution the numerical solution andthe analytical solution are close before x=05 and after thatthe analytical solution fluctuates

Meanwhile dynamic response of the lower beam is inves-tigated The displacement results of analytical and numericalsolution of lower beam differentiate obviously as shown inFigure 9 Figure 10 shows the displacement of lower beamchanging with time at x = 15m in which similar to upperbeam the analytical solution and the numerical solutionare in good agreement in trend In Figures 11 and 12 both

analytical and numerical solution trend of bending momentand stress of lower beam follow the same patterns of upperbeam in Figures 7 and 8

It is apparent that the peak values of dynamic responseincluding displacement bending moment and stress ofupper beam are bigger than that of lower beam becauseof upper beam dissipating majority of the vibration energydue to the moving load There are some analytical solu-tion fluctuations compared with numerical solution in Fig-ures 5 7 8 9 11 and 12 which may be aroused byneglecting the layer damping In summary the analyticaland numerical solutions of the dynamic response of theupper and lower beams in the above mentioned figuresare similar in the overall trend despite some acceptabledeviation

In order to investigate the properties of analytical andnumerical solutions more specifically we compare the peakvalues of displacement bending moment and stress (Figures5ndash12) of the upper and lower beams of dynamic response asshown in Table 1

It is clear that the relative error of the peak values betweenthe analytical solution and the numerical solution obtainedby ANSYS is within a reasonable range and fluctuates around5

10 Mathematical Problems in Engineering

0 05 1 15 2 25 3t (s)

minus3minus2minus1

01234

v=1000msx=15m Position

Matlab ResultANSYS Result

times10-3

W1

(mm

)

Figure 6 Upper beam x = 15m comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus700minus600minus500minus400minus300minus200minus100

0100200300

M1

(Nlowast

m)

Matlab ResultANSYS Result

l1=05m

Figure 7 Comparison of analytical solution with numerical solution for bending moment distribution of upper beam

0 05 1 15 2 25 3x (m)

minus400

minus300

minus200

minus100

0

100

200

Matlab ResultANSYS Result

l1=05m

1(MPa

)

Figure 8 Comparison of analytical solution with numerical solution of stress distribution on upper beam

0 05 1 15 2 25 3x (m)

minus2minus15minus1

minus050

051

152

W2

(mm

)

Matlab ResultANSYS Result

Figure 9 Comparison of analytical solution with numerical solution for deflection of lower beam

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Mathematical Problems in Engineering

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Page 5: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

Mathematical Problems in Engineering 5

11986211198723119894 sin119872119894119897 minus 11986221198723119894 cos119872119894119897 + 11986231198723119894 sh119872119894119897minus 11986241198723119894 ch119872119894119897 + 11986251198733119894 sin119873119894119897 minus 11986261198733119894 cos119873119894119897+ 11986271198733119894 sh119873119894119897 + 11986281198733119894 ch119873119894119897 = 0

minus 11986211198691198941198722119894 cos119872119894119897 minus 11986221198691198941198722119894 sin119872119894119897 + 11986231198691198941198722119894 ch119872119894119897+ 11986241198691198941198722119894 sh119872119894119897 minus 11986251198701198941198732119894 cos119873119894119897minus 11986261198701198941198732119894 sin119873119894119897 + 11986271198701198941198732119894 ch119873119894119897+ 11986281198701198941198732119894 sh119873119894119897 = 0

11986211198691198941198723119894 sin119872119894119897 minus 11986221198691198941198723119894 cos119872119894119897 + 11986231198691198941198723119894 sh119872119894119897minus 11986241198691198941198691198941198723119894 ch119872119894119897 + 11986251198701198941198733119894 sin119873119894119897minus 11986261198701198941198733119894 cos119873119894119897 + 11986271198701198941198733119894 sh119873119894119897+ 11986281198701198941198733119894 ch119873119894119897 = 0

(20)

From (19) we obtain 1198621 = minus11986231198622 = minus11986241198625 = minus11986271198626 = minus1198628

(21)

From (20) we obtainminus1198625 cos119873119894119897 minus 1198626 sin119873119894119897 + 1198627ch119873119894119897 + 1198628sh119873119894119897 = 0minus1198621 cos119872119894119897 minus 1198622 sin119872119894119897 + 1198623ch119872119894119897 + 1198624sh119872119894119897 = 01198625 sin119873119894119897 minus 1198626 cos119873119894119897 + 1198627sh119873119894119897 + 1198628ch119873119894119897 = 01198621 sin119872119894119897 minus 1198622 cos119872119894119897 + 1198623sh119872119894119897 + 1198624ch119872119894119897 = 0

(22)

Applying (21) into (22) we obtain

1198627 (cos119873119894119897 + ch119873119894119897) + 1198628 (sin119873119894119897 + sh119873119894119897) = 01198623 (cos119872119894119897 + ch119872119894119897) + 1198624 (sin119872119894119897 + sh119872119894119897) = 01198627 (minus sin119873119894119897 + sh119873119894119897) + C8 (cos119873119894119897 + ch119873119894119897) = 01198623 (minus sin119872119894119897 + sh119872119894119897) + C4 (cos119872119894119897 + ch119872119894119897) = 0

(23)

From (23) we obtain1198627 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198628 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198623 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 01198624 (119888ℎ119873119894119897 cos119873119894119897 + 1) = 0

(24)

From (24) we obtainch119873119894119897 cos119873119894119897 + 1 = 0ch119872119894119897 cos119872119894119897 + 1 = 0 (25)

Assume C3=1 C7=1 with1198621 = 1198625 = minus11198624 = minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 1198622 = sh119872119894119897 minus sin119872119894119897

ch119872119894119897 + cos119872119894119897 1198626 = sh119873119894119897 minus sin119873119894119897

ch119873119894119897 + cos119873119894119897 1198628 = minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897

(26)

Accordingly the mode function for upper and lower beamcan be obtained as follows119883119894 = minus cos119872119894119909 + ch119872119894119909

minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 (sh119872119894119909 minus sin119872119894119909)minus cos119873119894119909 + ch119873119894119909minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897 (sh119873119894119909 minus sin119873119894119909)

(27)

119885119894 = minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909minus sh119872119894119897 minus sin119872119894119897ch119872119894119897 + cos119872119894119897 119869119894 (sh119872119894119909 minus sin119872119894119909)minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909minus sh119873119894119897 minus sin119873119894119897ch119873119894119897 + cos119873119894119897119870119894 (sh119873119894119909 minus sin119873119894119909)

(28)

Then 1199081 and 1199082 can be expressed as

1199081 = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] (119860 119894sdot sin119875119894119905 + 119861119894 sin119875119894119905)

(29)

1199082 = infinsum119894=1

[minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)] (119860 119894 sin119875119894119905 + 119861119894 sin119875119894119905)

(30)

with 120589 = (sh119872119894119897minussin119872119894119897)(ch119872119894119897+cos119872119894119897) and 120585 = (sh119873119894119897minussin119873119894119897)(ch119873119894119897 + cos119873119894119897)23 Particular Solution of Upper Beam Assume f (x t) as theforce imposed on upper beam then substituting it into (2)and we obtain

11986411198681 120597411990811205971199094 + 1198981 120597211990811205971199052 + 1198881 (1199081 minus 1199082) = 119891 (119909 119905) (31)

6 Mathematical Problems in Engineering

Make 1199081 = suminfin119894=1119883119894(119909)119884119894(119905) and then we obtain

infinsum119894=1

[11986411198681119883119894(4)119884119894 + 119898111988410158401015840119894 119883119894 + 1198881 (119883119894119884119894 minus 1119869119883119894119884119894)]= 119891 (119909 119905)

(32)

Make (32) multiplied by 119883119895(119909) and integrate it in thewhole beam considering (27) and the orthogonality of modefunction of upper beam and then obtaining

Y10158401015840119894 + 1198752119894 119884119894 = 119876119894 (119905) (33)

where 119876119894(119905) = int1198970 119891(119909 119905)119883119894(119909)119889119909 named i-order generalizedloads The general solution of (33) is

Y119894 (119905) = 1119875119894 int119905

0119876119894 (120591) sin119875119894 (119905 minus 120591) d120591 (34)

Solving (34) then we can obtain the dynamic response ofupper beam under load as follows

1199081 (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minuscos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] int119905

0119891 (119909 119905)

sdot sin119875119894 (119905 minus 120591) 119889120591 119889119909(35)

24 Particular Solution of Lower Beam For lower beamthe load imposed on it is the pressure from upper beamAccording to (3) we obtain

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + (1198881 + 1198882) 1199082 minus 11988811199081 = 119891 (119909 119905) (36)

where 11988811199081 determined by (35) is the load imposed on lowerbeam Make 1199082 = suminfin119894=1 119885119894(119909)Y119894(119905) and we obtain

infinsum119894=1

[11986421198682119885119894(4)119884119894 + 119898211988410158401015840119894 119885119894 + (1198881 + 1198882) 119885119894119884119894] = 11988811199081 (37)

Make (37) multiplied by 119885119895(119909) and integrate it in thewhole beam considering (28) and the orthogonality of modefunction of lower beam and then obtaining

11988410158401015840 + (1198752119894 + 11988811198691198941198982)119884119894 = 119876119894 (119905) (38)

where 119876119894(119905) = (1198981199051198692119894 1198982) int1198970 11988811199081119885119894119889119909 is defined as gener-alized loads imposed on the lower beam Then make 119899119894 =(1198752119894 + 11988811198691198941198982)12 and the general solution of (38) is

119884119894 (119905) = 1119899119894 int119905

0119876119894 (120591) sin 119899119894 (119905 minus 120591) d120591 (39)

Substituting 119876119894(119905) (28) and (39) into 1199082 = suminfin119894=1 119885119894(119909)119884119894(119905)we obtain the dynamic response of lower beam under load asfollows

1199082= infinsum119894=1

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894

sdot 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)

minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]sdot int11990501199081 (119909 120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(40)

where 1199081 is the dynamic response of upper beam under loadobtained from (35)

25 Expression of Uniformly Distributed Load According tothe loading status of upper and lower beam the functionfor variable uniformly distributed load can be expressed asfollows

119901119902(119909119905) = minus119902119867 (V119905 minus 119909) (0 le 119905 le 119897V) (41)

where 119902 is the distributed electromagnetic force between therails and119867(119909) is the Heaviside Function3 Displacement of Double-Layer Beam underVariable Uniformly Distributed Load

31 Derivation of Dynamic Deflection Curve Equations ofUpper Beam Based on (31) we can obtain the dynamic

Mathematical Problems in Engineering 7

equation of upper beam attributed to uniformly distributedload as follows 11986411198681 120597411990811205971199094 + 1198981 120597

211990811205971199052 + 1198881 (1199081 minus 1199082) = 119901119902(119909119905) (42)

Substituting 119901119902(119909119905) into (42) we obtain

1199081119902(119909119905) (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)]

sdot int1199050119901119902(119909120591) sin119875119894 (119905 minus 120591) 119889120591 119889119909

(43)

Considering the property of Heaviside Function and sub-stituting 119901119902(119909119905) = minus119902119867(V119905 minus 119909) into (43) we can obtainedthe dynamic response of upper beam under uniformly dis-tributed load q through the integral as follows

1199081119902(119909119905) (119909 119905)= infinsum119894=1

minus119902[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 )+ 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 )

minus 1198722119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 )+ 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 )minus 120585 [1198732119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(44)

32 Bending Moment and Stress of Upper Beam Having gotthe dynamic deflection curve equations and then consideringthe expression of1198721 and 1205901 we can directly obtain from (44)

1198721 = 11986411198681 119889211990811199021198891199092 =infinsum119894=1

minus11986411198681119902 [1198721198942 cos119872119894119909 +1198721198942ch119872119894119909 minus 1205891198721198942 (sh119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942ch119873119894119909 minus 1205851198731198942 (sh119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(45)

8 Mathematical Problems in Engineering

1205901 = 1198641ℎ12 119889211990811199021198891199092 =

infinsum119894=1

minus1198641ℎ11199022 [1198721198942 cos119872119894119909 +1198721198942119888ℎ119872119894119909 minus 1205891198721198942 (119904ℎ119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942119888ℎ119873119894119909 minus 1205851198731198942 (119904ℎ119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ] (46)

33 Derivation of Dynamic Deflection Curve Equations ofLower Beam According to the determined 1199081119902(xt) similarlywe obtain

1199082119902(119909119905) = infinsum119894=0

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894+ 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]

sdot int11990501199081119902(119909120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(47)

Substituting (44) into (47) and simplifying the solutionwe can obtain

1199082119902(119909119905) = minus 119902119888111989811990511989911989411989821198692119894 (1198831 + 1198832) (1198671 + 1198672) (1198791 + 1198792) (48)

where the expression of11986711198672 1198791 and 1198792 can be reviewedin the appendix

34 Bending Moment and Stress of Lower Beam

1198722= minus11990211988811198981199051198641119868111989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792) (49)

1205902= minus11990211988811198981199051198641ℎ2211989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792)

(50)

where119883110158401015840 = 1198722119894 119869119894(cosh119872119894119909+ cos119872119894119909)minus 1205891198722119894 119869119894(sinh119872119894119909+sin119872119894119909) and 119883210158401015840 = 1198732119894 119870119894(cosh119873119894119909 + cos119873119894119909) minus1205851198732119894 119870119894(sinh119873119894119909 + sin119873119894119909)

4 Instance Analysis

41 Parameters The dynamic model for electromagneticlaunching rail due to uniformly distributed load is shownin Figure 4 The length of the beam is l=3000mm Thesection sizes of upper beam are 1198671=45mm and ℎ1=15mmrespectively The section sizes of lower beam are 1198672=75mmand ℎ2=30mm respectively The material of upper beamis copper of which the elastic modulus is 1198641=110GPa andthe mass density is 1205881=8290kgm3 the material of lowerbeam is copper of which the elastic modulus is 1198642 =283GPaand the mass density is 1205882 =980kgm3 The elastic constantbetween upper and lower beam is 1198881=3MPa the elasticconstant between lower beamand foundation is 1198882=6MPaTheuniformly distributed force is119902=3000NmThevelocity of thearmature is assumed as1000ms

42 Consequence and Analysis In the case of given motionparameters and structure parameters we have calculated thedisplacement bending moment and stress of double-layerelastic cantilever foundation beam and obtained analyticalsolutions and numerical results got from ANSYS which areshown in Figures 5ndash12

Mathematical Problems in Engineering 9

RailInsulation

qq

H 1H2

h2 h1 h2h1b

Cover plate

Figure 4 The cross-section of electromagnetic rail launcher

0 05 1 15 2 25 3x (m)

minus14minus12minus10minus8minus6minus4minus2

024

W1

(mm

)

l1=05m

Matlab ResultANSYS Result

Figure 5 Comparison of analytical solution with numerical solution for deflection of upper beam

Figures 5ndash8 show the dynamic response of the upperbeam attributed to moving load Analytical solution andnumerical solution for deflection of upper beam are depictedin Figure 5 in which peak values of displacement of analyticaland numerical solution are similar and the values occur atclose positions However the analytical solution begins tofluctuate near x=05m after which the numerical solutionremains stable Figure 6 shows the changing displacementof upper beam at x = 15m in which the analytical solutionand the numerical solution are in good agreement Figures7 and 8 depict the bending moment and stress of upperbeam at different positions due to the motion of load Dueto the linear relationship between the bending moment andthe stress both of them are identical in trend Similar to thechange of displacement solution the numerical solution andthe analytical solution are close before x=05 and after thatthe analytical solution fluctuates

Meanwhile dynamic response of the lower beam is inves-tigated The displacement results of analytical and numericalsolution of lower beam differentiate obviously as shown inFigure 9 Figure 10 shows the displacement of lower beamchanging with time at x = 15m in which similar to upperbeam the analytical solution and the numerical solutionare in good agreement in trend In Figures 11 and 12 both

analytical and numerical solution trend of bending momentand stress of lower beam follow the same patterns of upperbeam in Figures 7 and 8

It is apparent that the peak values of dynamic responseincluding displacement bending moment and stress ofupper beam are bigger than that of lower beam becauseof upper beam dissipating majority of the vibration energydue to the moving load There are some analytical solu-tion fluctuations compared with numerical solution in Fig-ures 5 7 8 9 11 and 12 which may be aroused byneglecting the layer damping In summary the analyticaland numerical solutions of the dynamic response of theupper and lower beams in the above mentioned figuresare similar in the overall trend despite some acceptabledeviation

In order to investigate the properties of analytical andnumerical solutions more specifically we compare the peakvalues of displacement bending moment and stress (Figures5ndash12) of the upper and lower beams of dynamic response asshown in Table 1

It is clear that the relative error of the peak values betweenthe analytical solution and the numerical solution obtainedby ANSYS is within a reasonable range and fluctuates around5

10 Mathematical Problems in Engineering

0 05 1 15 2 25 3t (s)

minus3minus2minus1

01234

v=1000msx=15m Position

Matlab ResultANSYS Result

times10-3

W1

(mm

)

Figure 6 Upper beam x = 15m comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus700minus600minus500minus400minus300minus200minus100

0100200300

M1

(Nlowast

m)

Matlab ResultANSYS Result

l1=05m

Figure 7 Comparison of analytical solution with numerical solution for bending moment distribution of upper beam

0 05 1 15 2 25 3x (m)

minus400

minus300

minus200

minus100

0

100

200

Matlab ResultANSYS Result

l1=05m

1(MPa

)

Figure 8 Comparison of analytical solution with numerical solution of stress distribution on upper beam

0 05 1 15 2 25 3x (m)

minus2minus15minus1

minus050

051

152

W2

(mm

)

Matlab ResultANSYS Result

Figure 9 Comparison of analytical solution with numerical solution for deflection of lower beam

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Page 6: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

6 Mathematical Problems in Engineering

Make 1199081 = suminfin119894=1119883119894(119909)119884119894(119905) and then we obtain

infinsum119894=1

[11986411198681119883119894(4)119884119894 + 119898111988410158401015840119894 119883119894 + 1198881 (119883119894119884119894 minus 1119869119883119894119884119894)]= 119891 (119909 119905)

(32)

Make (32) multiplied by 119883119895(119909) and integrate it in thewhole beam considering (27) and the orthogonality of modefunction of upper beam and then obtaining

Y10158401015840119894 + 1198752119894 119884119894 = 119876119894 (119905) (33)

where 119876119894(119905) = int1198970 119891(119909 119905)119883119894(119909)119889119909 named i-order generalizedloads The general solution of (33) is

Y119894 (119905) = 1119875119894 int119905

0119876119894 (120591) sin119875119894 (119905 minus 120591) d120591 (34)

Solving (34) then we can obtain the dynamic response ofupper beam under load as follows

1199081 (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minuscos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)] int119905

0119891 (119909 119905)

sdot sin119875119894 (119905 minus 120591) 119889120591 119889119909(35)

24 Particular Solution of Lower Beam For lower beamthe load imposed on it is the pressure from upper beamAccording to (3) we obtain

11986421198682 120597411990821205971199094 + 1198982 120597211990821205971199052 + (1198881 + 1198882) 1199082 minus 11988811199081 = 119891 (119909 119905) (36)

where 11988811199081 determined by (35) is the load imposed on lowerbeam Make 1199082 = suminfin119894=1 119885119894(119909)Y119894(119905) and we obtain

infinsum119894=1

[11986421198682119885119894(4)119884119894 + 119898211988410158401015840119894 119885119894 + (1198881 + 1198882) 119885119894119884119894] = 11988811199081 (37)

Make (37) multiplied by 119885119895(119909) and integrate it in thewhole beam considering (28) and the orthogonality of modefunction of lower beam and then obtaining

11988410158401015840 + (1198752119894 + 11988811198691198941198982)119884119894 = 119876119894 (119905) (38)

where 119876119894(119905) = (1198981199051198692119894 1198982) int1198970 11988811199081119885119894119889119909 is defined as gener-alized loads imposed on the lower beam Then make 119899119894 =(1198752119894 + 11988811198691198941198982)12 and the general solution of (38) is

119884119894 (119905) = 1119899119894 int119905

0119876119894 (120591) sin 119899119894 (119905 minus 120591) d120591 (39)

Substituting 119876119894(119905) (28) and (39) into 1199082 = suminfin119894=1 119885119894(119909)119884119894(119905)we obtain the dynamic response of lower beam under load asfollows

1199082= infinsum119894=1

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894

sdot 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)

minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]sdot int11990501199081 (119909 120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(40)

where 1199081 is the dynamic response of upper beam under loadobtained from (35)

25 Expression of Uniformly Distributed Load According tothe loading status of upper and lower beam the functionfor variable uniformly distributed load can be expressed asfollows

119901119902(119909119905) = minus119902119867 (V119905 minus 119909) (0 le 119905 le 119897V) (41)

where 119902 is the distributed electromagnetic force between therails and119867(119909) is the Heaviside Function3 Displacement of Double-Layer Beam underVariable Uniformly Distributed Load

31 Derivation of Dynamic Deflection Curve Equations ofUpper Beam Based on (31) we can obtain the dynamic

Mathematical Problems in Engineering 7

equation of upper beam attributed to uniformly distributedload as follows 11986411198681 120597411990811205971199094 + 1198981 120597

211990811205971199052 + 1198881 (1199081 minus 1199082) = 119901119902(119909119905) (42)

Substituting 119901119902(119909119905) into (42) we obtain

1199081119902(119909119905) (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)]

sdot int1199050119901119902(119909120591) sin119875119894 (119905 minus 120591) 119889120591 119889119909

(43)

Considering the property of Heaviside Function and sub-stituting 119901119902(119909119905) = minus119902119867(V119905 minus 119909) into (43) we can obtainedthe dynamic response of upper beam under uniformly dis-tributed load q through the integral as follows

1199081119902(119909119905) (119909 119905)= infinsum119894=1

minus119902[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 )+ 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 )

minus 1198722119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 )+ 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 )minus 120585 [1198732119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(44)

32 Bending Moment and Stress of Upper Beam Having gotthe dynamic deflection curve equations and then consideringthe expression of1198721 and 1205901 we can directly obtain from (44)

1198721 = 11986411198681 119889211990811199021198891199092 =infinsum119894=1

minus11986411198681119902 [1198721198942 cos119872119894119909 +1198721198942ch119872119894119909 minus 1205891198721198942 (sh119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942ch119873119894119909 minus 1205851198731198942 (sh119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(45)

8 Mathematical Problems in Engineering

1205901 = 1198641ℎ12 119889211990811199021198891199092 =

infinsum119894=1

minus1198641ℎ11199022 [1198721198942 cos119872119894119909 +1198721198942119888ℎ119872119894119909 minus 1205891198721198942 (119904ℎ119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942119888ℎ119873119894119909 minus 1205851198731198942 (119904ℎ119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ] (46)

33 Derivation of Dynamic Deflection Curve Equations ofLower Beam According to the determined 1199081119902(xt) similarlywe obtain

1199082119902(119909119905) = infinsum119894=0

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894+ 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]

sdot int11990501199081119902(119909120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(47)

Substituting (44) into (47) and simplifying the solutionwe can obtain

1199082119902(119909119905) = minus 119902119888111989811990511989911989411989821198692119894 (1198831 + 1198832) (1198671 + 1198672) (1198791 + 1198792) (48)

where the expression of11986711198672 1198791 and 1198792 can be reviewedin the appendix

34 Bending Moment and Stress of Lower Beam

1198722= minus11990211988811198981199051198641119868111989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792) (49)

1205902= minus11990211988811198981199051198641ℎ2211989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792)

(50)

where119883110158401015840 = 1198722119894 119869119894(cosh119872119894119909+ cos119872119894119909)minus 1205891198722119894 119869119894(sinh119872119894119909+sin119872119894119909) and 119883210158401015840 = 1198732119894 119870119894(cosh119873119894119909 + cos119873119894119909) minus1205851198732119894 119870119894(sinh119873119894119909 + sin119873119894119909)

4 Instance Analysis

41 Parameters The dynamic model for electromagneticlaunching rail due to uniformly distributed load is shownin Figure 4 The length of the beam is l=3000mm Thesection sizes of upper beam are 1198671=45mm and ℎ1=15mmrespectively The section sizes of lower beam are 1198672=75mmand ℎ2=30mm respectively The material of upper beamis copper of which the elastic modulus is 1198641=110GPa andthe mass density is 1205881=8290kgm3 the material of lowerbeam is copper of which the elastic modulus is 1198642 =283GPaand the mass density is 1205882 =980kgm3 The elastic constantbetween upper and lower beam is 1198881=3MPa the elasticconstant between lower beamand foundation is 1198882=6MPaTheuniformly distributed force is119902=3000NmThevelocity of thearmature is assumed as1000ms

42 Consequence and Analysis In the case of given motionparameters and structure parameters we have calculated thedisplacement bending moment and stress of double-layerelastic cantilever foundation beam and obtained analyticalsolutions and numerical results got from ANSYS which areshown in Figures 5ndash12

Mathematical Problems in Engineering 9

RailInsulation

qq

H 1H2

h2 h1 h2h1b

Cover plate

Figure 4 The cross-section of electromagnetic rail launcher

0 05 1 15 2 25 3x (m)

minus14minus12minus10minus8minus6minus4minus2

024

W1

(mm

)

l1=05m

Matlab ResultANSYS Result

Figure 5 Comparison of analytical solution with numerical solution for deflection of upper beam

Figures 5ndash8 show the dynamic response of the upperbeam attributed to moving load Analytical solution andnumerical solution for deflection of upper beam are depictedin Figure 5 in which peak values of displacement of analyticaland numerical solution are similar and the values occur atclose positions However the analytical solution begins tofluctuate near x=05m after which the numerical solutionremains stable Figure 6 shows the changing displacementof upper beam at x = 15m in which the analytical solutionand the numerical solution are in good agreement Figures7 and 8 depict the bending moment and stress of upperbeam at different positions due to the motion of load Dueto the linear relationship between the bending moment andthe stress both of them are identical in trend Similar to thechange of displacement solution the numerical solution andthe analytical solution are close before x=05 and after thatthe analytical solution fluctuates

Meanwhile dynamic response of the lower beam is inves-tigated The displacement results of analytical and numericalsolution of lower beam differentiate obviously as shown inFigure 9 Figure 10 shows the displacement of lower beamchanging with time at x = 15m in which similar to upperbeam the analytical solution and the numerical solutionare in good agreement in trend In Figures 11 and 12 both

analytical and numerical solution trend of bending momentand stress of lower beam follow the same patterns of upperbeam in Figures 7 and 8

It is apparent that the peak values of dynamic responseincluding displacement bending moment and stress ofupper beam are bigger than that of lower beam becauseof upper beam dissipating majority of the vibration energydue to the moving load There are some analytical solu-tion fluctuations compared with numerical solution in Fig-ures 5 7 8 9 11 and 12 which may be aroused byneglecting the layer damping In summary the analyticaland numerical solutions of the dynamic response of theupper and lower beams in the above mentioned figuresare similar in the overall trend despite some acceptabledeviation

In order to investigate the properties of analytical andnumerical solutions more specifically we compare the peakvalues of displacement bending moment and stress (Figures5ndash12) of the upper and lower beams of dynamic response asshown in Table 1

It is clear that the relative error of the peak values betweenthe analytical solution and the numerical solution obtainedby ANSYS is within a reasonable range and fluctuates around5

10 Mathematical Problems in Engineering

0 05 1 15 2 25 3t (s)

minus3minus2minus1

01234

v=1000msx=15m Position

Matlab ResultANSYS Result

times10-3

W1

(mm

)

Figure 6 Upper beam x = 15m comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus700minus600minus500minus400minus300minus200minus100

0100200300

M1

(Nlowast

m)

Matlab ResultANSYS Result

l1=05m

Figure 7 Comparison of analytical solution with numerical solution for bending moment distribution of upper beam

0 05 1 15 2 25 3x (m)

minus400

minus300

minus200

minus100

0

100

200

Matlab ResultANSYS Result

l1=05m

1(MPa

)

Figure 8 Comparison of analytical solution with numerical solution of stress distribution on upper beam

0 05 1 15 2 25 3x (m)

minus2minus15minus1

minus050

051

152

W2

(mm

)

Matlab ResultANSYS Result

Figure 9 Comparison of analytical solution with numerical solution for deflection of lower beam

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Mathematical Problems in Engineering

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Page 7: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

Mathematical Problems in Engineering 7

equation of upper beam attributed to uniformly distributedload as follows 11986411198681 120597411990811205971199094 + 1198981 120597

211990811205971199052 + 1198881 (1199081 minus 1199082) = 119901119902(119909119905) (42)

Substituting 119901119902(119909119905) into (42) we obtain

1199081119902(119909119905) (119909 119905) = infinsum119894=1

[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894 minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot int1198970[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909) minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)]

sdot int1199050119901119902(119909120591) sin119875119894 (119905 minus 120591) 119889120591 119889119909

(43)

Considering the property of Heaviside Function and sub-stituting 119901119902(119909119905) = minus119902119867(V119905 minus 119909) into (43) we can obtainedthe dynamic response of upper beam under uniformly dis-tributed load q through the integral as follows

1199081119902(119909119905) (119909 119905)= infinsum119894=1

minus119902[minus cos119872119894119909 + ch119872119894119909 minus 120589 (sh119872119894119909 minus sin119872119894119909)119875119894minus cos119873119894119909 + ch119873119894119909 minus 120585 (sh119873119894119909 minus sin119873119894119909)119875119894 ]sdot minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 )+ 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 )

minus 1198722119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 )+ 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 )minus 120585 [1198732119894 v2 (cos119875119905 minus 1) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 v2 (cos119875119905 minus 1) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(44)

32 Bending Moment and Stress of Upper Beam Having gotthe dynamic deflection curve equations and then consideringthe expression of1198721 and 1205901 we can directly obtain from (44)

1198721 = 11986411198681 119889211990811199021198891199092 =infinsum119894=1

minus11986411198681119902 [1198721198942 cos119872119894119909 +1198721198942ch119872119894119909 minus 1205891198721198942 (sh119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942ch119873119894119909 minus 1205851198731198942 (sh119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ]

(45)

8 Mathematical Problems in Engineering

1205901 = 1198641ℎ12 119889211990811199021198891199092 =

infinsum119894=1

minus1198641ℎ11199022 [1198721198942 cos119872119894119909 +1198721198942119888ℎ119872119894119909 minus 1205891198721198942 (119904ℎ119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942119888ℎ119873119894119909 minus 1205851198731198942 (119904ℎ119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ] (46)

33 Derivation of Dynamic Deflection Curve Equations ofLower Beam According to the determined 1199081119902(xt) similarlywe obtain

1199082119902(119909119905) = infinsum119894=0

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894+ 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]

sdot int11990501199081119902(119909120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(47)

Substituting (44) into (47) and simplifying the solutionwe can obtain

1199082119902(119909119905) = minus 119902119888111989811990511989911989411989821198692119894 (1198831 + 1198832) (1198671 + 1198672) (1198791 + 1198792) (48)

where the expression of11986711198672 1198791 and 1198792 can be reviewedin the appendix

34 Bending Moment and Stress of Lower Beam

1198722= minus11990211988811198981199051198641119868111989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792) (49)

1205902= minus11990211988811198981199051198641ℎ2211989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792)

(50)

where119883110158401015840 = 1198722119894 119869119894(cosh119872119894119909+ cos119872119894119909)minus 1205891198722119894 119869119894(sinh119872119894119909+sin119872119894119909) and 119883210158401015840 = 1198732119894 119870119894(cosh119873119894119909 + cos119873119894119909) minus1205851198732119894 119870119894(sinh119873119894119909 + sin119873119894119909)

4 Instance Analysis

41 Parameters The dynamic model for electromagneticlaunching rail due to uniformly distributed load is shownin Figure 4 The length of the beam is l=3000mm Thesection sizes of upper beam are 1198671=45mm and ℎ1=15mmrespectively The section sizes of lower beam are 1198672=75mmand ℎ2=30mm respectively The material of upper beamis copper of which the elastic modulus is 1198641=110GPa andthe mass density is 1205881=8290kgm3 the material of lowerbeam is copper of which the elastic modulus is 1198642 =283GPaand the mass density is 1205882 =980kgm3 The elastic constantbetween upper and lower beam is 1198881=3MPa the elasticconstant between lower beamand foundation is 1198882=6MPaTheuniformly distributed force is119902=3000NmThevelocity of thearmature is assumed as1000ms

42 Consequence and Analysis In the case of given motionparameters and structure parameters we have calculated thedisplacement bending moment and stress of double-layerelastic cantilever foundation beam and obtained analyticalsolutions and numerical results got from ANSYS which areshown in Figures 5ndash12

Mathematical Problems in Engineering 9

RailInsulation

qq

H 1H2

h2 h1 h2h1b

Cover plate

Figure 4 The cross-section of electromagnetic rail launcher

0 05 1 15 2 25 3x (m)

minus14minus12minus10minus8minus6minus4minus2

024

W1

(mm

)

l1=05m

Matlab ResultANSYS Result

Figure 5 Comparison of analytical solution with numerical solution for deflection of upper beam

Figures 5ndash8 show the dynamic response of the upperbeam attributed to moving load Analytical solution andnumerical solution for deflection of upper beam are depictedin Figure 5 in which peak values of displacement of analyticaland numerical solution are similar and the values occur atclose positions However the analytical solution begins tofluctuate near x=05m after which the numerical solutionremains stable Figure 6 shows the changing displacementof upper beam at x = 15m in which the analytical solutionand the numerical solution are in good agreement Figures7 and 8 depict the bending moment and stress of upperbeam at different positions due to the motion of load Dueto the linear relationship between the bending moment andthe stress both of them are identical in trend Similar to thechange of displacement solution the numerical solution andthe analytical solution are close before x=05 and after thatthe analytical solution fluctuates

Meanwhile dynamic response of the lower beam is inves-tigated The displacement results of analytical and numericalsolution of lower beam differentiate obviously as shown inFigure 9 Figure 10 shows the displacement of lower beamchanging with time at x = 15m in which similar to upperbeam the analytical solution and the numerical solutionare in good agreement in trend In Figures 11 and 12 both

analytical and numerical solution trend of bending momentand stress of lower beam follow the same patterns of upperbeam in Figures 7 and 8

It is apparent that the peak values of dynamic responseincluding displacement bending moment and stress ofupper beam are bigger than that of lower beam becauseof upper beam dissipating majority of the vibration energydue to the moving load There are some analytical solu-tion fluctuations compared with numerical solution in Fig-ures 5 7 8 9 11 and 12 which may be aroused byneglecting the layer damping In summary the analyticaland numerical solutions of the dynamic response of theupper and lower beams in the above mentioned figuresare similar in the overall trend despite some acceptabledeviation

In order to investigate the properties of analytical andnumerical solutions more specifically we compare the peakvalues of displacement bending moment and stress (Figures5ndash12) of the upper and lower beams of dynamic response asshown in Table 1

It is clear that the relative error of the peak values betweenthe analytical solution and the numerical solution obtainedby ANSYS is within a reasonable range and fluctuates around5

10 Mathematical Problems in Engineering

0 05 1 15 2 25 3t (s)

minus3minus2minus1

01234

v=1000msx=15m Position

Matlab ResultANSYS Result

times10-3

W1

(mm

)

Figure 6 Upper beam x = 15m comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus700minus600minus500minus400minus300minus200minus100

0100200300

M1

(Nlowast

m)

Matlab ResultANSYS Result

l1=05m

Figure 7 Comparison of analytical solution with numerical solution for bending moment distribution of upper beam

0 05 1 15 2 25 3x (m)

minus400

minus300

minus200

minus100

0

100

200

Matlab ResultANSYS Result

l1=05m

1(MPa

)

Figure 8 Comparison of analytical solution with numerical solution of stress distribution on upper beam

0 05 1 15 2 25 3x (m)

minus2minus15minus1

minus050

051

152

W2

(mm

)

Matlab ResultANSYS Result

Figure 9 Comparison of analytical solution with numerical solution for deflection of lower beam

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Page 8: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

8 Mathematical Problems in Engineering

1205901 = 1198641ℎ12 119889211990811199021198891199092 =

infinsum119894=1

minus1198641ℎ11199022 [1198721198942 cos119872119894119909 +1198721198942119888ℎ119872119894119909 minus 1205891198721198942 (119904ℎ119872119894119909 + sin119872119894119909)119875119894+ 1198731198942 cos119873119894119909 + 1198731198942119888ℎ119873119894119909 minus 1205851198731198942 (119904ℎ119873119894119909 + sin119873119894119909)119875119894 ] minus (119875119894 sin119872119894V119905 minus 119872119894V sin119875119894119905)119872119894 (minus1198722119894 v2 + 1198752119894 ) + 119875119894sh119872119894V119905 minus 119872119894V sin119875119894119905119872119894 (1198722119894 v2 + 1198752119894 )minus 120589 [1198722119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119872119894V119905 minus 1)1198721198941198752119894 (1198722119894 v2 + 1198752119894 ) minus 1198722119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119872119894V119905 minus 1)119872119894 (minusP1198941198722119894 v2 + 1198753119894 ) ]+ minus (119875119894 sin119873119894V119905 minus 119873119894V sin119875119894119905)119873119894 (minus1198732119894 v2 + 1198752119894 ) + 119875119894sh119873119894V119905 minus 119873119894V sin119875119894119905119873119894 (1198732119894 v2 + 1198752119894 ) minus 120585 [1198732119894 (v2 cos119875119905 minus v2) + 1198752119894 (ch119873119894V119905 minus 1)1198731198941198752119894 (1198732119894 v2 + 1198752119894 )minus 1198732119894 (v2 cos119875119905 minus v2) minus 1198752119894 (cos119873119894V119905 minus 1)119873119894 (minusP1198941198732119894 v2 + 1198753119894 ) ] (46)

33 Derivation of Dynamic Deflection Curve Equations ofLower Beam According to the determined 1199081119902(xt) similarlywe obtain

1199082119902(119909119905) = infinsum119894=0

119898119905 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909)]11989911989411989821198692119894+ 119898119905 [minus119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]11989911989411989821198692119894sdot int11989701198881 [minus119869119894 cos119872119894119909 + 119869119894ch119872119894119909 minus 120589119869119894 (sh119872119894119909 minus sin119872119894119909) minus 119870119894 cos119873119894119909 + 119870119894ch119873119894119909 minus 120585119870119894 (sh119873119894119909 minus sin119873119894119909)]

sdot int11990501199081119902(119909120591) sin 119899119894 (119905 minus 120591) 119889120591 119889119909

(47)

Substituting (44) into (47) and simplifying the solutionwe can obtain

1199082119902(119909119905) = minus 119902119888111989811990511989911989411989821198692119894 (1198831 + 1198832) (1198671 + 1198672) (1198791 + 1198792) (48)

where the expression of11986711198672 1198791 and 1198792 can be reviewedin the appendix

34 Bending Moment and Stress of Lower Beam

1198722= minus11990211988811198981199051198641119868111989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792) (49)

1205902= minus11990211988811198981199051198641ℎ2211989911989411989821198692119894 (119883110158401015840 + 119883210158401015840) (1198671 + 1198672) (1198791 + 1198792)

(50)

where119883110158401015840 = 1198722119894 119869119894(cosh119872119894119909+ cos119872119894119909)minus 1205891198722119894 119869119894(sinh119872119894119909+sin119872119894119909) and 119883210158401015840 = 1198732119894 119870119894(cosh119873119894119909 + cos119873119894119909) minus1205851198732119894 119870119894(sinh119873119894119909 + sin119873119894119909)

4 Instance Analysis

41 Parameters The dynamic model for electromagneticlaunching rail due to uniformly distributed load is shownin Figure 4 The length of the beam is l=3000mm Thesection sizes of upper beam are 1198671=45mm and ℎ1=15mmrespectively The section sizes of lower beam are 1198672=75mmand ℎ2=30mm respectively The material of upper beamis copper of which the elastic modulus is 1198641=110GPa andthe mass density is 1205881=8290kgm3 the material of lowerbeam is copper of which the elastic modulus is 1198642 =283GPaand the mass density is 1205882 =980kgm3 The elastic constantbetween upper and lower beam is 1198881=3MPa the elasticconstant between lower beamand foundation is 1198882=6MPaTheuniformly distributed force is119902=3000NmThevelocity of thearmature is assumed as1000ms

42 Consequence and Analysis In the case of given motionparameters and structure parameters we have calculated thedisplacement bending moment and stress of double-layerelastic cantilever foundation beam and obtained analyticalsolutions and numerical results got from ANSYS which areshown in Figures 5ndash12

Mathematical Problems in Engineering 9

RailInsulation

qq

H 1H2

h2 h1 h2h1b

Cover plate

Figure 4 The cross-section of electromagnetic rail launcher

0 05 1 15 2 25 3x (m)

minus14minus12minus10minus8minus6minus4minus2

024

W1

(mm

)

l1=05m

Matlab ResultANSYS Result

Figure 5 Comparison of analytical solution with numerical solution for deflection of upper beam

Figures 5ndash8 show the dynamic response of the upperbeam attributed to moving load Analytical solution andnumerical solution for deflection of upper beam are depictedin Figure 5 in which peak values of displacement of analyticaland numerical solution are similar and the values occur atclose positions However the analytical solution begins tofluctuate near x=05m after which the numerical solutionremains stable Figure 6 shows the changing displacementof upper beam at x = 15m in which the analytical solutionand the numerical solution are in good agreement Figures7 and 8 depict the bending moment and stress of upperbeam at different positions due to the motion of load Dueto the linear relationship between the bending moment andthe stress both of them are identical in trend Similar to thechange of displacement solution the numerical solution andthe analytical solution are close before x=05 and after thatthe analytical solution fluctuates

Meanwhile dynamic response of the lower beam is inves-tigated The displacement results of analytical and numericalsolution of lower beam differentiate obviously as shown inFigure 9 Figure 10 shows the displacement of lower beamchanging with time at x = 15m in which similar to upperbeam the analytical solution and the numerical solutionare in good agreement in trend In Figures 11 and 12 both

analytical and numerical solution trend of bending momentand stress of lower beam follow the same patterns of upperbeam in Figures 7 and 8

It is apparent that the peak values of dynamic responseincluding displacement bending moment and stress ofupper beam are bigger than that of lower beam becauseof upper beam dissipating majority of the vibration energydue to the moving load There are some analytical solu-tion fluctuations compared with numerical solution in Fig-ures 5 7 8 9 11 and 12 which may be aroused byneglecting the layer damping In summary the analyticaland numerical solutions of the dynamic response of theupper and lower beams in the above mentioned figuresare similar in the overall trend despite some acceptabledeviation

In order to investigate the properties of analytical andnumerical solutions more specifically we compare the peakvalues of displacement bending moment and stress (Figures5ndash12) of the upper and lower beams of dynamic response asshown in Table 1

It is clear that the relative error of the peak values betweenthe analytical solution and the numerical solution obtainedby ANSYS is within a reasonable range and fluctuates around5

10 Mathematical Problems in Engineering

0 05 1 15 2 25 3t (s)

minus3minus2minus1

01234

v=1000msx=15m Position

Matlab ResultANSYS Result

times10-3

W1

(mm

)

Figure 6 Upper beam x = 15m comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus700minus600minus500minus400minus300minus200minus100

0100200300

M1

(Nlowast

m)

Matlab ResultANSYS Result

l1=05m

Figure 7 Comparison of analytical solution with numerical solution for bending moment distribution of upper beam

0 05 1 15 2 25 3x (m)

minus400

minus300

minus200

minus100

0

100

200

Matlab ResultANSYS Result

l1=05m

1(MPa

)

Figure 8 Comparison of analytical solution with numerical solution of stress distribution on upper beam

0 05 1 15 2 25 3x (m)

minus2minus15minus1

minus050

051

152

W2

(mm

)

Matlab ResultANSYS Result

Figure 9 Comparison of analytical solution with numerical solution for deflection of lower beam

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Page 9: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

Mathematical Problems in Engineering 9

RailInsulation

qq

H 1H2

h2 h1 h2h1b

Cover plate

Figure 4 The cross-section of electromagnetic rail launcher

0 05 1 15 2 25 3x (m)

minus14minus12minus10minus8minus6minus4minus2

024

W1

(mm

)

l1=05m

Matlab ResultANSYS Result

Figure 5 Comparison of analytical solution with numerical solution for deflection of upper beam

Figures 5ndash8 show the dynamic response of the upperbeam attributed to moving load Analytical solution andnumerical solution for deflection of upper beam are depictedin Figure 5 in which peak values of displacement of analyticaland numerical solution are similar and the values occur atclose positions However the analytical solution begins tofluctuate near x=05m after which the numerical solutionremains stable Figure 6 shows the changing displacementof upper beam at x = 15m in which the analytical solutionand the numerical solution are in good agreement Figures7 and 8 depict the bending moment and stress of upperbeam at different positions due to the motion of load Dueto the linear relationship between the bending moment andthe stress both of them are identical in trend Similar to thechange of displacement solution the numerical solution andthe analytical solution are close before x=05 and after thatthe analytical solution fluctuates

Meanwhile dynamic response of the lower beam is inves-tigated The displacement results of analytical and numericalsolution of lower beam differentiate obviously as shown inFigure 9 Figure 10 shows the displacement of lower beamchanging with time at x = 15m in which similar to upperbeam the analytical solution and the numerical solutionare in good agreement in trend In Figures 11 and 12 both

analytical and numerical solution trend of bending momentand stress of lower beam follow the same patterns of upperbeam in Figures 7 and 8

It is apparent that the peak values of dynamic responseincluding displacement bending moment and stress ofupper beam are bigger than that of lower beam becauseof upper beam dissipating majority of the vibration energydue to the moving load There are some analytical solu-tion fluctuations compared with numerical solution in Fig-ures 5 7 8 9 11 and 12 which may be aroused byneglecting the layer damping In summary the analyticaland numerical solutions of the dynamic response of theupper and lower beams in the above mentioned figuresare similar in the overall trend despite some acceptabledeviation

In order to investigate the properties of analytical andnumerical solutions more specifically we compare the peakvalues of displacement bending moment and stress (Figures5ndash12) of the upper and lower beams of dynamic response asshown in Table 1

It is clear that the relative error of the peak values betweenthe analytical solution and the numerical solution obtainedby ANSYS is within a reasonable range and fluctuates around5

10 Mathematical Problems in Engineering

0 05 1 15 2 25 3t (s)

minus3minus2minus1

01234

v=1000msx=15m Position

Matlab ResultANSYS Result

times10-3

W1

(mm

)

Figure 6 Upper beam x = 15m comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus700minus600minus500minus400minus300minus200minus100

0100200300

M1

(Nlowast

m)

Matlab ResultANSYS Result

l1=05m

Figure 7 Comparison of analytical solution with numerical solution for bending moment distribution of upper beam

0 05 1 15 2 25 3x (m)

minus400

minus300

minus200

minus100

0

100

200

Matlab ResultANSYS Result

l1=05m

1(MPa

)

Figure 8 Comparison of analytical solution with numerical solution of stress distribution on upper beam

0 05 1 15 2 25 3x (m)

minus2minus15minus1

minus050

051

152

W2

(mm

)

Matlab ResultANSYS Result

Figure 9 Comparison of analytical solution with numerical solution for deflection of lower beam

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Page 10: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

10 Mathematical Problems in Engineering

0 05 1 15 2 25 3t (s)

minus3minus2minus1

01234

v=1000msx=15m Position

Matlab ResultANSYS Result

times10-3

W1

(mm

)

Figure 6 Upper beam x = 15m comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus700minus600minus500minus400minus300minus200minus100

0100200300

M1

(Nlowast

m)

Matlab ResultANSYS Result

l1=05m

Figure 7 Comparison of analytical solution with numerical solution for bending moment distribution of upper beam

0 05 1 15 2 25 3x (m)

minus400

minus300

minus200

minus100

0

100

200

Matlab ResultANSYS Result

l1=05m

1(MPa

)

Figure 8 Comparison of analytical solution with numerical solution of stress distribution on upper beam

0 05 1 15 2 25 3x (m)

minus2minus15minus1

minus050

051

152

W2

(mm

)

Matlab ResultANSYS Result

Figure 9 Comparison of analytical solution with numerical solution for deflection of lower beam

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Mathematical Problems in Engineering

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Page 11: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

Mathematical Problems in Engineering 11

0 05 1 15 2 25 3t (s)

minus35minus3

minus25minus2

minus15minus1

minus050

05

Matlab ResultANSYS Result

W2

(mm

)times10

-3

Figure 10 Lower beam x = 15m Comparison of analytical solution with numerical solution for response

0 05 1 15 2 25 3x (m)

minus30

minus20

minus10

0

10

20

30

Matlab ResultANSYS Result

l1=05m

M2

(Nlowast

m)

Figure 11 Comparison of analytical solution with numerical solution for bending moment distribution of lower beam

0 05 1 15 2 25 3x (m)

minus25minus20minus15minus10minus5

05

10152025

Matlab ResultANSYS Result

2(MPa

)

l1=05m

Figure 12 Comparison of analytical solution with numerical solution of stress distribution on lower beam

t (s)

v = 1000ms

x (m)

minus153

minus10

25

minus5

32

0

2515 2

5

15

10

1 105 050 0

W1

(mm

)

times10-3

Figure 13 Surface changes of upper beam deflection with time and position

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Page 12: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

12 Mathematical Problems in Engineering

Table1Analysis

ofcalculationresults

Figure

Maxim

umof

numerical

solutio

n

Maxim

umof

analytical

solutio

n

Errors

Minim

umof

numerical

solutio

n

Minim

umof

analytical

solutio

n

Errors

Relativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

nRe

lativ

enum

erical

solutio

nRe

lativ

eanalytic

alsolutio

n5

02536

mdashmdash

mdashmdash

-1298

-1362

49

47

63431

3593

47

45

-1969

-2078

55

52

72447

2578

54

51

-6546

-6242

46

49

81453

1527

51

48

-3885

-3699

48

50

90

1884

mdashmdash

mdashmdash

0-2009

mdashmdash

mdashmdash

100

00

0-2939

-3086

50

48

112639

2511

49

51

-2319

-2444

54

51

122345

2232

48

51

-2059

-217

355

52

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Mathematical Problems in Engineering

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Page 13: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

Mathematical Problems in Engineering 13

v=1000msminus1500

3

minus1000

25

minus500

32

0

25t (s)

15 2

500

x (m)15

1000

1 105 050 0M

1(N

lowastm

)

times10-3

Figure 14 Surface changes of upper beam bending moment with time and position

t (s)

v = 1000ms

x (m)

minus15003

minus1000

25

minus500

32

0

2515 2

500

15

1000

1 105 050 0

2 5

1(MPa

)

times10-3

Figure 15 Surface changes of upper beam stress with time and position

minus353minus3

minus25

25

minus2

3

minus15

2

minus1

25

minus05

t (s)

0

15 2

05

x (m)15

1

1 105 050 0

v = 1000mstimes10

-3

Figure 16 Surface changes of lower beam deflection with time and position

v=1000ms

minus6003

minus500minus400

25

minus300

3

minus200

2

minus100

25

0

15

100

2

200

15

300

1 105 050 0

v = 1000mstimes10

-3

Figure 17 Surface changes of lower beam bending moment with time and position

In addition to better illustrate the analytical dynamicresponse of double-layer elastic cantilever foundation beamsunder moving loads the deflection bending moment andstress of upper and lower beams are given in Figures 13 14and 15 and Figures 16 17 and 18 respectively

5 Conclusion

Based on the double-layer elastic foundation beam theorythis paper establishes the dynamic equations considering theboundary conditions where one end of beam is constrainedand another end is free and investigates the homogeneous

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

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Mathematical Problems in Engineering

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Page 14: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

14 Mathematical Problems in Engineering

x (m)

v = 1000ms

t (s)

minus503

minus40minus30

25

minus20

3

minus10

2 25

010

15 2

20

15

30

1 105 050 0times10

-3

2(M

Pa)

Figure 18 Surface changes of lower beam stress with time and position

solution for displacement of double-layer elastic cantileverfoundation beamwithout loading Introducing the HeavisideFunction we obtain the electromagnetic force functionThen the deflection curves equation and the expression ofbending moment and stress for upper and lower beam underload are derived We use the model to calculate a specificexample(1)Through an instance the analytical solution of deflec-tion stress and bending moment of the upper and lowerbeams is obtained with given geometric parameters motionparameters and armature force Using ANSYS the paperobtains the numerical solution and simultaneously comparesthe analytic solution with the numerical solution of thedynamic response of the double-layer elastic foundationbeams The comparison shows that both solutions are notcompletely compatible but the trends of analytical solutionand numerical solution are basically in good agreementdespite some acceptable numerical differencesThe reliabilityof the analytical model is verified

(2)Thepeak values of dynamic response including deflec-tion bending moment and stress are very important in thedesign of electromagnetic railgun In this paper the relativeerror between the peak values of analytical solution and thatof numerical solution is within a reasonable rangeThereforethe analytical model of double-layer elastic cantilever beamcould be used to guide the actual design of electromagneticgun(3) The oscillation of analytical solutions in this papermay be due to the lack of consideration of interlayer dampingwhich needs to be further improved in the future research(4) The derived analytical solution of the dynamicresponse of the double-layer elastic cantilever beam canimprove the theoretical research on the electromagneticlaunch device and can be used for reference of the engineer-ing design in related fields

Appendix

1198791 = 119875119894Im 11 +119872119894VIm 12119872119894 (1198751198942 minus1198721198942V2) + 119875119894Im 21 +119872119894VIm 22119872119894 (1198751198942 +1198721198942V2) + 120589(1198721198942V2Im 31 + 1198751198942Im 321198721198941198751198942 (1198751198942 +1198721198942V2) + 119872119894

2V2Im 41 + 1198751198942Im 42119872119894 (1198751198943 minus 1198751198941198721198942V2) )1198792 = 119875119894In11 + 119873119894VIn12119873119894 (1198751198942 minus 1198731198942V2) + 119875119894In21 + 119873119894VIn22119873119894 (1198751198942 + 1198731198942V2) + 120585(119873119894

2V2In31 + 1198751198942In321198731198941198751198942 (1198751198942 + 1198731198942V2) + 1198731198942V2In41 + 1198751198942In42119873119894 (1198751198943 minus 1198751198941198731198942V2) )

1198671 = 119869119894119875119894 (11988611 + 11988621 + 11988631 + 11988641 + 11988811 + 11988821 + 11988831 + 11988841)and 1198672 = 119870119894119875119894 (11988711 + 11988721 + 11988731 + 11988741 + 11988911 + 11988921 + 11988931 + 11988941)with 11988611 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894 11988612 = 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988621 = minus120589(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988622 = 1198972 + sin 21198721198941198974119872119894 minus 120589sin

21198721198941198972119872119894

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 15: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

Mathematical Problems in Engineering 15

11988631 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058922119872119894 + 120589(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988632 = 1205892 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120589sinh

2119872119894119897211987211989411988641 = 1205892 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120589sinh

21198721198941198972119872119894 11988642 = 1205892 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120589( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988711 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

21198721198941198972119872119894 11988712 = 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin119872119894119897211987211989411988721 = minus120585(cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 minus 1198972119872119894) minus cos119872119894119897 sinh1198721198941198972119872119894 minus cosh119872119894119897 sin1198721198941198972119872119894 11988722 = 1198972 + sin 21198721198941198974119872119894 minus 120585sin

2119872119894119897211987211989411988731 = (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897) 12058522119872119894 + 120585(cos119872119894119897 cosh119872119894119897 + sin119872119894119897 sinh119872119894119897 minus 12119872119894 ) 11988732 = 1205852 ( sinh 21198721198941198974119872119894 minus 1198972) minus 120585sinh

2119872119894119897211987211989411988741 = 1205852 ( 1198972 minus sinh 21198721198941198974119872119894 ) minus 120585sinh

21198721198941198972119872119894 11988742 = 1205852 (cos119872119894119897 sinh119872119894119897 minus cosh119872119894119897 sin119872119894119897)2119872119894 + 120585( 12119872119894 minus cos119872119894119897 cosh119872119894119897 minus sin119872119894119897 sinh1198721198941198972119872119894 )11988812 = 120585 (119873119894 (cos119872119894119897 cos119873119894119897 minus 1) +119872119894 sin119872119894119897 sin119873119894119897) minus (119872119894 cosh119873119894119897 sin119872119894119897 + 119873119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988821 = minus (119872119894 cosh119873119894119897 sinh119872119894119897 + 119873119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cosh119872119894119897 cos119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sin119873119894119897)1198721198942 + 119873119894211988822 = 119872119894 cosh119873119894119897 sinh119872119894119897 minus 119873119894 cosh119872119894119897 sinh119873119894119897 + 120585 (119873119894 (cosh119872119894119897 cosh119873119894119897 minus 1) minus119872119894 sinh119872119894119897 sinh119873119894119897)1198721198942 minus 119873119894211988831 = 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sin119873119894119897) minus 120589120585 (119872119894 cosh119872119894119897 sin119873119894119897 minus 119873119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988832 = 120589120585 (119872119894 cosh119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sinh119872119894119897) minus 120589 (119872119894 (cosh119872119894119897 cos119873119894119897 minus 1) + 119873119894 sinh119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988841 = 120589120585( sin 119897 (119872119894 minus 119873119894)2119872119894 minus 2119873119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120589 (119872119894 (cos119872119894119897 cos119873119894119897 minus 1) + 119873119894 sin119872119894119897 sin119873119894119897)1198721198942 minus 119873119894211988842 = 120589120585 (119872119894 cos119872119894119897 sinh119873119894119897 minus 119873119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cos119872119894119897 cosh119873119894119897 minus 1) + 119873119894 sin119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988911 = sin 119897 (119873119894 minus119872119894)2119872119894 minus 2119873119894 + sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 minus 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897)1198721198942 minus 1198731198942

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 16: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

16 Mathematical Problems in Engineering

11988912 = 120589 (119872119894 (cos119873119894119897 cos119872119894119897 minus 1) + 119873119894 sin119873119894119897 sin119872119894119897) minus (119873119894 cosh119872119894119897 sin119873119894119897 + 119872119894 cos119873119894119897 sinh119872119894119897)1198721198942 + 119873119894211988921 = minus (119873119894 cos119872119894119897 sinh119873119894119897 + 119872119894 cosh119873119894119897 sin119872119894119897) minus 120589 (119872119894 (cosh119873119894119897 cos119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sin119872119894119897)1198721198942 + 119873119894211988922 = (119873119894 cosh119872119894119897 sinh119873119894119897 minus 119872119894 cosh119873119894119897 sinh119872119894119897) + 120589 (119872119894 (cosh119873119894119897 cosh119872119894119897 minus 1) minus 119873119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988931 = 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sin119872119894119897) minus 120589120585 (119873119894 cosh119873119894119897 sin119872119894119897 minus 119872119894 cos119872119894119897 sinh119873119894119897)1198721198942 + 119873119894211988932 = 120589120585 (119873119894 cosh119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sinh119873119894119897) minus 120585 (119873119894 (cosh119873119894119897 cos119872119894119897 minus 1) +119872119894 sinh119873119894119897 sinh119872119894119897)1198731198942 minus119872119894211988941 = 120589120585( sin 119897 (119873119894 minus119872119894)2119873119894 minus 2119872119894 minus sin 119897 (119872119894 + 119873119894)2119872119894 + 2119873119894 ) + 120585 (119873119894 (cos119873119894119897 cos119872119894119897 minus 1) +119872119894 sin119873119894119897 sin119872119894119897)1198731198942 minus119872119894211988942 = 120589120585 (119873119894 cos119873119894119897 sinh119872119894119897 minus 119872119894 cosh119872119894119897 sin119873119894119897) minus 120585 (119873119894 (cos119873119894119897 cosh119872119894119897 minus 1) +119872119894 sin119873119894119897 sinh119872119894119897)1198721198942 + 1198731198942

Im 11 = 119872119894V sin 119899119894119905 minus 119899119894 sin119872119894V1199051198991198942 minus1198721198942V2 Im 12 = 119875119894 sin 119899119894119905 minus 119899119894 sin1198751198941199051198751198942 minus 1198991198942 Im 21 = 119899119894 sin119872119894V119905 minus 119872119894V sin 1198991198941199051198991198942 +1198721198942V2 Im 22 = minus Im 12Im 31 = 119899119894 (cos119875119894119905 minus cos 119899119894119905)1198751198942 minus 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 32 = minus119899119894 (cosh119875119894119905 minus cos 119899119894119905)1198751198942 + 1198991198942 minus cos 119899119894119905 minus 1119899119894 Im 41 = minus Im 31Im 42 = 119899119894 (cos 119899119894119905 minus cos119872119894V119905)1198991198942 minus1198721198942V2 minus cos 119899119894119905 minus 1119899119894 In11 = 119873119894V sin 119899119894119905 minus 119899119894 sin119873119894V1199051198991198942 minus 1198731198942V2 In12 = Im 12In21 = 119899119894 sin119873119894V119905 minus 119873119894V sin 1198991198941199051198991198942 + 1198731198942V2 In22 = minus Im 12In31 = Im 31In32 = Im 32In41 = minus Im 31

and In42 = 119899119894 (cos 119899119894119905 minus cos119873119894V119905)1198991198942 minus 1198731198942V2 minus cos 119899119894119905 minus 1119899119894 (A1)

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 17: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

Mathematical Problems in Engineering 17

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The authors acknowledge the financial support from theNatural Science Foundation of Hebei (Grant E2016203147)the construction of Science and Technology Research PlanProject of Hebei (Grant 2016-124) and Dr Fund of Yanshanuniversity (B791)

References

[1] A J Johnson and F C Moon ldquoElastic waves and solidarmature contact pressure in electromagnetic launchersrdquo IEEETransactions on Magnetics vol 42 no 3 pp 422ndash429 2006

[2] S Satapathy and C Persad ldquoThermal stresses in an activelycooled two-piece rail structurerdquo IEEE Transactions on Magnet-ics vol 39 no 1 pp 468ndash471 2003

[3] J T Tzeng and W Sun ldquoDynamic response of cantilevered railguns attributed to projectilegun interaction - Theoryrdquo IEEETransactions on Magnetics vol 43 no 1 pp 207ndash213 2007

[4] J T Tzeng ldquoStructuralmechanics for electromagnetic railgunsrdquoIEEE Transactions on Magnetics vol 41 no 1 pp 246ndash2502005

[5] J T Tzeng ldquoDynamic response of electromagnetic railgun dueto projectile movementrdquo IEEE Transactions on Magnetics vol41 no 1 pp 472ndash475 2003

[6] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985

[7] F V Filho ldquoLiterature review finite element analysis of struc-tures under moving loadsrdquo13e Shock and Vibration Digest vol10 no 8 pp 27ndash35 1978

[8] T-N Chen C-Y Bai Y-N Zhang and X-Z Bai ldquoDynamicresponse of electromagnetic railgun due to armature move-mentrdquo Journal of Dynamical and Control Systems vol 8 no 4pp 360ndash364 2010 (Chinese)

[9] L Jin B Lei Z Li and Q Zhang ldquoDynamic response of railsdue to armature movement for electromagnetic railgunsrdquo Inter-national Journal of Applied Electromagnetics andMechanics vol47 no 1 pp 75ndash82 2015

[10] Y Che W Yuan W Xu W Cheng Y Zhao and P Yan ldquoTheinfluence of different constraints and pretightening force onvibration and stiffness in railgunrdquo IEEE Transactions on PlasmaSciences vol 45 no 7 pp 1154ndash1160 2017

[11] B Cao X Ge W Guo et al ldquoAnalysis of rail dynamic defor-mation during electromagnetic launchrdquo IEEE Transactions onPlasma Sciences vol 45 no 7 pp 1269ndash1273 2017

[12] Y-H Lee S-H Kim S An Y Bae and B Lee ldquoDynamicresponse of an electromagnetic launcher accelerating a C-shaped armaturerdquo IEEE Transactions on Plasma Sciences vol45 no 7 pp 1639ndash1643 2017

[13] Z-G Tian X-Y Meng X-Y An and X-Z Bai ldquoDynamicresponse of composite rail during launch process of electromag-netic railgunrdquo Binggong XuebaoActa Armamentarii vol 38 no4 pp 651ndash657 2017

[14] M E Hassanabadi N K Attari A Nikkhoo and S MarianildquoResonance of a rectangular plate influenced by sequentialmoving massesrdquo Coupled Systems Mechanics vol 5 no 1 pp87ndash100 2016

[15] A Nikkhoo A Farazandeh and M Ebrahimzadeh Hassan-abadi ldquoOn the computation of moving massbeam interactionutilizing a semi-analytical methodrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 38 no 3pp 761ndash771 2016

[16] A Nikkhoo A Farazandeh M Ebrahimzadeh Hassanabadiand S Mariani ldquoSimplified modeling of beam vibrationsinduced by amovingmass by regression analysisrdquoActaMechan-ica vol 226 no 7 pp 2147ndash2157 2015

[17] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo 13in-Walled Structures vol 62 pp 53ndash64 2013

[18] W He and X-Z Bai ldquoDynamic responses of rails and panels ofrectangular electromagnetic rail launcherrdquo Journal of Vibrationand Shock vol 32 no 15 pp 144ndash148 2013 (Chinese)

[19] W He and X Bai ldquoThe influence of physical parameters forsquare electromagnetic launcher on system dynamic responserdquoYingyong Lixue XuebaoChinese Journal of Applied Mechanicsvol 30 no 5 pp 680ndash686 2013 (Chinese)

[20] W He and X-Z Bai ldquoThe influence of physical parameter forelectromagnetic launchers track and panel on its mechanicalpropertyrdquo Journal of Mechanical Strength vol 36 no 1 pp 92ndash97 2014 (Chinese)

[21] W He and X-Z Bai ldquoDynamic responses for rails and panelsof electromagnetic rail launcher in the electromagnetic forcerdquoApplied Mechanics and Materials vol 328 pp 515ndash525 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 18: Dynamic Response of Double Elastic Cantilever Beam ...downloads.hindawi.com/journals/mpe/2019/2657271.pdf · Dynamic Response of Double Elastic Cantilever Beam Attributed to Variable

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Geometric Nonlinear Analysis of Cantilever Beam

Geometric Nonlinear Analysis of Cantilever Beam

Beam Cantilever com

Beam Cantilever com

Forced Vibration of Multi-dof Cantilever Beam

Forced Vibration of Multi-dof Cantilever Beam

Vibrations of Cantilever Beam (Continuous System) (2)

Vibrations of Cantilever Beam (Continuous System) (2)

Analysis of Cantilever-Beam Bending Stress Relaxation ... · Analysis of Cantilever-Beam Bending Stress Relaxation ... Cantilever-beam ... to accurately predict response of a constantly

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Deflection of Cantilever Beam 3

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Dynamic Response Of Uniform Cantilever Beams On Elastic ...

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Propped Cantilever Beam Tutorial Abaqus Final

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Cantilever Beam Rigid Body

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I-Beam Cantilever Rack

Cantilever Beam in Patran

Cantilever Beam in Patran

I Beam Cantilever - Steel King

I Beam Cantilever - Steel King

Elastic Beam Model

Elastic Beam Model

Cantilever Beam Level 2 (Podium)

Cantilever Beam Level 2 (Podium)

I-Beam Cantilever Racks

I-Beam Cantilever Racks

Cantilever Beam tyu

Cantilever Beam tyu

Shock Spectra of Cantilever Beam

Shock Spectra of Cantilever Beam

Beam Elastic

Beam Elastic

Analisis Cantilever Beam (Teoretik, FEM, ANSYS)

Analisis Cantilever Beam (Teoretik, FEM, ANSYS)

Fabrication Process for Cantilever Beam Micromechanical ...

Fabrication Process for Cantilever Beam Micromechanical ...