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CHINESE JOURNAL OF SHIP RESEARCHVOL15NO1FEB 2020

Vibrational characteristics of amulti-span beam with elastic transversesupports of different shaped sections

BAO SiyuanZHOU JingSchool of Civil EngineeringSuzhou University of Science and TechnologySuzhou 215011China

Abstract［Objectives］In order to overcome the difficulties caused by boundary and elastic transverse supports instudying the vibrational characteristics of a continuous multi-span beamthis paper establishes an analytical model ofthe free vibration of a multi-span beam based on the Euler-Bernoulli beam theory［Methods］Firsta new improvedFourier series form is constructed to represent the lateral displacement function of the multi-span beam over the entiresegment Secondthe series expression of the displacement function is substituted into the Lagrangian function andcombined with the Rayleigh-Ritz methodand the problem of free vibration is transformed into the form of eigenvaluesof a standard matrix The natural frequencies of an elastically supported multi-span beam can be solved［Results］Inselected numerical examplesby changing the value of the translational elastic stiffness at the elastic supportthevibrational characteristics of the multi-span beam with any elastic support can be obtained The feasibility andaccuracy of the proposed method are fully verified by a comparison with existing results in the literature［Conclusions］Through the numerical simulation of the vibrational characteristics of a multi-span beam based on animproved fourier series method （IFSM） this paper provides effective preliminary assessment for the dynamicperformance of a multi-span beamKey wordsmulti-span beamelastic supportnatural frequencyimproved fourier series method（IFSM）CLC number U66144

To cite this articleBAO S Y ZHOU J Vibrational characteristics of a multi-span beam with elastic transverse supports ofdifferent shaped sections [JOL] Chinese Journal of Ship Research 2020 15(1) httpwwwship-researchcomENY2020V15I1162

DOI1019693jissn1673-3185 01569

Received2019 - 04 - 11 Accepted2019 - 09 - 09Supported byNational Natural Science Foundation of China (11202146)Authors BAO Siyuan male born in 1980 PhD associate professor Research interests structural vibration spectro-geometric

method symplectic method E-mail bsiyuan126comZHOU Jing male born in 1994 master degree candidate Research interest application of improved Fourier seriesmethod in structural vibration E-mail 1150123198qqcom

Corresponding authorBAO Siyuan

0 Introduction

The beam structures are widely used in variousprojects such as construction aerospace and shipping fields A ship is composed of plates and beamsIn the vibration analysis of a ship the vibrationalcharacteristics of beams are needed Huang et al [1]

simplified the hull into a beam of variable cross section with complete freedom at both ends uneven distribution of mass and stiffness along the length andstudied its elastic deformation during vibration

Many scholars have studied and analyzed the vibration of the beam structure Abbas[2] used the finiteelement method to solve the free vibration problem

of Timoshenko beam with an elastic boundaryChung [3] combined the Fourier series and Lagrangemultipliers to propose a calculation method of thenatural frequency and mode of beams under the classical boundary condition Zou et al [4] solved theproblem of transforming the vibration of a single-span beam into a system of ordinary differentialequations by using the quasi wavelet-precisetime-integration method Zeng et al [5] proposed toanalyze the vibration equation of a beam in the formof multi-symplectic Hamilton and finally explainedthe correctness of the theoretical analysis with numerical examples Wang et al [6] used the differencemethod to deduce the basic vibrational characteris

84

downloaded from wwwship-researchcom

tics of natural vibration of the difference discrete system of multi-span beams Liu et al [7] established thefree vibration models of three kinds of commonbeams (Euler beam Rayleigh beam and Timoshenko beam) analyzed them with the method of variation of parameters and deduced the frequency equation of the free vibration Zhang et al [8] establishedthe hencky bar-chain model (HBM) based on the finite difference model and derived the frequency formula of the HBM method The HBM method and finite difference method (FDM) were first proposed toanalyze the buckling load and vibration frequency ofthe internal elastic spring Zhang et al [9] used themethod of variable separation and transfer matrix toderive the characteristic equation of a beam and elaborated the geometric characteristics of the beam withunequal span variable cross section and arbitrarydiscontinuities Chen et al [10] established the vibration differential equation of the discontinuous Eulerbeam under the elastic foundation of the axial forcein the generalized function space and studied the influence of the rotational inertia of the additionalmass block on the beam-mass block system

However most of the methods of free vibrationproblems studied above have certain limitations Forexample some can only be used for qualitative research and some can only solve specific boundaryconditions Li [11] proposed an improved Fourier series form of function namely that four sine functionswere added to the traditional Fourier series formThe advantage of this form is that it can completelyeliminate the discontinuity of a function and its derivatives at the end point Zhou et al [12] analyzed thevibrational characteristics of the continuousmulti-span beam structure based on the improvedFourier series method (IFSM) Zhou et al [13] studiedthe lateral vibrational characteristics of the shaft system using the IFSM Shi et al [14] studied the lateralfree vibration of the orthotropic thin plate based onIFSM Bao et al [15-16] studied the in-plane free vibration of annular sector plate and rectangular platebased on IFSM

In order to study the free vibration of multi-span

beams this paper intends to propose a new IFSM torepresent the displacement function of beams whichis different from that in Reference [12] to avoid thediscontinuity at the end of multi-span beams underelastic boundary Firstly the displacement functionis substituted into the Lagrangian equation and thefree vibration problem is transformed into the eigenvalue form of the standard matrix by using the Rayleigh-Ritz method Then the frequency and modeshape of each order are obtained by programmingwith software Mathematica1 Calculation model of vibration

of multi-span beam with elasticsupport

11 Geometric model

Fig 1 shows the calculation model of vibration ofmulti-span beam with elastic support inside thespans The total length of the beam is L and the number of spans is p The transverse spring and the rotating spring are set at the left and right ends of thebeam to simulate the boundary conditions and thestiffness coefficient of the transverse spring is set tosimulate the intermediate elastic support conditionsThe stiffness coefficients of the transverse spring andthe rotating spring at the left end boundary are k1and K1 respectively the stiffness coefficients of thetransverse spring and the rotating spring at the rightend boundary are kp + 1 and Kp + 1 respectively and thestiffness coefficients of the transverse spring elastically supported at the middle from left to right are k2k3 kp When the boundary condition is a fixedboundary the stiffness values of the transversespring and the rotating spring should be set to infinity at the same time (for example take 1014EI whereEI is the bending stiffness of the section) When theboundary condition is a free boundary the stiffnessvalues of the transverse spring and the rotatingspring are taken as 0 When the stiffness coefficientsof the transverse spring and the rotating spring aretaken as finite values the elastic constraint boundary condition can be simulated

Fig1 Multi-span beam model with elastic supports

K1

L1

k1 k2 ki ki + 1 kp + 1kp

Li Lp

BAO S Y et al Vibrational characteristics of a multi-span beam with elastic transverse supports of differentshaped sections 85



12 Expression of the displacementfunction

For the beam structure with elastic support sincethe vibration equation of the beam is a fourth-orderdifferential equation the displacement function expressed by the traditional Fourier series will have adiscontinuity problem at the boundary In order toeliminate the discontinuity of the mechanical variables at the end of the beam Reference [12] usedthe improved Fourier series to express the bendingdisplacement function based on the superposition offour sine functions with infinite cosine functions

Different from Reference [12] this paper adoptsthe form of infinite sine function superimposed byfour cosine functions namely that a new improvedFourier series suitable for beam deflection under anyelastic support condition is

（1）where xisin[0 L] an is the undetermined constant λn =nπL

Eq (1) can be called the improved Fourier sine series but the existing literature has little research onthe improved Fourier sine series Reference [11]pointed out that the Fourier sine series or the Fouriercosine series are all convergent when the displacement function is expanded so this paper studies thevibration of multi-span beam based on the improvedFourier sine series

For the multi-span beam with p segments shownin Fig 1 the deflection function of each segment ofthe beam should be set as different deflection functions [7-912-13] but for the sake of simplicity the deflection function suitable for the whole segment of thebeam is adopted in this paper According to the analysis when Eq (1) is adopted the displacement androtation angle at the intermediate elastic support arecontinuous and when the stiffness EI of both sides ofthe intermediate elastic support is the same thebending moment and shear force are also continuous

In the physical sense due to the existence of transverse spring the shear force of the beam at the intermediate elastic support will be discontinuous Therefore in this paper the bending displacement function w(x) of the whole beam is introduced into themodel instead of the piecewise function which is anapproximate treatment If w(x) is continuous everywhere on the whole beam its first derivative (corresponding to the cross-sectional rotating angle) andsecond derivative (corresponding to the cross-sec

tional bending moment) can be continuous everywhere on the whole beam which is consistent withthe actual situation In fact the third derivative ofw(x) in the model (corresponding to the cross-sectional shear force) is also continuous everywhere butthe beam shear force at the intermediate elastic support has a sudden change which is inconsistent withthe actual situation resulting in a small error in thefrequency value The results of the following examples can also reflect that the deflection function suitable for the entire beam can better solve the free vibration problem of multi-span beams

For the elastically supported multi-span beamstructure shown in Fig 1 there are three parts of potential energy as follows

（2）

（3）

（4）where Vp is the strain energy of the multi-span beamstructure Vs1 is the elastic potential energy of thesimulated spring at the boundary of the multi-spanbeam structure Vs2 is the elastic potential energy ofthe support spring at the intermediate elastic supportof the multi-span beam structure E is the elasticmodulus I is the cross-sectional moment of inertia

Regardless of the mass of the constraint springthe maximum kinetic energy of the multi-span beamwith elastic support is

（5）where S is the cross-sectional area of the beam ρ isthe mass density of the beam ω is the circular frequency

The Lagrangian function of multi-span beamstructure [17] is defined as

（6）where Vmax is the maximum value of total potential energy of multi-span beam structure and Tmax is themaximum value of total kinetic energy of multi-spanbeam structure

Eq (1)-Eq (5) are substituted into the Lagrangianfunction According to the Rayleigh-Ritz methodthe Lagrangian function should take the extreme val

86


ues of each undetermined coefficient in Eq (1) ie（7）

In practical calculation the displacement seriesin Eq (1) cannot be infinite The maximum value ofn in Eq (1) is taken as m According to Eq (7) m+5linear equations can be obtained and marticulated

（8）where K is the stiffness matrix M is the mass matrixA is the column vector composed of unknown coefficients in the new improved Fourier series shown inEq (1) namely

The condition of a nonzero solution of Eq (8) is（9）

By solving the eigenvalue problem of the matrixwe can obtain the natural frequency of themulti-span beam with spring support under anyboundary constraint By substituting the eigenvectorcorresponding to each natural frequency into Eq (1)we can obtain the modes of the multi-span beam2 Numerical calculation and anal-

ysis

Software Mathematica is used for programming Inthe following description the simply supportedboundary is denoted as S the free boundary is denoted as F and the clamped boundary is denoted as C

21 Convergence of single-span beamunder new improved Fourier series

Example 1 In the new improved Fourier series itis impossible to take infinity in the calculation of displacement series so the value of truncation numberm is related to the convergence of the results in thispaper In order to ensure the convergence this paperselects the single-span beam structure under S-Sboundary for the convergence analysis in which thedimensionless natural frequency is defined as follows

（10）Table 1 shows the first six dimensionless frequen

cies of a single-span beam with the change of thetruncation number m According to Table 1 Fig 2shows the line graph between the truncation numberand the frequency It can be seen from Fig 2 that inorder to ensure the convergence of the method in thispaper we generally take the truncation number of asingle-span beam as m=8 It can be seen from thefollowing examples that the reasonable value of thetruncation number is related to the span number pand the frequency order and its value increases withthe increase of the span number and the frequency ofthe order In the following examples m=10 is adopted for three-span beams m=12 is adopted for5-span beams m=18 is adopted for 8-span beamsand m=22 is adopted for ten-span beams

Table 1 The first six frequencies for different truncation numbers under the S-S boundary

Fig2 Line chart between truncation numbers and frequencies5 10 15 20

m

80

60

40

20

ω6

ωiHz

ω5

ω4

ω3

ω2ω1

22 Analysis of vibrational characteris-tics of double-span beams with dif-ferent sections

Example 2 It is considered to set up a supporteddouble-span cantilever beam at the midpoint wherep = 2 and the model diagram is shown in Fig 3 Thelength of the beam is 1 m the section of the beam isa solid rectangular section and the width and heightof the section are b times h=01 m times 01 m The material




density is 7 850 kgm3 and the elastic modulus is E=206 times1011 Pa In order to simulate the chain bar inEq (4) we take the stiffness value of the supportingtransverse spring as 1014 The first nine natural frequencies with the intermediate chain bar support arecalculated as shown in Table 2 and are comparedwith the numerical solution in Reference [7] The results show that the error is within 15 which verifies the correctness of the method in this paper Fig 4

shows the diagram of the first two modes of the double-span cantilever beam supported by the intermediate chain bar

05 m05 mFig3 Double-span beam model of rectangular section

Table 2 The first nine natural frequencies of C-F boundary beam with intermediate chain support

10

05

-05

-10

02 04 06 08 10

2nd mode

1st mode

Fig4 The modes of double-span cantilever beamIn addition to investigate the vibrational charac

teristics of beams with other shapes of sectionsI-shaped section and T-shaped section are selectedas shown in Fig 5 and Fig 6 respectively Whenthere are B times H = 005 m times 012 m b times h = 004 m times008 m for I-shaped section and there are B=005 mH=012 m b2=002 m h=002 m for T-shaped section the first nine frequencies are also listed in Table 2 It can be seen from Eq (10) that if thecross-section shape of a multi-span beam structurecomposed of the same material is changed the natural frequency of its vibration is directly proportionalto IS

When the intermediate chain bar support of thedouble-span cantilever beam moves from the leftmost end to the rightmost end namely that the distance a between the chain bar support and the leftmost end boundary changes from 0 m to 1 m thechange of the first two natural frequencies is shownin Table 3 and is compared with the existing results

(calculated according to the analytic frequency equation provided in Reference [7]) The error is withinthe allowable range It can be seen from Table 3 thatwith the increase of a the first-order natural frequency of the cantilever beam increases continuouslyWhen a increases to be close to the right end boundary the natural frequency begins to decrease gradually The second-order natural frequency increasescontinuously before the support reaches the middleposition and decreases gradually after the support

H

b2

h

B

b2

Fig5 I-shaped section

H

b2h

B

b2

Fig6 T-shaped section

88


passes the middle positionTable 3 Variation of the first two natural frequencies of

C-F boundary double-span beam with thelocation of chain bar

Table 4 shows the change of the first two naturalfrequencies of the support beams with intermediatechain bar when the support beam changes with thechain bar position a under the S-S and C-C boundaries Fig 7 shows the fold line of the free vibrationof the first two natural frequencies of double-spanbeams under the above two boundaries with thechange of the support position of the chain

Table 4 Variation of the first two natural frequencies ofS-S and C-C boundaries double-span beamwith location of chain bar

It can be seen from Fig 7 that for beams underS-S and C-C boundaries the first natural frequencyincreases with the increase of a which is similar tocantilever beams but when a gt 05 m the first natural frequency decreases with the increase of a How

ever the second natural frequency increases firstand then decreases with the increase of a when thechain bar is on the left side When the chain bar ison the right side of the beam the second natural frequency increases first and then decreases with the increase of a23 Analysis of vibrational characteris-

tics of multi-span beams with differ-ent sections

Example 3 The natural frequencies of multi-span beams with intermediate elastic supports arestudied The boundary condition is selected as anS-S boundary The total length of the beam is L=1 mthe diameter of the constant section is d=001 m themass density is ρ=05 kgm3 the elastic modulus isE=202 times 1011 Pa The numbers of elastic supportsprings are respectively taken as 2 4 7 9 and thebeam is just divided into several equal sections Fig 8shows the schematic diagram of a 10-span beam Table 5 shows several natural frequencies of the S-Sboundary beam where the transverse spring stiffnessvalues of the elastic support are all 107 At the sametime Table 5 also shows the first six natural frequencies when p = 3 or 5 and the first ten natural frequencies when p = 8 or 10 Through comparison it can beseen that the results shown in Table 5 are consistentwith the frequency results in Reference [18] The section size of Example 2 is selected for the I-shapedsection and T-shaped section Fig 9 and Fig 10show the first two modes of the circular section of thethree-span beam and five-span beam respectively

10 mFig8 The model of a ten-span beam with circular section

Fig 7 Variation of the first two natural frequencies ofdouble-span beam with location of chain support

02 04 06 08 10am

2 5002 0001 5001 000500

ωHz

S-S 1st frequencyS-S 2nd frequency

C-C 1st frequencyC-C 2nd frequency




The data of the beam in Example 3 are used Forthe multi-span beam under the C-C boundary whenp=3 5 8 10 the first two natural frequencies changing with the stiffness value of elastic support areshown in Table 6 It can be seen from Table 6 thatthe first two frequencies of any p-span beam in

crease with the increase of the stiffness value of theintermediate elastic support When the increasereaches a certain limit value (ie chain bar support)the frequency basically remains unchanged Withthe increase of span number p the first two naturalfrequencies of the elastic support beam gradually de

Table 5 The first ten natural frequencies of S-S boundary multi-span beam

10

05

-05

-10

1 2 3 4 5

2nd mode

1st mode

Fig 10 The modes of a five-span beam

10

05

-05

-10

05 10 15 20 25 30

2nd mode 1st mode

Fig 9 The modes of a three-span beam

Table 6 The first two natural frequencies of C-C boundary multi-span beam with different elastic support stiffness values

90


crease3 Conclusions

In this paper based on the improved Fourier series method the vibrational characteristics ofmulti-span beams with multiple elastic supports aresolved by the Rayleigh-Ritz method The proposedmethod has the following characteristics

1) In this paper a new form of Fourier series ofbeam displacement function is adopted namely thatseveral cosine functions are superposed on the basisof the Fourier sine series

2) The advantage of the method in this paper isthat there is no need to assume the deflection function for each piecewise sub-beam and only an approximate displacement function needs to be assumed for the whole beam which greatly simplifiesthe calculation during vibration analysis

3) The method in this paper is not limited to a specific boundary but also applicable to beams underany elastic boundary

The method proposed in this paper has simpleprogramming and high calculation accuracy and theerror with the results of the existing literature is generally within 15 which has good reference valuefor the analysis of the vibrational characteristics ofmulti-span beams in engineering applicationsReferences[1] HUANG Q LI H REN H L et al Elastic deformation

prediction method of hull based on hull girder [J] ShipEngineering 2017 39 (8) 13-17 88 (in Chinese)

[2] ABBAS B A H Vibrations of Timoshenko beams withelastically restrained ends [J] Journal of Sound and Vibration 1984 97 (4) 541-548

[3] CHUNG H Analysis method for calculating vibrationcharacteristics of beams with intermediate supports [J]Nuclear Engineering and Design 1981 63 (1) 55-80

[4] ZOU P QU X G Quasi wavelet-precise time-integration method for solving the vibration problems of beam[J] Journal of Shaanxi University of Science amp Technology 2011 29 (6) 140ndash143 (in Chinese)

[5] ZENG W P ZHENG X H Multi-symplectic methodsfor the vibration equation of beams [J] Journal of Zhangzhou Teachers College (Natural Science) 2003 16 (4)1-5 8 (in Chinese)

[6] WANG Q S WU L WANG D J Some qualitative properties of frequency spectrum and modes of differencediscrete system of multibearing beam [J] Chinese Jour

nal of Theoretical and Applied Mechanics 2009 41 (6)947-952 (in Chinese)

[7] LIU X Y NIE H WEI X H The transverse free-vibration model of three multi-span beams [J] Journal of Vibration and Shock 2016 35 (8) 21-26 (in Chinese)

[8] ZHANG H WANG C M CHALLAMEL N Bucklingand vibration of Hencky bar-chain with internal elasticsprings [J] International Journal of Mechanical Sciences 2016 119 383-395

[9] ZHANG Z G WANG J ZHANG Z Y et al Vibrationsof multi-span non-uniform beams with arbitrary discontinuities and complicated boundary conditions [J] Journal of Ship Mechanics 2014 18 (9) 1129-1141 (inChinese)

[10] CHEN X C MAO Q B XUE X L Free Vibration analysis of elastic foundation Euler beams with different discontinuities based on generalized functions [J] Applied Mathematics and Mechanics 2014 35 (1)81-91 (in Chinese)

[11] LI W L Free vibrations of beams with general boundary conditions [J] Journal of Sound and Vibration2000 237 (4) 709-725

[12] ZHOU B SHI X J Vibration analysis of multi-spanbeam system [J] Machinery Design amp Manufacture2017 (8) 43-46 (in Chinese)

[13] ZHOU H J LYU B L DU J T et al Transverse vibration analysis of shafting based on an improved Fourierseries method [J] Noise and Vibration Control 201131 (4) 68-72 (in Chinese)

[14] SHI D Y WANG Q S SHI X J et al Free vibrationanalysis of orthotropic thin plates in general boundaryconditions [J] Journal of Shanghai Jiao Tong Univerisity 2014 48 (3) 434-438 444 (in Chinese)

[15] BAO S Y WANG S D WANG B An improved Fourier-Ritz method for analyzing in-plane free vibration ofsectorial Plates [J] Journal of Applied Mechanics2017 84 (9) 091001

[16] BAO S Y WANG S D A generalized solution procedure for in-plane free vibration of rectangular platesand annular sectorial plates [J] Royal Society OpenScience 2017 4 (8) 170484

[17] Department of Theoretical Mechanics of Harbin Institute of Technology Theoretical mechanics ( Ⅱ) [M]8th ed Beijing Higher Education Press 2016 (in Chinese)

[18] YAN B H A theoretical model for the vibration of fuelrod with multi spans supported by springs [J] Annalsof Nuclear Energy 2018 119 257-263




不同截面形状下弹性支撑多跨梁振动特性分析

鲍四元周静苏州科技大学土木工程学院江苏苏州 215011

摘要［目的目的］为克服边界及弹性横向支撑对连续多跨梁振动特性研究的束缚基于欧拉梁理论建立一种多

跨梁自由振动的分析模型［方法方法］首先构造新型改进傅里叶级数形式用以表示多跨梁在整段上的横向位移

函数其次将位移函数的级数表达式代入拉格朗日函数中结合瑞利mdash里兹法将自由振动问题变为标准矩阵

特征值形式以求解带有弹性支撑的多跨梁固有频率［结果结果］通过在算例部分改变弹性支撑处的横向弹簧刚度

值即可获得中间含任意弹性支撑多跨梁的振动特性所得结果与已有文献结果的比较充分验证了所提方法可

行且正确［结论结论］基于改进傅里叶级数法（IFSM）多跨梁振动特性的数值模拟可为多跨梁动态性能提供有效

的前期预测手段

关键词多跨梁弹性支撑固有频率改进傅里叶级数方法

核电平台连接机构设计与运动响应分析

李想李红霞黄一大连理工大学船舶工程学院辽宁大连 116024

摘要［目的目的］为满足深海冰区海洋核反应堆安全工作的要求设计冰区核电平台与弹簧阻尼连接机构［方方

法法］利用三维势流理论及刚体动力学理论建立平台与连接机构的仿真模型计算平台所受弹簧阻尼力研究连

接机构刚度阻尼系数特性选择最佳方案应用离散元法进行冰载荷数值模拟通过计算试验椎体所受冰载

荷验证该方法的准确性研究浪风流或海冰风流环境载荷联合作用下平台的运动响应［结果结果］结果显

示平台系泊于深海冰区可远离海啸的影响环境承载平台能较好抵抗冰载荷在连接机构与系泊系统的作用

下核堆支撑平台可抵御福岛核泄漏事故最大海啸波高与 17 级超强台风的联合作用在北海万年一遇风暴作

用下核堆支撑平台的水平位移与水深之比垂荡与纵摇响应及垂向加速度均小于海上浮动核电平台

（OFNP）［结论结论］核电平台与连接机构的设计可保证应用于深海冰区的核堆的安全稳定

关键词核电平台连接机构冰载荷减振运动响应系泊

[Continued from page 83]

10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

92


tics of natural vibration of the difference discrete system of multi-span beams Liu et al [7] established thefree vibration models of three kinds of commonbeams (Euler beam Rayleigh beam and Timoshenko beam) analyzed them with the method of variation of parameters and deduced the frequency equation of the free vibration Zhang et al [8] establishedthe hencky bar-chain model (HBM) based on the finite difference model and derived the frequency formula of the HBM method The HBM method and finite difference method (FDM) were first proposed toanalyze the buckling load and vibration frequency ofthe internal elastic spring Zhang et al [9] used themethod of variable separation and transfer matrix toderive the characteristic equation of a beam and elaborated the geometric characteristics of the beam withunequal span variable cross section and arbitrarydiscontinuities Chen et al [10] established the vibration differential equation of the discontinuous Eulerbeam under the elastic foundation of the axial forcein the generalized function space and studied the influence of the rotational inertia of the additionalmass block on the beam-mass block system

However most of the methods of free vibrationproblems studied above have certain limitations Forexample some can only be used for qualitative research and some can only solve specific boundaryconditions Li [11] proposed an improved Fourier series form of function namely that four sine functionswere added to the traditional Fourier series formThe advantage of this form is that it can completelyeliminate the discontinuity of a function and its derivatives at the end point Zhou et al [12] analyzed thevibrational characteristics of the continuousmulti-span beam structure based on the improvedFourier series method (IFSM) Zhou et al [13] studiedthe lateral vibrational characteristics of the shaft system using the IFSM Shi et al [14] studied the lateralfree vibration of the orthotropic thin plate based onIFSM Bao et al [15-16] studied the in-plane free vibration of annular sector plate and rectangular platebased on IFSM

In order to study the free vibration of multi-span

beams this paper intends to propose a new IFSM torepresent the displacement function of beams whichis different from that in Reference [12] to avoid thediscontinuity at the end of multi-span beams underelastic boundary Firstly the displacement functionis substituted into the Lagrangian equation and thefree vibration problem is transformed into the eigenvalue form of the standard matrix by using the Rayleigh-Ritz method Then the frequency and modeshape of each order are obtained by programmingwith software Mathematica1 Calculation model of vibration

of multi-span beam with elasticsupport

11 Geometric model

Fig 1 shows the calculation model of vibration ofmulti-span beam with elastic support inside thespans The total length of the beam is L and the number of spans is p The transverse spring and the rotating spring are set at the left and right ends of thebeam to simulate the boundary conditions and thestiffness coefficient of the transverse spring is set tosimulate the intermediate elastic support conditionsThe stiffness coefficients of the transverse spring andthe rotating spring at the left end boundary are k1and K1 respectively the stiffness coefficients of thetransverse spring and the rotating spring at the rightend boundary are kp + 1 and Kp + 1 respectively and thestiffness coefficients of the transverse spring elastically supported at the middle from left to right are k2k3 kp When the boundary condition is a fixedboundary the stiffness values of the transversespring and the rotating spring should be set to infinity at the same time (for example take 1014EI whereEI is the bending stiffness of the section) When theboundary condition is a free boundary the stiffnessvalues of the transverse spring and the rotatingspring are taken as 0 When the stiffness coefficientsof the transverse spring and the rotating spring aretaken as finite values the elastic constraint boundary condition can be simulated

Fig1 Multi-span beam model with elastic supports

K1

L1

k1 k2 ki ki + 1 kp + 1kp

Li Lp













（2）

（3）







86







ysis








m

80

60

40

20

ω6

ωiHz

ω5

ω4

ω3

ω2ω1










10

05

-05

-10

02 04 06 08 10

2nd mode

1st mode





H

b2

h

B

b2


H

b2h

B

b2


88












02 04 06 08 10am

2 5002 0001 5001 000500

ωHz









10

05

-05

-10

1 2 3 4 5

2nd mode

1st mode


10

05

-05

-10

05 10 15 20 25 30

2nd mode 1st mode



90


crease3 Conclusions


















































10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

92












（2）

（3）







86







ysis








m

80

60

40

20

ω6

ωiHz

ω5

ω4

ω3

ω2ω1










10

05

-05

-10

02 04 06 08 10

2nd mode

1st mode





H

b2

h

B

b2


H

b2h

B

b2


88












02 04 06 08 10am

2 5002 0001 5001 000500

ωHz









10

05

-05

-10

1 2 3 4 5

2nd mode

1st mode


10

05

-05

-10

05 10 15 20 25 30

2nd mode 1st mode



90


crease3 Conclusions


















































10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

92







ysis








m

80

60

40

20

ω6

ωiHz

ω5

ω4

ω3

ω2ω1










10

05

-05

-10

02 04 06 08 10

2nd mode

1st mode





H

b2

h

B

b2


H

b2h

B

b2


88












02 04 06 08 10am

2 5002 0001 5001 000500

ωHz









10

05

-05

-10

1 2 3 4 5

2nd mode

1st mode


10

05

-05

-10

05 10 15 20 25 30

2nd mode 1st mode



90


crease3 Conclusions


















































10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

92







10

05

-05

-10

02 04 06 08 10

2nd mode

1st mode





H

b2

h

B

b2


H

b2h

B

b2


88












02 04 06 08 10am

2 5002 0001 5001 000500

ωHz









10

05

-05

-10

1 2 3 4 5

2nd mode

1st mode


10

05

-05

-10

05 10 15 20 25 30

2nd mode 1st mode



90


crease3 Conclusions


















































10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

92












02 04 06 08 10am

2 5002 0001 5001 000500

ωHz









10

05

-05

-10

1 2 3 4 5

2nd mode

1st mode


10

05

-05

-10

05 10 15 20 25 30

2nd mode 1st mode



90


crease3 Conclusions


















































10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

92






10

05

-05

-10

1 2 3 4 5

2nd mode

1st mode


10

05

-05

-10

05 10 15 20 25 30

2nd mode 1st mode



90


crease3 Conclusions


















































10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

92


crease3 Conclusions


















































10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

92

























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

92


BAO Siyuan ZHOU Jingjournal16.magtechjournal.com/jwk_zgjcyj/fileup/PingShen/...CHINESE JOURNAL OF...

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