Research Article Hopf Bifurcation Characteristics of...

Research ArticleHopf Bifurcation Characteristics of Dual-Front Axle Self-ExcitedShimmy System for Heavy Truck considering Dry Friction

Daogao Wei1 Ke Xu1 Yibin Jiang1 Changhe Chen1 Wenjing Zhao1 and Fugeng Zhou2

1School of Mechanical and Automotive Engineering Hefei University of Technology Hefei 230009 China2China Anhui Jianghuai Automobile Co Ltd Hefei 230022 China

Correspondence should be addressed to Daogao Wei weidaogaohfuteducn

Received 19 April 2015 Accepted 13 August 2015

Academic Editor Marcello Vanali

Copyright copy 2015 Daogao Wei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Multiaxle steering is widely used in commercial vehicles However the mechanism of the self-excited shimmy produced by themultiaxle steering system is not clear until now This study takes a dual-front axle heavy truck as sample vehicle and considers theinfluences of mid-shift transmission and dry friction to develop a 9 DOF dynamics model based on Lagrangersquos equation Based onthe Hopf bifurcation theorem and center manifold theory the study shows that dual-front axle shimmy is a self-excited vibrationproduced from Hopf bifurcationThe numerical method is adopted to determine how the size of dry friction torque influences theHopf bifurcation characteristics of the system and to analyze the speed range of limit cycles and numerical characteristics of theshimmy system The consistency of results of the qualitative and numerical methods shows that qualitative methods can predictthe bifurcation characteristics of shimmy systems The influences of the main system parameters on the shimmy system are alsodiscussed Improving the steering transition rod stiffness and dry friction torque and selecting a smaller pneumatic trail and casterangle can reduce the self-excited shimmy reduce tire wear and improve the driving stability of vehicles

1 Introduction

In recent years with the rapid development of the trans-portation industry high-speed and multiaxle heavy truckswith dual-front axles have become widely used for their loadcapacity high performance-price ratio adaptability and highhorsepowerThe dual-front axle steering system is a relativelyadvanced steering system given its low cost simple handlingsteering safety and stability under heavy load and the lessharm it causes on the road surface during driving Howeverthe shimmy problem in its dual-front axle steering systemleads to abnormal tire wear (especially the tire on the secondaxle) off-tracking and shaking of the steering wheel [1ndash3]

Research on the shimmy of single-axle vehicles can betraced back to 80 years ago and can be classified into forcedshimmy and self-excited shimmy which is a Hopf bifurcationphenomenon Extensive research has been conducted onthis field and can be used to solve the shimmy problemsin engineering [4ndash7] Given the growth in the demand formultiaxle heavy trucks current research on the dual-axlesteering system shimmy has demonstrated its significance

Considering its sophisticated mechanism the shimmy ofa dual-axle steering system differs from that of a single-axle steering system Therefore many scholars have studiedthe problem extensively Watanabe et al [1] studied theeffect of the number and position of driving wheels on thesteering performance of dual-axle steering vehicles Gu et al[8] demonstrated the content and method in the designof heavy trucks with a multiaxle steering system analyzedthe main problems in this research field in China andproposed a design method for a steering system based onintegrated and optimized matching platform for a chassissystem Hou et al [9] established the kinematics model and amathematical optimization model for the multiaxle steeringsystem of 10 times 8 heavy-duty vehicle and designed a newweight function considering the probability of steering angleThe parameters of multiaxle steering system were optimizedThe result showed that the result with weight functionhad better effect than other conditions Wang et al [10]applied a robust design theory with design parameters andnoise factors following a normal distribution in a dual-frontaxle steering system He combined reliability optimization

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 839801 20 pageshttpdxdoiorg1011552015839801

2 Shock and Vibration

with robust design and built a mathematical model forthe robust reliability optimization of dual-front axles withclearances Xu et al [11] developed a steering wheel shimmymodel for a four-axis steering crane without considering thenonlinear factors and their impact on the four-axis steeringcrane shimmy Nisonger and Wormley [12] compared thetransverse dynamic characteristics of single- and dual-frontaxle steering systems using a nonlinear model with threedegrees of freedom Wu and Lin [13] found that the double-front axle can improve a carrsquos yaw stability Williams [14]extended the dual-axle model for vehicles to a multiaxlemodel and analyzed its steering ability and handling stabilityBy analyzing a linear dual-front axle yaw dynamics modelDemic [15] analyzed the influence of structural parameterson the front wheel shimmy of a heavy vehicle and foundthat the vibration of suspension can cause front wheelshimmy Cole and Cebon [16] developed a vibration modelfor the pneumatic suspension of a heavy truck to reducethe vibration in and damage to the road surface caused bytrucks through suspension parameter optimization Chen[17] concluded that two reasonsmdashinternal and externalmdashaccount for abnormal tire wear proposed improvementmeasures based on his practical experience and pointed outdirections for future research Liao [18] and Li et al [19]explored the application of multirigid body theory in thedynamic simulation of a dual-front axle steering system andproposed the condition underwhich tires bear heavy load in adual-front axle vehicle They also built a spatial model for thedual-front axle based onADAMSand conducted a simulationanalysis Using TruckSim a dynamics analysis software forvehicles Li [20] studied how to determine and describe theseven characteristic parameters of a heavy truck steeringsystem namely axle and suspension transmission tire bodybrake system and aerodynamics and laid down a solidfoundation for the study ofmultiaxle steering technology andthe improvement of its handling stability

Most of the aforementioned studies focus on the con-structional parameter analysis and simulation of dual-frontaxle steering system however there is little research onsteering system shimmy especially there is little research onthe mechanism of Hopf bifurcation produce self-shimmyBut it is not yet clear about Hopf bifurcation characteristicof dual-front axle self-shimmy for heavy truck Thereforethe research in this area is very important In this studywe took a JAC heavy truck as an example and establishednine degrees of freedom equations for the dual-front axlesteering systembased onLagrangersquos equationWedeterminedthe existence and the value of Hopf bifurcation with the useof the Hurwitz criterion [21] and Hopf bifurcation theoremTo obtain the manifold for the system in two-dimensionalspace we reduced the dimension of the differential equationsof state using center manifold theory and found the Hopfparadigm in the polar coordinates with the use of normativetheory so that the Hopf bifurcation characteristics of theshimmy system of the dual-front axles can be fully analyzedBased on the above analysis we then propose a methodof controlling such shimmy As such this study establishesthe theoretical foundation for research on the mechanismof the dual-front axle self-excited shimmy and provides

V

1205791

1205792

1205793

1205794

Figure 1 Schematic of dual-front axle steering system

Steeringgear

Firsttransition

rod1205751

r1

g1r2

1205752 g2

r3

1205753 g3

Anteriorlongitudinal

rod

Intermediaterocker

Secondtransition

rod

Firstrocker

Posteriorlongitudinal rod

Secondrocker

sim

Figure 2 Schematic of intermediate steering transmission mecha-nism

a reference on the design improvement of steering systemsThe contribution of this study is the finding that the self-excited shimmy phenomenon of the dual-front axle of aheavy truck is generated by Hopf bifurcation

2 Dynamics Model for a Dual-Front AxleShimmy System

21 Mechanical Model The schematic of the structure ofthe widely used dual-front axle steering system is shown inFigure 1 the diagram of the steering intermediate transmis-sion is shown in Figure 2 To facilitate the development ofthe mathematical model of the dual-front axle self-excitedshimmy system the mechanics model of the dual-front axleself-excited shimmy system is developed based on Figure 2with the dual-front axle steering system of a JAC heavy truckas prototype the schematics of which are shown in Figure 3Figure 3 illustrates the double-front axle steering principlethe steering force exerted by the driver is passed throughthe steering operating mechanism which is composed of thesteering shaft transmission shaft and steering joints towardsthe steering gear (2)The torque is then transmitted to the firstrocker (3) after its torque increases and its speed decreasesThe first rocker (3) then drives the steering knuckle arm (5)

of the first steering bridge through the anterior longitudinalrod (4) to turn the left wheel to rotate around the kingpinwhich is installed in the first steering bridge At the sametime the right wheel driven by the torque which passes

Shock and Vibration 3

1

2

34

5

67

1314 8

15

910

11

12

kc

kz1cz1

kt1ct1 kh1

ch1

1205931

cz2kz2

kh2

ch2

kt2

ct2

1205932

(1) Steering wheel(2) Steering gear(3) First rocker(4) Anterior longitudinal rod(5) Steering knuckle arm(6) First transition rod(7) Intermediate rocker(8) Tie rod

(9) Second transition rod(10) Second rocker(11) Steering power cylinder(12) Posterior longitudinal rod(13) Tire(14) Left trapezoid arm(15) Right trapezoid arm

Figure 3 Dual-front axle shimmy system mechanics model

through the left trapezoid arm (14) to the tie rod (8) to theright trapezoid arm (15) rotates The second bridge steeringis consistent with the first bridge steering and the steeringforce is passed through the first transition rod (6) to theintermediate rocker (7) to the second transition rod (9) to thesecond rocker (10) to the posterior longitudinal rod (12) andto the second bridge

In the mechanical model the elasticity of the rod isconsidered it is equivalent to a spring-damper unit In linewith the law of the right-hand coordinate system the centerof the mass of the vehicle is deemed as the coordinate originthe vehicle forward direction is for 119909-axis the vehicle leftdirection is for 119910-axis and perpendicular to the ground updirection is for 119911-axis Figure 3 shows that the system hasnearly 20 rotating hinges To facilitate the dynamic analysisand mathematical modeling and to highlight the effect ofmultisport hinges dry friction and other parameters ofthe self-excited shimmy system as shown in Figure 3 thefollowing assumptions are made to establish a dynamicsmodel for the dual-front axle steering shimmy system (1)The steering wheel is immobile (2) The impact of the forceof air is ignored (3) The various parts associated with thevibration and their couplings are simplified according to themoments of inertia springs and dampers (4) The anglebetween the steering trapezoid plane and 119883119884 plane and theangle between the steering linkage and the 119883119885 plane areignored (5)Thedirection and speed of the vehicle is constant(6) Longitudinal and lateral slips do not occur in the vehicle(7)The dry friction in the kinematic pairs is equivalent to thekingpin of the first or second bridge

22 The Equation for the Motion of the Shimmy SystemAccording to the dynamics model in Figure 3 we established

the equations for the motion of the dual-front axle steeringshimmy system of heavy vehicles The shimmy system hasnine degrees of freedom 120579

1is the swing angle at which the

left wheel of the first bridge rotates around the kingpin 1205792is

the swing angle at which the right wheel of the first bridgerotates around the kingpin 120593

1is the lateral swing angle of the

first bridge 1205793is the swing angle at which the left wheel of

the second bridge rotates around the kingpin 1205794is the swing

angle at which the right wheel of the second bridge rotatesaround the kingpin 120593

2is the lateral swing angle of the second

bridge 1205751is the swing angle of the first rocking arm 120575

2is the

swing angle of the intermediate steering arm 1205753is the swing

angle of the second rocking armIn this study the mathematical model of the shimmy

system of the sample vehicle is established using Lagrangersquosequations which can be expressed as

119889

119889119905

(

120597119879

120597119896

) minus

120597119879

120597119902119896

+

120597119880

120597119902119896

+

120597119863

120597119896

= 119876119896

(119896 = 1 2 3 9)

(1)

where 119902119896represents nine degrees of freedom of the system

119879 represents the systemrsquos kinetic energy 119880 represents thesystemrsquos potential energy119863 represents the systemrsquos dissipatedenergy and 119876

119896represents the nine generalized forces to

which the system is subjectedAccording to Figure 3 kinetic energy potential energy

dissipated energy and the nine generalized forces of the dual-front axle steering shimmy system are obtained as follows


The kinetic energy of the shimmy system is

119879 =

1

2

1198681(1205791

2

+1205792

2

) +

1

2

11986911

2+

1

2

1198682(1205793

2

+1205794

2

) +

1

2

11986922

2

+

1

2

1198681198881

1205751

2

+

1

2

1198681198882

1205752

2

+

1

2

1198681198883

1205753

2

(2)

where 119868119894(119894 = 1 2) is the moment of inertia of the wheel

around the kingpin in 119894th bridge 119868119888119894(119894 = 1 3) is the moment

of inertia of 119894th rocker 1198681198882

is the moment of inertia of theintermediate rocker and 119869

119894(119894 = 1 2) is the moment of inertia

of 119894th bridge around its side off-axisThe potential energy of the shimmy system is

119880 =

1

2

119896ℎ1(1205791minus 1205792)2

+

1

2

1198961198991205931

2+

1

2

119896ℎ2(1205793minus 1205794)2

+

1

2

1198961198991205932

2+

1

2

1198961199051205751

2+

1

2

1198961199051(11989211205751minus 11990321205752)2

+

1

2

1198961199052(11989221205752minus 11989231205753)2

+

1

2

1198961199111(11990311205751+ 11988611205792)2

+

1

2

1198961199112(11990331205753+ 11988621205794)2

(3)

where 119896ℎ119894(119894 = 1 2) is the stiffness of the tie rod converted

into the stiffness around the kingpin 119896119899is the equivalent

angle stiffness of the suspension converted into the side swingcenter 119896

119905is the inverse stiffness of the first rocking arm

to the steering wheel 119896119905119894(119894 = 1 2) is the stiffness of 119894th

transition rod 119896119911119894(119894 = 1 2) is the stiffness of the front and rear

longitudinal rods 1199031is the effective length of the first arm 119903

2is

the effective length of themiddle arm 1199033is the effective length

of the second arm11989211198922 and119892

3are the distances between the

pivot point of the rocker and the transition lever and the fixedend of the rocker arm and 119886

119894(119894 = 1 2) is the distance between

the hinge point of 119894th bridge knuckle arm and vertical rod andkingpin

The dissipated energy of the shimmy system is

119863 =

1

2

1198881198971(1205791+

1205792) +

1

2

119888ℎ1(1205791minus

1205792)

2

+

1

2

1198881198991

2

+

1

2

119888ℎ2(1205793minus

1205794)

2

+

1

2

1198881198972(1205793+

1205794) +

1

2

1198881198992

2

+

1

2

1198881199051205751

2

+

1

2

1198881199051(11989211205751minus 11990321205752)

2

+

1

2

1198881199052(11989221205752minus 11989231205753)

2

+

1

2

1198881199111(11990311205751+ 11988611205792)

2

+

1

2

1198881199112(11990331205753+ 11988621205794)

2

+11987222

1205794+11987221

1205793+11987212

1205792

+11987211

1205791

(4)

where 119888119897119894(119894 = 1 2) is the equivalent damping of 119894th wheel

around the kingpin 119888ℎ119894(119894 = 1 2) is the damping of the tie

rod converted into the damping around the kingpin 119888119899is the

equivalent damping of the suspension converted into the sideswing center 119888

119905is the inverse damping of the first rocking

arm to the steering wheel 119888119905119894(119894 = 1 2) is the damping of

119894th transition rod 119888119911119894(119894 = 1 2) is the damping of the front

and rear longitudinal rods 11987211

is the equivalent frictiontorque of the first bridge at the right wheel kingpin119872

12is the

equivalent friction torque of the first bridge at the left wheelkingpin 119872

21is the equivalent friction torque of the second

bridge at the right wheel kingpin and 11987222

is the equivalentfriction torque of the second bridge at the left wheel kingpin

The nine generalized forces of the shimmy system are asfollows for details about the procedure for calculation of thegeneralized forces see appendix

1198761= minus1198941198961

V119877

1+ 1198651(119877120574 + 119890)

minus (120574120588ℎ1119877 +

119896119888

2

11986111198971(minus119891 + 120574))120593

1

1198762= minus1198941198961

V119877

1+ 1198652(119877120574 + 119890)

minus (120574120588ℎ1119877 +

119896119888

2

11986111198971(minus119891 + 120574))120593

1

1198763= minus2120588119877ℎ

11205931+ 1198941198961

V119877

(1205791+

1205792) minus (119865

1+ 1198652) 119877

1198764= minus1198941198962

V119877

2+ 1198653(119877120574 + 119890)

minus (120574120588ℎ2119877 +

119896119888

2

11986121198972(minus119891 + 120574))120593

2

1198765= minus1198941198962

V119877

2+ 1198654(119877120574 + 119890)

minus (120574120588ℎ2119877 +

119896119888

2

11986121198972(minus119891 + 120574))120593

2

1198766= minus2120588119877ℎ

21205932+ 1198941198962

V119877

(1205793+

1205794) minus (119865

3+ 1198654) 119877

1198767= 0

1198768= 0

1198769= 0

(5)


around its own axis of rotation in 119894th bridge V is the vehiclespeed 119877 is the rolling radius of the tire 119865

1is the first bridge

right wheel subjected to lateral force 1198652is the first bridge left

wheel subjected to lateral force 1198653is the second bridge right

wheel subjected to lateral force 1198654is the second bridge left

wheel subjected to lateral force 120574 is the kingpin caster angle ofthe wheel 119890 is the pneumatic trail 120588 is the tire lateral stiffnessℎ119894is the height of 119894th suspension roll center (119894 = 1 2) 119896

119888

is the tire vertical stiffness 119861119894(119894 = 1 2) is the tread in 119894th

bridge 119897119894(119894 = 1 2) is the distance between the point of the

kingpin extension linewith the ground intersection and planeof symmetry of thewheel119891 is the friction coefficient betweenthe tire and the ground and 120590 is the tire relaxation length

According to (2) to (5) the kinetic equations for thesystem are derived from Lagrangersquos equations as given by (1)


Equations for the motion of the right wheel of the firstbridge around the kingpin are as follows

11986811205791+ 1198881198971

1205791+ 119896ℎ1(1205791minus 1205792) + 119888ℎ1(1205791minus

1205792) +

1198941198961V

119877

1

+ (120588119877ℎ1120574 +

1

2

11989611988811989711198611(120574 minus 119891)) 120593

1minus 1198651(119877120574 + 119890)

+11987211= 0

(6)

Equations for the motion of the left wheel of the first bridgearound the kingpin are as follows

11986811205792+ 1198881198971

1205792+ 119896ℎ1(1205792minus 1205791) + 119888ℎ1(1205792minus

1205791)

+ 11989611991111198861(11990311205751+ 11988611205792) + 11988811991111198861(11990311205751+ 11988611205792)

+

1198941198961V

119877

1+ (120588119877ℎ

1120574 +

1

2

11989611988811989711198611(120574 minus 119891)) 120593

1

minus 1198652(119877120574 + 119890) +119872

12= 0

(7)

Lateral swing equations for the motion of the first bridge areas follows

11986911+ 11988811989911+ (

1

2

1198961198881198611

2+ 119896119899+ 2120588119877ℎ

1)1205931

minus

1198941198961V

119877

(1205791+

1205792) + (119865

1+ 1198652) ℎ1= 0

(8)

Equations for the motion of the right wheel of the secondbridge around the kingpin are as follows

11986821205793+ 1198881198972

1205793+ 119896ℎ2(1205793minus 1205794) + 119888ℎ2(1205793minus

1205794) +

1198941198962V

119877

2

+ (120588119877ℎ2120574 +

1

2

11989611988811989721198612(120574 minus 119891)) 120593

2minus 1198653(119877120574 + 119890)

+11987221= 0

(9)

Equations for themotion of the leftwheel of the secondbridgearound the kingpin are as follows

11986821205794+ 1198881198972

1205792+ 119896ℎ2(1205794minus 1205793) + 119888ℎ2(1205794minus

1205793)

+ 11989611991121198862(11990331205753+ 11988621205794) + 11988811991121198862(11990331205753+ 11988621205794)

+

1198941198962V

119877

2+ (120588119877ℎ

2120574 +

1

2

11989611988811989721198612(120574 minus 119891)) 120593

2

minus 1198654(119877120574 + 119890) +119872

22= 0

(10)

Lateral swing equations for the motion of the second bridgeare as follows

11986922+ 11988811989922+ (

1

2

1198961198881198612

2+ 119896119899+ 2120588119877ℎ

2)1205932

minus

1198941198962V

119877

(1205793+

1205794) + (119865

3+ 1198654) ℎ2= 0

(11)

Magic formulaCubic formula

minus6 minus4 minus2minus8 2 4 60 8

Side-slip angle 120572 (∘)

minus4000

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

Tire

forc

eF(N

)

Figure 4 Curve of relationship between 119865119894and 120572

119894

Swing equations for the motion of the first rocking arm are asfollows

1198681198881

1205751+ 1198881199051205751+ 1198961199051205751+ 11988811990511198921(11989211205751minus 11990321205752)

+ 11989611990511198921(11989211205751minus 11990321205752) + 11988811991111199031(11990311205751+ 11988611205792)

+ 11989611991111199031(11990311205751+ 11988611205792) = 0

(12)

Swing equations for the motion of the intermediate steeringarm are as follows

1198681198882

1205752+ 11988811990511199032(11990321205752minus 11989211205751) + 11989611990511199032(11990321205752minus 11989211205751)

+ 11988811990521198922(11989221205752minus 11989231205753) + 11989611990521198922(11989221205752minus 11989231205753)

= 0

(13)

Swing equations for themotion of the second rocking arm areas follows

1198681198883

1205753+ 11988811990521198923(11989231205753minus 11989221205752) + 11989611990521198923(11989231205753minus 11989221205752)

+ 11988811991121199033(11990331205753+ 11988621205794) + 11989611991121199033(11990331205753+ 11988621205794) = 0

(14)

23 Tire Model Selection Several nonlinear tire models arecommonly used in the simulation of vehicle dynamics suchas Pacejkarsquos magic formula cubemodel Guo Konghuirsquos semi-empirical tire theoretical model and Gimrsquos tire models [22ndash25] Equations (15) and (16) are the mathematical expressionsof the cube and magic formula models respectively Thecornering force curves of these two tire models are shownin Figure 4 which shows that two cornering force curveshave the same trend and that the cube model is similar tothe magic formula Moreover the cube model is simple itdoes not require a considerable amount of experimental datacan precisely reveal the performance characteristic trends of


Table 1 Parameters of tire model

1198621119891

1198623119891

1198861

1198862

1198863

1198864

1198865

1198866

1198867

1198868

32740 481770 minus221 1011 1078 182 0208 0 minus0354 0707

the tire force and does not contain trigonometric functionsthereby facilitating qualitative analysis As such we choosethe cube model as the tire model in this study The cubemodel is the third-order truncation of themagic formula [23]The coefficients (119861

119894 119863119894 and 119864

119894) of magic formula take into

account the impact of the vertical load119865119885119894 So the cubemodel

cube considers the vertical force 119865119885119894

119865119894= minus (119862

1119891120572119894minus 1198623119891120572119894

3) (119894 = 1 2 3 4) (15)

119865119894= 119863119894sin (119862

119894

times arctan (119861119894119886119894minus 119864119894(119861119894119886119894minus arctan (119861

119894119886119894))))

(16)

In this model

119861119894=

1198863sdot sin (119886

4sdot (arctan (119886

5sdot 119865119885119894)))

119862119894sdot 119863119894

119862119894= 13

119863119894= 1198861sdot 119865119885119894

2+ 1198862sdot 119865119885119894

119864119894= 1198866sdot 119865119885119894

2+ 1198867sdot 119865119885119894+ 1198868

(119894 = 1 2 3 4)

(17)

where 1198621119891

and 1198623119891

are the fitting coefficients of the tire120572119894(119894 = 1 2 3 4 denote the first bridge right wheel the first

bridge left wheel the second bridge right wheel and thesecond bridge left wheel resp) is the side-slip angle of thetire 119861

119894 119862119894 119863119894 and 119864

119894denote the stiffness shape crest and

curvature factors respectively 1198861 1198862 1198863 1198864 1198865 1198866 1198867 and 119886

8

are obtained by test fitting 119865119885119894

is the vertical forces on thetires For details see appendix about 119865

119885119894 1198621119891 and 119862

3119891 The

required values [26 27] are shown in Table 1The relationship curve between the tire force 119865

119894and the

side-slip angle 120572119894is shown in Figure 4

The following are the tire rolling nonholonomic con-straint equations [28] namely the relationship between slipangle and the shimmy angle

119894+

V120590

120572119894+

V120590

120579119894minus

119890119894

120590

120579119894 119894 = 1 2 3 4 (18)

24 Dry Friction Model Selection In this study the dryfrictions in the suspension and steeringmechanism ball kine-matic pairs and steering gear kinematic pairs are consideredSuspension and steering systems are complex self-excitedshimmy systems thus we made the kinematic pairs dryfriction of the suspension and the steering system exceptfor the dry friction in the kingpin equivalent to that ofthe kingpin because the dry friction in the kingpin is themain component of sports vice dry friction We studied onlythe effect of the value of dry friction torque on the Hopf

bifurcation of the shimmy system and therefore we changeonly the dry friction torque 119872 when making numericalcalculations

We selected the Coulomb model as the ideal frictionmodel in this study It has a constant value and is alwaysin the opposite direction of the relative motion This kindof friction always impedes movement and is always in theopposite direction of the movement speed Its mathematicalexpression is

119872119894119895= 119872 sgn (

120579119894119895) (119894 119895 = 1 2)

|119872| =

120583119904119873119903 (

120579119894119895= 0)

120583119873119903 (120579119894119895

= 0)

(119894 119895 = 1 2)

(19)

where 120583 is the coefficient of sliding friction 120583119904is the

coefficient of static friction 119873 denotes a positive pressureand 119903 is the main pin radius

The parameter values of the sample vehicle are shown inTable 2

3 Hopf Bifurcation Qualitative Analysis ofthe Shimmy System

31 Existence Analysis of Hopf Bifurcation The existenceanalysis and stability assessment of Hopf bifurcation in anonlinear system are the most fundamental and importantassessments in studying the dynamics of a nonlinear systemHurwitz criterion is applied to investigate the existence ofHopf bifurcation in the dual-front axle shimmy system Thecenter manifold approach is applied to reduce and simplifythe nonlinear system and obtain a two-dimensional systemThe Hopf bifurcation paradigm is then applied to investigatethe stability of the bifurcation of limit cycles

Make 1199091= 1205791 1199092=

1205791 1199093= 1205792 1199094=

1205792 1199095= 1205931 1199096= 1

1199097= 1205793 1199098=

1205793 1199099= 1205794 11990910

=1205794 11990911

= 1205932 11990912

= 2

11990913

= 1205751 11990914

=1205751 11990915

= 1205752 11990916

=1205752 11990917

= 1205753 11990918

=1205753

11990919

= 1205721 11990920

= 1205722 11990921

= 1205723 and 119909

22= 1205724 Assume 119883 =

(1199091 1199092 1199093 1199094 1199095 1199096 1199097 1199098 1199099 11990910 11990911 11990912 11990913 11990914 11990915 11990916

11990917 11990918 11990919 11990920 11990921 11990922) according to the theory of

nonlinear dynamics the equilibrium point1198830is obtained by

solving the system static equations Equations (6) to (14) canbe expressed as the following static equations

= 119860119883 + 119865 (119883) 119883 isin 11987722 (20)

where 119860 is the Jacobian matrix in the equilibrium pointvicinity of the system and 119865(119883) contains quadratic and cubicnonlinear terms

According to Hurwitz criterion the equilibrium point ofthe system is stable when V = V

1or when V = V

2 whereas

the equilibrium point of the system is unstable when V1le

V le V2and when the critical speeds are V

1= 156 kmh and

V2= 408 kmh The eigenvalues of the Jacobian matrix 119860 are

shown in Table 3According to Table 3 the Jacobian matrix119860 of the system

has a pair of purely imaginary eigenvalues when V = V1and

V = V2 and the other eigenvalues have negative real parts


Table 2 Sample car parameter values

Parameter Value1198681 1198682kgsdotm2sdotradminus1 168

1198691 1198692kgsdotm2sdotradminus1 2108

1198681198881kgsdotm2sdotradminus1 038




119896ℎ1 119896ℎ2Nsdotradminus1 37790

119888ℎ1 119888ℎ2Nsdotmsdotssdotradminus1 25

1198881198971 1198881198972Nsdotmsdotssdotradminus1 42

119896119899Nsdotradminus1 31360

119888119899Nsdotmsdotssdotradminus1 1027

1205741 1205742∘ 25

1198971 1198972m 007

1199031m 034

1199033m 047

1198922m 026

1198861m 021

1198961199111 1198961199112Nsdotradminus1 36000


1198961199051 1198961199052Nsdotmminus1 37200

1198881199051 1198881199052Nsdots 15

119896119888Nsdotradminus1 1060000

119891 0015119896119905Nsdotradminus1 31000


120588Nsdotradminus1 765000119877m 0511198901 1198902m 007

ℎ1 ℎ2m 045

1198611 1198612m 206

1199032m 042

1198921m 017

1198923m 022

1198862m 021

Using theHopf bifurcation theorem as basis we can concludethat the critical speeds V

1= 156 kmh and V

2= 408 kmh are

the bifurcation points of the systems and a two-dimensionalcenter manifold exists When V

1le V le V

2 positive real part

eigenvalues exist and the original shimmy system has self-excited vibrations and produces limit cycles When V lt V

1

or V gt V2 all the eigenvalues of 119860 have negative real parts

and the original shimmy system is stable asymptotically andeventually becomes balanced

32 Stability Analysis of Limit Cycles The center manifoldapproach is applied to determine the stability of the originalsystem in low-dimensional systems and to reduce the originalsystem Assuming 120583

119896= VminusV

119896(119896 = 1 2) 120583

119896is an increment in

the speed bifurcation parameter at the bifurcation point andhas a minimum value Using the nonsingular transformation119883 = 119875119884 where 119884 has the same dimensions as 119883 119875

Table 3 Eigenvalues of Jacobian matrix 119860 corresponding to thecritical speed

V1corresponding eigenvalues V

2corresponding eigenvalues

1205821

0 + 1211119894 1205821

0 + 1456119894

1205822

0 minus 1211119894 1205822

0 minus 1456119894

1205823

minus16544 + 78083119894 1205823

minus16544 + 78083119894

1205824

minus16544 minus 78083119894 1205824

minus16544 minus 78083119894

1205825

minus7140 + 54908119894 1205825

minus7140 + 54908119894

1205826

minus7140 minus 54908119894 1205826

minus7140 minus 54908119894

1205827

minus437 + 6315119894 1205827

minus425 + 6316119894

1205828

minus437 minus 6315119894 1205828

minus425 minus 6316119894

1205829

minus434 + 6178119894 1205829

minus422 + 6179119894

12058210

minus434 minus 6178119894 12058210


12058211

minus218 + 2688119894 12058211

minus162 + 2773119894

12058212

minus218 minus 2688119894 12058212


12058213

minus224 + 2681119894 12058213

minus179 + 2746119894

12058214

minus224 minus 2681119894 12058214


12058215

minus122 + 1621119894 12058215

minus098 + 1765119894

12058216

minus122 minus 1621119894 12058216


12058217

minus24378 + 91155119894 12058217

minus24378 + 91155119894

12058218

minus24378 minus 91155119894 12058218

minus24378 minus 91155119894

12058219

minus892 12058219

minus1514

12058220

minus664 12058220

minus1358

12058221

minus355 12058221

minus909

12058222

minus354 12058222

minus910

is composed of the real and imaginary parts of all theeigenvectors andΛ = 119875

minus1119860119875 is 22times22 diagonal matrix (20)

is converted into

= Λ119884 + 119875minus1119892 (119884 120583

119896) (21)

Using center manifold theory to reduce dimension thecenter manifold 119882

119888 of the expansion system represented by(21) is tangent to the plane of (119910

1 1199102 120583119896) at the singular point

(1198830 V0) assuming the center manifold

119910119894= ℎ119894(1199101 1199102 120583119896)

= ℎ11989411199101

2+ ℎ11989421205831198961199101+ ℎ11989431205831198961199102+ ℎ11989441199102

2

119894 = 3 4 21 22

(22)

119894=

120597ℎ119894(1199101 1199102 120583119896)

1205971199101

1+

120597ℎ119894(1199101 1199102 120583119896)

1205971199102

2

+

120597ℎ119894(1199101 1199102 120583119896)

120597120583119896

120583119896

(23)

Equation (22) is substituted into (21) and is then com-bined with (23) after which the coefficients of the same itemon both sides of the equation are compared using the softwareMaple In solving the linear equations the coefficients ofℎ119894(1199101 1199102 V) (119894 = 3 4 21 22) can be obtained and brought

into the previous two equations of (21) After simplifying


the equation reduction equations can be obtained when V1=

156 kmh or V2= 408 kmh at the center manifold

When V = V1= 156 kmh the reduction equation is

1= 121 times 10119910

2+ 0762119910

21205831minus 0396119910

11205831minus 0169

times 10minus311991021205831

2minus 0156 times 10

minus41199102

3minus 0118

times 10minus311991011205831


minus311991011199102

2minus 066

times 10minus21199101

21199102minus 0261 times 10

minus11199101

3+ 0426


211991021205831+ 0464 times 10

minus41199101

31205831

2= minus121 times 10119910

1minus 0148119910

11205831+ 0853119910

21205831+ 0998

times 10minus411991011205831

2+ 0113 times 10

minus11199101

3+ 0429

times 10minus411991021205831

2+ 0286 times 10

minus21199101

21199102+ 0242

times 10minus311991011199102


minus41199101

31205831minus 0187


211991021205831

(24)


1= 146 times 10119910


minus21199101+ 0522119910

21205832

minus 055011991011205832minus 0784 times 10

minus21199102

3+ 0118

times 10minus411991011205832


minus11199102

21199101

minus 010111991021199101


minus11199101

3minus 0259

times 10minus411991021199101

21205832minus 0231 times 10

minus41199101

31205832

2= minus146 times 10119910


minus21199102+ 0275

times 10minus111991011205832+ 0261119910

21205832+ 0381

times 10minus411991011205832

2+ 0391 times 10

minus11199101

3minus 0531

times 10minus411991021205832

2+ 0565 times 10

minus111991021199101

2+ 0273


21199101+ 0439 times 10

minus21199102

3+ 0103


31205832+ 0116 times 10

minus411991021199101

21205832

(25)

According to the literature [29] the Hopf bifurcationparadigm under the polar coordinates of (24) to (25) can beobtained using Maple procedures

119903 = dur + 1198861199033+ hot

120579 = 120596 + 119887119903

2+ hot

(26)

where hot represents infinitesimals of higher orderAccording to (24)sim(28) we can obtain the following

F

120583sN120583N

r

minus120583Nminus120583sN

Figure 5 Shimmy system friction with the change of velocity curve

For V = V1= 156 kmh

119903 = (minus0751 times 10minus41205831+ 0229) 120583

1119903 + (minus0193

times 10minus6

1205831

2+ 0236 times 10

minus41205831minus 0106 times 10

minus1) 1199033

+ hot

(27)

For V = V2= 408 kmh

119903 = (minus0413 times 10minus4

1205832minus 0145) 120583

2119903 + (minus0495

times 10minus6

1205832


minus4

1205832minus 0476 times 10

minus1

) 1199033

+ hot

(28)

According to (27) and (28) we can obtain the bifurcationdiagram at equilibrium point119883

0of each critical velocity The

diagram is shown in Figure 6According to Figure 6 Hopf bifurcations are supercritical

at the critical speeds V1and V

2 When V lt V

1= 156 kmh

(ie 1205831lt 0) shimmy does not occur in the system that is

the system is stable and equilibrium point1198830is a stable focus

When V1lt V lt V

2(ie 120583

1gt 0 and 120583

2lt 0) shimmy occurs in

the system equilibrium point 1198830is an unstable focus and a

stable limit cycle appears The phenomenon of the mutationof a stable focus into a stable limit cycle is called the limit cyclehard to produce or stable hard lossWhen V gt V

2= 408 kmh

(ie 1205832gt 0) the equilibrium point turns into a stable focus

again the limit cycle disappears the shimmy phenomenondisappears and the system tends to be stableThus dual-frontaxle shimmy is a typical self-excited vibration

4 Calculation and Analysis of the HopfBifurcation in the Shimmy System

41 Numerical Calculation and Analysis Using the motionequations of the sample vehicle shimmy system and theRunge-Kutta method for the numerical calculation ofshimmy systems the bifurcation diagrams in Figure 7 showthat the left wheel swing angles of the first and second bridgesvary with speed


minus0015 minus001 minus0005 0 0005 001 0015 002minus002

Bifurcation parameter 1205831

minus08

minus06

minus04

minus02

0

02

04

06

08H

opf b

ifurc

atio

n br

anch

r

(a) Bifurcation diagram at V1

minus04

minus03

minus02

minus01

0

01

02

03

04



Hop

f bifu

rcat

ion

bran

chr

(b) Bifurcation diagram at V2

Figure 6 Bifurcation diagram of1198830at two critical speeds

20 40 60 80 100 120 1400Velocity V (kmh)

0

05

1

15

2

25

3

35

4

Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

(a) First bridge left wheel

20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12Se

cond

brid

ge le

ft w

heel

swin

g an

gle1205794

(∘)

(b) Second bridge left wheel

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

(c) The first bridge side pendulum

20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)

05

06

(d) The second bridge side pendulum

Figure 7 First and second bridge Hopf bifurcation with change of speed

According to Figure 7 when V lt 156 kmh and V gt

408 kmh the vibration of a dual-front axle system graduallystabilizes whereas when 156 kmh lt V lt 408 kmh periodicoscillations occur in the dual-front axle steering systemresulting in the limit cycle These results are consistent with

the qualitative calculation results in Section 31 which in turnshow that the theoretical qualitative methods are consistentwith the numerical methods

To better analyze the state changes in a dual-front axlesystem within the shimmy speed range we provide a swing


First bridgeSecond bridge

minus10minus15 minus5 10 1550

Left wheel swing angles 1205792 1205794 (∘)

minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(a) V = 20 kmh




minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(b) V = 25 kmh


minus150

minus100

minus50

0

50

100

150

minus5 0 5 10minus10


Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(c) V = 30 kmh

Figure 8 Phase diagram of the two front wheels at different speed

angle phase diagram of the left wheel of the first and secondbridges of heavy-duty vehicles under different speeds asshown in Figure 8

The conclusions on the shimmy characteristics and lawspertaining to the left wheel of the first and second bridgesunder different speeds as shown in Figures 8(a)ndash8(c) aredrawn The results are summarized in Table 4

(1) From Figures 8(a) to 8(c) in the bifurcation speedrange the shimmy of the steering wheel is the limitcycle vibration with a larger amplitude When thespeed increases the self-excited oscillation ampli-tude of the swing angle increases initially and thendecreases and the amplitude of the swing angle accel-eration has the same trend The maximum amplitude

Table 4 Shimmy characteristic at different speed

Vkmsdothminus1 20 25 301205792∘ 373 381 328

1205794∘ 1055 1092 947

Δ120579∘ 682 711 6191205792radsdotsminus1 083 086 0761205794radsdotsminus1 231 246 221

Δ120579radsdotsminus1 148 159 144

of the swing angle of the left wheel of the first bridge is381∘ and the maximum amplitude of the swing angleacceleration is 086 radsThemaximum amplitude ofthe swing angle of the left wheel of the second bridge



minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05

Side swing angles 1205931 1205932 (∘)

(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh

Figure 9 Phase diagram of the two bridges side pendulum at different speed

is 1092∘ and the maximum amplitude of the swingangle acceleration is 246 rads This phenomenonleads the vehicle to huntingmovement andmakes thedriver feel the tension and fatigue

(2) According to Figures 8(a)ndash8(c) and Table 4 the vari-ations in the left wheel swing angle differences andangular acceleration differences of the first and thesecond bridges are consistent with the variationsin the swing angles and angular accelerations Themaximum difference between the left wheel swingangles of the two bridges is 711∘ whereas the min-imum difference is 619∘ The maximum differenceof the angular acceleration is 159 rads whereas theminimum difference is 144 rads These results showthat the intensities of the tire shimmy of the first and

second bridges are in a state of serious imbalanceTheintensity of the tire shimmy at the second bridge issignificantly greater than that at the first bridge duringa shimmy These results coincide with the actual useof heavy-duty trucks in which the tire of the secondbridge experiences more severe wear than the tire ofthe first bridge

(3) According to Figures 8(a)ndash8(c) and Table 4 speed isan important bifurcation parameter in a dual-frontaxle shimmy systemThus driving speed should avoidthe bifurcation range as much as possible

Figure 9 shows the side pendulum phase diagrams ofthe first and second bridges of heavy-duty vehicles underdifferent speeds


Original systemReduced system

minus015 minus005minus01minus02 005 01501 020

First bridge right wheel swing angle 1205791 (∘)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)

(a) First bridge right wheel



First bridge left wheel swing angle 1205792 (∘)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)

(b) First bridge left wheel


0 05minus05

Second bridge right wheel swing angle 1205793 (∘)

minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)

(c) Second bridge right wheel


0 05minus05

Second bridge left wheel swing angle 1205794 (∘)

minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)

(d) Second bridge left wheel


minus0005 0 0005 001minus001

First bridge side swing angle 1205931 (∘)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)

(e) The first bridge side pendulum


minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002

Second bridge side swing angle 1205932 (∘)

(f) The second bridge side pendulum

Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)


First rocker swing angle 1205751 (∘)

(g) First rocker swing angle


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)


Intermediate rocker swing angle 1205752 (∘)

(h) Intermediate rocker swing angle


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)


Second rocker swing angle 1205753 (∘)

(i) Second rocker swing angle

Figure 10 For V = 156 kmh comparison of limit cycle of original systems and limit cycle of dimensionality reduction system

Figure 9 shows that the side pendulum phase diagram ofthe first bridge is elliptically regular indicating that the firstbridge side pendulum angle is at its maximum when theside pendulum angular velocity is at its minimum the sidependulum angle is at its minimum and the side pendulumangular velocity is at its maximum The pendulum phasediagram curve of the second bridge side is inwardly recessedalong 119910-axis direction indicating that the second bridge sidependulum angle is at its maximum when the side pendulumangular velocity is at its minimum and the side pendulumangle is at its minimum but the side pendulum angularvelocity is not at its maximum Taking Figure 9(b) as anexample when the angular velocity is less than 0 the slope ofthe first bridge phase diagram trajectory only changes oncefrom negative to positive By contrast the slope of the secondbridge phase diagram trajectory changes twice from negativeto positive and then becomes positive Figure 9 shows that

the value of the second bridge side pendulum is not onlygreater than that of the first bridge but also more complexthan that of the first bridge

42 Dimensionality Reduction System Limit Cycle Comparedwith the Original Systems When V = V

1= 156 kmh 120583

1=

0001V1 we generate the original system limit cycle phase

diagram with the use of the four- and five-order Runge-Kuttamethod and the dimensionality reduction system limit cyclephase diagram with the use of center manifold theory Thesediagrams are shown in Figure 10

When V = V2= 408 kmh 120583

2= 0001V

2 we generate

the original system limit cycle phase diagram with the useof the four- and five-order Runge-Kutta method and thedimensionality reduction system limit cycle phase diagramwith the use of center manifold theory These diagrams areshown in Figure 11



minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)

002 004 0060minus004 minus002minus006






minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)




minus015 minus01 minus005 0 005 01 015minus02


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 minus0005 0 0005 001 0015minus0015



Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03

minus04minus003 minus002 minus001


Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)




As shown in Figures 10 and 11 the dimension reductionsystem retains the bifurcation characteristics of the originalsystem at the bifurcation point Thus we can utilize the limitcycle phase diagram of the bifurcation characteristics of thedimensionality reduction system near the bifurcation point

43 Effect of Dry Friction Torque on Hopf Bifurcation Char-acteristics We formulate the shimmy system differentialequations considering the impact of kingpin clearance dryfriction Using the four- and five-order Runge-Kutta methodwe generate the bifurcation diagrams of the shimmy charac-teristics of the first and second bridge wheels in the dual-front axle system showing that the shimmy characteristicsvary with velocity (see Figure 5) These diagrams are shownin Figure 12

Figures 12(a) to 12(d) show that with an increase in dryfriction torque the shimmy system speed bifurcation range

bifurcation curve amplitude and the peak corresponding tothe limit cycle all decrease

Although increasing the dry friction can reduce theshimmy speed range and the system shimmy amplitudeit also reduces the steering performance and increases thewear between parts Therefore in designing suspensionand steering systems both steering agility and self-excitedshimmy characteristics of the vehicle should be consideredaside from dry friction torque

44 Effect of System Parameters on Shimmy A heavy truckusually works under the conditions of overload and poorroads Under these conditions the dual-front axle steeringsystem is subjected to tens of thousands119873 force and its levercomponents are usually up to a few meters If the stiffnessof the lever is designed unreasonably the lever of a dual-front axle steering system inevitably becomes significantly


60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0

(a) First bridge left wheelM = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

(c) The first bridge side pendulumM = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)


Figure 12 Influence of dry friction on the first and second bridge Hopf bifurcation characteristics

deformed in the process of turning and the deformationresults in the shimmy and wear of tires This phenomenonshould not be ignored The pneumatic trail 119890 and kingpincaster angle 120574 are also important factors that affect shimmyFigure 13 illustrates the influence factor of the first transitionrod rigidity 119896

1199051 second transition rod rigidity 119896

1199052 pneumatic

trail 119890 and kingpin caster angle 120574 of the first and secondbridge wheel shimmy characteristics when V = 30 kmh

Figure 13(a) shows that when the first transition rodstiffness increases from 20000Nm to 100000Nm the firstbridge left wheel swing angle increases from 235∘ to 347∘Δ1205792= 112∘ the second bridge left wheel swing angle decreases

from 1122∘ to 889∘ Δ1205794

= minus233∘ the first bridge side

pendulum angle increases from 015∘ to 019∘ Δ1205792= 004∘

the second bridge side pendulum angle decreases from 054∘

to 042∘ and Δ1205794= minus012 Figure 13(b) shows that when the

second transition rod stiffness increases from 20000Nm to100000Nm the first bridge left wheel swing angle increasesfrom 361∘ to 158∘ Δ120579

2= 203 the second bridge left wheel

swing angle decreases from 1236∘ to 672∘ Δ1205794= minus564∘ the

first bridge side pendulum angle increases from 011∘ to 018∘Δ1205792= 007∘ the second bridge side pendulum angle decreases

from063∘ to 032∘ andΔ1205794=minus031∘Thus with an increase in

transition rod stiffness 1198961199051and transition rod stiffness 119896

1199052 the

first bridge left wheel swing angle and side pendulum angleincrease the second bridge left wheel swing angle and sidependulum angle decrease and the second bridge left wheelswing angle and side pendulum angle significantly differ fromthose of the first bridge left wheel in amplitude In additionthe effects of the second transition rod stiffness on the left


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104

(1) First bridge left wheel (2) Second bridge left wheel

(3) The first bridge side pendulum (4) The second bridge side pendulum

22242628

332343638

3 4 5 6 7 8 9 102First transition rod stiffness Kt1 (Nm)

015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115

3 95 6 7 842 10First transition rod stiffness Kt1 (Nm)

(a) Effect of the first transition rod stiffness 1198961199051 on shimmy characteristics

Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102

Second transition rod stiffness Kt2 (Nm)



Second transition rod stiffness Kt2 (Nm)4 6 8 102 3 4 5 6 7 8 9 102

3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

(b) Effect of the second transition rod stiffness 1198961199052 on shimmy characteristics

Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25

005 01 015 02 0250Pneumatic trail e (m)


Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250

Pneumatic trail e (m)005 01 015 02 0250

Pneumatic trail e (m)

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1

(c) Effect of the pneumatic trail 119890 on shimmy characteristics

Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05

(d) Effect of caster angle on shimmy characteristics

Figure 13 Influence of system parameters


wheel swing angle and side pendulum angle magnitude ofboth bridges are significantly greater than those of the firsttransition rod stiffness

Figure 13(c) shows that when the pneumatic trail 119890

increases from the minimum to 015 the dual-front axleshimmy changes from steady into single-cycle limit cycleoscillation and ultimately to chaos and the vibration ampli-tude becomes increasingly larger Figure 13(d) shows thatwhen the caster angle 120574 is less than 17 by the initialexcitation the system eventually stabilizes however when 120574

increases to more than 17 the system state changes to thelimit cycle oscillation As 120574 increases the limit cycle increases

Thus to ensure that the tire wear in the two bridgesis small and uniform the stiffness of the first transitionand second transition rods should be enhanced Howeverconsidering the production process and production costsdesigning a system with large transition rod stiffness isimpossibleWe can reduce or even eliminate however systemshimmy by choosing a smaller pneumatic trail and kingpincaster angle

5 Conclusions

(1) Based on the current widely used dual-front axlesteering system in heavy trucks we established amechanics model for its dual bridge shimmy systemmechanics and equations for its differential motion

(2) Using Hopf bifurcation theorem and center manifoldtheory we were able to determine the existence andstability of the shimmy system limit cycle Usingnumerical method analysis stable limit cycle char-acteristics at the critical speed point and bifurcationrange were also determined In conclusion the sam-ple vehicle dual-front axle shimmy is a self-excitedvibration generated by Hopf bifurcation

(3) In the dual-front axle shimmy system of heavy trucksthe shimmy intensities of the wheels of the first andsecond bridges are in a state of serious imbalance theshimmy intensity at the second bridge is significantlygreater than that at the first bridge These findingscoincide with the fact that the tire wear of the secondbridge is always more severe than that of the firstbridge in practiceTherefore themechanicsmodel forthe dual-axle shimmy system has high credibility andcan simulate the sample vehicle during actual driving

(4) The results of the methods of qualitative theoryand those of the numerical methods have goodconsistency The bifurcation characteristics of theshimmy system can be predicted using the methodsof qualitative theory and can provide a theoreticalreference for improving the design of a double-frontaxle vehicle

(5) Speed is a bifurcation parameter of the vibrationsystem and dual-front axle steering transition rodstiffness is a sensitive parameter that affects sys-tem shimmy Improving transition rod stiffness andselecting a smaller pneumatic trail and kingpin caster

angle can reduce self-excited shimmy reduce tirewear and improve the driving stability and ridecomfort of a dual-front axle vehicle

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This project is supported by the National Natural ScienceFoundation of China (Grant no 51375130)

References

[1] K Watanabe J Yamakawa M Tanaka and T Sasaki ldquoTurningcharacteristics of multi-axle vehiclesrdquo Journal of Terramechan-ics vol 44 no 1 pp 81ndash87 2007

[2] C Mu J Yu Y Yang and K Wu ldquoDesign for dual-front axlesteering angle of the heavy truckrdquo in Proceedings of the Inter-national Conference on Educational and Network Technology(ICENT rsquo10) pp 185ndash187 June 2010

[3] J Stuart S Cassara B Chan and N Augustyniak ldquoRecentexperimental and simulation efforts to mitigate wobble andshimmy in commercial line haul vehiclesrdquo SAE InternationalJournal of Commercial Vehicles vol 7 no 2 pp 366ndash380 2014

[4] H B Pacejka Analysis of the Shimmy Phenomenon TechnischeHogeschool Delft 1966

[5] G Dihua H Zeming S Jian et al ldquoStudy on the vibration ofvehiclersquos steering wheelsrdquo Automotive Engineering vol 2 pp29ndash40 1984 (Chinese)

[6] S Li and Y Lin ldquoStudy on the bifurcation character of steeringwheel self-excited shimmy of motor vehiclerdquo Vehicle SystemDynamics vol 44 supplement 1 pp 115ndash128 2006

[7] G Jiang Study of the Effect of Dry Friction on Multiple LimitCycles in Shimmy Hefei University of Technology Hefei China2012 (Chinese)

[8] Y Gu Z Fang G Zhang and Y Qi ldquoDesign of Heavymdashdutytruck multi-axle steering systemrdquo Automobile Technology vol1 pp 1ndash5 2009 (Chinese)

[9] Y Hou Y Hu D Hu C Li and Y Hou ldquoSynthesis of multi-axlesteering system of heavy duty vehicle based on probability ofsteering anglerdquo SAE Technical Paper 2000-01-3434 2000

[10] L Wang X Liang and P Ji ldquoReliability-based robust opti-mization on double-front-axle steering mechanism of truckswith clearancesrdquoAutomobile Engineering vol 1 pp 90ndash93 2014(Chinese)

[11] Z Xu Y He H Yin et al ldquoStudy of steering wheel shimmyof LT1080 truck cranerdquo Engineering Machinery vol 3 pp 7ndash111994 (Chinese)

[12] R L Nisonger and D N Wormley ldquoDynamic performance ofautomated guideway transit vehicles with dual-axle steeringrdquoIEEE Transactions on Vehicular Technology vol 28 no 1 pp88ndash94 1979

[13] DHWu and J H Lin ldquoAnalysis of dynamic lateral response fora multi-axle-steering tractor and trailerrdquoHeavy Vehicle Systemsvol 10 no 4 pp 281ndash294 2003

[14] D E Williams ldquoGeneralised multi-axle vehicle handlingrdquoVehicle System Dynamics vol 50 no 1 pp 149ndash166 2012


[15] M Demic ldquoAnalysis of influence of design parameters onsteered wheels shimmy of heavy vehiclesrdquo Vehicle SystemDynamics vol 26 no 5 pp 343ndash379 1996

[16] D J Cole and D Cebon ldquoTruck suspension design to minimizeroad damagerdquo Proceedings of the Institution of MechanicalEngineers Part D vol 210 no 2 pp 95ndash107 1996

[17] Y Chen ldquoDouble axle steering auto tire abnormal wear oflsquointernal researchrsquo and lsquoexternal researchrsquordquo Equipment Manufac-turing Technology vol 11 pp 113ndash116 2011 (Chinese)

[18] D Liao Optimized Design and Simulation Study for Dual-Front Axle Steering System of Heavy Trucks Hunan UniversityChangsha China 2012 (Chinese)

[19] L Li C Zhang ZWang M Liu and J Shi ldquoMotion simulationand optimization design of double-front axle steering systembased on ADAMSrdquo in Proceedings of the IEEE TransportationElectrification Conference and Expo (ITEC rsquo14) pp 1ndash5 IEEEBeijing China September 2014

[20] H Li Steering and Handling Stability Simulation of Multi-Axle Wuhan University of Technology Wuhan China 2013(Chinese)

[21] J Q Shen and Z J Jing ldquoA new detectingmethod for conditionsof existence of Hopf bifurcationrdquoActaMathematicae ApplicataeSinica vol 11 no 1 pp 79ndash93 1995

[22] H Pacejka ldquoA new tyre model with an application in vehicledynamics studiesrdquo SAE Paper 89007 SAE International 1989

[23] HHu and ZWu ldquoStability andHopf bifurcation of four-wheel-steering vehicles involving driverrsquos delayrdquo Nonlinear Dynamicsvol 22 no 4 pp 361ndash374 2000

[24] K Guo D Lu S-K Chen W C Lin and X-P Lu ldquoTheUniTiremodel a nonlinear and non-steady-state tyremodel forvehicle dynamics simulationrdquo Vehicle System Dynamics vol 43supplement 1 pp 341ndash358 2005

[25] G Gim and P E Nikravesh ldquoAn analytical model of pneumatictyres for vehicle dynamic simulations Part 1 pure slipsrdquoInternational Journal of Vehicle Design vol 11 no 6 pp 589ndash618 1990

[26] DWei PWang Z Pan S Hu andH Xiao ldquoStudy of the effectsupon vehicle stability exerted by tie rod end clearance underslalom maneuverrdquo SAE International Journal of Passenger Carvol 7 no 1 pp 395ndash404 2014

[27] P Tao A Study on Control of Nonlinear Stability for Tractor-Semitrailer Jilin University Jilin China 2012 (Chinese)

[28] R S Sharp ldquoA comparison of tyre representation in a simplewheel shimmy problemrdquo Vehicle System Dynamics vol 9 no 1pp 45ndash47 1980

[29] A Y T Leung Z Qichang and C Yushu ldquoNormal formanalysis of hopf bifurcation exemplified by Duffingrsquos equationrdquoShock and Vibration vol 1 no 3 pp 233ndash240 1994

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Hindawi Publishing Corporation httpwwwhindawicom

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Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



with robust design and built a mathematical model forthe robust reliability optimization of dual-front axles withclearances Xu et al [11] developed a steering wheel shimmymodel for a four-axis steering crane without considering thenonlinear factors and their impact on the four-axis steeringcrane shimmy Nisonger and Wormley [12] compared thetransverse dynamic characteristics of single- and dual-frontaxle steering systems using a nonlinear model with threedegrees of freedom Wu and Lin [13] found that the double-front axle can improve a carrsquos yaw stability Williams [14]extended the dual-axle model for vehicles to a multiaxlemodel and analyzed its steering ability and handling stabilityBy analyzing a linear dual-front axle yaw dynamics modelDemic [15] analyzed the influence of structural parameterson the front wheel shimmy of a heavy vehicle and foundthat the vibration of suspension can cause front wheelshimmy Cole and Cebon [16] developed a vibration modelfor the pneumatic suspension of a heavy truck to reducethe vibration in and damage to the road surface caused bytrucks through suspension parameter optimization Chen[17] concluded that two reasonsmdashinternal and externalmdashaccount for abnormal tire wear proposed improvementmeasures based on his practical experience and pointed outdirections for future research Liao [18] and Li et al [19]explored the application of multirigid body theory in thedynamic simulation of a dual-front axle steering system andproposed the condition underwhich tires bear heavy load in adual-front axle vehicle They also built a spatial model for thedual-front axle based onADAMSand conducted a simulationanalysis Using TruckSim a dynamics analysis software forvehicles Li [20] studied how to determine and describe theseven characteristic parameters of a heavy truck steeringsystem namely axle and suspension transmission tire bodybrake system and aerodynamics and laid down a solidfoundation for the study ofmultiaxle steering technology andthe improvement of its handling stability

Most of the aforementioned studies focus on the con-structional parameter analysis and simulation of dual-frontaxle steering system however there is little research onsteering system shimmy especially there is little research onthe mechanism of Hopf bifurcation produce self-shimmyBut it is not yet clear about Hopf bifurcation characteristicof dual-front axle self-shimmy for heavy truck Thereforethe research in this area is very important In this studywe took a JAC heavy truck as an example and establishednine degrees of freedom equations for the dual-front axlesteering systembased onLagrangersquos equationWedeterminedthe existence and the value of Hopf bifurcation with the useof the Hurwitz criterion [21] and Hopf bifurcation theoremTo obtain the manifold for the system in two-dimensionalspace we reduced the dimension of the differential equationsof state using center manifold theory and found the Hopfparadigm in the polar coordinates with the use of normativetheory so that the Hopf bifurcation characteristics of theshimmy system of the dual-front axles can be fully analyzedBased on the above analysis we then propose a methodof controlling such shimmy As such this study establishesthe theoretical foundation for research on the mechanismof the dual-front axle self-excited shimmy and provides

V

1205791

1205792

1205793

1205794

Figure 1 Schematic of dual-front axle steering system

Steeringgear

Firsttransition

rod1205751

r1

g1r2

1205752 g2

r3

1205753 g3

Anteriorlongitudinal

rod

Intermediaterocker

Secondtransition

rod

Firstrocker

Posteriorlongitudinal rod

Secondrocker

sim

Figure 2 Schematic of intermediate steering transmission mecha-nism

a reference on the design improvement of steering systemsThe contribution of this study is the finding that the self-excited shimmy phenomenon of the dual-front axle of aheavy truck is generated by Hopf bifurcation

2 Dynamics Model for a Dual-Front AxleShimmy System

21 Mechanical Model The schematic of the structure ofthe widely used dual-front axle steering system is shown inFigure 1 the diagram of the steering intermediate transmis-sion is shown in Figure 2 To facilitate the development ofthe mathematical model of the dual-front axle self-excitedshimmy system the mechanics model of the dual-front axleself-excited shimmy system is developed based on Figure 2with the dual-front axle steering system of a JAC heavy truckas prototype the schematics of which are shown in Figure 3Figure 3 illustrates the double-front axle steering principlethe steering force exerted by the driver is passed throughthe steering operating mechanism which is composed of thesteering shaft transmission shaft and steering joints towardsthe steering gear (2)The torque is then transmitted to the firstrocker (3) after its torque increases and its speed decreasesThe first rocker (3) then drives the steering knuckle arm (5)

of the first steering bridge through the anterior longitudinalrod (4) to turn the left wheel to rotate around the kingpinwhich is installed in the first steering bridge At the sametime the right wheel driven by the torque which passes


1

2

34

5

67

1314 8

15

910

11

12

kc

kz1cz1

kt1ct1 kh1

ch1

1205931

cz2kz2

kh2

ch2

kt2

ct2

1205932

















2is the




119889

119889119905

(

120597119879

120597119896

) minus

120597119879

120597119902119896

+

120597119880

120597119902119896

+

120597119863

120597119896

= 119876119896

(119896 = 1 2 3 9)

(1)








119879 =

1

2

1198681(1205791

2

+1205792

2

) +

1

2

11986911

2+

1

2

1198682(1205793

2

+1205794

2

) +

1

2

11986922

2

+

1

2

1198681198881

1205751

2

+

1

2

1198681198882

1205752

2

+

1

2

1198681198883

1205753

2

(2)







119880 =

1

2

119896ℎ1(1205791minus 1205792)2

+

1

2

1198961198991205931

2+

1

2

119896ℎ2(1205793minus 1205794)2

+

1

2

1198961198991205932

2+

1

2

1198961199051205751

2+

1

2

1198961199051(11989211205751minus 11990321205752)2

+

1

2

1198961199052(11989221205752minus 11989231205753)2

+

1

2

1198961199111(11990311205751+ 11988611205792)2

+

1

2

1198961199112(11990331205753+ 11988621205794)2

(3)








2is








119863 =

1

2

1198881198971(1205791+

1205792) +

1

2

119888ℎ1(1205791minus

1205792)

2

+

1

2

1198881198991

2

+

1

2

119888ℎ2(1205793minus

1205794)

2

+

1

2

1198881198972(1205793+

1205794) +

1

2

1198881198992

2

+

1

2

1198881199051205751

2

+

1

2

1198881199051(11989211205751minus 11990321205752)

2

+

1

2

1198881199052(11989221205752minus 11989231205753)

2

+

1

2

1198881199111(11990311205751+ 11988611205792)

2

+

1

2

1198881199112(11990331205753+ 11988621205794)

2

+11987222

1205794+11987221

1205793+11987212

1205792

+11987211

1205791

(4)










12is the






1198761= minus1198941198961

V119877

1+ 1198651(119877120574 + 119890)

minus (120574120588ℎ1119877 +

119896119888

2

11986111198971(minus119891 + 120574))120593

1

1198762= minus1198941198961

V119877

1+ 1198652(119877120574 + 119890)

minus (120574120588ℎ1119877 +

119896119888

2

11986111198971(minus119891 + 120574))120593

1

1198763= minus2120588119877ℎ

11205931+ 1198941198961

V119877

(1205791+

1205792) minus (119865

1+ 1198652) 119877

1198764= minus1198941198962

V119877

2+ 1198653(119877120574 + 119890)

minus (120574120588ℎ2119877 +

119896119888

2

11986121198972(minus119891 + 120574))120593

2

1198765= minus1198941198962

V119877

2+ 1198654(119877120574 + 119890)

minus (120574120588ℎ2119877 +

119896119888

2

11986121198972(minus119891 + 120574))120593

2

1198766= minus2120588119877ℎ

21205932+ 1198941198962

V119877

(1205793+

1205794) minus (119865

3+ 1198654) 119877

1198767= 0

1198768= 0

1198769= 0

(5)








119888







11986811205791+ 1198881198971

1205791+ 119896ℎ1(1205791minus 1205792) + 119888ℎ1(1205791minus

1205792) +

1198941198961V

119877

1

+ (120588119877ℎ1120574 +

1

2

11989611988811989711198611(120574 minus 119891)) 120593

1minus 1198651(119877120574 + 119890)

+11987211= 0

(6)


11986811205792+ 1198881198971

1205792+ 119896ℎ1(1205792minus 1205791) + 119888ℎ1(1205792minus

1205791)

+ 11989611991111198861(11990311205751+ 11988611205792) + 11988811991111198861(11990311205751+ 11988611205792)

+

1198941198961V

119877

1+ (120588119877ℎ

1120574 +

1

2

11989611988811989711198611(120574 minus 119891)) 120593

1

minus 1198652(119877120574 + 119890) +119872

12= 0

(7)


11986911+ 11988811989911+ (

1

2

1198961198881198611

2+ 119896119899+ 2120588119877ℎ

1)1205931

minus

1198941198961V

119877

(1205791+

1205792) + (119865

1+ 1198652) ℎ1= 0

(8)


11986821205793+ 1198881198972

1205793+ 119896ℎ2(1205793minus 1205794) + 119888ℎ2(1205793minus

1205794) +

1198941198962V

119877

2

+ (120588119877ℎ2120574 +

1

2

11989611988811989721198612(120574 minus 119891)) 120593

2minus 1198653(119877120574 + 119890)

+11987221= 0

(9)


11986821205794+ 1198881198972

1205792+ 119896ℎ2(1205794minus 1205793) + 119888ℎ2(1205794minus

1205793)

+ 11989611991121198862(11990331205753+ 11988621205794) + 11988811991121198862(11990331205753+ 11988621205794)

+

1198941198962V

119877

2+ (120588119877ℎ

2120574 +

1

2

11989611988811989721198612(120574 minus 119891)) 120593

2

minus 1198654(119877120574 + 119890) +119872

22= 0

(10)


11986922+ 11988811989922+ (

1

2

1198961198881198612

2+ 119896119899+ 2120588119877ℎ

2)1205932

minus

1198941198962V

119877

(1205793+

1205794) + (119865

3+ 1198654) ℎ2= 0

(11)




minus4000

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

Tire

forc

eF(N

)


119894


1198681198881

1205751+ 1198881199051205751+ 1198961199051205751+ 11988811990511198921(11989211205751minus 11990321205752)

+ 11989611990511198921(11989211205751minus 11990321205752) + 11988811991111199031(11990311205751+ 11988611205792)

+ 11989611991111199031(11990311205751+ 11988611205792) = 0

(12)


1198681198882

1205752+ 11988811990511199032(11990321205752minus 11989211205751) + 11989611990511199032(11990321205752minus 11989211205751)

+ 11988811990521198922(11989221205752minus 11989231205753) + 11989611990521198922(11989221205752minus 11989231205753)

= 0

(13)


1198681198883

1205753+ 11988811990521198923(11989231205753minus 11989221205752) + 11989611990521198923(11989231205753minus 11989221205752)

+ 11988811991121199033(11990331205753+ 11988621205794) + 11989611991121199033(11990331205753+ 11988621205794) = 0

(14)




1198621119891

1198623119891

1198861

1198862

1198863

1198864

1198865

1198866

1198867

1198868

32740 481770 minus221 1011 1078 182 0208 0 minus0354 0707


119894 119863119894 and 119864




119865119894= minus (119862

1119891120572119894minus 1198623119891120572119894

3) (119894 = 1 2 3 4) (15)

119865119894= 119863119894sin (119862

119894


119894119886119894))))

(16)

In this model

119861119894=

1198863sdot sin (119886


5sdot 119865119885119894)))

119862119894sdot 119863119894

119862119894= 13

119863119894= 1198861sdot 119865119885119894

2+ 1198862sdot 119865119885119894

119864119894= 1198866sdot 119865119885119894

2+ 1198867sdot 119865119885119894+ 1198868

(119894 = 1 2 3 4)

(17)

where 1198621119891

and 1198623119891



119894 119862119894 119863119894 and 119864



8



119885119894 1198621119891 and 119862

3119891 The


119894and the



119894+

V120590

120572119894+

V120590

120579119894minus

119890119894

120590

120579119894 119894 = 1 2 3 4 (18)




119872119894119895= 119872 sgn (

120579119894119895) (119894 119895 = 1 2)

|119872| =

120583119904119873119903 (

120579119894119895= 0)

120583119873119903 (120579119894119895

= 0)

(119894 119895 = 1 2)

(19)






Make 1199091= 1205791 1199092=

1205791 1199093= 1205792 1199094=

1205792 1199095= 1205931 1199096= 1

1199097= 1205793 1199098=

1205793 1199099= 1205794 11990910

=1205794 11990911

= 1205932 11990912

= 2

11990913

= 1205751 11990914

=1205751 11990915

= 1205752 11990916

=1205752 11990917

= 1205753 11990918

=1205753

11990919

= 1205721 11990920

= 1205722 11990921

= 1205723 and 119909

22= 1205724 Assume 119883 =

(1199091 1199092 1199093 1199094 1199095 1199096 1199097 1199098 1199099 11990910 11990911 11990912 11990913 11990914 11990915 11990916




= 119860119883 + 119865 (119883) 119883 isin 11987722 (20)



1or when V = V

2 whereas



1= 156 kmh and


















1205741 1205742∘ 25

1198971 1198972m 007

1199031m 034

1199033m 047

1198922m 026

1198861m 021

1198961199111 1198961199112Nsdotradminus1 36000


1198961199051 1198961199052Nsdotmminus1 37200

1198881199051 1198881199052Nsdots 15

119896119888Nsdotradminus1 1060000

119891 0015119896119905Nsdotradminus1 31000



ℎ1 ℎ2m 045

1198611 1198612m 206

1199032m 042

1198921m 017

1198923m 022

1198862m 021


1= 156 kmh and V

2= 408 kmh are


1le V le V



1




119896= VminusV

119896(119896 = 1 2) 120583






1205821

0 + 1211119894 1205821

0 + 1456119894

1205822

0 minus 1211119894 1205822

0 minus 1456119894

1205823

minus16544 + 78083119894 1205823

minus16544 + 78083119894

1205824

minus16544 minus 78083119894 1205824

minus16544 minus 78083119894

1205825

minus7140 + 54908119894 1205825

minus7140 + 54908119894

1205826

minus7140 minus 54908119894 1205826

minus7140 minus 54908119894

1205827

minus437 + 6315119894 1205827

minus425 + 6316119894

1205828

minus437 minus 6315119894 1205828


1205829

minus434 + 6178119894 1205829

minus422 + 6179119894

12058210

minus434 minus 6178119894 12058210


12058211

minus218 + 2688119894 12058211

minus162 + 2773119894

12058212

minus218 minus 2688119894 12058212


12058213

minus224 + 2681119894 12058213

minus179 + 2746119894

12058214

minus224 minus 2681119894 12058214


12058215

minus122 + 1621119894 12058215

minus098 + 1765119894

12058216

minus122 minus 1621119894 12058216


12058217

minus24378 + 91155119894 12058217

minus24378 + 91155119894

12058218

minus24378 minus 91155119894 12058218

minus24378 minus 91155119894

12058219

minus892 12058219

minus1514

12058220

minus664 12058220

minus1358

12058221

minus355 12058221

minus909

12058222

minus354 12058222

minus910



is converted into

= Λ119884 + 119875minus1119892 (119884 120583

119896) (21)





119910119894= ℎ119894(1199101 1199102 120583119896)

= ℎ11989411199101

2+ ℎ11989421205831198961199101+ ℎ11989431205831198961199102+ ℎ11989441199102

2

119894 = 3 4 21 22

(22)

119894=

120597ℎ119894(1199101 1199102 120583119896)

1205971199101

1+

120597ℎ119894(1199101 1199102 120583119896)

1205971199102

2

+

120597ℎ119894(1199101 1199102 120583119896)

120597120583119896

120583119896

(23)







1= 121 times 10119910

2+ 0762119910

21205831minus 0396119910

11205831minus 0169

times 10minus311991021205831


minus41199102

3minus 0118

times 10minus311991011205831


minus311991011199102

2minus 066


21199102minus 0261 times 10

minus11199101

3+ 0426


211991021205831+ 0464 times 10

minus41199101

31205831

2= minus121 times 10119910

1minus 0148119910

11205831+ 0853119910

21205831+ 0998

times 10minus411991011205831

2+ 0113 times 10

minus11199101

3+ 0429

times 10minus411991021205831

2+ 0286 times 10

minus21199101

21199102+ 0242

times 10minus311991011199102


minus41199101

31205831minus 0187


211991021205831

(24)


1= 146 times 10119910


minus21199101+ 0522119910

21205832


minus21199102

3+ 0118

times 10minus411991011205832


minus11199102

21199101

minus 010111991021199101


minus11199101

3minus 0259

times 10minus411991021199101

21205832minus 0231 times 10

minus41199101

31205832

2= minus146 times 10119910


minus21199102+ 0275

times 10minus111991011205832+ 0261119910

21205832+ 0381

times 10minus411991011205832

2+ 0391 times 10

minus11199101

3minus 0531

times 10minus411991021205832

2+ 0565 times 10

minus111991021199101

2+ 0273


21199101+ 0439 times 10

minus21199102

3+ 0103


31205832+ 0116 times 10

minus411991021199101

21205832

(25)


119903 = dur + 1198861199033+ hot

120579 = 120596 + 119887119903

2+ hot

(26)


F

120583sN120583N

r



For V = V1= 156 kmh


1119903 + (minus0193

times 10minus6

1205831

2+ 0236 times 10


minus1) 1199033

+ hot

(27)

For V = V2= 408 kmh


1205832minus 0145) 120583

2119903 + (minus0495

times 10minus6

1205832


minus4

1205832minus 0476 times 10

minus1

) 1199033

+ hot

(28)





2 When V lt V

1= 156 kmh



When V1lt V lt V

2(ie 120583

1gt 0 and 120583




2= 408 kmh








minus08

minus06

minus04

minus02

0

02

04

06

08H

opf b

ifurc

atio

n br

anch

r


minus04

minus03

minus02

minus01

0

01

02

03

04



Hop

f bifu

rcat

ion

bran

chr



20 40 60 80 100 120 1400Velocity V (kmh)

0

05

1

15

2

25

3

35

4

Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12Se

cond

brid

ge le

ft w

heel

swin

g an

gle1205794

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)

05

06











minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(a) V = 20 kmh




minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(b) V = 25 kmh


minus150

minus100

minus50

0

50

100

150



Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(c) V = 30 kmh






Vkmsdothminus1 20 25 301205792∘ 373 381 328

1205794∘ 1055 1092 947






minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05


(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh











minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

2

34

5

67

1314 8

15

910

11

12

kc

kz1cz1

kt1ct1 kh1

ch1

1205931

cz2kz2

kh2

ch2

kt2

ct2

1205932

















2is the




119889

119889119905

(

120597119879

120597119896

) minus

120597119879

120597119902119896

+

120597119880

120597119902119896

+

120597119863

120597119896

= 119876119896

(119896 = 1 2 3 9)

(1)








119879 =

1

2

1198681(1205791

2

+1205792

2

) +

1

2

11986911

2+

1

2

1198682(1205793

2

+1205794

2

) +

1

2

11986922

2

+

1

2

1198681198881

1205751

2

+

1

2

1198681198882

1205752

2

+

1

2

1198681198883

1205753

2

(2)







119880 =

1

2

119896ℎ1(1205791minus 1205792)2

+

1

2

1198961198991205931

2+

1

2

119896ℎ2(1205793minus 1205794)2

+

1

2

1198961198991205932

2+

1

2

1198961199051205751

2+

1

2

1198961199051(11989211205751minus 11990321205752)2

+

1

2

1198961199052(11989221205752minus 11989231205753)2

+

1

2

1198961199111(11990311205751+ 11988611205792)2

+

1

2

1198961199112(11990331205753+ 11988621205794)2

(3)








2is








119863 =

1

2

1198881198971(1205791+

1205792) +

1

2

119888ℎ1(1205791minus

1205792)

2

+

1

2

1198881198991

2

+

1

2

119888ℎ2(1205793minus

1205794)

2

+

1

2

1198881198972(1205793+

1205794) +

1

2

1198881198992

2

+

1

2

1198881199051205751

2

+

1

2

1198881199051(11989211205751minus 11990321205752)

2

+

1

2

1198881199052(11989221205752minus 11989231205753)

2

+

1

2

1198881199111(11990311205751+ 11988611205792)

2

+

1

2

1198881199112(11990331205753+ 11988621205794)

2

+11987222

1205794+11987221

1205793+11987212

1205792

+11987211

1205791

(4)










12is the






1198761= minus1198941198961

V119877

1+ 1198651(119877120574 + 119890)

minus (120574120588ℎ1119877 +

119896119888

2

11986111198971(minus119891 + 120574))120593

1

1198762= minus1198941198961

V119877

1+ 1198652(119877120574 + 119890)

minus (120574120588ℎ1119877 +

119896119888

2

11986111198971(minus119891 + 120574))120593

1

1198763= minus2120588119877ℎ

11205931+ 1198941198961

V119877

(1205791+

1205792) minus (119865

1+ 1198652) 119877

1198764= minus1198941198962

V119877

2+ 1198653(119877120574 + 119890)

minus (120574120588ℎ2119877 +

119896119888

2

11986121198972(minus119891 + 120574))120593

2

1198765= minus1198941198962

V119877

2+ 1198654(119877120574 + 119890)

minus (120574120588ℎ2119877 +

119896119888

2

11986121198972(minus119891 + 120574))120593

2

1198766= minus2120588119877ℎ

21205932+ 1198941198962

V119877

(1205793+

1205794) minus (119865

3+ 1198654) 119877

1198767= 0

1198768= 0

1198769= 0

(5)








119888







11986811205791+ 1198881198971

1205791+ 119896ℎ1(1205791minus 1205792) + 119888ℎ1(1205791minus

1205792) +

1198941198961V

119877

1

+ (120588119877ℎ1120574 +

1

2

11989611988811989711198611(120574 minus 119891)) 120593

1minus 1198651(119877120574 + 119890)

+11987211= 0

(6)


11986811205792+ 1198881198971

1205792+ 119896ℎ1(1205792minus 1205791) + 119888ℎ1(1205792minus

1205791)

+ 11989611991111198861(11990311205751+ 11988611205792) + 11988811991111198861(11990311205751+ 11988611205792)

+

1198941198961V

119877

1+ (120588119877ℎ

1120574 +

1

2

11989611988811989711198611(120574 minus 119891)) 120593

1

minus 1198652(119877120574 + 119890) +119872

12= 0

(7)


11986911+ 11988811989911+ (

1

2

1198961198881198611

2+ 119896119899+ 2120588119877ℎ

1)1205931

minus

1198941198961V

119877

(1205791+

1205792) + (119865

1+ 1198652) ℎ1= 0

(8)


11986821205793+ 1198881198972

1205793+ 119896ℎ2(1205793minus 1205794) + 119888ℎ2(1205793minus

1205794) +

1198941198962V

119877

2

+ (120588119877ℎ2120574 +

1

2

11989611988811989721198612(120574 minus 119891)) 120593

2minus 1198653(119877120574 + 119890)

+11987221= 0

(9)


11986821205794+ 1198881198972

1205792+ 119896ℎ2(1205794minus 1205793) + 119888ℎ2(1205794minus

1205793)

+ 11989611991121198862(11990331205753+ 11988621205794) + 11988811991121198862(11990331205753+ 11988621205794)

+

1198941198962V

119877

2+ (120588119877ℎ

2120574 +

1

2

11989611988811989721198612(120574 minus 119891)) 120593

2

minus 1198654(119877120574 + 119890) +119872

22= 0

(10)


11986922+ 11988811989922+ (

1

2

1198961198881198612

2+ 119896119899+ 2120588119877ℎ

2)1205932

minus

1198941198962V

119877

(1205793+

1205794) + (119865

3+ 1198654) ℎ2= 0

(11)




minus4000

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

Tire

forc

eF(N

)


119894


1198681198881

1205751+ 1198881199051205751+ 1198961199051205751+ 11988811990511198921(11989211205751minus 11990321205752)

+ 11989611990511198921(11989211205751minus 11990321205752) + 11988811991111199031(11990311205751+ 11988611205792)

+ 11989611991111199031(11990311205751+ 11988611205792) = 0

(12)


1198681198882

1205752+ 11988811990511199032(11990321205752minus 11989211205751) + 11989611990511199032(11990321205752minus 11989211205751)

+ 11988811990521198922(11989221205752minus 11989231205753) + 11989611990521198922(11989221205752minus 11989231205753)

= 0

(13)


1198681198883

1205753+ 11988811990521198923(11989231205753minus 11989221205752) + 11989611990521198923(11989231205753minus 11989221205752)

+ 11988811991121199033(11990331205753+ 11988621205794) + 11989611991121199033(11990331205753+ 11988621205794) = 0

(14)




1198621119891

1198623119891

1198861

1198862

1198863

1198864

1198865

1198866

1198867

1198868

32740 481770 minus221 1011 1078 182 0208 0 minus0354 0707


119894 119863119894 and 119864




119865119894= minus (119862

1119891120572119894minus 1198623119891120572119894

3) (119894 = 1 2 3 4) (15)

119865119894= 119863119894sin (119862

119894


119894119886119894))))

(16)

In this model

119861119894=

1198863sdot sin (119886


5sdot 119865119885119894)))

119862119894sdot 119863119894

119862119894= 13

119863119894= 1198861sdot 119865119885119894

2+ 1198862sdot 119865119885119894

119864119894= 1198866sdot 119865119885119894

2+ 1198867sdot 119865119885119894+ 1198868

(119894 = 1 2 3 4)

(17)

where 1198621119891

and 1198623119891



119894 119862119894 119863119894 and 119864



8



119885119894 1198621119891 and 119862

3119891 The


119894and the



119894+

V120590

120572119894+

V120590

120579119894minus

119890119894

120590

120579119894 119894 = 1 2 3 4 (18)




119872119894119895= 119872 sgn (

120579119894119895) (119894 119895 = 1 2)

|119872| =

120583119904119873119903 (

120579119894119895= 0)

120583119873119903 (120579119894119895

= 0)

(119894 119895 = 1 2)

(19)






Make 1199091= 1205791 1199092=

1205791 1199093= 1205792 1199094=

1205792 1199095= 1205931 1199096= 1

1199097= 1205793 1199098=

1205793 1199099= 1205794 11990910

=1205794 11990911

= 1205932 11990912

= 2

11990913

= 1205751 11990914

=1205751 11990915

= 1205752 11990916

=1205752 11990917

= 1205753 11990918

=1205753

11990919

= 1205721 11990920

= 1205722 11990921

= 1205723 and 119909

22= 1205724 Assume 119883 =

(1199091 1199092 1199093 1199094 1199095 1199096 1199097 1199098 1199099 11990910 11990911 11990912 11990913 11990914 11990915 11990916




= 119860119883 + 119865 (119883) 119883 isin 11987722 (20)



1or when V = V

2 whereas



1= 156 kmh and


















1205741 1205742∘ 25

1198971 1198972m 007

1199031m 034

1199033m 047

1198922m 026

1198861m 021

1198961199111 1198961199112Nsdotradminus1 36000


1198961199051 1198961199052Nsdotmminus1 37200

1198881199051 1198881199052Nsdots 15

119896119888Nsdotradminus1 1060000

119891 0015119896119905Nsdotradminus1 31000



ℎ1 ℎ2m 045

1198611 1198612m 206

1199032m 042

1198921m 017

1198923m 022

1198862m 021


1= 156 kmh and V

2= 408 kmh are


1le V le V



1




119896= VminusV

119896(119896 = 1 2) 120583






1205821

0 + 1211119894 1205821

0 + 1456119894

1205822

0 minus 1211119894 1205822

0 minus 1456119894

1205823

minus16544 + 78083119894 1205823

minus16544 + 78083119894

1205824

minus16544 minus 78083119894 1205824

minus16544 minus 78083119894

1205825

minus7140 + 54908119894 1205825

minus7140 + 54908119894

1205826

minus7140 minus 54908119894 1205826

minus7140 minus 54908119894

1205827

minus437 + 6315119894 1205827

minus425 + 6316119894

1205828

minus437 minus 6315119894 1205828


1205829

minus434 + 6178119894 1205829

minus422 + 6179119894

12058210

minus434 minus 6178119894 12058210


12058211

minus218 + 2688119894 12058211

minus162 + 2773119894

12058212

minus218 minus 2688119894 12058212


12058213

minus224 + 2681119894 12058213

minus179 + 2746119894

12058214

minus224 minus 2681119894 12058214


12058215

minus122 + 1621119894 12058215

minus098 + 1765119894

12058216

minus122 minus 1621119894 12058216


12058217

minus24378 + 91155119894 12058217

minus24378 + 91155119894

12058218

minus24378 minus 91155119894 12058218

minus24378 minus 91155119894

12058219

minus892 12058219

minus1514

12058220

minus664 12058220

minus1358

12058221

minus355 12058221

minus909

12058222

minus354 12058222

minus910



is converted into

= Λ119884 + 119875minus1119892 (119884 120583

119896) (21)





119910119894= ℎ119894(1199101 1199102 120583119896)

= ℎ11989411199101

2+ ℎ11989421205831198961199101+ ℎ11989431205831198961199102+ ℎ11989441199102

2

119894 = 3 4 21 22

(22)

119894=

120597ℎ119894(1199101 1199102 120583119896)

1205971199101

1+

120597ℎ119894(1199101 1199102 120583119896)

1205971199102

2

+

120597ℎ119894(1199101 1199102 120583119896)

120597120583119896

120583119896

(23)







1= 121 times 10119910

2+ 0762119910

21205831minus 0396119910

11205831minus 0169

times 10minus311991021205831


minus41199102

3minus 0118

times 10minus311991011205831


minus311991011199102

2minus 066


21199102minus 0261 times 10

minus11199101

3+ 0426


211991021205831+ 0464 times 10

minus41199101

31205831

2= minus121 times 10119910

1minus 0148119910

11205831+ 0853119910

21205831+ 0998

times 10minus411991011205831

2+ 0113 times 10

minus11199101

3+ 0429

times 10minus411991021205831

2+ 0286 times 10

minus21199101

21199102+ 0242

times 10minus311991011199102


minus41199101

31205831minus 0187


211991021205831

(24)


1= 146 times 10119910


minus21199101+ 0522119910

21205832


minus21199102

3+ 0118

times 10minus411991011205832


minus11199102

21199101

minus 010111991021199101


minus11199101

3minus 0259

times 10minus411991021199101

21205832minus 0231 times 10

minus41199101

31205832

2= minus146 times 10119910


minus21199102+ 0275

times 10minus111991011205832+ 0261119910

21205832+ 0381

times 10minus411991011205832

2+ 0391 times 10

minus11199101

3minus 0531

times 10minus411991021205832

2+ 0565 times 10

minus111991021199101

2+ 0273


21199101+ 0439 times 10

minus21199102

3+ 0103


31205832+ 0116 times 10

minus411991021199101

21205832

(25)


119903 = dur + 1198861199033+ hot

120579 = 120596 + 119887119903

2+ hot

(26)


F

120583sN120583N

r



For V = V1= 156 kmh


1119903 + (minus0193

times 10minus6

1205831

2+ 0236 times 10


minus1) 1199033

+ hot

(27)

For V = V2= 408 kmh


1205832minus 0145) 120583

2119903 + (minus0495

times 10minus6

1205832


minus4

1205832minus 0476 times 10

minus1

) 1199033

+ hot

(28)





2 When V lt V

1= 156 kmh



When V1lt V lt V

2(ie 120583

1gt 0 and 120583




2= 408 kmh








minus08

minus06

minus04

minus02

0

02

04

06

08H

opf b

ifurc

atio

n br

anch

r


minus04

minus03

minus02

minus01

0

01

02

03

04



Hop

f bifu

rcat

ion

bran

chr



20 40 60 80 100 120 1400Velocity V (kmh)

0

05

1

15

2

25

3

35

4

Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12Se

cond

brid

ge le

ft w

heel

swin

g an

gle1205794

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)

05

06











minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(a) V = 20 kmh




minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(b) V = 25 kmh


minus150

minus100

minus50

0

50

100

150



Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(c) V = 30 kmh






Vkmsdothminus1 20 25 301205792∘ 373 381 328

1205794∘ 1055 1092 947






minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05


(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh











minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119879 =

1

2

1198681(1205791

2

+1205792

2

) +

1

2

11986911

2+

1

2

1198682(1205793

2

+1205794

2

) +

1

2

11986922

2

+

1

2

1198681198881

1205751

2

+

1

2

1198681198882

1205752

2

+

1

2

1198681198883

1205753

2

(2)







119880 =

1

2

119896ℎ1(1205791minus 1205792)2

+

1

2

1198961198991205931

2+

1

2

119896ℎ2(1205793minus 1205794)2

+

1

2

1198961198991205932

2+

1

2

1198961199051205751

2+

1

2

1198961199051(11989211205751minus 11990321205752)2

+

1

2

1198961199052(11989221205752minus 11989231205753)2

+

1

2

1198961199111(11990311205751+ 11988611205792)2

+

1

2

1198961199112(11990331205753+ 11988621205794)2

(3)








2is








119863 =

1

2

1198881198971(1205791+

1205792) +

1

2

119888ℎ1(1205791minus

1205792)

2

+

1

2

1198881198991

2

+

1

2

119888ℎ2(1205793minus

1205794)

2

+

1

2

1198881198972(1205793+

1205794) +

1

2

1198881198992

2

+

1

2

1198881199051205751

2

+

1

2

1198881199051(11989211205751minus 11990321205752)

2

+

1

2

1198881199052(11989221205752minus 11989231205753)

2

+

1

2

1198881199111(11990311205751+ 11988611205792)

2

+

1

2

1198881199112(11990331205753+ 11988621205794)

2

+11987222

1205794+11987221

1205793+11987212

1205792

+11987211

1205791

(4)










12is the






1198761= minus1198941198961

V119877

1+ 1198651(119877120574 + 119890)

minus (120574120588ℎ1119877 +

119896119888

2

11986111198971(minus119891 + 120574))120593

1

1198762= minus1198941198961

V119877

1+ 1198652(119877120574 + 119890)

minus (120574120588ℎ1119877 +

119896119888

2

11986111198971(minus119891 + 120574))120593

1

1198763= minus2120588119877ℎ

11205931+ 1198941198961

V119877

(1205791+

1205792) minus (119865

1+ 1198652) 119877

1198764= minus1198941198962

V119877

2+ 1198653(119877120574 + 119890)

minus (120574120588ℎ2119877 +

119896119888

2

11986121198972(minus119891 + 120574))120593

2

1198765= minus1198941198962

V119877

2+ 1198654(119877120574 + 119890)

minus (120574120588ℎ2119877 +

119896119888

2

11986121198972(minus119891 + 120574))120593

2

1198766= minus2120588119877ℎ

21205932+ 1198941198962

V119877

(1205793+

1205794) minus (119865

3+ 1198654) 119877

1198767= 0

1198768= 0

1198769= 0

(5)








119888







11986811205791+ 1198881198971

1205791+ 119896ℎ1(1205791minus 1205792) + 119888ℎ1(1205791minus

1205792) +

1198941198961V

119877

1

+ (120588119877ℎ1120574 +

1

2

11989611988811989711198611(120574 minus 119891)) 120593

1minus 1198651(119877120574 + 119890)

+11987211= 0

(6)


11986811205792+ 1198881198971

1205792+ 119896ℎ1(1205792minus 1205791) + 119888ℎ1(1205792minus

1205791)

+ 11989611991111198861(11990311205751+ 11988611205792) + 11988811991111198861(11990311205751+ 11988611205792)

+

1198941198961V

119877

1+ (120588119877ℎ

1120574 +

1

2

11989611988811989711198611(120574 minus 119891)) 120593

1

minus 1198652(119877120574 + 119890) +119872

12= 0

(7)


11986911+ 11988811989911+ (

1

2

1198961198881198611

2+ 119896119899+ 2120588119877ℎ

1)1205931

minus

1198941198961V

119877

(1205791+

1205792) + (119865

1+ 1198652) ℎ1= 0

(8)


11986821205793+ 1198881198972

1205793+ 119896ℎ2(1205793minus 1205794) + 119888ℎ2(1205793minus

1205794) +

1198941198962V

119877

2

+ (120588119877ℎ2120574 +

1

2

11989611988811989721198612(120574 minus 119891)) 120593

2minus 1198653(119877120574 + 119890)

+11987221= 0

(9)


11986821205794+ 1198881198972

1205792+ 119896ℎ2(1205794minus 1205793) + 119888ℎ2(1205794minus

1205793)

+ 11989611991121198862(11990331205753+ 11988621205794) + 11988811991121198862(11990331205753+ 11988621205794)

+

1198941198962V

119877

2+ (120588119877ℎ

2120574 +

1

2

11989611988811989721198612(120574 minus 119891)) 120593

2

minus 1198654(119877120574 + 119890) +119872

22= 0

(10)


11986922+ 11988811989922+ (

1

2

1198961198881198612

2+ 119896119899+ 2120588119877ℎ

2)1205932

minus

1198941198962V

119877

(1205793+

1205794) + (119865

3+ 1198654) ℎ2= 0

(11)




minus4000

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

Tire

forc

eF(N

)


119894


1198681198881

1205751+ 1198881199051205751+ 1198961199051205751+ 11988811990511198921(11989211205751minus 11990321205752)

+ 11989611990511198921(11989211205751minus 11990321205752) + 11988811991111199031(11990311205751+ 11988611205792)

+ 11989611991111199031(11990311205751+ 11988611205792) = 0

(12)


1198681198882

1205752+ 11988811990511199032(11990321205752minus 11989211205751) + 11989611990511199032(11990321205752minus 11989211205751)

+ 11988811990521198922(11989221205752minus 11989231205753) + 11989611990521198922(11989221205752minus 11989231205753)

= 0

(13)


1198681198883

1205753+ 11988811990521198923(11989231205753minus 11989221205752) + 11989611990521198923(11989231205753minus 11989221205752)

+ 11988811991121199033(11990331205753+ 11988621205794) + 11989611991121199033(11990331205753+ 11988621205794) = 0

(14)




1198621119891

1198623119891

1198861

1198862

1198863

1198864

1198865

1198866

1198867

1198868

32740 481770 minus221 1011 1078 182 0208 0 minus0354 0707


119894 119863119894 and 119864




119865119894= minus (119862

1119891120572119894minus 1198623119891120572119894

3) (119894 = 1 2 3 4) (15)

119865119894= 119863119894sin (119862

119894


119894119886119894))))

(16)

In this model

119861119894=

1198863sdot sin (119886


5sdot 119865119885119894)))

119862119894sdot 119863119894

119862119894= 13

119863119894= 1198861sdot 119865119885119894

2+ 1198862sdot 119865119885119894

119864119894= 1198866sdot 119865119885119894

2+ 1198867sdot 119865119885119894+ 1198868

(119894 = 1 2 3 4)

(17)

where 1198621119891

and 1198623119891



119894 119862119894 119863119894 and 119864



8



119885119894 1198621119891 and 119862

3119891 The


119894and the



119894+

V120590

120572119894+

V120590

120579119894minus

119890119894

120590

120579119894 119894 = 1 2 3 4 (18)




119872119894119895= 119872 sgn (

120579119894119895) (119894 119895 = 1 2)

|119872| =

120583119904119873119903 (

120579119894119895= 0)

120583119873119903 (120579119894119895

= 0)

(119894 119895 = 1 2)

(19)






Make 1199091= 1205791 1199092=

1205791 1199093= 1205792 1199094=

1205792 1199095= 1205931 1199096= 1

1199097= 1205793 1199098=

1205793 1199099= 1205794 11990910

=1205794 11990911

= 1205932 11990912

= 2

11990913

= 1205751 11990914

=1205751 11990915

= 1205752 11990916

=1205752 11990917

= 1205753 11990918

=1205753

11990919

= 1205721 11990920

= 1205722 11990921

= 1205723 and 119909

22= 1205724 Assume 119883 =

(1199091 1199092 1199093 1199094 1199095 1199096 1199097 1199098 1199099 11990910 11990911 11990912 11990913 11990914 11990915 11990916




= 119860119883 + 119865 (119883) 119883 isin 11987722 (20)



1or when V = V

2 whereas



1= 156 kmh and


















1205741 1205742∘ 25

1198971 1198972m 007

1199031m 034

1199033m 047

1198922m 026

1198861m 021

1198961199111 1198961199112Nsdotradminus1 36000


1198961199051 1198961199052Nsdotmminus1 37200

1198881199051 1198881199052Nsdots 15

119896119888Nsdotradminus1 1060000

119891 0015119896119905Nsdotradminus1 31000



ℎ1 ℎ2m 045

1198611 1198612m 206

1199032m 042

1198921m 017

1198923m 022

1198862m 021


1= 156 kmh and V

2= 408 kmh are


1le V le V



1




119896= VminusV

119896(119896 = 1 2) 120583






1205821

0 + 1211119894 1205821

0 + 1456119894

1205822

0 minus 1211119894 1205822

0 minus 1456119894

1205823

minus16544 + 78083119894 1205823

minus16544 + 78083119894

1205824

minus16544 minus 78083119894 1205824

minus16544 minus 78083119894

1205825

minus7140 + 54908119894 1205825

minus7140 + 54908119894

1205826

minus7140 minus 54908119894 1205826

minus7140 minus 54908119894

1205827

minus437 + 6315119894 1205827

minus425 + 6316119894

1205828

minus437 minus 6315119894 1205828


1205829

minus434 + 6178119894 1205829

minus422 + 6179119894

12058210

minus434 minus 6178119894 12058210


12058211

minus218 + 2688119894 12058211

minus162 + 2773119894

12058212

minus218 minus 2688119894 12058212


12058213

minus224 + 2681119894 12058213

minus179 + 2746119894

12058214

minus224 minus 2681119894 12058214


12058215

minus122 + 1621119894 12058215

minus098 + 1765119894

12058216

minus122 minus 1621119894 12058216


12058217

minus24378 + 91155119894 12058217

minus24378 + 91155119894

12058218

minus24378 minus 91155119894 12058218

minus24378 minus 91155119894

12058219

minus892 12058219

minus1514

12058220

minus664 12058220

minus1358

12058221

minus355 12058221

minus909

12058222

minus354 12058222

minus910



is converted into

= Λ119884 + 119875minus1119892 (119884 120583

119896) (21)





119910119894= ℎ119894(1199101 1199102 120583119896)

= ℎ11989411199101

2+ ℎ11989421205831198961199101+ ℎ11989431205831198961199102+ ℎ11989441199102

2

119894 = 3 4 21 22

(22)

119894=

120597ℎ119894(1199101 1199102 120583119896)

1205971199101

1+

120597ℎ119894(1199101 1199102 120583119896)

1205971199102

2

+

120597ℎ119894(1199101 1199102 120583119896)

120597120583119896

120583119896

(23)







1= 121 times 10119910

2+ 0762119910

21205831minus 0396119910

11205831minus 0169

times 10minus311991021205831


minus41199102

3minus 0118

times 10minus311991011205831


minus311991011199102

2minus 066


21199102minus 0261 times 10

minus11199101

3+ 0426


211991021205831+ 0464 times 10

minus41199101

31205831

2= minus121 times 10119910

1minus 0148119910

11205831+ 0853119910

21205831+ 0998

times 10minus411991011205831

2+ 0113 times 10

minus11199101

3+ 0429

times 10minus411991021205831

2+ 0286 times 10

minus21199101

21199102+ 0242

times 10minus311991011199102


minus41199101

31205831minus 0187


211991021205831

(24)


1= 146 times 10119910


minus21199101+ 0522119910

21205832


minus21199102

3+ 0118

times 10minus411991011205832


minus11199102

21199101

minus 010111991021199101


minus11199101

3minus 0259

times 10minus411991021199101

21205832minus 0231 times 10

minus41199101

31205832

2= minus146 times 10119910


minus21199102+ 0275

times 10minus111991011205832+ 0261119910

21205832+ 0381

times 10minus411991011205832

2+ 0391 times 10

minus11199101

3minus 0531

times 10minus411991021205832

2+ 0565 times 10

minus111991021199101

2+ 0273


21199101+ 0439 times 10

minus21199102

3+ 0103


31205832+ 0116 times 10

minus411991021199101

21205832

(25)


119903 = dur + 1198861199033+ hot

120579 = 120596 + 119887119903

2+ hot

(26)


F

120583sN120583N

r



For V = V1= 156 kmh


1119903 + (minus0193

times 10minus6

1205831

2+ 0236 times 10


minus1) 1199033

+ hot

(27)

For V = V2= 408 kmh


1205832minus 0145) 120583

2119903 + (minus0495

times 10minus6

1205832


minus4

1205832minus 0476 times 10

minus1

) 1199033

+ hot

(28)





2 When V lt V

1= 156 kmh



When V1lt V lt V

2(ie 120583

1gt 0 and 120583




2= 408 kmh








minus08

minus06

minus04

minus02

0

02

04

06

08H

opf b

ifurc

atio

n br

anch

r


minus04

minus03

minus02

minus01

0

01

02

03

04



Hop

f bifu

rcat

ion

bran

chr



20 40 60 80 100 120 1400Velocity V (kmh)

0

05

1

15

2

25

3

35

4

Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12Se

cond

brid

ge le

ft w

heel

swin

g an

gle1205794

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)

05

06











minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(a) V = 20 kmh




minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(b) V = 25 kmh


minus150

minus100

minus50

0

50

100

150



Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(c) V = 30 kmh






Vkmsdothminus1 20 25 301205792∘ 373 381 328

1205794∘ 1055 1092 947






minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05


(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh











minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











11986811205791+ 1198881198971

1205791+ 119896ℎ1(1205791minus 1205792) + 119888ℎ1(1205791minus

1205792) +

1198941198961V

119877

1

+ (120588119877ℎ1120574 +

1

2

11989611988811989711198611(120574 minus 119891)) 120593

1minus 1198651(119877120574 + 119890)

+11987211= 0

(6)


11986811205792+ 1198881198971

1205792+ 119896ℎ1(1205792minus 1205791) + 119888ℎ1(1205792minus

1205791)

+ 11989611991111198861(11990311205751+ 11988611205792) + 11988811991111198861(11990311205751+ 11988611205792)

+

1198941198961V

119877

1+ (120588119877ℎ

1120574 +

1

2

11989611988811989711198611(120574 minus 119891)) 120593

1

minus 1198652(119877120574 + 119890) +119872

12= 0

(7)


11986911+ 11988811989911+ (

1

2

1198961198881198611

2+ 119896119899+ 2120588119877ℎ

1)1205931

minus

1198941198961V

119877

(1205791+

1205792) + (119865

1+ 1198652) ℎ1= 0

(8)


11986821205793+ 1198881198972

1205793+ 119896ℎ2(1205793minus 1205794) + 119888ℎ2(1205793minus

1205794) +

1198941198962V

119877

2

+ (120588119877ℎ2120574 +

1

2

11989611988811989721198612(120574 minus 119891)) 120593

2minus 1198653(119877120574 + 119890)

+11987221= 0

(9)


11986821205794+ 1198881198972

1205792+ 119896ℎ2(1205794minus 1205793) + 119888ℎ2(1205794minus

1205793)

+ 11989611991121198862(11990331205753+ 11988621205794) + 11988811991121198862(11990331205753+ 11988621205794)

+

1198941198962V

119877

2+ (120588119877ℎ

2120574 +

1

2

11989611988811989721198612(120574 minus 119891)) 120593

2

minus 1198654(119877120574 + 119890) +119872

22= 0

(10)


11986922+ 11988811989922+ (

1

2

1198961198881198612

2+ 119896119899+ 2120588119877ℎ

2)1205932

minus

1198941198962V

119877

(1205793+

1205794) + (119865

3+ 1198654) ℎ2= 0

(11)




minus4000

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

Tire

forc

eF(N

)


119894


1198681198881

1205751+ 1198881199051205751+ 1198961199051205751+ 11988811990511198921(11989211205751minus 11990321205752)

+ 11989611990511198921(11989211205751minus 11990321205752) + 11988811991111199031(11990311205751+ 11988611205792)

+ 11989611991111199031(11990311205751+ 11988611205792) = 0

(12)


1198681198882

1205752+ 11988811990511199032(11990321205752minus 11989211205751) + 11989611990511199032(11990321205752minus 11989211205751)

+ 11988811990521198922(11989221205752minus 11989231205753) + 11989611990521198922(11989221205752minus 11989231205753)

= 0

(13)


1198681198883

1205753+ 11988811990521198923(11989231205753minus 11989221205752) + 11989611990521198923(11989231205753minus 11989221205752)

+ 11988811991121199033(11990331205753+ 11988621205794) + 11989611991121199033(11990331205753+ 11988621205794) = 0

(14)




1198621119891

1198623119891

1198861

1198862

1198863

1198864

1198865

1198866

1198867

1198868

32740 481770 minus221 1011 1078 182 0208 0 minus0354 0707


119894 119863119894 and 119864




119865119894= minus (119862

1119891120572119894minus 1198623119891120572119894

3) (119894 = 1 2 3 4) (15)

119865119894= 119863119894sin (119862

119894


119894119886119894))))

(16)

In this model

119861119894=

1198863sdot sin (119886


5sdot 119865119885119894)))

119862119894sdot 119863119894

119862119894= 13

119863119894= 1198861sdot 119865119885119894

2+ 1198862sdot 119865119885119894

119864119894= 1198866sdot 119865119885119894

2+ 1198867sdot 119865119885119894+ 1198868

(119894 = 1 2 3 4)

(17)

where 1198621119891

and 1198623119891



119894 119862119894 119863119894 and 119864



8



119885119894 1198621119891 and 119862

3119891 The


119894and the



119894+

V120590

120572119894+

V120590

120579119894minus

119890119894

120590

120579119894 119894 = 1 2 3 4 (18)




119872119894119895= 119872 sgn (

120579119894119895) (119894 119895 = 1 2)

|119872| =

120583119904119873119903 (

120579119894119895= 0)

120583119873119903 (120579119894119895

= 0)

(119894 119895 = 1 2)

(19)






Make 1199091= 1205791 1199092=

1205791 1199093= 1205792 1199094=

1205792 1199095= 1205931 1199096= 1

1199097= 1205793 1199098=

1205793 1199099= 1205794 11990910

=1205794 11990911

= 1205932 11990912

= 2

11990913

= 1205751 11990914

=1205751 11990915

= 1205752 11990916

=1205752 11990917

= 1205753 11990918

=1205753

11990919

= 1205721 11990920

= 1205722 11990921

= 1205723 and 119909

22= 1205724 Assume 119883 =

(1199091 1199092 1199093 1199094 1199095 1199096 1199097 1199098 1199099 11990910 11990911 11990912 11990913 11990914 11990915 11990916




= 119860119883 + 119865 (119883) 119883 isin 11987722 (20)



1or when V = V

2 whereas



1= 156 kmh and


















1205741 1205742∘ 25

1198971 1198972m 007

1199031m 034

1199033m 047

1198922m 026

1198861m 021

1198961199111 1198961199112Nsdotradminus1 36000


1198961199051 1198961199052Nsdotmminus1 37200

1198881199051 1198881199052Nsdots 15

119896119888Nsdotradminus1 1060000

119891 0015119896119905Nsdotradminus1 31000



ℎ1 ℎ2m 045

1198611 1198612m 206

1199032m 042

1198921m 017

1198923m 022

1198862m 021


1= 156 kmh and V

2= 408 kmh are


1le V le V



1




119896= VminusV

119896(119896 = 1 2) 120583






1205821

0 + 1211119894 1205821

0 + 1456119894

1205822

0 minus 1211119894 1205822

0 minus 1456119894

1205823

minus16544 + 78083119894 1205823

minus16544 + 78083119894

1205824

minus16544 minus 78083119894 1205824

minus16544 minus 78083119894

1205825

minus7140 + 54908119894 1205825

minus7140 + 54908119894

1205826

minus7140 minus 54908119894 1205826

minus7140 minus 54908119894

1205827

minus437 + 6315119894 1205827

minus425 + 6316119894

1205828

minus437 minus 6315119894 1205828


1205829

minus434 + 6178119894 1205829

minus422 + 6179119894

12058210

minus434 minus 6178119894 12058210


12058211

minus218 + 2688119894 12058211

minus162 + 2773119894

12058212

minus218 minus 2688119894 12058212


12058213

minus224 + 2681119894 12058213

minus179 + 2746119894

12058214

minus224 minus 2681119894 12058214


12058215

minus122 + 1621119894 12058215

minus098 + 1765119894

12058216

minus122 minus 1621119894 12058216


12058217

minus24378 + 91155119894 12058217

minus24378 + 91155119894

12058218

minus24378 minus 91155119894 12058218

minus24378 minus 91155119894

12058219

minus892 12058219

minus1514

12058220

minus664 12058220

minus1358

12058221

minus355 12058221

minus909

12058222

minus354 12058222

minus910



is converted into

= Λ119884 + 119875minus1119892 (119884 120583

119896) (21)





119910119894= ℎ119894(1199101 1199102 120583119896)

= ℎ11989411199101

2+ ℎ11989421205831198961199101+ ℎ11989431205831198961199102+ ℎ11989441199102

2

119894 = 3 4 21 22

(22)

119894=

120597ℎ119894(1199101 1199102 120583119896)

1205971199101

1+

120597ℎ119894(1199101 1199102 120583119896)

1205971199102

2

+

120597ℎ119894(1199101 1199102 120583119896)

120597120583119896

120583119896

(23)







1= 121 times 10119910

2+ 0762119910

21205831minus 0396119910

11205831minus 0169

times 10minus311991021205831


minus41199102

3minus 0118

times 10minus311991011205831


minus311991011199102

2minus 066


21199102minus 0261 times 10

minus11199101

3+ 0426


211991021205831+ 0464 times 10

minus41199101

31205831

2= minus121 times 10119910

1minus 0148119910

11205831+ 0853119910

21205831+ 0998

times 10minus411991011205831

2+ 0113 times 10

minus11199101

3+ 0429

times 10minus411991021205831

2+ 0286 times 10

minus21199101

21199102+ 0242

times 10minus311991011199102


minus41199101

31205831minus 0187


211991021205831

(24)


1= 146 times 10119910


minus21199101+ 0522119910

21205832


minus21199102

3+ 0118

times 10minus411991011205832


minus11199102

21199101

minus 010111991021199101


minus11199101

3minus 0259

times 10minus411991021199101

21205832minus 0231 times 10

minus41199101

31205832

2= minus146 times 10119910


minus21199102+ 0275

times 10minus111991011205832+ 0261119910

21205832+ 0381

times 10minus411991011205832

2+ 0391 times 10

minus11199101

3minus 0531

times 10minus411991021205832

2+ 0565 times 10

minus111991021199101

2+ 0273


21199101+ 0439 times 10

minus21199102

3+ 0103


31205832+ 0116 times 10

minus411991021199101

21205832

(25)


119903 = dur + 1198861199033+ hot

120579 = 120596 + 119887119903

2+ hot

(26)


F

120583sN120583N

r



For V = V1= 156 kmh


1119903 + (minus0193

times 10minus6

1205831

2+ 0236 times 10


minus1) 1199033

+ hot

(27)

For V = V2= 408 kmh


1205832minus 0145) 120583

2119903 + (minus0495

times 10minus6

1205832


minus4

1205832minus 0476 times 10

minus1

) 1199033

+ hot

(28)





2 When V lt V

1= 156 kmh



When V1lt V lt V

2(ie 120583

1gt 0 and 120583




2= 408 kmh








minus08

minus06

minus04

minus02

0

02

04

06

08H

opf b

ifurc

atio

n br

anch

r


minus04

minus03

minus02

minus01

0

01

02

03

04



Hop

f bifu

rcat

ion

bran

chr



20 40 60 80 100 120 1400Velocity V (kmh)

0

05

1

15

2

25

3

35

4

Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12Se

cond

brid

ge le

ft w

heel

swin

g an

gle1205794

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)

05

06











minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(a) V = 20 kmh




minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(b) V = 25 kmh


minus150

minus100

minus50

0

50

100

150



Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(c) V = 30 kmh






Vkmsdothminus1 20 25 301205792∘ 373 381 328

1205794∘ 1055 1092 947






minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05


(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh











minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198621119891

1198623119891

1198861

1198862

1198863

1198864

1198865

1198866

1198867

1198868

32740 481770 minus221 1011 1078 182 0208 0 minus0354 0707


119894 119863119894 and 119864




119865119894= minus (119862

1119891120572119894minus 1198623119891120572119894

3) (119894 = 1 2 3 4) (15)

119865119894= 119863119894sin (119862

119894


119894119886119894))))

(16)

In this model

119861119894=

1198863sdot sin (119886


5sdot 119865119885119894)))

119862119894sdot 119863119894

119862119894= 13

119863119894= 1198861sdot 119865119885119894

2+ 1198862sdot 119865119885119894

119864119894= 1198866sdot 119865119885119894

2+ 1198867sdot 119865119885119894+ 1198868

(119894 = 1 2 3 4)

(17)

where 1198621119891

and 1198623119891



119894 119862119894 119863119894 and 119864



8



119885119894 1198621119891 and 119862

3119891 The


119894and the



119894+

V120590

120572119894+

V120590

120579119894minus

119890119894

120590

120579119894 119894 = 1 2 3 4 (18)




119872119894119895= 119872 sgn (

120579119894119895) (119894 119895 = 1 2)

|119872| =

120583119904119873119903 (

120579119894119895= 0)

120583119873119903 (120579119894119895

= 0)

(119894 119895 = 1 2)

(19)






Make 1199091= 1205791 1199092=

1205791 1199093= 1205792 1199094=

1205792 1199095= 1205931 1199096= 1

1199097= 1205793 1199098=

1205793 1199099= 1205794 11990910

=1205794 11990911

= 1205932 11990912

= 2

11990913

= 1205751 11990914

=1205751 11990915

= 1205752 11990916

=1205752 11990917

= 1205753 11990918

=1205753

11990919

= 1205721 11990920

= 1205722 11990921

= 1205723 and 119909

22= 1205724 Assume 119883 =

(1199091 1199092 1199093 1199094 1199095 1199096 1199097 1199098 1199099 11990910 11990911 11990912 11990913 11990914 11990915 11990916




= 119860119883 + 119865 (119883) 119883 isin 11987722 (20)



1or when V = V

2 whereas



1= 156 kmh and


















1205741 1205742∘ 25

1198971 1198972m 007

1199031m 034

1199033m 047

1198922m 026

1198861m 021

1198961199111 1198961199112Nsdotradminus1 36000


1198961199051 1198961199052Nsdotmminus1 37200

1198881199051 1198881199052Nsdots 15

119896119888Nsdotradminus1 1060000

119891 0015119896119905Nsdotradminus1 31000



ℎ1 ℎ2m 045

1198611 1198612m 206

1199032m 042

1198921m 017

1198923m 022

1198862m 021


1= 156 kmh and V

2= 408 kmh are


1le V le V



1




119896= VminusV

119896(119896 = 1 2) 120583






1205821

0 + 1211119894 1205821

0 + 1456119894

1205822

0 minus 1211119894 1205822

0 minus 1456119894

1205823

minus16544 + 78083119894 1205823

minus16544 + 78083119894

1205824

minus16544 minus 78083119894 1205824

minus16544 minus 78083119894

1205825

minus7140 + 54908119894 1205825

minus7140 + 54908119894

1205826

minus7140 minus 54908119894 1205826

minus7140 minus 54908119894

1205827

minus437 + 6315119894 1205827

minus425 + 6316119894

1205828

minus437 minus 6315119894 1205828


1205829

minus434 + 6178119894 1205829

minus422 + 6179119894

12058210

minus434 minus 6178119894 12058210


12058211

minus218 + 2688119894 12058211

minus162 + 2773119894

12058212

minus218 minus 2688119894 12058212


12058213

minus224 + 2681119894 12058213

minus179 + 2746119894

12058214

minus224 minus 2681119894 12058214


12058215

minus122 + 1621119894 12058215

minus098 + 1765119894

12058216

minus122 minus 1621119894 12058216


12058217

minus24378 + 91155119894 12058217

minus24378 + 91155119894

12058218

minus24378 minus 91155119894 12058218

minus24378 minus 91155119894

12058219

minus892 12058219

minus1514

12058220

minus664 12058220

minus1358

12058221

minus355 12058221

minus909

12058222

minus354 12058222

minus910



is converted into

= Λ119884 + 119875minus1119892 (119884 120583

119896) (21)





119910119894= ℎ119894(1199101 1199102 120583119896)

= ℎ11989411199101

2+ ℎ11989421205831198961199101+ ℎ11989431205831198961199102+ ℎ11989441199102

2

119894 = 3 4 21 22

(22)

119894=

120597ℎ119894(1199101 1199102 120583119896)

1205971199101

1+

120597ℎ119894(1199101 1199102 120583119896)

1205971199102

2

+

120597ℎ119894(1199101 1199102 120583119896)

120597120583119896

120583119896

(23)







1= 121 times 10119910

2+ 0762119910

21205831minus 0396119910

11205831minus 0169

times 10minus311991021205831


minus41199102

3minus 0118

times 10minus311991011205831


minus311991011199102

2minus 066


21199102minus 0261 times 10

minus11199101

3+ 0426


211991021205831+ 0464 times 10

minus41199101

31205831

2= minus121 times 10119910

1minus 0148119910

11205831+ 0853119910

21205831+ 0998

times 10minus411991011205831

2+ 0113 times 10

minus11199101

3+ 0429

times 10minus411991021205831

2+ 0286 times 10

minus21199101

21199102+ 0242

times 10minus311991011199102


minus41199101

31205831minus 0187


211991021205831

(24)


1= 146 times 10119910


minus21199101+ 0522119910

21205832


minus21199102

3+ 0118

times 10minus411991011205832


minus11199102

21199101

minus 010111991021199101


minus11199101

3minus 0259

times 10minus411991021199101

21205832minus 0231 times 10

minus41199101

31205832

2= minus146 times 10119910


minus21199102+ 0275

times 10minus111991011205832+ 0261119910

21205832+ 0381

times 10minus411991011205832

2+ 0391 times 10

minus11199101

3minus 0531

times 10minus411991021205832

2+ 0565 times 10

minus111991021199101

2+ 0273


21199101+ 0439 times 10

minus21199102

3+ 0103


31205832+ 0116 times 10

minus411991021199101

21205832

(25)


119903 = dur + 1198861199033+ hot

120579 = 120596 + 119887119903

2+ hot

(26)


F

120583sN120583N

r



For V = V1= 156 kmh


1119903 + (minus0193

times 10minus6

1205831

2+ 0236 times 10


minus1) 1199033

+ hot

(27)

For V = V2= 408 kmh


1205832minus 0145) 120583

2119903 + (minus0495

times 10minus6

1205832


minus4

1205832minus 0476 times 10

minus1

) 1199033

+ hot

(28)





2 When V lt V

1= 156 kmh



When V1lt V lt V

2(ie 120583

1gt 0 and 120583




2= 408 kmh








minus08

minus06

minus04

minus02

0

02

04

06

08H

opf b

ifurc

atio

n br

anch

r


minus04

minus03

minus02

minus01

0

01

02

03

04



Hop

f bifu

rcat

ion

bran

chr



20 40 60 80 100 120 1400Velocity V (kmh)

0

05

1

15

2

25

3

35

4

Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12Se

cond

brid

ge le

ft w

heel

swin

g an

gle1205794

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)

05

06











minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(a) V = 20 kmh




minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(b) V = 25 kmh


minus150

minus100

minus50

0

50

100

150



Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(c) V = 30 kmh






Vkmsdothminus1 20 25 301205792∘ 373 381 328

1205794∘ 1055 1092 947






minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05


(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh











minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















1205741 1205742∘ 25

1198971 1198972m 007

1199031m 034

1199033m 047

1198922m 026

1198861m 021

1198961199111 1198961199112Nsdotradminus1 36000


1198961199051 1198961199052Nsdotmminus1 37200

1198881199051 1198881199052Nsdots 15

119896119888Nsdotradminus1 1060000

119891 0015119896119905Nsdotradminus1 31000



ℎ1 ℎ2m 045

1198611 1198612m 206

1199032m 042

1198921m 017

1198923m 022

1198862m 021


1= 156 kmh and V

2= 408 kmh are


1le V le V



1




119896= VminusV

119896(119896 = 1 2) 120583






1205821

0 + 1211119894 1205821

0 + 1456119894

1205822

0 minus 1211119894 1205822

0 minus 1456119894

1205823

minus16544 + 78083119894 1205823

minus16544 + 78083119894

1205824

minus16544 minus 78083119894 1205824

minus16544 minus 78083119894

1205825

minus7140 + 54908119894 1205825

minus7140 + 54908119894

1205826

minus7140 minus 54908119894 1205826

minus7140 minus 54908119894

1205827

minus437 + 6315119894 1205827

minus425 + 6316119894

1205828

minus437 minus 6315119894 1205828


1205829

minus434 + 6178119894 1205829

minus422 + 6179119894

12058210

minus434 minus 6178119894 12058210


12058211

minus218 + 2688119894 12058211

minus162 + 2773119894

12058212

minus218 minus 2688119894 12058212


12058213

minus224 + 2681119894 12058213

minus179 + 2746119894

12058214

minus224 minus 2681119894 12058214


12058215

minus122 + 1621119894 12058215

minus098 + 1765119894

12058216

minus122 minus 1621119894 12058216


12058217

minus24378 + 91155119894 12058217

minus24378 + 91155119894

12058218

minus24378 minus 91155119894 12058218

minus24378 minus 91155119894

12058219

minus892 12058219

minus1514

12058220

minus664 12058220

minus1358

12058221

minus355 12058221

minus909

12058222

minus354 12058222

minus910



is converted into

= Λ119884 + 119875minus1119892 (119884 120583

119896) (21)





119910119894= ℎ119894(1199101 1199102 120583119896)

= ℎ11989411199101

2+ ℎ11989421205831198961199101+ ℎ11989431205831198961199102+ ℎ11989441199102

2

119894 = 3 4 21 22

(22)

119894=

120597ℎ119894(1199101 1199102 120583119896)

1205971199101

1+

120597ℎ119894(1199101 1199102 120583119896)

1205971199102

2

+

120597ℎ119894(1199101 1199102 120583119896)

120597120583119896

120583119896

(23)







1= 121 times 10119910

2+ 0762119910

21205831minus 0396119910

11205831minus 0169

times 10minus311991021205831


minus41199102

3minus 0118

times 10minus311991011205831


minus311991011199102

2minus 066


21199102minus 0261 times 10

minus11199101

3+ 0426


211991021205831+ 0464 times 10

minus41199101

31205831

2= minus121 times 10119910

1minus 0148119910

11205831+ 0853119910

21205831+ 0998

times 10minus411991011205831

2+ 0113 times 10

minus11199101

3+ 0429

times 10minus411991021205831

2+ 0286 times 10

minus21199101

21199102+ 0242

times 10minus311991011199102


minus41199101

31205831minus 0187


211991021205831

(24)


1= 146 times 10119910


minus21199101+ 0522119910

21205832


minus21199102

3+ 0118

times 10minus411991011205832


minus11199102

21199101

minus 010111991021199101


minus11199101

3minus 0259

times 10minus411991021199101

21205832minus 0231 times 10

minus41199101

31205832

2= minus146 times 10119910


minus21199102+ 0275

times 10minus111991011205832+ 0261119910

21205832+ 0381

times 10minus411991011205832

2+ 0391 times 10

minus11199101

3minus 0531

times 10minus411991021205832

2+ 0565 times 10

minus111991021199101

2+ 0273


21199101+ 0439 times 10

minus21199102

3+ 0103


31205832+ 0116 times 10

minus411991021199101

21205832

(25)


119903 = dur + 1198861199033+ hot

120579 = 120596 + 119887119903

2+ hot

(26)


F

120583sN120583N

r



For V = V1= 156 kmh


1119903 + (minus0193

times 10minus6

1205831

2+ 0236 times 10


minus1) 1199033

+ hot

(27)

For V = V2= 408 kmh


1205832minus 0145) 120583

2119903 + (minus0495

times 10minus6

1205832


minus4

1205832minus 0476 times 10

minus1

) 1199033

+ hot

(28)





2 When V lt V

1= 156 kmh



When V1lt V lt V

2(ie 120583

1gt 0 and 120583




2= 408 kmh








minus08

minus06

minus04

minus02

0

02

04

06

08H

opf b

ifurc

atio

n br

anch

r


minus04

minus03

minus02

minus01

0

01

02

03

04



Hop

f bifu

rcat

ion

bran

chr



20 40 60 80 100 120 1400Velocity V (kmh)

0

05

1

15

2

25

3

35

4

Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12Se

cond

brid

ge le

ft w

heel

swin

g an

gle1205794

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)

05

06











minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(a) V = 20 kmh




minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(b) V = 25 kmh


minus150

minus100

minus50

0

50

100

150



Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(c) V = 30 kmh






Vkmsdothminus1 20 25 301205792∘ 373 381 328

1205794∘ 1055 1092 947






minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05


(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh











minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1= 121 times 10119910

2+ 0762119910

21205831minus 0396119910

11205831minus 0169

times 10minus311991021205831


minus41199102

3minus 0118

times 10minus311991011205831


minus311991011199102

2minus 066


21199102minus 0261 times 10

minus11199101

3+ 0426


211991021205831+ 0464 times 10

minus41199101

31205831

2= minus121 times 10119910

1minus 0148119910

11205831+ 0853119910

21205831+ 0998

times 10minus411991011205831

2+ 0113 times 10

minus11199101

3+ 0429

times 10minus411991021205831

2+ 0286 times 10

minus21199101

21199102+ 0242

times 10minus311991011199102


minus41199101

31205831minus 0187


211991021205831

(24)


1= 146 times 10119910


minus21199101+ 0522119910

21205832


minus21199102

3+ 0118

times 10minus411991011205832


minus11199102

21199101

minus 010111991021199101


minus11199101

3minus 0259

times 10minus411991021199101

21205832minus 0231 times 10

minus41199101

31205832

2= minus146 times 10119910


minus21199102+ 0275

times 10minus111991011205832+ 0261119910

21205832+ 0381

times 10minus411991011205832

2+ 0391 times 10

minus11199101

3minus 0531

times 10minus411991021205832

2+ 0565 times 10

minus111991021199101

2+ 0273


21199101+ 0439 times 10

minus21199102

3+ 0103


31205832+ 0116 times 10

minus411991021199101

21205832

(25)


119903 = dur + 1198861199033+ hot

120579 = 120596 + 119887119903

2+ hot

(26)


F

120583sN120583N

r



For V = V1= 156 kmh


1119903 + (minus0193

times 10minus6

1205831

2+ 0236 times 10


minus1) 1199033

+ hot

(27)

For V = V2= 408 kmh


1205832minus 0145) 120583

2119903 + (minus0495

times 10minus6

1205832


minus4

1205832minus 0476 times 10

minus1

) 1199033

+ hot

(28)





2 When V lt V

1= 156 kmh



When V1lt V lt V

2(ie 120583

1gt 0 and 120583




2= 408 kmh








minus08

minus06

minus04

minus02

0

02

04

06

08H

opf b

ifurc

atio

n br

anch

r


minus04

minus03

minus02

minus01

0

01

02

03

04



Hop

f bifu

rcat

ion

bran

chr



20 40 60 80 100 120 1400Velocity V (kmh)

0

05

1

15

2

25

3

35

4

Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12Se

cond

brid

ge le

ft w

heel

swin

g an

gle1205794

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)

05

06











minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(a) V = 20 kmh




minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(b) V = 25 kmh


minus150

minus100

minus50

0

50

100

150



Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(c) V = 30 kmh






Vkmsdothminus1 20 25 301205792∘ 373 381 328

1205794∘ 1055 1092 947






minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05


(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh











minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












minus08

minus06

minus04

minus02

0

02

04

06

08H

opf b

ifurc

atio

n br

anch

r


minus04

minus03

minus02

minus01

0

01

02

03

04



Hop

f bifu

rcat

ion

bran

chr



20 40 60 80 100 120 1400Velocity V (kmh)

0

05

1

15

2

25

3

35

4

Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12Se

cond

brid

ge le

ft w

heel

swin

g an

gle1205794

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)

05

06











minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(a) V = 20 kmh




minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(b) V = 25 kmh


minus150

minus100

minus50

0

50

100

150



Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(c) V = 30 kmh






Vkmsdothminus1 20 25 301205792∘ 373 381 328

1205794∘ 1055 1092 947






minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05


(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh











minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(a) V = 20 kmh




minus150

minus100

minus50

0

50

100

150

Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(b) V = 25 kmh


minus150

minus100

minus50

0

50

100

150



Ang

ular

vel

ociti

es120579998400 2120579

998400 4(∘

s)

(c) V = 30 kmh






Vkmsdothminus1 20 25 301205792∘ 373 381 328

1205794∘ 1055 1092 947






minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05


(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh











minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus5

0

5A

ngul

ar v

eloc

ities

120593998400 1120593

998400 2(∘

s)

0 05minus05


(a) V = 20 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(b) V = 25 kmh


minus6

minus4

minus2

0

2

4

6

0 05minus05

Ang

ular

vel

ociti

es120593998400 1120593

998400 2(∘

s)


(c) V = 30 kmh











minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25A

ngul

ar v

eloc

ity120579998400 1

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 2

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

8

Ang

ular

vel

ocity

120579998400 3

(∘s

)



0 05minus05


minus6

minus4

minus2

0

2

4

6

Ang

ular

vel

ocity

120579998400 4

(∘s

)



minus0005 0 0005 001minus001


minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Ang

ular

vel

ocity

120593998400 1

(∘s

)



minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

Ang

ular

vel

ocity

120593998400 2

(∘s

)

minus001 0 001 002minus002



Figure 10 Continued



minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus15

minus1

minus05

0

05

1

15A

ngul

ar v

eloc

ity120575998400 1

(∘s

)





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Ang

ular

vel

ocity

120575998400 2

(∘s

)





minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120575998400 3

(∘s

)








1= 156 kmh 120583

1=



When V = V2= 408 kmh 120583

2= 0001V

2 we generate




minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1A

ngul

ar v

eloc

ity120579998400 1

(∘s

)







minus08

minus06

minus04

minus02

0

02

04

06

08

Ang

ular

vel

ocity

120579998400 2

(∘s

)




minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 3

(∘s

)






minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Ang

ular

vel

ocity

120579998400 4

(∘s

)



0 2minus2minus4 4


minus006

minus004

minus002

0

002

004

006

Ang

ular

vel

ocity

120593998400 1

(∘s

)

times10minus3



minus02

minus015

minus01

minus005

0

005

01

015

Ang

ular

vel

ocity

120593998400 2

(∘s

)




Figure 11 Continued



minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05A

ngul

ar v

eloc

ity120575998400 1

(∘s

)




0 001 002 003

0

01

02

03

04


minus01

minus02

minus03



Ang

ular

vel

ocity

120575998400 2

(∘s

)


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1



Ang

ular

vel

ocity

120575998400 3

(∘s

)











60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










60 80 10040 120 140200Velocity V (kmh)

0

05

1

15

2

25

3

35

4Fi

rst b

ridge

left

whe

el sw

ing

angl

e1205792

(∘)

M = 10M = 5M = 0


20 40 60 80 100 120 1400Velocity V (kmh)

0

2

4

6

8

10

12

Seco

nd b

ridge

left

whe

el sw

ing

angl

e1205794

(∘)


M = 10M = 5M = 0

20 40 60 80 100 120 1400Velocity V (kmh)

0

002

004

006

008

01

012

014

016

018

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


20 40 60 80 100 120 1400Velocity V (kmh)

0

01

02

03

04

05

06

Seco

nd b

ridge

side

swin

g an

gle1205932

(∘)





1199052 pneumatic



from 1122∘ to 889∘ Δ1205794











1199052 the



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



22242628

332343638


015

016

017

018

019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)


04042044046048

05052054056058

06

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


85

9

95

10

105

11

115



Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)

times104

times104 times104

times104



3 4 5 6 7 8 9 102





3 95 6 7 842 1015

2

25

3

35

4

6

7

8

9

10

11

12

13

01011012013014015016017018019

02

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

03504

04505

05506

06507

angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)


Figure 13 Continued


Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Firs

t brid

ge le

ft w

heel

swin

g an

gle1205792

(∘)

swin

g an

gle

Seco

nd b

ridge

left

whe

el1205794

(∘)



0

5

10

15

20

25

0

5

10

15

20

25



Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘) 14

12

1

08

06

04

02

0005 01 015 02 0250



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

2

15

1

05

0

minus05

minus1


Firs

t brid

ge le

ft w

heel

swin

gan

gle1205792

(∘)

angl

eSe

cond

brid

ge le

ft w

heel

swin

g1205794

(∘)



angl

eSe

cond

brid

ge si

de sw

ing

1205932

(∘)

005

115

225

335

445

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

1 2 3 4 50Caster angle 120574 (∘)

0

5

10

15

025

Firs

t brid

ge si

de sw

ing

angl

e1205931

(∘)

02

015

01

005

0 0005

01015

02025

03035

04045

05









5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















5 Conclusions









Acknowledgment


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Hopf Bifurcation Characteristics of...

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