# Modeling and Analysis of Vehicle Shimmy with Consideration of the Coupling Effects of Vehicle Body

Authors: Xiaogao Li; Ning Zhang; Xianjian Jin; Nan Chen
Journal: Shock and Vibration
 (2019)

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## Abstract

Based on the Lagrange equation, a 9-degrees-of-freedom shimmy model with consideration of the coupling effects between the motions of vehicle body and the shimmy of front wheels and a 5-degrees-of-freedom shimmy model ignoring these coupling effects for a vehicle with double-wishbone independent front suspensions are presented here to study the problem of vehicle shimmy. According to the eigenvalue loci of system’s Jacobian matrix plotted on the complex plane, the Hopf bifurcation characteristics of nonlinear shimmy are studied and the conditions for the generation of limit cycle are analyzed. Numerical calculation and simulation are used to study the dynamic behavior of vehicle shimmy. By comparing the dynamic responses of two different shimmy models, the coupling effects of vehicle body on vehicle shimmy are studied. Finally, the relationship between the amplitude of each DoF and vehicle velocity and the influences of vehicle parameters such as the mass of vehicle body, the longitudinal position of the center of gravity of vehicle body, and the inclination angle of front suspension on shimmy are studied.


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## Body

## 1. Introduction

Shimmy phenomenon, which widely exists in vehicles, motorcycle and landing gear of airplane, is a problem of nonlinear dynamics [1–3]. Vehicle shimmy is a sustained oscillation of the steering wheels around the kingpin, combined with the lateral yaw motion caused by the interaction between the tire and the dynamic behavior of steering system. It is closely related to the control stability, ride performance, dynamic property, fuel economic, and safety of the vehicle. Thus, it is still a problem for engineers to eliminate shimmy completely in the design phase of vehicles.Since De Lavaud [4] published the first paper on wheel shimmy in 1920s, a large number of investigations have been conducted to study the phenomenon of vehicle shimmy in the past decades. Recent research studies indicate that the main factors which affect vehicle shimmy are (a) the dynamic response of tires, (b) the parameters of suspensions and steering system, and (c) the clearance and friction in mechanical structures. Von Schlippe and Dietrich [5] introduced the kinematics theory of pure rolling wheel and the finite contact length stretched string tire model in the early 1940s. Thereafter, more and more scholars focused on the problem of tire dynamics and set up several tire mathematical models for vehicle dynamics analysis. Pacejka et al. [6, 7] did a more comprehensive study on the problem of wheel shimmy, especially on constructing the mathematical model of tire, and developed the famous Magic Formula [8]. Gim and Nikravesh [9] studied the dynamic properties of tire due to pure slips and comprehensive slips and analytically derived the explicit formulations for the tire dynamic properties. Gim’s tire model can be used to calculate the forces and moments between the pneumatic tires and the road surface. Guo and Lu [10] established the UniTire model, which is a nonlinear and non-steady-state tire model for vehicle dynamics simulation and control. In recent years, because of the establishment of the above tire model and continuous improvement, great progress has been made in the research on the problem of vehicle shimmy. Takács and Stépan [11, 12] have made rich achievements in the research of the shimmy problem, involving modeling of tire, modeling of steering system, and modeling of suspension system. Ran et al. [13] studied the effects of the nonlinearities in tire on the stability of vehicle shimmy by the energy flow method. Wei et al. [14] developed a nine-degrees-of-freedom (DoF) dynamic model to study the shimmy of a dual front axle heavy truck. Lu et al. [15–17] performed a comprehensive study on vehicle shimmy in which the clearance in mechanism structure, such as in the steering linkage mechanism, the steering handling mechanism, and the cross shaft type universal joint, is taken into account, and the effect of the roll motion of vehicle body to vehicle shimmy is also considered. With the rapid development of new energy vehicle, the shimmy problem of electric vehicle stimulates the interest of some researchers. For instance, Mi et al. [18] paid more attention to the shimmy problem of electric vehicle with independent suspensions.Most of the above research studies are based on the assumption that vehicle body is stationary, and the effects of the motions of vehicle body, such as yaw, pitch, roll, and lateral and vertical motion, on the shimmy of front wheels are ignored. But in fact, the shimmy of front wheels around the kingpins will produce the forces acting on the vehicle body, and the motions of the vehicle body certainly affect the shimmy of the front wheels. Therefore, there are coupling effects between the shimmy of the front wheels and the motions of the vehicle body. However, it is still not clear how the motions of vehicle body affect the dynamic behavior of vehicle shimmy. Independent suspensions are widely used in vehicles, and the dynamic model of a vehicle with independent front suspension structure is presented here. Based on the numerical analysis, the coupling effects between vehicle shimmy and the motions of vehicle body are discussed with the time history response and the phase portrait of each DoF of vehicle shimmy. Meanwhile the relationships between the parameters of vehicle body and the shimmy of front wheels are presented here.The rest of this paper will be organized as follows. Mechanical and mathematical models for vehicle shimmy are established in Section2. The Hopf bifurcation characteristics of two systems are analyzed in Section 3. Numerical simulation is carried out in Section 4. The influences of parameters related to vehicle body and front suspensions on shimmy are studied in Section 5. Some conclusive remarks are given at the end.


## 2. Vehicle Shimmy Model

### 2.1. Mechanical Model of Vehicle Shimmy with Consideration of the Motions of Vehicle Body

The mechanical model of shimmy with consideration of the coupling effects of the motions of vehicle body for a vehicle with double-wishbone independent front suspensions is shown in Figure1. In line with the law of right-hand coordinate system, the coordinate axes ox, oy, and oz point to the front, the left, and the top of the vehicle, respectively.Figure 1
Mechanical model of vehicle shimmy with consideration of vehicle body. (a) Three-dimensional view. (b) Vertical view of steering system.To facilitate dynamic analysis and mathematical modeling, the following assumptions are made to establish a dynamics model for vehicle shimmy.(1)
The steering wheel is immobile(2)
The effect of air resistance is neglected(3)
The plane of steering trapezium is parallel to the horizontal planexoy(4)
The vehicle keeps moving at a constant velocity, and without cornering(5)
The effect of the caster angle is considered, while other alignment parameters are neglected(6)
Vehicle structure is bilaterally symmetrical(7)
The central lines of the spring of rear suspension are perpendicular to the horizontal plane, and the vertical height of front and the rear suspensions are the same(8)
The mass other than wheels and vehicle body will not be consideredParameters related to the structure size of the vehicle are marked on the mechanical model of vehicle shimmy, as shown in Figure1. In which, Wf and Wr are half of the distances between two connection points where the dampers of the front and rear suspensions connect with the vehicle body, respectively; Lf and Lr are the distances from vehicle body’s CoG to the connection line of two connection points where the dampers of front and rear suspensions connect with the vehicle body, respectively; la is the horizontal distance of connection points between the vehicle body and suspension arm, suspension spring; lb is the distance from the intersection point of the kingpin extension line and the ground to the symmetry plane of the wheel; lc is the horizontal distance between the two ends of the suspension spring; ld is the arm of force of steering rod acting on kingpin; lf is the lateral swing arm of front suspension; lg is the arm of force of steering tie rod acting on pitman arm; lh is the vertical height between the two ends of suspension spring, namely, the vertical height of front suspension springs and rear suspension springs.


### 2.2. Lagrange Equation

According to the mechanical model of vehicle shimmy with consideration of the coupling effects of the motions of the vehicle body, its mathematical model can be established. This shimmy model has nine DoFs:θ1,2 are the angles of front wheels swing that revolve around their respective kingpin (namely, the shimmy angles), θ3 is the angle of pitman arm’s swing in steering system, φ1,2 are the angles of front wheel axes’ lateral swing that revolve around their respective lateral swing centers which are parallel to the x axis, θr,p,ω are the roll, pitch, and yaw angle of vehicle body respectively, and z is the displacement of vehicle body along the z axis.The mathematical model of vehicle shimmy can be established based on the Lagrange equations which are expressed as(1)ddt∂T∂q˙i−∂T∂qi+∂U∂qi+∂D∂q˙i=Qi,i=1,2,3,…,9.where T represents the kinetic energy of the system, U represents the potential energy of the system, D represents the dissipative potential function of the system, Qi represents the generalized force to which each DoF of the system is subjected, and qi represents the generalized coordinate of the system, which is expressed as q1q2⋯q9=θ1θ2θ3φ1φ2θrθpθωz.The kinetic energy of the system can be given by(2)T=12Jdφ˙12+12J0vR−φ˙1θ12+12Jdθ˙1−φ˙1γ2+12mωv−lbθ˙12+12mωlfφ˙1−lbθ˙1γ2+12Jdφ˙22+12J0vR−φ˙2θ22+12Jdθ˙2−φ˙2γ2+12mωv−lbθ˙22+12mωlfφ˙2−lbθ˙2γ2+12J3θ˙32+12msv2+12msz˙2+12Jrθ˙r2+12Jpθ˙p2+12Jωθ˙ω2,where J0 and Jd are the moments of inertia of front wheels around spin axes and diameters, respectively; J3 is the moment of inertia of the pitman arm; Jr, Jp, and Jω are the moments of inertia of vehicle body around the x, y, and z axis, respectively; mw is the mass of wheels, and ms is the mass of vehicle body; γ is the caster angle of front wheels; R is the rolling radius of tires; and v is the vehicle velocity.The potential energy of the system is given by(3)U=12k1ldθ1−lgθ32+12k2lgθ3−ldθ22+12k3θ32+12k4Δfl2+Δfr2+12k5Δrl2+Δrr2+12kblfφ1−lbθ1γ2+12kblfφ2−lbθ2γ2+12kyRφ1−Rθ1γ2+12kyRφ2−Rθ2γ2,where k1 and k2 are the stiffness of left and right steering tie rods, respectively; k3 is the stiffness of pitman arm; k4 and k5 are the stiffness of front and rear suspensions, respectively; ky is tire lateral stiffness; kb is tire vertical stiffness; Δfl, Δfr, Δrl, and Δrr are the deformations of the front-left, front-right, rear-left, and rear-right suspensions, respectively, which are given by equation (8).The dissipative potential function is given by(4)D=12ceθ˙12+θ˙22+12c1ldθ˙1−lgθ˙32+12c2lgθ˙3−ldθ˙22+12c3θ˙32+12c4Δ˙fl2+Δ˙fr2+12c5Δ˙rl2+Δ˙rr2+fkblfφ1lbθ˙1+fkblfφ2lbθ˙2,where ce is the equivalent damping of front wheels around their kingpins; c1 and c2 are the damping of left and right steering tie rods, respectively; c3 is the damping of pitman arm; c4 and c5 are the damping of front and rear suspensions, respectively; and f is the friction coefficient between the tires and the ground.The generalized force in the system is expressed as(5)Qi=FY1∂∂q˙i−Rγ+eθ˙1+Rφ˙1+FY2∂∂q˙i−Rγ+eθ˙2+Rφ˙2.Each generalized force corresponding to the generalized coordinate in the system can be given by(6)Q1=−FY1Rγ+e,Q2=−FY2Rγ+e,Q3=0,Q4=FY1R,Q5=FY2R,Q6=0,Q7=0,Q8=0,Q9=0,where FY1 and FY2 are the lateral forces of front-left wheel and front-right wheel, respectively, and e is the pneumatic trail of tire.The displacement caused by angle change of each DoF is nonlinear. To simplify the model and calculation, assume the angles in equations (2)–(6) are small; thus, equation sinθ=θ holds, where θ∈θ1θ2θ3φ1φ2θrθpθω.


### 2.3. Calculation of Suspension Deformation

The diagram used to calculate the deformation of vehicle suspensions when shimmy occurs is shown in Figure2. Thus, the deformation of vehicle suspensions Δfl, Δfr, Δrl, and Δrr can be deduced by(7)Δfl=lsfl′−lsfl=lh−lacφ1+Wfθr−Lfθp+z2+lc+msinθa−msinθa+θω2+mcosθa−mcosθa+θω2−lh2+lc2,Δfr=lsfr′−lsfr=lh+lacφ1−Wfθr−Lfθp+z2+lc+msinθa−msinθa−θω2+mcosθa−mcosθa−θω2−lh2+lc2,Δrl=lsrl′−lsrl=lh+Wrθr+Lrθp+z2+nsinθb−nsinθb−θω2+ncosθb−ncosθb−θω2−lh,Δrr=lsrr′−lsrr=lh−Wrθr+Lrθp+z2+nsinθb−nsinθb+θω2+ncosθb−ncosθb+θω2−lh,where lsfl, lsfr, lsrl, and lsrr are the original lengths of front-left, front-right, rear-left, and rear-right suspension springs; lsfl′, lsfr′, lsfl′, and lsrr′ are the lengths of springs after deformation corresponding to lsfl, lsfr, lsrl, and lsrr; m and n are the distances from vehicle body’s CoG to the connection points between front suspensions and vehicle body and between rear suspensions and vehicle body; θa and θb are the angles between line m and x axis and between line n and x axis; θs is the inclination angle of front suspension, which is defined as the angle between the central line of spring in front suspension and the horizontal plane; and make lac=la+lc. These parameters are all marked on Figure 2.Figure 2
Diagram for calculating the deformation of front and rear suspensions when vehicle shimmy occurs.It can be seen clearly from equation (7) that the deformation of suspensions is nonlinear. Based on the above small angle change hypothesis, the linearized formulas for calculating the deformation of four vehicle suspensions when shimmy occurs are given as(8)Δfl=−khclacφ1+khcWfθr−khcLfθp−kccmcosθaθω+khcz,Δfr=khclacφ2−khcWfθr−khcLfθp+kccmcosθaθω+khcz,Δrl=Wrθr+Lrθp+z,Δrr=−Wrθr+Lrθp+z,where, khc and kcc can be given by(9)khc=lhlh2+lc2,kcc=lclh2+lc2.


### 2.4. Mathematical Model of Vehicle Shimmy with Consideration of the Motions of Vehicle Body

Based on the above analysis, the 9-DoF mathematical model of shimmy with consideration of the coupling effects of the motions of the vehicle body for a vehicle with double-wishbone independent front suspensions can be available, which is listed in equation (13).In equation (13), the equivalent moment of inertia Jα, Jβ, and Jγ are given by(10)Jα=Jd+mwlb2+mwlb2γ2,Jβ=Jd+Jdγ2+mwlf2,Jγ=Jd+mwlblfγ.The seventh and the eighth terms of the first formula in equation (13) are caused by the lateral swing angle of front-left wheel axis φ1, and they are torques acting on shimmy angle θ1. Because the wheel rotates around its spin axis, when the wheel axis revolves around another axis, it is similar to a high-speed gyroscope rotating around its spin axis. Thus, the seventh term is the coupling torque caused by the elastic characteristics of the tire ky and kb, and the eighth term is the gyroscopic moment generated by the rotation of front-left wheel. It shows that there is coupling effect between the shimmy of front-left wheel around its kingpin and the lateral swing of front-left wheel axis around its lateral swing center. The same coupling effect also exists in the shimmy of front-right wheel and the lateral swing of front-right wheel axis.The above conclusions can also be inferred from the first, second, fourth, and fifth formulas in equation (13), and another conclusion can be drawn from the sixth to the ninth formula in equation (13) that there are coupling effects between the motions of the vehicle body θr,p,ω and z, and the motion of front wheel axes’ lateral swing φ1,2. Therefore, there are coupling effects between the shimmy of the front wheels and the motions of the vehicle body, and these coupling effects will be studied later.


### 2.5. Mathematical Model of Vehicle Shimmy Ignoring the Motions of Vehicle Body

Ignoring the coupling effects of the motions of vehicle body on shimmy, it can be assumed that the body is fixed. Thus, there is no displacement in each DoF of the vehicle body, that is to say, equationsθr = 0, θp = 0, θω = 0, and z = 0 hold. Substituting them into equation (13), the 5-DoF mathematical model of shimmy, including the angles of front wheels’ shimmy θ1,2, the angle of pitman arm’s swing θ3, and the angles of front wheel axes’ lateral swing φ1,2, can be obtained, and its differential equation is listed in equation (14).


### 2.6. Tire Model and Tire Rolling Constraint Equation

Vehicle self-excited shimmy is a nonlinear dynamic bifurcation phenomenon, and it is one of the key factors that precisely expresses the cornering characteristics of the tire to accurately and effectively analyze the shimmy of the front wheels. As mentioned above, the nonlinear tire models that are commonly used in the simulation of vehicle dynamics mainly include Gim’s tire models, Pacejka’s Magic Formula, and Guokonghui’s semiempirical tire theoretical model. The Magic Formula was chosen as the tire model in this study [19], and the lateral force of the tire can be given as(11)FY=DsinCarctanBα−SH1−E+EarctanBα−SH+SV,where α is the side-slip angle of wheel, B is the stiffness factor, C is the shape factor, D is the peak value, E is the curvature factor, SH is the horizontal shift, and SV is the vertical shift, respectively. Here, we take SH=0 and SV=0, and B, C, D, E can be solved by(12)C=a0,D=a1Fz2+a2Fz,BCD=a3sin2arctanFza4×1−a5β,E=a6Fz+a7,(13)Jαθ¨1−Jγφ¨1+k1ld2+kyR2γ2+kblb2γ2θ1+ce+c1ld2θ˙1−k1ldlgθ3−c1ldlgθ˙3−kyR2γ+kblblfγ−fφ1+J0vRφ˙1=FY1Rγ+e,Jαθ¨2−Jγφ¨2+k2ld2+kyR2γ2+kblb2γ2θ2+ce+c2ld2θ˙2−k2ldlgθ3−c2ldlgθ˙3−kyR2γ+kblblfγ−fφ2+J0vRφ˙2=FY2Rγ+e,J3θ¨3−k1ldlgθ1−c1ldlgθ˙1−k2ldlgθ2−c2ldlgθ˙2+k1lg2+k2lg2+k3θ3+c1lg2+c4lg2+c5θ˙3=0,Jβφ¨1−Jγθ¨1−kyR2+kblblfγθ1−J0vRθ˙1+k4khc2lac2+kyR2+kblf2φ1+c4khc2lac2φ˙1−k4khc2lacWfθr−c4khc2lacWfθ˙r+k4khc2lacLfθp+c4khc2lacLfθ˙p+k4khckcclacmcosθaθω+c4khckcclacmcosθaθ˙ω−k4khc2lacz−c4khc2lacz˙=−FY1R,Jβφ¨2−Jγθ¨2−kyR2+kblblfγθ2−J0vRθ˙2+k4khc2lac2+kyR2+kblf2φ2+c4khc2lac2φ˙2−k4khc2lacWfθr−c4khc2lacWfθ˙r−k4khc2lacLfθp−c4khc2lacLfθ˙p+k4khckcclacmcosθaθω+c4khckcclacmcosθaθ˙ω+k4khc2lacz+c4khc2lacz˙=−FY2R,Jrθ¨r−k4khc2lacWfφ1−c4khc2lacWfφ˙1−k4khc2lacWfφ2−c4khc2lacWfφ˙2+2k4khc2Wf2+k5Wr2θr+2c4khc2Wf2+c5Wr2θ˙r−2k4khckccWfmcosθaθω−2c4khckccWfmcosθaθ˙ω=0,Jpθ¨p+k4khc2lacLfφ1+c4khc2lacLfφ˙1−k4khc2lacLfφ2−c4khc2lacLfφ˙2+2k4khc2Lf2+k5Lr2θp+2c4khc2Lf2+c5Lr2θ˙p−2k4khc2Lf−k5Lrz−2c4khc2Lf−c5Lrz˙=0,Jωθ¨ω+k4kcckhclacmcosθaφ1+c4kcckhclacmcosθaφ˙1+k4khckcclacmcosθaφ2+c4khckcclacmcosθaφ˙2−2k4khckccWfmcosθaθr−2c4khckccWfmcosθaθ˙r+2k4kcc2m2cos2θaθω+2c4kcc2m2cos2θaθ˙ω=0,msz¨−k4khc2lacφ1−c4khc2lacφ˙1+k4khc2lacφ2+c4khc2lacφ˙2−2k4khc2Lf−k5Lrθp−2c4khc2Lf−c5Lrθ˙p+2k4khc2+k5z+2c4khc2+c5z˙=0,(14)Jαθ¨1−Jγφ¨1+k1ld2+kyR2γ2+kblb2γ2θ1+ce+c1ld2θ˙1−k1ldlgθ3−c1ldlgθ˙3−kyR2γ+kblblfγ−fφ1+J0vRφ˙1=FY1Rγ+e,Jαθ¨2−Jγφ¨2+k2ld2+kyR2γ2+kblb2γ2θ2+ce+c2ld2θ˙2−k2ldlgθ3−c2ldlgθ˙3−kyR2γ+kblblfγ−fφ2+J0vRφ˙2=FY2Rγ+e,J3θ¨3−k1ldlgθ1−c1ldlgθ˙1−k2ldlgθ2−c2ldlgθ˙2+k1lg2+k2lg2+k3θ3+c1lg2+c4lg2+c5θ˙3=0,Jβφ¨1−Jγθ¨1−kyR2+kblblfγθ1−J0vRθ˙1+k4khc2lac2+kyR2+kblf2φ1+c4khc2lac2φ˙1=−FY1R,Jβφ¨2−Jγθ¨2−kyR2+kblblfγθ2−J0vRθ˙2+k4khc2lac2+kyR2+kblf2φ2+c4khc2lac2φ˙2=−FY2R,where Fz is the normal load acting on wheels, β is the camber angle of wheel, and the parameters a0,a1,…,a7 are all constants that have to be determined for each tire. In this study, the value of β is 0, and the values of a0,a1,…,a7 are listed in Table 1 (for choice of parameters, see Pacejka et al. [8, 19]).Table 1
Values of the coefficients of Magic Formula.
Parameter
Value
a
0
1.65000
a
1
1250.00
a
2
12.80
a
3
−0.02103
a
4
−34.0
a
5
3036.00
a
6
0.00501
a
7
0.77394It is assumed that the vehicle has no lateral acceleration; the normal load acting on the left wheelFz1 and the normal load acting on the right wheel Fz2 are given as(15)Fz1=Fz0−kblfφ1,Fz2=Fz0+kblfφ2,where Fz0 is the static normal load acting on front-left and front-right wheels, which is given by(16)Fz0=12Lr′Lf′+Lr′ms+4mwg,where Lf′ and Lr′ are the distances from the CoG of the vehicle to its front axle and rear axle, respectively.The constraint of the side-slip angle and the shimmy angle of front wheel can be given as(17)α˙1+vσα1+vσθ1−aσθ˙1=0,α˙2+vσα2+vσθ2−aσθ˙2=0,where σ is the relaxation length of tires and a is the half-length of the tire contact area. In this study, σ is 0.65 m and a is 0.2 m.


## 2.1. Mechanical Model of Vehicle Shimmy with Consideration of the Motions of Vehicle Body

The mechanical model of shimmy with consideration of the coupling effects of the motions of vehicle body for a vehicle with double-wishbone independent front suspensions is shown in Figure1. In line with the law of right-hand coordinate system, the coordinate axes ox, oy, and oz point to the front, the left, and the top of the vehicle, respectively.Figure 1
Mechanical model of vehicle shimmy with consideration of vehicle body. (a) Three-dimensional view. (b) Vertical view of steering system.To facilitate dynamic analysis and mathematical modeling, the following assumptions are made to establish a dynamics model for vehicle shimmy.(1)
The steering wheel is immobile(2)
The effect of air resistance is neglected(3)
The plane of steering trapezium is parallel to the horizontal planexoy(4)
The vehicle keeps moving at a constant velocity, and without cornering(5)
The effect of the caster angle is considered, while other alignment parameters are neglected(6)
Vehicle structure is bilaterally symmetrical(7)
The central lines of the spring of rear suspension are perpendicular to the horizontal plane, and the vertical height of front and the rear suspensions are the same(8)
The mass other than wheels and vehicle body will not be consideredParameters related to the structure size of the vehicle are marked on the mechanical model of vehicle shimmy, as shown in Figure1. In which, Wf and Wr are half of the distances between two connection points where the dampers of the front and rear suspensions connect with the vehicle body, respectively; Lf and Lr are the distances from vehicle body’s CoG to the connection line of two connection points where the dampers of front and rear suspensions connect with the vehicle body, respectively; la is the horizontal distance of connection points between the vehicle body and suspension arm, suspension spring; lb is the distance from the intersection point of the kingpin extension line and the ground to the symmetry plane of the wheel; lc is the horizontal distance between the two ends of the suspension spring; ld is the arm of force of steering rod acting on kingpin; lf is the lateral swing arm of front suspension; lg is the arm of force of steering tie rod acting on pitman arm; lh is the vertical height between the two ends of suspension spring, namely, the vertical height of front suspension springs and rear suspension springs.


## 2.2. Lagrange Equation

According to the mechanical model of vehicle shimmy with consideration of the coupling effects of the motions of the vehicle body, its mathematical model can be established. This shimmy model has nine DoFs:θ1,2 are the angles of front wheels swing that revolve around their respective kingpin (namely, the shimmy angles), θ3 is the angle of pitman arm’s swing in steering system, φ1,2 are the angles of front wheel axes’ lateral swing that revolve around their respective lateral swing centers which are parallel to the x axis, θr,p,ω are the roll, pitch, and yaw angle of vehicle body respectively, and z is the displacement of vehicle body along the z axis.The mathematical model of vehicle shimmy can be established based on the Lagrange equations which are expressed as(1)ddt∂T∂q˙i−∂T∂qi+∂U∂qi+∂D∂q˙i=Qi,i=1,2,3,…,9.where T represents the kinetic energy of the system, U represents the potential energy of the system, D represents the dissipative potential function of the system, Qi represents the generalized force to which each DoF of the system is subjected, and qi represents the generalized coordinate of the system, which is expressed as q1q2⋯q9=θ1θ2θ3φ1φ2θrθpθωz.The kinetic energy of the system can be given by(2)T=12Jdφ˙12+12J0vR−φ˙1θ12+12Jdθ˙1−φ˙1γ2+12mωv−lbθ˙12+12mωlfφ˙1−lbθ˙1γ2+12Jdφ˙22+12J0vR−φ˙2θ22+12Jdθ˙2−φ˙2γ2+12mωv−lbθ˙22+12mωlfφ˙2−lbθ˙2γ2+12J3θ˙32+12msv2+12msz˙2+12Jrθ˙r2+12Jpθ˙p2+12Jωθ˙ω2,where J0 and Jd are the moments of inertia of front wheels around spin axes and diameters, respectively; J3 is the moment of inertia of the pitman arm; Jr, Jp, and Jω are the moments of inertia of vehicle body around the x, y, and z axis, respectively; mw is the mass of wheels, and ms is the mass of vehicle body; γ is the caster angle of front wheels; R is the rolling radius of tires; and v is the vehicle velocity.The potential energy of the system is given by(3)U=12k1ldθ1−lgθ32+12k2lgθ3−ldθ22+12k3θ32+12k4Δfl2+Δfr2+12k5Δrl2+Δrr2+12kblfφ1−lbθ1γ2+12kblfφ2−lbθ2γ2+12kyRφ1−Rθ1γ2+12kyRφ2−Rθ2γ2,where k1 and k2 are the stiffness of left and right steering tie rods, respectively; k3 is the stiffness of pitman arm; k4 and k5 are the stiffness of front and rear suspensions, respectively; ky is tire lateral stiffness; kb is tire vertical stiffness; Δfl, Δfr, Δrl, and Δrr are the deformations of the front-left, front-right, rear-left, and rear-right suspensions, respectively, which are given by equation (8).The dissipative potential function is given by(4)D=12ceθ˙12+θ˙22+12c1ldθ˙1−lgθ˙32+12c2lgθ˙3−ldθ˙22+12c3θ˙32+12c4Δ˙fl2+Δ˙fr2+12c5Δ˙rl2+Δ˙rr2+fkblfφ1lbθ˙1+fkblfφ2lbθ˙2,where ce is the equivalent damping of front wheels around their kingpins; c1 and c2 are the damping of left and right steering tie rods, respectively; c3 is the damping of pitman arm; c4 and c5 are the damping of front and rear suspensions, respectively; and f is the friction coefficient between the tires and the ground.The generalized force in the system is expressed as(5)Qi=FY1∂∂q˙i−Rγ+eθ˙1+Rφ˙1+FY2∂∂q˙i−Rγ+eθ˙2+Rφ˙2.Each generalized force corresponding to the generalized coordinate in the system can be given by(6)Q1=−FY1Rγ+e,Q2=−FY2Rγ+e,Q3=0,Q4=FY1R,Q5=FY2R,Q6=0,Q7=0,Q8=0,Q9=0,where FY1 and FY2 are the lateral forces of front-left wheel and front-right wheel, respectively, and e is the pneumatic trail of tire.The displacement caused by angle change of each DoF is nonlinear. To simplify the model and calculation, assume the angles in equations (2)–(6) are small; thus, equation sinθ=θ holds, where θ∈θ1θ2θ3φ1φ2θrθpθω.


## 2.3. Calculation of Suspension Deformation

The diagram used to calculate the deformation of vehicle suspensions when shimmy occurs is shown in Figure2. Thus, the deformation of vehicle suspensions Δfl, Δfr, Δrl, and Δrr can be deduced by(7)Δfl=lsfl′−lsfl=lh−lacφ1+Wfθr−Lfθp+z2+lc+msinθa−msinθa+θω2+mcosθa−mcosθa+θω2−lh2+lc2,Δfr=lsfr′−lsfr=lh+lacφ1−Wfθr−Lfθp+z2+lc+msinθa−msinθa−θω2+mcosθa−mcosθa−θω2−lh2+lc2,Δrl=lsrl′−lsrl=lh+Wrθr+Lrθp+z2+nsinθb−nsinθb−θω2+ncosθb−ncosθb−θω2−lh,Δrr=lsrr′−lsrr=lh−Wrθr+Lrθp+z2+nsinθb−nsinθb+θω2+ncosθb−ncosθb+θω2−lh,where lsfl, lsfr, lsrl, and lsrr are the original lengths of front-left, front-right, rear-left, and rear-right suspension springs; lsfl′, lsfr′, lsfl′, and lsrr′ are the lengths of springs after deformation corresponding to lsfl, lsfr, lsrl, and lsrr; m and n are the distances from vehicle body’s CoG to the connection points between front suspensions and vehicle body and between rear suspensions and vehicle body; θa and θb are the angles between line m and x axis and between line n and x axis; θs is the inclination angle of front suspension, which is defined as the angle between the central line of spring in front suspension and the horizontal plane; and make lac=la+lc. These parameters are all marked on Figure 2.Figure 2
Diagram for calculating the deformation of front and rear suspensions when vehicle shimmy occurs.It can be seen clearly from equation (7) that the deformation of suspensions is nonlinear. Based on the above small angle change hypothesis, the linearized formulas for calculating the deformation of four vehicle suspensions when shimmy occurs are given as(8)Δfl=−khclacφ1+khcWfθr−khcLfθp−kccmcosθaθω+khcz,Δfr=khclacφ2−khcWfθr−khcLfθp+kccmcosθaθω+khcz,Δrl=Wrθr+Lrθp+z,Δrr=−Wrθr+Lrθp+z,where, khc and kcc can be given by(9)khc=lhlh2+lc2,kcc=lclh2+lc2.


## 2.4. Mathematical Model of Vehicle Shimmy with Consideration of the Motions of Vehicle Body

Based on the above analysis, the 9-DoF mathematical model of shimmy with consideration of the coupling effects of the motions of the vehicle body for a vehicle with double-wishbone independent front suspensions can be available, which is listed in equation (13).In equation (13), the equivalent moment of inertia Jα, Jβ, and Jγ are given by(10)Jα=Jd+mwlb2+mwlb2γ2,Jβ=Jd+Jdγ2+mwlf2,Jγ=Jd+mwlblfγ.The seventh and the eighth terms of the first formula in equation (13) are caused by the lateral swing angle of front-left wheel axis φ1, and they are torques acting on shimmy angle θ1. Because the wheel rotates around its spin axis, when the wheel axis revolves around another axis, it is similar to a high-speed gyroscope rotating around its spin axis. Thus, the seventh term is the coupling torque caused by the elastic characteristics of the tire ky and kb, and the eighth term is the gyroscopic moment generated by the rotation of front-left wheel. It shows that there is coupling effect between the shimmy of front-left wheel around its kingpin and the lateral swing of front-left wheel axis around its lateral swing center. The same coupling effect also exists in the shimmy of front-right wheel and the lateral swing of front-right wheel axis.The above conclusions can also be inferred from the first, second, fourth, and fifth formulas in equation (13), and another conclusion can be drawn from the sixth to the ninth formula in equation (13) that there are coupling effects between the motions of the vehicle body θr,p,ω and z, and the motion of front wheel axes’ lateral swing φ1,2. Therefore, there are coupling effects between the shimmy of the front wheels and the motions of the vehicle body, and these coupling effects will be studied later.


## 2.5. Mathematical Model of Vehicle Shimmy Ignoring the Motions of Vehicle Body

Ignoring the coupling effects of the motions of vehicle body on shimmy, it can be assumed that the body is fixed. Thus, there is no displacement in each DoF of the vehicle body, that is to say, equationsθr = 0, θp = 0, θω = 0, and z = 0 hold. Substituting them into equation (13), the 5-DoF mathematical model of shimmy, including the angles of front wheels’ shimmy θ1,2, the angle of pitman arm’s swing θ3, and the angles of front wheel axes’ lateral swing φ1,2, can be obtained, and its differential equation is listed in equation (14).


## 2.6. Tire Model and Tire Rolling Constraint Equation

Vehicle self-excited shimmy is a nonlinear dynamic bifurcation phenomenon, and it is one of the key factors that precisely expresses the cornering characteristics of the tire to accurately and effectively analyze the shimmy of the front wheels. As mentioned above, the nonlinear tire models that are commonly used in the simulation of vehicle dynamics mainly include Gim’s tire models, Pacejka’s Magic Formula, and Guokonghui’s semiempirical tire theoretical model. The Magic Formula was chosen as the tire model in this study [19], and the lateral force of the tire can be given as(11)FY=DsinCarctanBα−SH1−E+EarctanBα−SH+SV,where α is the side-slip angle of wheel, B is the stiffness factor, C is the shape factor, D is the peak value, E is the curvature factor, SH is the horizontal shift, and SV is the vertical shift, respectively. Here, we take SH=0 and SV=0, and B, C, D, E can be solved by(12)C=a0,D=a1Fz2+a2Fz,BCD=a3sin2arctanFza4×1−a5β,E=a6Fz+a7,(13)Jαθ¨1−Jγφ¨1+k1ld2+kyR2γ2+kblb2γ2θ1+ce+c1ld2θ˙1−k1ldlgθ3−c1ldlgθ˙3−kyR2γ+kblblfγ−fφ1+J0vRφ˙1=FY1Rγ+e,Jαθ¨2−Jγφ¨2+k2ld2+kyR2γ2+kblb2γ2θ2+ce+c2ld2θ˙2−k2ldlgθ3−c2ldlgθ˙3−kyR2γ+kblblfγ−fφ2+J0vRφ˙2=FY2Rγ+e,J3θ¨3−k1ldlgθ1−c1ldlgθ˙1−k2ldlgθ2−c2ldlgθ˙2+k1lg2+k2lg2+k3θ3+c1lg2+c4lg2+c5θ˙3=0,Jβφ¨1−Jγθ¨1−kyR2+kblblfγθ1−J0vRθ˙1+k4khc2lac2+kyR2+kblf2φ1+c4khc2lac2φ˙1−k4khc2lacWfθr−c4khc2lacWfθ˙r+k4khc2lacLfθp+c4khc2lacLfθ˙p+k4khckcclacmcosθaθω+c4khckcclacmcosθaθ˙ω−k4khc2lacz−c4khc2lacz˙=−FY1R,Jβφ¨2−Jγθ¨2−kyR2+kblblfγθ2−J0vRθ˙2+k4khc2lac2+kyR2+kblf2φ2+c4khc2lac2φ˙2−k4khc2lacWfθr−c4khc2lacWfθ˙r−k4khc2lacLfθp−c4khc2lacLfθ˙p+k4khckcclacmcosθaθω+c4khckcclacmcosθaθ˙ω+k4khc2lacz+c4khc2lacz˙=−FY2R,Jrθ¨r−k4khc2lacWfφ1−c4khc2lacWfφ˙1−k4khc2lacWfφ2−c4khc2lacWfφ˙2+2k4khc2Wf2+k5Wr2θr+2c4khc2Wf2+c5Wr2θ˙r−2k4khckccWfmcosθaθω−2c4khckccWfmcosθaθ˙ω=0,Jpθ¨p+k4khc2lacLfφ1+c4khc2lacLfφ˙1−k4khc2lacLfφ2−c4khc2lacLfφ˙2+2k4khc2Lf2+k5Lr2θp+2c4khc2Lf2+c5Lr2θ˙p−2k4khc2Lf−k5Lrz−2c4khc2Lf−c5Lrz˙=0,Jωθ¨ω+k4kcckhclacmcosθaφ1+c4kcckhclacmcosθaφ˙1+k4khckcclacmcosθaφ2+c4khckcclacmcosθaφ˙2−2k4khckccWfmcosθaθr−2c4khckccWfmcosθaθ˙r+2k4kcc2m2cos2θaθω+2c4kcc2m2cos2θaθ˙ω=0,msz¨−k4khc2lacφ1−c4khc2lacφ˙1+k4khc2lacφ2+c4khc2lacφ˙2−2k4khc2Lf−k5Lrθp−2c4khc2Lf−c5Lrθ˙p+2k4khc2+k5z+2c4khc2+c5z˙=0,(14)Jαθ¨1−Jγφ¨1+k1ld2+kyR2γ2+kblb2γ2θ1+ce+c1ld2θ˙1−k1ldlgθ3−c1ldlgθ˙3−kyR2γ+kblblfγ−fφ1+J0vRφ˙1=FY1Rγ+e,Jαθ¨2−Jγφ¨2+k2ld2+kyR2γ2+kblb2γ2θ2+ce+c2ld2θ˙2−k2ldlgθ3−c2ldlgθ˙3−kyR2γ+kblblfγ−fφ2+J0vRφ˙2=FY2Rγ+e,J3θ¨3−k1ldlgθ1−c1ldlgθ˙1−k2ldlgθ2−c2ldlgθ˙2+k1lg2+k2lg2+k3θ3+c1lg2+c4lg2+c5θ˙3=0,Jβφ¨1−Jγθ¨1−kyR2+kblblfγθ1−J0vRθ˙1+k4khc2lac2+kyR2+kblf2φ1+c4khc2lac2φ˙1=−FY1R,Jβφ¨2−Jγθ¨2−kyR2+kblblfγθ2−J0vRθ˙2+k4khc2lac2+kyR2+kblf2φ2+c4khc2lac2φ˙2=−FY2R,where Fz is the normal load acting on wheels, β is the camber angle of wheel, and the parameters a0,a1,…,a7 are all constants that have to be determined for each tire. In this study, the value of β is 0, and the values of a0,a1,…,a7 are listed in Table 1 (for choice of parameters, see Pacejka et al. [8, 19]).Table 1
Values of the coefficients of Magic Formula.
Parameter
Value
a
0
1.65000
a
1
1250.00
a
2
12.80
a
3
−0.02103
a
4
−34.0
a
5
3036.00
a
6
0.00501
a
7
0.77394It is assumed that the vehicle has no lateral acceleration; the normal load acting on the left wheelFz1 and the normal load acting on the right wheel Fz2 are given as(15)Fz1=Fz0−kblfφ1,Fz2=Fz0+kblfφ2,where Fz0 is the static normal load acting on front-left and front-right wheels, which is given by(16)Fz0=12Lr′Lf′+Lr′ms+4mwg,where Lf′ and Lr′ are the distances from the CoG of the vehicle to its front axle and rear axle, respectively.The constraint of the side-slip angle and the shimmy angle of front wheel can be given as(17)α˙1+vσα1+vσθ1−aσθ˙1=0,α˙2+vσα2+vσθ2−aσθ˙2=0,where σ is the relaxation length of tires and a is the half-length of the tire contact area. In this study, σ is 0.65 m and a is 0.2 m.


## 3. Existence Analysis of Hopf Bifurcation

According to equations (10)–(13) and equations (15)–(17), the state equation of the 9-DoF system with consideration of the motions of the vehicle body can be written as follows:(18)x˙=fδ,v,x,where x∈R [19] is the state vector of the 9-DoF system and δ is the parameter of the 9-DoF system. It is obvious that the system represented by equation (18) is a nonlinear autonomous system.Makingx0 = 0 ∈ R [19] and substituting it into equation (13), the following equation can be derived.(19)x˙i=fiδ,v,x0=0,i=1,2,…,20.Consequently,x0 is one of the equilibrium points of the system represented by equation (18).Whenx varies in the vicinity of x0, the nonlinear differential equation (18) can be rewritten as following.(20)x˙=Aδ,vx+gδ,v,x.In which,Aδ,v is the Jacobian matrix of equation (18) in the equilibrium point x0 and gδ,v,x is the infinitesimal of higher order. Therefore, Aδ,v and gδ,v,x are the linear part and the nonlinear part of equation (18), respectively.The Jacobian matrixAδ,v can be solved by(21)Aδ,vi×j=∂fi∂xjx=x0,i,j=1,2,…,20.The Jacobian matrixAδ,v is a function of δ and v; thus, if δ and v are fixed, Aδ,v can be written as a constant matrix A.In the same way, according to equations (10)–(12) and equations (14)–(17), the state equation of the 5-DoF system ignoring the motions of body can be written as the sum of linear part and nonlinear part, as follows:(22)x˙′=A′δ′,vx′+g′δ′,v,x′.In which,x′∈R [11] is the state vector of the 5-DoF system; δ′ is the parameters of 5-DoF system; A′δ′,v is the Jacobian matrix of 5-DoF system state equation in the equilibrium point x0′; and nonlinear part g′δ′,v,x′ is the infinitesimal of higher order.According to the nonlinear system theory, the Jacobian matrix of the nonlinear system in the equilibrium point can be used to analyze the stability of the system. It can be seen from equation (21) that the matrix A varies with velocity v; thus, the eigenvalues of the Jacobian matrix vary with vehicle velocity. For clearly illustrating the relationships between the eigenvalues of the Jacobian matrix and vehicle velocity, the eigenvalue loci of the Jacobian matrix A varying with vehicle velocity v are shown in Figure 3, and the eigenvalue loci of the Jacobian matrix A′ of the 5-DoF system varying with vehicle velocity v are also drawn on.Figure 3
Eigenvalue loci of 9-DoF shimmy and 5-DoF shimmy.It is obvious that the eigenvalue loci of the system are symmetrical on the real axis. The Jacobian matrixA of the 9-DoF system has 20 curves of eigenvalue locus, and the Jacobian matrix A′ of the 5-DoF system has 12 curves of eigenvalue locus. With the increase of v from 0 m/s to 40 m/s, eight curves of eigenvalue locus of both systems vary obviously, among which, four eigenvalue loci pass through the imaginary axis from the left half of the complex plane to the right, and then come back to the left again, but the rest nearly have no change, which assemble into a point on the complex plane. The red points in the ellipse in Figure 3 are eigenvalue loci which relate to the vehicle body. It can be seen from Figure 3 that the eigenvalue loci of matrix A and A′ have the same trend when vehicle velocity v increases, in addition to few differences.Here, the Jacobian matrixA of the 9-DoF nonlinear system is used as an example to study the Hopf bifurcation characteristics of vehicle shimmy. The velocity at which the Jacobian matrix A has two conjugate purely imaginary eigenvalues, and the real parts of other eigenvalues are all negative, is called the critical velocity of the system. Figure 3 shows the system has two critical velocities v1 and v2 (v1<v2). When v<v1 or v>v2, the real parts of all eigenvalues in matrix A are negative, and the system is stable asymptotically. When v1<v<v2, matrix A has positive real part eigenvalues, and the original system has self-excited oscillation and produces limit cycles. When v varies at the vicinity of critical velocity, the eigenvalue loci of the Jacobian matrix A pass through the imaginary axis of the complex plane, so that the stability of vehicle shimmy changes. While vehicle velocity v increases or decreases to the critical velocity, that is, when the eigenvalue loci of the system pass through the imaginary axis, the nonlinear system (18) will produce the Hopf bifurcation, and the critical velocity is the bifurcation point. Thus, shimmy occurs only when the Jacobian matrix A of the system has positive real part eigenvalues, and it is obvious that vehicle shimmy occurs in the moderate velocity range. The 5-DoF system has the similar Hopf bifurcation characteristic as the 9-DoF system, and its Hopf bifurcation characteristic and stability can also be analyzed in the same way.


## 4. Numerical Simulation

Based on the mathematic model of vehicle shimmy, the dynamic response of each DoF can be discussed by means of numerical simulation, and the coupling effects between the motions of the vehicle body and the shimmy of front wheels will be studied. It is assumed that a 0.01 rad angular displacement is exerted on the front-left wheel as initial excitation. Based on equations (10)–(17) and the parameters listed in Tables 1 and 2 (for choice of parameters in Table 2, see Wei et al. [14] and Mi et al. [18]), numerical simulation can be carried out. The dynamic responses of front wheels’ shimmy, pitman arm’s swing, and front wheel axes’ lateral swing in two systems at the velocity of 10 m/s are shown as Figures 4–6, where the time history response and steady-state phase portrait of each DoF are drawn on. It is assumed that the time is t = 0 s when the initial excitation is exerted.Table 2
Values of the parameters of vehicle shimmy.
Parameter
Value
J
0
8 (kg·m2)
J
d
6 (kg·m2)
J
3
3 (kg·m2)
m
w
60 (kg)
m
s
1248 (kg)
J
r
325.52 (kg·m2)
J
p
1229.82 (kg·m2)
J
ω
1255.82 (kg·m2)
L
f
′
1.220 (m)
L
r
′
1.473 (m)
L
f
1.220 (m)
L
r
1.473 (m)
W
f
0.370 (m)
W
r
0.440 (m)
k
1
2240 (kN·m−1)
k
2
2240 (kN·m−1)
k
3
60 (kN·m−1)
k
4
200 (kN·m−1)
k
5
220 (kN·m−1)
c
1
630 (N·s·m−1)
c
2
630 (N·s·m−1)
c
3
100 (N·s·m−1)
c
4
6560 (N·s·m−1)
c
5
7120 (N·s·m−1)
c
e
44 (N·s·m)
k
y
68 (kN·m−1)
k
b
360 (kN·m−1)
F
0.015
γ
0.06 (rad)
R
0.40 (m)
E
0.07 (m)
l
a
0.140 (m)
l
b
0.200 (m)
l
c
0.160 (m)
l
d
0.126 (m)
l
f
0.612 (m)
l
g
0.100 (m)
l
h
0.400 (m)Figure 4
Dynamic responses of front wheels’ shimmy and pitman arm’s swing. (a) Front-left wheel. (b) Front-right wheel. (c) Pitman arm.
(a)
(b)
(c)Figure 5
Dynamic responses of front wheel axes’ lateral swing. (a) Front-left wheel axis. (b) Front-right wheel axis.
(a)
(b)Figure 6
Dynamic responses of the motions of vehicle body. (a) Roll motion. (b) Pitch motion. (c) Yaw motion. (d) Vertical motion.
(a)
(b)
(c)
(d)The dynamic responses of front wheels’ shimmy and pitman arm’s swing are shown in Figure4. It shows that the shimmy of front wheels and the swing of pitman arm step into the steady state at about 3 s, and their steady-state phase portraits are definitely limit cycles. The amplitudes of front wheels’ shimmy in the steady state are far beyond the initial excitation which is exerted on it.The dynamic responses of front wheel axes’ lateral swing are shown in Figure5. Because of the coupling effects, the lateral swing of front wheel axes gradually steps into stable oscillation with the shimmy of the front wheels at the same time, but their phase portraits are not symmetrical. There are little deviations between the equilibrium positions of the lateral swing and their corresponding original positions, and their equilibrium positions are drawn on with a green line.Due to the coupling effects, the body vibrates with the shimmy of the front wheels. The dynamic responses of the motions of the vehicle body are shown in Figure6. The dynamic courses of the pitch and vertical motions of vehicle body show as violent vibrations. Similar to the lateral swing of front wheel axes, there are deviations between the equilibrium positions of the pitch and vertical motions and their corresponding original positions when the vehicle body vibrates, and their equilibrium positions are drawn on with a green line. Moreover, it can be seen from the sixth and the eighth formulas in equation (13) that there is coupling effect between the roll and yaw motions, and it also can be seen from the seventh and the ninth formulas in equation (13) that the coupling effect between pitch motion and vertical motion exists. Therefore, as shown in Figure 6, the roll motion has similar dynamic behavior as the yaw motion, and the pitch motion has similar dynamic behavior as the vertical motion.Figures4 and 5 show that the dynamic behavior of each DoF in the above two systems are similar, but the amplitude of each DoF in a 9-DoF system is bigger than that in a 5-DoF system when v = 10 m/s.


## 5. Influences of Vehicle Parameters on Vehicle Shimmy

The dynamic behavior of vehicle shimmy depends on vehicle parameters and velocity, thus the relationship between the amplitude of each DoF and vehicle velocity, the influences of vehicle parameters such as the mass of vehicle body, the longitudinal position of vehicle body’s CoG and the inclination angle of front suspension on the amplitude of each DoF, and the influences of theses parameters on velocity range of shimmy are studied.


### 5.1. Relationship between the Amplitude of Each DoF and Vehicle Velocity

The relationship between the amplitude of each DoF in a 9-DoF system and a 5-DoF system and vehicle velocity is studied, and the results are shown in Figure7. Conclusions can be drawn as follows:(1)
The curves around which the amplitudes of front wheels’ shimmy and pitman arm’s swing vary with vehicle velocity are arched, which is consistent with the conclusion in the study of Lu et al. [17] and Mi et al. [18].(2)
The curves around which the amplitudes of front wheel axes’ lateral swing and the motions of the vehicle body vary with vehicle velocity present saddle shape whose local maximum is found in the high-velocity region and the low-velocity region, and the local minimum is found in the medium-velocity region.(3)
If vehicle velocity is in the medium-velocity region, the amplitudes of front wheels’ shimmy and pitman arm’s swing are relatively big, and the amplitudes of other DoFs are relatively small.(4)
The amplitude of each DoF in the 9-DoF system is bigger than that in the 5-DoF system in the low-velocity region and smaller in the high-velocity region.(5)
The critical velocities of the 9-DoF system are all smaller than those of the 5-DoF system, thus its velocity range of shimmy moves to the low-velocity region.Figure 7
Relationship between the amplitude of each DoF and vehicle velocity. (a) The amplitudes ofθ1,2,3. (b) The amplitudes of φ1,2. (c) The amplitudes of θr,p,ω, and z.
(a)
(b)
(c)


### 5.2. Influences of the Mass of Vehicle Body

Since the forces acting on suspensions and tires vary withms, the dynamic response of each DoF of vehicle shimmy is likely to be affected by it. So the influences of ms on the amplitude of each DoF are studied when vehicle shimmies, and results are shown in Figure 8. Conclusions can be made as follows:(1)
Ifms is big enough, the amplitudes of front wheels’ shimmy, pitman arm’s swing and front wheel axes’ lateral swing all increase with the increment of ms, and the increments are obviously big.(2)
Ifms is relatively small, the amplitudes of the motions of vehicle body increase with the increment of ms, But when ms is relatively big, the influences of the increment of ms on the amplitudes of the motions of vehicle body are not distinct.(3)
Ifms is small, the normal loads acting on front wheels are too small to produce enough torque acting on the kingpin to keep vehicle shimmy.Figure 8
Influences ofms on the amplitude of each DoF. (a) The amplitudes of θ1,2,3. (b) The amplitudes of φ1,2. (c) The amplitudes of θr,p,ω, and z.
(a)
(b)
(c)


### 5.3. Influences of the Longitudinal Position of Vehicle Bodyʼs CoG

It is assumed that the CoG of the vehicle body coincides with the CoG of the whole vehicle, then equationsLf=Lf′ and Lr=Lr′ hold, and the ratio of Lf and Lr (Lf : Lr) corresponds to the longitudinal position of the vehicle body’s CoG. When the vehicle body’s CoG moves backward, Lf increases and Lr decreases, and the value of Lf : Lr increases. The influences of the longitudinal position of vehicle body’s CoG on the amplitude of each DoF are studied when vehicle shimmies, and results are shown in Figure 9. When the vehicle body’s CoG moves backward, conclusions can be drawn as follows:(1)
The amplitudes of front wheels’ shimmy, pitman arm’s swing, and front wheel axes’ lateral swing all gradually decrease, and the decrement is relatively large.(2)
The amplitudes of the motions of the vehicle body gradually decrease. The decrement of the amplitudes of the roll motion and the vertical motion are relatively bigger, but the decrements of the amplitudes of the pitch motion and the yaw motion are relatively smaller.(3)
The normal load acting on the front wheels decreases. Thus, the torque acting on the kingpin decreases, and the amplitude of front wheels’ shimmy decreases. Because of the coupling effects, the amplitudes of pitman arm’s swing and front wheel axes’ lateral swing and the motions of vehicle body decrease.Figure 9
Influences ofLf : Lr on the amplitude of each DoF. (a) The amplitudes of θ1,2,3. (b) The amplitudes of φ1,2. (c) The amplitudes of θr,p,ω, and z.
(a)
(b)
(c)


### 5.4. Influences of the Inclination Angles of Front Suspensions

The forces acting on the vehicle body and front suspensions vary withθs, so its influences on the amplitude of each DoF are studied when vehicle shimmies, and the results are shown in Figure 10. When θs increases, the following conclusions can be drawn:(1)
The amplitudes of front wheels’ shimmy and pitman arm’s swing all increase. But the increments of the amplitudes are small, and nearly cannot be seen.(2)
The amplitude of front wheel axes’ lateral swing gradually decreases, but the decrement is small, and it is not obvious.(3)
The amplitudes of the roll motion, the pitch motion, and the vertical motion of the vehicle body increase, and the amplitude of the yaw motion of the vehicle body decreases to zero finally, but the increments of the pitch motion and the vertical motion are very small, and nearly cannot be seen.(4)
The normal forces acting on the front wheels have little changes, thus they have little influences on the amplitudes of front wheels’ shimmy, pitman arm’s swing, and front wheel axes’ lateral swing.(5)
The vertical component force acting on vehicle body increases and the lateral component force decreases. Thus, the amplitude of pitch motion increases and the amplitude of the yaw motion decreases. But the amplitudes of the pitch motion and the vertical motion have little changes.Figure 10
Influences ofθs on the amplitude of each DoF. (a) The amplitudes of θ1,2,3. (b) The amplitudes of φ1,2. (c) The amplitudes of θr,p,ω, and z.
(a)
(b)
(c)


### 5.5. Influence on Velocity Range of Shimmy

As analyzed above, parameters such asms, the longitudinal position of vehicle body’s CoG and θs influence the amplitude of each DoF of vehicle shimmy, thus the velocity range of shimmy must be influenced by these parameters. The influences of parameters on the velocity range of shimmy can be seen in Figure 11. Conclusions can be made as follows:(1)
The velocity range of shimmy will be enlarged with the increasing ofms, in other words, if ms increases, v1 will decrease and v2 will increase.(2)
The velocity range of shimmy will be narrowed if vehicle body’s CoG moves back.(3)
The effect ofθs on velocity range of shimmy is insignificant, and it nearly cannot be seen.Figure 11
Influences of vehicle parameters on the critical velocity of vehicle shimmy. (a) Influences ofms. (b) Influences of Lf : Lr. (c) Influences of θs.
(a)
(b)
(c)


## 5.1. Relationship between the Amplitude of Each DoF and Vehicle Velocity

The relationship between the amplitude of each DoF in a 9-DoF system and a 5-DoF system and vehicle velocity is studied, and the results are shown in Figure7. Conclusions can be drawn as follows:(1)
The curves around which the amplitudes of front wheels’ shimmy and pitman arm’s swing vary with vehicle velocity are arched, which is consistent with the conclusion in the study of Lu et al. [17] and Mi et al. [18].(2)
The curves around which the amplitudes of front wheel axes’ lateral swing and the motions of the vehicle body vary with vehicle velocity present saddle shape whose local maximum is found in the high-velocity region and the low-velocity region, and the local minimum is found in the medium-velocity region.(3)
If vehicle velocity is in the medium-velocity region, the amplitudes of front wheels’ shimmy and pitman arm’s swing are relatively big, and the amplitudes of other DoFs are relatively small.(4)
The amplitude of each DoF in the 9-DoF system is bigger than that in the 5-DoF system in the low-velocity region and smaller in the high-velocity region.(5)
The critical velocities of the 9-DoF system are all smaller than those of the 5-DoF system, thus its velocity range of shimmy moves to the low-velocity region.Figure 7
Relationship between the amplitude of each DoF and vehicle velocity. (a) The amplitudes ofθ1,2,3. (b) The amplitudes of φ1,2. (c) The amplitudes of θr,p,ω, and z.
(a)
(b)
(c)


## 5.2. Influences of the Mass of Vehicle Body

Since the forces acting on suspensions and tires vary withms, the dynamic response of each DoF of vehicle shimmy is likely to be affected by it. So the influences of ms on the amplitude of each DoF are studied when vehicle shimmies, and results are shown in Figure 8. Conclusions can be made as follows:(1)
Ifms is big enough, the amplitudes of front wheels’ shimmy, pitman arm’s swing and front wheel axes’ lateral swing all increase with the increment of ms, and the increments are obviously big.(2)
Ifms is relatively small, the amplitudes of the motions of vehicle body increase with the increment of ms, But when ms is relatively big, the influences of the increment of ms on the amplitudes of the motions of vehicle body are not distinct.(3)
Ifms is small, the normal loads acting on front wheels are too small to produce enough torque acting on the kingpin to keep vehicle shimmy.Figure 8
Influences ofms on the amplitude of each DoF. (a) The amplitudes of θ1,2,3. (b) The amplitudes of φ1,2. (c) The amplitudes of θr,p,ω, and z.
(a)
(b)
(c)


## 5.3. Influences of the Longitudinal Position of Vehicle Bodyʼs CoG

It is assumed that the CoG of the vehicle body coincides with the CoG of the whole vehicle, then equationsLf=Lf′ and Lr=Lr′ hold, and the ratio of Lf and Lr (Lf : Lr) corresponds to the longitudinal position of the vehicle body’s CoG. When the vehicle body’s CoG moves backward, Lf increases and Lr decreases, and the value of Lf : Lr increases. The influences of the longitudinal position of vehicle body’s CoG on the amplitude of each DoF are studied when vehicle shimmies, and results are shown in Figure 9. When the vehicle body’s CoG moves backward, conclusions can be drawn as follows:(1)
The amplitudes of front wheels’ shimmy, pitman arm’s swing, and front wheel axes’ lateral swing all gradually decrease, and the decrement is relatively large.(2)
The amplitudes of the motions of the vehicle body gradually decrease. The decrement of the amplitudes of the roll motion and the vertical motion are relatively bigger, but the decrements of the amplitudes of the pitch motion and the yaw motion are relatively smaller.(3)
The normal load acting on the front wheels decreases. Thus, the torque acting on the kingpin decreases, and the amplitude of front wheels’ shimmy decreases. Because of the coupling effects, the amplitudes of pitman arm’s swing and front wheel axes’ lateral swing and the motions of vehicle body decrease.Figure 9
Influences ofLf : Lr on the amplitude of each DoF. (a) The amplitudes of θ1,2,3. (b) The amplitudes of φ1,2. (c) The amplitudes of θr,p,ω, and z.
(a)
(b)
(c)


## 5.4. Influences of the Inclination Angles of Front Suspensions

The forces acting on the vehicle body and front suspensions vary withθs, so its influences on the amplitude of each DoF are studied when vehicle shimmies, and the results are shown in Figure 10. When θs increases, the following conclusions can be drawn:(1)
The amplitudes of front wheels’ shimmy and pitman arm’s swing all increase. But the increments of the amplitudes are small, and nearly cannot be seen.(2)
The amplitude of front wheel axes’ lateral swing gradually decreases, but the decrement is small, and it is not obvious.(3)
The amplitudes of the roll motion, the pitch motion, and the vertical motion of the vehicle body increase, and the amplitude of the yaw motion of the vehicle body decreases to zero finally, but the increments of the pitch motion and the vertical motion are very small, and nearly cannot be seen.(4)
The normal forces acting on the front wheels have little changes, thus they have little influences on the amplitudes of front wheels’ shimmy, pitman arm’s swing, and front wheel axes’ lateral swing.(5)
The vertical component force acting on vehicle body increases and the lateral component force decreases. Thus, the amplitude of pitch motion increases and the amplitude of the yaw motion decreases. But the amplitudes of the pitch motion and the vertical motion have little changes.Figure 10
Influences ofθs on the amplitude of each DoF. (a) The amplitudes of θ1,2,3. (b) The amplitudes of φ1,2. (c) The amplitudes of θr,p,ω, and z.
(a)
(b)
(c)


## 5.5. Influence on Velocity Range of Shimmy

As analyzed above, parameters such asms, the longitudinal position of vehicle body’s CoG and θs influence the amplitude of each DoF of vehicle shimmy, thus the velocity range of shimmy must be influenced by these parameters. The influences of parameters on the velocity range of shimmy can be seen in Figure 11. Conclusions can be made as follows:(1)
The velocity range of shimmy will be enlarged with the increasing ofms, in other words, if ms increases, v1 will decrease and v2 will increase.(2)
The velocity range of shimmy will be narrowed if vehicle body’s CoG moves back.(3)
The effect ofθs on velocity range of shimmy is insignificant, and it nearly cannot be seen.Figure 11
Influences of vehicle parameters on the critical velocity of vehicle shimmy. (a) Influences ofms. (b) Influences of Lf : Lr. (c) Influences of θs.
(a)
(b)
(c)


## 6. Conclusion

The coupling effects between the motions of the vehicle body and the shimmy of the front wheels are usually ignored in previous research studies about vehicle shimmy. In this paper, a 9-DoF shimmy model of vehicle with consideration of the coupling effect of the motions of the vehicle body is established, and a 5-DoF shimmy model of vehicle ignoring these coupling effects is established according to the 9-DoF model. The dynamic behaviors of shimmy in two systems are compared. According to the above theoretical analyses and numerical simulation result, conclusions can be drawn as follows:(1)
If the effects of the motions of vehicle body on shimmy are take into account, the dynamic behaviors of front wheels, front wheel axes, and pitman arm are similar as that of ignoring the effects of the motions of the vehicle body, but the amplitude of each DoF in this system is bigger in low-velocity region and is smaller in high-velocity region.(2)
Because of the symmetry of vehicle mechanism, the shimmies of the front-left wheel and the front-right wheel and the lateral swings of the front-left wheel axis and the front-right wheel axis have the same dynamic performance.(3)
Because of the coupling effect, the roll motion and the yaw motion of vehicle body have similar dynamic behaviors, and the pitch motion and the vertical motion of vehicle body have similar dynamic behaviors.(4)
The amplitudes of the front wheels’ shimmy and the motions of the vehicle body are all affected by the vehicle velocity.(5)
The mass of the vehicle body and the longitudinal position of the vehicle body’s CoG have great influences on vehicle shimmy because they can change the normal forces acting on wheels.(6)
The inclination angle of front suspension has great influences on the roll motion and the yaw motion of vehicle body, but has little influences on other DoFs.(7)
Compared with the 5-DoF system, the velocity range of shimmy of the 9-DoF system moves to the low-velocity region, in other words, shimmy is more likely to occur in the low-speed region for a 9-DoF system.(8)
The mass of the vehicle body and the longitudinal position of vehicle body’s CoG have obvious influences on velocity range of shimmy, but the inclination angle of front suspension has little influence.



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*Source: 3707416-2019-03-24.xml*