Self-excited vibration of whole vehicle with multiple limit cycles induced by shimmy of front wheels

Abstract

Front wheel shimmy may induce the self-excited vibration of whole vehicle with multiple limit cycles. This kind of shimmy is difficult to control and can do great harm to the vehicle. In order to obtain the mechanism of this phenomenon, a 7-DOF dynamic model of whole vehicle self-excited vibration induced by the shimmy of front wheels which take the nonlinear factor of tire lateral force and the dry friction force in suspension and steering system into consideration was established. By using Hopf bifurcation theorem and central manifold theorem, the existence and stability of limit cycles are qualitatively determined. By means of numerical analysis, the self-excited vibration with multiple limit cycles is found. The results show that in the speed range of 50–185 km/h, front wheels shimmy induced the self-excited vibration of whole vehicle and the amplitude of system’s vibration decreases with the increase of vehicle speed; in the speed range of 50–83 km/h and 149–185 km/h, the self-excited vibration with multiple limit cycles occurs; in the speed range of 83–149 km/h, the self-excited vibration with single limit cycle occurs.

Keywords

Shimmy of front wheels multiple limit cycles Hopf bifurcation center manifold theorem self-excited vibration

Introduction

Shimmy of front wheels is the phenomenon of front wheels which continue to vibrate around the kingpin when the vehicle is running on a flat road. It will cause the car body to sway, resulting in whole vehicle self-excited vibration, and the vehicle wound appear serpentine trajectory. This will seriously affect the vehicle’s handling stability, ride comfort and safety. The self-excited vibration with multiple limit cycles means that the induced vibration’s amplitude may be different under different initial excitations.

Over the years, scholars from all over the world have conducted extensive and in-depth research on the shimmy of front wheels from a linear perspective, and have achieved rich results.^1–3

However, the vehicle is a multiple degree-of-freedom (DOF) nonlinear system, and its nonlinear dynamic behavior becomes more complicated as the vehicle speed increases. Therefore, in recent years, some scholars have studied the mechanism of shimmy of front wheels and its effect on motion stability of the vehicle from a nonlinear perspective. A three-DOF shimmy model of steering system in dependent suspension vehicle was established in literature,⁴ which took clearance and dry friction into consideration. The numerical calculations showed that both factors would induce front wheels shimmy system to experience multiple limit cycles vibration. A front wheel shimmy model of independent suspension vehicle was established in literature,⁵ which took steering clearance and dry friction into consideration. The numerical calculations showed that the clearance and dry friction induced the system to experience supercritical Hopf bifurcation. A front wheels shimmy model of independent suspension vehicle was established in literature,⁶ which took dry friction and nonlinear characteristics of the tires into consideration. The numerical calculations showed that dry friction induced the front wheels to experience shimmy with multiple limit cycles. Motion differential equations of motorcycle were established in literature,⁷ in which shimmy of front wheels and its effect on the motion stability of motorcycle were studied. Based on a certain type of trailer, a 24-DOF dynamic model was established in literature,⁸ in which the dynamic behavior of vehicle instability in a high speed was studied. Stability equations of motion of trailers were established in literature,⁹ in which the effect of system parameters on high-speed shimmy and handling stability of vehicle were studied. Equations of motion of semi-trailers were established in literature,¹⁰ in which the transverse self-excited vibration of semi-trailers and its effect on handling stability of vehicles were studied. Considering the shimmy of steering system, a 3-DOF dynamic equations of vehicle motion stability were established in literature,¹¹ in which the effect of steering system’s parameters on handling stability of vehicles were studied. The results show that the shimmy of front wheels induced vehicle system to experience a supercritical Hopf bifurcation behavior. But the author did not find the self-excited vibration with multiple limit cycles. We preliminarily studied the self-excited vibration and multiple limit cycles shimmy of front wheels in the early research,^12–14 but we did not study the self-excited vibration of whole vehicle.

In summary, the above research on shimmy can be divided into two categories. One is that dynamic model was confined to the shimmy system of front wheels and the shimmy system of whole vehicle was not studied. The other is that the dynamic model of whole vehicle shimmy system was established but the self-excited vibration with multiple limit cycles is not found, which may be related to the matching of system parameters and the selection of dry friction model. Based on the above deficiencies, this paper establishes a 7-DOF dynamic model of whole vehicle self-excited vibration induced by the shimmy of front wheels. The qualitative analysis method is used to determine the Hopf bifurcation of the system, and the numerical analysis method is used to find the multiple limit cycles of the system, which enriches the knowledge of dynamics of the vehicle shimmy system.

Establish the dynamic model of vehicle shimmy

Dynamic model of vehicle

Choose a certain vehicle as the sample vehicle. The vehicle’s front suspension is McPherson suspension. The steering system is a disconnected steering mechanism with rockers and it has steering gears of recirculating ball type, as shown in Figure 1(a).

Figure 1.

Mechanical model of vehicle steering. (a) Front suspension and steering mechanism; (b) planar motion and (c) roll motion.

The whole vehicle is simplified as a 7-DOF mechanical model, which is shown in Figure 1 (a) to (c). Seven degrees of freedom are, respectively, the left and right wheels swing angle around its kingpin θ₁ and θ₂, the intermediate rocker arm angle around its axis of rotation θ₃, the steering wheel angle θ₄, sideslip angle of centroid β, yaw speed ω, and the body roll angle ϕ. In the model, O₁ is barycenter and O₂ is roll center. The following assumptions were made before differential equations of vehicle self-excited shimmy system were established:

The steering system of the vehicle uses an off-ackermann steering linkage with an intermediate steering arm. The left and right trapezoidal arms, suspension and tire structures are axisymmetrical with respect to X-axis. Therefore, it is assumed that the rotational inertia, mass, stiffness, damping and other structural parameters of the left and right wheels and the left and right trapezoidal arms are the same, and the lateral forces of the left and right rear wheels are approximately equal.

Only the dry friction in the motion pair of the suspension and the steering system are considered, and the collision force in the motion pair clearance is ignored. Only the dry friction at the kingpin is considered and the dry friction in the motion pair of the suspension and the steering system are equivalent to be at the kingpin.

The vehicle’s front suspension is McPherson suspension. When the wheel is bouncing up and down, the displacement of the wheel camber freedom is small, and the corresponding gyroscopic effect is weak. Therefore, only the freedom of swinging around the kingpin is considered for the two front wheels.

The vehicle runs on a straight road at a constant velocity.

Differential equations of vehicle self-excited vibration system

Based on mechanical model in Figure 1(a) to (c) and the above-mentioned assumptions, differential equations of vibration of steering system and vehicle motion were established as follows by using Lagrange’s equation:

Shimmy equation of left wheel around kingpin

\begin{array}{l} J_{1} {\ddot{θ}}_{1} + k_{1} θ_{1} - k_{1} k_{α} θ_{3} + (c_{1 l} + c_{1}) {\dot{θ}}_{1} - c_{1} k_{α} {\dot{θ}}_{3} + (F_{z 1} - m_{1} g) [\sin α (l \cos γ - R \sin γ) \cos θ_{1} \\ + \sin^{2} α (l \sin γ + R \cos γ) \sin θ_{1}] + F_{y 1} [D_{1} \cos α + \sin α (l \sin γ + R \cos γ)] \\ + F_{x 1} (l \cos γ - R \cos γ) \cos α + M_{f 1} = 0 \end{array}

(1)

Shimmy equation of right wheel around kingpin

\begin{array}{l} J_{2} {\ddot{θ}}_{2} + k_{2} θ_{2} - k_{2} k_{α} θ_{3} + (c_{2 r} + c_{2}) {\dot{θ}}_{1} - c_{2} k_{α} {\dot{θ}}_{3} + (F_{z 2} - m_{2} g) [\sin α (l \cos γ - R \sin γ) \cos θ_{2} \\ + \sin^{2} α (l \sin γ + R \cos γ) \sin θ_{2}] + F_{y 2} [D_{1} \cos α + \sin α (l \sin γ + R \cos γ)] \\ + F_{x 2} (l \cos γ - R \cos γ) \cos α + M_{f 2} = 0 \end{array}

(2)

Shimmy equation of rocker

\begin{array}{l} J_{3} {\ddot{θ}}_{3} + (k_{2} k_{α}^{2} + k_{1} k_{α}^{2} + k_{3}) θ_{3} + k_{1} k_{α} θ_{1} - k_{2} k_{α} θ_{2} + (c_{1} k_{α}^{2} + c_{2} k_{α}^{2} + c {}_{3 y}+ c_{3}) {\dot{θ}}_{3} - c_{1} k_{α} {\dot{θ}}_{1} \\ - c_{2} k_{α} {\dot{θ}}_{2} - k_{3} k_{β} θ_{4} - c_{3} k_{β} {\dot{θ}}_{4} = 0 \end{array}

(3)

Shimmy equation of steering assembly

J_{4} {\ddot{θ}}_{4} + (k_{3} k_{β}^{2} + k_{4}) θ_{4} - k_{3} k_{β} θ_{3} - c_{3} k_{β} {\dot{θ}}_{3} + (c_{3} k_{β}^{2} + c {}_{4 z}+ c_{4}) {\dot{θ}}_{4} = 0

(4)

Vehicle sliding sideways equation

mu (\dot{β} + ω) - m_{s} h_{s} \ddot{ϕ} = F_{y 1} \cos θ_{1} + F_{y 2} \cos θ_{2} + 2 F_{yr}

(5)

Motion equation of vehicle yaw

I_{z} \dot{ω} = l_{1} F_{y 1} \cdot \cos θ_{1} + l_{1} F_{y 2} \cdot \cos θ_{2} - 2 l_{2} F_{yr}

(6)

Motion equation of vehicle roll

I_{x} \ddot{φ} + c_{φ} \dot{φ} + k_{φ} φ = m_{s} h_{s} u (\dot{β} + ω) + m_{s} g h_{s} φ

(7)

In these equations, J₁ and J₂ are moments of inertia of left and right front wheel assembly around the kingpin, respectively; J₃ is equivalent moment of inertia of intermediate rocker arm assembly about its axis of rotation; J₄ is equivalent moment of inertia of steering gear assembly converting to its input; F_z1 and F_z2 are perpendicular forces from the ground on the left and right front wheel; F_x1 and F_x2 are longitudinal forces from the ground on the left and right front wheel; F_y1 and F_y2 are transverse forces from the ground on the left and right front wheel; M_f1 and M_f2 are equivalent dry friction torques of left and right front wheel around the kingpin; c_1l and c_2r are equivalent angles damping coefficient of left and right front wheel assemblies around the kingpin; c₃ is the equivalent angle damping coefficient of intermediate rocker arm assembly about its axis of rotation; c₄ is the equivalent angle damping coefficient of steering gear assembly converting to its input; k₁ and k₂ are equivalent connection angular stiffness between left and right front wheel and intermediate rocker arm; k₃ is equivalent connection angular stiffness between intermediate rocker arm and steering gear; k₄ is equivalent connection angular stiffness between steering gear and steering wheel; c₁ and c₂ are equivalent connection angular damping coefficients between left and right front wheels and intermediate rocker arm; c_3y is equivalent connection angular damping coefficient between intermediate rocker arm and steering gear; c_4z is equivalent connection angular damping coefficient between steering gears and steering wheel; m₁g and m₂g are loads of left and right wheels; f is rolling resistance coefficient of the tire; γ is wheel camber; R is rolling radius of the wheel; α is inclination angle of kingpin; k_α is amplification coefficient between left and right front wheels and intermediate rocker arm; k_β is amplification coefficient between intermediate rocker arm and steering gears; D₁ is pneumatic trail; l is effective length of steering knuckle; l₁ is the front wheelbase; l₂ is the rear wheelbase; m is mass of the vehicle; u is vehicle speed; m_s is mass on sprung; δ_f rotary angle of front wheel; δ_r is rotary angle of rear wheel; k_p is scaling coefficient of rotary angle of front and rear wheels; F_yr is lateral force of- rear wheel; I_z is moment of inertia of the vehicle around the yaw axis(Z); I_x is moment of inertia of the vehicle around the roll axis(X); c_ϕ is Roll angle damping; h_s is the distance of centroid of mass on sprung to the roll axis and it is the difference between the centroid height of mass on sprung h and the height of the roll axis h_ϕ; and k_ϕ is angle stiffness of roll.

Dynamics model of tire

Vertical load model of tire

According to the elastic energy change when the tire has a radial deformation, perpendicular force from the ground on the left and right front wheel is deduced when steering system experiences shimmy.

{\begin{array}{l} F_{z 1} = m_{1} g - k_{5} (l \cos γ - R \sin γ) \sin γ \sin θ_{1} + k_{5} \sin^{2} α (l \sin γ + R \cos γ) (1 - \cos θ_{1}) \\ F_{z 2} = m_{2} g - k_{5} (l \cos γ - R \sin γ) \sin γ \sin θ_{2} + k_{5} \sin^{2} α (l \sin γ + R \cos γ) (1 - \cos θ_{2}) \end{array}

(8)

In the equation, k₅ is vertical stiffness of the tire; F_z1 is vertical load of left front wheel; F_z2 is vertical load of right front wheel. The rear vertical load is assumed approximately unchanged and it is defined as F_zr.

Selection of lateral force model of tire

Lateral force of tire is the main aspect affecting the vehicle’s transverse stability. Pacejka’s magic formula tire model has some advantages. For example, its form is simple, it is easy to calculate, and it has commonality among different tire fitting parameters and so on. In this paper, the reduced magic formula is chosen for lateral force of tire¹⁵

F_{yi} = D_{i} \sin (C_{i} \cdot \arctan (B_{i} α_{i} - E_{i} (B_{i} α_{i} - \arctan (B_{i} α_{i}))))

(9)

Among it

{\begin{array}{l} B_{i} = \frac{a_{3} \cdot \sin (a_{4} \cdot (\arctan (a_{5} \cdot F_{Zi})))}{C_{i} \cdot D_{i}} \\ C_{i} = 1.3 (i = 1, 2, r) \\ D_{i} = a_{1} \cdot F_{Zi}^{2} + a_{2} \cdot F_{Zi} \\ E_{i} = a_{6} \cdot F_{Zi}^{2} + a_{7} \cdot F_{Zi} + a_{8} \end{array}

(10)

In the equation, α_i ((i = 1, 2, r) separately represent left front wheel, right front wheel, rear wheel)) represents sideslip angle of left front wheel, right front wheel and rear wheel. B_i_, C_i_, D_i and E_i represent separately stiffness factor, shape factor, crest factor, and curvature factor. a₁, a₂, a₃, a₄, a₅, a₆, a₇ and a₈ are parameters obtained by experiment fitting. The value is shown as Table 1. Figure 2 shows the curve which is between cornering force and sideslip angle of front and rear tire under an approximately static equilibrium state, and calculated by equation (9) and Table 1.

Table 1.

Fitting parameters of tire.

Parameter	a₁	a₂	a₃	a₄	a₅	a₆	a₇	a₈
Value	−22.1	1011	1078	1.82	0.208	0	−0.354	0.707

Figure 2.

Curve between cornering force and sideslip angle of front and rear tire.

The tire’s sideslip angle, which is required when calculating how front wheel shimmy induces vehicle self-excited vibration, is given by equations (11) and (12).

The front wheel will have self-excited vibration when vehicle runs because of the presence of dry friction in joint clearance of suspension and steering system. The sideslip angle of tire cannot be expressed by the vehicle’s motion state directly. Sideslip characteristic of front wheel is described by choosing a first-order approximation tension wire theory. The rolling constraint equation of left and right tires of front axle was established.⁴

{\begin{array}{l} {\dot{α}}_{1} + \frac{u}{σ} * α_{1} + \frac{u}{σ} * θ_{1} - \frac{a}{σ} * {\dot{θ}}_{1} = 0 (i = 1, 2 expresses left and right separately) \\ {\dot{α}}_{2} + \frac{u}{σ} * α_{2} + \frac{u}{σ} * θ_{2} - \frac{a}{σ} * {\dot{θ}}_{2} = 0 \end{array}

(11)

a is semi-length of tire blot; σ is relaxation length of tire.

Considering the vehicle’s motion state, the expression of sideslip angle of rear wheel is as follows

α_{r} = \arctan (\frac{β - l_{2} * ω}{u}) (l_{2} is the rear wheelbase)

(12)

Selection of dry friction model

In this paper, we choose to use the Stefanski-Wojewoda static dry friction model.¹⁶ The dry friction models commonly used in vehicle shimmy system include Coulomb dry friction model¹⁷ and Stribeck dry friction model,¹⁸ because these two models are simple in mathematical form (the friction coefficient in Coulomb dry friction model is constant and the Stribeck dry friction model is in exponent form), which is convenient for computer numerical calculation. Stefanski-Wojewoda dry friction model has been neglected by scholars because of its complex mathematical form (piecewise function form), which results in large amount of numerical calculation. But in this paper, we mainly consider dry friction of ball joints between suspension and steering mechanism and dry friction of steering system, and the relative speed in the kinematic pair is close to zero. When the relative speed of two objects is close to zero, frictional lag and negative slope of friction occur, which are called frictional lag effect and Stribeck effect. These two effects are widely considered to be closely related to unstable phenomena such as shimmy in engineering. Compared with other static dry friction models, the Stefanski-Wojewoda dry friction model can well describe both frictional lag effect and Stribeck effect when the relative speed is close to zero. Therefore, this paper uses Stefanski-Wojewoda dry friction model.

The mathematical expression of Stefanski-Wojewoda dry friction model is as follows

M_{f} = M_{c} \cdot f

(13)

f = {\begin{array}{l} f_{st} s ign (v), f_{st} < f_{d +} \cup sign (\frac{dv}{dt} \cdot v) > 0 \\ f_{d +} s ign (v), f_{st} > f_{d -} \cup sign (\frac{dv}{dt} \cdot v) > 0 \\ f_{d -} s ign (v), sign (\frac{dv}{dt} \cdot v) < 0 \end{array}

(14)

f_{st} = \frac{1}{2} \frac{k}{N_{c}} \frac{v^{2}}{| v^{'} |} - f_{o}; f_{o} = 2 f_{c} - f_{s}; f_{d +} = f_{c} \cdot (1 + \frac{f_{s} (v) - f_{c}}{f_{c}} \cdot g (v, \frac{dv}{dt})); v = \dot{θ} \cdot r;

f_{s} (v) = f_{s} + Δ f_{s} \frac{| v_{A} |}{| v_{A} | + | v |}; g (v, \frac{dv}{dt}) = \frac{1}{1 + {(\frac{v - τ \cdot dv / dt}{v_{A}})}^{2}}; f_{d -} = f_{c} \cdot (1 - \frac{f_{s} - f_{c}}{f_{c}} \cdot g (v, \frac{dv}{dt}))

In the equation, v is relative velocity, f_c is dynamic friction coefficient, f_s is static friction factor, Δf_s is adjustment parameter of maximum static friction force, v_A is adjustment parameter of average speed, τ is hysteresis time, M_c is amplitude coefficient of equivalent dry friction torque. The curve between dry friction coefficient f and relative velocity v is shown as Figure 3 and it is calculated by Stefanski-Wojewoda dry friction model and parameter in Table 2.

Figure 3.

Graph of Stefanski-Wojewoda dry friction model.

Table 2.

Parameters for calculating dry friction.

Parameter	f_s	f_c	τ	v_A	Δf_s
Value	0.6	0.4	0.002	0.2	0.003

The parameter values of the sample vehicle are shown in Table 3.

Table 3.

The vehicle parameter values.

parameter	Value	parameter	value	parameter	value
J₁, J₂	15 (kg.m²/rad)	c_3y	10 (N.m.s/rad)	J₃	6 (kg.m²/rad)
c_4z	10 (N.m.s/rad)	J₄	12 (kg.m²/rad)	a	0.08 (m)
c_1l, c_2r	50 (N.m.s/rad)	R	0.35 (m)	c₃	20 (N.m.s/rad)
f	0.015	c₄	60 (N.m.s/rad)	k_α	1.37
k₁, k₂	30000 (N.m/rad)	k _β	0.049	k₃	60000 (N.m/rad)
D₁	0.06 (m)	k₄	70000 (N.m/rad)	l	0.15 (m)
k₅	1200000 (N.m/rad)	m₁g, m₂g	3500 (N)	c₁, c₂	20 (N.m.s/rad)
σ	0.22 (m)	α	4(°)	h _ϕ	0.087 (m)
m	1340(kg)	I_x	584 (kg.m²/rad)	m_s	1200(kg)
I_z	2048.1(kg.m²/rad)	k _ϕ	48900(N.m/rad)	F_zr	3200 (N)
c _ϕ	4476 (N.m.s/rad)	M_c	20(N.m)	h	0.542(m)
l ₁	1.21(m)	l₂	1.47(m)

Hopf bifurcation qualitative analysis of the shimmy system

The existence analysis and stability analysis of Hopf bifurcation in a nonlinear system are the most fundamental and important method in studying the dynamics of a nonlinear system. In this section, Hurwitz criterion¹⁹ is applied to investigate the existence of Hopf bifurcation in the shimmy system of the whole vehicle. The center manifold approach is applied to reduce the original system from a 14-dimensional system to a 2-dimensional system. And the Hopf bifurcation normal form is then applied to investigate the stability of the bifurcation of limit cycles.

Existence analysis of Hopf bifurcation

Make

\begin{array}{l} x_{1} = θ_{1}, x_{2} = {\dot{θ}}_{1}, x_{3} = θ_{2}, x_{4} = {\dot{θ}}_{2}, x_{5} = θ_{3}, x_{6} = {\dot{θ}}_{3}, x_{7} = θ_{4}, x_{8} = {\dot{θ}}_{4}, \\ x_{9} = β, x_{10} = ω, x_{11} = ϕ, x_{12} = \dot{ϕ}, x_{13} = α_{1}, x_{14} = α_{2} \end{array}

Assume X=(x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀, x₁₁, x₁₂, x₁₃, x₁₄). According to the theorem of nonlinear dynamics,²⁰ the equilibrium point of system X₀ is (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). If the vehicle speed u is taken as a bifurcation parameter, equations (1) to (14) can be expressed as the following static equations

\dot{X} = A X + G (X), X \in R^{14}

(15)

where A is the Jacobian matrix in the equilibrium point vicinity of the system, and G(X) contains quadratic and cubic nonlinear terms.

According to Hurwitz criterion, the equilibrium point of the system is stable when u=u₁ or when u=u₂. Whereas the equilibrium point of the system is unstable when u₁<u<u₂. And when the critical speeds are u₁= 50.2 km/h and u₂= 185.3 km/h, the eigenvalues of the Jacobian matrix are shown in Table 4.

Table 4.

Eigenvalues of Jacobi matrix A corresponding to the critical vehicle speeds.

Eigenvalues corresponding to u₁				Eigenvalues corresponding to u₂
λ₁	1.24i	λ₈	−2.92 + 76.4i	λ₁	0.64i	λ₈	−2.92 + 76.4i
λ₂	−1.24i	λ₉	−2.92 − 76.4i	λ₂	−0.64i	λ₉	−2.92-76.4i
λ₃	−14.2 + 171i	λ₁₀	−1.66 + 53.9i	λ₃	−28.7	λ₁₀	−2.87 + 54.9i
λ₄	−14.2 − 171i	λ₁₁	−1.66 − 53.9i	λ₄	−14.3 + 26.8i	λ₁₁	−2.87-54.9i
λ₅	−35.4 + 153i	λ₁₂	−6.2 + 8.86i	λ₅	−14.3-26.8i	λ₁₂	−6.2 + 9i
λ₆	−35.4-153i	λ₁₃	−6.2 − 8.86i	λ₆	−10.4 + 167i	λ₁₃	−6.2-9i
λ₇	−81.1	λ₁₄	−4.39	λ₇	−10.4 − 167i	λ₁₄	−4.57

According to Table 4, the Jacobian matrix of the system has a pair of purely imaginary eigenvalues when u=u₁ and u=u₂, and the other eigenvalues have negative real parts. According to the Hopf bifurcation theorem, we can get the critical speeds u₁ = 50.2 km/h and u₂ = 185.3 km/h as the bifurcation points of the systems, and a two-dimensional center manifold exists. When u₁<u<u₂, positive real part eigenvalues exist, and the shimmy occurs in the original system and produces limit cycles. When u<u₁ or u>u₂, all the eigenvalues of A have negative real parts, and the original shimmy system is stable asymptotically and eventually becomes balanced.

Stability analysis of limit cycles

The center manifold approach is applied to reduce the original system from a 14-dimensional system to a two-dimensional system and to determine the stability of the original system in two-dimensional systems. Assume μ_k=u-u_k (k = 1,2), and μ _k is an increment in the speed bifurcation parameter at the bifurcation point and has a minimum value. Using the non-singular transformation X=PU, where U has the same dimensions as X, P is composed of the real and imaginary parts of all the eigenvectors, and Λ=P⁻¹AP is a 14 × 14 diagonal matrix. Equation (15) is converted into

\dot{U} = Λ U + P^{- 1} G (PU, μ_{k})

(16)

Using the center manifold theorem to reduce dimension, the center manifold W^c of the expansion system represented by equation (16) is tangent to the plane of (y₁, y₂, μ_k) at the singular point (X₀_, u₀), assuming the center manifold

y_{i} = h_{i} (y_{1}, y_{2}) = h_{i 1} y_{1}^{2} + h_{i 2} y_{1} y_{2} + h_{i 3} y_{2}^{2} + h_{i 4} y_{1}^{3} + h_{i 5} y_{1}^{2} y_{2} + h_{i 6} y_{1} y_{2}^{2} + h_{i 7} y_{2}^{3}, i = 3, 4, \dots, 13, 14

(17)

{\dot{y}}_{i} = \frac{\partial h_{i} (y_{1}, y_{2}, μ_{k})}{\partial y_{1}} {\dot{y}}_{1} + \frac{\partial h_{i} (y_{1}, y_{2}, μ_{k})}{\partial y_{2}} {\dot{y}}_{2} + \frac{\partial h_{i} (y_{1}, y_{2}, μ_{k})}{\partial μ_{k}} {\dot{μ}}_{k}

(18)

Substitute equation (17) into equation (16) and then combine with equation (18), after which the coefficients of the same item on both sides of the equation are compared using the software Maple. In solving the linear equations, the coefficients of h_i (y₁, y₂, μ_k) (i = 3, 4,…13, 14) can be obtained and brought into the previous two equations of equation (16). After simplifying the equation, reduction equations can be obtained when u₁= 50.2 km/h or u₂= 185.3 km/h at the center manifold.

In the vicinity of u₁=50.2km/h, the reduced equation is

\begin{array}{l} {\dot{y}}_{1} = - 9.3767 y_{2} - 0.6205 μ_{1} y_{1} + 0.4936 μ_{1} y_{2} - 0.01265 μ_{1} y_{1}^{3} - 0.008495 μ_{1} y_{1}^{2} y_{2} \\ - 0.001426 μ_{1} y_{1} y_{2}^{2} + 1.9378 e^{- 8} μ_{1} y_{2}^{3} - 0.0233 y_{1}^{3} - 0.04624 y_{1}^{2} y_{2} - 0.02721 y_{1} y_{2}^{2} \\ - 0.03849 μ_{1}^{2} y_{1} - 0.001737 y_{2}^{3} + 0.0004941 μ_{1}^{2} y_{2} \\ {\dot{y}}_{2} = 9.3767 y_{1} - 0.4825 μ_{1} y_{1} - 0.6323 μ_{1} y_{2} + 0.003823 μ_{1} y_{1}^{3} + 0.002567 μ_{1} y_{1}^{2} y_{2} \\ + 0.0004308 μ_{1} y_{1} y_{2}^{2} - 4.5380 e^{- 9} μ_{1} y_{2}^{3} + 0.1387 y_{1}^{3} + 0.2397 y_{1}^{2} y_{2} + 0.004690 y_{1} y_{2}^{2} \\ + 0.0005100 μ_{1}^{2} y_{1} + 0.1274 y_{2}^{3} - 0.03940 μ_{1}^{2} y_{2} \end{array}

(19)

In the vicinity of u₂=185.3 km/h, the reduced equation is

\begin{array}{l} {\dot{y}}_{1} = - 11.3334 y_{2} - 0.5596 μ_{2} y_{1} + 0.3117 μ_{2} y_{2} - 0.005651 μ_{2} y_{1}^{3} - 0.02198 μ_{2} y_{1}^{2} y_{2} \\ - 0.006328 μ_{2} y_{1} y_{2}^{2} - 3.6681 e^{- 7} μ_{2} y_{2}^{3} - 0.02072 y_{1}^{3} - 0.2903 y_{1}^{2} y_{2} - 0.02579 y_{1} y_{2}^{2} \\ - 0.008508 μ_{2}^{2} y_{1} - 0.07096 y_{2}^{3} - 0.0003041 μ_{2}^{2} y_{2} \\ {\dot{y}}_{2} = 11.3334 y_{1} - 0.3217 μ_{2} y_{1} - 0.5487 μ_{2} y_{2} + 0.001943 μ_{2} y_{1}^{3} + 0.001994 μ_{2} y_{1}^{2} y_{2} \\ + 0.002056 μ_{2} y_{1} y_{2}^{2} + 3.7521 e^{- 8} μ_{2} y_{2}^{3} + 0.009987 y_{1}^{3} + 0.2852 y_{1}^{2} y_{2} + 0.03355 y_{1} y_{2}^{2} \\ + 0.0003161 μ_{2}^{2} y_{1} + 0.1332 y_{2}^{3} - 0.008703 μ_{2}^{2} y_{2} \end{array}

(20)

According to the Hopf bifurcation theorem of planar system,²¹ equations (19) and (20) can be turned into the Hopf bifurcation paradigm under the polar coordinates

{\begin{array}{l} \dot{ρ} = c μ ρ + a ρ^{3} \\ \dot{θ} = ω + e μ + b ρ^{2} \end{array}

(21)

when c = 0, the Hopf bifurcation is the first type of degenerate Hopf bifurcation which mean that the existence of limit cycle may not be unique in the system.²¹

According to equations (19) to (21), we can get the following:

In the vicinity of u=u₁=50.2 km/h

\dot{ρ} = (- 0.004601 μ_{1} - 0.0656) ρ^{3} - 0.6205 μ_{1} ρ

(22)

In the vicinity of u=u₂=185.3 km/h

\dot{ρ} = (- 0.002661 μ_{2} + 0.7457) ρ^{3} - 0.5596 μ_{2} ρ

(23)

The bifurcation diagram of the system equilibrium point X₀ at each critical velocity is obtained from equations (22) and (23), as shown in Figure 4.

Figure 4.

Bifurcation diagram of equilibrium point X₀ corresponding to critical speed.

According to Figure 4, Hopf bifurcations are supercritical at the critical speeds u₁ and u₂. When u < u₁= 50.2 km/h (that μ₁<0), shimmy does not occur in this system, which mean that the system is stable and equilibrium point X₀ is a stable focus. When u₁<u<u₂ (that μ₁>0 and μ₂<0), shimmy occurs in the system, the equilibrium point X₀ is has unstable focus, and a stable limit cycle appears. The phenomenon of the mutation of a stable focus into a stable limit cycle is called the limit cycle hard to produce or stable hard loss. When u > u₂ = 185.3 km/h (that μ₂>0), the equilibrium point turns into a stable focus again, the limit cycle disappears, the shimmy phenomenon disappears, and the system tends to be stable. Therefore, the front wheels shimmy is a typical self-excited vibration.

According to equations (22) and (23), we can get u = u₁ = 50.2 km/h, c = −0.6205 ≠ 0; and when u = u₂ = 185.3 km/h, c=−0.5596 ≠ 0. This means that the Hopf bifurcation is not a degenerate Hopf bifurcation. Therefore, through the qualitative analysis above, we only found that the system has a single limit cycle self-excited vibration. However, in the third section of the paper, we found that the system has the self-excited vibration with multiple limit cycles. The possible reasons will be analyzed after the third section of numerical analysis.

Calculation and analysis of multiple limit cycles characteristic of the system

Calculation and analysis of bifurcation characteristic of the system

Based on the above-established dynamic differential equations (1) to (14), the paper used four-order Runge-Kutta to make numerical calculation aimed at searching multiple limit cycles bifurcation characteristic of self-excited vibration.

In order to study the characteristic of vehicle self-excited vibration with multiple limit cycles under different speeds, the paper used forward speed of the vehicle as bifurcation parameters. The paper made numerical calculation when the initial excitations of front wheel’s angle are 0.2° and 10°, respectively. Through the numerical calculation, we can get that the shimmy of front wheel induced the self-excited vibration of the whole vehicle with multiple limit cycles, as shown in Figure 5(a1), (b1), (c1), and (d1).

Figure 5.

Self-excited vibration speed bifurcation diagram of the whole vehicle and front wheels. (a1) Bifurcation diagram of θ₁ about vehicle speed; (a2) three-dimensional phase diagram of θ₁ − θ′₁ − u; (b1) bifurcation diagram of ω about vehicle speed; (b2) three-dimensional phase diagram of ω − ω′ − u; (c1) bifurcation diagram of β about vehicle speed; (c2) three-dimensional phase diagram of β − β′ − u; (d1) bifurcation diagram of ϕ about vehicle speed; (d2) three-dimensional phase diagram of ϕ − ϕ′ − u.

Figure 5(a2), (b2), (c2), (d2) shows the lateral sectional view of Figure 5(a1), (b1), (c1), (d2) at speeds of 30 km/h, 60 km/h, 90 km/h, 120 km/h, 150 km/h and 180 km/h, respectively.

Table 5 shows the speed range of no limit cycle, single limit cycle and multiple limit cycles in Figure 5. It can be seen from Table 5 that the speed ranges of front wheel shimmy (θ₁) are completely consistent with the speed range of whole vehicle self-excited vibration (ω, β, ϕ). Therefore, it can be considered that the self-excited vibration of whole vehicle (ω, β, ϕ) was induced by front wheels shimmy (θ₁). And the self-excited vibration of system occurs only in a certain speed range: In the range of vehicle speed u = 2–50 km/h and u > 185 km/h, the front wheels shimmy and the induced vehicle self-excited vibration’s amplitude are very small, so it can be considered that there is no self-excited vibration at this time; when u = 50–83 km/h and u = 149–185 km/h, the self-excited vibration with multiple limit cycles occurred; when u = 83–149 km/h, the self-excited vibration with single limit cycle occurred. In the speed ranges of self-excited vibration occurring (u = 50–185 km/h), the amplitude of system’s self-excited vibration decreases with the increase of vehicle speed.

Table 5.

The speed range of no limit cycle, single limit cycle and multiple limit cycles.

	The speed range of no limit cycle (km/h)	The speed range of single limit cycle (km/h)	The speed range of multiple limit cycles(km/h)
θ₁	2–50 and >185	83–149	50–83 and 149–185
ω	2–50 and >185	83–149	50–83 and 149–185
β	2–50 and >185	83–149	50–83 and 149–185
ϕ	2–50 and >185	83–149	50–83 and 149–185

The critical bifurcation speed is just the actual speed of the self-excited vibration of the vehicle with multiple degrees of freedom. When the vehicle is at the upper critical bifurcation speed, the Hopf bifurcation occurs in the whole vehicle system, which is represented by the vibration with limit cycles. As to θ₁, front wheels have a constant amplitude swing around kingpin. As to ω, the vehicle yaw angle has a constant amplitude swing around yaw axis. As to β, the vehicle has a constant amplitude swing around Y-axis. As to ϕ, the suspended portion has a constant amplitude swing around the rolling axis. When the speed of the vehicle increases to the lower critical bifurcation speed, the limit cycles disappears, the self-excited vibration of the vehicle with multiple degrees of freedom disappears, and the vehicle returns to stable driving conditions.

Multiple limit cycles characteristic of the system

To further analyze the characteristic of self-excited vibration with multiple limit cycles induced by front wheels shimmy, the paper made calculations and analysis for the vibration with multiple limit cycles’ characteristic of main variables which describe the characteristic of vehicle self-excited vibration when u = 60 km/h. As a result, multiple limit cycles’ phase diagrams of each variable are shown as Figure 6 and the comparison of its multiple limit cycles’ amplitude are shown in Table 6.

Figure 6.

When Initial excitation θ₁=0.2° or θ₁=10° and u = 60 km/h, the phase diagram of the self-excited vibration with multiple limit cycles.

Table 6.

The comparison of limit cycles’ amplitude with different initial excitation.

speed	Initial excitation= 0.2°				Initial excitation= 10°
speed	θ₁	ω	β	ϕ	θ₁	ω	β	ϕ
u=60km/h	0.122°	0.071°/s	0.013°	0.002°	5.223°	4.094°/s	1.027°	0.344°

According to Figure 6(a), it can be got that the front wheel shimmy produces three limit cycles with different amplitudes when the front wheel is under different initial excitation: a large stable limit cycle and a small stable limit cycle, and there is an unstable limit cycle between the two stable limit cycles. And the unstable limit cycle was obtained by the numerical approximation method. According to Figure 6(b) to (d), the phase diagrams of ω, β and ϕ show two stable limit cycles with different amplitudes. There must be one unstable limit cycle between the two stable limit cycles. But the unstable limit cycle cannot be solved by knowing methods because in this model the initial excitation is from the front wheel. When the vehicle is subjected to an excitation’s amplitude smaller than the excitation that causes the unstable limit cycle self-excited vibration, the system do self-excited vibration with the smaller limit cycle; when the system is subjected to an excitation’s amplitude greater than the excitation, that causes an unstable limit cycle self-excited vibration, the system has self-excited vibration with the bigger limit cycle.

In this section, the self-excited vibration with multiple limit cycles of the system was found by numerical analysis. However, in the qualitative analysis of the Hopf bifurcation qualitative analysis of the shimmy system section, we did not find evidence of the degenerate Hopf bifurcation occurring which is inconsistent with the conclusions in this section. The possible reasons are as follows.

The existence, stability and numerical characteristics of limit cycles have been well studied, but it is still a difficult problem to determine the number of limit cycles. (The second part of Hilbert’s 16th Problem²² is concerned with the number and relative distributions of limit cycles of planar polynomial systems. Smale²³ in his paper titled ‘Mathematical problems for the 21th century’ posed the problem again.) At present, the number of limit cycles can only be judged for some low-dimensional systems.²⁴There are two main strong non-linear factors in the vehicle shimmy system: (1)The strong non-linearity of tires; (2) the strong non-linearity of dry friction in suspension/steering system. In the process of qualitative analysis, it is necessary to reduce the dimension of the original system. It is necessary to carry out Taylor series transformation on these two nonlinear factors, and discard the high-order terms, thus losing some nonlinear factors.²⁵ The occurrence of the limit cycle is very dependent on the nonlinear factors of the system itself.²⁴ After losing this part of the nonlinear factors, the system is likely to be reduced from the original system with multiple limit cycles to the system with single limit cycle. However, in the numerical analysis, we did not lose the nonlinear components of tire lateral force and dry friction of suspension/steering system in the model, and found the phenomenon of multiple limit cycles. It can be seen that the reason why the degenerate Hopf bifurcation is not found in Section 2 may indeed be that the system reduces the dimension and loses the high-order component of the two nonlinear factors.

Conclusion

Considering the nonlinear factor of tire lateral force and the dry friction force of suspension/steering system, a mechanical model of vehicle self-excited vibration and 7-DOF system differential equations was established.

Reducing the 14-dimensional system into 2-dimensional system by means of center manifold theorem. By using Hopf bifurcation theorem, the existence and stability of limit cycles are qualitatively determined.

By means of numerical analysis, the self-excited vibration with multiple limit cycles is found. The results show that: in the speed range of 50–185 km/h, front wheels shimmy induce the self-excited vibration of the whole vehicle and the amplitude of system’s shimmy decreases with the increase of vehicle speed; in the speed range of 50–83 km/h and 149–185 km/h, the self-excited vibration with multiple limit cycles occurs; in the speed range of 83–149 km/h, the self-excited vibration with single limit cycle occurs.

The paper provides a theoretical reference for the vehicle development process to predict and avoid vehicle self-excited vibration induced by front wheels shimmy. The main content of next step is to find accurate numerical solution methods of unstable limit cycle when vehicle has self-excited vibrations with multiple limit cycles and to inhibit this self-excited vibration phenomenon by matching the vehicle system parameters.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Financial support for this research was provided by the National Natural Science Foundation of China (51375130, 51875154).

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