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In this paper, the stability of vehicle concerning the slow-varying sprung mass is analyzed based on two degrees of freedom quarter-car model. A mathematical model of vehicle is established, the nonlinear vibration caused by sprung mass vibration is solved, and frequency curve is obtained. The characteristics of a stable solution and the parameters affecting the stability are analyzed. The numeric solution shows that a slow-varying sprung mass is equivalent to adding a negative damping coefficient to the suspension system, making the effective damping coefficient change from negative to positive. Such changing parameters lead to Hopf bifurcation and a shrinking limit cycle. The simulation results indicate the existence of static as well as dynamic bifurcation and the result is a change in the final stable vibration of the suspension. Even the tiny vibration of the sprung mass will lead to amplitude mutation, leading to the sprung mass instability.

The stability and bifurcation of a nonlinear system are closely related. Bifurcations that include both static and dynamic bifurcations affect the stability of the system. Static bifurcation such as saddle node bifurcation occurs mainly due to the nonlinear stiffness, discussed in vibration absorbers [

Research on static bifurcation varies. Static bifurcation caused by maneuvering load of an aircraft is analyzed in [

Considering the case of high-speed rail CRH3 in China as the object, the critical speed leading to Hopf bifurcation between the wheel and the rail, which is critical for the safety of high-speed trains, is analyzed [

Slowly varying parameter defines the parameter which demands one-tenth time or less compared with other parameters with same self-variation in one system. Vehicle suspension is a system where the sprung mass is slowly varying. The fast-varying parameters like velocity or acceleration vary back and forth quickly. They are considered as variables in system equations. Different from that, the sprung mass is usually considered as constant. Actually, it decreases slowly, adding a slowly varying parameter to a suspension system, which changes the form of the equations. This is always ignored by researchers. For example, an adaptive control of nonlinear uncertain active suspension systems is analyzed in [

This paper is organized as follows. In Section

Leaving the turn of the vehicle out of account, the suspension model can be described in Figure

Nonlinear quarter-car model.

According to Newton’s second law, the governing equation is

Owing to the fact that the slow time satisfies

Comparing to suspension system with constant sprung mass, this system contains an additional damping term

The sprung mass vibration makes the whole suspension in a forced vibration state. Equations (

The sprung mass varies due to the change of load. When

Because the exact solution of (

Because the stiffness of the tire

Substituting (

The term

According to the method of multiscale, the approximate solution can be expressed as

Under 3 : 1 internal resonance condition,

There are differential operators like these

Substituting (

The general solution of (

Substituting (

Change

Substituting (

Since sprung mass

The number of solutions concerning (

As can be seen from (

Combining (

From (

When the suspension is in the uncoupled case, the steady statement of sprung mass vibration is that of autonomous system in the singularity

Linearizing (

Use

Expand (

As for

In the slow time scale

Focus on (

Then we get two solution curves intersecting at (0, 0). That is,

They are point and circle in polar coordinates, which correspond to the equilibrium point and limit cycle of the two-dimensional systems (

For trivial solution

For nontrivial solution

For the nonlinear suspension system, the sprung mass

Let

According to (

Figure

The effect of varying acceleration

Figures

The effect of varying nonlinear damping coefficient

The effect of varying nonlinear stiffness

From Figure

The effect of varying force frequency

Figures

The effect of varying force frequency

The effect of varying force frequency

Let the nonlinear stiffness

The effect of varying nonlinear damping coefficient

The effect of varying nonlinear stiffness

The effect of varying force frequency

Let the nonlinear damping coefficient

The effect of varying force frequency

When

The critical lower limit frequency is

Evolution of the phase plane with the force frequency: (a) 6.81, (b) 6.82, (c) 6.85, and (d) 6.88 rad/s.

When

Phase portrait and time-domain vibration amplitude of Hopf bifurcation.

This process changes the initial value. Since the nonlinear vibration is closely related to initial state, it will change the nature of the vibration.

In Figure

Different solution induced by Hopf bifurcation.

In this paper, the stability of vehicle concerning the slow-varying sprung mass is analyzed based on two-degree-of-freedom quarter-car model. The vibration amplitude solution is solved using multiscale method, followed by numerical validation. After analyzing amplitude-frequency curve, phase plane of steady-state response and phase portrait, and time-domain vibration amplitude of Hopf bifurcation, the main points of our work can be concluded as follows:

(a) There exist rich mechanical phenomena, which include internal resonance, jumping phenomenon, and Hopf bifurcation.

When the unsprung mass

(b) The essential of the effect of slowly varying sprung mass is changing the damping coefficient.

The numeric solution shows that a slow-varying sprung mass is equivalent to adding a negative damping coefficient to the suspension system, making the effective damping coefficient change from negative to positive. Such changing parameters lead to Hopf bifurcation and a shrinking limit cycle.

(c) The stability of nonlinear suspension is closely related to damping coefficient, nonlinear stiffness, and a slowly varying sprung mass.

Because of the slowly varying process, vibration with different initial states converges in the stable limit cycle in the end. They cannot result in final vibration with different states. Since the nonlinear vibrations are closely related to the initial states, Hopf bifurcation can change the final stable state of the vibration.

All of the above observations indicate the existence of static as well as dynamic bifurcation and the result is a change in the final stable vibration of the suspension. Even the tiny vibration of the sprung mass will lead to amplitude mutation, leading to the sprung mass instability.

The authors declare that no conflicts of interest exist regarding the publication of this paper.