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This study investigated the dynamics and control of a nonlinear suspension system using a quarter-car model that is forced by the road profile. Bifurcation analysis used to characterize nonlinear dynamic behavior revealed codimension-two bifurcation and homoclinic orbits. The nonlinear dynamics were determined using bifurcation diagrams, phase portraits, Poincaré maps, frequency spectra, and Lyapunov exponents. The Lyapunov exponent was used to identify the onset of chaotic motion. Finally, state feedback control was used to prevent chaotic motion. The effectiveness of the proposed control method was determined via numerical simulations.

The suspension system of vehicles is meant to isolate the vehicle body and passengers from oscillations imposed by road irregularities and maintain continuous contact between the tires and surface of the road. Suspension systems provide a smoother ride and help the operator to maintain control of the vehicle over rough terrain or during sudden stops [

Nonlinear phenomena of codimension bifurcations have been studied extensively, particularly in engineering systems [

The fact that hysteresis is not a smooth process renders the characteristics of a suspension system with hysteretic damping inherently nonlinear. This can greatly complicate the design of a controller. Accurately predicting and/or controlling the performance of a system require that the effects of nonlinear phenomena be taken into account. The problem of rough road profiles and unwanted vehicle vibrations produced by kinematic excitation make it necessary to form accurate predictions of dynamic behavior over a range of operating conditions in the design of suspension systems. Bifurcation and chaos are crucial concepts in the study of stability in nonlinear systems, and numerous researchers have addressed issues pertaining to nonlinear dynamics in suspension systems [

In this study, we examined a suspension system presenting codimension-two bifurcation and homoclinic orbits, both of which determine the nature of the dynamic behavior and the existence of strange attractors. We sought to calculate the strange attractor using numerical simulation in which external forces from a road profile are added to the system as a parameter with a specific range of values. We employed bifurcation diagrams, phase portraits, Poincaré maps, frequency spectra, and Lyapunov exponents to explain periodic and chaotic motions in a vehicle suspension system that exerts hysteretic nonlinear damping forces. Algorithms that computes Lyapunov exponents for smooth dynamic systems [

Chaotic behavior is normally considered undesirable due to the restrictions it imposes on the operating range of electrical and mechanical devices. Many engineering problems can only be resolved by driving a chaotic attractor in a periodic orbit. Numerous methods have been proposed for the control of chaos [

This paper is organized as follows. Section

Figure

Schematic diagram of the quarter-car model.

The states are chosen such that

The amplitude of the road profile excitation is

Nomenclature and parameters.

Notation | Value | Unit | Description |
---|---|---|---|

| 240 | kg | The body mass |

| 16 ^{4} | N/m | The suspension stiffness coefficient |

| 30 ^{4} | N/m^{3} | The nonlinear hysteretic suspension stiffness coefficient |

| 250 | Ns/m | The linear damping coefficient |

| −25 | Ns^{3}/m^{3} | The nonlinear hysteretic suspension viscous damping coefficient |

| 0.05 | m | The amplitude of road excitation |

| m | The road excitation | |

| m | The body’s vertical displacement | |

| rad/s | The frequency of road excitation |

Our main objective in this analysis is to observe the qualitative behavior of the system and how it changes as system parameters are varied. Changes occurring near the equilibrium point of the system are referred to as local bifurcation. The codimension of bifurcation refers to the minimum number of parameters that must be varied in order to observe a specific type of bifurcation [

Nonlinear analysis must be based on linear analysis. Bifurcation analysis is based on the Jacobian matrix in (

The blow-up technique is used to check whether the cubic terms in (

Stability analysis reveals that the system has two degenerate equilibrium points (0, 0) and (0,

After dividing by the unaffected common component,

For the other equilibrium point, (0,

The term “bifurcation set” refers to a set in parameter space on which the system bifurcates. It is obtained from unfolded equations in the normal form.

There are three equilibrium points in this case:

(0, 0) for all

(

By checking the stability of these equilibrium points, we obtain the following:

Let

Thus, after rescaling, (

After a series of operations, the following expression is obtained:

For the following change in coordinates,

Let

Clearly, it can be seen that

According to the Hopf bifurcation theorem [

Numerical simulations using (

The bifurcation sets that are associated with typical phase portraits in each region, where

Numerical simulation for region (I), for (

Numerical simulation for SC, for (

Numerical simulation for region (II), for (

Numerical simulation for region (III), for (

Numerical simulation for region (V), for (

Region (I) includes sources and a saddle. If the initial conditions are in the neighborhood of the two sources, then the mass is distant from the equilibrium points (Figure

In terms of the equilibria and Figure

In (

Bifurcation diagram for the system when the amplitude of the road excitation,

The period-one orbit at

Phase portrait: (a)

Quasiperiodic orbit at

Quasiperiodic orbit at

Quasiperiodic orbit at

Chaotic motion at

The analysis presented in Section

Figure

Evolution of the largest Lyapunov exponent.

Improving the performance of a dynamic system and/or avoiding chaotic behavior require periodic motion, which is more important when working under specific conditions. Cai et al. [

System (

The bifurcation diagram for a system with state feedback control, where

Transforming chaotic motion to a desired period-one orbit for

In many general systems that involve chaotic dynamics, this state feedback technique does not require preliminary or on-line analysis of the system dynamics and can be implemented by adding the feedback of a suitable variable to the original system with sufficient control gain to suppress the development of chaos. In this regard, the state feedback control technique is simple and effective approach to chaos suppression.

This study examined the nonlinear dynamic behavior in a vehicle suspension system with hysteretic damping. Our objective was to develop a method to control chaotic vibration. Dynamic behavior was observed over the entire range of parameter values of the bifurcation sets and bifurcation diagram. Using bifurcation analysis, we identified a codimension-two bifurcation structure with double-zero degeneracy in the system. The bifurcation diagram indicates the existence of quasiperiodic orbit and chaotic motion, thereby demonstrating the effectiveness of using Lyapunov exponents as a tool for identifying chaotic motion. Finally, we proposed a state feedback control method to suppress chaotic motions and improve the ride provided by an automotive suspension system. Simulation results confirm the feasibility of the proposed method. Our results provide insight into the operating conditions under which undesirable dynamic motion takes place in vehicle suspension systems. Our results also provide a useful reference for engineers involved in the design of control systems for vehicle suspensions. It should be noted that in practical engineering, phenomena such as actuator faults, time-varying delays, unmodeled dynamics, and dynamics disturbances often occur simultaneously, and these problems are beyond the scope of the current study. In future work, we will seek to control nonlinear systems with time-varying delays.

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

The authors would like to thank the Ministry of Science and Technology of the Republic of China, Taiwan, for financially supporting this research under Contract no. MOST 105-2221-E-212-016.

_{∞}control to active suspension systems on half-car models