Suppression of chaotic vibrations in suspension system of vehicle dynamics using chattering-free optimal sliding mode control

  • S. Mahdi AbtahiEmail author
Technical Paper


In this wok, chaos analysis along with chaos control is studied in vertical model of the vehicle system. The chaotic behavior has been demonstrated in the suspension system under the specific initial conditions, values of parameters, and profile of road roughness. In order to analyze chaos in the dynamical model, the power spectrum density and Lyapunov exponent methods are used. Moreover, the phase portrait and Poincare’ sections of the simulations verify numerically chaos in uncontrolled system. For the purpose of chaos control, a novel optimal sliding mode control strategy is designed for stabilization of the system’s behavior via the semi-active suspension using MR fluid damper. As results, the optimal sliding mode control eliminates the chattering phenomenon in the responses and suppresses the chaotic oscillations in comparison with ordinary sliding mode control system. Responses of the feedback system depict the far-better performance of the proposed optimal sliding mode control from the viewpoint of reducing the settling time, overshoot, energy, and amortization in the suspension system.


Chaos Power spectrum density Optimal sliding mode control Control Lyapunov function Semi-active suspension 

List of symbols

x(t), θ(t)

Vertical displacement and pitch angular motion of sprung mass

x1(t), x2(t)

Heave motion of the unsprung masses

m, I

Mass and inertia moment of chassis

\(m_{1}\), \(m_{2}\)

Masses of front and rear unsprung

\(l_{1}\), \(l_{2}\)

Length of front and rear axels

\(x_{{{\text{rp}}1}}\), \(x_{\text{rp2}}\)

Applied excitation displacement from the road surface on the front and rear tires,

\(K_{\text{t1}}\), \(K_{\text{t2}}\)

Front and rear tire stiffness

\(C_{\text{t1}}\), \(C_{\text{t2}}\)

Front and rear tire damping coefficient

\(f_{\text{s}}\), \(\Delta s\)

Dynamic force and change of length of springs


Stiffness of the springs


Dissipative force in the MR fluid damper

\(C_{\text{d}}\), \(C_{0}\)

Viscous damping at low and high velocity,


Piston relative displacement


Scaling value for the Bouc–Wen hysteresis loop



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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Faculty of Industrial and Mechanical Engineering, Qazvin BranchIslamic Azad UniversityQazvinIran

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