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A stochastic wheelset model with a nonlinear wheel-rail contact relationship is established to investigate the stochastic stability and stochastic bifurcation of the wheelset system with the consideration of the stochastic parametric excitations of equivalent conicity and suspension stiffness. The wheelset is systematized into a one-dimensional (1D) diffusion process by using the stochastic average method, the behavior of the singular boundary is analyzed to determine the hunting stability condition of the wheelset system, and the critical speed of stochastic bifurcation is obtained. The stationary probability density and joint probability density are derived theoretically. Based on the topological structure change of the probability density function, the stochastic Hopf bifurcation form and bifurcation condition of the wheelset system are determined. The effects of stochastic factors on the hunting stability and bifurcation characteristics are analyzed, and the simulation results verify the correctness of the theoretical analysis. The results reveal that the boundary behavior of the diffusion process determines the hunting stability of the stochastic wheelset system, and the left boundary characteristic value cL = 1 is the critical state of hunting stability. Besides, stochastic D-bifurcation and P-bifurcation will appear in the wheelset system, and the critical speeds of the two kinds of stochastic bifurcation decrease with the increase in the stochastic parametric excitation intensity. [ABSTRACT FROM AUTHOR]

The forced vibration of a single-degree-of-freedom piecewise linear system containing fractional time-delay feedback was investigated. The approximate analytical solution of the system was obtained by employing an averaging method. A frequency response equation containing time delay was obtained by studying a steady-state solution. The stability conditions of the steady-state solution, the amplitude–frequency results, and the numerical solutions of the system under different time-delay parameters were compared. Comparison results indicated a favorable goodness of fit between the two parameters and revealed the correctness of the analytical solution. The effects of the time-delay and fractional parameters, piecewise stiffness, and piecewise gap on the principal resonance and bifurcation of the system were emphasized. Results showed that fractional time delay occurring in the form of equivalent linear dampness and stiffness under periodic variations in the system and influenced the vibration characteristic of the system. Moreover, piecewise stiffness and gap induced the nonlinear characteristic of the system under certain parameters. [ABSTRACT FROM AUTHOR]

In this paper, a dynamic model of piecewise nonlinear system with fractional-order time delay is simplified. The amplitude frequency response equation of the dynamic model of piecewise nonlinear system with fractional-order time delay under periodic excitation is obtained by using the average method. It is found that the amplitude of the system changes when the external excitation frequency changes. At the same time, the amplitude frequency response characteristics of the system under different time delay parameters, different fractional-order parameters, and coefficient are studied. By analyzing the amplitude frequency response characteristics, the influence of time delay and fractional-order parameters on the stability of the system is analyzed in this paper, and the bifurcation equations of the system are studied by using the theory of continuity. The transition sets under different piecewise states and the constrained bifurcation behaviors under the corresponding unfolding parameters are obtained. The variation of the bifurcation topology of the system with the change of system parameters is given. [ABSTRACT FROM AUTHOR]

To overcome the shortage of traditional rear axle compliance steering (RACS) technology, a kind of viscoelastic smart material is introduced into the rear suspension of a vehicle to construct rear wheel semiactive steering system. This article focuses on the nonlinear dynamic behavior of the vehicle with RACS incorporating viscoelastic smart material. First of all, considering the tire nonlinearity and the fractional derivative constitutive relation of the viscoelastic material, the nonlinear dynamic model of the vehicle with RACS is formulated. Then, the lateral dynamic behavior of the vehicle with RACS is demonstrated through numerical experiments. Finally, some factors that influence shimmy of the compliance steering wheel are investigated. Numerical results demonstrate the Hopf bifurcation characteristics of the vehicle with RACS and disclose the influence factors of Hopf bifurcation characteristics for the vehicle with RACS, which lay the theoretical foundation for the development of the rear wheel semiactive steering technology. [ABSTRACT FROM AUTHOR]

Van der Pol (VDP) oscillator under forced excitation has complicated vibration modes, and its harm to the system cannot be ignored. The vibration reduction mechanism of VDP oscillator by means of nonlinear energy sink (NES) is explored under low-frequency forced excitation. Firstly, a mechanical model of VDP oscillator coupled NES is established by utilizing Newton's law. Then the time history diagrams under different excitation amplitudes are compared to measure the vibration reduction effect of the coupled system. Finally, the stability and Hopf bifurcation behavior of the corresponding autonomous system are discussed, and the mechanism of cluster phenomenon and the mechanism of vibration reduction are revealed. It is interesting that, the vibration reduction effect is optimal when cluster phenomenon occurs, which is a typical vibration model in slow–fast system. The real part of eigenvalues and hysteresis time are critical factors affecting vibration reduction. These researches provide a theoretical basis for the vibration reduction of VDP oscillator coupled NES. • A dynamic model of Van der Pol oscillator coupled NES is established. • The stability and Hopf bifurcation behavior of the corresponding autonomous system are discussed. • The mechanism of cluster phenomenon is revealed by the slow–fast analysis method. • The vibration reduction mechanism is revealed from the real part of eigenvalue and hysteresis time. [ABSTRACT FROM AUTHOR]

The stability and bifurcation behaviors of fractional-order vibro-impact system are investigated, where a two-degrees-of-freedom vibro-impact oscillator excited by external harmonic excitation is considered. The approximate analytical solution of the fractional-order vibro-impact system is acquired based on the approximate equivalent integer-order system of fractional-order system obtained by the averaging method. Poincaré mapping of the system is established, the linearized matrix of Poincaré mapping is obtained according to the approximate analytical solution, and the stability of n -1 periodic motion is analyzed. The bifurcation behaviors of the two-degrees-of-freedom vibro-impact system are investigated by numerical solutions. Under different system parameters, the bifurcation behaviors of the system are analyzed in detail when the excitation frequency and fractional order change. It is found that there are Hopf bifurcation, period doubling bifurcation, quasi periodic motion and chaotic motion in the two-degrees-of-freedom vibro-impact system with fractional-order derivative, and there are two routes to chaos in the system when the excitation frequency changes. • We studied the stability and bifurcation of the fractional-order 2DOF vibro-impact system. • The analytical approximation is obtained by averaging method and its equivalent system. • The stability of n-1 periodic motion of the 2DOF vibro-impact system is analyzed. • The bifurcation behaviors of the 2DOF vibro-impact system are analyzed in detail. [ABSTRACT FROM AUTHOR]