Analysis of super-harmonic resonance and periodic motion transition of fractional nonlinear vibration isolation system

The precision instruments and equipment are often utilized in low-frequency and micro-amplitude vibration systems, in which many vibration isolators of rubber materials are widely used. Ignoring the low-frequency amplitude will result in errors in the fatigue life design of the vibration isolators and predicting the dynamic response of each frequency band accurately becomes necessary. However, integer-order models cannot describe the frequency dependence of rubber materials, while the fractional-order models can describe it instead. On the other hand, the elastic restoring force is strongly nonlinear under large deformation, and the vibration of the nonlinear system contains multiple harmonic components. In order to solve those issues, the fractional nonlinear Nishimura model is used to characterize the constitutive relation of vibration isolators such as air springs, which are made of carbon black filled natural rubber. The high-order harmonic balance method is used to obtain the steady-state response of the vibration system, while the fourth-order Runge–Kutta method is applied to simulate the dynamic response of the system in the low-frequency region, and the Lyapunov exponent is used to determine the stability of the system. Furthermore, the influence of parameters on the amplitude–frequency characteristics of different frequency bands is also studied, and a method to solve the optimal damping coefficient is proposed based on the primary resonance amplitude–frequency curves. The results show that there is a diversity of periodic motions in the process of adjacent super-harmonic resonance transition. Numerical simulations also demonstrate that multi-periodic motions coexist in the system. The motion transition law between the polymorphic coexistence region and its adjacent regions is summarized.