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A simple dynamic model of a vibro-impact system subjected to harmonic excitation with two asymmetric clearances is considered. The Semi-Analytical Method for getting periodic solutions of the vibro-impact system is proposed. Diversity and evolution of the fundamental periodic impact motions are analyzed. The formation mechanism of the complete chatting-impact periodic motion with sticking motion by the influence of gazing bifurcation is analyzed. The transitional law of periodic motions in the periodical inclusions area is presented. The coexistence of periodic motions and the extreme sensitivity of the initial value within the high frequency region are studied. The global distribution of the periodic and chaos motions of the system is obtained by the state-parameter space co-simulation method which very few have considered before. The distribution of the attractor and the corresponding attracting domain corresponding to different periodic motions are also studied. [ABSTRACT FROM AUTHOR]

• The global distribution offers detailed insight into motions of the molding machine. • The polymorphic coexistence by grazing and saddle-node bifurcations is revealed. • The results enables the consideration of the design of the molding machine. Global behavior of a simplified non-smooth dynamic model for a micro-vibration molding machine in parameter-state space is explored. Solutions for the periodic motions of the impact oscillator are obtained by the semi-analytical method. Conditions for the grazing motions and sticking motions of the model are deduced. The motions distribution and transition in the fundamental periodic motion regions and the periodic inclusions areas are studied. Transition of the motions due to the singularity of the grazing bifurcation is analyzed. Polymorphic coexistence under the effect of the grazing bifurcation and the saddle-node bifurcation are revealed. Through the global analysis method in parameter-state space, the global distribution map in high frequency region can be acquired, which is rarely considered before. Under the guidance of the global motions distribution, the evolution of the polymorphic coexistence areas can be further discussed. The results can provide some references and ideas for the dynamic design and motion control of the micro-vibration molding machine. [ABSTRACT FROM AUTHOR]

In order to reflect the dynamic response of the rubber vibration isolation system more accurately in medium and low frequency ranges, the nonlinear Zener model is used to characterize the mechanical properties of viscoelastic materials like rubber. The approximate analytical solution of the system response is obtained by harmonic balance method, and its correctness is verified through numerical method and Universal Mechanism (UM) software simulation. The quantitative relationship between mass displacement and node displacement is calculated, and a new method to obtain hysteresis characteristics of rubber vibration isolation system based on approximate analytical solution of the model is presented. Under the influence of symmetry and hysteresis, the system possesses bifurcation, chaos, polymorphic coexistence and other complex nonlinear dynamical behaviors. Then, with the help of global bifurcation diagram, phase diagram, Poincaré map and cell mapping, the formation mechanism of polymorphic coexisting motion induced by pitchfork bifurcation, saddle-node bifurcation, period-doubling bifurcation and boundary crisis is comprehensively analyzed. Based on this, the formation mechanism of bursting oscillation is revealed. Finally, the influence of nonlinear stiffness on the transition law of the system periodic motion is analyzed, and the distribution law of the coexisting attractors and the variation of the attracting domain in various of polymorphic coexisting motions are presented. The research results of the complex dynamic behavior of the nonlinear Zener vibration isolation system in this paper can provide some theoretical guidance for the optimal design of the viscoelastic vibration isolation system. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]