Research On Dynamic Characteristics Of Fractional Nonlinear Vibration Isolation System

The equivalent mechanical model of traditional viscoelastic materials is described by the integer order model.With the rapid development of fractional order calculus,the fractional order model can not only describe the viscoelastic properties of materials,but also describe the memory restoring properties of rubber materials.Among them,rubber has superelasticity under large deformation,and this super-elastic property leads to strong nonlinear characteristics in mechanical models.In the field of rail transit,the design of structural parameters of air spring will affect the dynamic performance of vehicle system.For a vibration isolation system,fatigue failure is related to amplitude of different frequencies so that it is necessary to accurately predict the dynamic response of vibration isolation system in different frequency bands.Firstly,a fractional nonlinear Zener model is established to describe the constitutive relation of rubber materials,and the dynamic differential equation of vibration isolation system is obtained.The first-order approximate analytical solution is solved by using the first-order harmonic balance method.The amplitude-frequency response of linear system under multiple parameters and the dynamic response of nonlinear system under different parameters are discussed respectively.The numerical method and virtual simulation experiment are combined to compare the approximate solutions,and then the force transfer formula of vibration isolation system is derived.The amplitude-frequency characteristics of the system are analyzed from the perspective of frequency domain,and the influence of different system parameters on the coexistence frequency band of amplitude-frequency response polymorphic solutions is discussed.In addition,the high order harmonic response of the system is obtained by using the high order harmonic balance method,the effect of super-harmonic resonance on the force transfer of vibration isolation system is discussed by comparing the high order harmonic response solutions with a variety of numerical methods.On the basis of discussing the amplitude-frequency response of vibration isolation system,the variation rules of the amplitude-frequency response and the periodic motion transition of the system are also discussed.The transition rule of periodic motion leading to chaos in the range of middle and low frequency is analyzed under the induction of various bifurcation types.During the numerical simulation,it is found that the system has periodic motion and chaotic coexistence,and the motion law of the multi-state coexistence region and its adjacent region is summarized.In addition,the fractional nonlinear Nishimura model is used to characterize the constitutive relation of the air spring,and the dynamic differential equation of the vibration isolation system is obtained and the steady-state response of the system is solved.The shape and energy loss of hysteresis loops are compared under different parameters.The steady-state dynamic response of the system is obtained by means of the high-order harmonic balance method.The effects of different parameters on the main resonance and the super-harmonic resonance of the vibration isolation system are discussed by comparing the results with the numerical simulation,and a method based on the primary resonance amplitude-frequency curve for solving the optimal damping coefficient is proposed.Then,the dynamic behavior of the system in the low frequency region is obtained by numerical simulation.The transition rule from periodic motion to chaos induced by bifurcation types in the low frequency range of the vibration isolation system is discussed.The dynamic stability of the system is determined by Lyapunov exponent.In the numerical simulation,it is also found that there is a phenomenon of multi-period motion coexistence in the system,and the motion transition of the multi-period coexistence region and its adjacent region is summarized,as well as the distribution variation of the multi-period coexistence attraction region during the transition process.Finally,the equivalent mechanical model of the suspension system is established and the dynamic differential equation of the suspension system under the foundation excitation is obtained.The dynamic response of the suspension system amplitude under different foundation excitation amplitudes is analyzed,and the variation law of the jumping phenomenon and the diversity of periodic motion in the amplitude curve with multiple values is summarized.The numerical simulation also found that when the gap of the suspension system changes,the system has the phenomenon of multi-periodic motion coexistence.The distribution of different attractors on the attractor domain of periodic motion is discussed by combining with cell mapping method,and the transition rule of different excitation amplitudes on the periodic motion is summarized.The dynamic stability of the suspension system is analyzed by combining with Lyapunov exponent.