| A class of dynamic systems that arise due to the existence of various nonsmooth factors such as collisions and dry friction is called nonsmooth systems.These systems are ubiquitous in various engineering fields.Additionally,because fractional calculus can more accurately describe the dynamic response of dynamical systems,the application of fractional calculus in dynamical systems is becoming increasingly widespread.Therefore,the dynamical system combining non-smooth factors and fractional order derivatives is one of the hot spots of current research.So,it is of great significance to study in-depth the dynamic response,stochastic bifurcation,and chaotic motion of fractional-order vibro-impact systems under random excitation.Firstly,based on the traditional stochastic averaging method and nonsmooth transformation,the steady-state response of a single-degree-of-freedom single-side vibro-impact system under fractional-order joint random excitation was studied.According to the analytical FPK equation obtained,the steady-state probability density functions of the system’s amplitude,displacement,and velocity were obtained.The steady-state probability density maps,time-displacement maps,and time-velocity maps of the amplitude,displacement,and velocity were obtained based on the system’s probability density function.Subsequently,the accuracy of the method was verified by numerical simulation.The influence of each parameter on the steady-state response of the system was discussed.Secondly,the stochastic response of a single-side vibro-impact system under fractional-order random parametric excitation was studied using the energy enveloping stochastic averaging method and nonsmooth transformation.Under the assumption of small damping coefficient,low energy loss,and high-intensity random parametric excitation,the probability density equations of the system’s energy,displacement,and velocity,as well as the joint probability density equations of displacement and velocity,were obtained based on the energy perspective.Furthermore,the effects of the fractional-order order,fractional-order coefficient,restitution coefficient,and random parametric excitation on the steady-state stochastic response of the vibro-impact system were discussed.It was found that changes in some of the system’s parameters can lead to stochastic bifurcation.Thirdly,in response to the discovery of stochastic bifurcations in the system,the problem of stochastic bifurcations in a single-sided fractional-order Rayleigh oscillator vibro-impact system under multiplicative random excitation is studied in detail using traditional stochastic averaging methods.Fractional-order derivatives are equivalent to corresponding damping and restoring forces,and the original system is transformed into a system without velocity jumps using non-smooth transformations.The system’s probability density function and its steady-state solution are obtained using stochastic averaging methods.The critical parameter condition expression for the system undergoing stochastic P-bifurcations is derived using catastrophe theory,and the effects of the fractional-order coefficient,fractional-order order,restoring coefficient,and other main parameters on the occurrence of bifurcations in the fractional-order Rayleigh oscillator vibro-impact system are analyzed.Finally,the chaos of the collision vibration system under fractional-order periodic and random excitations was studied using the Melnikov method.Based on the Melnikov method under harmonic periodic excitation,the Melnikov function of the vibro-impact system under random excitation was obtained,which consists of deterministic terms and random terms.The variance and mean-square criterion formula of the steady-state response function were derived by transforming it accordingly.Thus,the chaos threshold is a function of both the noise intensity and the restoring coefficient,and the variation of the noise intensity and restoring coefficient has a certain impact on the threshold.The results were verified using phase portraits,Poincaré maps,time displacement graphs,and time velocity graphs.It was found that the change in noise intensity can cause the system to generate chaos to a certain extent,enhance chaos,or suppress chaos... |
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