Research On The Stochastic Dynamics For A Kind Of Typical Vibroimpact System

Vibroimpact system,as a typical nonsmooth system,is ubiquitous in practical engineering.Researches on theoretical analysis,numerical calculation and application of vibroimpact system have become an international topic in the field of dynamics at present.Given that vibroimpact system is inevitably affected by random excitations in the process of vibration,it is of important theoretical and realistic significance to study the dynamical behaviors of the vibroimpact system under random excitations.In this thesis,vibroimpact system under special continuous random excitation(Gaussian white noise)and jump random excitation(Poisson white noise)are considered,respectively.The corresponding analytical methods are developed,and effects of the restitution coefficient and random excitation intensity on the system dynamical behaviors are analyzed.Firstly,the random vibration problems of Rayleigh-Van der Pol oscillator are considered by means of a single-degree-of-freedom system with a unilateral rigid barrier under different Gaussian white noise parametric excitations.The nonsmooth transformation method is adopted to convert the vibroimpact system to an equivalent system without velocities jump.The modified stochastic averaging technique for energy envelope is used to deal with the transformed system and the averaged It? stochastic differential equation is obtained under the assumption that the damping,the energy loss and the random excitations are weak.The analytical solution of the response of the original vibroimpact system is derived by solving the corresponding Fokker-Planck-Kolmogorov(FPK)equation.The effects of restitution coefficient and parametric Gaussian white noises on the response and bifurcation of the system are investigated.In addition,the critical condition of stochastic bifurcations is also explored.Typical points near the critical condition curve are chosen to verify the effectiveness of the analytical expression of the critical condition.Secondly,stochastic responses and bifurcations of the single-degree-of-freedom vibroimpact systems under the external and parametric excitations of the Gaussian white noises are studied.Using the nonsmooth transformation and Dirac function,the original vibroimpact system is transformed to an approximately equivalent system.Then,an energy dependent system that has exact steady-state solution is used to approximate the transformed system.The equivalent principle is that they have the same averaged energy envelope.The probability density function of the steady-state response is obtained by solving the equivalent nonlinear system.After that,several common used vibroimpact system models in engineering are solved using the above method,and the analytical solutions of the steady-state responses are compared with numerical simulation results to verify the effectiveness of the presented method.Finally,the stochastic P-bifurcation of Duffing-Rayleigh vibroimpact system under random parametric excitation is worked out analytically.The analytical expression of critical condition of stochastic P-bifurcation is obtained.Results show that the changes of parametric Gaussian white noise intensity can induce the occurrence of the stochastic P-bifurcation.Thirdly,given that many random excitations are jump random excitations in nature,the stochastic responses of quasilinear vibroimpact system driven by Poisson white noise,a typical representation of jumping random excitation model,are studied.With the help of a nonsmooth transformation,the original vibroimpact system under Poisson white noise excitation is converted into an approximately equivalent system without impact.The corresponding It? stochastic differential equation is derived by using It? stochastic differential rule for Poisson white noise excitation.Then,the steady-state responses of vibroimpact system are obtained by means of the stochastic averaging method and perturbation method.After that,the effectiveness of the procedure presented above is verified through several examples.In addition,the stochastic bifurcation of a vibroimpact system is analyzed according to the calculation results.At last,the Duffing vibroimpact system with two rigid barriers that are symmetrical with respect to the equilibrium point is considered for the cases of external and parametric Gaussian white noise random excitations.Based on the initial energy value,the motions of the unperturbed vibroimpact system are categorized in two ways: oscillations without impacts and oscillations with alternate impacts on both sides.Then,under the assumption that the vibroimpact Duffing system is quasi-conservative,the stochastic averaging method for energy envelope is applied to obtain the averaged drift and diffusion coefficients for the two types of motions,respectively.The probability density functions of stationary responses are derived by solving the corresponding FPK equation.After that,results obtained from the proposed procedure are validated by numerical simulation.Meanwhile,the effects of the position of bilateral barriers and the random excitations on the probability density functions of the stationary responses are also discussed.