Study Of Random Response Of Nonlinear Vibro-impact System With Bilateral Barriers

In recent years,vibro-impact system has gradually become one of the important research topics in the field of disaster prevention and reduction in civil engineering.On one hand,vibro-impact is one of the important causes of damage in many practical engineering fields,for example,the gap of the reserved expansion joint cannot meet the relative displacement,which causes the crack of the beam end during the earthquake.On the other hand,many machines or components designed by the principle of vibration dynamics can also well reflect the use of vibro-impact,such as particle dampers.Therefore,based on the harm and utilization brought by the vibro-impact,the study of the relevant dynamic characteristics has very important theoretical significance and practical application value.However,compared with the abundant results of deterministic vibration,there are few studies on random vibro-impact systems.This paper applies the iterative method of weighted residual to obtain the random response approximate closed-form solution of single-degree-of-freedom(SDOF)nonlinear vibro-impact system with bilateral barriers based on Hertz contact theory model and instantaneous impact model under random excitation.Aiming at the Hertz contact theory model,the response of approximate closed solution of a single-degree-of-freedom(SDOF)strongly nonlinear vibro-impact system based on under Gaussian white noise excitation is solved.First,the approximate expression of the steady-state solution of Fokker-Planck-Kolmogorov(FPK)equation is constructed by the concepts of probability circulation and probability potential flow.Second,the undetermined coefficients in the approximate expressions are obtained by using the method of weighted residuals.Finally,an iterative technique is utilized to gain the probability density estimate of given precision obtained with the method of weighted residual.As the illustrative examples,theDuffing vibro-impact system and dry friction vibro-impact system are worked out in detail,and the analytical results are compared with the Monte Carlo simulation data.It is shown that the analytical solutions are coincident with Monte Carlo simulation.In addition,the random response of vibro-impact system with bilateral barriers based on the instantaneous impact model under the Gaussian white noise excitation is studied.First,a period approximate transformation is employed to convert the equations of the motion to a continuous form.The probabilistic description of the system is subsequently defined through the corresponding Fokker-Planck-Kolmogorov(FPK)equation.The closed-form stationary probability density function(PDF)of the response is obtained by solving the reduced FPK equation using the iterative method of weighted residual together with the concepts of the circulatory probability flow and the potential probability flow.Finally,the versatility of the proposed approach is demonstrated by its application to two typical examples.The results of above the two examples show that iterative method of weighted residual method can still achieve satisfactory results.The results obtained in this paper are all approximate closed-form solutions with high precision,which can be directly used in the structural optimization design of practical correlation systems.