Research On The Steady-state Probabilistic Response Of Nonlinear Stochastic Constrained Systems

Nonlinear systems with constraint exist in many fields,especially in the equipment manufactur-ing industry.However,most of the studies focus on the probable response of nonlinear vibro-impact systems subject to the Gaussian white noise,which has some limitations.In this paper,a new s-tochastic averaging method is proposed to obtain the probability response of a nonlinear stochastic vibro-impact system under the Gaussian colored noise.In the first chapter,the research trends at home and abroad of the nonlinear stochastic con-strained system are introduced.Also,the research background,the relevant basic knowledge and the main contents of the article are expounded.In the second chapter,a new stochastic averaging method is developed,which combines the quasi-conservative stochastic averaging method and the energy loss.In the case of the system with unilateral constraints,the stochastic vibro-impact system is divided into two types:free motion of system without energy loss and vibro-impact system with energy loss due to collision,through the relationship between the total energy of the system and the potential energy of the system in the con-strained position.Based on these relationships,the quasi-conservative stochastic averaging method combining Fourier series expansion and energy loss is developed,and the corresponding non-smooth mean Fokker-Planck-Kilomogorov equation is obtained.Then the analytical expression of the prob-ability response of a strong nonlinear stochastic system with unilateral constraint is obtained.At the end of this chapter,vibro-impact Duffing system with a unilateral constrained driven by Gaus-sian color noise as an example is provided to verify the proposed method,and the validity of the conclusion is verified by Monte-Carlo simulation.In the third chapter,the method which has been presented in the previous chapter is extended to a stochastic Duffing system with bilateral constraints driven by Gaussian color noise.The analytic expression of the probability response of the stochastic vibro-impact system is obtained.On the basis of this,the article analyzes the effects of the four physical quantities of stochastic vibro-impact system in detail,including vibro-impact position,the restitution coefficient,and the noise intensity and correlate time of the Gaussian colored noise on the probability response of stochastic vibro-impact system.The effectiveness of the proposed method is verified by Monte-Carlo simulation.The fourth chapter summarizes the conclusions of this paper and the application prospects of the new method.