Response And Reliability Of The Vibro-impact System With Stochastic Excitations

Collision,impact and dry friction widely exist in mechanical systems which are affected by stochastic factors.They can dramatically change the dynamical behaviors and even induce the structural insecurity.The existence of friction in the vibro-impact makes these systems show strong nonlinear and singular characteristics,which make the dynamic behavior of the system more complex and richer.In this thesis,the dynamical system with non-smooth factors such as impact and friction under stochastic are investigated.The analytical methods for studying the stochastic response and reliability of these systems are developed.Further,the influence of non-smooth factors and random excitation on the response and reliability of the impact vibration system is analyzed.Thesis is organized as follows.1.The response and bifurcation of Duffing van der Pol collision system with viscoelastic term under wide-band noise excitation are studied.By using the approximate method,the original system is transformed into an equivalent system without viscoelastic force,and the discrete collision mapping of the system is transformed into an equivalent system without velocity jump by means of non-smooth transformation.At this time,the original collision system is transformed into an approximate smooth system without viscoelastic force.Then,the transformed system is analyzed analytically by means of the stochastic average method based on the generalized function,and the averaged it(?) stochastic differential equation is obtained.Further,the corresponding Fokker Planck Kolmogorov(FPK)equation is solved to obtain the probability density function of the system.The excellent agreement between analytical result and Monte Carlo simulation result shows that the effectiveness of proposed method.Moreover,the effects of viscoelastic parameters and random excitation on the system response are discussed.2.The stochastic response of the vibro-impact system with friction term under broadband noise excitation is studied.By means of non-smooth transformation,the equivalent approximate system is obtained.Then,by virtue of the stochastic averaging method based on generalized harmonic function,the averaged It(?) differential equation of system amplitude is obtained,by which its probability density function is obtained.Taking Duffing van der Pol collision friction system and Rayleigh van der Pol collision friction system as examples,the effectiveness of the theoretical method described in this chapter is verified.The effects of the bandwidth and noise intensity of the stochastic excitation on the response of the system are discussed.At the same time,the effects of two non-smooth factors,such as the restitution coefficient and the friction coefficient,on the response and bifurcation of the system are analyzed.Finally,it is found that the changes of the restitution coefficient and the friction parameter can cause the stochastic p-bifurcation of the system.3.The reliability of elastic vibro-impact system with friction term excited by Gaussian white noise is studied.Combined with non-smooth properties of the impact and friction,the stochastic averaging method of energy envelope is improved to obtain the It(?) stochastic differential equation of system energy.By virtue of the theory of diffusion process,the Back Kolmogrov(BK)equation satisfied by reliability function and the Generalized Pontryagin equation satisfied by the mean first passage time are obtained.The Duffing Van der Pol elastic vibro-impact system with friction and the Rayleigh Van der Pol elastic vibro-impact with friction are adopted to verify the validity of the proposed method.The influence of system parameters on system reliability is analyzed,respectively.The results show that the friction parameter,the noise intensity and the damping coefficient of the linear spring can improve the reliability of the elastic vibro-impact system with friction term.4.The reliability and the optimal bounded control to improve the reliability of vibro-impact system under the excitation of additive and multiplicative white Gaussian noise are studied.The improved stochastic averaging method of energy envelope is proposed,by which the corresponding backward Kolmogorov equation about the conditional reliability function and Pontryagin equation about the first passage time can be obtained based on the It(?) stochastic differential equation of system energy.For the controlled system,the optimal bounded control is derived by the dynamic programming principle and performance index about the first passage.The related analytical results of the first passage of the system under stochastic optimal control can be obtained by applying the method of the first passage of the research above.It is found that reducing the recovery coefficient or the intensity of random excitation can improve the reliability of the system.Moreover,the optimal bounded control can effectively improve the reliability and the average first passage time of the impact vibration system.