Research On Dynamics Of A Non-smooth Stochastic System

In engineering,the vibro-impact system with clearance has strong nonlinear and discontinuity because of non-smooth and random factors.It has practical meaning,study the dynamic behaviors of the non-smooth stochastic system and chaos control.In this thesis the bifurcation,the route to chaos and chaos control are studied.The main contents of this thesis are as following:Firstly,the development history and the current study of the stochastic dynamic systems and the chaotic vibration and chaos control are introduced.A nonlinear dynamical model of the single-degree-of-freedom vibro-impact system with random clearance is established;the analytic expression of Poincar(?) map of the system is derived.The stability and bifurcation of the periodic motions are analyzed by calculating the eigenvalue of Jacobi matrix of Poincar(?) map.The analysis shows that the periodic motions will be unstable because of random variable clearance,and the bifurcation phenomena will be happened.These are helpful to optimize system parameters of a practical mechanical vibration system with clearances.By introducing the mean impact plane and the mean shock equation,the stochastical vibro-impact system is changed into the equivalent deterministic vibro-impact system using the method of Chebyshev polynomials.Furthermore,the response of the deterministic system is solved and the period-doubling bifurcation,grazing bifurcation and chaos are analyzed. Numerical results show that in some points of parameters,the two systems are in good coincidence.This shows Chebyshev polynomial approximation is an effective method in the study of the non-smooth stochastic system.We also find that in the non-smooth stochastic system the routes to chaos via period-doubling bifurcation or grazing bifurcation as same as in the non-smooth system.A one-degree-of-freedom vibro-impact system with clearance is established.The analytic expression of Poincar(?) map of the system is derived,and the spectrum of Lyapunov exponents of the system is calculated numerically,the effects of the dynamical behavior of vibro-impact system with random disturbance are analyzed.At last,through the largest Lyapunov exponent,the stochastic bifurcation of random non-smooth system is studied. Numerical simulations show that period-doubling bifurcation also exists in the random non-smooth system,but different from that in the deterministic system.The adaptive impulsive control method is developed to stabilize the chaotic motions in a class of one-degree-of-freedom vibro-impact system.Just when the impact occurs,the adaptive pulses is implemented to the variable quantity of system.Numerical simulations show that the behavior of chaos in system can be controlled effectively by this control method, and also show that this method is robust even for high levels of multiplicative noise or additive noise.In the last,the work is concluded and some aspects to be further studied are also pointed out.