Research On Response Of A Type Of Vibro-impact System

There are many impact and dry friction nonsmooth factors in practical world. The systemis called nonsmooth system if it contains nonsmooth factors. Compared with smooth system,it shows strong nonlinearity and singularity. Vibro-impact system is an important type ofnonsmooth system, which appears rather complex dynamic behavior because of the impact.And some interesting phenomena, such as grazing bifurcation, torus bifurcation, chatter andsticking motion, are observed in this type of systems because of the presence of the impact.Although the vibro-impact system has received much attention in recent years, the theory ofit is far from perfect, such as the response of a vibro-impact system with random noiseexcitation. On the one hand, the stochastic averaging method is used to obtain theapproximate analytical response of the system; on the other hand, the generalized cellmapping method is applied to calculating the numerical response of the system in thedissertation. Moreover, the control of sticking motion is also discussed.Firstly, the stochastic response of a vibro-impact system excited by different noises isstudied. By the aid of the nonsmooth transformation, the vibro-impact system is firstlytransformed into a new system which can be considered as a quasi-smooth system. Thestochastic averaging method can be used to obtain the stochastic response of the new systemwhen the vibro-impact system owns weak damping and weak excitation. Then the stochasticresponse of the original system can be derived by using the aforementioned transformation.The correlated Gaussian white noise and Gaussian colored noise are studied, respectively. Inthe case of the Gaussian colored noise, the Fourier series and the concept of residual phaseare respectively used to obtain the approximated analytical results. Some examples are givento illustrate the application of the proposed procedure. It is shown that the results of theseexamples by using the proposed procedure agree very well with those from Monte Carlosimulation results of the corresponding systems. Furthermore, stochastic P-bifurcations arealso explored.Secondly, the numerical method is considered to calculate the stochastic response of a vibro-impact system with noise excitation. The stochastic averaging method becomesimpractical for most problems of random system with impacts. It is especially true for avibro-impact system with high impact losses or with impacts at a rigid barrier at the system’snon-equilibrium position. So the numerical method is the only tool to be used to derive thestochastic response. The generalized cell mapping method is extended to analyze theresponse of a vibro-impact system with noise excitation. And three kinds of excitation, whichare Gaussian white noise, harmonic and Gaussian white noise and Lévy noise, arerespectively discussed. Some examples are given to illustrate the application of thegeneralized cell mapping method. And the accuracy of the method is verified by thecorresponding Monte Carlo results. Moreover, the stationary response is used to discuss thestochastic P-bifurcations.Finally, the impulsive control method is used to study the sticking motion which appears ina vibro-impact system. A Van de Pol one-sided constraint system is given to indicate theaccuracy of the method. It is found that the impulsive control method can not only restrainsticking motion but also induce sticking motion. Numerical results show the validity andstability of the method. And the method is stable even for high levels of multiplicative noiseor additive noise.