Non-smooth System Of A Preliminary Study

There are a plenty of random and non-smooth factors in actual engineering and mechanics, such as impacts, collisions, dry frictions, variable stiffness, switch, threshold, impulse control, numerical control and so on. The study of non-smooth dynamics is becoming an important and challenge one. In the paper some theories about smooth dynamics have been extended to the study of the non-smooth dynamics, and then, the dynamic behaviors were explored, including several aspects as following:Firstly, the conditions of constraint are transacted by means of the mean constraint plane and the mean jump equation, by which the method of the Chebyshev polynomial approximation used to explore period-doubling bifurcation of stochastic smooth systems, is operated in stochastic non-smooth systems. Numerical simulations show that period-doubling bifurcation exists in stochastic Duffing one-side constraint system as same as in smooth stochastic Duffing system, furthermore, Chebyshev polynomial approximation is a effective method in exploring dynamical actions of stochastic non-smooth system. There are difference between stochastic systems and deterministic systems because of the random factor.Secondly, the stationary responses of non-linear systems with unilateral constraints are studied by using the quasi-conservative averaging method. By means of a non-smooth variable transformation and the Dirac delta function, the response probability density functions are obtained analytically. Meanwhile, the transformation method was validated comparing with numerical results.Thirdly, based on the literature [8], the Poincaré map of non-smooth system was constructed by the discontinuous map, and then, the problem of period solutions of non-smooth system with unilateral constraints was transformed to the boundary value problem, the dynamic behaviors including period solutions , stabilization and...