| There are many uncertain factors in the clearance between the components of mechanical system,and hence it is more suitable to simulate these uncertain factors by randomness.This thesis regards cantilever beam systems as the research object.Two kinds of cantilever beam system model under random boundary constraint are established by considering the tiny random process of boundary constraint.The period-doubling bifurcation,the grazing bifurcation and the influence of stochastic constraints on the system stability are studied.Numerical simulations verify the theoretical deduction.The main contents are as follows:Based on Ornstein-Uhlenbeck process,a class of second-order differentiable stochastic differential equations and corresponding stochastic differential equations of velocity and acceleration are derived.The explicit solution of Ito-form and the corresponding expectation and variance are obtained.Grazing dynamics of a single-degree-of-freedom cantilever beam vibration system with random boundary rigid constraints are studied.Based on the definition of grazing point and the dynamic equation of impact cantilever beam,the existence of periodic grazing motion is analyzed.The random local discontinuous mapping with parameters near the grazing motion is derived by selecting fixed phase as Poincaré section.The random composite piecewise normal mapping is established by combining the smooth flow mapping and discontinuous mapping.Based on random normal form mapping,the unique banded bifurcation structure and the dynamic behaviors near the grazing point of the collision beam system under random constraints are revealed,and the effect of random perturbation scale on the grazing behavior is analyzed.Results verify the effectiveness of random normal form mapping.The local dynamic behaviors of periodic orbits of a two-degree-of-freedom cantilever beam system with random boundary rigid constraints are studied.Firstly,aming at the case of transverse contact between periodic orbit and constraint boundary,the linearization matrix with parameters at fixed point and the random linear matrix at fixed points on the basis of the extending state space of system are deduced.The period-doubling bifurcation of system is studied by the linear matrix and the applicability of the theory is verified by numerical simulations.The effects of different intensity random disturbances on the bifurcation behavior of the system are also investigated.Aming at the case of grazing contact between periodic orbit and constraint boundary,the random composite piecewise normal mapping with parameters is derived by a kind of improved random local discontinuous mapping method and the practicability of the theory is verified by numerical simulations. |
