Bifurcation Characteristics and Optimal Control of Magnetic Shape Memory Alloy Beam Subjected to Stochastic Excitation

Abstract

Nonlinear dynamic characteristics and optimal control of magnetic shape memory alloy (MSMA) cantilever beam subjected to axial stochastic excitation is studied in this paper. Nonlinear difference item is introduced to interpret the hysteretic phenomena of MSMA, and the nonlinear dynamic model of MSMA cantilever beam subjected to axial stochastic excitation is developed. The steady-state probability density function of the dynamic response of the system is obtained, and the conditions of stochastic Hopf bifurcation are analyzed. The reliability function of the system is solved, and then the probability density of the first-passage time is obtained. Finally, the optimal control strategy is proposed in stochastic dynamic programming method. Numerical simulations show that the stability of the trivial solution varies with bifurcation parameter, and stochastic Hopf bifurcation appears in the process; the reliability of the system is proved by stochastic optimal control, and the first-passage time is delayed. The results are helpful to engineering application of MSMA cantilever beam.