四边简支矩形薄板的双Hopf分岔分析

基于奇异性理论,研究了主参数共振-1∶3内共振情形下参数激励与外激励联合作用下四边简支矩形薄板的双Hopf分岔问题.考虑弱阻尼和弱激励的情形,得到了四边简支矩形薄板的分岔方程,给出了四边简支矩形薄板在参数平面μ-σ1上的分岔图.对参数激励与外激励联合作用下四边简支矩形薄板的阻尼系数、外激励、参数激励以及调谐参数进行不同的取值,通过数值模拟得到了四边简支矩形薄板平衡解将发生Hopf分岔,并分岔出周期解,薄板系统的非线性振动形式为周期运动.当四边简支矩形薄板的参数满足给定条件时,我们得到薄板的1∶3共振双Hopf分岔.随后,四边简支矩形薄板将会呈现概周期振动.

Double hopf bifurcations of rectangular thin plates simply supported four edges

Based on the singularity theory,this paper studies the double Hopf bifurcation problem of one rectangular thin plate with simply supported four edges under the combined action of parametric excitation and external excitation in the cases of primary parametric resonance and 1∶3 internal resonance.The bifurcation equation of rectangular thin plate with simple supported edges is obtained by considering the case of weak damping and weak excitation,and the bifurcation diagram of the rectangular thin plate is also given.Taking the damping coefficient,external excitation,excitation parameters and tuning parameters of rectangular thin plate as different values,the equilibrium solutions of thin plate generate Hopf bifurcation,and bifurcate to periodic solutions.The nonlinear vibration form of thin plate system is periodic motion.When the values of other parameters for the rectangular plate satisfy the given conditions,1∶3 resonant double Hopf bifurcation of the thin plate can be obtained.Subsequently,the four edges-simply supported rectangular plate also show almost periodic vibration.