复合边界条件下功能梯度板1:1内共振的周期与混沌运动

PERIODIC AND CHAOTIC MOTION OF MEXIED BOUNDARY FGM PLATE WITH 1:1 INTERNAL RESONANCE

以两对边简支另两对边自由的功能梯度材料板为研究对象,首先建立了考虑材料物性参数与温度相关的、在热/机械载荷共同作用下的几何非线性动力学方程,采用渐进摄动法对系统在1:1内共振-主参数共振-1/2亚谐共振情况下的非线性动力学行为进行了摄动分析,得到系统的四自由度平均方程,并对平均方程进行数值计算,分析外激励对系统非线性动力学行为的影响,发现在一定条件下通过改变外激励可以改变系统的运动形式,产生混沌运动.另外,第二阶模态的幅值远比第一阶模态的幅值大,这应该是两阶模态耦合产生内共振的结果,因此,研究该类结构的非线性动力学行为时不应该只考虑一阶模态,而应考虑到前两阶甚至更多阶模态的相互作用,以便于更好地利用或控制其运动形式.

This paper studied the nonlinear vibration of FGM plate with two simply supported opposite and two free edges subjected to the in-plane and transversal excitations.The geometrical nonlinear system,whose material properties of the FGM plate are assumed to be temperature dependent,was used.The resonant case considered here is 1：1 internal resonance and principal parametric resonance-1/2 subharmonic resonance.The asymptotic perturbation method was utilized to obtain four-dimensional nonlinear averaged equation.Numerical method was used to find the nonlinear dynamic responses of the FGM rectangular plate.It is found that there exist periodic,quasiperiodic solutions and chaotic motions for the plates under certain conditions.It is thought that the forcing excitations can change the form of motions for the FGM rectangular plate.It is also found from the numerical simulations that the amplitude of the first mode for the FGM plate is much less than that of the second mode in the given parameters.It means that we must consider the second mode when the internal resonance occurs.