1:1内共振条件下矩形薄板的全局分叉和多脉冲混沌动力学

首次利用广义Melnikov方法研究了一个四边简支矩形薄板的全局分叉和多脉冲混沌动力学．矩形薄板受面外的横向激励和面内的参数激励．利用von Krmn模型和Galerkin方法得到一个二自由度非线性非自治系统用来描述矩形薄板的横向振动．在1∶1内共振条件下，利用多尺度方法得到一个四维的平均方程．通过坐标变换把平均方程化为标准形式，利用广义Melnikov方法研究该系统的多脉冲混沌动力学，并且解释了矩形薄板模态间的相互作用机理．在不求同宿轨道解析表达式的前提下，提供了一个计算Melnikov函数的方法．进一步得到了系统的阻尼、激励幅值和调谐参数在满足一定的限制条件下，矩形薄板系统会存在多脉冲混沌运动．数值模拟验证了该矩形薄板的确存在小振幅的多脉冲混沌响应．

Global Bifurcations and Multi-Pulse Chaotic Dynamics of a Rectangular Thin Plate With One-to-One Internal Resonance

Global bifurcations and multi-pulse chaotic dynamics for a simply supported rectangular thin plate were studied using the extended Melnikov method for the first time.The rectangular thin plate was subjected to transversal and in-plane excitations.A two-degree-of-freedom nonlinear non-autonomous system governing equations of motion for the rectangular thin plate was derived using the von Krmn type equation and the Galerkin’s approach.The resonant case considered here is 1 ∶ 1 internal resonance.The averaged equation was obtained by the method of multiple scales.After transforming the averaged equation into a standard form,the extended Melnikov method was employed to show the existence of multi-pulse chaotic dynamics,which coudle be applied to explain the mechanism of modal interactions of thin plates.A skill for calculating the Melnikov function was given without the explicit analytical expression of homoclinic orbits.Furthermore,the restrictions on the damping,excitations and the detuning parameters were obtained,under which multi-pulse chaotic dynamics was expected.The results of numerical simulations are also given to indicate the existence of small amplitude multi-pulse chaotic responses for the rectangular thin plate.