以航空发动机压气机叶片为实际工程背景,将叶片简化为功能梯度材料的旋转悬臂板模型。基于Reddy的高阶剪切变形理论和von Karman的大变形理论,考虑了变转速和离心力的作用,由一阶活塞理论得到气动力的表达式,利用Hamilton原理建立了系统的非线性动力学方程。应用Galerkin离散法进行二阶离散得到系统的常微分控制方程。考虑系统1:1内共振和主参数共振的情况,利用渐进摄动法得到了旋转悬臂板系统四维直角坐标形式的平均方程。通过数值仿真研究了变转速旋转悬臂板结构的复杂非线性振动响应。结果表明,叶片转速的变化对系统动力学特性有着重要的影响,在不同的转速下,系统存在着周期运动、多倍周期运动和混沌运动等多种复杂非线性动力学行为。

In the study of the aero-engine compressor blades, the blade is simplified as a rotating cantilever FGM plate with varying rotating speed. Based on the Reddy's high-order shear deformation theory and the von Karman type equations for the geometric nonlinearity, the nonlinear governing partial differential equations of motion are derived by using the Hamilton's principle. The aerodynamic load is determined by using the first-order piston theory. The Galerkin approach is used to transform the nonlinear partial differential governing equations of motion into a two-degree-of-freedom nonlinear system. The principal parametric resonance and the 1:1 internal resonance are considered. The asymptotic perturbation method is used to obtain a four-dimensional nonlinear averaged equation. The numerical method is used to find the nonlinear dynamic responses of the rotating cantilever FGM plate. It is found the rotating speed has an important influence on its nonlinear dynamic behavior. It is shown that, at different rotating speeds, there exist the chaotic, periodic and quasi-periodic motions for the rotating cantilever FGM plate.