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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。
Taking aeroengine compressor blade as the actual engineering background, the blade is simplified as a rotating cantilever plate model with functionally graded material. Based on Reddy’s theory of high-order shear deformation and von Karman’s theory of large deformation, the effect of variable speed and centrifugal force was considered. The expression of aerodynamic force was obtained from first-order piston theory. The nonlinear dynamic equation . The Galerkin discretization method is used to make the second-order dispersion to obtain the system’s ordinary differential governing equations. Considering the resonance of 1: 1 system and the resonance of the main parameters, the average equation in the form of rectangular Cartesian coordinates of rotating cantilever system is obtained by the method of progressive perturbation. The complex nonlinear vibration response of a variable speed rotating cantilever plate structure is studied by numerical simulation. The results show that the change of blade rotation speed has an important influence on the system dynamics. Under different rotation speeds, the system has many complicated nonlinear dynamic behaviors such as periodic motion, periodic motion and chaotic motion.