采用非线性Galerkin方法的柔性梁模型降阶研究

Study on the Model Order Reduction of Flexible Beam Based on Nonlinear Galerkin Method

针对采用Galerkin方法获取的结构动力学降阶模型精度不高的问题,以考虑几何非线性的两端固支柔性梁作为研究对象,建立了两端固支柔性梁非线性动力学模型。首先采用Galerkin方法将原系统降阶,得到单自由度、三自由度和五自由度系统,再采用非线性Galerkin方法将二自由度和三自由度系统降阶为单自由度系统。通过分析降阶模型的非线性动力学行为,得到系统响应随外荷载幅值变化的分岔图,给出系统做周期运动、倍周期运动和混沌时的时程曲线、相图与庞加莱映射图。在特定频率下,通过改变外荷载幅值来控制系统进出混沌状态。与Galerkin方法得到的降阶模型进行了对比,通过比较进出混沌状态时的外荷载幅值,分析了两种方法的降阶效果。结果表明:非线性Galerkin方法能够有效提高降阶模型的精度,更接近真实模型;采用非线性Galerkin方法降阶得到相同自由度的低阶系统时,原模型阶数越高,得到的低阶模型越精确。

A nonlinear dynamics model of a flexible beam clamped at both ends is established, and the order of the model is reduced using nonlinear Galerkin method to solve the problem of low accuracy using common Galerkin method. On this basis, the nonlinear dynamical behaviors of the reduced-order model are analyzed, and the bifurcation diagram of system response is obtained when the external load amplitude changes. The time-displacement curves, phase diagrams and Poincare maps are drawn when the system is put in periodic motion, quasi-periodic motion and chaotic motion. Experimental results show that the reduced-order model using nonlinear Galerkin method has higher accuracy than common Galerkin method, especially when the original model has a relatively high order.