一种求解非线性模态的改进Galerkin法

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振动与冲击 ›› 2022, Vol. 41 ›› Issue (18) : 157-165.

论文

李诚1，李鸿光2

作者信息 +

Improved Galerkin method for computing nonlinear normal modes

LI Cheng1,LI Hongguang2

Author information +

文章历史 +

摘要

提出了一种改进的 Galerkin方法以计算非线性系统的非线性模态在不变流形定义下的解曲面。在已有的两种非线性模态Galerkin解法的基础上，该方法在假设函数中引入了有限元形式的展开式，并利用该问题中特定的Jacobian矩阵的稀疏性加速待求系数的非线性代数方程组的求解。以一种非线性双级隔振器为例，采用了这三种方法计算其主共振对应的非线性模态，并将解进行了比较。本方法在求解域较大时，仍能获得较为准确的解。本方法进一步与已有的Galerkin法进行了综合，在保证求解精度的基础上加速了计算。
关键词：非线性模态；不变流形；Galekin法；稀疏雅可比矩阵

Abstract

An improved Galerkin method is proposed to solve the nonlinear normal mode of the nonlinear system under the definition of invariant manifold. On the basis of two existing Galerkin methods for the nonlinear normal mode solution, this method introduces the finite element form of shape functions into the solution expansions, and applies a corresponding strategy for the approximation of the specific sparse Jacobian matrix to accelerate the calculation of the expansion coefficients. A nonlinear two-stage vibration isolator is considered. Its nonlinear normal mode corresponding to the primary resonance is solved, and the solutions obtained by these three methods are compared. The proposed method can yield more accurate solutions in large domains. Then this method is integrated with the existing Galerkin method, and the calculation is further accelerated to obtain an accurate solution.
Key words: nonlinear normal mode; invariant manifold; Galerkin method; sparse Jacobian matrix

导出引用

李诚1，李鸿光2. 一种求解非线性模态的改进Galerkin法[J]. 振动与冲击, 2022, 41(18): 157-165

LI Cheng1,LI Hongguang2. Improved Galerkin method for computing nonlinear normal modes[J]. Journal of Vibration and Shock, 2022, 41(18): 157-165

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