采用谱单元Galerkin法求解非线性模态  

Calculation of Nonlinear Normal Modes using Galerkin Method with Spectral Elements

作　　者：李诚[1] 李鸿光[1] LI Cheng;LI Hongguang(State Key Laboratory of Mechanical System and Vibration,Shanghai Jiaotong University,Shanghai 200240,China)

机构地区：[1]上海交通大学机械系统与振动国家重点实验室,上海200240

出　　处：《噪声与振动控制》2022年第4期25-31,37,共8页Noise and Vibration Control

基　　金：国家自然科学基金资助项目(11972222)。

摘　　要：为进一步提高非线性振动系统在不变流形定义下的非线性模态的求解精度,采用一种基于谱单元的Galerkin求解方案。不同于已有的非线性模态Galerkin分片求解方法,该方案选取第二类Chebyshev多项式的零点构造单元的Lagrange插值函数,将其与谐波函数一起作为基函数对整个求解域进行Galerkin离散。在展开系数的迭代求解中,Jacobian矩阵的稀疏性因选取的谱单元阶数不同而不同。采用该方法与分片求解法分别计算一个非线性振动系统的非线性模态并进行比较。结果表明该方法在求解域较大时仍可获得较为准确的解。A Galerkin solution scheme based on spectral elements is applied to improve the accuracy of nonlinear normal modes solution of the nonlinear vibration system under the definition of invariant manifolds.Different from the existing Galerkin piecewise segment method,in this scheme the basis functions for each element are constructed by the harmonic functions and the Lagrange interpolations on the zeros of Chebyshev polynomials of the second kind.The Galerkin discretization is implemented on the entire domain.In the iterative process of expansion coefficients,the sparsity of the Jacobian matrix varies with the order of the polynomials in spectral elements.The proposed method and the piecewise segment method are respectively used for solving the nonlinear normal modes of a nonlinear vibration system.Comparison of the results shows that the proposed method can yield more accurate solutions in large domains.

关 键 词：振动与波 非线性模态 不变流形 GALERKIN法 谱单元 

分 类 号：O322[理学—一般力学与力学基础]