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 引用本文: 金栋平. 非线性振动系统的多项式向量方法. 力学学报, 2023, 55(10): 2373-2380.
Jin Dongping. Method of polynomial vectors for nonlinear vibration systems. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(10): 2373-2380.
 Citation: Jin Dongping. Method of polynomial vectors for nonlinear vibration systems. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(10): 2373-2380.

## METHOD OF POLYNOMIAL VECTORS FOR NONLINEAR VIBRATION SYSTEMS

• 摘要: 对于常微分方程描述的非线性振动系统, 当采用摄动方法求近似解时, 先是给出满足各阶近似解的二阶常微分方程组, 继而依次对每一个常微分方程进行求解, 以致多自由度非线性振动系统的求解过程相当繁琐. 文章针对常微分方程表示的非线性振动系统, 提出了一种求解非线性振动系统近似解的多项式向量方法, 该方法将二阶常微分方程组表示成一阶状态方程组, 将非线性部分写成常数矩阵和多项式向量之积的形式. 然后, 采用直接摄动方法, 获得每个幂次近似解所满足的一组状态方程, 此时状态方程的非线性部分成为常数矩阵和前一幂次近似解作为元素组成的多项式向量的乘积. 进一步, 借助Toeplitz矩阵将多项式向量之乘法表示成矩阵形式, 以解决多项式相乘带来的幂次方系数的确定问题, 再根据一阶非齐次方程组的求解方法, 获得状态方程组的全部近似解析解. 多项式向量方法将二阶常微分描述的非线性振动求解过程转换为一阶非齐次状态方程组的求解问题, 计算过程主要是矩阵和向量之间乘法运算, 提高了计算效率和程序化水平.

Abstract: The perturbation methods for nonlinear vibration systems make it necessary to solve a set of second-order ordinary differential equations (ODEs), which are obtained by equating the like power of the perturbation solutions respectively. One of the main drawbacks of the ODEs-based methods is of low efficiency, especially for nonlinear vibration systems of multiple degrees of freedom. In this paper, a method of polynomial vectors for solving the approximate solution of nonlinear vibration systems is proposed. The second order ordinary differential equations are written in a set of state equations of the first order first, wherein the nonlinear terms of the state equations are expressed as the products of a constant matrix and a polynomial vector with the like power. By using the direct perturbation method, the linear non-homogeneous equations are obtained for the like power approximations, while the nonlinear terms are written as the products of the constant matrix and the polynomial vector with the previous approximate solutions as its element. Furthermore, the multiplication of polynomials in the polynomial vector is expressed in matrix form via Toeplitz matrix, and then all approximate analytical formulas of the state equations are determined by the first-order non-homogeneous equations. Results show that the proposed method based upon the state equations yields a concise calculation for nonlinear vibration systems of multiply degrees of freedom.

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