非线性振动分析的切比雪夫谱元法

Chebyshev Spectral Element Method for Analysis of Nonlinear Vibration Problems

为了进一步探索Chebyshev时间谱元法求解非线性的振动问题,从Bubnov-Galerkin方法出发,在第二类Chebyshev正交多项式极点处;用重心Lagrange插值来构造节点基函数及其特性,推导了非线性振动问题的伽辽金谱元离散方案,借助Newton-Raphson法求解非线性方程组。对于非线性单摆,还需要将二分法和重心Lagrange插值结合求解角频率。以Duffing型非线性振动和非线性单摆振动问题为例,验证了此方法具有现实可行和高精度的优点。

The solution of nonlinear vibration problems was studied by using Chebyshev spectral elements method.The node-based functions were constructed by barycentric Lagrange interpolation at the pole points of Chebyshev orthogonalpolynomials of the 2nd kind which characteristics were analyzed by using Bubnov-Galerkin method. Galerkin discretizationscheme for the nonlinear vibration problems was derived. Finally, the nonlinear equations were solved by Newton-Raphsonmethod. For nonlinear single pendulums, the angular frequencies were solved using the combination of the dichotomywith the barycentric Lagrange interpolation. Two examples of Duffing-type vibration equations and nonlinear vibration ofpendulums were employed to illustrate the feasibility and advantages of high-precision of the proposed method.