非线性梁振动偏微分方程求解的L-稳定方法

L-Stable Method for Solving Partial Differential Equations of Nonlinear Beam Vibration

摘要: 基于高阶非线性梁振动偏微分方程的一般形式，构造了数值求解的L-稳定格式。首先，选取三角插值基函数，基于插值定理进行空间离散，将带有初边值条件的偏微分方程求解问题转化为微分–代数方程求解。然后在时间区间上构造L-稳定求解格式进行求解。以无轴向运动简支梁在外部激励下的强迫振动方程为例进行数值仿真，对梁的位移轨迹、边界条件及系统能量进行探究，并与龙格–库塔法、微分求积法进行对比，结果表明，L-稳定方法可以在较大步长下满足边界，位移轨迹与模型方程一致，在计算精度和稳定性上都有较好的体现。

Abstract: The general form of higher order nonlinear partial differential equation of beam vibration was given, and the solution method based on L-stable scheme was studied. Firstly, the triangular interpolation basis function was selected to discretize the space based on the interpolation theorem, and the problem of solving partial differential equations with initial boundary conditions was transformed into solving differential-algebraic equation. Then the L-stable solution scheme was constructed in the time interval. The forced vibration equation of a simply supported beam without axial motion subject to external excitation was taken as an example; the displacement trajectory, boundary conditions and system energy of the beam were studied, and compared with Runge-Kutta method and differential quadrature method. The results show that the L-stable method can satisfy the boundary conditions in large step size, the displacement trajectory is consistent with the model equation, and it has good performance in calculation accuracy and stability.