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Advances in Applied Mathematicsis used to discretize the space,
the discrete variational method is constructed on the time interval, and the Euler-Lagrange equa-
tion is variational. The simulation experiment uses MATLAB for numerical calculation. Taking the
forced vibration equation of a simply supported beam without axial motion under external excita-
tion as an example, the effects of the type of interpolation basis function, time step, interpolation
node type and simulation time on the solution are studied. The numerical results show that the
constraint and energy stability of the discrete variational method in a short time are better than
those of the classical Runge-Kutta method; under long-time simulation, the accuracy of the dis-
crete variational method is higher than that of Runge-Kutta method, and can maintain the stability
of constraints.
Keywords
Partial Differential Equation of Beam Vibration, Differential Algebraic Equation, Discrete
Variational Method, Stability
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Open Access
1. 引言
梁振动方程在土木工程、电子、桥梁建设等领域有着广泛的应用[1] [2]，该方程属于高阶偏微分方程，
对梁振动方程精确稳定的数值求解具有重要意义[3]。求解梁振动方程的数值方法有有限差分法、伽辽金
方法、微分求积法等[4]。微分求积法具有数学原理简单、易于实现和计算高效性等优点[5] [6] [7]。王波
和陈立群等[8]用微分求积法分析了在复杂边界下轴向变速黏弹性梁的稳态幅频响应，并和解析近似解进
行对比。唐媛[9]应用微分求积研究在不同边界条件下梁的力学响应问题，分析了功能梯度参数对梁力学
响应的影响。传统微分求积法在处理约束时对约束单独处理，约束的稳定性不能得到很好的保持。
为了提高精度和稳定性，在处理约束时把约束看作代数方程组，将偏微分方程转化为微分代数方程
组，本文采用课题组的离散变分方法加以求解。近年来，离散变分方法已经在多体系统动力学中得到了