高阶梁振动偏微分方程离散变分方法

Discrete Variational Method for Higher Order Beam Vibration Partial Differential Equation

摘要: 针对高阶梁振动偏微分方程这类求解问题，研究了离散变分方法。首先运用微分求积法离散空间，在时间区间上构造离散变分方法，对离散后的欧拉–拉格朗日方程进行变分。仿真实验运用MATLAB进行数值计算。以无轴向运动简支梁在外部激励下的强迫振动方程为例研究了插值基函数的种类、时间步长、插值节点类型与仿真时间等对求解的影响。数值结果表明，短时间内离散变分法的约束和能量稳定性优于经典龙格–库塔法；长时间仿真下，离散变分法的结果精度高于龙格–库塔法，并且可以很好地保持约束的稳定性。

Abstract: The discrete variational method is studied for the solution of high-order beam vibration partial differential equations. Firstly, the differential quadrature method is used to discretize the space, the discrete variational method is constructed on the time interval, and the Euler-Lagrange equation 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 excitation 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 discrete variational method is higher than that of Runge-Kutta method, and can maintain the stability of constraints.