动力学方程的解析逐步积分法

AN ANALYTICAL STEP-BY-STEP INTEGRAL PROCEDURE OF DYNAMICS EQUATIONS

提出了求解动力学方程的一个新型的逐步积分法。基于动力学方程的解析齐次解，构造出动力学方程解的一般积分表达式，借助于显式、自起动、预测－校正的单步四阶精度的积分算法，离散方程右端的等价荷载项，给出了一个新的解析逐步积分方法格式。如果用分块求解，其刚度阵、质量阵等将有较小的规模，将使计算效率更高。算例表明本文方法比中心差分法、Ｎｅｗｍａｒｋ、Ｗｉｌｓｏｎ－θ、Ｈｏｕｂｏｌｔ法等有较高的精度，本文结果更接近解析解。本文方法也适用于非线性，因为本计算格式是显示，因此不需要迭代求解。

A new step-by-step integral procedure of dynamics equations is presented. The general expression of solution of dynamics equations is obtained on the basis of the homogenous analytical solutions of dynamics equations. The explicit analytical integration algorithm, which is characterized by fourth-order accuracy, self-starting and self-correcting, is employed to discretize the equivalent load terms at the right-hand terms of the equations. The transfer matrix is not limited by time step size. If the group solution method is used, the size of stiffness matrix and mass matrix will be reduced and the method will be more cost-effective. Numerical examples show that the results are highly accurate in comparison with those of Newmark, Wilson- Houbolt and central difference method. Moreover, the present results are closer to the exact solution, than those of other methods. In addition, the method is also suitable for treatment of nonlinear dynamics problems since no iterative procedure is needed in the present algorithm.