杆系结构自由振动精确求解的理论和算法

杆系结构的自由振动特性对结构的抗震设计至关重要。与常规有限元方法采用近似形函数将原问题化为线性特征值问题不同,本文的精确方法从杆件精确的形函数出发获得精确的动力刚度,将原问题化为非线性特征值问题。已有的Wittrick-Willliams算法很好地解决了该问题的频率求解。在此基础上,进一步提出了求解该非线性问题的导护型Newton法格式,并优化了各个算法环节。该法能同时求出频率和振型,求解结果精确可靠且具有二阶收敛速度,是一种快速精确、可靠实用的工程计算方法。

THEORY AND ALGORITHM OF THE EXACT METHOD FOR FREE VIBRATION PROBLEMS OF SKELETAL STRUCTURES

The solution of free vibration problem of skeletal structures provides important data for engineering design. This paper surveys a new exact method which uses exact dynamic stiffnesses and therefore results in a nonlinear eigenvalue problem in contrast to the traditional finite element method which reduces the problem into a linear algebraic eigenvalue problem by using approximate shape functions. The well-established Wittrick-Williams algorithm can be used to calculate the frequencies of the nonlinear problem elegantly. Recently the authors presented a new Newton-type method for this nonlinear eigenvalue problem, incorperating many optimum techniques. The recursive use of the proposed method employs the Wittrick-Williams algorithm to guide and guard each Newton correction and therefore gives secure second order convergence on both natural frequencies and mode vectors. Numerical examples show that this method is not only exact and reliable but also very economical and efficient, and therefore is well applicable to large-scale engineering problems.