二端连续元件动力学特性的矩阵描述法

Investigation on dynamic characteristics of two poles continuous elements by matrix description approach

在元件坐标系下,分别推导了简谐激励下横向振动的欧拉-伯努利梁和纵向振动的杆的输入端状态向量及输出端状态向量与微分方程的解的关系。在状态向量正方向的定义相同时,给出了向前传递矩阵、向后传递矩阵、阻抗矩阵和导纳矩阵的解析公式,研究了四种矩阵的特性。结合欧拉梁和杆的相关公式,得到了一般二维梁的传递矩阵和广义阻抗矩阵的解析公式。数值算例表明:在激励频率较低时,该方法与有限元法的结果基本相同。最后,讨论了动力学中的矩阵描述法(阻抗矩阵法、导纳矩阵法和传递矩阵法)与静力学中的矩阵描述法(位移法、力法和混合法)的对应关系。矩阵描述法有利于利用计算机编程计算动力学问题,所推导的公式可用于连续系统的动力学分析,计算工作量不大且精度较高。

Relationships between state vectors and solutions of differential equations of an Euler-Bernoulli beam and of a rod were deduced respectively in element coordinating system.Analytical formulae of forward transfer matrix,reverse transfer matrix,impedance matrix and mobility matrix were derived separately with the assumption of the same positive definition of state vectors.The characteristics of the four matrices were studied.Analytical formulae of the four matrices of a general beam were given by combining the corresponding formulae of Euler-Bernoulli beam with those of the rod.Numerical examples show that the results of the present method and those of the finite element method are nearly the same.The relationship between dynamic matrix description method and static matrix description method was discussed.It is also shown that matrix description methods facilitate solving dynamic questions by using computer programming.Analytical formulae in the paper can be used in dynamic analyses of continuous systems to achieve good precision with less computation.