Computerized Symbolic Manipulation in ViBration Analysis

在振动分析中很少发现计算机操作程序，因此在基本算法、编程和应用限制方面存在许多不明确的问题。因此，本报告旨在开发求解振动问题的符号操作算法，并统一数值方法和符号方法，以简化分析过程。本文首先讨论了计算机符号操作程序在振动分析中的应用潜力。计算机符号处理可以有效地完成以下任务：(1)求解线性振动系统的控制方程；(2)求解多自由度的固有频率和模态；(3)求解非线性系统的近似方法，如摄动法和渐近法。采用一种特殊的一阶微分方程分解算法，对控制方程的求解有了重要的改进。为了简化多自由度质量矩阵和刚度矩阵的推导，实现了梯度函数。本文还指出，将符号操作程序应用于实验模态分析，大大提高了编程效率和灵活性。利用符号数学系统进行了实验模态分析，成功地分析了曲率半径对制冷机管道系统动态特性的影响。

Computerized manipulation procedure is seldom found in the vibration analysis, and hence there are a number of unclarified points regarding fundamental algorithms, programming, and limitations in applications. The present report therefore aims at developing symbolic manipulation algorithms for solution of vibration problems and unifying numerical methods and symbolic methods in order to simplifying analysis process.The potential of using computerized symbolic manipulation procedure in vibration analysis is discussed at first in the present report. Tasks which can be efficiently performed using computerized symbolic manipulation are as follows : (1) solving governing equations in linear vibration systems, (2) natural frequencies and modes in multiple degrees of freedom, (3) approximation methods in nonlinear systems, such as perturbation method and asymptotic method. By applying a special algorithm decomposing into simultaneous first order differential equations, an important improvement is obtained in the solution of governing equations. The gradient function is implemented for simplifying the derivation of the mass and stiffness matrices of multiple degrees of freedom.In the present report, it is also pointed out that both programming efficiency and flexibility are greatly improved by applying the symbolic manipulation procedure to the experimental modal analysis. Effects of radius of curvature on the dynamic characteristics of piping systems in refrigerator can be analyzed successfully by the experimental modal analysis using the symbolic mathematical system.