Computerized Symbolic Manipulation in ViBration Analysis

在振动分析中，计算机处理程序很少被发现，因此在基本算法、编程和应用限制方面存在许多不清楚的地方。因此，本报告的目的是发展符号处理算法的振动问题的解决和统一的数值方法和符号方法，以简化分析过程，使用计算机符号处理程序在振动分析的潜力进行了讨论。利用计算机符号处理可以有效地完成的任务如下：（1）求解线性振动系统的控制方程，（2）多自由度的固有频率和模态，（3）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组分解算法，对控制方程的求解方法进行了改进。采用梯度函数简化了多自由度质量矩阵和刚度矩阵的推导过程，并指出将符号处理方法应用于实验模态分析，大大提高了编程效率和灵活性。采用符号数学系统进行实验模态分析，可以成功地分析曲率半径对电冰箱管路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.