A Rational Design of Helical Spring

为了使螺旋弹簧的设计更加合理，在分析螺旋弹簧的动态特性和应力特性时，必须考虑螺距角和端部条件的影响。近年来，作者证明了基于有限元方法的解析法对各种弹簧特性的评估是有效的。然而，将有限元方法应用于螺旋弹簧的设计时，需要进行繁琐的计算以获得弹簧单元的形函数，需要大量的数值分析知识和熟练的编程技术。因此，在本报告中，为了避免上述困难并简化编程，在求取形函数和推导弹簧单元矩阵所需的计算处理中使用了符号数学系统REDUE。由于符号运算需要大量的计算机内存，因此在进行符号积分以获得弹簧单元矩阵时，需要仔细编程以节省内存。提出了一种新的符号积分算法，并成功地得到了单元矩阵。作为上述计算方法的应用，本文还讨论了螺旋弹簧固有频率对横向稳定性的影响。分析表明，当平移振型对应的固有频率等于横向振型对应的两倍固有频率时，横向失稳现象明显。

In order to rationalyze the helical spring design, it is essential to take the effects of the pitch angle and end conditons into account in the analysis of the dynamic characteristics and stress characteristics. Recently, the authors showed that the analytical approach based on the finite element method proves effective for evaluating various kinds of spring characteristics. However, when applying the finite elemnt method to the helical spring design, cumbersome caluculations for obtaining the shape function for the spring element, abundant store of knowledge about numerical analysis and skillful programming techniques are reqired. In the present report, therefore, in order to avoid the above mentioned difficulties and simplify programming, the symbolic mathematic system REDUCE is used in computational processing reqired for obtaining the shape function and deriving spring element matrices. As symbolic manipulation reclaims large amounts of computer memory, careful programming for memory savings is required in the symbolic integration obtaining spring element matrices. New algorithm for symbolic integraton is developed and element matrices are obtained successfully.The pressent report also deals with the effect of natural frequencies of coil spring on lateral stability as an application of the above calculation method. Analysis shows that lateral instability apperars clearly, when natural freqency ,corresponding to translational mode, equals twice natural frequency, corresponding to transverse mode.