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，必须考虑螺旋角和端部条件的影响。最近，作者表明，基于有限元法的分析方法被证明是有效的评估各种弹簧特性。然而，将有限元法应用于螺旋弹簧设计时，弹簧单元形函数的计算繁琐，需要有丰富的数值分析知识和熟练的编程技术。因此，为了避免上述困难和简化程序设计，本文采用符号数学系统REDUCE进行求形函数和弹簧单元矩阵的计算处理。由于符号运算需要占用大量的计算机内存，因此在符号积分获得弹簧单元矩阵时需要仔细编程以节省内存。本文提出了一种新的符号积分算法，成功地求出了单元矩阵，并作为上述计算方法的应用，讨论了螺旋弹簧固有频率对横向稳定性的影响。分析表明，当对应于平移模态的固有频率等于对应于横向模态的固有频率的两倍时，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.