根據Wittrick的微分方程式並且加入溫度效應，本文利用Rayleigh-Ritz方法建立了圓柱形螺旋彈簧的一維有限元素運動方程式。對圓柱形螺旋彈簧靜態問題，本文修正之Wittrick的微分方程式可解得其正解，由此正解之位移函數(displacement functions)可求得本文彈簧元素之內插函數，以使用在本文建立之有限元素方程式中求取元素之質量矩陣、勁度矩陣與熱負荷向量等。本文建立之有限元素分析模式分別使用於圓柱形螺旋彈簧的靜態問題及自然頻率與模態分析。本文靜態問題分析中圓柱形螺旋彈簧除承受機械力外另考慮承受溫度變化。本文有限元素分析結果並與正解和文獻中其他彈簧元素的結果比較。由數值結果可展現本文圓柱形螺旋彈簧元素的準確性與效率。

Following the Wittrick’s differential equations modified with including the temperature effect, the present study uses the Rayleigh-Ritz method to develop the one-dimensional finite element equations of motion for the cylindrical helical springs. For the static problems of cylindrical helical springs, the modified Wittrick’s differential equations can be solved for the exact solutions. The displacement functions in the exact solutions are used to derive the interpolation functions which are used in the computation of the mass, stiffness matrices and thermal load of the present spring element. The present finite element model is applied in the static problems and free vibration analyses of cylindrical helical springs, respectively. In the present static analysis, the cylindrical helical springs subjected to uniform temperature change are also considered. The present model performance is compared with that of exact solutions and other finite elements. The accuracy and efficiency of the present spring element are demonstrated through the numerical results.

1. J. H. Michell, “The small deformation of curves and surfaces with
application to the vibrations of a helix and a circular ring”, Mess. Math. pp. 19-68, (1890).
2. A. E. H. Love, “The propagation of waves of elastic displacement along a helical wire”, Trans. Camb. phil. Soc. pp. 18-36, (1899)
3. S. P. Timoshenko, “Theory of Elastic Stability”, New York and London: McGraw Hill. First edition, (1936)
4. Y. Yoshimura and Y. Murata, “On the elastic waves propagated along coil springs”, Rep. Inst. Sci. Technol., Tokyo, pp. 6-27, (1952)
5. W. H. Wittrick, “On elastic wave propagation in helical springs”,
International Journal of Mechanical Sciences 8, pp. 25–47, (1965)
6. Y. Kagawa, “On the dynamical properties of helical springs of finite length with small pitch”, J. Sound Vib. 8, 1, (1968)
7. J. W. Phillips and G. A. Costello, “Large deflections of impacted helical springs”, J. Acoust. Soc. Am. 51, pp.967-973, (1972)
8. G. A. Costello, “Radial expansion of impacted helical springs”, J. Appl. Mech. Trans. ASME 42, pp. 789-792, (1975)
9. S. K. Sinsha and G. A. Costello, “The numerical solution of the dynamic response of helical springs”, IJNME 12, pp. 949-961, (1978)
10. JE. Mottershead, “Finite elements for dynamical analysis of helical rods”, International Journal of Mechanical Sciences. 22, pp. 267–83, (1980)
11. L. Della Pietra and S. Della Valle, “On the dynamic behaviour of axially excited helical springs”, Meccanica, 17, pp. 31-43, (1982)
12. D. Pearson, “The transfer matrixmethod for the vibration of compressed helical springs”, Journal of Mechanical Engineering Science 24, pp. 163-171, (1982)
13. D. Pearson and WH. Wittrick, “An exact solution for the vibration of helical springs using a Bernoulli–Euler model”, International Journal of Mechanical Sciences 28, pp.83–96, (1986)
14. Y. Lin. and A. P. Pisano, “General Dynamic Equations of Helical Springs With Static Solution and Experimental Verification”, ASME J. Appl. Mech., 54, pp. 910–916, (1987)
15. W. Jiang W. Jones, K. Wu and T. Wang , “Non-linear and linear, static, and dynamic analyses of helical springs”, 1989 Proceedings of the 30th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Mobile, pp. 386-395, (1989)
16. W. Jiang, W.K. Jones, T. L. Wang and K. H.Wu , “Free vibration of helical springs”, transactions of the ASME 58, pp. 222-228, (1991)
17. L. E. Becker and W. L. Cleghorn, “On the buckling of helical
compression springs”, International Journal of MechanicalSciences,
34, pp. 275, (1992)
18. V. Yildirim ,“Investigation of parameters a7ecting free vibration frequency of helical springs”, International Journal for Numerical Methods in Engineering 39, pp.99–114, (1996)
19. GG. Chassie, LE. Becker, and WL. Cleghorn ,“On the buckling of helical springs under combined compression and torsion”, International Journal of Mechanical Sciences 39, pp. 697–704, (1996)

20. Kim. Dojoong, “Development of a finite element program for dynamic analysis of helical springs”, IEEE Mechanics, KORUS’99, pp. 309-314, (1999)
21. J. N. Reddy, “An introduction to the Finite Element Method” , Second Edition., New York：McGraw Hill,(1993)
22. Mott 編著，黃仁明 編譯，“機械元件設計”，全華科技圖書股份有限公司 印行。
23.黃祖德 主編，林金填 陳常成 戴義國 校訂，“機械元件設計”，文京圖書股份有限公司，總經銷大揚出版社。
24.楊政達 張嘉峰 楊智旭 編著，“MATLAB進階與工程問題應用”，全威圖書股份有限公司 出版。
25.George Lindfield and Dr. John Penny編著，黃俊銘 編譯，“數值方法-使用MATLAB程式語言” Second Edition.，全華科技圖書股份有限公司 印行。