根據Wittrick的12條微分方程式並且加入預扭角效應，本文利用Rayleigh-Ritz方法建立了空間曲樑結構的一維有限元素運動方程式，以使用在具有預扭角螺旋彈簧的分析。對具預扭角螺旋彈簧問題，本文修正之Wittrick的12條微分方程式可利用轉置矩陣法(transfer matrix method)解得其正解。由此正解中之位移函數(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與勁度矩陣。本文建立之有限元素分析模式使用於具預扭角螺旋彈簧的自然頻率與模態分析時，為簡化分析，因此本文將具預扭角螺旋彈簧的每個元素之扭轉角度固定，但彼此間並不相同。當使用的元素數目增加，此種簡化作法所導致的幾何誤差可以減小。本文有限元素分析結果並和文獻中不同作法的結果作比較。由數值結果可展現本文具預扭角螺旋彈簧元素的準確性與效率。同時本文亦探討一些參數，如邊界條件、彈簧圈數、矩形剖面長寬比與材料性質等，對具預扭角螺旋彈簧自然頻率與模態的影響。

Following the Wittrick’s twelve differential equations modified with including the pre-twisted angles effect, the present study uses the Rayleigh-Ritz method to develop the one-dimensional finite element equations of motion for the spatial curved beams which can be used in the analysis of pre-twisted helical springs. For the static problems of pre-twisted helical springs, the modified Wittrick’s twelve differential equations can be solved for the exact solutions by the transfer matrix method. The displacement functions in the exact solutions are used to derive the interpolation functions which are used in the computation of the mass and stiffness matrices of the present spring element. The present finite element model is applied in the free vibration analyses of pre-twisted helical springs. To simplify the analysis, constant but different pre-twisted angle is assumed for each element of a pre-twisted helical spring, and the geometric error incurred can be reduced if the number of elements is increased. The present model performance is compared with the analytic solutions of different approaches in the literature. The accuracy and efficiency of the present pre-twisted helical spring element are demonstrated through the numerical results. Also the effects of some parameters, such as boundary condition, number of coils, aspect ratio of the rectangular cross-section and material properties, on the natural frequencies and modes of pre-twisted helical springs are investigated.

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