根據Wittrick的12條微分方程式並且加入溫度效應，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，以使用在螺旋彈簧的分析。對圓柱形螺旋彈簧靜態問題，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與非圓柱形螺旋彈簧的靜態問題及自然頻率與模態分析。為簡化分析，本文以圓柱形螺旋彈簧元素來模擬非圓柱形螺旋彈簧。因此模擬非圓柱形螺旋彈簧的每個元素之曲率為固定，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。本文靜態問題分析中圓柱形與非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螺旋彈簧元素的準確性與效率。同時本文亦探討ㄧ些參數對三種非圓柱形螺旋彈簧自然振動頻率的影響。

Following the Wittrick’s twelve 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 spatial curved beams which can be used in the analysis of helical springs. For the static problems of cylindrical 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, 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 and non-cylindrical helical springs, respectively. To simplify the analysis, the non-cylindrical helical springs are modeled with the cylindrical helical elements attached sequentially to each other. Hence, constant but different curvature is assumed for each element of a non-cylindrical helical spring, and the geometric error incurred can be reduced if the number of elements is increased. In the present static analysis, the cylindrical and non-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 element in the literature. The accuracy and efficiency of the present spring element are demonstrated through the numerical results. Also the effects of some parameters on the natural frequencies of three types of non-cylindrical helical springs and investigated.

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