錐狀蛇行彈簧具有獨特的變形行為，空間使用較有效率，適合使用在微機電元件之中。本研究利用卡氏第二定理推導出錐狀蛇行彈簧在同平面與出平面方向的彈簧常數的理論公式，可供設計者快速計算錐狀蛇行彈簧在不同軸向的彈簧常數。本研究依針對錐狀蛇行彈簧的變形特性，對彈簧的剛性和彈簧覆蓋面積之間的關係作一完整的分析與討論，並定義一個新的參數K/A，作為比較彈簧特性的性能指標，K/A表示在單位面積下提供的彈簧常數。當K/A較低時，顯示在相同的單位面積下，彈簧可提供較低的彈簧常數，使元件具有較大的位移變化量。本研究亦分析錐狀蛇行彈簧在同平面與出平面運動時，幾何尺寸的參數對K/A的影響。從理論分析結果發現，錐狀彈簧在出平面與同平面運動時，K/A的效能均優於傳統蛇行彈簧。當錐狀蛇行彈簧在具有更多的折數、較長的總長度、較大的張角角度、較細的彈簧樑寬度以及較薄的彈簧厚度等情況下會有較小的K/A值，亦即錐狀蛇行彈簧在單位面積下可提供更多的變形與位移。本研究成功的製造同平面以及出平面運動的錐狀蛇行彈簧，由實驗結果顯示，實驗所得的彈簧常數和理論分析推導所得的彈簧常數相當吻合，誤差值在10% 以下。此外，本研究進一步分析彈簧結構的厚度與寬度比對同平面及出平面運動之彈簧常數之影響。結果顯示，彈簧的寬度與厚度比(w/h)大於1時，平面的彈簧常數(ky)會大於出平面方向的彈簧常數(kz)。反之，彈簧寬度與厚度比小於1時，出平面方向的彈簧常數(kz)會大於平面運動方向的彈簧常數(ky)。
本文進一步討論幾何尺寸對於錐狀蛇行彈簧非線性行為的影響。分析結果顯示，當彈簧折數越多、角度越大、寬度越細且厚度越薄的情形下，彈簧的非線性變形行為容易發生。利用線性迴歸，將達芬方程式(Duffing equation)中的彈簧常數k1及k3表示成彈簧的角度、折數、寬度以及厚度等幾何參數組成之迴歸方程式。本文定義一非線性轉折點作為非線性發生的參考值，討論錐狀蛇行彈簧在非線性轉折點時的最大應力值和幾何尺寸的關係。由結果顯示，彈簧的張角角度以及彈簧樑的寬度，對於彈簧達到非線性行為有明顯影響。幾何非線性變形行為容易在彈簧具有大張角角度及細的彈簧寬度的情況下發生。

The awl-shaped serpentine microspring plays a important role in MEMS devices due to the advantages of unique deformation behavior and more effective spatial usage. The analytic solutions of spring constants of awl-shaped serpentine microspring for in-plane and out-of-plane motion are theoretically analyzed by using Castigliano’s theorem. Using the theoretical solutions, the spring constant of microspring can be calculated quicker and the awl-shaped serpentine microsprings be easierly designed in MEMS devices. Moreover, the effect of spring constant and layout area on deformation performance of awl-shaped serpentine microspring are discussed. A parameter of spring constant to layout area ratio (K/A) is defined to be used as the index for comparing spring constants under the same unit area. A smaller K/A value would induce larger deformation under the same applying force and layout area. From the theoretical results, the awl-shaped serpentine microspring has a lower K/A value than the traditional serpentine microspring for in-plane and out-of-plane motion with the same total effective length and folds. Hence, the awl-shaped serpentine microsprings can induce larger deformation than traditional serpentine microsprings under the same applying force and layout area. The effect of the size and geometry of the awl-shaped serpentine microspring on the spring performance was investigated. With a greater taper angle, a longer total effective length, more folds, a smaller beam width and lower beam thickness, the awl-shaped serpentine microspring will produce a smaller K/A to achieve a larger displacement under the same layout area.The proposed awl-shaped serpentine microspring was successfully fabricated for in-plane and out-of-plane motion. Experiments results were conducted to compare the theoretical and numerical results, which were in close agreement. The error was less than 10%. Furthermore, the spring constants for in-plane and out-of-plane motion are compared and discussed. As w∕h being >1, the spring constant of in-plane motion ky is always larger than that of out-of-plane motion kz. If w∕h is <1, the spring constant kz would be larger than ky.
Moreover, the effect of geometric sizes on the nonlinear deformation of microspring is investigated too. It is found that the nonlinear deformation is easier found for more folds, a greater taper angle, a smaller beam width, and lower beam thickness of awl-shaped serpentine microspring. Using the linear regression method, the linear spring constant k1 and cubic spring constant k3 of the Duffing equation could be determined and expressed in terms of N, ϕ, w, and h by the regression equation. Therefore, a critical nonlinear point is defined as the beginning of nonlinear deformation behavior of the awl-shaped serpentine microspring. The effect of the maximum stress and geometric sizes on the critical nonlinear point of awl-shaped serpentine microspring is also discussed. The analytic results illustrate that the taper angle and beam width of awl-shaped serpentine microspring is the significant afactors on the nonlinear deformation behavior.

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