渦旋彈簧之靜態變形與穩定性分析

Deformation and vibration of a spiral spring

陳逸軒(I-Shein Chen)

指導教授 ： 陳振山

國立臺灣大學/工學院/機械工程學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.00819

摘要

在本文中，使用空間彈性理論來分析渦旋彈簧的靜態變形與振動頻率，首先考慮圓形截面的渦旋彈簧，渦旋彈簧內外有兩端點，其中一端夾持於一固定端，另一端夾持在一可旋轉的圓形轉盤上。由靜態分析中可以得知，渦旋彈簧受扭轉一定角度之後，彈簧發生挫曲，形狀由平面變成三維變形。從扭力與旋轉角的關係圖中，可知彈簧會在轉兩圈後回到相同變形。在振動分析中，求得平面變形有平面及空間兩種模態振形，空間振動模態所相對應的自然頻率比平面振動模態所對應的自然頻率先下降至零，可以由此預測彈簧將以此模態振形發生空間變形。為驗證理論分析的正確性，本研究設計了一個初始圈數為五圈、圓形截面的渦旋彈簧，以及實驗機構，量測其扭力與旋轉角的關係與自然頻率，實驗結果與理論分析相符。接下來考慮矩形截面的渦旋彈簧，在扭力與旋轉角的關係圖中，可以看出矩形截面的長寬比不影響其平面變形曲線，但是會影響到發生挫曲時的旋轉角，以及空間變形曲線。最後，本論文利用振動法求出來的挫曲角度與矩形截面的長寬比的關係，也與前人利用平衡法的做出的結果做了比較。

關鍵字

渦旋彈簧 ； 彈性圓桿理論 ； 振動

並列摘要

In this paper we use elastica theory to study the deformation and natural frequencies of a spiral spring. We first consider a spiral spring with circular cross section. Static analysis shows that the spring undergoes planar deformation first and buckles into spatial deformation at a bifurcation point. The load-deflection curve repeats itself after two full turns. Therefore, the spring returns to its initial shape after two full turns. The planar deformation has two types of mode shapes; i.e., in-plane and out-of-plane. If one proceeds from initial shape to the bifurcation point, the natural frequency of one of the out-of-plane modes reduces to zero. This out-of-plane mode shape will be the buckling mode when planar deformation buckles into spatial deformation. Experiments are conducted on a custom made spiral spring with circular cross section. The measured deformations and natural frequencies agree with theoretical predictions very well. We next proceed to consider a spiral spring with rectangular cross section. It is found that the planar deformation is independent of the stiffness ratio between the two principal directions of the cross section. However, the spatial deformation and the bifurcation point depend on the stiffness ratio. Finally, the relation between the critical angle and the stiffness ratio is obtained via vibration method and compared with previous work of others based on equilibrium method.

並列關鍵字

Spiral spring ； Elastica ； Vibration

參考文獻

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