非對稱彎曲旋轉圓盤之變形與反挫曲

On the Deformation and Reverse Snapping Speed of a Spinning Disk with Unsymmetrical Initial Shape

張永燁(YUNG-YEH CHANG)

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(2005年)

https://doi.org/10.6342/NTU.2005.01709

摘要

於本文將探討非對稱彎曲旋轉圓盤於轉速下的穩態變形與反挫曲現象。圓盤的初始形狀與變形均由軸對稱與非對稱模態所組成。另外，於軸對稱彎曲旋轉圓盤之非對稱變形的分析，可推斷不需考慮非對稱變形，因為非對稱模態的加入而多出的非對稱平衡解均為不穩定，且不改變軸對稱的解與穩定性。
軸對稱圓盤若加入一小非對稱模態於初始形狀時，則將探討此非對稱模態對軸對稱圓盤的變形與穩定性所造成的影響。於非對稱圓盤的分析可知，若軸對稱圓盤加入一個節徑的非對稱模態，則輕微的增加反挫曲臨界轉速。另一方面，若加入兩個節徑以上的非對稱模態，則顯著降低反挫曲臨界轉速。
實驗將以非對稱彎曲的紅銅片為樣本，其中初始形狀由軸對稱與兩個節徑的非對稱模態所組成。由實驗量測結果可驗證理論的預測

關鍵字

非對稱旋轉圓盤 ； 葛拉金法 ； 折斷式挫曲

並列摘要

In this paper we study the steady state deflection and reverse snapping phenomenon of a spinning non-flat disk. Both the initial and the deformed shapes of the disk are allowed to have axisymmetrical and unsymmetrical components. For the analysis of a rotating axisymmetrical disk, we conclude that there is no need to include the unsymmetrical assumed modes in the expansion because all the unsymmetrical deformations are unstable and the stability of the axisymmetrical positions will not be changed. We then consider the case when the initial shape of the disk contains a dominant axisymmetrical component and a smaller unsymmetrical component. Attention is focused on the effect of this unsymmetrical component on the overall deformation and stability of the spinning disk. It is found that the unsymmetrical component with one nodal diameter tends to slightly defer the occurrence of the reverse snapping phenomenon. On the other hand, all other unsymmetrical components with more than one nodal diameter tend to reduce dramatically the reverse snapping speed. Experiment is conducted on a non-flat copper disk containing an axisymmetrical component and an unsymmetrical component with two nodal-diameters. The experimental measurement confirms the theoretical prediction.

並列關鍵字

Unsymmetrically Spinning Disk ； Snap Through

參考文獻

[2] Capino, M., 1991, “The Effect of Initial Curvature in a Flexible Disk Rotating Near a Flat Plate,” ASME Journal of Tribology, 113, pp. 355-360.

[3] Benson, R.C., and Cole, K.A., 1991, “Transverse “Runout” of a Nonflat Spinning Disk,” Tribology Transaction, 34, pp. 545-552.

[4] Jia, H.S., 2000, “Analysis of Transverse Runout in Rotating Flexible Disks by Using Galerkin’s Method,” International Journal of Mechanical Sciences, 42, pp. 237-248.

[5] Chen, J.-S., and Lin, C.-C., 2005, “Axisymmetrical snapping of a spinning non-flat disk,” ASME Journal of Applied Mechanics, in press.

[6] Nowinski, J.L., 1964, “Nonlinear Transverse Vibrations of a Spinning Disk,” ASME Journal of Applied Mechanics, 31, pp. 72-78.