Vibration of a Long Heavy Elastica on Rigid Foundation

本文研究一個受重力影響的彈性板條置於平坦的剛性基底上，在兩端點受軸向力的挫曲變形與振動分析。變形的種類可能為對稱或非對稱，也有可能發生彈性板條的自我接觸。這個力學問題常見於深海油管及電纜鋪設的施工中。前人的研究中，侷限於靜態變形的計算。由於大變形產生的非線性，使得在指定的受力以及幾何條件下，會有多個變形同時存在。實際上只有穩定情況下的變形才是實際觀察的到。前人透過實驗，觀察在尚未自我接觸的彈性板條遇到對稱分支點(symmetric-breaking)時，不穩定的情況將會產生。至於如何判斷哪些靜態變形是穩定的，尚無方法加以預測。振動法判別穩定性的主要難度在於振動時，彈性板條自我接觸點以及與剛性基底接觸點的分析。我們將原本的Lagrangian描述式座標轉換成Eulerian描述式，在與彈性板條的靜態變形很相近的地方線性化。本文利用振動法求解特定靜態變形之自然頻率，從中可以判斷其穩定性。A long heavy elastica resting on a horizontal rigid foundation and under the action of a pair of equal and opposite compressive force tends to buckle away from the foundation. The deformation may be symmetric or asymmetric. Self contact may also occur. Domokos et al. (2003) demonstrated through experimental observation that the symmetric non-self-contact deformation becomes unstable when a symmetry breaking bifurcation occurs. After the symmetry-breaking bifurcation, the elastica branches to a non-self-contact asymmetric deformation when the end shortening continues to increase. In this paper we present a vibration method which is capable of predicting the stability of an elastica with self contact. The main difficulty of the vibration analysis is the variation of contact points between the elastica and the foundation, and the variation of the self-contact point during vibration. After transforming the governing equations from the original Lagrangian description to an Eulerian one, the equations are linearized near the neighborhood of the static deformation. From the calculated natural frequencies, one can determine the stability of the long heavy elastica. The stability predicted by Domokos et al. (2003) is confirmed theoretically