On the Use of Energy Method to Determine the Stability of a Constrained Elastica

受拘束的彈性板條，在受到邊界推力下，有許多種靜態平衡狀態。由於出現單向的拘束面，一般不容易判斷平衡位置的穩定性。本文提出之能量法為解決此問題。為了解決單向的拘束面帶來的問題，我們允許彈性板條與拘束面的接觸點能夠在加上虛位移前後發生一點改變。值得注意的是，若平衡位置為對稱的變形，為了得到總位能的極值，接觸點的改變量必須要受到一些限制。拘束方程式在線性化之後，可以得到系統總位能的二階導數矩陣。彈性板條各種變形的穩定性可由此矩陣的特徵值決定。本文先介紹討論單點接觸變形以及單一線接觸的變形，在介紹兩點接觸以及三點接觸的變形，其他的變形可以類似的方法進行分析。前人發展之振動法允許接觸點在振動時可以變換，本文之能量法能夠給予補充。A constrained elastica under edge thrust may have multiple static equilibrium positions. It is in general difficult to determine the stability of these equilibrium positions because of the presence of unilateral constraints. In this paper we propose an energy method for this purpose. To deal with the unilateral constraints in question, we allow the contact point on the elastica to be slightly different before and after superposing virtual displacements. It is noted that certain restrictions must be imposed on the contact point change in order for the total potential to be stationary if the equilibrium position is symmetric. After linearizing the constraint equations, the matrix associated with the second variation of the total potential can be established. From the eigenvalues of this matrix, the stability of the constrained elastica can be determined. One-point-contact and one-line-contact deformations are discussed in detail. Other deformation patterns can be analyzed in a similar manner. This energy method supplements the vibration method proposed earlier by the first author, in which the contact point is allowed to change during vibration