In this paper, we study the deformation of a thin ela.itic rod constrained inside a cylindrical tube and under the action of an end twisting moment. The ends of the rod are clamped in the lateral direction. Unlike the previous works of others, in which only the fully developed line-contact spiral was considered, we present a complete analysis on the deformation when the dimensionless twisting moment M^. is increased from zero. It is found that the straight rod buckles into a spiral shape and touches the inner wall of the tube at the midpoint when M^ reaches 8.987. As A/j increases to 11.472, the contact point in the middle splits into two, leaving the midpoint floating in the air. As M^ increases to 13.022, the midpoint returns to touch the tube wall and the two-point-contact deformation evolves to a three-point-contact deformation. Starting from A/, = 13.098, the point contact in the middle evolves to a line contact, cmd the deformation becomes a pointUne-point contact conflguration and remains so thereafter. In the case when the linecontact pattern is fully developed, it is possible to predict the spiral shape analytically. The numerical results are found to agree very well with those predicted analytically. Finally, an experimental setup is constructed to observe the deformation evolution of the constrained rod under end twist.
a b s t r a c tIn this paper, we study the deformation and stability of a heavy elastica resting symmetrically on two frictionless point supports on the same horizontal level. The static analysis finds multiple equilibria when the distance of the two point supports is smaller than a certain value. In order to determine whether these equilibria are stable, a dynamic analysis is conducted to calculate their natural frequencies. In order to take into account the sliding between the elastica and the smooth point supports during vibration, a dynamic analysis based on an Eulerian description is adopted. It is found that stable equilibrium can exist only when the half support span a is between two limits a ðsÞ min and a ðsÞ max . This range depends on a dimensionless ratio between the weight density and the flexural rigidity of the elastica. When an a between a ðsÞ min and a ðsÞ max decreases and approaches a ðsÞ min , the elastica will slip away from the side. On the other hand, when an a between a ðsÞ min and a ðsÞ max increases and approaches a ðsÞ max , the elastica will slip through between the two supports.
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