This paper studies, both theoretically and experimentally, the deformation, the vibration, and the stability of a buckled elastic strip (also known as an elastica) constrained by a space-fixed point in the middle. One end of the elastica is fully clamped while the other end is allowed to slide without friction and clearance inside a rigid channel. The point constraint is located at a specified height above the clamping plane. The elastic strip buckles when the pushing force reaches the conventional buckling load. At this buckling load, the elastica jumps to a symmetric configuration in contact with the point constraint. As the pushing force increases, a symmetry-breaking bifurcation occurs and the elastica evolves to one of a pair of asymmetric deformations. As the pushing force continues to increase, the asymmetric deformation experiences a second jump to a self-contact configuration. A vibration analysis based on an Eulerian description taking into account the sliding between the elastica and the point constraint is described. The natural frequencies and the stability of the calculated equilibrium configurations can then be determined. The experiment confirms the two jumps and the symmetry-breaking bifurcation predicted theoretically.
a b s t r a c tThis paper studies the vibration and stability of an elastica constrained by a pair of symmetrically placed parallel plane walls. One end of the elastica is fully clamped, while the other end is allowed to slide through a rigid channel under edge thrust. In order to take into account the variation of the contact points between the elastica and the walls during vibration, an Eulerian version of the equations of motion is adopted. It is found that the lowest few natural frequencies approach and remain degenerately zero when point-contact deformations evolve to line-contact patterns. As a consequence, the stability of all line-contact deformations before secondary buckling cannot be determined from the linear vibration analysis. A load-controlled experiment was conducted to find that the elastica jumped from one-point to twopoint, and then to three-point contact with the walls without going through any line-contact deformations. These experimental observations are different from the results reported previously by others with different set-ups, in which line-contact deformations did exist. Explanations based on experimental evidences and theoretical analyses are provided to confirm the validity of these previous investigations and clarify the cause of the difference.
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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