Vibration of a spatial elastica constrained inside a straight tube

• Eulerian version of the equations of motion is formulated.
• Director theory is adopted for analysis.
• Both force and displacement controls are studied.
• Focus is placed on the vibration of the planar and spatial one-point-contact deformations.
• Natural frequencies and mode shapes are calculated with shooting method.

In this paper we study the dynamic behavior of a clamped–clamped spatial elastica under edge thrust constrained inside a straight cylindrical tube. Attention is focused on the calculation of the natural frequencies and mode shapes of the planar and spatial one-point-contact deformations. The main issue in determining the natural frequencies of a constrained rod is the movement of the contact point during vibration. In order to capture the physical essence of the contact-point movement, an Eulerian description of the equations of motion based on director theory is formulated. After proper linearization of the equations of motion, boundary conditions, and contact conditions, the natural frequencies and mode shapes of the elastica can be obtained by solving a system of eighteen first-order differential equations with shooting method. It is concluded that the planar one-point-contact deformation becomes unstable and evolves to a spatial deformation at a bifurcation point in both displacement and force control procedures.