Nonlinear vibration of buckled beams: some exact solutions

In this paper, exact solutions are obtained for the dynamics of buckled beams with different types of end conditions. The differential equation for the vibrations of buckled beams, with restrictions on axial stretch, is nonlinear. Using the modes of the corresponding linear problem, which readily satisfy the boundary conditions, the natural frequencies are obtained for the nonlinear problem by using Jacobi elliptic functions. Analytical solutions for the dynamics of buckled beams, with various rotational restraints, are presented. (C) 2001 Published by Elsevier Science Ltd.