| Abstract: | A Ritz finite element approach is used here to study the large amplitude free flexural vibrations of beams with immovable ends. The formulation is based on Lagrange's equation of motion with the definition of the time function at an instant corresponding to the point of reversal. The element displacement vector is chosen as a combination of inplane and transverse displacements. The nonlinear stiffness is written as a combination of the bending–membrane interaction and bending stiffness. The solution for nonlinear equations is sought by using an algorithm—the direct iteration technique—suitably modified for eigenvalue problems. Convergence is checked using the displacement norms on eigen-modes, and frequency norms for eigenvalues. The nonlinear frequencies, and modeshapes for transverse and longitudinal displacements, are determined for the simply-supported, clamped–clamped and simply-supported–clamped beams. Results are presented in the form of tables. In almost all the cases the nonlinear frequency values are found to be the lower bound like the earlier Galerkin finite element method. |
