Relatively simple finite element formulation for the large amplitude free vibrations of uniform beams

Large amplitude free vibration analysis of uniform, slender and isotropic beams is investigated through a relatively simple finite element formulation, applicable to homogenous cubic nonlinear temporal equation (homogenous Duffing equation). All possible boundary conditions where the von-Karman type nonlinearity is applicable, where the ends are axially immovable are considered. The finite element formulation begins with the assumption of the simple harmonic motion and is subsequently corrected using the harmonic balance method and is general for the type of the nonlinearity mentioned earlier. The nonlinear stiffness matrix derived in the present finite element formulation leads to symmetric stiffness matrix as compared to other recent formulations. Empirical formulas for the nonlinear to linear radian frequency ratios, for the boundary conditions considered, are presented using the least square fit from the solutions of the same obtained for various central amplitude ratios. Numerical results using the empirical formulas compare very well with the results available from the literature for the classical boundary conditions such as the hinged–hinged, clamped–clamped and clamped–hinged beams. For the beams with nonclassical boundary conditions such as the hinged–guided and clamped–guided, the numerical results obtained, apparently for the first time and are in line with the physics of the problem.