Large-Amplitude Free Vibrations of a Beam

Free vibrations of an initially straight elastic bar having free-free or clamped-free ends are analyzed by assuming that the bar is undergoing inextensional motion and that shearing effects and rotatory inertia are negligible. If Hamilton’s principle is applied, an isoperimetric variational problem is reached. Upon elimination of the subsidiary condition, the remaining Euler equation (a highly nonlinear partial differential equation) can be reduced to an ordinary nonlinear differential equation by means of an extension of Galerkin’s procedure (Bubnov method). Because of the particular type of the reduced equation, use can be made of Atkinson’s superposition method for the frequency of the periodic solutions of the equation. Finally, a plot shows the approximate non-linear period versus the initial amplitude, both rendered nondimensional by introducing proper linear expressions.