The large amplitude vibration of hinged beams

The large amplitude free vibrations of a simply-supported beam with ends kept a constant distance apart is studied using the actual nonlinear equilibrium equations (i.e. specification of loads in terms of the deformed coordinates of the beam) and the exact nonlinear expression for curvature in addition to the nonlinearity arising from the axial force. A variable separable assumption, together with certain assumptions as to the behaviour of the time function defines an eigenvalue characteristic of the vibration. A numerically exact successive integration and iterative technique establishes the dependence of this quantity on the amplitude of vibrations. The hardening effect of nonlinearity is then interpreted in terms of the variation of this quantity with the amplitude of vibration. This new criteria to define nonlinearity, is compared with several existing in the literature. The present analysis allows the separation of the effects of stretching and large deflection equations on the nonlinear behaviour and the conclusion can be made, based on numerical evidence, that the predominant nonlinearity is due to stretching. The axial force at any station in the beam and the bending stress can also be computed in a numerically exact sense, at the point of maximum amplitude. © 1978.