Large amplitude free vibration of axially loaded beams resting on variable elastic foundation

In the present study, large amplitude free vibration of beams resting on variable elastic foundation is investigated. The Euler–Bernoulli hypothesis and the Winkler model have been applied for beam and elastic foundation, respectively. The beam is axially loaded and is restrained by immovable boundary conditions, which yields stretching during vibrations. The energy method and Hamilton’s principle are used to derive equation of motion, where after decomposition an ordinary differential equation with cubic nonlinear term is obtained. The second order homotopy perturbation method is applied to solve nonlinear equation of motion. An explicit amplitude-frequency relation is achieved from solution with relative error less than 0.07% for all amplitudes. This solution is applied to study effects of variable elastic foundation, amplitude of vibration and axial load on nonlinear frequency of beams with simply supported and fully clamped boundary conditions. Proposed formulation is capable to dealing with any arbitrary distribution of elastic foundation.