A Note on Free Vibration of a Double-beam System with Nonlinear Elastic Inner Layer

In this note, small amplitude free vibration of a double-beam system in presence of inner layer nonlinearity is investigated. The nonlinearity is due to inner layer material and is not related to large amplitude vibration. At first, frequencies of a double-beam system with linear inner layer are studied and categorized as synchronous and asynchronous frequencies. It is revealed that the inner layer does not affect higher modes significantly and mainly affects the first frequency. Then, equation of motion in the presence of cubic nonlinearity in the inner layer is derived and transformed to the form of Duffing equation. Using an analytical solution, the effect of nonlinearity on the frequency for simply-supported and clamped boundary conditions is analyzed. Results show that the nonlinearity effect is not significant and, in small amplitude free vibration analysis of a double-beam system, the material nonlinearity of the inner layer could be neglected

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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

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