Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia

This paper is concerned with analytical treatment of non-linear oscillations of planar, flexural large amplitude free vibrations of a slender, inextensible cantilever beam carrying a lumped mass with rotary inertia at an intermediate position along its span. An analytic approximate technique, namely Optimal Homotopy Asymptotic Method (OHAM) is employed for this purpose. It is proved that OHAM provide accurate solutions for large amplitudes and large modal constants in the considered nonlinear equations, when other classical methods fail. Our procedure provides us with a convenient way to optimally control the convergence of solution, such that the accuracy is always guaranteed. An excellent agreement of the approximate frequencies and periodic solutions with the numerical results and published results has been demonstrated. Two examples are given and the results reveal that this procedure is very effective, simple and accurate. This paper demonstrates the general validity and the great potential of the OHAM for solving strongly nonlinear problems.