This study analyzes the fourth-order nonlinear free vibration of a Timoshenko beam. We discretize the governing differential equation by Galerkin's procedure and then apply the homotopy analysis method (HAM) to the obtained ordinary differential equation of the generalized coordinate. We derive novel analytical solutions for the nonlinear natural frequency and displacement to investigate the effects of rotary inertia, shear deformation, pre-tensile loads and slenderness ratios on the beam. In comparison to results achieved by perturbation techniques, this study demonstrates that a first-order approximation of HAM leads to highly accurate solutions, valid for a wide range of amplitude vibrations, of a highorder strongly nonlinear problem.
This study investigates forced nonlinear vibrations of a simply supported Euler-Bernoulli beam on a nonlinear elastic foundation with quadratic and cubic nonlinearities. Applying the homotopy analysis method (HAM) to the spatially discretized governing equation, we derive novel analytical solutions and discuss their convergence to present nonlinear frequency responses with varying contributions of the nonlinearity coefficients. A comparison with numerical solutions is conducted and nonlinear time responses and phase planes are compared to the results from linear beam theory. The study demonstrates that apart from nonlinear problems of free vibrations, HAM is equally capable of solving strongly nonlinear problems of forced vibrations.
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