Study on asymptotic analytical solutions using HAM for strongly nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass

Presented herein is to establish the asymptotic analytical solutions for the fifth-order Duffing type temporal problem having strongly inertial and static nonlinearities. Such a problem corresponds to the strongly nonlinear vibrations of an elastically restrained beam with a lumped mass. Taking into consideration of the inextensibility condition and using an assumed single mode Lagrangian method, the single-degree-of-freedom ordinary differential equation can be derived from the governing equations of the beam model. Various parameters of the nonlinear unimodal temporal equation stand for different vibration modes of inextensible cantilever beam. By imposing the homotopy analysis method (HAM), we establish the asymptotic analytical approximations for solving the fifth-order nonlinear unimodal temporal problem. Within this research framework, both the frequencies and periodic solutions of the nonlinear unimodal temporal equation can be explicitly and analytically formulated. For verification, numerical comparisons are conducted between the results obtained by the homotopy analysis and numerical integration methods. Illustrative examples are selected to demonstrate the accuracy and correctness of this approach. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions.