The Homotopy-Analysis Approach for the Dynamical Study of a Microbeam Modeled on the Basis of the Strain-Gradient Theory

The purpose of this work is to investigate the nonlinear dynamics of a slender microbeam, modeled within the framework of the strain-gradient elasticity, adopting the homotopy analysis method (HAM). The microbeam is fixed at both edges and a geometric nonlinearity is also present accounting for the axial stretch. To attain an accurate and reliable model, so that the error is spread smoothly over the domain, a Chebyshev approximation for the nonlinear electric actuation term is introduced. A reduced-order model for the governing equation of motion, represented by an high-order nonlinear partial differential equation, is obtained. Then, the single-degree-of-freedom model is studied to find an analytical approximated solution. The free vibrations of the beam are investigated and the effects of several parameters, such as the applied axial load, are analyzed. Particular attention is also paid to find the influence of the high-order length scale material parameters, introduced by the non-classical theory, that progressively modify the oscillating behaviour. The results on the nonlinear phenomena, show both an hardening and a softening behaviour, in competition between them, varying the beam parameters. A numerical solution, obtained by a 4th order Runge Kutta algorithm, is also proposed as a benchmark for the analytical results.