Nonlinear free vibration of a microscale beam based on modified couple stress theory

This paper presents a nonlinear free vibration analysis of the microbeams based on the modified couple stress Euler–Bernoulli beam theory and von Kármán geometrically nonlinear theory. The governing differential equations are established in variational form from Hamilton principle, with a material length scale parameter to interpret the size effect. These partial differential equations are reduced to corresponding ordinary ones by eliminating the time variable with the Kantorovich method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem in spatial variable, are then solved numerically by shooting method, and the size-dependent characteristic relations of nonlinear vibration frequency vs. central amplitude of the microbeams are obtained successfully. The comparisons with available published results show that the current approach and algorithm are of good practicability. A parametric study is conducted involving the dependency of the frequency on the length scale parameter along with Poisson ratio, which shows that the nonlinear vibration frequency predicted by the current model is higher than that by the classical one.

Graphical abstractThe nonlinear vibration of a microbeam exhibits a hardening spring behavior, and the modified couple stress theory models the beams stiffer than does the classical beam theory.Figure optionsDownload full-size imageDownload as PowerPoint slideHighlights► Mathematical model for nonlinear vibration of size-dependent beam is developed. ► Governing equation is reduced by Kantorovich method. ► Nonlinear vibration frequency vs. amplitude is obtained by shooting method. ► Current approach and algorithm are validated. ► Dependency of frequency on length scale parameter is studied.