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This paper presents a new semianalytical approach for geometrically nonlinear vibration analysis of Euler-Bernoulli beams with different boundary conditions. The method makes use of Linstedt-Poincaré perturbation technique to transform the nonlinear governing equations into a linear differential equation system, whose solutions are then sought through the use of differential quadrature approximation in space domain and an analytical series expansion in time domain. Validation of the present method is conducted in numerical examples through direct comparisons with existing solutions, showing that the proposed semianalytical method has excellent convergence and can give very accurate results at a long time interval.

In this study, nonlinear couple stress–strain constitutive relationships in the modified couple stress theory (MCST) are derived on the basis of previous classical stress–strain constitutive relationships of nonlinear elasticity materials. Hamilton's principle is employed to obtain higher-order nonlinear governing equations within the framework of the updated MCST, von Kármán geometric nonlinearity and Bernoulli–Euler beam theory. These mathematical formulations are solved numerically by the differential quadrature method together with an iterative algorithm to determine the nonlinear dynamic features of microbeams with four groups of boundary conditions. A detailed parametric study is conducted to analyze the influences of nonlinear elasticity properties on the nonlinear free vibration characteristics of the microbeams. Results show that these microbeams exhibiting nonlinear couple constitutive relationships have lower frequencies than their approximately simplified linear couple constitutive relationships. In addition, the frequencies of microbeams with nonlinear elasticity properties decrease as the vibration amplitude increases.

This paper investigates the dynamic pull-in instability of a micro-actuator made from nonlinear elasticity materials. The theoretical formulations are based on Bernoulli–Euler beam theory and include the effects of both material nonlinearity and mid-plane stretching due to large deformation. By employing the Galerkin method, the nonlinear partial differential governing equation is decoupled into a set of nonlinear ordinary differential equations which are then solved using the Runge–Kutta method. Numerical results show that the linear constitutive relationship used in previous studies is valid for small deformation only, whereas for large deformation the nonlinear elasticity constitutive relationship must be used for accurate analysis. The effects of material nonlinearity, initial gap, beam length, and beam width on the pull-in instability of the micro-actuator are studied.