Vibration analysis of variable cross-section microbeams using perturbation and generalized differential quadrature methods

Abstract This study aims to analyze the linear and nonlinear dynamic responses of variable cross-section microbeams. To represent microscale structural behavior, Hamilton’s principle is applied to obtain the governing equations, while the Modified Couple Stress Theory was incorporated into the formulation. To solve the nonlinear differential equations, the Generalized Differential Quadrature Method and the Multi-Time Scale Perturbation Method are employed in conjunction. While spatial discretization is performed with high numerical accuracy using GDQM, the MTS method ensured that the nonlinear behavior developing over time is captured analytically. The analysis reveals that small-scale effects in the microstructure play a significant role in determining natural frequencies and that, when MCST is considered, the stiffness of the system increases, thereby raising the natural frequencies. Additionally, it is determined that a decrease in the radius ratio leads to a reduction in natural frequencies, while the mode shapes become more complex due to microscale effects. The findings emphasize the importance of considering nonlinear dynamics in the design and analysis of applications such as micro-electromechanical systems.