A Fourier series-based approach for the size-dependent free vibration of viscoelastic microbeams

Abstract In this study, the free vibration behavior of viscoelastic microbeams was investigated by combining the Kelvin–Voigt viscoelastic model with the Modified Couple Stress theory. A semi-analytical method based on the Fourier series and Stokes transformations was developed for the analysis. The proposed approach provides highly accurate results under various boundary conditions due to its structure, which requires solving only a single eigenvalue problem. The convergence analysis of the method was performed to determine the optimal number of terms, and calculations were performed with the number of terms that would yield the most reliable results. Furthermore, the results were compared with studies in literature and confirmed to exhibit high accuracy at the micro scale. Furthermore, the relationship between frequency and damping according to the Kelvin–Voigt model was examined in detail; it was found that an increment in the viscous damping coefficient causes damping to rise linearly and the natural frequency to decrease. It was observed that the damping effect also became more pronounced due to the increase in micro-level stiffness in the system caused by the boost in the material length scale parameter. In conclusion, the developed method and the findings obtained provide important engineering data for the dynamic analysis of viscoelastic microbeams and the design of micro-scale structural elements, while also contributing to a deeper understanding of the frequency–damping interaction in viscoelastic systems.