Abstract In this study, the free vibration behavior of perforated viscoelastic nonlocal Timoshenko beams has been investigated using a semi analytic method based on the Fourier series and the Stokes transformation. First, the geometric structure of the perforated beam was defined; then, by combining the Kelvin–Voigt viscoelastic model with the theory of nonlocal elasticity, the equations of motion describing the system were derived. During the solution phase, the boundary conditions of the beam were reduced to an eigenvalue problem using a semi analytic approach, with spring elements restricting rotation at both ends, and high-accuracy results have been obtained under various boundary conditions. The findings indicate that an increase in the nonlocal scale parameter has a softening effect on structural stiffness, thereby reducing vibration characteristics. Additionally, it is observed that an increase in the number of holes enhances discontinuity along the beam, leading to significant changes in system behavior. It is determined that the viscoelastic damping effect becomes more dominant, particularly under rigid boundary conditions, but weakens as the perforation density increases. The study also reveals that the void ratio, shear deformation, and boundary conditions are in strong interaction with one another. The proposed method offers an effective and reliable methodological approach for the dynamic analysis of perforated nanostructures, thanks to both its high convergence performance and its ability to model different boundary conditions within the same formulation.
Kadioglu et al. (2026) studied this question.