Fourier series in simulation of dynamic response of arbitrary restrained FG microbeam rested on viscoelastic foundation

In this study, a semi-analytical method based on Fourier series is proposed to investigate the dynamic behavior of functionally graded microbeams resting on a viscoelastic foundation constrained by elastic springs at both ends. The theoretical basis of the study begins with obtaining the equations of motion for a microbeam resting on a viscoelastic foundation and is completed by applying a functionally graded process as the material property. In the solution phase, the partial differential equations obtained under the assumption of simple harmonic motion have been reduced to ordinary differential equations, and the vertical displacement function has been expressed using a Fourier sine series. The higher-order derivatives of this function have been obtained using the Stokes' transformation. The obtained series have been substituted into the motion equation to determine the Fourier coefficients, and an eigenvalue problem has been formulated by defining the boundary conditions. By solving this eigenvalue problem, the frequencies of functionally graded microbeams resting on a viscoelastic foundation have been calculated for various boundary conditions. The obtained results have been verified by comparing them with similar studies in literature; thus, it has been demonstrated that the proposed method provides a flexible and highly accurate solution without the need to create a separate eigenvalue problem for each boundary condition. All findings have been discussed in detail through tables and graphs. It has also been determined that the beam mass is the key parameter determining the effects of the viscoelastic properties of foundation on dynamic behavior; while the scale parameter increases the frequency by enhancing stiffness, foundation stiffness is the parameter controlling this effect. In this context, it is believed that the method proposed in this study and the results obtained will provide important information to the literature on the design and optimization of micro/nano-scale structures.