Free transverse vibration analysis of orthotropic multi-viscoelastic microplate system embedded in visco-Pasternak medium via modified strain gradient theory

In this paper, an analytical process is proposed to investigate the size-dependent free vibration of orthotropic multi-viscoelastic microplate systems (OMVMPS) embedded in Kelvin-Voigt visco-Pasternak medium according to the modified strain gradient theory. Governing equations of motion in the partial form and the related boundary conditions are derived by utilizing the Kirchhoff plate theory and Hamilton's variational principle. The two different sorts of "chain" boundary conditions like "clamped Chain" and "free chain" systems are considered for the ends of microplate system. Navier's method, which convinces that the simply supported boundary conditions and trigonometric methods are applied to analytically investigate the size effect of the natural frequencies of OMVMPS. The numerical outcomes are offered to report the variation of OMVMPS natural frequencies with the numerous amounts of the microplate numbers, the length scale parameter, aspect ratio, visco-Pasternak foundation parameters, the thickness of microplate, and higher modes number. Several numerical outcomes of this research depict that when the number of microplates is low, there is a significant distinction between natural frequencies achieved for "clamped chain" and "free chain" systems. Also, it is demonstrated that by increasing the number of microplates, the effect of the visco-Pasternak substrate on the natural frequency of system vibration decreases.