Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium

Modeling of nano-structure systems using nonlocal theories has received a great attention in recent years. However, there are not so many papers giving the exact solution for the free vibration problem of multiple nano-structure systems especially when damping properties of the system are considered. Here, we analyzed the free transverse vibration of a viscoelastic multi-nanoplate system (MNPS) embedded in a viscoelastic medium taking into account small-scale effects by using the nonlocal theory of Eringen. The modified Kelvin–Voigt viscoelastic constitutive relation was used to model material and nano-scale characteristics of nanoplates. Based on the Kirchhoff–Love plate theory, D’ Alembert’s principle and viscoelastic constitutive relation, we derived the homogeneous system of m partial differential equations of motion coupled through the viscoelastic interaction. The closed form solutions of undamped natural frequencies and modal damping factor were obtained utilizing the Navier’s method and trigonometric method for the case of “Cantilever-Chain” system. The proposed analytical results are verified with the results available in the literature and excellent agreement is achieved. In addition, we investigated the influences of a nonlocal and viscoelastic parameter, plate aspect ratio and coefficients of a viscoelastic medium on the complex eigenvalues through numerical simulations.