A rigorously analytical exploration of vibrations of arbitrarily shaped multi-layered nanomembranes from different materials

Abstract

The chief aim of this paper is to present a novel analytical solution for vibrational scrutiny of multi-layered composite nanomembranes of any shape. Both the nonlocality and surface effects are incorporated into the governing equations based on the idea of the deflected nanomembrane’s curves with identical deflections. Irrespective of the regular equations of motion of nanomembranes whose stiffness terms are generally expressed as the partial differential of the unknown fields to the independent components of the coordinates system, the newly developed ones are ordinary differential equations, which are exclusively stated as a function of a single parameter. This is the main characteristic of the curves with equal deflection, and thereby, the established methodology is called isodeflection curve approach. Through presenting several examples, the characteristic equations associated with annular, elliptic, and parabolic multi-layered nanomembranes are derived and solved for the natural frequencies of synchronous vibration modes. With some efforts, the proposed analytical solution can also be extended to free transverse vibrations of nanoplates as well as other complex multi-layered nanocomposite structures.