Analysing regular nonlinear vibrations of nano/micro plates based on the nonlocal theory and combination of reduced order modelling and multiple scale method

Abstract

We study the geometrically nonlinear vibrations of rectangular micro/nanoplates with an account of small-scale effects in a framework of the nonlocal elasticity theory. Hamilton’s principle yields the governing system of nonlinear partial differential equations (PDEs) based on the Kirchhoff-Love hypotheses and geometric nonlinearity introduced through the von Kármán theory.Next, we employed a strategy to get coupled nonlinear and non-autonomous system of ordinary differential equations (ODEs) due to the concept of both reduced order modelling (ROM) truncated to the double-mode approximation of the plate deflection and the Bubnov-Galerkin procedure. The latter one allowed for the transformation of the initial problem to that with separated time and position functions and the used approach was validated.Then we have studied the reduced model of second-order nonlinear ODEs with coupled geometric nonlinearities through the multiple scales method (MSM) in time domain. Both resonant and non-resonant vibrations have been considered with emphasis put on the small-scale effects and the approximation accuracy. In spite of numerous novel results regarding parametric analysis carried out by combined analytical-numerical approach, we have detected ambiguous and unambiguous back-bone curves which allow one to predict and possibly avoid the pull-in phenomena related to the primary problem governed by the nonlinear vibrations of micro/nanoplates.