Non-linear vibration and instability of multi-phase composite plate subjected to non-uniform in-plane parametric excitation: Semi-analytical investigation

Abstract

Non-linear vibration and instability of a randomly distributed carbon nanotube fiber-reinforced composite (CNTFRC) plate under the action of different types of non-uniform in-plane periodic loadings are presented in this study. The composite plates are modeled considering von Kàrmàn non-linearity and Higher-order shear deformation theory (HSDT). The analytical expression for stresses (σxx,σyy,andτxy) distribution within the CNTFRC plate due to non-uniform loads is developed by solving the in-plane elasticity problem using Airy’s stress approach. Using these stresses, Hamilton’s principle is applied to derive the non-linear partial differential equations for dynamic instability and non-linear vibration of CNTFRC plates. Employing the Galerkin method, the non-linear partial differential equations are transformed into a set of non-linear ordinary differential (Mathieu type) equations. After dropping the nonlinearity terms, the linear ordinary differential equations are solved by using Bolotin’s method to trace the boundaries of the dynamic instability region corresponding to periods 2T and T. In the end, the non-linear ordinary differential equations are solved by using the Incremental Harmonic Balance method (IHB) for analyzing the non-linear vibration behavior of the CNTFRC plate. The result obtained from the current work will help in the appropriate design of the CNTFRC plate against stability and vibration in presence of non-uniform in-plane loadings.