Implementation of the Nodal Discontinuous Galerkin Method for the Plate Vibration Problem using Linear Elasticity Equations

This work presents a numerical solution of the forced plate vibration problem using the nodal discontinuous Galerkin (DG) method. The plate is modelled as a three-dimensional domain, and its vibration is governed by the linear elasticity equations. The nodal DG method discretises the spatial domain and computes the spatial derivatives of the equations element-wise, while the time integration is conducted using the Runge-Kutta method. This method is in particular of interest as it is very favourable to carry out the computation by a parallel implementation. Several aspects regarding the numerical implementation such as the plate boundary conditions, the point force excitation, and the upwind numerical flux are presented. The numerical results are validated for rectangular concrete plates with different sets of boundary conditions and thicknesses, by a comparison with the exact mobilities that are derived from the classical plate theory (CPT) and the first order shear deformation theory (FSDT). The plate thickness is varied to understand its effect regarding the comparison with the CPT. An excellent agreement between the numerical solution and the FSDT was found. The agreement with the CPT occurs only at the first couple of resonance frequencies, and as the plate is getting thinner. Furthermore, the numerical example is extended to an L-shaped concrete plate. The mobility is then compared with the mobilities obtained by the CPT, FSDT, and linear elasticity equations.

Large-amplitude vibration of imperfect angleply laminated rectangular plates with four boundary conditions

This paper solves the modified-Duffing ordinary differential equation for the largeamplitude vibration problem of imperfect angle-ply laminated rectangular plate. Two inplane and two out-of-plane constraints are considered to form four boundary conditions. The initial condition is chosen to be initial vibration amplitude. To solve this angle-ply laminated rectangular plate vibration problem, Lindstedt’s perturbation technique and Runge-Kutta method are applied. The solution from both methods are plotted and compared for a validity check. Lindstedt’s perturbation technique is proved to be accurate for a sufficiently small vibration amplitude especially when imperfection exists. The results from Runge-Kutta method are plotted to form the typical backbone curves.

Vibration Analysis of Carbon Nanotube Reinforced Composite Plates

This paper presents an investigation on the free vibration of rectangular nanocomposite plates reinforced by aligned single-walled carbon nanotubes (SWCNTs). The CNT reinforcement may be uniformly distributed (UD) or functionally graded (FG) over the thickness direction of a plate. The material properties of the CNT composite are determined through a micromechanical model. The eigenvalue equation governing the plate vibration problem is derived by the p-Ritz method through minimizing the virtual strain and kinetic energies of a CNT composite plate. The influences of CNT distribution and reinforcing angle, plate thickness ratio, aspect ratio and support conditions on the vibration behaviour of the plates are discussed.

A Finite Element Modeling of Higher Order Shear Deformation Theory on a Plate Vibration Problem

Higher order Shear Defamation Plate Theories (HSDPT) are improved theories over Mindlin plate theory because their assumptions are closer to reality. However, they are seldom used in solving ordinary engineering works. This is due to the fact that mathematical formulations and computations are so lengthy that time and efforts required are close to solving a exact 3-D model. For problems involving sharp stress variation, higher order theories are anticipated to give better results. The combination of HSDPT and Finite Element Modelings are especially attractive because a finite element modeling is much simpler. The current research develops a finite element model on the higher order shear deformation theory. A plate vibration problem was solved. The plates contain square interior cutout. Stress distributions are much complicated than whole plates. Results of HSDPT are compared with FSDPT (First order Shear Deformation Plate Theories) and CPT (Classical Plate Theory). Better accuracies are obtained by using the HSDPT.