Abstract
The Dynamic Finite Element (DFE) formulation is a superconvergent, semianalytical method used to perform vibration analysis of structural components during the early stages of design. It was presented as an alternative to analytical and numerical methods that exhibit various drawbacks, which limit their applicability during the preliminary design stages. The DFE method, originally developed by the second author, has been exploited heavily to study the modal behaviour of beams in the past. Results from these studies have shown that the DFE method is capable of arriving at highly accurate results with a coarse mesh, thus, making it an ideal choice for preliminary stage modal analysis and design of structural components. However, the DFE method has not yet been extended to study the vibration behaviour of plates. Thus, the aim of this study is to develop a set of frequency-dependent, trigonometric shape functions for a 4-noded, 4-DOF per node element as a basis for developing a DFE method for thin rectangular plates. To this end, the authors exploit a distinct quasi-exact solution to the plate governing equation and this solution is then used to derive the new, trigonometric basis and shape functions, based on which the DFE method would be developed.
Highlights
Thin plates have been used to model the vibration behaviour of low curvature two-dimensional structures having very small thicknesses compared to the other dimensions
The results have consistently shown the Dynamic Finite Element (DFE) method to have a higher accuracy and rate of convergence compared to conventional finite element method (FEM) owing to the increased efficiency of the frequency-dependent, trigonometric shape functions based on the exact solutions to the governing equation which the DFE method employs
The primary objective of this study is to develop a set of viable and robust frequency-dependent, trigonometric shape functions for a 4-noded, 4-DOF per node element and to this end a quasi-exact solution to the thin rectangular plate governing equation was pursued by following a unique approach
Summary
Thin plates have been used to model the vibration behaviour of low curvature two-dimensional structures having very small thicknesses compared to the other dimensions. At the onset of airframe design, engineers are required to simulate the vibration behaviour of the airframe component to determine the operational range of frequencies and the mode shapes that arise under real-life conditions When such a modal analysis is carried out, the effects of loading and boundary conditions and the contributions from any nearby vibrating entities are incorporated as these factors could modify the vibration characteristics of the entire system. Among the many methods available for vibration analysis, the analytical methods yield the highest accuracy but one major hurdle in using these methods is that they require the closed form solution to the governing partial differential equation This can be a very tedious process, if at all a tractable one. These exact methods are only applicable to specific plate shapes and plates subjected to certain boundary conditions, as briefly discussed below
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