A Dynamic Coefficient Matrix Method for the Free Vibration of Thin Rectangular Isotropic Plates

Shock and Vibration, Jul 2018

The free flexural vibration of thin rectangular plates is revisited. A new, quasi-exact solution to the governing differential equation is formed by following a unique method of decomposing the governing equation into two beam-like expressions. Using the proposed quasi-exact solution, a Dynamic Coefficient Matrix (DCM) method is formed and used to investigate the free lateral vibration of a rectangular thin plate, subjected to various boundary conditions. Exploiting a special code written on MATLAB®, the flexural natural frequencies of the plate are found by sweeping the frequency domain in search of specific frequencies that yield a zero determinant. Results are validated extensively both by the limited exact results available in the open literature and by numerical studies using ANSYS® and in-house conventional FEM programs using both 12- and 16-DOF plate elements. The accuracy of all methods for lateral free vibration analysis is assessed and critically examined through benchmark solutions. It is envisioned that the proposed quasi-exact solution and the DCM method will allow engineers to more conveniently investigate the vibration behaviour of two-dimensional structural components during the preliminary design stages, before a detailed design begins.

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A Dynamic Coefficient Matrix Method for the Free Vibration of Thin Rectangular Isotropic Plates

International Journal of A Dynamic Coefficient Matrix Method for the Free Vibration of Thin Rectangular Isotropic Plates Supun Jayasinghe 0 Seyed M. Hashemi 0 Lorenzo Dozio 0 Department of Aerospace Engineering, Ryerson University , Toronto , Canada The free flexural vibration of thin rectangular plates is revisited. A new, quasi-exact solution to the governing differential equation is formed by following a unique method of decomposing the governing equation into two beam-like expressions. Using the proposed quasi-exact solution, a Dynamic Coefficient Matrix (DCM) method is formed and used to investigate the free lateral vibration of a rectangular thin plate, subjected to various boundary conditions. Exploiting a special code written on MATLAB , the flexural natural frequencies of the plate are found by sweeping the frequency domain in search of specific frequencies that yield a zero determinant. Results are validated extensively both by the limited exact results available in the open literature and by numerical studies using ANSYS and in-house conventional FEM programs using both 12- and 16-DOF plate elements. The accuracy of all methods for lateral free vibration analysis is assessed and critically examined through benchmark solutions. It is envisioned that the proposed quasi-exact solution and the DCM method will allow engineers to more conveniently investigate the vibration behaviour of two-dimensional structural components during the preliminary design stages, before a detailed design begins. 1. Introduction Many vibrating airframe structural components could be modelled as thin plates. Not only that do these structural elements transmit various internal and external loads that may affect their stiffness but they are also frequently in close proximity to vibrating components such as engines. Therefore, it is of utmost importance to device and develop solution techniques to study the vibrational characteristics of these structures during preliminary design stages. Such vibrational analyses would allow the designers to investigate the effects of various boundary conditions the structural elements would be subjected to during its operation and the vibrational characteristics of the component before progressing to advanced stages of design. Using these results designers could alter the geometry or the materials used to avoid resonance and gain a favourable outcome. Among the many methods available for vibration analysis, the analytical and semianalytical 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. To circumvent this problem, many simplifying assumptions have been incorporated into the existing exact methods and as a result they exhibit many limitations. Having lost their generality, these exact methods are then only applicable to specific plate shapes, geometries, and those subjected to certain boundary conditions. The orthogonality, completeness, and stability of Fourier series expansions have resulted in their frequent application to plate vibration problems [1]. The Navier [2] and Levy methods [3, 4] are two of the most common analytical procedures available for plate vibration analysis that incorporate such Fourier series expansions, where the former exploits a double Fourier series to solve the governing differential equation, the latter is based on a single Fourier series. However, both methods have a common drawback in that they are only applicable to plates having at least two simply supported boundaries. In addition, the Levy method is also limited to rectangular shaped plate configurations and is incapable of taking into account the effects of bending-twisting coupling. In addition to the above weaknesses, all methods that are based on conventional Fourier series expansions consist of a convergence problem along the boundaries arising as a result of discontinuities in displacement and its derivatives [1]. Therefore, both of these methods are unsuitable for most aerospace applications as they could only tackle simple and special cases. In order to overcome the discontinuity in displacement and its derivatives along the boundaries, the Improved Fourier Series Method (IFSM) [5] was later proposed. Although IFSM possesses a higher rate of convergence and is more readily applicable to a host of plate configurations and boundary types, it is still inadequate to study problems comprising material and geometric nonlinearity. The Rayleigh-Ritz method is another very popular exact method that has been exploited by many researchers in the past. It was first introduced by Rayleigh [6] and later improved by Ritz [7] by assuming a set of admissible trial functions, each of which had independent amplitude coefficients; thus, it is termed the Rayleigh-Ritz method or Ritz method. Young [8] and Warburton [9] used the Ri (...truncated)


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Supun Jayasinghe, Seyed M. Hashemi. A Dynamic Coefficient Matrix Method for the Free Vibration of Thin Rectangular Isotropic Plates, Shock and Vibration, 2018, 2018, DOI: 10.1155/2018/1071830