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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)