Effects of Damaged Boundaries on the Free Vibration of Kirchhoff Plates: Comparison of Perturbation and Spectral Collocation Solutions

Abstract

In order to compare numerical and analytical results for the free vibration analysis of Kirchhoff plates with both partially and completely damaged boundaries, the Chebyshev collocation and perturbation methods are utilized in this paper, where the damaged boundaries are represented by distributed translational and torsional springs. In the Chebyshev collocation method, the convergence studies are performed to determine the sufficient number of the grid points used. In the perturbation method, the small perturbation parameter is defined in terms of the damage parameter of the plate, and a sequence of recurrent linear boundary value problems is obtained which is further solved by the separation of variables technique. The results of the two methods are in good agreement for small values of the damage parameter as well as with the results in the literature for the undamaged case. The case of mixed damaged boundary conditions is also treated by the Chebyshev collocation method.

Free vibration analysis of non-local annular sector Mindlin plates

“…Furthermore, the Chebyshev collocation method (as a numerical technique) was not applied to analyze the free vibration problem of micro and nanostructures. The Chebyshev collocation method was successfully employed to carry out the free vibration analysis of local continuous systems with different shapes, geometries, and boundary conditions [19][20][21][22].…”

An exact spectral dynamic stiffness theory for composite plate-like structures with arbitrary non-uniform elastic supports, mass attachments and coupling constraints

“…1(b)) and elastic coupling constraints ( Fig.1(c)) has always been a challenging problem in many engineering areas. Such an analysis is important to avoid resonance or undesirable dynamic response, control acoustic emission and power flow [1], analyse the effects of cracks [2] or damaged boundaries [3] for identifying the location and destructiveness of these cracks and damages, and many others. The applicability of the proposed theory include, but not limited to, buildings, bridges, ships, aeroplanes, space structures, armoured vehicles, automobiles, machines, robots, optical beam pointing systems and so on.…”

“…The treatment of non-uniform elastic supports is obviously much more demanding when using any established method. Understandably, only a few research has attempted such problems [3,[13][14][15][16][17][18]. However, most of the published methods have limitations of different natures.…”

“…Some methods can only be applied to plates with restricted boundary conditions. For example, the Spectral Collocation [3] and the Discrete Singular Convolution [17] methods appear to be limited to non-uniform rotational elastic supports and they have problems with non-uniform translational elastic supports and/or when any free edge is encountered. Moreover, the results computed by most of these methods do not seem to be sufficiently accurate.…”

“…This is mainly due to the numerical instability which generally occurs and prevents these methods from computing more accurate results by using higher order basis functions. Additionally, most of the existing methods are only applied to single plates [3,[13][14][15][16][17][18][19] and very few are applied to plates with intermediate uniform elastic supports [6,7]. The application to more complex plate assemblies with non-uniform elastic supports is no doubt a difficult task when using most of the existing analytical methods.…”