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Research Papers

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

[+] Author and Article Information
Ma’en Sari, Morad Nazari

Department of Mechanical and Aerospace Engineering,  New Mexico State University, Las Cruces, NM 88003

Eric A. Butcher1

Department of Mechanical and Aerospace Engineering,  New Mexico State University, Las Cruces, NM 88003eab@nmsu.edu

1

Corresponding author.

J. Comput. Nonlinear Dynam 7(1), 011011 (Oct 05, 2011) (11 pages) doi:10.1115/1.4004808 History: Received March 25, 2011; Revised July 26, 2011; Published October 05, 2011; Online October 05, 2011

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.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 3

General form of the meshgrid for Chebyshev collocation formulation

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Figure 5

An illustrative scheme for the perturbation method. The red line corresponds to the perturbation method used in Ref. [18], whereas the orange line corresponds to the perturbation method used here.

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Figure 6

(a) SDSD and (b) SCSD plates, where D is a damaged clamped boundary

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Figure 7

Model of a partially damaged boundary using translational and torsional springs

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Figure 8

The first three natural frequencies versus α = αL  = αT for a DCCC square plate via the Chebyshev collocation method. The dashed lines represent the frequencies of a free boundary.

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Figure 9

The first three natural frequencies versus α = αL  = αT for a SCSD square plate via the Chebyshev collocation method. The dashed lines represent the frequencies of a free boundary.

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Figure 10

The first three natural frequencies versus α = αL  = αT for a SDSD square plate via the Chebyshev collocation method. The dashed lines represent the frequencies of a free boundary.

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Figure 4

7 × 7 Chebyshev grid

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Figure 11

The second and third mode shapes before and after the veering of the SCSD square plate in Fig. 9 via the Chebyshev collocation method; (a) αL  = αT  = 0.02, (b) αL  = αT  = 0.03

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Figure 13

Perturbation method (dashed line) versus Chebyshev collocation method (solid line) for the first three natural frequencies of SDSD plate

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Figure 14

Perturbation method (dashed line) versus Chebyshev collocation method (solid line) for the first three natural frequencies of SCSD plate

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Figure 15

The effect of the partially damaged boundary of a DCCC square plate on the first natural frequency

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Figure 16

The effect of the partially damaged boundary of a DCCC square plate on the second natural frequency

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Figure 17

The effect of the partially damaged boundary of a DCCC square plate on the third natural frequency

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Figure 2

Chebyshev points numbered from right to left

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Figure 1

Model of a damaged boundary using translational and torsional springs

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Figure 12

The second and third mode shapes before and after the veering of the SDSD square plate in Fig. 1 via the Chebyshev collocation method; (a) αL  = αT  = 0.005, (b) αL  = αT  = 0.015

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