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Research Papers

A Spectral-Tchebychev Solution for Three-Dimensional Vibrations of Parallelepipeds Under Mixed Boundary Conditions

[+] Author and Article Information
Sinan Filiz, Bekir Bediz

Department of Mechanical Engineering,  Carnegie Mellon University, Pittsburgh, PA 15213

L. A. Romero

 Sandia National Laboratories, Albuquerque, NM 87185

O. Burak Ozdoganlar1

Department of Mechanical Engineering,  Carnegie Mellon University, Pittsburgh, PA 15213ozdoganlar@cmu.edu

1

Corresponding author.

J. Appl. Mech 79(5), 051012 (Jun 29, 2012) (11 pages) doi:10.1115/1.4006256 History: Received September 08, 2010; Revised February 27, 2012; Posted March 03, 2012; Published June 28, 2012; Online June 29, 2012

Vibration behavior of structures with parallelepiped shape—including beams, plates, and solids—are critical for a broad range of practical applications. In this paper we describe a new approach, referred to here as the three-dimensional spectral-Tchebychev (3D-ST) technique, for solution of three-dimensional vibrations of parallelepipeds with different boundary conditions. An integral form of the boundary-value problem is derived using the extended Hamilton’s principle. The unknown displacements are then expressed using a triple expansion of scaled Tchebychev polynomials, and analytical integration and differentiation operators are replaced by matrix operators. The boundary conditions are incorporated into the solution through basis recombination, allowing the use of the same set of Tchebychev functions as the basis functions for problems with different boundary conditions. As a result, the discretized equations of motion are obtained in terms of mass and stiffness matrices. To analyze the numerical convergence and precision of the 3D-ST solution, a number of case studies on beams, plates, and solids with different boundary conditions have been conducted. Overall, the calculated natural frequencies were shown to converge exponentially with the number of polynomials used in the Tchebychev expansion. Furthermore, the natural frequencies and mode shapes were in excellent agreement with those from a finite-element solution. It is concluded that the 3D-ST technique can be used for accurate and numerically efficient solution of three-dimensional parallelepiped vibrations under mixed boundary conditions.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Generic description of parallelepiped geometry

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Figure 4

Convergence of the nondimensional natural frequencies for the rectangular cross-sectioned beam with both top and bottom sides fixed

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Figure 5

Mode shapes of the fixed-fixed rectangular beam calculated from the 3D-ST solution

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Figure 6

Convergence of the nondimensional natural frequencies for the unconstrained plate in terms of the logarithm of the convergence ratio for the (a) AA1, (b) SS1, and (c) SA1 modes

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Figure 7

Mode shapes of the unconstrained plate calculated from the 3D-ST solution

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Figure 8

Convergence of the nondimensional natural frequencies for the constrained plate in terms of the logarithm of the convergence ratio for the (a) AA1, (b) AA2, and (c) AA3 modes

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Figure 9

Mode shapes of the constrained plate (two sides fixed) calculated from the 3D-ST solution

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Figure 10

Convergence of the nondimensional natural frequencies for the unconstrained thick plate in terms of the logarithm of the convergence ratio for the (a) AA1, (b) SS1, and (c) SA1 modes

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Figure 11

Mode shapes of the unconstrained thick plate calculated from the 3D-ST solution

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Figure 12

Convergence of the nondimensional natural frequencies for the unconstrained solid structure (cube) in terms of the logarithm of the convergence ratio for the (a) AA1, (b) SS1, and (c) SA2 modes

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Figure 13

The convergence plots for the solid structure using the same number of polynomials in each direction for the (a) AA1, (b) SS1, and (c) SA1 modes. The exponential convergence is clear from the graphs.

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Figure 14

Mode shapes of the unconstrained solid structure (cube) calculated from the 3D-ST solution

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Figure 3

Mode shapes of the square cross-section beams calculated from the 3D-ST solution under different boundary conditions

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Figure 2

Convergence of the nondimensional natural frequencies for the square cross-sectioned beam in terms of log10 (Δn ) for the AA1, SS1, and SA1 modes with (a)–(c) free, (d)–(f) fixed bottom, and (g)–(i) fixed top and bottom boundary conditions

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