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Research Papers

Optimization of Boundary Supports for Sound Radiation Reduction of Vibrating Structures

[+] Author and Article Information
H. Denli

Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716

J. Q. Sun1

Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716

1

Corresponding author.

J. Vib. Acoust 130(1), 011007 (Nov 12, 2007) (7 pages) doi:10.1115/1.2776345 History: Received June 08, 2006; Revised October 17, 2006; Published November 12, 2007

The purpose of this research is to design optimal boundary supports for minimum structural sound radiation. The influence of the boundary conditions on the structural dynamics of a cantilever beam is first examined to motivate the research. The boundary supports constraining both the in- and out-of-plane degrees of freedom of the plate are considered as the design parameters. The fixed and free boundary degrees of freedom are represented by a continuous function with the help of homogenization. Analytical expressions of sensitivity functions are employed in the optimization, leading to more efficient and accurate numerical solutions. The sensitivity expressions are based on the linear equation system obtained with the finite element method. Numerical examples of single frequency and broadband optimizations are presented. The sensitivity of the optimal design parameters with respect to small random perturbations is also studied. The examples demonstrate that an encouraging reduction of sound radiation as measured by the mean square normal velocity can be achieved with the optimal boundary conditions as compared with the base line structure.

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Figures

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Figure 1

A uniform beam with a spring support at one end and clamped on the other

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Figure 2

Variation of the first mode shape of the beam with the spring constant of the support. The effective boundary condition varies from a free end to a fixed end. The arrow indicates the increasing direction of the support stiffness (from the free to fixed boundary).

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Figure 3

Variation of (a) the resonant frequencies and (b) the sensitivity function of the resonant frequencies with respect to the support stiffness. The order of the eigenvalue increases from the bottom up in (a) and from the left to the right in (b).

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Figure 4

The domain of an elastic body with boundary coordinates

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Figure 5

The variation of the fundamental frequency of a square plate with q and pi of the boundary parameters. (a) The variation with the translational boundary support only, which spans from free to pinned conditions. (b) The variation with the rotational boundary support, which spans from the pinned to clamped conditions. The arrows point to the increasing direction of q from 2 to 6.

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Figure 6

The frequency response of the mean square normal velocity of the plate with uniform boundary conditions. Solid line: the cantilever plate. Dot-dashed line: two opposite sides pinned. Dashed line: all edges pinned. Dotted lines: completely clamped boundary.

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Figure 7

Comparison of the optimal sound radiation (solid line) from the square plate with that from the base line plate (dotted line). (a) Tonal optimization at 500Hz. The vertical dashed line marks 500Hz. (b) Broadband optimization from 10Hzto1000Hz.

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Figure 8

Optimal boundary conditions for the square plate. Square, circle, and triangle markers on the boundary refer to the supports constraining θx, θy, and w. The color scheme depicts the level of the support design parameter pi. (a) Tonal optimization. (b) Broadband optimization.

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Figure 9

(a) Comparison of the optimal sound radiation (solid line) from the square plate with that from the base line plate (dotted line). (b) Variation of the radiated power with respect to the random perturbation of the optimal design parameters.

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Figure 10

Comparison of the optimal sound radiation (solid line) from the circular plate with that from the base line plate (dotted line). (a) Tonal optimization at 500Hz. The vertical dashed line marks 500Hz. (b) Broadband optimization from 10Hzto1000Hz.

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Figure 11

Optimal boundary conditions for the circular plate. Square, circle, and triangle markers on the boundary refer to the supports constraining θx, θy, and w. The color scheme depicts the level of the support design parameter pi. (a) Tonal optimization. (b) Broadband optimization.

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