A comparison of sound radiation behavior of plate in air medium with attached discrete patches/point masses having different thickness variations with different taper ratio of 0.3, 0.6, and 0.9 is analysed. Finite element method is used to find the vibration characteristics while Rayleigh integral is used to predict the sound radiation characteristics. Minimum peak sound power level obtained is at a taper ratio of 0.6 with parabolic increasing-decreasing thickness variation for plate with four discrete patches. At higher taper ratio, linearly increasing-decreasing thickness variation is another alternative for minimum peak sound power level suppression with discrete patches. It is found that, in low frequency range, average radiation efficiency remains almost the same, but near first peak, four patches or four point masses cause increase in average radiation efficiency; that is, redistribution of point masses/patches does have effect on average radiation efficiency at a given taper ratio.
Vibration and sound radiation problems of plates with and without attached discrete mass/patch are used in many engineering application. Generally, variable thickness is used to alter the resonant frequency and to reduce weight and size of structure. The stiffness is a mechanical property of the material. The thickness will influence the dynamic stiffness (D = EI) because the inertia will change. Several researchers investigated the free vibration analysis of plate with variable thickness without attached discrete mass/patch. Appl and Byers [
Structural vibration of plate causes radiation of sound. The prediction of sound radiated from vibration of plate is important to control the noise generated from vibrating structure. Several researchers investigated sound radiation behavior of isotropic and composite plates with uniform thickness subjected to time varying harmonic excitation. Petrone et al. [
This paper presents a comparison of vibroacoustic response of isotropic plate with attached discrete patches and point masses having different thickness variation with different taper ratios studied. The effects of redistribution of discrete patch/mass on acoustic radiation behavior of plate with varying thickness having different taper ratios of 0.3, 0.6, and 0.9 in one direction are investigated.
The strain energy of plate
The kinetic energy for the plate is given by [
The natural frequency and mode shape of the plate are obtained by solving the following eigenvalue problem for
It is assumed that the rectangular thin plate, simply supported along four edges in flexural vibration, is set on a flat rigid baffle of infinite extent as shown in Figure
Coordinate system of a vibrating plate [
By integrating the far-field acoustic intensity over a hemisphere of radius
For the plate with simply supported edges, the mode shape function
The modal velocity amplitude
The modal mass for the simply supported boundary [
The natural frequency
By substituting (
An analytical solution of
The radiation efficiency “
The dimension and the physical properties of plate and patch.
Dimensions | Plate without patches/masses | Single patch | Two patches | Four patches |
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(Figure |
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Length [m] |
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Width [m] |
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Thickness [m] |
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Density [kg/m3] | 7850 | 7850 | 7850 | 7850 |
Young’s modulus [N/m2] |
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Poisson ratio | 0.3 | 0.3 | 0.3 | 0.3 |
(a) Plate without any discrete patch/point mass, (b) plate with single patch, mass of patch = 0.1 times plate mass, (c) plate with two discrete patches, mass of each patch = 0.05 times plate mass, and (d) plate with four discrete patches, mass of each patch = 0.025 times plate mass.
Four different types of variation in thickness [
Plate with different variable thickness [
The equations of different types of thickness variation are given by [
For validation of natural frequency of a rectangular isotropic plate and sound power radiation, the published result of S. Li and X. Li [
Comparison and validation of eigenfrequency (
Mode | (S. Li and X. Li [ |
Present work | % variation in natural frequency |
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(1,1) | 87 | 86.989 | 0.012 |
(2,1) | 193 | 192.32 | 0.352 |
(1,2) | 241 | 240.81 | 0.078 |
(2,2) | 345 | 344.53 | 0.191 |
Comparison of sound power level with that of S. Li and X. Li [
The effects on natural frequencies due to attached point masses and discrete patches are investigated for different cases keeping the mass of the “plate + patches” and “plate + point mass” constant.
Tables
Comparison of first four eigenfrequencies (
Mode | Unloaded plate (Figure |
Plate with one discrete patch (Figure |
% variation in natural frequency | Plate with two discrete patches (Figure |
% variation in natural frequency | Plate with four discrete patches (Figure |
% variation in natural frequency |
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1st | 87 | 86.942 | 0.066 | 85.865 | 1.304 | 85.102 | 2.181 |
2nd | 193 | 192.74 | 0.134 | 191.84 | 0.601 | 191.19 | 0.937 |
3rd | 241 | 239.43 | 0.651 | 237.88 | 1.294 | 235.43 | 2.311 |
4th | 345 | 341.47 | 1.023 | 340.54 | 1.292 | 337.57 | 2.153 |
Comparison of first four eigenfrequencies (
Mode | Unloaded plate (Figure |
Plate with one discrete point mass (Figure |
% variation in natural frequency | Plate with two discrete point masses (Figure |
% variation in natural frequency | Plate with four discrete point masses (Figure |
% variation in natural frequency |
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1st | 87 | 73.022 | 16.066 | 84.497 | 2.877 | 83.558 | 3.956 |
2nd | 193 | 189.87 | 1.621 | 177.85 | 7.849 | 174.55 | 9.559 |
3rd | 241 | 235.14 | 2.431 | 229.32 | 4.846 | 224.63 | 6.792 |
4th | 345 | 310.88 | 9.889 | 284.54 | 17.524 | 281.54 | 18.394 |
Figure
Comparison of natural frequency variation of plate with attached four patches with different taper ratio (
Case | Mode | Natural frequency | |||
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I | I | 85.102 | 83.311 | 82.302 | 78.65 |
II | 191.19 | 188.29 | 180.85 | 166.04 | |
III | 235.43 | 220.02 | 219.96 | 206.45 | |
IV | 337.57 | 332.58 | 330.42 | 306.92 | |
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II | I | 85.102 | 83.978 | 83.580 | 82.809 |
II | 191.19 | 188.99 | 185.83 | 176.58 | |
III | 235.43 | 230.01 | 229.39 | 209.41 | |
IV | 337.57 | 335.47 | 332.40 | 325.37 | |
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III | I | 85.102 | 83.571 | 83.571 | 82.894 |
II | 191.19 | 189.38 | 187.23 | 169.45 | |
III | 235.43 | 225.16 | 220.41 | 203.41 | |
IV | 337.57 | 330.37 | 326.45 | 301.69 | |
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IV | I | 85.102 | 84.84 | 84.79 | 83.581 |
II | 191.19 | 190.23 | 188.27 | 174.10 | |
III | 235.43 | 227.61 | 222.26 | 206.30 | |
IV | 337.57 | 334.06 | 329.15 | 312.30 |
Comparison of natural frequency variation of plate with attached four point masses with different taper ratio (
Case | Mode | Natural frequency | |||
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I | I | 83.558 | 83.096 | 81.579 | 71.519 |
II | 174.55 | 169.98 | 161.80 | 139.18 | |
III | 224.63 | 219.30 | 196.24 | 152.52 | |
IV | 281.54 | 272.39 | 267.29 | 255.22 | |
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II | I | 83.558 | 82.27 | 80.301 | 72.754 |
II | 174.55 | 170.66 | 167.22 | 159.54 | |
III | 224.63 | 217.61 | 200.72 | 165.97 | |
IV | 281.54 | 274.27 | 270.98 | 262.24 | |
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III | I | 83.558 | 83.455 | 82.42 | 79.912 |
II | 174.55 | 171.38 | 170.42 | 150.89 | |
III | 224.63 | 219.32 | 205.04 | 194.90 | |
IV | 281.54 | 272.21 | 261.36 | 254.90 | |
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IV | I | 83.558 | 83.091 | 82.913 | 81.649 |
II | 174.55 | 167.05 | 162.18 | 158.97 | |
III | 224.63 | 218.05 | 204.16 | 185.41 | |
IV | 281.54 | 276.36 | 268.62 | 259.12 |
Comparison of the first four natural frequencies.
However, it is noticed from Table
Figures
Mode variation of plate with attached four discrete patches with different taper ratios for case 1.
Mode variation of plate with attached four discrete point masses with different taper ratios for case 1.
The sound power level (dB, reference = 10−12 W) of simply supported plate with attached point masses and patches with different taper parameter (
Comparison of sound power level (dB) of plate with different tapers (
Comparison of sound power level (dB) of plate with different tapers (
Comparison of sound power level (dB) of plate with different tapers (
Comparison of sound power level (dB) of plate with different tapers (
From Figures
Comparison of sound power level (dB) of plate with attached four patches and four point masses at taper ratios (
The 10 randomly selected locations of the point on the plate were used to obtain radiation characteristics. The average radiation efficiency of plate with attached one point mass and patch for different taper ratio is also plotted to investigate the effect of taper ratio on sound radiation characteristics as shown in Figures
Comparison of radiation efficiency of plate with different tapers (
Comparison of radiation efficiency of plate with different tapers (
Comparison of radiation efficiency of plate with taper (
Comparison of radiation efficiency of plate with taper (
Figures
(a) Variation of peak sound power level without taper (
A comparison of sound radiation behavior of plate in air medium with attached discrete patches/point masses having different taper ratio is analysed. The masses of the “plate + patches” and “plate + point masses” are kept equal. It has been observed that redistribution of discrete point masses affects the eigenvalues more significantly in comparison to that of patches in eigenvalue suppression. There is significant variation in minimum peak sound power level behavior of plates with parabolic increasing-decreasing thickness variation. It is noticed that higher taper ratio of
The authors declare that there is no conflict of interests regarding the publication of this paper.