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The effects of distributed masses on acoustic radiation behavior of plates
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Applied Acoustics 69 (2008) 272–279 www.elsevier.com/locate/apacoust
Technical note
The effects of distributed masses on acoustic radiation behavior of plates Sheng Li *, Xianhui Li Department of Naval Architecture, Dalian University of Technology, Dalian 116024, People’s Republic of China Received 26 April 2006; received in revised form 13 September 2006; accepted 2 November 2006 Available online 17 January 2007
Abstract The acoustic radiation behavior of a plate with a distributed mass loading is studied. A set of in vacuo normal modes or fluid-loaded undamped normal modes are used for modal analysis of the acoustic radiation from a plate in air or in water. Modal radiation efficiency, modal volume displacement, modal input energy and sound power level are computed to show the effects of size and location of the mass loading on the acoustic radiation of the plate. It is observed that the acoustic radiation behavior of a mode in both cases will have relatively larger changes if the mass loading is placed on an antinode of the mode shape or the mass loading is more concentrated. The acoustic radiation behavior of a mode and the radiated power of the plate in water have less change than those in air with the same mass loading due to the added mass of the water, especially for the first few modes. 2006 Elsevier Ltd. All rights reserved. Keywords: Distributed mass; Acoustic radiation; Fluid-loaded modes; Modal analysis; Sound power level
1. Introduction
2. Theory
Vibration and acoustic radiation problem of plates with a distributed mass loading is very common in engineering applications. Refs. [1,2] show that a mass loading can significantly affect the natural frequencies and mode shapes of in vacuo modes of a plate. Ref. [3] uses point masses to change a mode shape of a plate into a ‘‘weak radiator’’ in air and illustrates a design strategy to minimize the sound power output from a structure based on material tailoring. While there are many researches focusing on the case in air, few works deal with the case in water. In this paper, a set of in vacuo normal modes or fluid-loaded undamped normal modes are used for modal analysis of the acoustic radiation from a baffled plate surrounded by air or water. The effects of size and location of the mass loading on the acoustic radiation behavior of the plate are investigated.
2.1. Quadratic form of acoustic power
*
Corresponding author. E-mail address: [email protected] (S. Li).
0003-682X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2006.11.004
Consider a vibrating structure with time harmonic dependency for the structural and acoustic quantities. The total acoustic power radiated by the structure is found by taking the real part of an integral in terms of surface quantities Z 1 W ¼ Reðpvn Þ dS; ð1Þ 2 S where p is the acoustic pressure and vn is the normal velocity on the surface S of the structure, * denotes the complex conjugate. For a plate with a planar surface extends over an infinite half-space, discretisizing the plate surface into elements and interpolating the structural normal velocity and surface pressure over each element allow the Rayleigh integral on the plate surface to be written in terms of the nodal normal velocity {vn} and surface pressure {p} as fpg ¼ ½Zfvn g;
ð2Þ
S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
where [Z] is the frequency-dependent acoustic impedance matrix, {p} and {vn} are the vectors of length I consisting of the field values at the nodal locations of a grid defining the surface of the structure for the surface acoustic pressure and normal velocity, I is the number of nodes. Substituting Eq. (2) into Eq. (1) leads to a matrix multiplication form [4–7] 1 H H W ¼ Re fvn g ½A½Zfvn g ¼ fvn g ½Rfvn g; ð3Þ 2 R T where ½A ¼ S ½N ½N dS, [N] is the matrix of interpolation functions, the matrix [A][Z] is symmetric based on an acoustic reciprocal principle [8,9], [R] = ([A]/2)Re([Z]) is a purely real symmetric matrix. The matrix [R] must be positive definite on physical grounds since the power output must be greater than zero unless the normal velocity is zero (W > 0"{vn} 6¼ 0) [5,10]. 2.2. Natural frequencies and mode shapes 2.2.1. In air For a plate in an infinite half-space, the air-loaded modes are reduced to the standard in vacuo modes because of the light fluid loading, the natural frequencies xi and mode shapes {/i} (i = 1, . . ., N, N is the number of structural finite element degrees of freedom) of in vacuo modes of a structure can be determined by solving a generalized eigenproblem, ½Kf/g x2 ½Mf/g ¼ 0;
ð4Þ
where [K] and [M] are the stiffness and mass matrices of the plate with mass loading, respectively. A set of normal modes (normalized with respect to [M]) with normal mode shapes {Ui} are known as f/i g fUi g ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : f/i gT fMgf/i g
ð5Þ
2.2.2. In water The water-loaded modes cannot be adequately represented by the above in vacuo modes due to heavy fluid loading. The problem of determining fluid-loaded modes has been addressed by a number of publications [11–16]. State-space coupling methods [11–13], singular value decomposition [14] and modal reduction method [15] are used to compute fluid-loaded modes. The fluid-loaded modes are complex modes. The fluid-loaded natural frequencies and modal damping ratios are obtained from the complex eigenvalues and the fluid-loaded mode shapes are given by the complex eigenvectors. A complex mode shape can be presented separately by the real and imaginary plots or the magnitude and phase plots. However, the above methods have some difficulties to get normal modes because the added mass of the acoustic fluid is frequency dependent and there are no explicit mass matrices in the coupled equations. For a fluid-loaded mode with
273
natural frequency xi, the correct added mass matrix [Ma] is given as [16] T
½M a ¼
½G ½A½Zðxi Þ½G ; xi
ð6Þ
where [G] is a matrix to transform a vector of forces to a vector of normal forces. In this paper, we first use the method described in Ref. [12] to determine the natural frequencies xi and modal damping ratio of water-loaded plate, then we determine the normal mode shapes {Ui} corresponding to xi from the stiffness matrix [K] and the mass matrix [[M] + [Ma]]. It should be noted that the effect of the radiation damping is ignored in deriving the normal modes, hence the mode shapes obtained are fluid-loaded undamped real mode shapes. As we know, while an in vacuo undamped mode is in the synchronous motion, a damped complex mode may not. However, if the radiation damping is small, a fluid-loaded damped mode can be well approximated by the fluid-loaded undamped real mode. 2.3. Modal analysis of acoustic radiation 2.3.1. Modal volume displacement Volume displacement or volume velocity is an index to indicate the capability of sound radiation from a structure. A weak radiator is a structure which radiates sound very inefficiently due to a correspondingly low net volume velocity [3,12]. The volume velocity ^v is directly related to the volume displacement d^ of the structure by [3] Z Z ^ ^v ¼ vn dS ¼ ix d n dS ¼ ixd; ð7Þ where dn is the normal displacement on the surface S. Since the components of {Ui} are proportional to the nodal displacements of the ith mode, {Ui} is used to compute the modal volume displacement of the ith mode according to Eq. (7). 2.3.2. Modal radiation efficiency The radiation efficiency is used to describe the capability of a structure to radiate sound and is generally defined as r¼
W ; qcS 0 hv2n i
ð8Þ
where W is the radiated sound power, q is the density of the acoustic medium, c is the sound speed in the acoustic medium, S0 is the total surface area of the structure, and hv2n i is space average mean-square velocity defined as Z 1 2 hv2n i ¼ jvn j dS: ð9Þ 2S 0 The vector of normal velocity {vn} can be written in terms of the vector of modal velocity {r} as fvn g ¼ ½G½Ufrg: Substitution of Eq. (10) into Eq. (3) gives [4,17]
ð10Þ
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H
W ¼ frg ½T frg ¼
N X N X i¼1
T ij ðri rj Þ ¼
j¼1
N X N X i¼1
W ij ;
ð11Þ
j¼1
in which T
T
½T ¼ ½U ½G ½R½G½U;
ð12Þ
[T] is also symmetric due to symmetric [R] and must be positive definite on physical grounds since W > 0, "{r} 6¼ 0. It can be seen from Eq. (11) that the modes do not radiate independently and the off-diagonal elements of [T] indicate acoustic interaction between modes. It also should be noted that Wii = Tiijrij2 > 0, "{ri} 6¼ 0 since [T] is positive definite. In terms of Eq. (8), the self-radiation efficiency of the ith mode can be written as ri ¼ 2T ii =qcS 0 ;
ð13Þ
and the mutual-radiation efficiency of the ith mode and jth mode can be defined as rij ¼ j2T ij =qcS 0 j:
ð14Þ
It can be seen that the modal radiation efficiency can be used to evaluate the power radiated by a mode. 2.3.3. Modal input energy The modal volume velocity and the modal radiation efficiency describe the ability of a mode to radiate sound, but the acoustic radiation behavior of a mode also largely depends on the driving force f. As we know, when a force acts on a node or a nodal line of a mode, this mode will not be driven and then no peak of radiated sound power is generated at the resonance frequency of the mode. Modal input energy or modal input power can be used to indicate the energy or the power input to a mode. The modal input energy ^e is directly related to the modal input power ^p by ^ p ¼ f jvn j ¼ f jixd n j ¼ x^e: ð15Þ 3. Numerical results and discussion The effects of size and location of distributed mass loading on vibration behavior of a plate are investigated in Ref. [1] and similar configurations of distributed mass loadings are used in this paper. The four different cases of mass loading are shown in Fig. 1. The additional mass loading in loading cases 1–3 is 10% of the mass of the unloaded
plate and in loading case 4 is 50%. The dimensions of the simply supported plate are Lx = 0.455 m, Ly = 0.379 m, and thickness h = 0.003 m. The plate is made of steel with density qs = 7850 kg/m3, Young’s modulus E = 2.1 · 1011 N/m2, and Poison’s ratio m = 0.3. The plate is assumed to be vibrating either in air with a density q = 1.21 kg/m3 and a sound speed c = 343 m/s or in water with a density q = 1000 kg/m3 and a sound speed c = 1500 m/s. The critical frequency of the plate is about 4.01 kHz in air and 76.84 kHz in water. To show the damping levels caused by the radiation loading, no structural damping is considered in determining the natural frequencies and modal damping ratios of the water-loaded modes. In the forced response analysis of the plate, a structural damping coefficient of 0.01 is used for the plate in water and a modal damping ratio of 0.01 is assumed for all modes of the plate in air. The plate is modeled by 16 · 16 four-noded quadrilateral isoparametric elements and this structural finite element model is valid up to 1 kHz according to the commonly applied rule of thumb to use six linear elements per structural wavelength. The finite element is based on Mindlin plate theory (first-order shear deformation theory.). The Rayleigh integral on the plate surface is calculated by discretisizing the plate surface with the same mesh as the plate finite element mesh. The total number of structural DOFs is N = 803 and the total number of nodes is I = 289. Water-loaded natural frequencies and modal damping ratios in the frequency band (10, 560) Hz are obtained using 12 evenly distributed interpolation frequencies and five probing vectors based on the model reduction method [15]. The changes of the first 10 in vacuo natural frequencies for the four loading cases are given in Table 1. The first 10 water-loaded natural frequencies and modal damping ratios are shown in Table 2. It can be seen that the natural frequencies of water-loaded plate have relatively less change than those of the air-loaded plate with the same mass loading due to the large added mass of the water. Also it is shown that the radiation damping caused by the water is small and the mode with high radiation efficiency has high modal damping ratio. That is, the odd– odd modes with high radiation efficiencies have large damping ratios, the even–even modes with low radiation efficiencies have small damping ratios, the radiation efficiencies and the damping ratios of the even–odd or
Fig. 1. Rectangular plates with distributed mass loadings of different size and location.
Table 1 Natural frequencies of the plate loaded with different distributed masses in vacuo Mode
Loading case 1
Loading case 2
Loading case 3
Loading case 4
Modal indices (m,n)
Frequency (Hz)
Modal indices (m, n)
Frequency (Hz)
Modal indices (m, n)
Frequency (Hz)
Modal indices (m, n)
Frequency (Hz)
Modal indices (m, n)
Frequency (Hz)
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
86.7 193.4 241.0 344.6 373.6 501.3 519.7 600.0 631.2 766.7
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (4, 2)
77.1 179.3 223.3 328.3 351.9 472.7 500.1 570.3 596.5 733.7
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
73.5 190.6 237.5 324.8 344.2 463.6 512.8 580.0 611.7 712.6
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
82.3 173.1 226.0 308.3 366.2 476.2 501.6 576.5 612.0 730.5
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (4, 1) (3, 2) (2, 3) (3, 3)
49.8 179.4 223.6 272.6 342.7 446.4 461.0 477.1 606.6 674.5
Table 2 Natural frequencies and modal damping rations of the plate loaded with different distributed masses in water Mode
1 2 3 4 5 6 7 8 9 10
Unloaded plate
Loading case 1
Loading case 2
Loading case 3
Loading case 4
Modal indices (m, n)
Frequency (Hz)
Damping ratio (%)
Modal indices (m, n)
Frequency (Hz)
Damping ratio (%)
Modal indices (m, n)
Frequency (Hz)
Damping ratio (%)
Modal indices (m, n)
Frequency (Hz)
Damping ratio (%)
Modal indices (m, n)
Frequency (Hz)
Damping ratio (%)
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
30.4 93.9 121.6 193.6 210.3 295.5 320.7 376.3 401.3 511.0
0.60198 0.00226 0.00380 0.00007 0.42042 0.44250 0.00839 0.02386 0.03540 0.12139
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
30.0 92.2 119.2 190.7 205.0 288.1 315.6 367.7 390.2 503.0
0.57445 0.00200 0.00343 0.00002 0.47378 0.44265 0.00903 0.03571 0.02042 0.14833
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
29.7 93.6 121.2 193.5 195.3 281.6 319.0 372.2 395.2 481.7
0.56430 0.00218 0.00373 0.00008 0.61393 0.30631 0.00880 0.03342 0.02068 0.20338
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
30.2 91.7 118.9 184.0 208.5 291.7 312.3 367.7 396.3 501.2
0.59085 0.00313 0.00426 0.00162 0.38214 0.33365 0.05194 0.03141 0.04091 0.04667
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (3, 2) (4, 1) (2, 3) (3, 3)
27.3 92.4 119.5 154.4 193.3 265.0 311.2 339.2 386.6 440.7
0.42890 0.00205 0.00339 1.00770 0.00008 0.11186 0.01059 0.03669 0.00078 0.31850
S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
1 2 3 4 5 6 7 8 9 10
Unloaded plate
275
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S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
tion efficiency or the volume displacement. It is observed from Fig. 2 that the radiation efficiency in water is much lower than that in air and the acoustic radiation behavior of a mode in water has relatively less change than that in air with the same mass loading. It can be concluded, at least for the particular cases be studied, that the acoustic radiation behavior of a plate is more sensitive to mass loading in air than in water. The mutual-radiation efficiencies of mode (1, 1) and mode (3, 1) in air and in water are plotted in Fig. 3. It can be seen that the mutual radiation between these modes has very high efficiency both in air and in water and the mass loading significantly affects the mutual-radiation efficiency. At resonance, the acoustic radiation is dominated by one mode and the mutual radiation between modes at
odd–even modes are between the above two kinds of mode groups. The self-radiation efficiencies of the odd–odd modes (1, 1), (3, 1) and (1, 3) in air and in water are plotted in Fig. 2. The modal volume displacements of the first 10 modes in air and in water are shown in Tables 3 and 4, respectively. It can be seen that the modal radiation efficiency of a mode is mainly influenced by its volume displacement. The volume displacement show the same change trends for these four loading cases in both air and water. Relatively large changes are observed in the acoustic radiation behavior of a mode both in air and in water when the mass loading is placed on an antinode of the mode shape or the mass loading is more concentrated. However, it should be noted that the mass loading may decrease (e.g. for mode (1, 1)) or increase (e.g. for mode (3, 1)) the radia-
0.01
0.1
Radiation efficiency
Radiation efficiency
1E-3
0.01
1E-3
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-4
0
100
200
300
400
500
600
700
1E-4
1E-5
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-6
1E-7
800
0
50
100 150 200
(a) mode (1,1) in air
(b) mode (1,1) in water
1E-3
1E-4
0.01
Radiation efficiency
Radiation efficiency
0.1
1E-3
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-4
1E-5
0
100
200
300
400
500
600
700
1E-5
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-6
1E-7
1E-8
800
0
50
100 150 200 250 300 350 400 450 500 550
Frequency (Hz)
Frequency (Hz)
(d) mode (3,1) in water
(c) mode (3,1) in air 0.1
1E-3
0.01
1E-4
Radiation efficiency
Radiation efficiency
250 300 350 400 450 500 550
Frequency (Hz)
Frequency (Hz)
1E-3
1E-4
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-5
1E-5
1E-6
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-7
1E-8
1E-6 0
100
200
300
400
500
Frequency (Hz)
(e) mode (1,3) in air
600
700
800
0
50
100
150
200 250
300 350 400
450
Frequency (Hz)
(f) mode (1,3) in water
Fig. 2. The self-radiation efficiency of mode (1, 1), (3, 1) and (3, 1) in air and in water.
500 550
Table 3 Modal volume displacement and modal input energy of the plate loaded with different distributed masses in air Mode
Loading case 1
Loading case 2
Loading case 3
Loading case 4
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
6.89E 02 2.41E 06 2.50E 06 7.80E 11 2.24E 02 2.24E 02 1.56E 06 2.08E 06 1.25E 05 7.26E 03
1.00E + 00 6.73E 03 8.07E 03 5.60E 05 1.00E + 00 1.00E + 00 8.19E 03 6.97E 03 1.27E 02 1.00E + 00
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (4, 2)
6.11E 02 2.78E 06 2.84E 06 2.88E 10 2.72E 02 2.45E 02 6.63E 06 1.17E 05 1.36E 05 1.27E 08
8.88E 01 6.43E 03 7.74E 03 5.70E 05 9.12E 01 8.37E 01 7.95E 03 1.17E 02 6.36E 03 1.19E 04
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
5.78E 02 1.07E 05 1.31E 05 4.24E 02 4.72E 08 2.00E 02 4.22E 05 4.66E 05 2.60E 05 8.54E 03
8.64E 01 6.95E 03 8.42E 03 8.39E 01 5.70E 05 4.77E 01 8.36E 03 1.52E 02 6.73E 03 5.96E 01
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
6.52E 02 1.25E 02 5.36E 03 1.21E 03 1.99E 02 8.56E 03 1.63E 02 8.27E 03 1.86E 03 3.82E 03
9.30E 01 2.44E 01 1.63E 01 3.70E 01 9.35E 01 5.65E 01 7.07E 01 2.80E 01 8.78E 02 5.68E 01
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (4, 1) (3, 2) (2, 3) (3, 3)
3.81E 02 5.83E 05 9.96E 05 6.07E 02 2.18E 07 1.61E 02 1.09E 04 2.83E 04 2.45E 05 5.11E 03
5.99E 01 7.45E 03 9.36E 03 4.08E 01 6.90E 05 1.69E 01 1.54E 02 9.61E 03 1.20E 03 1.81E 01
Table 4 Modal volume displacement and modal input energy of the plate loaded with different distributed masses in water Mode
1 2 3 4 5 6 7 8 9 10
Unloaded plate
Loading case 1
Loading case 2
Loading case 3
Loading case 4
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
2.43E 02 3.81E 07 7.68E 07 4.15E 09 7.85E 03 6.88E 03 9.73E 07 6.49E 07 2.34E 06 2.98E 03
3.45E 01 3.20E 03 3.98E 03 2.80E 05 6.47E 01 6.02E 01 5.11E 03 5.46E 03 7.46E 03 6.97E 01
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
2.39E 02 4.22E 07 7.94E 07 2.62E 12 8.37E 03 6.86E 03 1.31E 06 1.39E 06 2.50E 06 3.17E 03
3.40E 01 3.17E 03 3.94E 03 2.82E 05 6.40E 01 5.59E 01 5.13E 03 6.85E 03 5.80E 03 6.90E 01
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
2.37E 02 7.12E 07 1.16E 06 2.05E 07 9.81E 03 5.83E 03 4.20E 06 4.56E 06 5.73E 06 3.96E 03
3.39E 01 3.23E 03 4.03E 03 1.45E 05 6.70E 01 4.69E 01 5.23E 03 7.45E 03 6.85E 03 6.13E 01
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
2.41E 02 6.04E 04 4.61E 04 4.64E 04 7.52E 03 6.00E 03 2.19E 03 1.18E 03 3.72E 04 1.82E 03
3.42E 01 1.31E 02 2.35E 02 1.18E 01 6.36E 01 5.57E 01 1.61E 01 7.52E 02 1.68E 03 5.43E 01
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (3, 2) (4, 1) (2, 3) (3, 3)
2.17E 02 3.30E 06 5.73E 06 1.41E 02 3.34E 08 3.59E 03 1.47E 05 2.70E 05 1.11E 05 5.11E 03
3.17E 01 3.36E 03 4.27E 03 5.29E 01 3.24E 05 2.09E 01 6.06E 03 1.10E 02 2.83E 03 3.59E 01
S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
1 2 3 4 5 6 7 8 9 10
Unloaded plate Modal indices (m, n)
277
278
S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
1E-3
1E-4
Radiation efficiency
Radiation efficiency
0.1
0.01
1E-3
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-4
1E-5
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-6
1E-7 0
100
200
300
400
500
600
700
0
800
50
100 150 200 250 300 350 400 450 500 550
Frequency (Hz)
Frequency (Hz)
(a) in air
(b) in water
Fig. 3. The mutual-radiation efficiency of mode (1, 1) and (3, 1) in air and in water.
resonance is negligible. At off-resonance, the contribution of the mutual radiation may be of importance to the radiated power. The sound power level (dB, re:1012 W) of the plate excited by a concentrated force is shown in Fig. 4, which shows the effects of the distributed mass loading on the forced response. The modal input energy of the first 10 modes in air and in water is shown in Tables 3 and 4, respectively. The concentrated force is located at the plate center with amplitude 1 N. From Tables 3, 4, and Fig. 4, it can be observed that for the mode (1, 3) in the fourth loading case, the mass loading decreases the modal input energy a lot and reduces radiated sound power. For the third loading case the modes other than odd-odd modes are also excited due to a change of location of the node lines caused by the mass loading. It is clear that the radiated power at resonance frequencies depends on both the modal radiation efficiency and the modal input energy. An optimization procedure can be applied to implement an effective design strategy to determine the size and location of the mass loading to suppress the acoustic radiation [3]. It is also clear that the radiated power in water has less change than in air for different loading cases, especially for the first few modes due to the large added mass of the water. However, it also can be seen that the effects of mass loading on acoustic radiation behavior in water will be more obvious
with increasing frequency due to the decreasing effect of the added mass of the water. 4. Conclusions The effects of the distributed mass loading on the acoustic radiation from a plate are investigated. The finite element method is employed for discretisizing the structure. The Rayleigh integral is used for modeling the acoustic fluid. Modal radiation efficiency, modal volume displacement, modal input energy and sound power level are computed to show the effects of size and location of the mass loading on the acoustic radiation of a baffled plate both in air and in water. The normal mode shapes of the fluidloaded undamped modes are used to compute the modal radiation efficiency, modal volume displacement, modal input energy of the plate in water. It is observed both in air and in water that the acoustic radiation behavior of a mode will have relatively larger changes if the mass loading is placed on an antinode of the mode shape or the mass loading is more concentrated. The mass loading may decrease or increase the acoustic radiation from the plate at resonance frequency and the radiated power at resonance frequencies depend on both the modal radiation efficiency and the modal input energy. At off-resonance frequency, the added mass generally reduces the radiated
100 100 90 90
Sound power level (dB)
Sound power level (dB)
80 70 60 50
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
40 30 20
80 70 60
40 30
0
100
200
300
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
50
400
500
600
700
800
0
50
100
150
200
250
300
350
Frequency (Hz)
Frequency (Hz)
(b) in water
(a) in air
Fig. 4. The sound power level.
400
450
500
550
S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
power. However, the acoustic radiation behavior of a mode and the total radiated power in water is less sensitive than in air to the mass loading due to the large added mass of the water, especially for the first few modes. Acknowledgement The authors are grateful for the support of the National Natural Science Foundation of China under Grant No. 10402004. References [1] Wong WO. The effects of distributed mass loading on plate vibration behavior. J Sound Vib 2002;252:577–83. [2] Kopmaz O, Telli S. Free vibration of a rectangular plate carrying a distributed mass. J Sound Vib 2002;251:39–57. [3] Koopmann GH, Fahnline JB. Designing quiet structures-a sound power minimization approach. Academic press; 1997. [4] Elliott SJ, Johnson ME. Radiation modes and the active control of sound power. J Acoust Soc Am 1993;94:2194–204. [5] Johnson ME, Elliott SJ. Active control of sound radiation using volume velocity cancellation. J Acoust Soc Am 1995;98:2174–86. [6] Cunefare KA, Koopmann GH. A boundary element approach to optimization of active noise control sources on three-dimensional structures. ASME Trans, J Vib Acoust 1991;113:387–94.
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[7] Cunefare KA, Koopmann GH. Global optimum active noise control: surface and far-field effects. J Acoust Soc Am 1991;90:365–73. [8] Chen PT, Ginsberg JH. Complex power, reciprocity, and radiation modes for submerged bodies. J Acoust Soc Am 1995;98:3343–51. [9] Chen PT, Ju SH, Cha KC. A symmetric formulation of coupled BEM/FEM in solving responses of submerged elastic structures for large degrees of freedom. J Sound Vib 2000;233:407–22. [10] Nelson PA, Curtis ARD, Elliott SJ, Bullmore AJ. The minimum power output of free field point sources and the active control of sound. J Sound Vib 1987;116:397–414. [11] Giordano JA, Koopmann GH. State-space boundary element-finite element coupling for fluid-structure interaction analysis. J Acoust Soc Am 1995;98:363–72. [12] Cunefare KA, De Rosa S. An improved state-space method for coupled fluid-structure interaction analysis. J Acoust Soc Am 1999;105:206–10. [13] Li S. A state-space coupling method for fluid-structure interaction analysis of plates. J Acoust Soc Am 2005;118:800–5. [14] Fahnline JB. Computing fluid-coupled resonance frequencies, mode shapes and damping loss factors using singular value decomposition. J Acoust Soc Am 2004;115:1474–82. [15] Li X, Li S. Modal parameters estimation for fluid-loaded structures from reduced order models. J Acoust Soc Am 2006;120:1996–2003. [16] McCollum MD, Siders CM. Modal analysis of a structure in a compressible fluid using a finite element/boundary element approach. J Acoust Soc Am 1996;99:1949–57. [17] Cunefare KA. Effect of modal interaction on sound radiation from vibrating structures. AIAA J 1992;30:2819–28.
Applied Acoustics 69 (2008) 272–279 www.elsevier.com/locate/apacoust
Technical note
The effects of distributed masses on acoustic radiation behavior of plates Sheng Li *, Xianhui Li Department of Naval Architecture, Dalian University of Technology, Dalian 116024, People’s Republic of China Received 26 April 2006; received in revised form 13 September 2006; accepted 2 November 2006 Available online 17 January 2007
Abstract The acoustic radiation behavior of a plate with a distributed mass loading is studied. A set of in vacuo normal modes or fluid-loaded undamped normal modes are used for modal analysis of the acoustic radiation from a plate in air or in water. Modal radiation efficiency, modal volume displacement, modal input energy and sound power level are computed to show the effects of size and location of the mass loading on the acoustic radiation of the plate. It is observed that the acoustic radiation behavior of a mode in both cases will have relatively larger changes if the mass loading is placed on an antinode of the mode shape or the mass loading is more concentrated. The acoustic radiation behavior of a mode and the radiated power of the plate in water have less change than those in air with the same mass loading due to the added mass of the water, especially for the first few modes. 2006 Elsevier Ltd. All rights reserved. Keywords: Distributed mass; Acoustic radiation; Fluid-loaded modes; Modal analysis; Sound power level
1. Introduction
2. Theory
Vibration and acoustic radiation problem of plates with a distributed mass loading is very common in engineering applications. Refs. [1,2] show that a mass loading can significantly affect the natural frequencies and mode shapes of in vacuo modes of a plate. Ref. [3] uses point masses to change a mode shape of a plate into a ‘‘weak radiator’’ in air and illustrates a design strategy to minimize the sound power output from a structure based on material tailoring. While there are many researches focusing on the case in air, few works deal with the case in water. In this paper, a set of in vacuo normal modes or fluid-loaded undamped normal modes are used for modal analysis of the acoustic radiation from a baffled plate surrounded by air or water. The effects of size and location of the mass loading on the acoustic radiation behavior of the plate are investigated.
2.1. Quadratic form of acoustic power
*
Corresponding author. E-mail address: [email protected] (S. Li).
0003-682X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2006.11.004
Consider a vibrating structure with time harmonic dependency for the structural and acoustic quantities. The total acoustic power radiated by the structure is found by taking the real part of an integral in terms of surface quantities Z 1 W ¼ Reðpvn Þ dS; ð1Þ 2 S where p is the acoustic pressure and vn is the normal velocity on the surface S of the structure, * denotes the complex conjugate. For a plate with a planar surface extends over an infinite half-space, discretisizing the plate surface into elements and interpolating the structural normal velocity and surface pressure over each element allow the Rayleigh integral on the plate surface to be written in terms of the nodal normal velocity {vn} and surface pressure {p} as fpg ¼ ½Zfvn g;
ð2Þ
S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
where [Z] is the frequency-dependent acoustic impedance matrix, {p} and {vn} are the vectors of length I consisting of the field values at the nodal locations of a grid defining the surface of the structure for the surface acoustic pressure and normal velocity, I is the number of nodes. Substituting Eq. (2) into Eq. (1) leads to a matrix multiplication form [4–7] 1 H H W ¼ Re fvn g ½A½Zfvn g ¼ fvn g ½Rfvn g; ð3Þ 2 R T where ½A ¼ S ½N ½N dS, [N] is the matrix of interpolation functions, the matrix [A][Z] is symmetric based on an acoustic reciprocal principle [8,9], [R] = ([A]/2)Re([Z]) is a purely real symmetric matrix. The matrix [R] must be positive definite on physical grounds since the power output must be greater than zero unless the normal velocity is zero (W > 0"{vn} 6¼ 0) [5,10]. 2.2. Natural frequencies and mode shapes 2.2.1. In air For a plate in an infinite half-space, the air-loaded modes are reduced to the standard in vacuo modes because of the light fluid loading, the natural frequencies xi and mode shapes {/i} (i = 1, . . ., N, N is the number of structural finite element degrees of freedom) of in vacuo modes of a structure can be determined by solving a generalized eigenproblem, ½Kf/g x2 ½Mf/g ¼ 0;
ð4Þ
where [K] and [M] are the stiffness and mass matrices of the plate with mass loading, respectively. A set of normal modes (normalized with respect to [M]) with normal mode shapes {Ui} are known as f/i g fUi g ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : f/i gT fMgf/i g
ð5Þ
2.2.2. In water The water-loaded modes cannot be adequately represented by the above in vacuo modes due to heavy fluid loading. The problem of determining fluid-loaded modes has been addressed by a number of publications [11–16]. State-space coupling methods [11–13], singular value decomposition [14] and modal reduction method [15] are used to compute fluid-loaded modes. The fluid-loaded modes are complex modes. The fluid-loaded natural frequencies and modal damping ratios are obtained from the complex eigenvalues and the fluid-loaded mode shapes are given by the complex eigenvectors. A complex mode shape can be presented separately by the real and imaginary plots or the magnitude and phase plots. However, the above methods have some difficulties to get normal modes because the added mass of the acoustic fluid is frequency dependent and there are no explicit mass matrices in the coupled equations. For a fluid-loaded mode with
273
natural frequency xi, the correct added mass matrix [Ma] is given as [16] T
½M a ¼
½G ½A½Zðxi Þ½G ; xi
ð6Þ
where [G] is a matrix to transform a vector of forces to a vector of normal forces. In this paper, we first use the method described in Ref. [12] to determine the natural frequencies xi and modal damping ratio of water-loaded plate, then we determine the normal mode shapes {Ui} corresponding to xi from the stiffness matrix [K] and the mass matrix [[M] + [Ma]]. It should be noted that the effect of the radiation damping is ignored in deriving the normal modes, hence the mode shapes obtained are fluid-loaded undamped real mode shapes. As we know, while an in vacuo undamped mode is in the synchronous motion, a damped complex mode may not. However, if the radiation damping is small, a fluid-loaded damped mode can be well approximated by the fluid-loaded undamped real mode. 2.3. Modal analysis of acoustic radiation 2.3.1. Modal volume displacement Volume displacement or volume velocity is an index to indicate the capability of sound radiation from a structure. A weak radiator is a structure which radiates sound very inefficiently due to a correspondingly low net volume velocity [3,12]. The volume velocity ^v is directly related to the volume displacement d^ of the structure by [3] Z Z ^ ^v ¼ vn dS ¼ ix d n dS ¼ ixd; ð7Þ where dn is the normal displacement on the surface S. Since the components of {Ui} are proportional to the nodal displacements of the ith mode, {Ui} is used to compute the modal volume displacement of the ith mode according to Eq. (7). 2.3.2. Modal radiation efficiency The radiation efficiency is used to describe the capability of a structure to radiate sound and is generally defined as r¼
W ; qcS 0 hv2n i
ð8Þ
where W is the radiated sound power, q is the density of the acoustic medium, c is the sound speed in the acoustic medium, S0 is the total surface area of the structure, and hv2n i is space average mean-square velocity defined as Z 1 2 hv2n i ¼ jvn j dS: ð9Þ 2S 0 The vector of normal velocity {vn} can be written in terms of the vector of modal velocity {r} as fvn g ¼ ½G½Ufrg: Substitution of Eq. (10) into Eq. (3) gives [4,17]
ð10Þ
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H
W ¼ frg ½T frg ¼
N X N X i¼1
T ij ðri rj Þ ¼
j¼1
N X N X i¼1
W ij ;
ð11Þ
j¼1
in which T
T
½T ¼ ½U ½G ½R½G½U;
ð12Þ
[T] is also symmetric due to symmetric [R] and must be positive definite on physical grounds since W > 0, "{r} 6¼ 0. It can be seen from Eq. (11) that the modes do not radiate independently and the off-diagonal elements of [T] indicate acoustic interaction between modes. It also should be noted that Wii = Tiijrij2 > 0, "{ri} 6¼ 0 since [T] is positive definite. In terms of Eq. (8), the self-radiation efficiency of the ith mode can be written as ri ¼ 2T ii =qcS 0 ;
ð13Þ
and the mutual-radiation efficiency of the ith mode and jth mode can be defined as rij ¼ j2T ij =qcS 0 j:
ð14Þ
It can be seen that the modal radiation efficiency can be used to evaluate the power radiated by a mode. 2.3.3. Modal input energy The modal volume velocity and the modal radiation efficiency describe the ability of a mode to radiate sound, but the acoustic radiation behavior of a mode also largely depends on the driving force f. As we know, when a force acts on a node or a nodal line of a mode, this mode will not be driven and then no peak of radiated sound power is generated at the resonance frequency of the mode. Modal input energy or modal input power can be used to indicate the energy or the power input to a mode. The modal input energy ^e is directly related to the modal input power ^p by ^ p ¼ f jvn j ¼ f jixd n j ¼ x^e: ð15Þ 3. Numerical results and discussion The effects of size and location of distributed mass loading on vibration behavior of a plate are investigated in Ref. [1] and similar configurations of distributed mass loadings are used in this paper. The four different cases of mass loading are shown in Fig. 1. The additional mass loading in loading cases 1–3 is 10% of the mass of the unloaded
plate and in loading case 4 is 50%. The dimensions of the simply supported plate are Lx = 0.455 m, Ly = 0.379 m, and thickness h = 0.003 m. The plate is made of steel with density qs = 7850 kg/m3, Young’s modulus E = 2.1 · 1011 N/m2, and Poison’s ratio m = 0.3. The plate is assumed to be vibrating either in air with a density q = 1.21 kg/m3 and a sound speed c = 343 m/s or in water with a density q = 1000 kg/m3 and a sound speed c = 1500 m/s. The critical frequency of the plate is about 4.01 kHz in air and 76.84 kHz in water. To show the damping levels caused by the radiation loading, no structural damping is considered in determining the natural frequencies and modal damping ratios of the water-loaded modes. In the forced response analysis of the plate, a structural damping coefficient of 0.01 is used for the plate in water and a modal damping ratio of 0.01 is assumed for all modes of the plate in air. The plate is modeled by 16 · 16 four-noded quadrilateral isoparametric elements and this structural finite element model is valid up to 1 kHz according to the commonly applied rule of thumb to use six linear elements per structural wavelength. The finite element is based on Mindlin plate theory (first-order shear deformation theory.). The Rayleigh integral on the plate surface is calculated by discretisizing the plate surface with the same mesh as the plate finite element mesh. The total number of structural DOFs is N = 803 and the total number of nodes is I = 289. Water-loaded natural frequencies and modal damping ratios in the frequency band (10, 560) Hz are obtained using 12 evenly distributed interpolation frequencies and five probing vectors based on the model reduction method [15]. The changes of the first 10 in vacuo natural frequencies for the four loading cases are given in Table 1. The first 10 water-loaded natural frequencies and modal damping ratios are shown in Table 2. It can be seen that the natural frequencies of water-loaded plate have relatively less change than those of the air-loaded plate with the same mass loading due to the large added mass of the water. Also it is shown that the radiation damping caused by the water is small and the mode with high radiation efficiency has high modal damping ratio. That is, the odd– odd modes with high radiation efficiencies have large damping ratios, the even–even modes with low radiation efficiencies have small damping ratios, the radiation efficiencies and the damping ratios of the even–odd or
Fig. 1. Rectangular plates with distributed mass loadings of different size and location.
Table 1 Natural frequencies of the plate loaded with different distributed masses in vacuo Mode
Loading case 1
Loading case 2
Loading case 3
Loading case 4
Modal indices (m,n)
Frequency (Hz)
Modal indices (m, n)
Frequency (Hz)
Modal indices (m, n)
Frequency (Hz)
Modal indices (m, n)
Frequency (Hz)
Modal indices (m, n)
Frequency (Hz)
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
86.7 193.4 241.0 344.6 373.6 501.3 519.7 600.0 631.2 766.7
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (4, 2)
77.1 179.3 223.3 328.3 351.9 472.7 500.1 570.3 596.5 733.7
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
73.5 190.6 237.5 324.8 344.2 463.6 512.8 580.0 611.7 712.6
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
82.3 173.1 226.0 308.3 366.2 476.2 501.6 576.5 612.0 730.5
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (4, 1) (3, 2) (2, 3) (3, 3)
49.8 179.4 223.6 272.6 342.7 446.4 461.0 477.1 606.6 674.5
Table 2 Natural frequencies and modal damping rations of the plate loaded with different distributed masses in water Mode
1 2 3 4 5 6 7 8 9 10
Unloaded plate
Loading case 1
Loading case 2
Loading case 3
Loading case 4
Modal indices (m, n)
Frequency (Hz)
Damping ratio (%)
Modal indices (m, n)
Frequency (Hz)
Damping ratio (%)
Modal indices (m, n)
Frequency (Hz)
Damping ratio (%)
Modal indices (m, n)
Frequency (Hz)
Damping ratio (%)
Modal indices (m, n)
Frequency (Hz)
Damping ratio (%)
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
30.4 93.9 121.6 193.6 210.3 295.5 320.7 376.3 401.3 511.0
0.60198 0.00226 0.00380 0.00007 0.42042 0.44250 0.00839 0.02386 0.03540 0.12139
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
30.0 92.2 119.2 190.7 205.0 288.1 315.6 367.7 390.2 503.0
0.57445 0.00200 0.00343 0.00002 0.47378 0.44265 0.00903 0.03571 0.02042 0.14833
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
29.7 93.6 121.2 193.5 195.3 281.6 319.0 372.2 395.2 481.7
0.56430 0.00218 0.00373 0.00008 0.61393 0.30631 0.00880 0.03342 0.02068 0.20338
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
30.2 91.7 118.9 184.0 208.5 291.7 312.3 367.7 396.3 501.2
0.59085 0.00313 0.00426 0.00162 0.38214 0.33365 0.05194 0.03141 0.04091 0.04667
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (3, 2) (4, 1) (2, 3) (3, 3)
27.3 92.4 119.5 154.4 193.3 265.0 311.2 339.2 386.6 440.7
0.42890 0.00205 0.00339 1.00770 0.00008 0.11186 0.01059 0.03669 0.00078 0.31850
S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
1 2 3 4 5 6 7 8 9 10
Unloaded plate
275
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S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
tion efficiency or the volume displacement. It is observed from Fig. 2 that the radiation efficiency in water is much lower than that in air and the acoustic radiation behavior of a mode in water has relatively less change than that in air with the same mass loading. It can be concluded, at least for the particular cases be studied, that the acoustic radiation behavior of a plate is more sensitive to mass loading in air than in water. The mutual-radiation efficiencies of mode (1, 1) and mode (3, 1) in air and in water are plotted in Fig. 3. It can be seen that the mutual radiation between these modes has very high efficiency both in air and in water and the mass loading significantly affects the mutual-radiation efficiency. At resonance, the acoustic radiation is dominated by one mode and the mutual radiation between modes at
odd–even modes are between the above two kinds of mode groups. The self-radiation efficiencies of the odd–odd modes (1, 1), (3, 1) and (1, 3) in air and in water are plotted in Fig. 2. The modal volume displacements of the first 10 modes in air and in water are shown in Tables 3 and 4, respectively. It can be seen that the modal radiation efficiency of a mode is mainly influenced by its volume displacement. The volume displacement show the same change trends for these four loading cases in both air and water. Relatively large changes are observed in the acoustic radiation behavior of a mode both in air and in water when the mass loading is placed on an antinode of the mode shape or the mass loading is more concentrated. However, it should be noted that the mass loading may decrease (e.g. for mode (1, 1)) or increase (e.g. for mode (3, 1)) the radia-
0.01
0.1
Radiation efficiency
Radiation efficiency
1E-3
0.01
1E-3
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-4
0
100
200
300
400
500
600
700
1E-4
1E-5
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-6
1E-7
800
0
50
100 150 200
(a) mode (1,1) in air
(b) mode (1,1) in water
1E-3
1E-4
0.01
Radiation efficiency
Radiation efficiency
0.1
1E-3
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-4
1E-5
0
100
200
300
400
500
600
700
1E-5
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-6
1E-7
1E-8
800
0
50
100 150 200 250 300 350 400 450 500 550
Frequency (Hz)
Frequency (Hz)
(d) mode (3,1) in water
(c) mode (3,1) in air 0.1
1E-3
0.01
1E-4
Radiation efficiency
Radiation efficiency
250 300 350 400 450 500 550
Frequency (Hz)
Frequency (Hz)
1E-3
1E-4
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-5
1E-5
1E-6
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-7
1E-8
1E-6 0
100
200
300
400
500
Frequency (Hz)
(e) mode (1,3) in air
600
700
800
0
50
100
150
200 250
300 350 400
450
Frequency (Hz)
(f) mode (1,3) in water
Fig. 2. The self-radiation efficiency of mode (1, 1), (3, 1) and (3, 1) in air and in water.
500 550
Table 3 Modal volume displacement and modal input energy of the plate loaded with different distributed masses in air Mode
Loading case 1
Loading case 2
Loading case 3
Loading case 4
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
6.89E 02 2.41E 06 2.50E 06 7.80E 11 2.24E 02 2.24E 02 1.56E 06 2.08E 06 1.25E 05 7.26E 03
1.00E + 00 6.73E 03 8.07E 03 5.60E 05 1.00E + 00 1.00E + 00 8.19E 03 6.97E 03 1.27E 02 1.00E + 00
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (4, 2)
6.11E 02 2.78E 06 2.84E 06 2.88E 10 2.72E 02 2.45E 02 6.63E 06 1.17E 05 1.36E 05 1.27E 08
8.88E 01 6.43E 03 7.74E 03 5.70E 05 9.12E 01 8.37E 01 7.95E 03 1.17E 02 6.36E 03 1.19E 04
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
5.78E 02 1.07E 05 1.31E 05 4.24E 02 4.72E 08 2.00E 02 4.22E 05 4.66E 05 2.60E 05 8.54E 03
8.64E 01 6.95E 03 8.42E 03 8.39E 01 5.70E 05 4.77E 01 8.36E 03 1.52E 02 6.73E 03 5.96E 01
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
6.52E 02 1.25E 02 5.36E 03 1.21E 03 1.99E 02 8.56E 03 1.63E 02 8.27E 03 1.86E 03 3.82E 03
9.30E 01 2.44E 01 1.63E 01 3.70E 01 9.35E 01 5.65E 01 7.07E 01 2.80E 01 8.78E 02 5.68E 01
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (4, 1) (3, 2) (2, 3) (3, 3)
3.81E 02 5.83E 05 9.96E 05 6.07E 02 2.18E 07 1.61E 02 1.09E 04 2.83E 04 2.45E 05 5.11E 03
5.99E 01 7.45E 03 9.36E 03 4.08E 01 6.90E 05 1.69E 01 1.54E 02 9.61E 03 1.20E 03 1.81E 01
Table 4 Modal volume displacement and modal input energy of the plate loaded with different distributed masses in water Mode
1 2 3 4 5 6 7 8 9 10
Unloaded plate
Loading case 1
Loading case 2
Loading case 3
Loading case 4
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
Modal indices (m, n)
Volume displacement
Input energy
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
2.43E 02 3.81E 07 7.68E 07 4.15E 09 7.85E 03 6.88E 03 9.73E 07 6.49E 07 2.34E 06 2.98E 03
3.45E 01 3.20E 03 3.98E 03 2.80E 05 6.47E 01 6.02E 01 5.11E 03 5.46E 03 7.46E 03 6.97E 01
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
2.39E 02 4.22E 07 7.94E 07 2.62E 12 8.37E 03 6.86E 03 1.31E 06 1.39E 06 2.50E 06 3.17E 03
3.40E 01 3.17E 03 3.94E 03 2.82E 05 6.40E 01 5.59E 01 5.13E 03 6.85E 03 5.80E 03 6.90E 01
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
2.37E 02 7.12E 07 1.16E 06 2.05E 07 9.81E 03 5.83E 03 4.20E 06 4.56E 06 5.73E 06 3.96E 03
3.39E 01 3.23E 03 4.03E 03 1.45E 05 6.70E 01 4.69E 01 5.23E 03 7.45E 03 6.85E 03 6.13E 01
(1, 1) (2, 1) (1, 2) (2, 2) (3, 1) (1, 3) (3, 2) (2, 3) (4, 1) (3, 3)
2.41E 02 6.04E 04 4.61E 04 4.64E 04 7.52E 03 6.00E 03 2.19E 03 1.18E 03 3.72E 04 1.82E 03
3.42E 01 1.31E 02 2.35E 02 1.18E 01 6.36E 01 5.57E 01 1.61E 01 7.52E 02 1.68E 03 5.43E 01
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (1, 3) (3, 2) (4, 1) (2, 3) (3, 3)
2.17E 02 3.30E 06 5.73E 06 1.41E 02 3.34E 08 3.59E 03 1.47E 05 2.70E 05 1.11E 05 5.11E 03
3.17E 01 3.36E 03 4.27E 03 5.29E 01 3.24E 05 2.09E 01 6.06E 03 1.10E 02 2.83E 03 3.59E 01
S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
1 2 3 4 5 6 7 8 9 10
Unloaded plate Modal indices (m, n)
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S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
1E-3
1E-4
Radiation efficiency
Radiation efficiency
0.1
0.01
1E-3
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-4
1E-5
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
1E-6
1E-7 0
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500
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0
800
50
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Frequency (Hz)
Frequency (Hz)
(a) in air
(b) in water
Fig. 3. The mutual-radiation efficiency of mode (1, 1) and (3, 1) in air and in water.
resonance is negligible. At off-resonance, the contribution of the mutual radiation may be of importance to the radiated power. The sound power level (dB, re:1012 W) of the plate excited by a concentrated force is shown in Fig. 4, which shows the effects of the distributed mass loading on the forced response. The modal input energy of the first 10 modes in air and in water is shown in Tables 3 and 4, respectively. The concentrated force is located at the plate center with amplitude 1 N. From Tables 3, 4, and Fig. 4, it can be observed that for the mode (1, 3) in the fourth loading case, the mass loading decreases the modal input energy a lot and reduces radiated sound power. For the third loading case the modes other than odd-odd modes are also excited due to a change of location of the node lines caused by the mass loading. It is clear that the radiated power at resonance frequencies depends on both the modal radiation efficiency and the modal input energy. An optimization procedure can be applied to implement an effective design strategy to determine the size and location of the mass loading to suppress the acoustic radiation [3]. It is also clear that the radiated power in water has less change than in air for different loading cases, especially for the first few modes due to the large added mass of the water. However, it also can be seen that the effects of mass loading on acoustic radiation behavior in water will be more obvious
with increasing frequency due to the decreasing effect of the added mass of the water. 4. Conclusions The effects of the distributed mass loading on the acoustic radiation from a plate are investigated. The finite element method is employed for discretisizing the structure. The Rayleigh integral is used for modeling the acoustic fluid. Modal radiation efficiency, modal volume displacement, modal input energy and sound power level are computed to show the effects of size and location of the mass loading on the acoustic radiation of a baffled plate both in air and in water. The normal mode shapes of the fluidloaded undamped modes are used to compute the modal radiation efficiency, modal volume displacement, modal input energy of the plate in water. It is observed both in air and in water that the acoustic radiation behavior of a mode will have relatively larger changes if the mass loading is placed on an antinode of the mode shape or the mass loading is more concentrated. The mass loading may decrease or increase the acoustic radiation from the plate at resonance frequency and the radiated power at resonance frequencies depend on both the modal radiation efficiency and the modal input energy. At off-resonance frequency, the added mass generally reduces the radiated
100 100 90 90
Sound power level (dB)
Sound power level (dB)
80 70 60 50
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
40 30 20
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40 30
0
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300
Unloaded plate Loading case 1 Loading case 2 Loading case 3 Loading case 4
50
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800
0
50
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350
Frequency (Hz)
Frequency (Hz)
(b) in water
(a) in air
Fig. 4. The sound power level.
400
450
500
550
S. Li, X. Li / Applied Acoustics 69 (2008) 272–279
power. However, the acoustic radiation behavior of a mode and the total radiated power in water is less sensitive than in air to the mass loading due to the large added mass of the water, especially for the first few modes. Acknowledgement The authors are grateful for the support of the National Natural Science Foundation of China under Grant No. 10402004. References [1] Wong WO. The effects of distributed mass loading on plate vibration behavior. J Sound Vib 2002;252:577–83. [2] Kopmaz O, Telli S. Free vibration of a rectangular plate carrying a distributed mass. J Sound Vib 2002;251:39–57. [3] Koopmann GH, Fahnline JB. Designing quiet structures-a sound power minimization approach. Academic press; 1997. [4] Elliott SJ, Johnson ME. Radiation modes and the active control of sound power. J Acoust Soc Am 1993;94:2194–204. [5] Johnson ME, Elliott SJ. Active control of sound radiation using volume velocity cancellation. J Acoust Soc Am 1995;98:2174–86. [6] Cunefare KA, Koopmann GH. A boundary element approach to optimization of active noise control sources on three-dimensional structures. ASME Trans, J Vib Acoust 1991;113:387–94.
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