Surface contributions to radiated sound power

Surface contributions to radiated sound power Steffen Marburga) LRT4-Institute of Mechanics, Universit€ at der Bundeswehr M€ unchen, D-85579 Neubiberg, Germany

€sche Eric Lo MTU Friedrichshafen GmbH, 88045 Friedrichshafen, Germany

Herwig Peters and Nicole Kessissoglou School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, New South Wales 2052, Australia

(Received 10 August 2012; revised 26 March 2013; accepted 28 March 2013) This paper presents a method to identify the surface areas of a vibrating structure that contribute to the radiated sound power. The surface contributions of the structure are based on the acoustic radiation modes and are computed for all boundaries of the acoustic domain. The surface contributions are compared to the acoustic intensity, which is a common measure for near-field acoustic energy. Sound intensity usually has positive and negative values that correspond to energy sources and sinks on the surface of the radiating structure. Sound from source and sink areas partially cancel each other and only a fraction of the near-field acoustic energy reaches the far-field. In contrast to the sound intensity, the surface contributions are always positive and no cancelation effects exist. The technique presented here provides a method to localize the relevant radiating surface areas on a vibrating structure. To illustrate the method, the radiated sound power from a baffled square plate is C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4802741] presented. V

I. INTRODUCTION

Identification of the contribution to radiated sound from individual components of a vibrating structure is important in applications such as aircraft, ships and vehicles, where it is desirable to reduce both interior and exterior structure-borne sound. For interior noise problems, a method to predict the contribution of each node of a boundary element model to the total sound pressure was first described by Ishiyama et al.1 In the 1990s, many commercial boundary element codes for acoustic simulations started to provide a feature called panel acoustic contribution analysis (PACA).2–4 Mohanty et al. applied PACA to a truck cabin interior to determine the areas that contribute most to the sound pressure level at the driver’s ear.5 The placement of sound absorbing material at these predetermined locations helped to reduce the interior structure-borne noise. Marburg and Hardtke minimized the sound pressure level at the driver’s ear due to the spare wheel well in the trunk of a passenger car.6 Contribution coefficients were used to visualize the difference of surface contribution patterns before and after the structural-acoustic optimization of the spare wheel well. For exterior noise problems, the sound intensity is commonly used to analyze contributions of vibrating surfaces to the radiated sound power. Assuming the simplest case of constant boundary elements, the nodal contribution to sound power is the product of nodal sound intensity and element area. Earlier methods to identify acoustic energy source areas on a vibrating structure are inverse boundary element a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

3700

J. Acoust. Soc. Am. 133 (6), June 2013

Pages: 3700–3705

techniques7 and the near-field acoustic holography (NAH).8 NAH is an experimental technique that is capable of computing the vector field of acoustic energy between the source and the far-field based on a microphone measurement over a two dimensional surface in the acoustic nearfield. Subsequently, the concept of the supersonic acoustic intensity was introduced by Williams to identify only those components of a structure that radiate energy to the acoustic far-field.9,10 Since subsonic wave components of the vibrating structure only contribute to evanescent acoustic energy in the near-field, these wave components are filtered out. Only the remaining supersonic wave components, which correspond to the resistive part of sound intensity, radiate acoustic energy to the far-field. Magelhaes and Tenenbaum11 extended the supersonic acoustic intensity technique to consider arbitrarily shaped sources. The Kirchoff-Helmholtz equation was discretized on the source boundary and then singular value decomposition was used to remove the inefficient radiation modes. A particular characteristic of the aforementioned intensity based methods is that a vibrating structure can have positively and negatively contributing areas, which may lead to cancelation effects with respect to the total radiated sound power. This paper presents a new method to compute the surface contributions to the radiated sound power from a vibrating structure. The surface contributions are based on the acoustic radiation modes.12,13 The surface contributions are computed for every node of a boundary element mesh of the radiator. In contrast to the sound intensity, the surface contributions are always positive and cancelation effects are avoided. The surface contributions for a baffled square plate are compared to results obtained for the acoustic intensity.

0001-4966/2013/133(6)/3700/6/$30.00

C 2013 Acoustical Society of America V

Author's complimentary copy

PACS number(s): 43.20.Rz, 43.40.Rj [EGW]

The radiated sound power is defined as P¼

ð

1 I  n dC ¼ < 2 C

ð C

 pvn dC ;

(7)

where I ¼ ð1=2Þ p and v ¼ /> v yields

FIG. 1. Exterior acoustic problem with acoustic domain X, complementary domain Xc , boundary C, and normal direction n on C.

II. SOUND POWER

An exterior acoustic problem as shown in Fig. 1 is considered. Given a harmonic time dependence eixt , the sound pressure in 3D space is (1)

Using Eq. (1), the acoustic wave equation is simplified to the well-known Helmholtz equation x2X;

(2)

where x is the angular frequency, i is the imaginary unit and k is the acoustic wave number. The Neumann boundary condition is given by @pðyÞ ¼ avðyÞ; @n

y2C;

(3)

where v is the particle velocity, a ¼ ix.0 and .0 is the average density of the fluid. Using Eq. (3), Eq. (2) can be rewritten in terms of a Kirchhoff–Helmholtz integral equation as14 ð

@Gðx; yÞ pðxÞdCðxÞ cðyÞpðyÞ þ C @nðxÞ ð ¼ Gðx; yÞavðxÞdCðxÞ; y2C;

(4)

where cðyÞ is a constant depending on the local geometry of C at y and G is the free-space Green’s function. Discretization of this integral equation by collocation or the Galerkin method and omission of the arguments leads to the following linear system of equations: (5)

where G and H are the boundary element matrices. Given a vector of nodal values for the particle velocity v on the boundary C, the nodal values for sound pressure p are obtained from p ¼ H1 Gv: J. Acoust. Soc. Am., Vol. 133, No. 6, June 2013

C



1 ¼ Hv g; 2

(8)

P¼

N X

Pk ¼

N ð X k¼1

Ik  nk dCk :

(9)

C

The nodal contributions in terms of the sound power Pk or the sound intensity Ik can be either positive or negative which results in cancelation effects of energy on the boundary C. Thus, Pk and Ik are not suitable to visualize the surface contributions to the radiated sound power from a vibrating structure. Substitution of the expression for p given by Eq. (6) into Eq. (8) leads to an expression for sound power in terms of the particle velocity vector 1 1 P ¼ G> H> Hv g ¼ Zv g; 2 2

(10)

where Z ¼ G> H> H is the acoustic impedance matrix.13,16 The sound power can be further rewritten as13

C

Hp ¼ Gv;

p> //> v dC

where ðÞ> is the matrix transpose, / is a columnÐ matrix containing the interpolation functions15 and H ¼ C //> dC is the boundary mass matrix. The boundary mass matrix is evaluated in a similar way to a mass matrix in the finite element method but the integration area is instead just the boundary and not the domain.14 The discretized sound power can be written as a sum of all nodal sound power contributions by

k¼1

DpðxÞ þ k2 pðxÞ ¼ 0;

ð

(6)

1 1 P ¼ v> ZR v ; 2 2

(11)

where ZR is the resistive impedance matrix. ZR is real, symmetric and has full rank.13 III. SURFACE CONTRIBUTION TO SOUND POWER

In what follows, the radiated sound power is described in terms of the sum of only positive sound power contributions of the radiating surface. When all the contributions are positive, the cancelation effects observed in Eq. (9) are eliminated, thus delivering a tool to visualize surface contributions to the radiated sound power. Defining the surface contribution to the radiated sound power as g, the total radiated sound power is expressed by the following boundary surface integral Marburg et al.: Surface contributions to radiated sound power

3701

Author's complimentary copy

p~ðx; tÞ ¼ ðxÞb and omitting the arguments yields 1 P¼ 2

ð

a¼ 1 b // b dC ¼ b> Hb ; 2 C >

> 

(13)

where b is a vector without physical significance. For any interpolation node xk on the boundary C, gk is given by gk ¼ bk bk :

(14)

It is obvious that gk is always real and positive for any complex b. Before gk can be computed, b must be expressed in terms of known boundary values. One possibility to compute b is via the acoustic radiation modes W which diagonalize the boundary mass matrix H and the resistive acoustic impedance matrix ZR as follows: W> HW ¼ Id ;

(15)

W> ZR W ¼ K;

(16)

where Id is the identity matrix and K is the diagonal matrix of the eigenvalues kl . The eigenvalues K ¼ diagfkl g and eigenvectors W ¼ ½w1 ; w2 ; …; wn  are found by solving the following algebraic eigenvalue problem ZR w ¼ kHw:

(17)

A particle velocity distribution can be expressed through a weighted sum of acoustic radiation modes as follows: v¼

n X

wk fk ¼ Wf;

(18)

k¼1

where f is the vector of the modal contribution factors fk . Left multiplication of Eq. (18) with W> H and consideration of Eq. (15) yields f ¼ W> Hv:

(19)

Substitution of Eq. (18) into Eq. (11) and consideration of Eq. (16) yields 1 1 pffiffiffiffipffiffiffiffi P ¼ f> Kf ¼ f> K Kf : 2 2

(20)

Using Eq. (15), Eq. (20) can be rewritten as pffiffiffiffi 1 pffiffiffiffi P ¼ f> KW> HW Kf ; 2

pffiffiffiffi pffiffiffiffi b ¼ W Kf ¼ W KW> Hv: J. Acoust. Soc. Am., Vol. 133, No. 6, June 2013

pffiffiffiffi Hb:

(23)

Substitution of Eq. (23) into Eq. (13) yields 1 1X 1X P ¼ a> a ¼ ak ak ¼ l; 2 2 k 2 k k

(24)

where, analogous to Eq. (14), the discrete surface contribution of an arbitrary interpolation node to the radiated sound power is given by lk ¼ ak ak :

(25)

The relation between discrete and continuous surface contributions is similar to that between nodal surface forces and surface pressure, respectively.17 The discrete surface contribution l contains area information and thus depends on the boundary element mesh. Thus a regular mesh of n nodes, a uniform distribution of lk and a radiated sound power of P yields lk ¼ P=n. Doubling the number of nodes has no effect on P but results in lk being halved. For non-uniform meshes, the value of lk depends on the topology of the surface mesh and is directly associated to node k. However, while the continuous surface contribution g is computed using a discretized surface, its magnitude is independent of the underlying surface mesh (assuming a fully converged solution). The physical unit of g and l is watts. For visualization purposes, g is preferred to eliminate the dependency on the boundary element mesh. To illustrate the difference between the use of the sound intensity and the continuous surface contribution to the radiated sound power from a vibrating structure, a numerical example is presented in what follows. IV. IMPLEMENTATION

The method has been implemented using the boundary element code Akusta developed by one of the authors.18,19 Additional subroutines are written using the programming language FORTRAN 90. Eigenvalue problems are solved by using the ARPACK routines or using a simple simultaneous vector iteration procedure which has been programmed by the authors. A residual tolerance of 105 was required. Constant discontinuous boundary elements have been used for the model. Evaluation of the impedance matrix Z requires an explicit inversion of matrix H, cf. Eq. (10), which was achieved using an ordinary Gaussian elimination algorithm with row pivoting.

(21)

V. BAFFLED PLATE

(22)

A baffled structure comprising a steel plate (1 mm thick) of dimensions 1 m  1 m is examined in what follows. A modal analysis is initially conducted from which the mode shapes are obtained. Certain mode shapes are selected to

which yields

3702

Equation (22) shows that only the particle velocity vector v is needed as input data in order to find b. All other required information, are known properties of the boundary element domain. A different formulation of the surface contribution is achieved through the introduction of a new factor

Marburg et al.: Surface contributions to radiated sound power

Author's complimentary copy

P¼

FIG. 2. (Color online) Normalized sound radiation for excitation of the particle velocity ð0  0Þ mode at 100 Hz (top) and 1000 Hz (bottom).

column shows the predicted pattern of the sound pressure at the different frequencies for the given particle velocity distribution. For the ð0  0Þ mode shown in Fig. 2, no significant differences between the patterns of the normal sound intensity (third column) and the surface contribution (fourth column) are observed. The maxima of the normal sound intensity and surface contribution are in the middle of the plate, while the minima are in the corners of the plate. At the lower frequency of 100 Hz, the distributions of the normal sound intensity and the surface contribution differ slightly. No differences exist at the higher frequency of 1000 Hz. In Figs. 3 and 4 for the ð1  1Þ and ð2  2Þ modes, respectively, the results for the intensity and surface contribution at the lower frequency of 100 Hz are notably different. While the normal sound intensity indicates a distribution of energy similar to normal particle velocity and sound pressure, the surface contribution shows a distribution where only the corners of the plate are significantly contributing to the radiated sound. This result is consistent with the well-known

FIG. 3. (Color online) Normalized sound radiation for excitation of the particle velocity ð1  1Þ mode at 100 Hz (top) and 1000 Hz (bottom).

J. Acoust. Soc. Am., Vol. 133, No. 6, June 2013

Marburg et al.: Surface contributions to radiated sound power

3703

Author's complimentary copy

represent a velocity distribution pattern for the plate. The structure is then “discarded” and the remaining pattern at the fluid boundary (of the chosen velocity distribution pattern) is retained. The velocity patterns are equal to the first four ðm  mÞ modes of the clamped plate. The frequency is then increased for that chosen velocity distribution pattern to observe the acoustic intensity and surface contributions at different frequencies. The baffled plate is modeled in air with 48  48 ¼ 2304 constant boundary elements. Density of . ¼ 1:3 kg=m3 and speed of sound of c ¼ 340 m=s are assumed for air. Damping exists in the form of radiation damping. The normal sound intensity In ¼ I  n is compared to the continuous surface contribution g for four different normal particle velocity patterns vn ¼ v  n. Figures 2–4 show the given particle velocity pattern, the corresponding magnitude of the sound pressure, sound intensity, and continuous surface contribution to radiated sound power. In the first column of Fig. 2, the particle velocity is given in the form of a ð0  0Þ mode pattern. The second

FIG. 4. (Color online) Normalized sound radiation for excitation of the particle velocity ð2  2Þ mode at 100 Hz (top) and 1000 Hz (bottom).

theory of edge and corner mode radiation.20–24 The theory of radiation by corner monopoles at low frequencies was introduced by Maidanik.20 He showed that at low frequencies and low radiation efficiency, radiation from rectangular plates only occurs due to corner effects or corner monopoles since dipoles and higher order multipoles are very inefficient sound radiators. Williams10 and Fernandez-Grande et al.24 presented

results for a baffled plate using the supersonic acoustic intensity to illustrate corner mode radiation. Their results are confirmed by the surface contribution analysis results described here. In order to clearly observe the changes in sound pressure, intensity and surface contributions with increasing frequency for a given mode, Fig. 5 presents results for

3704

J. Acoust. Soc. Am., Vol. 133, No. 6, June 2013

Marburg et al.: Surface contributions to radiated sound power

Author's complimentary copy

FIG. 5. (Color online) Normalized sound radiation for excitation of the particle velocity ð3  3Þ mode at 100; 500; 600; 800; 900; and 1200 Hz (top to bottom).

VI. CONCLUSIONS

A method to identify the surface contributions to the radiated sound power of a vibrating structure has been presented. The surface contributions to the far-field radiated sound power can be observed at the fluid boundary on the surface of the structure. An expression for the sound power is derived in terms of the acoustic radiation modes. The surface contributions are then computed for every node of a boundary element mesh of the radiator. In contrast to the sound intensity, using surface contributions, the radiated sound power is described as the sum of only positive sound power contributions of the vibrating surface, thus avoiding cancelation effects. To illustrate the method, the surface contributions of a baffled square plate have been compared to results for the acoustic intensity. Using the surface contribution method, corner monopoles from the simple vibrating plate at low frequencies can be observed. Results for the surface contribution and sound intensity were shown to converge as the frequency increased, highlighting that the acoustic intensity on the surface of a vibrating structure does not directly correspond to the intensity that radiates to the acoustic far-field. The technique presented here provides a new method to localize the relevant radiating surface areas on a vibrating structure. 1

S. Ishiyama, M. Imai, S. Maruyama, H. Ido, N. Sugiura, and S. Suzuki, “The applications of ACOUST/BOOMa noise level predicting and reducing computer code,” in Proceedings of the Seventh International

J. Acoust. Soc. Am., Vol. 133, No. 6, June 2013

Conference on Vehicle Structural Mechanics (1988), Society of Automotive Engineers, Warrendale, PA, pp. 195–205. 2 LMS International, SYSNOISE Rev 5.6 Release Notes & Getting Started Manual, 1st ed. (LMS International, Leuven, Belgium, 2003), pp. 123–156. 3 Automated Analysis Corporation, COMET/ACOUSTICS User’s manual, version 2.1 (Automated Analysis Corporation, Ann Arbor, Michigan, 1994). 4 Computational Mechanics BEASY, BEASY User Guide (Computational Mechanics BEASY, Southampton, England, 1994). 5 A. Mohanty, B. Pierre, and P. Suruli-Narayanasami, “Structure-borne noise reduction in a truck cab interior using numerical techniques,” Appl. Acoust. 59, 1–17 (2000). 6 S. Marburg and H. Hardtke, “Investigation and optimization of a spare wheel well to reduce vehicle interior noise,” J. Comput. Acoust. 11(3), 425–450 (2003). 7 J.-G. Ih, “Inverse boundary element techniques for the holographic identification of vibro-acoustic source parameters,” in Computational Acoustics of Noise Propagation in Fluids. Finite and Boundary Element Methods, edited by S. Marburg and B. Nolte (Springer, Berlin, 2008), Chap. 20, pp. 547–572. 8 J. Maynard, E. Williams, and Y. Lee, “Nearfield acoustic holography: I. Theory of generalized holography and the development of NAH,” J. Acoust. Soc. Am. 78(4), 1395–1413 (1985). 9 E. Williams, “Supersonic acoustic intensity,” J. Acoust. Soc. Am. 97(1), 121–127 (1995). 10 E. Williams, “Supersonic acoustic intensity on planar sources,” J. Acoust. Soc. Am. 104(5), 2845–2850 (1998). 11 M. B. S. Magelhaes and R. B. Tenenbaum, “Supersonic acoustic intensity for arbitrarily shaped sources,” Acta Acust. Acust. 92, 189–201 (2006). 12 K. A. Cunefare and M. N. Currey, “On the exterior acoustic radiation modes of structures,” J. Acoust. Soc. Am. 96, 2302–2312 (1994). 13 P. T. Chen and J. H. Ginsberg, “Complex power, reciprocity, and radiation modes for submerged bodies,” J. Acoust. Soc. Am. 98, 3343–3351 (1995). 14 S. Marburg and B. Nolte, “A unified approach to finite and boundary element discretization in linear acoustics,” in Computational Acoustics of Noise Propagation in Fluids. Finite and Boundary Element Methods, edited by S. Marburg and B. Nolte (Springer, Berlin, 2008), Chap. 0, pp. 1–34. 15 H. Peters, S. Marburg, and N. Kessissoglou, “Structural-acoustic coupling on non-conforming meshes with quadratic shape functions,” Int. J. Numer. Methods Eng. 91(1), 27–38 (2012). 16 K. Naghshineh and G. Koopmann, “Active control of sound power using acoustic basis functions as surface velocity filters,” J. Acoust. Soc. Am. 93(5), 2740–2752 (1993). 17 S. Marburg, “Efficient optimization of a noise transfer function by modification of a shell structure geometry. Part I: Theory,” Struct. Multidiscip. Optim. 24, 51–59 (2002). 18 S. Marburg and S. Schneider, “Influence of element types on numeric error for acoustic boundary elements,” J. Comput. Acoust. 11, 363–386 (2003). 19 S. Marburg and S. Amini, “Cat’s eye radiation with boundary elements: Comparative study on treatment of irregular frequencies,” J. Comput. Acoust. 13, 21–45 (2005). 20 G. Maidanik, “Response of ribbed panels to reverberant acoustic fields,” J. Acoust. Soc. Am. 34, 809–826 (1962). 21 E. Williams, Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography (Academic, London, 1999), pp. 67–77. 22 C. Deffayet and P. Nelson, “Active control of low-frequency harmonic sound radiated by a finite panel,” J. Acoust. Soc. Am. 84, 2192–2199 (1988). 23 C. Fuller, S. J. Elliott, and P. A. Nelson, Active Control of Vibration (Academic, London,1996), pp. 299–300. 24 E. Fernandez-Grande, F. Jacobsen, and Q. Leclere, “Direct formulation of the supersonic acoustic intensity in space domain,” J. Acoust. Soc. Am. 131(1), 186–193 (2012). 25 E. Williams, “Imaging the sources on a cylindrical shell from far-field pressure measured on a semicircle,” J. Acoust. Soc. Am. 99(4), 2022–2032 (1996). 26 A. Norris, “Far-field acoustic holography onto cylindrical surfaces using pressure measured on semicircles,” J. Acoust. Soc. Am. 102(4), 2098–2107 (1997). 27 S. Wu, “Methods for reconstructing acoustic quantities based on acoustic pressure measurements,” J. Acoust. Soc. Am. 124(5), 2680–2697 (2008).

Marburg et al.: Surface contributions to radiated sound power

3705

Author's complimentary copy

excitation of the ð3  3Þ mode for six different frequencies. The first column corresponds to the chosen particle velocity distribution pattern used for the pattern at the fluid boundary, and hence is the same for all frequencies. The second and third columns correspond to the real and imaginary parts of the sound pressure, respectively. It is observed that the surface contribution gradually aligns with the normal sound intensity with increasing frequency until both are qualitatively equal (at 1200 Hz). This is due to the decreasing wavelength which makes it more difficult for adjacent areas that have areas with sound intensities of opposite signs to exchange acoustic energy. It can also be observed that the real part of sound pressure forms patterns similar to the surface contribution for all frequencies considered. When comparing the real part, imaginary part and magnitude of sound pressure, it is observed that the magnitude is dominated by the imaginary part of sound pressure at frequencies up to 900 Hz for this case. Above 900 Hz, the magnitude is dominated by the real part of the sound pressure due to decreasing wavelength and increasing radiation efficiency. The results obtained using the surface contribution method shows similarities with those obtained from acoustic holography using far-field data, as described by Williams25 and Norris.26 Using acoustic holography, the supersonic intensity is shown to identify regions of structural vibration which are only relevant for far-field radiation. However, in contrast to near-field acoustic holography and the surface contribution method presented here, far-field acoustic holography is generally not able to identify the entire vibration pattern of a radiator, as discussed in the article by Wu.27