# Estimation of Radiated Sound Power: A Case Study on Common Approximation Methods

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4 ACTA ACUSTICA UNITED WITH ACUSTICA Figure 2. Floor panel: sound power in terms of frequency. 3. Radiating floor panel Figure 1. Floor panel: original finite element model (upper) and boundary element half space model (lower) Volume velocity The volume velocity u is defined as the integral of the particle velocity over the radiating surface [1] N e u = v dγ = v µ S µ. (18) µ=1 Avolume velocity based radiated sound power P VV is formulated as P VV = k2 f c f 4π uu. (19) The authors are not aware of areference about sound power estimation based on volume velocity. However, since uu can be rewritten as N e N e uu = v µ vν Sµ S ν, (20) µ=1ν=1 this sound power estimate can be understood as areduction of P LP M,cf. Equation (16) with the weighting by the Green s function G µν and afactor of k/2π. Note that this factor is equal 1for µ = ν,i.e. x y,cf. [1]. The sound power estimation based on volume velocity contains local acoustic effects based on antiphase vibration of the sources (dipole effects) but requires the evaluation of a single sum only, ifequation (19) is solved on the basis of (18). The floor panel of asedan s bodywork accounts for the first example. The original FE model of this structural component is shown in the upper part of Figure 1. In order to compute a coupled FE BE solution, we need to enhance this model. The MLFMA realization of the boundary element solution requires a closed fluid surface, so that we introduce auxiliary elements creating aclosed obstacle with the half space plane, i.e. the half space plane is used as a symmetry plane. The structure consisting of multi layered sheets of steel is reduced to its outer surface. The final fluid surface model can be found in the lower part of Figure 1. The radiating elements of the FE structural model match with the elements of the fluid model so that acoupling is easily generated. The auxiliary elements are assumed to have zero particle velocity. The FE structural model s surface consists of nodes with adof of 6each and linear continuous shell elements. In the simulation the entire structural model is solved but only the elements on its surface are taken into account for the subsequent coupled acoustical simulation. The fluid boundary element model contains elements and nodes. Since constant elements are preferred for the BE solution, we end up with unknowns for the half space solution. The vibration of the model is excited by two single harmonic point forces at the rear part of the floor panel. The sound power solutions which are gained by the previously explained methods are computed over a frequency range of 10 to 200 Hz. They are shown in Figure 2. As expected, the ERP overestimates the actual radiated sound power. Moreover, it contains more peaks than the other curves because acoustic short circuit effects are not considered in this function. As one example, the peak at 55 Hz in the LPM or PVV functions is not as much developed as in the ERP function. The structural mode shape belonging to this peak defines atorsional vibration, i.e. quadrupole so- 836

5 ACTA ACUSTICA UNITED WITH ACUSTICA lution. Such amode shape is well known to have lowradiation efficiency. The LPM and PVV solutions give higher values than the BEM reference solution in the frequency range up to approximately 80 Hz. Above this frequency they seem to normally underestimate the sound power. This is supposedly due to the Rayleigh integral assumption, which is quite contrary to the tub like shape of the floor panel. If we compare the LPM and the PVV functions, they show high discrepancy in the frequency range from 130 Hz to 180 Hz. The interaction between the discrete sources on the surface (i.e. the elements), which is only included in the LPM value, affects the sound power significantly in this frequency range. The error for LPM, PVV and ERP compared to the BEM solution is plotted in Figure 3. If the BE solution is understood as the reference solution, the lumped parameter model becomes highly interesting due to its efficient computation and good agreement. One has to admit and consequently be aware of the fact, that there is no error estimate for the applied plain radiator assumption yet. However, the ERP and PVV can be computed even faster, but the implemented approximative assumptions can lead to questionable results. It is suggested to use both values (P VV and P ERP )incombination to qualitatively estimate the radiated sound power,in particular to identify resonance peaks and to distinguish between resonances with lowand with high radiation efficiency Δ P in [db] PERP PBEM PVV PBEM PLP M PBEM f in Hz Figure 3. Floor panel: absolute error of sound power estimates in terms of frequency. 4. Radiating diesel engine The second example investigates the sound radiation of a diesel engine. Herein, the sound power solutions of the BEM, the lumped parameter model and the ERP approximation are compared. Again, the finemesh of the structure is directly used for the fluid. The fluid surface model contains nodes and constant elements. The problem is solvefor the frequency range up to 3000 Hz. The model is presented in Figure 4. In what follows, we consider one case of realistic excitation and further cases of artificial excitations using the entire engine as an elementary radiator, i.e. monopole or dipole source. The excitation of the acoustic field is applied by defining the particle velocity over the surface at each investigated frequency Realistic excitation The particle velocity distribution over the engine s surface for a certain operations condition was computed and provided by the AVL/ACC Graz (Austria). In Figure 4, the geometry mesh of the surface is shown. Originally, the particle velocity was given on the mesh of linear continuous elements. The piecewise constant particle velocity data which is used for our simulations can be understood as an average of the normal velocity on each element. To provide the reader with avivid impression and a comparison of the level distributions of the particle velocity, the sound pressure and the sound intensity, these data Figure 4. Diesel engine: boundary element model. are visualized for two specified frequencies, i.e. 503 Hz, cf. Figure 5, and 2196 Hz, cf. Figure 6. It can be realized at the lower frequency of503 Hz that the intensity contribution does not match with the velocity contribution. At 2196 Hz, the contributions show less differences. This means that the intensity is dominated by the velocity contribution. The same effect is revealed by the sound power spectrum in Figure 7. There, the ERP values agree with the BEM solution better and better the higher the frequencies are. The differences between the ERP and LPM solutions with respect to the BEM reference are shown in Figure 8. Actually, it is quite surprising that the simple approximations of ERP and LPM catch the behaviour of the reference solution with this accuracy, not only in the high frequency range. Even in the lower frequencyrange, the difference of approximate solutions and reference are a couple of decibel. However, peaks and valleys in the curves are found at the same frequencies. 837

6 ACTA ACUSTICA UNITED WITH ACUSTICA Figure 5. Diesel engine: surface distribution of particle velocity, sound pressure and sound intensity at 503 Hz Engine as an elementary radiator According to the Rayleigh integral assumptions the engine can not be understood as a plain radiator. Although we can simply model the engine as a cuboid by six half space plains it is questionable to explain the good agreement between the LPM and the BE solutions. Therefore, two simple cases of the engine s surface vibrations are investigated. Firstly, we exercise a harmonically pulsating vibration of the engine, i.e. a constant unit particle velocity is applied to all elements. This is similar to a monopole radiation. The resulting sound power levels are shown in Figure 9. The second case investigates the harmonic rigid body translation of the engine which can be interpreted as 838 a dipole source. Furthermore, we divide this case into two subcases which comprise two directions of motion. These are the two horizontal directions x and y, cf. Figure 4. We assume a unit vibration amplitude of the rigid body motion. Figures 10 and 11 show the resulting sound power levels. In all cases, the ERP estimation yields a constant sound power level. It provides a good approximation for higher frequencies. At low frequencies the ERP approximate is not able to represent the acoustic elimination eﬀects. However, the LPM fails to give a good estimation of the radiated sound power for the entire frequency range. This is caused by the incorrect calculation of the distances Rµν between the source and receiver elements. Again, this is

7 ACTA ACUSTICA UNITED WITH ACUSTICA Figure 6. Diesel engine: surface distribution of particle velocity, sound pressure and sound intensity at 2196 Hz. due to the violation of the Rayleigh integral presumption. For concave ﬂuid boundaries such as for the engine model, this problem can be avoided by the visibility test. This test checks the visibility between source and receiver point. Unfortunately, the test becomes quite expensive for large scale models [31]. (It is only required once, though.) When carrying out the visibility check, only the elements which have an unobstructed connection to each other are considered for the summations. This means, that each of the six sides of the cuboid like engine is separately investigated and then summed up to the overall radiated sound power value. Alternatively, an eﬀective distance could be calculated via the periphery of the engine. This means, that the distance of the sound wave around the obstacle is taken into account. However, it is assumed that such an additional algorithm will be computationally ineﬃcient and, thus, destroying the eﬃciency impact of the LPM compared to the MLFMA. 5. Interpretation of the results According to piecewise constant elements of the radiating surface, the previously presented sound power approximations can be reformulated in the following way, cf. [29] P = Ne \$ Ne \$ µ=1 ν=1 Pµν = Ne \$ µ=1 Pµµ + 2 N e 1 \$ Ne \$ Pµν. (21) µ=1 ν=µ+1 839

9 ACTA ACUSTICA UNITED WITH ACUSTICA Table I. Overview onthe three sound power approximations. Method estimated time mesh refinement frequencyrange ERP O(N e ) no effect no prediction available PVV O(N e ) no effect low frequencies LPM O(Ne 2 ) convergence for finer mesh low/mid frequency range ERP PVV LPM σ σ µµ =1 σ σ µ ν σ µµ = k 2 S ν 2π R µν µ ν σ µµ = k 2 S ν 2π R µν µ ν Figure 12. Radiation efficiencyofthe three approximation methods (schematic visualization). information but subdivision of elemental data. This means if the genuine mesh is refined by subdividing the elements, i.e. no additional information for velocity contribution is created, the ERP and the PVV will each provide the results of the coarser mesh again. The LPM depends on the mesh refinement since it shows convergence behavior for finer meshes in this case. The radiation efficiency, cf. Figure 12, in terms of R µν can be approximated more accurately for finer meshes. If the sound power is estimated from avery coarse mesh, the ERP or PVV solution can provide sufficient results. R µν R R R on the structural vibration. Consequently, the coupling between the structure and the fluid is modeled unidirectionally. To determine the sound power, the velocity distribution on the structure s surface acts as input information of the acoustic field. The particle velocity is approximated by piecewise constant interpolation. Two ofthe methods, the ERP and the PVV, require evaluation of asingle summation of order O(N e )ofthe velocity distribution over the N e constant elements only. Since therefore these methods are computationally very fast, theycan be used in combination to estimate the sound radiation in the lower frequency mainly in aqualitative manner. The lumped parameter model contains a double summation of order O(N 2 e )overall N e elements since the interaction between the discretized piston sources is considered. This model is based on the Rayleigh integral but even may provide acceptable results if the Rayleigh integral assumption is violated. The LPM will mostly fail if the structural vibration contains rigid body motion or if no phase information is included in the velocity distribution. The general statements about the estimated computation time, the dependency on mesh refinement and the recommended frequencyrange of the approximation methods are compiled in Table I. It was shown in this article that these approximation methods can be successfully used for realistic problems. An expensive and highly detailed boundary element computation wascarried out to provide areference solution for the investigated examples. Acknowledgments These investigations were extracted from the research project P 579. The research project P 579 " Minimum sound emission of steel plates" was carried out by the Institut für Festkörpermechanik with technical and scientific support by the FOSTA Research Association for Steel Application, Düsseldorf (Germany), with funds of the Stiftung Stahlanwendungsforschung, Essen (Germany). Wefurther thank the Audi AG Ingolstadt (Germany) and AVL/ACC in Graz (Austria). The computation was run on the SGI Orig0 at the Zentrum für Hochleistungsrechnen of the Technische Universität Dresden. 6. Conclusions The presented sound power approximations have in common that the acoustic field is not solved to estimate the radiated sound power of astructural component. It was presumed in these methods that the acoustic field has no effect References [1] G. H. Koopmann, J. B. Fahnline (eds.): Designing quiet structures: A sound power minimization approach. Academic Press, San Diego, London, [2] S. Marburg: Developments in structural acoustic optimization for passive noise control. Archives of Computational 841

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