Sound Radiation of an Expansion Chamber due to Pressure Induced Structural

Vibrations

Michael Junge1, Falko Schube2, Lothar Gaul1

1 Institut für Angewandte und Experimentelle Mechanik, Universität Stuttgart, Deutschland, Email: junge@iam.uni-stuttgart.de2 Friedrich Boysen GmbH & Co. KG, 72206 Altensteig, Deutschland, Email: falko.schube@boysen-online.de

Introduction

Exhaust systems are exposed to large pressure pulsationsdue to the periodically blown out exhaust gas. Theselarge pulsations may lead to structural vibrations of theexhaust system by the transfer of energy from the fluid tothe structural parts of the exhaust system. Furthermore,the vibrating structural parts of the exhaust system sig-nificantly contribute to the sound radiation of the system.This phenomenon of the so called surface radiated soundis reported in [1, 2, 3] and is experimentally investigatedon a production-model main muffler of which the innerstructural parts are removed.

Experimental Analysis

The experimental investigation is divided into threesteps: In the first step, an experimental modal analy-sis (EMA) is carried out to determine the fundamentalvibration mode shapes and corresponding frequencies ofthe production-model main muffler depicted in Fig. 1,of which the inner structural parts are removed. In thenext step, in order to quantify the excitability of the ex-haust system by the acoustic path, a test setup is cre-ated, where the acoustic pressure and the specific acous-tic impedance on the inlet as well as the structural vibra-tion on the shell of the exhaust system can be measured.In the last step, the acoustic pressure is measured on fieldpoints close to the surface of the upper shell for an har-monic acoustic excitation at the inlet to characterize thesurface sound radiation.

For the EMA the main focus is set to the upper shellof the main muffler: Transfer functions are measuredbetween 375 points and two reference points by use ofmulti–reference fitting algorithms. Contour plots of twoexperimental mode shapes are depicted in Fig. 2. Themode shape in Fig 2(a) is a typical breathing mode, whilethe mode shape shown in Fig. 2(b) looks analogue to a(2,1)–mode of a rigidly fixed plate, though it is morecomplex due to the given geometry.

Figure 1: Test object: Production series main muffler

(a) f2 = 393 Hz (b) f5 = 479 Hz

Figure 2: Second and fifth experimental mode shape of themain muffler.

In the next step, the two–microphone–method (TMM)is employed to determine both the specific acousticimpedance as well as the pressure on the inlet of the ex-haust system. The TMM makes use of two microphoneslocated at fixed known positions x1 and x2 in a duct asshown in Fig. 3. Under the assumption of plane wavesin the duct, which for the presented setup is guaranteedup to a frequency of fu ≈ 2.8 kHz, the acoustic pressuremeasured by the two microphones, p1 and p2, is used toback–out the reflection coefficient and thus the specificacoustic impedance at the specified cross–section:

R =H12 − HIHR − H12

e2iκx1 , (1)

with H12 as the transfer function between the micro-phones one and two, and the time delay functions HI =e−iκ(x1−x2) and HR = e

iκ(x1−x2) . In Eq. (1) the onlyunknown is the transfer function H12. In the experi-mental setup a third microphone is used allowing thecomputation of three different transfer functions Hij inorder to cross–check the quality of the measurement. Themagnitude of the reflection coefficient at the inlet of themain muffler is depicted in Fig. 4 up to a frequency off = 1500 Hz making use of all three transfer functionsHij , showing an excellent consistency of the measure-ments. The vertical dashed lines in the same plot repre-sent the eigen frequencies of the upper shell. It can beseen that the reflection coefficient strongly varies at manyof these frequencies, demonstrating the strong influence

Micro 1Micro 2 acoustic source

exhaust

systemx1

x2

pI

pRpT

Figure 3: Measurement configuration for the two–microphone–method (TMM).

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500 1000 150000

0.2

0.4

0.6

0.8

1

f [Hz]

|R|

121323

Figure 4: Magnitude of measured reflection coefficient andstructural resonance frequencies of the upper shell of the mainmuffler (vertical dashed lines)

.

300 350 400 450 500 550 600

-3

-2

-1

10

10

10

10

0

|Hptv

i|

frequency [Hz]

Hptv3287Hptv3290Hptv3756Hptv3765

Figure 5: Normalized magnitude of transfer function Hptvibetween the transmitted pressure pT and the normal velocityvi on the upper shell.

of the structural dynamics on the interior acoustical be-havior of the system.

The magnitude and phase of the pressure that is trans-mitted into the exhaust system pT can be computed by

pT = (1 − R)1 − H12HIHR − HI

e−κx2 p1 (2)

This allows to determine the transfer function HpTvi be-tween pT and the normal velocity on the surface of theupper shell vi. Figure 5 shows the normalized magnitudeHpTvi in a frequency range between 300 Hz and 600 Hzfor four points located on the upper shell as shown inFig. 6. All transfer functions show a high level close tothe resonance frequencies. The transfer functions addi-tionally indicate that the operational deflection shapesare dominated by the mode shapes, e.g. at frequenciesclose to 390 Hz the normal velocities in the middle of thestructure (points 3290 and 3756) are much higher thanthe normal velocities at the outer part of the structure(points 3287 and 3765) correlating well with the resultsdepicted in Fig. 2(a). The plot shows that there is a sig-nificant excitability of the structure by the acoustic path.Thus both the acoustic and the structural domain mutu-ally influence each other necessitating the use of a fullycoupled simulation method.

In the last step, the surface radiation of the main muf-fler is investigated. The measurement of the acousticpressure on so called field points provides informationabout the sound emission characteristics of the systemat a given harmonic excitation frequency fh. Consider-ing the results of the EMA the excitation frequency ischosen at fh = 478 Hz. The 175 field points are locatedon a vertical plane arranged in an equidistant 25x7 gridwith a mesh size of 0.05 m as depicted in Fig. 6. The re-

Figure 6: Selected nodes on upper shell and field point plane

Figure 7: Sound pressure level (SPL) at the field points indB at fh = 478 Hz.

sults of the field point measurement are shown in Fig. 7.It can be seen that the contour lines of regions with thesame sound pressure level (SPL) are approximately of anelliptical shape. There are two regions with high values,which are colored red and magenta and reach a SPL up to90 dB. Going back to Fig 2(b) it becomes clear that themaxima are caused by the sound emission of the “(2,1)-mode”. Since the two regions with maximal displacementare phase-shifted by 180◦ the sound field is expected tobe similar to the sound field of a dipole. This also implies,that in-between the two maximum regions the SPL willbe low due to destructive interference. As expected, thegreen and blue colors indicate low acoustical pressures,with a minimum value of 62 dB.

Conclusion

This work systematically identified the excitation ofstructural vibrations by the acoustic path and showed thesurface radiation on the example of a main muffler. Theresults demonstrate the need for an effective simulationtool for the prediction of this phenomenon as proposedin [2].

Acknowledgement

Funding of this project by the Friedrich-und-Elisabeth-Boysen-Stiftung is gratefully acknowledged.

References

[1] J.-F. Brand and D. Wiemeler. Surface radiated noise ofexhaust systems – Structural Transmission Loss Test rig,part 1. In CFA/DAGA, Strasbourg, March 22– 25 2004.

[2] M. Junge, M. Fischer, M. Maess, and L. Gaul. Akustis-che Simulation von Abgasanlagen mittels gekoppelter FE-Methodik und Fast-BEM. In Deutsche Jahrestagung fürAkustik, München, March 14– 17 2005.

[3] P. Garcia, D. Wiemeler, and J.-F. Brand.Oberflächenschallabstrahlung von Abgasanlagen – CAE-Methode und Entwicklungsprozess. MTZ, 76(11):852–859,2006.

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