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Figure 20. Sound pressure level , SPL= 128 (dB), M=0.21, 100 averages obtained using: —, STDA, —— FDA, -----, CSFDA.

Figure 20 Sound pressure level , SPL= 128 (dB), M=0.21, 100 averages obtained using: —, STDA, —— FDA, -----, CSFDA.

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where a and b refers to the two sides of the two-port, the plus and minus signs indicate propagation outwards and into the two-port respectively, the superscript s refers to source generated sound, and p describes the reflection and r the transmission of incoming waves. The definition of propagation directions and other indices are shown in Fig. 1. Throughout this paper, an exp(imt) time dependence is assumed. Figure 1 Schematic of a two-port illustrating the convention for positive directions used in this paper. 3.2 Wave decomposition methods
where a and b refers to the two sides of the two-port, the plus and minus signs indicate propagation outwards and into the two-port respectively, the superscript s refers to source generated sound, and p describes the reflection and r the transmission of incoming waves. The definition of propagation directions and other indices are shown in Fig. 1. Throughout this paper, an exp(imt) time dependence is assumed. Figure 1 Schematic of a two-port illustrating the convention for positive directions used in this paper. 3.2 Wave decomposition methods
which is solved by the Moore-Penrose pseudo-inverse [63]. Just as before, the system will be solvable when at Fujimori et al [61] formulated the problem as a least squares problem in which the Euclidian norm of f' is to be minimized, a problem equivalent to solving the over-determined system
which is solved by the Moore-Penrose pseudo-inverse [63]. Just as before, the system will be solvable when at Fujimori et al [61] formulated the problem as a least squares problem in which the Euclidian norm of f' is to be minimized, a problem equivalent to solving the over-determined system
which can be solved analytically if the wave numbers are known. In their work they tested up to six microphone positions per half a wavelength, using three different arrays of microphones. They also concluded that arrays with equidistant microphone spacing yield the best results, and that their method is superior to the two microphone method if more than two microphone positions are used.
which can be solved analytically if the wave numbers are known. In their work they tested up to six microphone positions per half a wavelength, using three different arrays of microphones. They also concluded that arrays with equidistant microphone spacing yield the best results, and that their method is superior to the two microphone method if more than two microphone positions are used.
where the matrix Inversion should be interpreted as a Moore-Penrose pseudo-inversion if the matrix is no square. The idea of this over-determination was first suggested by Abom [55], and tested by Allam et al [70] for an orifice at M@ < 0.1, demonstrating that the results could be significantly improved using five differen oudspeaker combinations. As previously discussed, in order for Eq. (25) to be valid, it is required that at leas wo among the pressure amplitude vector sets are linearly independent. If more than two are available all may be used. However, another approach is to discard some of he measurements based upon the coherence function.
where the matrix Inversion should be interpreted as a Moore-Penrose pseudo-inversion if the matrix is no square. The idea of this over-determination was first suggested by Abom [55], and tested by Allam et al [70] for an orifice at M@ < 0.1, demonstrating that the results could be significantly improved using five differen oudspeaker combinations. As previously discussed, in order for Eq. (25) to be valid, it is required that at leas wo among the pressure amplitude vector sets are linearly independent. If more than two are available all may be used. However, another approach is to discard some of he measurements based upon the coherence function.
A transformed source vector can then be defined using the primed quantities above, and then transformed back to the original cross section, via In order to combine the data at two different cross-sections transformation equations are required. The transformation of the scattering and the reflection matrix from the reference section to a new (primed) section is given by Since Eq. (32) contains auto spectra, uncorrelated sound will still influence the results. Lavrentjev et al. [18] suggested using microphones at two different cross sections at each side to estimate the source data. The method is based upon the fact that if the distance between two microphones is long enough, the turbulent flow noise will be uncorrelated. This distance can, using the model of Corcos, be estimated from [35]
A transformed source vector can then be defined using the primed quantities above, and then transformed back to the original cross section, via In order to combine the data at two different cross-sections transformation equations are required. The transformation of the scattering and the reflection matrix from the reference section to a new (primed) section is given by Since Eq. (32) contains auto spectra, uncorrelated sound will still influence the results. Lavrentjev et al. [18] suggested using microphones at two different cross sections at each side to estimate the source data. The method is based upon the fact that if the distance between two microphones is long enough, the turbulent flow noise will be uncorrelated. This distance can, using the model of Corcos, be estimated from [35]
for holder 3 and X =3 mm for holder 4. Y can be changed Figure 7. Slit in the pipe used for holder configuration 5.
for holder 3 and X =3 mm for holder 4. Y can be changed Figure 7. Slit in the pipe used for holder configuration 5.
Figure 6. Cross section of the holder used for configuration 2.
Figure 6. Cross section of the holder used for configuration 2.
Figure 5. Construction drawing for the reference holder.
Figure 5. Construction drawing for the reference holder.
Figure 9 Loudspeaker connection configurations tested: a) loudspeaker connected using an open hole, b) as a) but with the hole filled with absorbing material, c) loudspeaker connected using a perforate pipe with 50% porosity.
Figure 9 Loudspeaker connection configurations tested: a) loudspeaker connected using an open hole, b) as a) but with the hole filled with absorbing material, c) loudspeaker connected using a perforate pipe with 50% porosity.
Figure 8 Slit (30 x 2) and © 6.4 mm hole used for holder configuration 6.
Figure 8 Slit (30 x 2) and © 6.4 mm hole used for holder configuration 6.
Figure 10. Picture for the expansion used in the experiments.
Figure 10. Picture for the expansion used in the experiments.
Table 1. Dimension of the expansion chamber components. Figure 11. Picture for the commercial muffler used in the experiments.
Table 1. Dimension of the expansion chamber components. Figure 11. Picture for the commercial muffler used in the experiments.
Figure 12. Test object (a triangular plate) used in the experiments a) Vortex mixer plate dimensions b) Vortex mixer angle of attack when mounted
Figure 12. Test object (a triangular plate) used in the experiments a) Vortex mixer plate dimensions b) Vortex mixer angle of attack when mounted
Figure 13. Difference in signal-to-noise ratio compared to the reference holder at M=0.16. —, reference holder; —--—, holder 2; -----, holder 3; ----, holder 5. AS a first step the holder configurations expected to give similar results were tested against each other. First comparisons were made between holders with similar design such as holder 3 and 4, with a 1,5 mm and 3 mm long cavity respectively. Since the smaller cavity, holder 3 gave a slightly better result it was used for subsequent comparisons. When comparing holder 5, axial slit, and 6, axial slit with a 6.4 mm diameter hole it was found that holder 5 is slightly better for low frequencies while holder 6 is slightly better for high frequencies. It was decided to use holder 5 for subsequent comparisons. Fig. 13 shows the difference in signal-to-noise ratio for holders 2, 3 and 5 compared to the reference holder for M=0.16. The result is similar for other Mach-numbers. A negative S/N ratio in dB means that the results are worse han for the reference holder and a positive S/N ratio that hey are better. It was concluded that the reference holder configuration gave the best signal-to-noise ratio. All subsequent tests of signal enhancement techniques were therefore made using this holder.
Figure 13. Difference in signal-to-noise ratio compared to the reference holder at M=0.16. —, reference holder; —--—, holder 2; -----, holder 3; ----, holder 5. AS a first step the holder configurations expected to give similar results were tested against each other. First comparisons were made between holders with similar design such as holder 3 and 4, with a 1,5 mm and 3 mm long cavity respectively. Since the smaller cavity, holder 3 gave a slightly better result it was used for subsequent comparisons. When comparing holder 5, axial slit, and 6, axial slit with a 6.4 mm diameter hole it was found that holder 5 is slightly better for low frequencies while holder 6 is slightly better for high frequencies. It was decided to use holder 5 for subsequent comparisons. Fig. 13 shows the difference in signal-to-noise ratio for holders 2, 3 and 5 compared to the reference holder for M=0.16. The result is similar for other Mach-numbers. A negative S/N ratio in dB means that the results are worse han for the reference holder and a positive S/N ratio that hey are better. It was concluded that the reference holder configuration gave the best signal-to-noise ratio. All subsequent tests of signal enhancement techniques were therefore made using this holder.
Figure 14 Signal-to-noise ratio at the six microphone positions using the loudspeaker mounting configuration according to Figure 9 a) with the source on the up-stream side using 1000 averages at M=0.3. —, Mic.1; ---, Mic. 2: -.-.-., Mic. 3:....., Mic. 4: ++" *. Mic. 5: ++++ Mic. 6.
Figure 14 Signal-to-noise ratio at the six microphone positions using the loudspeaker mounting configuration according to Figure 9 a) with the source on the up-stream side using 1000 averages at M=0.3. —, Mic.1; ---, Mic. 2: -.-.-., Mic. 3:....., Mic. 4: ++" *. Mic. 5: ++++ Mic. 6.
Figure 15. Signal-to-noise ratio at the six microphone positions using the loudspeaker mounting configuration according to Figure 9 b) with the source on the down-stream side using 1000 averages at M=0.3. —,
Figure 15. Signal-to-noise ratio at the six microphone positions using the loudspeaker mounting configuration according to Figure 9 b) with the source on the down-stream side using 1000 averages at M=0.3. —,
Figure 16 Signal-to-noise ratio at the six microphone positions using the loudspeaker mounting configuration according to Figure 9 c) with the source on the down-stream side using 1000 averages at M=0.3. —, Mic.1; ---, Mic. 2; -.-.-., Mic. 3; ..... , Mic. 4; + +++, Mic. 5; ++++ Mic. 6.
Figure 16 Signal-to-noise ratio at the six microphone positions using the loudspeaker mounting configuration according to Figure 9 c) with the source on the down-stream side using 1000 averages at M=0.3. —, Mic.1; ---, Mic. 2; -.-.-., Mic. 3; ..... , Mic. 4; + +++, Mic. 5; ++++ Mic. 6.
Figure 17. Sound pressure level for the acoustic signal and flow noise using frequency domain averaging, SPL =128dB at M=0.21. —, acoustic signal; flow noise: — - —,1 average; ——-, 10 averages; -----,100 averages. The test setup was the same as described in the previous section. The signal-to-noise ratio was also in this case determined by first measuring the noise generated by the loudspeaker at the microphone position without flow. Then the loudspeaker was turned off and the microphone signal was measured again with flow. A deterministic signal with a noise free reference for the synchronisation of the averaging was needed in this case. A saw-tooth signal was used with a fundamental frequency of 40 Hz. Measurements were taken for different flow Mach numbers from M = 0.07 to M = 0.21 and with different sound pressure levels at the microphone position (98, 116. 128, 136, 142 dB).
Figure 17. Sound pressure level for the acoustic signal and flow noise using frequency domain averaging, SPL =128dB at M=0.21. —, acoustic signal; flow noise: — - —,1 average; ——-, 10 averages; -----,100 averages. The test setup was the same as described in the previous section. The signal-to-noise ratio was also in this case determined by first measuring the noise generated by the loudspeaker at the microphone position without flow. Then the loudspeaker was turned off and the microphone signal was measured again with flow. A deterministic signal with a noise free reference for the synchronisation of the averaging was needed in this case. A saw-tooth signal was used with a fundamental frequency of 40 Hz. Measurements were taken for different flow Mach numbers from M = 0.07 to M = 0.21 and with different sound pressure levels at the microphone position (98, 116. 128, 136, 142 dB).
Figure 18. Sound pressure level for the signal and flow noise using time domain averaging, SPL= 128 (dB), M=0.21. —, signal; -----, | average; -——, 10 averages;— -—, 50 averages;—— +, 100 averages.
Figure 18. Sound pressure level for the signal and flow noise using time domain averaging, SPL= 128 (dB), M=0.21. —, signal; -----, | average; -——, 10 averages;— -—, 50 averages;—— +, 100 averages.
result indicating that the flow effects mentioned above influencing the sound pressure level at the microphone position are small compared to the signal-to-noise ratio increase obtained.
result indicating that the flow effects mentioned above influencing the sound pressure level at the microphone position are small compared to the signal-to-noise ratio increase obtained.
Figure 21. Sound pressure level at microphone position obtained using 1000 averages for; —, acoustic excitation only; ——-, flow noise only.
Figure 21. Sound pressure level at microphone position obtained using 1000 averages for; —, acoustic excitation only; ——-, flow noise only.
Figure 23. Real part of first element of the acoustic two port for straight pipe at M=0.24 and T=293° K. ---, theory; UU, random excitation 10000 averages (CSFDA); 0000, stepped sine excitation 400 averages (CSFDA); ++++, saw tooth excitation 10000 averages (STDA).
Figure 23. Real part of first element of the acoustic two port for straight pipe at M=0.24 and T=293° K. ---, theory; UU, random excitation 10000 averages (CSFDA); 0000, stepped sine excitation 400 averages (CSFDA); ++++, saw tooth excitation 10000 averages (STDA).
If a noise free reference signal is not available an improvement can be obtained by using the microphone with the highest signal to noise ratio as the reference as discussed in section 2. Fig. 25 shows a comparison of the result obtained using microphone 4 as the fixed reference microphone and using the microphone with the highest signal-to-noise ratio for each frequency as the reference. The comparison is made using stepped sine excitation and 400 cross-spectrum based frequency domain averages for M=0.32. It can be seen that at this high flow velocity a slight improvement is obtained when the highest signal-to-noise ratio microphone is used as reference. This technique can therefore give an extra improvement in the most difficult measurement situations.
If a noise free reference signal is not available an improvement can be obtained by using the microphone with the highest signal to noise ratio as the reference as discussed in section 2. Fig. 25 shows a comparison of the result obtained using microphone 4 as the fixed reference microphone and using the microphone with the highest signal-to-noise ratio for each frequency as the reference. The comparison is made using stepped sine excitation and 400 cross-spectrum based frequency domain averages for M=0.32. It can be seen that at this high flow velocity a slight improvement is obtained when the highest signal-to-noise ratio microphone is used as reference. This technique can therefore give an extra improvement in the most difficult measurement situations.
Figure 24 Transmission loss versus frequency for different type of excitations at M=0.3. ----, random excitation with 10000 averages (CSFDA); 0000 stepped sine with 400 averages (CSFDA); ++++, saw tooth excitation with 10000 averages (STDA).
Figure 24 Transmission loss versus frequency for different type of excitations at M=0.3. ----, random excitation with 10000 averages (CSFDA); 0000 stepped sine with 400 averages (CSFDA); ++++, saw tooth excitation with 10000 averages (STDA).
Figure 26. Transmission loss versus frequency for different type of excitations at M=0.2. Random excitation; ---- 10000 averages (CSFDA); +++1000 averages (FDA); 0000 stepped sine excitation with 400 averages (CSFDA). This commercial automotive muffler gives a high transmission loss which makes it a more difficult measurement object compared to the straight duct and he simple expansion chamber studied in the previous sections. Transmission loss has been measured for three different flow velocities (M=0, 0.2, and 0.26). For the no flow case all excitation signals and signal processing echniques gave identical results. Fig. 26 shows results for the ordinary frequency domain averaging (FDA) and cross-spectrum based frequency domain averaging CSFDA) at M=0.2. The results show that random excitation with FDA gives results which are not in agreement with the reference case, stepped sine excitation and 400 averages, at all frequencies. This is the case even if a very high level input signal is used. CSFDA with 10000 averages gives the same result as the stepped sine excitation. Fig. 27 show results for saw tooth excitation and synchronised time domain averaging (STDA) with 1000 averages and 10000 averages at M = 0.26. For this high flow velocity 1000 averages was not sufficient to give a good result but with 10000 averages a result in agreement with the stepped sine result was obtained.
Figure 26. Transmission loss versus frequency for different type of excitations at M=0.2. Random excitation; ---- 10000 averages (CSFDA); +++1000 averages (FDA); 0000 stepped sine excitation with 400 averages (CSFDA). This commercial automotive muffler gives a high transmission loss which makes it a more difficult measurement object compared to the straight duct and he simple expansion chamber studied in the previous sections. Transmission loss has been measured for three different flow velocities (M=0, 0.2, and 0.26). For the no flow case all excitation signals and signal processing echniques gave identical results. Fig. 26 shows results for the ordinary frequency domain averaging (FDA) and cross-spectrum based frequency domain averaging CSFDA) at M=0.2. The results show that random excitation with FDA gives results which are not in agreement with the reference case, stepped sine excitation and 400 averages, at all frequencies. This is the case even if a very high level input signal is used. CSFDA with 10000 averages gives the same result as the stepped sine excitation. Fig. 27 show results for saw tooth excitation and synchronised time domain averaging (STDA) with 1000 averages and 10000 averages at M = 0.26. For this high flow velocity 1000 averages was not sufficient to give a good result but with 10000 averages a result in agreement with the stepped sine result was obtained.
Figure 27. Transmission losses versus frequency for different type of excitations at M=0.26. ----, random excitation with 10000 averages (CSFDA); 0000 stepped sine excitation with 400 averages (CSFDA); +++4, saw tooth excitation with 10000 averages (STDA).
Figure 27. Transmission losses versus frequency for different type of excitations at M=0.26. ----, random excitation with 10000 averages (CSFDA); 0000 stepped sine excitation with 400 averages (CSFDA); +++4, saw tooth excitation with 10000 averages (STDA).
radians, i.e. to a much more accurate value. However, for the phase of t,, this temperature change will decrease the phase at the same frequency from -0.19 to -0.47, i.e. increasing the error. This is due to the fact that for a fixed Mach number a temperature change will affect both wavenumbers in the same way, while a Mach number modification (for a fixed temperature) will increase one of the wavenumbers and decrease the other, thus allowing for both the phase of t,, and of 1, to approach zero as the Mach number is modified.
radians, i.e. to a much more accurate value. However, for the phase of t,, this temperature change will decrease the phase at the same frequency from -0.19 to -0.47, i.e. increasing the error. This is due to the fact that for a fixed Mach number a temperature change will affect both wavenumbers in the same way, while a Mach number modification (for a fixed temperature) will increase one of the wavenumbers and decrease the other, thus allowing for both the phase of t,, and of 1, to approach zero as the Mach number is modified.
Figure 34The magnitude without the plate, obtained with microphones fulfilling of the scattering matrix using over-determination, Eq. (17) used; (a), upstream reflection coefficient p,, (b), downstream to upstream transmission coefficient 7,,, transmission coefficient 7,,, c), upstream to downstream (d), downstream reflection coefficient p,. Solid line: Mach number from acoustic measurements (M= 0.207 used, dashed line: Mach number from Prandtl pipe measurements (VM = 0.201) used.
Figure 34The magnitude without the plate, obtained with microphones fulfilling of the scattering matrix using over-determination, Eq. (17) used; (a), upstream reflection coefficient p,, (b), downstream to upstream transmission coefficient 7,,, transmission coefficient 7,,, c), upstream to downstream (d), downstream reflection coefficient p,. Solid line: Mach number from acoustic measurements (M= 0.207 used, dashed line: Mach number from Prandtl pipe measurements (VM = 0.201) used.
Figure 36. The diagonal elements of the source cross
Figure 36. The diagonal elements of the source cross
determined using microphones 1-3 and 6-8 for the first vector and 4, 5, 9 and 10 for the second.
determined using microphones 1-3 and 6-8 for the first vector and 4, 5, 9 and 10 for the second.
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