1. Current techniques for measuring
Muffler Transmission Loss

2. Why Transmission Loss?
To understand how effectively an acoustical treatment can block the incident sound when designing a mechanical system
Transmission Loss quantifies the acoustical treatment for engineering application

3. Parameters for acoustic performance of a Muffler
Noise reduction (NR):SPL difference across the muffler
Insertion loss (IL):SPL difference outside the system with and without the muffler present
Transmission Loss (TL):Sound power level difference between the incident and the transmitted wave assuming anechoic termination given by =

4. Methods for measuring TL:
Decomposition method
Two Source method using 4 microphones
Two Load method

5. Decomposition Method:
= Incident Auto Power
Spectrum
= Reflected Auto power
spectrum
S33 = Transmitted Auto Power
Spectrum

6. Contd.. Decomposition Method:
Muffler Transmission Loss is given by:
Where Wi = Incident sound power
Wt = Transmitted sound power
Inlet Sound Pressure can be decomposed into incident and reflected wave and respectively
Using Decomposition theory:

7. Contd.. Decomposition Method:
where and are the Auto spectra of the total acoustic pressure at 1 & 2 resp.
& are the real and imag part of cross spectrum between points 1 & 2
K = wave number
is the distance between the two microphones
The rms amplitude of incident sound wave and transmitted wave is given by:
pt =
where incident & transmitted are the rms pressure amplitudes
and S33 is the auto power spectrum

8. Contd.. Decomposition Method:
Now the sound power of incident and reflected wave is given by:
and
where are the muffler inlet and outlet tube areas
Therefore ,Transmission Loss is given by:

9. Contd.. Implementation of Decomposition Method
Find auto power & cross power spectrum at the two microphones 1 & 2
Utilize decomposition theory to find incident auto power spectra
Find rms amplitude by taking the square root of the incident auto power spectra and transmitted auto power spectra (measured directly from the microphone 3)
Plug in the calculated rms values in the TL equation

10. Two Source Method:

11. Contd.. Two Source Method:
Based on Transfer matrix approach-relation between input pressure & velocity to the output pressure and velocity
Any acoustical element can be modeled by its four pole parameters which is given by:
where are the sound pressure amplitudes at the inlet & outlet
are the particle velocity amplitudes at the inlet and the outlet
are the 4 pole parameters of the system

12. Contd.. Two Source Method:
For Configuration ‘a’:
The 4 pole equation for the element 2-3 is given by
where subscript ‘a’ refers to configuration a
Also, 4 pole equations for elements 1-2 & 3-4 is given by
where are the microphone spacings for elements 1-2 & 3-4

13. Contd.. Two Source Method:
Combining all the equations for 1-2,3-4 & 2-3 gives
For Configuration ‘b’: Moving sound source to the other side
For this configuration, the equation for element 3-2 is given by:
where

14. Contd.. Two Source Method:
Now the combined equation for configuration ‘b’ is given by:
Now using these equations one can obtain 4 –pole parameters given by:

15. Two Source Method: Contd..
Also, transfer function ( ) is the ratio of cross and auto spectrums
Therefore, Transmission Loss can be expressed by:

16. Contd.. Implementation of Two Source Method:
Use Two microphones with random excitation (or white noise)
Obtain all transfer functions by moving one microphone and using the other as a reference
Put the obtained transfer functions in the equations shown above to find the 4 pole parameters of the transformation matrix
Find the TL by plugging the 4 pole parameters and the measured cross sectional areas of the tube in the equation given above

17. Two Load Method Setup:

18. Contd.. Two Load method:
Similar to the Two Source method with little changes as follows:
Instead of moving the sound source, two end conditions are applied to find 2 additional equations in order to solve the complete transfer matrix
Changing end conditions changes the impedance at the termination from to
Two loads can be 2 different length tubes, a single tube with & without absorbing material or even 2 different mufflers

19. Contd.. Measurement setup for Two Load method

20. Contd.. Calculations using Cross Spectrums for Two Load method :
Involves two basic measurements with two different terminations
Terminations must be very different
Generally open ended or anechoic (o) & closed ended or reverberant (c) terminations are used
The equations utilized are as follows:
and
where A & B and C & D are the forward and backward complex pressure amplitudes wave in the source tube and receiving tube respectively with ‘a’ & ‘b’ denoting two different end conditions

21. Contd.. Two Load Method:
The 4 coefficients A,B,C,D can be calculated by the following relations:
where P1,P2,P3,P4 are the measured Sound Pressures
Now, using Cross spectrum and FRF’s to minimize noise in the signal, we can obtain the following equations:
where are the cross power spectrums using ‘o’ as a reference

22. Contd.. Two Load Method:
The Transmission Loss coefficient is given by the ratio of amplitude A of incident wave and amplitude C of the transmitted wave assuming no reflection i.e D = 0
Therefore, from the above relations
where is the first element of the transfer matrix

23. Contd.. Limitations of Cross Spectrums calculations and two load method:
Must require two different terminations
Difficult to obtain ideal anechoic termination
There seems to be a flaw in the calculations of TL since they assume that the reflection to be zero i.e D =0 but we are also using the same equation with reverberant termination.
Prior to taking actual measurements, it requires complete measurements without inserting the acoustical material to assure that residual TL is much less than the measured TL
If the numerical value of the difference in denominator becomes smaller than the absolute value of the absolute nos., then the solution becomes unstable

24. Comparison of all 3 Methods

25. Advantages of Two source method over other methods
The above results indicate the limitation of decomposition method in the absence of anechoic termination
Decomposition method does not lead to 4 pole parameters of muffler
Unlike two load method, two source method does not require any termination material at the end
Although Two load method is easier to employ but better results require two different loads

26. General Procedure:
Impedance tube with small diameter (29mm) can be utilized for measuring TL of Briggs & Stratton Muffler
The apparatus is designed to measure TL and other acoustics properties using the following ASTM E-1050 standard:
Working frequency range: where
size and spacing of microphones:
Location of microphones: minimum of 3 tube diameter from source to first microphone
Sound source- type & signal: random noise having uniform spectral density
Length of tube: should be large for plane wave propagation
Determining the individual microphone sensitivity (mV/Pa)
Calibrate the microphones correctly
Means of correcting the measured transfer function data for mismatch in both amplitude and phase responses of measurement channels

27. Calibration of microphones setup

28. contd.. Calculations for Calibration of microphones
Place a highly absorptive material to prevent strong acoustics reflections and to obtain most accurate correction factor possible
Measure Transfer functions in 2 configuration 1 and 2 as shown
Compute the calibration factor representing the amplitude and phase mismatches
where

29. Schematic diagram for STL measurement

30. Impedance Tube setup
Small Impedance Tube Setup with 4 microphones

31. References:
Z. Tao and Seybert, A.F., “A review of current techniques for measuring muffler TL”
Seybert, A.F. and Ross, D.F., “Experimental Determination of Acoustic Properties Using a Two microphone Random Excitation Technique,” J. Acoust. Soc. Am., 61, (1977)
Munjal, M.L. and Doige A.G., “Theory of a Two Source-location Method for Direct Experimental Evaluation of the Four-pole Parameters of an Aeroacoustic Element,” Journal of Sound and Vibration, 141(2), (1990)
ASTM standard, E , “Standard Test Method for Impedance and Absorption of Acoustical Material Using a Tube, Two Microphones and a Digital Frequency Analysis System,” (1998)

32. ?