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**Current techniques for measuring**

Muffler Transmission Loss

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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

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**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 =

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**Methods for measuring TL:**

Decomposition method Two Source method using 4 microphones Two Load method

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**Decomposition Method:**

= Incident Auto Power Spectrum = Reflected Auto power spectrum S33 = Transmitted Auto Power Spectrum

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**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:

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**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

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**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:

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**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

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Two Source Method:

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**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

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**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

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**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

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**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:

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**Two Source Method: Contd..**

Also, transfer function ( ) is the ratio of cross and auto spectrums Therefore, Transmission Loss can be expressed by:

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**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

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Two Load Method Setup:

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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

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**Contd.. Measurement setup for Two Load method**

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**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

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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

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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

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**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

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**Comparison of all 3 Methods**

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**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

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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

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**Calibration of microphones setup**

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**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

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**Schematic diagram for STL measurement**

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Impedance Tube setup Small Impedance Tube Setup with 4 microphones

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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)

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