HW3_2007_Solution - Homework 3_rev Acoustics I due date November 19 2007(1 An expansion chamber muffler is shown below If S = a 2 a is 2 cm S1 = 20

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Homework 3_rev Acoustics I due date: November 19 2007 (1) An expansion chamber muffler is shown below. If, 2 Sa π = , a is 2 cm, , temp = 80 1 20 S = S B 0 C, calculate and plot the power transmission coefficients T for the muffler with an anechoic termination shown below for the frequency range 0-3,000 Hz. Π i P r P S S 1 S L 1 A B Fig. 1. Simple expansion chamber L 1 was not given. One can choose 0.3 m that was given in version 1 or 0.55 m, the total length of the muffler given in problem 2. The overall four pole equation of the system with an anechoic termination always becomes 1 AB TT B a Z CD ⎧⎫ ⎡⎤ == ⎨⎬ ⎢⎥ ⎣⎦ ⎩⎭ QQ P PP Q . (1) where, the particle velocity B o c ρ = P U at B , BB o S S c QU P ; therefore o B a B c Z S P Q in this particular case. There is only one element between the input and output ports; therefore, 11 1 1 1 1 sin cos sin cos o o j Sk L kL c jc k L kL S = . (2) Because and Ai =+ r P ( o S c =− ) r P P Q , Eq. (1) can be re-written; ( o ir T T a c Z S ) B −= + Q (3) ( ) T T a Z += + Q B (4) Dividing Eq. (3) by Eq. (4), we obtain 1 1 p oT T a T T a cA BZ SC DZ p −+ ++ R R + . (5) 1
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R p is obtained as 1 1 p x x = + R (6) where, oT T a TT cA BZ x SC DZ a ρ + = + . Finally, 2 1 p T Π =− R . T Π with the given configuration is obtained as in Fig. 2. Each peak in Fig. 2 corresponds to the resonance frequency of the both-end closed pipe of length L 1 (30 cm). 0 500 1000 1500 2000 2500 3000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 frequency (Hz) Power Trans. Coeff. Fig. 2 Power transmission coefficient of the system in Fig. 1. More often transmission loss (TL) is used, which is defined as 10 1 10log TL T Π ⎛⎞ = ⎝⎠ . (7) TL is plotted in Fig. 3. 0
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This note was uploaded on 10/19/2009 for the course MECH 666 taught by Professor Kim during the Fall '07 term at University of Cincinnati.

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HW3_2007_Solution - Homework 3_rev Acoustics I due date November 19 2007(1 An expansion chamber muffler is shown below If S = a 2 a is 2 cm S1 = 20

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