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Research Papers

Assessing the Effects of Impedance Modifications Using the Moebius Transformation

[+] Author and Article Information
Y. Zhang

Department of Mechanical Engineering,
University of Kentucky,
151 RGAN Building,
Lexington, KY 40506-0503
e-mail: yitian.zhang@uky.edu

D. W. Herrin

Department of Mechanical Engineering,
University of Kentucky,
151 RGAN Building,
Lexington, KY 40506-0503
e-mail: dherrin@engr.uky.edu

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 13, 2014; final manuscript received November 18, 2014; published online January 27, 2015. Assoc. Editor: Liang-Wu Cai.

J. Vib. Acoust 137(3), 031011 (Jun 01, 2015) (9 pages) Paper No: VIB-14-1303; doi: 10.1115/1.4029214 History: Received August 13, 2014; Revised November 18, 2014; Online January 27, 2015

The Moebius transformation maps straight lines or circles in one complex domain into straight lines or circles in another. It has been observed that the equations relating acoustic or mechanical impedance modifications to responses under harmonic excitation are in the form of the Moebius transformation. Using the properties of the Moebius transformation, the impedance modification that will minimize the response at a particular frequency can be predicted provided that the modification is between two positions. To prove the utility of this method for acoustic and mechanical systems, it is demonstrated that the equations for calculation of transmission and insertion loss of mufflers, insertion loss of enclosures, and insertion loss of mounts are in the form of the Moebius transformation for impedance modifications. The method is demonstrated for enclosure insertion loss by adding a short duct in a partition introduced to an enclosure. In a similar manner, it is shown that the length or area of a bypass duct in a muffler can be tuned to maximize the transmission loss. In the final example, the insertion loss of an isolator system is improved at a particular frequency by adding mass to one side of the isolator.

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References

Figures

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Fig. 1

Schematic of a mechanical–acoustical system with inputs and impedance modification (Adapted from Ref. [4])

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Fig. 2

Schematic showing enclosure with partition and bypass duct. Sides with sound absorption are shaded (unit: m).

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Fig. 3

Comparison of insertion loss between original and optimized enclosure (lz: longest dimension of enclosure)

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Fig. 4

p3 due to variations in the diameter (d in mm) of the bypass duct

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Fig. 5

Schematic showing box enclosure with single outlet. Sides having sound absorption are shaded (unit: m).

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Fig. 6

Insertion loss comparison for outlet lengths (lz: longest dimension of enclosure)

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Fig. 7

p2 as the outlet length is varied for klz=3. Dashed circle indicates limits of desired range for p2 (unit for length l is m).

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Fig. 8

Photograph of muffler and bypass duct

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Fig. 9

Schematic showing muffler with bypass duct. The muffler dimensions are: lw = 0.3 m, ll = 0.35 m, lp = 0.2 m, ls = 0.17 m, di = do = 0.05 m, and height = 0.15 m.

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Fig. 10

Transmission loss for original muffler and muffler with bypass duct added. Measured and plane wave simulation curves are shown.

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Fig. 11

Effect of changing length of bypass duct on the transmission loss vector STL. Dots indicate STL values for different lengths and the large dot indicates the optimal solution.

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Fig. 12

Transmission loss for muffler with bypass duct at the limits of the feasible range. The case with no bypass duct is included for reference.

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Fig. 13

Schematic showing plate with geometric parameters used for calculation of the impedance

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Fig. 14

Effect of changing imaginary part of the foundation impedance on SIL (small dots indicate SIL values for different modifications; large dot indicates the optimum point furthest from the origin).

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Fig. 15

Insertion loss comparison before and after optimization, where f1 indicates the first resonant frequency for the original system

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