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Research Papers

Separation of Traveling and Standing Waves in a Rigid-Walled Circular Duct Containing an Intermediate Impedance Discontinuity

[+] Author and Article Information
Yongxiong Xiao

College of Mechanical Engineering,
Key Laboratory for Special Purpose Equipment
and Advanced Manufacturing Technology,
Ministry of Education and Zhejiang Province,
Zhejiang University of Technology,
18 Chaowang Road,
Hangzhou 310014, China
e-mail: ethanxiao_svlab@163.com

Antoine Blanchard

Department of Aerospace Engineering,
University of Illinois at Urbana-Champaign,
104 South Wright Street,
Urbana, IL 61801
e-mail: ablancha@illinois.edu

Yao Zhang

College of Mechanical Engineering,
Key Laboratory for Special Purpose Equipment
and Advanced Manufacturing Technology,
Ministry of Education and Zhejiang Province,
Zhejiang University of Technology,
18 Chaowang Road,
Hangzhou 310014, China
e-mail: zhangyao_svlab@163.com

Huancai Lu

Mem. ASME
College of Mechanical Engineering,
Key Laboratory for Signal Processing
of Zhejiang Province,
Zhejiang University of Technology,
18 Chaowang Road,
Hangzhou 310014, China
e-mail: huancailu@zjut.edu.cn

D. Michael McFarland

Mem. ASME
Department of Aerospace Engineering,
University of Illinois at Urbana-Champaign,
104 South Wright Street,
Urbana, IL 61801
e-mail: dmmcf@illinois.edu

Alexander F. Vakakis

Fellow ASME
Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign,
1208 West Green Street,
Urbana, IL 61801
e-mail: avakakis@illinois.edu

Lawrence A. Bergman

Fellow ASME
Department of Aerospace Engineering,
University of Illinois at Urbana-Champaign,
104 South Wright Street,
Urbana, IL 61801
e-mail: lbergman@illinois.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 21, 2016; final manuscript received April 7, 2017; published online July 26, 2017. Assoc. Editor: Stefano Lenci.

J. Vib. Acoust 139(6), 061001 (Jul 26, 2017) (8 pages) Paper No: VIB-16-1408; doi: 10.1115/1.4036866 History: Received August 21, 2016; Revised April 07, 2017

In this paper, we study the phenomenon of separation of traveling and standing waves in a one-dimensional rigid-walled circular duct. The underlying mechanism for separation, mode complexity, is linear and introduced here by a damped side branch representing an impedance discontinuity. The left end of the duct is driven at a single frequency by a harmonic acoustic source, and the right end is a rigid termination. The position and impedance of the side branch are independent parameters in the analysis. Sufficient conditions for acoustic wave separation in the duct are derived analytically and employed in a three-dimensional finite element analysis to verify the theoretical result. A physical experiment, consisting of a circular duct with a damped side branch, was constructed based on analytical predictions, the physical parameters were measured or identified, and its performance was documented. These experimental parameters were employed in a second three-dimensional finite element analysis to obtain a direct comparison with experimental results. The comparison reveals the extent to which higher-order (unmodeled) effects degrade the separation phenomenon. It is demonstrated that an intermediate damped side branch used as a nonresonant device can be predictively designed to achieve nearly ideal separation of traveling and standing waves in a rigid-walled circular duct in order to direct and control acoustic energy transmission through the duct system.

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Figures

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Fig. 1

Schematic representation of the acoustic duct with an asymmetrically placed side branch. One end is subjected to harmonic excitation, and the other is a rigid termination.

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Fig. 2

Steady-state response of the acoustic duct system: (a) Evolution of the normalized spatial phase distribution in the two regions of the duct partitioned by the side branch, and (b) steady-state acoustic pressure at different instants of normalized time; ω=5.7π and x0=0.7 are specified, and (Rb,Xb)=(0.3757,−0.4843) are determined according to Eqs. (24) and (25)

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Fig. 3

Spatiotemporal evolution of (a) potential energy and (b) total energy, for ω=5.7π and x0=0.7 specified and (Rb,Xb)=(0.3757,−0.4843) determined according to Eqs. (24) and (25)

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Fig. 4

Evolution of the normalized instantaneous acoustic intensity at the left limit of the side branch, x0− (solid line), dissipated acoustic intensity (dashed line), and the sum of the intensities at the right limit of the side branch, x0+, and acoustic intensity at the input of the side branch, x0b (circular dots), at steady-state, when a traveling wave is formed; ω=5.7π and x0=0.7 are specified, and (Rb,Xb)=(0.3757,−0.4843) are determined according to Eqs. (24) and (25)

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Fig. 5

The acoustic pressure amplitude reflection coefficient plotted with respect to (a) normalized frequency of the sound source when x0=0.7 and (b) normalized position of the side branch when ω=5.7

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Fig. 6

The experimental apparatus

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Fig. 7

Three-dimensional finite element model showing (a) the impedance discontinuity modeled by an impedance sheet and (b) the impedance discontinuity modeled by a side branch (a portion of the model is shown)

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Fig. 8

Sound pressure amplitude distribution in the duct for f=1940 Hz and x̃0=367 mm: Comparison between theoretical analysis (solid line), finite element analysis using an ideal impedance discontinuity represented by an impedance sheet (dotted line), measurements from the physical experiment (circles), and finite element analysis using measured and computed parameters from the physical experiment (dashed line)

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Fig. 9

Spatiotemporal evolution of the potential energy from the finite element analysis using measured and computed parameters from the physical experiment; f=1940 Hz and x̃0=367 mm

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Fig. 10

Sound pressure level distribution on the cutting plane in the vicinity of the side branch from the FE analysis using measured and computed parameters from the physical experiment; f=1940 Hz and x̃0=367 mm

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