Research Papers

Movable Rigid Scatterer Model for Flexural Wave Scattering on Thin Plates

[+] Author and Article Information
Liang-Wu Cai

Department of Mechanical and
Nuclear Engineering,
Kansas State University,
Manhattan, KS 66506
e-mail: cai@ksu.edu

Stephen A. Hambric

Applied Research Laboratory,
Pennsylvania State University,
State College, PA 16801

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 30, 2015; final manuscript received February 19, 2016; published online April 15, 2016. Assoc. Editor: Ronald N. Miles.

J. Vib. Acoust 138(3), 031016 (Apr 15, 2016) (10 pages) Paper No: VIB-15-1414; doi: 10.1115/1.4033060 History: Received September 30, 2015; Revised February 19, 2016

Rigid scatterers are fundamentally important in the study of scattering of many types of waves. However, in the recent literature on scattering of flexural waves on thin plates, a “rigid scatterer” has been used to represent a clamped boundary. Such a model physically resembles riveting the plate to a fixed structure. In this paper, a movable model for a rigid scatterer that allows rigid-body motion is established. It is shown that, when the mass density of the movable rigid scatterer is much greater than that of the host plate and at high frequencies, the movable rigid scatterer approaches the limiting case that is the riveted rigid scatterer. The single- and multiple-scattering by such scatterers are examined. Numerical examples show that, at the extreme end of lower frequencies, the scattering cross section for the movable model vanishes while that of the riveted models approaches infinity. An array of such movable rigid scatterers can form a broad and well-defined stop band for flexural wave transmission. With a volume fraction above 50%, the spectrum is rather clean: consisting of only an extremely broad stop band and two groups of higher frequency Perot–Fabry resonance peaks. Increasing either scatterer’s mass density or the lattice spacing can compress the spectral features toward lower frequencies.

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Cai, L.-W. , and Hambric, S. A. , 2016, “ Multiple Scattering of Flexural Waves on Thin Plates,” ASME J. Vib. Acoust., 138(1), p. 011009. [CrossRef]
Norris, A. N. , and Vemula, C. , 1995, “ Scattering of Flexural Waves on Thin Plates,” J. Sound Vib., 181(1), pp. 115–125. [CrossRef]
Matus, V. V. , Kunets, Y. I. , Porochovs’kyj, V. V. , and Mishchenko, V. O. , 2007, “ Scattering of Flexural Waves by a Rigid Inclusion in Thin Plate,” 12th International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-2007), Lviv, Ukraine, Sept. 17–20, pp. 195–198.
Chan, K. L. , Smith, B. , and Wester, E. , 2009, “ Flexural Wave Scattering in a Quarter-Infinite Thin Plate With Circular Scatterers,” Int. J. Solids Struct., 46(20), pp. 3669–3676. [CrossRef]
Matus, V. V. , and Emets, V. F. , 2010, “ T-Matrix Method Formulation Applied to the Study of Flexural Waves Scattering From a Through Obstacle in a Plate,” J. Sound Vib., 329(14), pp. 2843–2850. [CrossRef]
Hickling, R. , and Wang, N. M. , 1966, “ Scattering of Sound by a Rigid Movable Sphere,” J. Acoust. Soc. Am., 39(2), pp. 276–279. [CrossRef]
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Pao, Y. H. , and Mow, C. C. , 1973, Diffraction of Elastic Waves and Dynamic Stress Concentration, Crane Russak, New York.
Chou, C.-F. , 1966, “ Diffraction of Flexural Wave by Circular Rigid Inclusion in an Elastic Plate,” M.S. thesis (in Chinese), National Taiwan University, Taiwan, China.
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Fig. 1

Rigid-body motions in rigid scatterer. Left: Φ in y–z plane. Right: Ψ in x–z plane.

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Fig. 2

Shear force Vr and bending moment Mr across the thickness direction (shown only those at angle θ) along the rim of the rigid scatterer

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Fig. 3

Distribution of displacement amplitude in the vicinity of three scatterers. At the top is the riveted right scatterer, and the two below are the movable rigid scatterers.

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Fig. 4

Distribution of real and imaginary parts of displacement along the x-axis. Vertical thin dotted lines demarcate the scatterer–host interfaces.

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Fig. 5

Distribution of real and imaginary parts of displacement along the y-axis. Vertical thin dotted lines demarcate the scatterer–host interfaces.

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Fig. 6

Polar plot of scattering form factor |f(θ)|/a at ka = 0.5

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Fig. 7

Polar plot of scattering form factor |f(θ)|/a at ka = 1

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Fig. 8

Polar plot of scattering form factor |f(θ)|/a at ka = 5

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Fig. 9

Amplitude of scattered wave in backward direction at (x,y)=(−10,0) m

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Fig. 10

Normalized scattering cross section for three rigid scatterer models

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Fig. 11

Configurations of 3 × 7 and 5 × 7 square lattice arrangements. The line segments, which is located a lattice constant from the centers of the right-most column, indicate the locations where displacement amplitude is averaged.

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Fig. 12

Transmission spectra for sonic crystals of 3 × 7 and 5 × 7 arrangements comprising the moveable rigid scatterers

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Fig. 13

Transmission spectra by the 5 × 7 crystal comprising the movable rigid scatterers for mass density ratios ρ1/ρ=0.01, 0.1, 1, 10, and 100

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Fig. 14

Comparison of transmission spectra by the 5 × 7 crystal comprising of the movable rigid scatterers for lattice constant d = 60 mm, 80 mm, 100 mm, and 120 mm

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Fig. 15

Key frequencies of the main band gap produced by the four different lattice constants, which is converted into the volume fraction vf occupied by the scatterer



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