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Research Papers

Multiple Scattering of Flexural Waves 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 16804

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 15, 2015; final manuscript received August 31, 2015; published online October 26, 2015. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 138(1), 011009 (Oct 26, 2015) (10 pages) Paper No: VIB-15-1219; doi: 10.1115/1.4031535 History: Received June 15, 2015; Revised August 31, 2015

In this paper, the scattering of flexural waves on a thin Kirchhoff plate by an ensemble of through-thickness circular scatterers is formulated by using the concept of the T-matrix in a generalized matrix notation, with a focus on deterministic numerical computations. T-matrices for common types of scatterers, including the void (hole), rigid, and elastic scatterers, are obtained. Wave field properties in the multiple-scattering setting, such as the scattering amplitude, and scattering cross section, as well as properties of the T-matrix due to the energy conservation are discussed. After an extensive validation, numerical examples are used to explore the band gap formation due to different types of scatterers. One of the interesting observations is that a type of inclusion commonly referred to as the “rigid inclusion” in fact represents a clamped boundary that is closer to a riveted confinement than a rigid scatterer; and an array of such scatterers can block the wave transmission at virtually all frequencies.

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References

Figures

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Fig. 1

Geometries in multiple scattering: (a) geometry for Graf's addition theorem for Bessel functions and (b) geometric relations between two local coordinate systems

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Fig. 2

Amplitude of deflection in thin plate embedded with four different circular inclusions, from quadrant I–IV: void, rigid, rubber, and steel. For the cases of no wave fields inside the inclusion (void and rigid), the deflection is set to zero.

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Fig. 3

Distribution of the deflection amplitude along the two diagonal lines of Fig. 2. Dashed vertical lines demarcate the interfaces between host plate and scatterers.

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Fig. 4

Configuration of ensemble of inclusions forming a rectangular lattice arrangement. The line segment to the right indicates the location where deflection is averaged.

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Fig. 5

Spectra of averaged deflection amplitude in forward direction for sonic crystals of four different types of inclusions

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Fig. 6

Amplitude distribution near sonic crystals at f = 213 kHz (ka = 1.3077). Left: void inclusions and right: rubber inclusions.

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Fig. 7

Amplitude distribution near sonic crystals at f = 1 MHz (ka = 2.8334). Left: void inclusions and right: rubber inclusions.

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