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Research Papers

Stable and Accurate Computation of Dispersion Relations for Layered Waveguides, Semi-Infinite Spaces and Infinite Spaces

[+] Author and Article Information
Q. Gao

Faculty of Vehicle Engineering and Mechanics,
State Key Laboratory of Structural Analysis for
Industrial Equipment,
Department of Engineering Mechanics,
Dalian University of Technology,
Dalian 116023, China
e-mail: qgao@dlut.edu.cn

Y. H. Zhang

Faculty of Vehicle Engineering and Mechanics,
State Key Laboratory of Structural Analysis for
Industrial Equipment,
Department of Engineering Mechanics,
Dalian University of Technology,
Dalian 116023, China
e-mail: zyh0826@mail.dlut.edu.cn

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January 12, 2018; final manuscript received January 14, 2019; published online March 4, 2019. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 141(3), 031012 (Mar 04, 2019) (16 pages) Paper No: VIB-18-1017; doi: 10.1115/1.4042708 History: Received January 12, 2018; Revised January 14, 2019

This paper studies the dispersion characteristics of guided waves in layered finite media, surface waves in layered semi-infinite spaces, and Stoneley waves in layered infinite spaces. Using the precise integration method (PIM) and the Wittrick–Williams (W-W) algorithm, three methods that are based on the dynamic stiffness matrix, symplectic transfer matrix, and mixed energy matrix are developed to compute the dispersion relations. The dispersion relations in layered media can be reduced to a standard eigenvalue problem of ordinary differential equations (ODEs) in the frequency-wavenumber domain. The PIM is used to accurately solve the ODEs with two-point boundary conditions, and all of the eigenvalues are determined by using the eigenvalue counting method. The proposed methods overcome the difficulty of seeking roots from nonlinear transcendental equations. In theory, the three proposed methods are interconnected and can be transformed into each other, but a numerical example indicates that the three methods have different levels of numerical stability and that the method based on the mixed energy matrix is more stable than the other two methods. Numerical examples show that the method based on the mixed energy matrix is accurate and effective for cases of waves in layered finite media, layered semi-infinite spaces, and layered infinite spaces.

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Figures

Grahic Jump Location
Fig. 1

Layered models for (a) finite waveguides, (b) semi-infinite spaces, and (c) infinite spaces

Grahic Jump Location
Fig. 2

Relative errors in the phase velocity of the proposed methods based on the dynamic stiffness matrix (squares), the symplectic transfer matrix (triangles), and the mixed energy matrix (circles) for (a) κ=1000, (b) κ=10,000, (c) κ=25,000, and (d) κ=100,000 rad/m

Grahic Jump Location
Fig. 3

Dispersion curves for first ten modes of (a) SH and (b) Lamb waves

Grahic Jump Location
Fig. 4

Relative errors in the phase velocity of Lamb waves for the SAFE method with (a) Δh= 0.025, (b) Δh= 0.01, (c) Δh= 0.0025 mm, and for (d) the proposed method

Grahic Jump Location
Fig. 5

The eigenfunctions that correspond to the components (a) u and (b) w of Lamb wave for the first mode (solid line), second mode (dashed-dotted line), third mode (dashed line), and fourth mode (dotted line) at κ=5000 rad/m

Grahic Jump Location
Fig. 6

Dispersion curves for first ten modes of (a) Love and (b) Rayleigh waves

Grahic Jump Location
Fig. 7

The eigenfunctions that correspond to the components (a) u and (b) w of Rayleigh wave for the first mode (solid line), second mode (dashed-dotted line), third mode (dashed line), and fourth mode (dotted line) at κ=0.3 rad/m

Grahic Jump Location
Fig. 8

Relative errors in the phase velocity of Rayleigh waves for the SAFE method with (a) hl+1=10 m (dashed lines for Δh= 0.25 m and solid lines for Δh= 0.025 m), (b) hl+1=50 m (dashed lines for Δh= 0.25 m and solid lines for Δh= 0.025 m), and for (c) the proposed method

Grahic Jump Location
Fig. 9

Dispersion curves for first ten modes of (a) SH and (b) P-SV Stoneley waves

Grahic Jump Location
Fig. 10

The eigenfunctions that correspond to the components (a) u and (b) w of P-SV Stoneley waves for the first mode (solid line), second mode (dash-dotted line), third mode (dashed line), and fourth mode (dotted line) at κ=0.3 rad/m

Grahic Jump Location
Fig. 11

Relative errors in the phase velocity of P-SV Stoneley waves for the SAFE method with (a) h0=hl+1=10 m (dashed lines for Δh= 0.25 m and solid lines for Δh= 0.025 m), (b) h0=hl+1=50 m (dashed lines for Δh= 0.25 m and solid lines for Δh= 0.025 m), and for (c) the proposed method

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