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Research Papers

Analysis of Damped Bloch Waves by the Rayleigh Perturbation Method

[+] Author and Article Information
A. Srikantha Phani

Department of Mechanical Engineering,
The University of British Columbia,
2069-6250 Applied Science Lane,
Vancouver, BC V6T 1Z4, Canada
e-mail: srikanth@mech.ubc.ca

Mahmoud I. Hussein

Department of Aerospace Engineering Sciences,
University of Colorado Boulder,
Boulder, CO 80309
e-mail: mih@colorado.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 5, 2012; final manuscript received April 29, 2013; published online June 6, 2013. Assoc. Editor: Massimo Ruzzene.

J. Vib. Acoust 135(4), 041014 (Jun 06, 2013) (11 pages) Paper No: VIB-12-1139; doi: 10.1115/1.4024397 History: Received May 05, 2012; Revised April 29, 2013

Bloch waves in viscously damped periodic material and structural systems are analyzed using a perturbation method originally developed by Rayleigh for vibration analysis of finite structures. The extended method, called the Bloch–Rayleigh perturbation method here, utilizes the Bloch waves of an undamped unit cell as basis functions to provide approximate closed-form expressions for the complex eigenvalues and eigenvectors of the damped unit cell. In doing so, we circumvent the solution of a quadratic Bloch eigenvalue problem and subsequent computationally intensive transformation to first order/state-space form. Dispersion curves of a one-dimensional damped spring-mass chain and a two-dimensional phononic crystal with square inclusions are calculated using the state-space method and the proposed method. They are compared and found to be in excellent quantitative agreement for both proportional and nonproportional viscous damping models. The perturbation method is able to capture anomalous dispersion phenomena—branch overtaking, branch cut-on/cut-off, and frequency contour transformation—in parametric ranges where state-space formulations encounter numerical issues. Generalization to other linear nonviscous damping models is permissible.

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Figures

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Fig. 1

A finite two degree-of-freedom system

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Fig. 2

An infinite two degree-of-freedom system; the unit cell is shown in the box

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Fig. 3

Dispersion curves for the stiffness-proportional damped chain at relatively low damping levels: Bloch–Rayleigh perturbation method versus state-space method

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Fig. 4

Dispersion curves for the stiffness-proportional damped chain at relatively high damping levels

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Fig. 5

Dispersion curves for the chain with general viscous damping

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Fig. 6

(a) Unit cell of a two-phase phononic crystal with square-lattice symmetry, and (b) the corresponding first Brillouin zone with irreducible Brillouin zone and high symmetry points marked

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Fig. 7

Dispersion curves for the two-phase phononic crystal with different stiffness-proportional damping levels. The acoustic and optical branches are indicated by the solid and dashed lines, respectively.

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Fig. 8

Dispersion curves for the two-phase phononic crystal with different mass-proportional damping levels. The acoustic and optical branches are indicated by the solid and dashed lines, respectively.

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Fig. 9

Definition of the wavenumber bandgaps for stiffness-proportional (left) and mass-proportional (right) damping for a phononic crystal with square-lattice symmetry

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Fig. 10

Frequency (top row) and wavenumber (bottom row) bandgap maps of the proportionally damped two-phase phononic crystal with the damping matrix C = pM + qK. The wavenumber band gaps are evaluated in a manner as defined in Fig. 9.

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Fig. 11

Evolution of the acoustic branch dispersion surfaces for the two-phase phononic crystal with damping. The constants p and q, respectively, denote the mass- and stiffness-proportional damping parameters.

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Fig. 12

Dispersion curves for the two-phase phononic crystal with general viscous damping. Nonproportionality is introduced by changing the Lamé constants λdamp and μdamp in the definition of the C matrix in comparison to the λ and μ values of a proportionally damped system with the parameters p = 0 and q = 0.05. Here, μdamp is varied for each case presented while setting λdamp=0.6λ.

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Fig. 13

Dispersion curves for the two-phase phononic crystal with general viscous damping. Nonproportionality is introduced by changing the Lamé constants λdamp and μdamp in the definition of the C matrix in comparison to the λ and μ values of a proportionally damped system with the parameters p = 0 and q = 0.05. Here, λdamp is varied for each case presented while setting μdamp=1.4μ.

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