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Research Papers

An Investigation of Vibrational Power Flow in One-Dimensional Dissipative Phononic Structures

[+] Author and Article Information
H. Al Ba'ba'a

Mechanical and Aerospace
Engineering Department,
University at Buffalo (SUNY),
Buffalo, NY 14260

M. Nouh

Mechanical and Aerospace
Engineering Department,
University at Buffalo (SUNY),
Buffalo, NY 14260
e-mail: mnouh@buffalo.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 28, 2016; final manuscript received October 5, 2016; published online February 3, 2017. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 139(2), 021003 (Feb 03, 2017) (10 pages) Paper No: VIB-16-1148; doi: 10.1115/1.4035108 History: Received March 28, 2016; Revised October 05, 2016

Owing to their ability to block propagating waves at certain frequencies, phononic materials of self-repeating cells are widely appealing for acoustic mitigation and vibration suppression applications. The stop band behavior achieved via Bragg scattering in phononic media is most commonly evaluated using wave propagation models which predict gaps in the dispersion relations of the individual unit cells for a given frequency range. These models are in many ways limited when analyzing phononic structures with dissipative constituents and need further adjustments to account for viscous damping given by complex elastic moduli and frequency-dependent loss factors. A new approach is presented which relies on evaluating structural intensity parameters, such as the active vibrational power flow in finite phononic structures. It is shown that the steady-state spatial propagation of vibrational power flow initiated by an external disturbance reflects the wave propagation pattern in the phononic medium and can thus be reverse engineered to numerically predict the stop band frequencies for different degrees of damping via a stop band index (SBI). The treatment is shown to be very effective for phononic structures with viscoelastic components and provides a clear distinction between Bragg scattering effects and wave attenuation due to material damping. Since the approach is integrated with finite element methods, the presented analysis can be extended to two-dimensional lattices with complex geometries and multiple material constituents.

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Figures

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Fig. 1

Periodicity in phononic structures achieved via discontinuities in (a) material, (b) geometric, and (c) boundary conditions

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Fig. 2

(a) Cantilevered 1D phononic rod of ten self-repeating two-material cells and (b) axial deformation and forces at the cell boundaries and at the material discontinuity

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Fig. 3

Eigenvalues of the transfer matrix [Tc] capturing the band structure of the phononic rod unit cell (top). The frequency response of the attenuation constant α obtained using an analytical solution for [Tc], an finite element methods solution of 20 elements per cell, and for a uniform rod showing the lack of phononic stop bands (bottom).

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Fig. 4

Frequency response of the axial deformation of the free end of the phononic rod compared to a uniform rod (of material A only)

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Fig. 5

Wave propagation parameters of a unit cell of a viscously damped phononic rod

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Fig. 6

2DOF spring–mass–damper system

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Fig. 7

(a) Active and (b) reactive vibrational power components for a 2DOF spring–mass–damper system

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Fig. 8

Total (a) active and (b) reactive vibrational power versus frequency for the phononic and uniform rods

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Fig. 9

(a) Active and (b) reactive vibrational power versus frequency at the first and tenth cells for the phononic and uniform rods

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Fig. 10

Variation of active vibrational power along the length of a uniform ((a)–(c)) and a phononic ((d)–(f)) rod at 3, 4.2, and 5.8 kHz excitations

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Fig. 11

Active power flow maps (top) and mode shapes of axial deformation (bottom) for a one-material uniform rod at the (a) first, (b) second, and (c) third vibration modes

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Fig. 12

Active power flow maps (top) and mode shapes of axial deformation (bottom) for a phononic rod at the (a) first, (b) second, and (c) third vibration modes

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Fig. 13

Active power flow maps for a uniform and a phononic rod at different frequencies spanning the 3–5.8 kHz range

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Fig. 14

α versus frequency and the corresponding active power flow maps at (a) 3, (b) 3.075, and (c) 3.1 kHz for a phononic rod

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Fig. 15

SBI of an undamped phononic rod

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Fig. 16

Frequency response of the axial deformation of the free end of a phononic rod with a frequency-dependent damping compared to a uniform rod (top). Active power flow maps for the phononic rod at (a) 3, (b) 4.4, and (c) 5.8 kHz.

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Fig. 17

Frequency response of the axial deformation of the free end of a uniform rod and a damped phononic rod with different loss factors

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Fig. 18

Damping effect on the active power flow propagation in a phononic rod during pass band (3 kHz) and stop band (3.6 kHz) frequencies

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Fig. 19

Active power flow propagation in a damped phononic rod (ηB=0.025) with a midpoint excitation at a pass band (3 kHz) and a stop band (3.6 kHz) frequency. The (×) marker indicates the location of the exciting force.

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