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Research Papers

Acoustic Non-Reciprocity in Lattices With Nonlinearity, Internal Hierarchy, and Asymmetry: Computational Study

[+] Author and Article Information
Matthew D. Fronk

School of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: mfronk3@gatech.edu

Sameh Tawfick

Assistant Professor
Department of Mechanical Science and Engineering,
University of Illinois, Urbana—Champaign,
Urbana, IL 61801
e-mail: tawfick@illinois.edu

Chiara Daraio

Professor
Division of Engineering and Applied Science,
California Institute of Technology,
Pasadena, CA 91125
e-mail: daraio@caltech.edu

Shuangbao Li

College of Science,
Civil Aviation University of China,
Tianjin, China
e-mail: shuangbaoli@yeah.net

Alexander Vakakis

Professor
Department of Mechanical Science and Engineering,
University of Illinois, Urbana—Champaign,
Urbana, IL 61801
breake-mail: avakakis@illinois.edu

Michael J. Leamy

Professor
School of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: michael.leamy@me.gatech.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 January 23, 2018; final manuscript received April 29, 2019; published online June 11, 2019. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 141(5), 051011 (Jun 11, 2019) (11 pages) Paper No: VIB-18-1035; doi: 10.1115/1.4043783 History: Received January 23, 2018; Accepted April 30, 2019

Reciprocity is a property of linear, time-invariant systems whereby the energy transmission from a source to a receiver is unchanged after exchanging the source and receiver. Nonreciprocity violates this property and can be introduced to systems if time-reversal symmetry and/or parity symmetry is lost. While many studies have induced nonreciprocity by active means, i.e., odd-symmetric external biases or time variation of system properties, considerably less attention has been given to acoustical structures that passively break reciprocity. This study presents a lattice structure with strong stiffness nonlinearities, internal scale hierarchy, and asymmetry that breaks acoustic reciprocity. Macroscopically, the structure exhibits periodicity yet asymmetry exists in its unit cell design. A theoretical study, supported by experimental validation, of a two-scale unit cell has revealed that reciprocity is broken locally, i.e., within a single unit cell of the lattice. In this work, global breaking of reciprocity in the entire lattice structure is theoretically analyzed by studying wave propagation in the periodic arrangement of unit cells. Under both narrowband and broadband excitation, the structure exhibits highly asymmetrical wave propagation, and hence a global breaking of acoustic reciprocity. Interpreting the numerical results for varying impulse amplitude, as well as varying harmonic forcing amplitude and frequency/wavenumber, provides strong evidence that transient resonant capture is the driving force behind the global breaking of reciprocity in the periodic structure. In a companion work, some of the theoretical results presented herein are experimentally validated with a lattice composed of two-scale unit cells under impulsive excitation.

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Figures

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Fig. 1

Isolated unit cell of the asymmetrical, hierarchical lattice structure: (a) proposed fabrication method in which nonlinear coupling is achieved by elastomeric bumpers and (b) the equivalent spring-mass-damper model of the unit cell

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Fig. 2

Hierarchical unit cell with labeled parameter notation

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Fig. 3

Lattice structure containing unit cells with internal hierarchy, nonlinearity, and asymmetry: (a) the elastomeric bumper design is extended periodically and (b) spring-mass-damper representation of the lattice

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Fig. 4

Hierarchical unit cell consisting of two scales

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Fig. 5

Response of the two-scale system to different impulse magnitudes: (a) LS excitation and (b) SS excitation

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Fig. 6

Effect of varying dimensionless parameters on local nonreciprocity: varying Πm (a), Πk3(b), Πc1 (c), and Πc2(d). When not varied, Πm = 0.05 (b,c,d), Πk3 = 1 (a,c,d), Πc1 = 0.001 (a,b,d), and Πc2 = 0.001 (a,b,c).

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Fig. 7

Impulsive excitation of the large scale of the unit cell at the chain's center in a nonlinear (a) and linear (b) lattice. Note the preferential propagation of energy from left-to-right for the case with nonlinear interactions: mouter = 1, mmiddle = 0.05, minner = 0.005, k1 = 1, k1,middle = 0, k1,inner = 0, k3,middle = 1, k3,inner = 0.1, couter = 0.002, cmiddle = 0.002, cinner = 0.0002, I0/Inorm = 7.36. For (b), k3,middle and k3,inner govern linear restoring forces.

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Fig. 8

Response of a 1D lattice to various impulse amplitudes applied to the large scale of the unit cell at its center. Nonreciprocal behavior occurs at specific impulse magnitudes: mouter = 1, mmiddle = 0.05, minner = 0.005, k1 = 1, k1,middle = 0.05, k1,inner = 0.05, k3,middle = 1, k3,inner = 0.1, couter = 0.002, cmiddle = 0.002, cinner = 0.0002.

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Fig. 9

Impulsive excitation of the large scale of the unit cells at the chain's boundaries. Energy propagates rightward when initiated at the left boundary (a) but localizes when initiated at the right boundary (b): mouter = 1, mmiddle = 0.05, minner = 0.005, k1 = 1, k1,middle = 0.05, k1,inner = 0.05, k3,middle = 1, k3,inner = 0.1, couter = 0.002, cmiddle = 0.002, cinner = 0.0002, I0/Inorm = 0.1.

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Fig. 10

Response of a 1D lattice to various impulse amplitudes applied to the large scale of the unit cells at the left (a) and right (b) boundaries. The preferential energy propagation occurs at various impulse amplitudes: mouter = 1, mmiddle = 0.05, minner = 0.005, k1 = 1, k1,middle = 0.05, k1,inner = 0.05, k3,middle = 1, k3,inner = 0.1, couter = 0.002, cmiddle = 0.002, cinner = 0.0002.

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Fig. 11

Acoustics of the 21-unit cell nonlinear lattice for relatively low-intensity excitation of the large scale of the left cell (a,d), middle cell (b,e), and right cell (c,f). Upper plots (a,b,c) depict the variations of the instantaneous energies of selected unit cells and lower plots (d,e,f) the spatiotemporal variations of the instantaneous cell energies of the lattice: mouter = 1.0, minner = 0.05, k1 = 1.0, k3,inner = 1.0, couter = 0.002, cinner = 0.002.

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Fig. 12

Acoustics of the 21-unit cell nonlinear lattice for higher-intensity excitation of the large scale of the left cell (a,d), middle cell (b,e), and right cell (c,f). Upper plots (a,b,c) depict the variations of the instantaneous energies of selected unit cells and lower plots (d,e,f) the spatiotemporal variations of the instantaneous cell energies of the lattice: mouter = 1.0, minner = 0.05, k1 = 1.0, k3,inner = 1.0, couter = cinner = 0.002.

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Fig. 13

Effect of nesting different numbers of small scales in each unit cell on global nonreciprocity. An increased number of nested masses enhances the nonreciprocal behavior over a range of impulse amplitudes

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Fig. 14

Internal resonator design: (a) elastomeric bumper fabrication strategy and (b) equivalent spring-mass-damper representation

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Fig. 15

Response to various impulse amplitudes on the left (a) and right (b) boundaries of the internal resonator design. Nonreciprocity occurs over a finite range of amplitudes. mouter = 1, mmiddle = 0.05, minner = 0.005, k1 = 1, k3,middle = 1, k3,inner = 0.1, couter = 0.002, cmiddle = 0.002, cinner = 0.0002.

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Fig. 16

Harmonic excitation of the lattice at the large scale of the unit cell in its center. Energy primarily distributes at the right side of the forcing: mouter = 1, mmiddle = 0.05, minner = 0.005, k1 = 1, k1,middle = 0.05, k1,inner = 0.05, k3,middle = 1, k3,inner = 0.1, couter = 0.002, cmiddle = 0.002, cinner = 0.0002, F0 = 4.37, ω = 1.6.

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Fig. 17

Harmonic excitation of the lattice at the large scale of the unit cell on the left (a) and right (b) boundaries of the lattice. Propagating waves are generated from forcing at the left boundary while evanescent waves are generated from forcing at the right boundary: mouter = 1, mmiddle = 0.05, minner = 0.005, k1 = 1, k1,middle = 0.05, k1,inner = 0.05, k3,middle = 1, k3,inner = 0.1, couter = 0.002, cmiddle = 0.002, cinner = 0.0002, F0 = 6.67, ω = 1.6.

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Fig. 18

Dispersion analysis for three degree-of-freedom lattice structure. Nonreciprocal propagation occurs at the band edge when the coupling between nested masses is predominantly nonlinear. Asymmetrical propagation is evident by comparing the real part of the wavenumber for each forcing frequency (a,b,c,d) to the imaginary part (e,f) of the wavenumber for leftward and rightward waves. mouter = 1, mmiddle = 0.05, minner = 0.005, k1 = 1, k1,middle = 0.05, k1,inner = 0.05, k3,middle = 1, k3,inner = 0.1, couter = 0.002, cmiddle = 0.002, cinner = 0.0002, F0 = 6.67.

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