Tunable Acoustic Nonreciprocity in Strongly Nonlinear Waveguides with Asymmetry

Alireza Mojahed, Jonathan Bunyan, Sameh Tawfick, and Alexander F. Vakakis

Phys. Rev. Applied 12, 034033 – Published 17 September 2019

Abstract

In linear acoustics, the reciprocal behavior of waves traveling in periodic materials can be manipulated by imposing external or configurational biases to the system. However, the nonreciprocity observed in linear systems is energetically expensive. We demonstrate that strongly nonlinear, asymmetric lattices can be designed to exhibit strong nonreciprocity that is passively adaptive and tunable. An alternative class of waveguides consisting of two coupled nonlinear lattices, one stiffer than the other, allows wave propagation preferentially in one direction at certain energy ranges. This “giant” nonreciprocal behavior is obtained passively by tuning the propagation zones of these lattices in the frequency-energy domain. We present numerical simulations corroborated by experiments to show an instance of this alternative class of nonlinear waveguides. Specifically, at low input energy, wave packets generated by an applied impulse at the lateral boundaries of the waveguide are blocked at the interface of the two lattices. However, at intermediate energy ranges, wave packets initiated at the free boundary of the softer lattice propagate through the waveguide, whereas wave packets initiated at the free boundary of the stiffer lattice are blocked at the interface. The nonreciprocal acoustics persist until at the critical level of input energy, above which waves propagate in both directions within the waveguide. The range of energy over which nonreciprocal wave transmission occurs is passively tunable by appropriately manipulating the nonlinear propagation zones of the lattices in the frequency-energy domain. The nonreciprocity concept is applicable to materials and systems capable of exhibiting strongly nonlinear behavior and can find broad applications in cases where passive targeted (directed) energy transfer in space and/or frequency is a desired outcome. For example, nonlinear nonreciprocal metamaterials can be used in passive acoustic isolation designs with the capacity for unidirectional sound transmission, thus eliminating their “acoustic signature”; in ultrasonics, to yield better wave focusing at preferential frequencies, and thus improved signal-to-noise ratios; in shock isolation systems, for example, by rapid nonreciprocal low-to-high nonlinear energy transfers, yielding fast structural response attenuation; or in networks of coupled oscillators enabling passive, irreversible energy transmission in preferential directions. Clearly, such capabilities for passive nonreciprocity are not attainable in linear systems.

Received 2 April 2019
Revised 3 June 2019
Corrected 10 July 2020

DOI:https://doi.org/10.1103/PhysRevApplied.12.034033

Physics Subject Headings (PhySH)

Collective dynamics Coupled oscillators Nonlinear acoustics Nonlinear waveguides Ultrasonics

Diodes

T-symmetry

Resonance techniques

Nonlinear Dynamics

Corrections

10 July 2020

Correction: Equation (1) contained an error and has been fixed.

Authors & Affiliations

Alireza Mojahed^*, Jonathan Bunyan^†, Sameh Tawfick^‡, and Alexander F. Vakakis^§

Department of Mechanical Science and Engineering, University of Illinois, Urbana, Illinois 61801, USA

^*mojahed2@illinois.edu
^†jbunyan2@illinois.edu
^‡tawfick@illinois.edu
^§avakakis@illinois.edu

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Vol. 12, Iss. 3 — September 2019

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Images

Figure 1
Nonreciprocal 10-cell waveguide composed of two dissimilar lattices: (a) Reduced order model of the waveguide with uniform nonlinear coupling elements—the “stiff” lattice is composed of cells 1 to 5 with springs and viscous dampers k_g_,1 and d_g_,1, respectively, whereas the “soft” lattice consists of cells 6 to 10 with parameters k_g_,2 and d_g_,2. (b) Breather transmission in the propagation zones (PZs) of the two lattices, with the breather frequency band represented by arrows for weak (blue), critical (green), and strong (orange) impulsive excitation applied to the free end of the stiff (darker shade) or soft (lighter shade) lattice—forks in the arrows indicate successful initiation of a wave packet in the PZ of the downstream lattice. (c-e) Schematic representation of breather propagation in the waveguide for weak (c), critical (d), and strong (e) excitation, respectively—darker colored arrows denote stiff-soft, and lighter colored arrows soft-stiff direction of propagation.
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Figure 2
Computational spatio-temporal evolution of the total instantaneous energy in the waveguide for an impulse applied to the stiff (left column) and soft (right column) lattice, with red squares indicating the cell where the impulsive force, measured from the experiments, is applied, and darker shades corresponding to higher energy levels. (a) Weak impulses—16.75 N impulse amplitude for the stiff and 15.64 N for the soft lattice. (b) Critical-energy impulses—31.96 N for the stiff and 31.47 N for the soft lattice. (c) Strong impulses—55.18 N for the stiff and 53.04 N for the soft lattice (for clarity and to account for viscous dissipation, at each time instant, the energy is normalized with respect to the maximum instantaneous energy at that time instant).
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Figure 3
Waveguide composed of 10 unit cells: (a) Schematic top view of unit cells 4-7 (top) and experimental realization (bottom); the waveguide is excited with a modal hammer fitted with a force transducer, and the response of each unit cell is measured by means of ten accelerometers attached to the cells. (b) Detailed isometric view of the schematic design and the experimental realization of the first unit cell showing the aluminum mass (yellow) grounded to a rigid frame (gray) through linear flexure springs (green) whose stiffness is tuned through the geometry of the flexures; adjacent unit cells are coupled through thin wires (black) attached by clamps (red) and bolts (purple) —this provides the essentially nonlinear stiffness. (c) Schematic representation of the force-extension relationships of the different stiffness elements of the waveguide—stiff linear grounding k_g_,1 (dotted line), soft linear grounding k_g_,2 (dashed line), and nonlinear coupling k_nl (solid line), which are responsible for the frequency-energy tunability of the two PZs of the constituent lattices of the waveguide.
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Figure 4
Experimental spatio-temporal evolution of the total instantaneous energy in the waveguide for an impulse applied to the stiff (left column) and soft (right column) lattices, with red squares indicating the cell where the impulse is applied, and darker shades corresponding to higher energy levels. (a) Weak impulses, (b) critical-energy impulses, and (c) strong impulses—in all cases the impulse magnitudes are identical to those of the numerical simulations of Fig. 2 (for clarity and to account for viscous dissipation, at each time instant, the energy of each cell is normalized with respect to the maximum instantaneous energy of the waveguide at that time instant).
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Figure 5
Transient frequency content of the experimentally obtained velocities, one upstream and two downstream cells relative to the interface—cells 5-6-7 in the case of stiff-soft and cells 6-5-4 in the case of soft-stiff. Cell numbers are on the top right corner of each plot. Horizontal lines indicate the lower boundaries of the PZ of the stiff (black dashed) and soft (gray dash-dotted) lattices. Darker regions on the color map indicate a larger amplitude. Red triangles point to the frequency content in the primary wave as it propagates through each cell. (a) weak excitations of 16.75 N and 15.64 N applied to the stiff and soft lattices, respectively. (b) Critical excitations of 31.96 N and 31.47 N applied to the stiff and soft lattices, respectively. (c) Strong excitations of 55.18 N and 53.04 N applied to the stiff and soft lattices, respectively.
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Physical Review Applied