Wave transmission in time- and space-variant helicoidal phononic crystals

F. Li, C. Chong, J. Yang, P. G. Kevrekidis, and C. Daraio

Phys. Rev. E 90, 053201 – Published 4 November 2014

Abstract

We present a dynamically tunable mechanism of wave transmission in one-dimensional helicoidal phononic crystals in a shape similar to DNA structures. These helicoidal architectures allow slanted nonlinear contact among cylindrical constituents, and the relative torsional movements can dynamically tune the contact stiffness between neighboring cylinders. This results in cross-talking between in-plane torsional and out-of-plane longitudinal waves. We numerically demonstrate their versatile wave mixing and controllable dispersion behavior in both wavenumber and frequency domains. Based on this principle, a suggestion toward an acoustic configuration bearing parallels to a transistor is further proposed, in which longitudinal waves can be switched on and off through torsional waves.

Received 26 August 2014

DOI:https://doi.org/10.1103/PhysRevE.90.053201

Authors & Affiliations

F. Li¹, C. Chong^2,3,*, J. Yang^1,†, P. G. Kevrekidis², and C. Daraio^3,4

¹Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195-2400, USA
²Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
³Department of Mechanical and Process Engineering (D-MAVT), Swiss Federal Institute of Technology (ETH), 8092 Zurich, Switzerland
⁴Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA

^*cchong@ethz.ch
^†jkyang@aa.washington.edu

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Vol. 90, Iss. 5 — November 2014

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Figure 1
(a) Stiffness and sensitivity of the cylindrical Hertzian contact as a function of alignment angles. A schematic of the HPC composed of a 1D chain of helicoidally stacked cylindrical particles under precompression is shown in the inset. (b) The stiffness coefficient $k_{cyl}$ (solid blue line) and its single harmonic approximation based on its Taylor expansion (green markers). The parameter values in this case are $α_{0} = 10^{\circ}$ , $A_{α} = 1^{\circ}$ , and $f_{α} = 3$ kHz.
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Figure 2
(a) Superposition of (i) Floquet exponents $σ (k) / (2 π)$ (solid line) and a first-order harmonic $σ (k) / 2 π + f_{α}$ [dashed line, see Eq. (10)] versus the wavenumber $k$ [parameter values are $α_{0} = 10$ , $A_{α} = 5$ , and $f_{α} = 40$ kHz, which satisfy the stability condition Eq. (8)], and (ii) PSD [dB] of velocity component of the solution of Eq. (1) obtained by imposing a chirped pulse (0 to 30 kHz) with an amplitude of 5 nm on the left end of a resting chain. (b) PSD [arbitary units] of velocity component of the solution with the unstable parameter values $α_{0} = 10$ , $A_{α} = 5$ , and $f_{α} = 30$ kHz. The vertical dashed line corresponds to the wavenumber yielding the Floquet exponent with the largest real part. The horizontal line is the corresponding imaginary part of that exponent. The chain was excited in the same way as described for panel (a). (c) Same as described for panel (a), but with the parameter values $α_{0} = 10$ , $A_{α} = 1$ , and $f_{α} = 3$ kHz, which do not satisfy the stability condition Eq. (8). However, in this case, the wavenumbers do not fall into regions of instability of the Mathieu equation due to the finite nature of the simulations (see text), and thus the observed dynamics are stable. The chain was excited in the same way as described for panel (a).
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Figure 3
(a) Dispersion relation in the case of space-dependent torsional waves. Shown is the superposition of (i) $N = 3$ dispersion curves (solid lines) versus the wavenumber $k \in [0, π / N]$ (parameter values are $α_{0} = 10^{\circ}$ and $A_{α} = 1^{\circ}$ ) and (ii) PSD [dB] of velocity component of the solution of Eq. (1) obtained by imposing a chirped pulse (0 to 30 kHz) with an amplitude of 5 nm on the left end of a resting chain. (b)–(d) Same as described for panel (a) but with (b) $N = 5$ , (c) $N = 7$ , and (d) $N = 10$ .
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Figure 4
(a) Dispersion relation in the case of space- and time-dependent torsional waves for $α_{0} = 10^{\circ}$ , $A_{α} = 2^{\circ}$ , $N = 3$ and $f_{α} = 3$ kHz. Shown is the superposition of (i) $N = 3$ Floquet exponents (solid lines) and the first order harmonic shifts $σ (k) / 2 π \pm f_{α}$ (dashed lines), and (ii) PSD [dB] of velocity component of the solution of Eq. (1) obtained by imposing a chirped pulse (0 to 30 kHz) with an amplitude of 5 nm on the left end of a resting chain. (b) PSD [arbitrary units] of velocity component of the solution with unstable parameter values $α_{0} = 10$ , $A_{α} = 5$ , $N = 3$ and $f_{α} = 40$ kHz (note this parameter set is stable for $N = 1$ , see e.g. Fig. 2). In this case, the Floquet multiplier with the largest modulus occurs for $k \approx 0$ . The horizontal line is the imaginary part of the corresponding Floquet exponent. The chain was excited in the same way as in panel (a).
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Figure 5
(a) The transistor effect of HPCs. The black curve denotes the Gaussian input signal, while the red (blue) curve corresponds to transmitted signals under the switch off (on) state. $f_{L}$ and $f_{cutoff}$ are the input and cutoff frequencies. (b) Variation of the transmission of the acoustic wave at frequency $f_{L}$ as a function of the gate frequency $f_{α}$ .
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Figure 6
(a) PSD of the displacements $u_{n}$ of the solution shown in Fig. 2 for the wavenumber $k = 1$ . The spectral peaks corresponding to $a_{- 1}, a_{0}$ and $a_{1}$ are shown as red points. (b) The decay $a_{1} / a_{0}$ (green dashed line with markers) and $a_{- 1} / a_{0}$ (blue line with markers) versus the wavenumber $k$ . The analytical predictions from Eq. (18) (black dashed line) and Eq. (19) (solid black line) are also shown.
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