Dynamics of time-modulated, nonlinear phononic lattices

The propagation of acoustic and elastic waves in time-varying, spatially homogeneous media can exhibit different phenomena when compared to traditional spatially varying, temporally homogeneous media. In the present work, the response of a one-dimensional phononic lattice with time-periodic elastic properties is studied with experimental, numerical and theoretical approaches in both linear and nonlinear regimes. The system consists of repelling magnetic masses with grounding stiffness controlled by electrical coils driven with electrical signals that vary periodically in time. For small-amplitude excitation, in agreement with linear theoretical predictions, wave-number band gaps emerge. The underlying instabilities associated to the wave-number band gaps are investigated with Floquet theory and the resulting parametric amplification is observed in both theory and experiments. In contrast to genuinely linear systems, large-amplitude responses are stabilized via the nonlinear nature of the magnetic interactions of the system, and results in a family of nonlinear time-periodic states. The bifurcation structure of the periodic states is studied. It is found the linear theory correctly predicts parameter values from which the time-periodic states bifurcate from the zero state. In the presence of an external drive, the parametric amplification induced by the wave-number band gap can lead to bounded and stable responses that are temporally quasiperiodic. Controlling the propagation of acoustic and elastic waves by balancing nonlinearity and external modulation offers a new dimension in the realization of advanced signal processing and telecommunication devices. For example, it could enable time-varying, cross-frequency operation, mode- and frequency-conversion, and signal-to-noise ratio enhancements.

Figure 1

(a) Photo of the experimental apparatus, with electromagnetic coils corresponding to grounding springs (modulating coils) and ring magnets inside each coil (not fully visible) sliding freely on a low-friction rod. (b) Schematic of the mass-spring lattice model with nonlinear coupling stiffness $k_c$, time-varying grounding stiffness $k_g\left(t\right)$, and viscous damping $c$; right panel shows the harmonic modulation form of the time-varying grounding stiffness. (c) Force-distance measurement (blue line) [15] and fit (red dashed line) of repulsive magnetic force between neighboring masses (d) Measured (blue shaded region) and numerically simulated (red dashed lines) nodal velocity envelope, used for viscous damping parameter fitting by matching spatial decay.

Figure 2

Dispersion relation and transmission response in unmodulated and modulated lattice at $f_{\mathrm{mod}}=40$ Hz. (a) Numerical dispersion reconstruction for the unmodulated lattice. The analytical predication is shown by the red curve. (b) Numerical dispersion reconstruction with modulation frequency $f_{\mathrm{mod}}=40$ Hz. The real part (red curve) and imaginary part (blue curve) of the analytical approximation is also shown. (c) Expanded view of the band gap from the gray dashed window in panel (b). (d) Experimentally measured dispersion reconstruction for the unmodulated lattice. (e) Experimentally measured dispersion reconstruction with modulation frequency $f_{\mathrm{mod}}=40$ Hz. The frequency $f_{\mathrm{mod}}/2=20$ Hz is indicated by the gray dash-dotted line. The arrow highlights amplitude peak in dispersion branch. (f) Numerically simulated frequency transmission spectra for the unmodulated lattice (black dashed curve) and modulated lattice (red curve) with $f_{\mathrm{mod}}=40$ Hz. The frequency $f_{\mathrm{mod}}/2=20$ Hz is indicated by the gray dash-dotted line. (g) Same as panel (f), but for the experimentally measured frequency transmission spectra.

Figure 3

Stability of modulation parameters. (a) The black curve shows the analytical prediction of the stability boundary based on condition (15). Shaded region indicates unstable solutions for modulation parameter combinations as determined by the Floquet analysis. The white circles indicate parameter combinations for which the fully nonlinear simulation exhibits a nondecaying, modulation-driven response to an initial impulse. The black star indicates the parameters shown in panel (b). (b) Numerically simulated velocity output time series for the parameters indicated by the black star in panel (a). The fully nonlinear simulation (black line) has a bounded response, while the linearized simulation (gray line) exhibits exponential growth. (c) Experimental nondecaying parameter combinations (white squares). The black curve shows the analytical prediction of the stability boundary based on condition Eq. (15). Shaded region indicates unstable solutions for modulation parameter combinations as determined by the Floquet analysis.

Figure 4

Time-periodic solutions of Eq. (1) with $f_{\mathrm{mod}}=41.6$ Hz and $A_{\mathrm{mod}}=70$ $\mathrm{N}{\mathrm{m}}^{-1}$. (a) Spatial profile of a stable solution. The inset shows the Floquet multiplers. (b) Spatial profile of an unstable stable solution. (c) Intensity plot of the spatiotemporal evolution of the solution shown in panel (a) for one period of motion (color intensity corresponds to displacement). (d) Intensity plot of the spatiotemporal evolution of the solution shown in panel (b).

Figure 5

(a) Bifurcation diagram with $f_{\mathrm{mod}}=41.6$ Hz fixed showing how time-periodic states bifurcate from the zero state. (b) The frequency response of the experimental lattice for fixed amplitude harmonic driving and increasing harmonic modulation. At a critical amplitude, the lattice transitions from a driving-dominated response to a high-amplitude, modulation-dominated response.

Figure 6

Nonlinear lattice dynamics (a) Fourier amplitudes for $f_{dr}$ (blue/squares) and $f_{\mathrm{mod}}/2$ [black (triangles)] vs modulation amplitude for experimental (markers) and numerical simulation (lines). Note that error bars are also shown. (b) Hysteresis of $f_{\mathrm{mod}}/2$ Fourier component around mode transition, with slowly increasing (black triangles) and decreasing (gray upside-down triangles) modulation amplitude. Numerical simulation shown in dashed lines. (c) Poincaré section of the output response of the numerical simulation, with sampling period $T=1/f_{dr}$. Pre- and postforward sweep transition modulation amplitude responses are shown as blue squares and black triangles, respectively. (d) Same as panel (c) for the experiment.

Figure 7

Bifurcation diagram with $f_{\mathrm{mod}}=41.6$ Hz fixed showing how time-periodic states bifurcate from the zero state in the fully modulated lattice.

Figure 8

(a), (b) Forward and backward sweep for driving and modulation frequency combinations that are not (a) and are (b) rational multiples. (c), (d) Poincaré sections for low (black triangles) and high (blue squares) amplitude regime with same frequency combinations as (a) and (b), respectively.