Parametric Instability in Discrete Models of Spatiotemporally Modulated Materials

TL;DR

This work analyzes parametric instability in a 1-D discrete model of spatiotemporally modulated materials using Floquet theory and a perturbation framework. It shows that unstable modulation frequencies (UMFs) occur at combinations of two unmodulated natural frequencies divided by a natural number $\beta$, with $\beta=1$ causing inevitable instability in the undamped case and higher $\beta$ depending on the modulation amplitude. The analysis demonstrates that spatial modulation via the phase $\phi$ profoundly shapes stability, while damping sets finite thresholds that can suppress many unstable tongues, especially in longer systems. The results yield practical guidance for designing stable, high-amplitude, slowly modulated devices by exploiting wide stable regions and phase control in the stability diagrams.

Abstract

We investigate the phenomenon of parametric instability in discrete models of spatiotemporally modulated materials. These materials are celebrated in part because they exhibit nonreciprocal transmission characteristics. However, parametric instability may occur for strong modulations, or occasionally even at very small modulation amplitudes, and prevent the safe operation of spatiotemporally modulated devices due to an exponential growth in the response amplitude. We use Floquet theory to conduct a detailed computational investigation of parametric instability. We explore the roles of modulation parameters (frequency, amplitude, wavenumber), the number of modulated units, and damping on the stability of the system. We highlight the pivotal role of spatial modulation in parametric instability, a feature that is predominantly overlooked in this context. We use the perturbation method to obtain analytical expressions for modulation frequencies at which the response becomes unstable. We hope that our findings enable and inspire new applications of spatiotemporally modulated materials that operate at higher amplitudes.

Figures (14)

• Figure 1: Schematic representation of the modulated system with $n$ DoF.
• Figure 2: Displacements of the first masses in two modulated systems with $K_c=0.6$, $\phi=0.5\pi$ and $\zeta=0$. (a) $n=3$, $\Omega_m=2.6$ and $K_m=0.2$; this scenario falls inside a stable region in Fig. \ref{['fig_STB_1']}(b). (b) $n=5$, $\Omega_m=2.6$ and $K_m=0.1$; this scenario falls inside an unstable region in Fig. \ref{['fig_STB_1']}(b). The initial conditions for both examples are: $\dot{x}_n\left(0\right)=0.1$, $x_n\left(0\right)=x_p\left(0\right)=\dot{x}_p\left(0\right)=0$ for $1\leq p \leq n-1$.
• Figure 3: Stability diagrams for $K_c=0.6$, $\phi=0.5\pi$, $\zeta=0$ and different numbers of modulated units: (a) $n=2$, (b) $n=3$ and (c) $n=5$. Grey regions represent unstable response, white regions represent stable response. Red dashed lines indicate UMFs obtained from perturbation analysis.
• Figure 4: Stability diagrams for $K_c=0.6$ and $\phi=0.5\pi$. (a) $n=2$, (b) $n=3$ and (c) $n=5$. Red dashed lines indicate UMFs with $\beta \geq 2$, predicted by perturbation analysis.
• Figure 5: Stability diagrams for $K_c=0.6$ and $\phi=0.5\pi$. (a) $n=2$, (b) $n=3$ and (c) $n=5$. Red dashed lines indicate the UMFs predicted by perturbation analysis.