Collective vibrational resonance and mode selection in nonlinear resonator arrays

TL;DR

The paper addresses active spectral control in a 1D array of weakly nonlinear resonators by applying a uniform high-frequency drive that renormalizes onsite stiffness to $\tilde{\omega}_n^2$ and damping to $\gamma_{\mathrm{eff}}$, thereby reshaping the dispersion relation and moving normal modes into the parametric-resonance window. It combines Blekhman's direct partition of motion with a two-time-scale expansion to derive slow envelope dynamics and a complex amplitude flow for modal amplitudes, predicting a HF-driven dispersion shift that selects which normal modes enter resonance. Analytical results provide explicit relations such as $\gamma_{\mathrm{eff}} = \gamma + \frac{\eta f_n^2}{2}$ and $\tilde{\omega}_n^2 = \omega_0^2 + \frac{3}{2}\epsilon\alpha f_n^2$, along with a dispersion condition $\tilde{\omega}_n^2 = \omega_p^2/4 + 2\epsilon d\sin^2(q_m/2)$ that together with the mode-index relation $m(g_{\text{res}})$ explains selective excitation, all validated by numerical simulations. The findings demonstrate a practical, site-independent method to steer energy into bulk modes, with potential impact on programmable phononic crystals and MEMS/NEMS devices.

Abstract

This article investigates how a uniform high frequency (HF) drive applied to each site of a weakly-coupled discrete nonlinear resonator array can modulate the onsite natural stiffness and damping and thereby facilitate the active tunability of the nonlinear response and the phonon dispersion relation externally. Starting from a canonical model of parametrically excited \textit{van der Pol-Duffing} chain of oscillators with nearest neighbor coupling, a systematic two-widely separated time scale expansion (\textit{Direct Partition of Motion}) has been employed, in the backdrop of Blekhman's perturbation scheme. This procedure eliminates the fast scale and yields the effective collective dynamics of the array with renormalized stiffness and damping, modified by the high-frequency drive. The resulting dispersion shift controls which normal modes enter the parametric resonance window, allowing highly selective activation of specific bulk modes through external HF tuning. The collective resonant response to the parametric excitation and mode-selection by the HF drive has been analyzed and validated by detailed numerical simulations. The results offer a straightforward, experimentally tractable route to active control of response and channelize energy through selective mode activation in MEMS/NEMS arrays and related resonator platforms.

Figures (5)

• Figure 1: A schematic model of a coupled parametrically excited van der Pol-Duffing resonator array with fixed ends. Each site is excited by an additive high-frequency drive $g_n\cos(\Omega t)$.
• Figure 2: FIG. 2. HF–driven tuning of collective mode selection in the nonlinear resonator array. (a) Variation of the resonant mode index $m$ with the HF resonant drive amplitude $g_{res}$ by Eq. \ref{['eq:g_peak']}. Also describing the minimum HF threshold $g_{th}$ required to activate bulk modes for different drive frequencies $\Omega$. (b) Mode-index colormap in the $g_{res}\text{--}\Omega$ plane by Eq. \ref{['eq:mode_index']}, showing how the HF-induced dispersion shift selectively brings the specific modes into the parametric resonance window. (c) Dependence of the resonant mode number on the HF resonant amplitude for arrays of different sizes $N$, showing that the percentile mode position $m/N$ remains nearly invariant across system sizes, highlighting the collective nature of mode selection.The parameters are taken as $\epsilon = 0.1;\omega_0= 0.4;\omega_p= 1.0;d= 2.0;\alpha= 1.0;\gamma=2.0;\eta=2.0;h=3.9$.
• Figure 3: Response intensity $a_1$ and $a_2$ as a function of HF drive strength $g$ for two oscillators $N=2$ is shown.The stable and unstable single-mode solutions are obtained from Eqs. \ref{['eq:a_1_single_mode']} and \ref{['eq:a_2_single_mode']}. The parameters are fixed at $\epsilon = 0.1;\omega_0= 0.4;\omega_p= 1.0;d= 2.0;\alpha= 1.0;\gamma=2.0;\eta=2.0;h=6.0$.
• Figure 4: The collective parametric response in the nonlinear resonator array ($N=120$) as a function of the high-frequency drive amplitude ($g$) is validated numerically. (a) RMS intensity (I) against $g$. The critical hysteretic region is depicted in the plot. While the blue line (up sweep) jumps up at the threshold ($g_{up}=g_{res} \approx 290$),the down sweep occurs at nearly $g=130$.(b) Shows the energy intensity localized in each normal mode ($m$) throughout the $g$ sweep. At near resonant drive $g_{res}=290$, the response is highly selective, confining almost all energy to a narrow band of bulk propagating modes ($m \approx 20-25$), confirming the close approximation of mode tuning mechanism through Eq.\ref{['eq:mode_index']}. (c) The bifurcation plot of the mean displacement of the central node ($n=60$) is non-zero, suggesting the existence of jump stability during up sweep near $g_{res}=290$.The dynamically oscillating state is represented by the area between the UP and DOWN curves. (d) The amplitude distribution at the maximum response peak $g_{res}=290$ across all lattice sites ($n$). The near-uniform amplitude validates the collective nature of the resonance by confirming the selective excitation of a bulk propagating mode ($m=31$) with minimal boundary effects. The other related parameters are used as:$\epsilon = 0.1;\omega_0= 0.4;\omega_p= 1.0;d= 2.0;\alpha= 1.0;\gamma=2.0;\eta=2.0;h=3.9$
• Figure 5: The collective parametric response in the nonlinear resonator array ($N=60$) as a function of the high-frequency drive amplitude ($g$) is validated numerically. (a) RMS intensity (I) against $g$. The critical hysteretic region is depicted in the plot. While the blue line (up sweep) jumps up at the threshold ($g_{up}=g_{res} \approx 290$),the down sweep occurs at nearly $g=130$.(b) Shows the energy intensity localized in each normal mode ($m$) throughout the $g$ sweep. At near resonant drive $g_{res}=290$, the response is highly selective, confining almost all energy to a narrow band of bulk propagating modes ($m \approx 10-15$), confirming the close approximation of mode tuning mechanism through Eq.\ref{['eq:mode_index']}. (c) The bifurcation plot of the mean displacement of the central node ($n=30$) is non-zero, suggesting the existence of jump stability during the up sweep near $g_{res}=290$. The dynamically oscillating state is represented by the area between the UP and DOWN curves. (d) The amplitude distribution at the maximum response peak $g_{res}=290$ across all lattice sites ($n$). The near-uniform amplitude validates the collective nature of the resonance by confirming the selective excitation of a bulk propagating mode ($m=16$) with minimal boundary effects. The other related parameters are used as:$\epsilon = 0.1;\omega_0= 0.4;\omega_p= 1.0;d= 2.0;\alpha= 1.0;\gamma=2.0;\eta=2.0;h=3.9$