Collective vibrational resonance and mode selection in nonlinear resonator arrays

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 $\mathit{\text{van}}\mathit{\text{der}}\mathit{\text{Pol-Duffing}}$ chain of oscillators with nearest neighbor coupling, a systematic two-widely separated time scale expansion ($\mathit{\text{Direct}}\mathit{\text{Partition}}\mathit{\text{of}}\mathit{\text{Motion}}$) 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.