Effective quality factor of mechanical resonators under complex-frequency excitations

TL;DR

This work tackles the challenge of improving detection sensitivity and resolution in resonant systems by boosting the effective quality factor $Q_{eff}$ without altering structural properties. It introduces complex-frequency excitations, and shows that via a transform $x(t)=e^{-\omega_i t} z(t)$ the response maps to a harmonic steady state, with an optimal decay $\omega_i^* = \zeta \omega_n$ that narrows the amplitude–frequency curve toward an undamped limit. The theory identifies complex poles $\omega^* = \omega_r^*+i\omega_i^*$, and extends from SDOF to MDOF systems, with numerical verification and experimental demonstrations on aluminum and acrylic cantilevers. Experiments show $Q_{eff}$ enhancements up to 54x in acrylic and 4.5x in aluminum, validating the method's practical feasibility and its potential to enhance sensing in SHM and MEMS/NEMS without structural changes. The approach opens pathways for higher-resolution measurements while maintaining bounded actual displacements even at the transfer function poles.

Abstract

We investigate the dynamics of mechanical resonators subject to excitations comprising of an oscillating or harmonic part, whose amplitude decays exponentially in time. We call these complex frequency excitations and show that the resulting response is quasi-steady, i.e. after an appropriate transform, the response of the new variable corresponds to the steady state behavior under a harmonic excitation. A procedure is presented to determine the amplitude-frequency response and effective quality factor based on this steady-state behavior. Optimal excitations are identified for both single and multi-degree of freedom systems that result in the amplitude-frequency response approaching that of an undamped system. The feasibility of the proposed method is verified through numerical simulations. Experiments with cantilever beams made of acrylic show a 54-fold increase in the effective quality factor. Our method does not involve any structural modifications and opens avenues for improving detection sensitivity in nondestructive testing and enhancing resolution in micro- and nano-electromechanical sensors.

Figures (12)

• Figure 1: Overview of proposed concept. Under harmonic excitations, an under-damped system's frequency response curve is wide. Under complex‑frequency excitations, the curve becomes narrower, approaching that of an undamped system.
• Figure 2: Complex-frequency excitation procedure and framework. Operator $T$ modifies the target excitation $F_{ori}(t)$ and an excitation $F(t)$ is applied to the system. Its actual displacement is $x(t)$; applying operator $T^{-1}$ yields the target displacement $z(t)=T^{-1}[x(t)]$. Fourier transforms of $z(t)$ and $F_{ori}(t)$ are used to determine the frequency response and $Q_{eff}$.
• Figure 3: Numerical simulation results: (a) normalized amplitude–frequency response curves and (b) $Q_{eff}$ of the SDOF system. The response curves progressively sharpen as $\omega_i$ gets closer to $\omega_i^*$. $Q_{eff}$ attains its peak value precisely at $\omega_i=\omega_i^*$.
• Figure 4: Computed $Q_{eff}$ at $\ (\omega_i = \omega_i^*)$ versus total simulation time $T_{final}$. Effective quality factor $Q_{eff}$ at the optimal decay exponent ($\omega_i=\omega_i^{*}$) increases with $T_{final}$, indicating that $Q_{eff}\to\infty$ as $T_{final} \to \infty$, similar to an undamped resonator.
• Figure 5: Actual displacement $x$ from transient simulation. The Hilbert envelope of $x(t)$ confirms the analytical bounds: maximum actual amplitude $X_{\max}\approx0.368$ mm occurs at $t_{\max}=0.2\;$s, consistent with $X_{\max}=F_{0}/(2e m\omega_r^{*}\omega_i^{*})$ and $t_{\max}=1/\omega_i^{*}$.