A validated lumped-element model for bioinspired acoustic flow sensing toward the performance limit

TL;DR

This work addresses how design parameters govern the performance of bioinspired acoustic flow sensors. It introduces a lumped-element model grounded in Euler–Bernoulli beam theory with impedance $R(\omega)$ to capture broadband motion of slender cantilevers in fluids and derives transfer functions $H_j(\omega)$ for mode-by-mode dynamics, ultimately predicting flow response and thermomechanical noise. Experimental validation across multiple cantilever geometries confirms accurate predictions of velocity response, noise floors, and minimum detectable signal from $100~\text{Hz}$ to $10~\text{kHz}$ in air, outperforming prior continuum models. The model highlights design levers—enhanced damping and reduced modal complexity within the sensing band—to improve sensitivity toward the thermomechanical limit, enabling high-fidelity vector-sound detection with micro- and nanoscale devices.

Abstract

Flow sensing is fundamental to both biological survival and technological innovation. Inspired by biological mechanoreceptors, artificial flow sensors detect subtle fluid motion using slender, viscous-driven structures. Among these, acoustic flow sensors that mimic nature's velocity-sensitive ears have the potential to transform vector sound detection. Yet, despite their potential, understanding of how design parameters determine ultimate sensor performance remains limited. To effectively guide flow sensor design, we develop and experimentally validate a lumped-element model that captures the broadband motion of slender microcantilevers immersed in fluid, combining analytical simplicity with quantitative accuracy. The model predicts flow-induced motion, thermomechanical noise, and the minimum detectable signal level, showing strong agreement with experimental measurements in air over a broad frequency range from 100 Hz to 10,000 Hz. This validated model provides a straightforward theoretical framework for designing high-performance micro- and nanomechanical sensors for flow and vector sound detection.

Figures (4)

• Figure 1: Acoustic flow sensing with slender microcantilevers demonstrating high fidelity and intrinsic directionality.a, Silicon nitride microcantilevers fabricated using a mass-producible semiconductor process. The top optical micrograph shows a fabricated chip (10 mm $\times$ 10 mm) with microcantilevers arranged within a central opening (5 mm $\times$ 5 mm). The bottom scanning electron micrograph shows a slender microcantilever ($L = 1000~\mu$m, $b = 2~\mu$m, $h = 780~\text{nm}$). b, Schematic of the experimental setup. The sound field was generated by a loudspeaker in an anechoic chamber. The sound pressure near the chip was measured using a calibrated pressure microphone (Mic), while the tip motion of a microcantilever was measured using a laser Doppler vibrometer (LDV). c, Time-domain waveforms and corresponding spectrograms in response to a speech signal, measured by the microphone (top) and the laser Doppler vibrometer (bottom, at the microcantilever tip shown in a). A recorded speech signal (“Hello, One, Two, Three, Four, Five”) was reproduced by a loudspeaker positioned 3 m from the chip, with sound propagating normal to the chip surface ($\theta = 0^{\circ}$). d, Directional response of the microcantilever to a 1000 Hz tone at 67 dB SPL, generated by a loudspeaker positioned 0.5 m from the chip at various sound incidence angles ($\theta$).
• Figure 2: Sound-induced velocity responses in slender microcantilevers. Sound waves were generated using loudspeakers placed 3 m from the microcantilever chip and propagated perpendicularly to the chip surface. a, Acoustic particle velocity within the chip frame at 100 Hz, obtained from finite element analysis (FEA) under an acoustic pressure of 1 Pa, corresponding to a planar-wave particle velocity of $U = 2.4~\text{mm/s}$. The inset shows the velocity field amplitude across the chip thickness along line A–B. b, Velocity amplitude profile along line A–B, as indicated in a. c, d, Velocity responses of microcantilevers with varying length ($b = 2~\mu$m) and varying width ($L = 500~\mu$m), respectively. Results from three approaches show excellent agreement: theoretical predictions from Eq. (\ref{['eq:transfer function']}) (dashed lines), experimental measurements from the laser Doppler vibrometer (solid lines), and FEA simulations (dotted lines).
• Figure 3: Thermomechanical motion in slender microcantilevers.a, b, Square root of the single-sided velocity power spectral density, $G^{1/2}_{\dot{z}\dot{z}}(f)$, for microcantilevers with varying length ($b = 2~\mu\mathrm{m}$) and varying width ($L = 500~\mu\mathrm{m}$), respectively. $G^{1/2}_{\dot{z}\dot{z}}(f)$ is related to the two-sided spectral density by $G_{\dot{z}\dot{z}}(f) = 4\pi S_{\dot{z}\dot{z}}(\omega)$. Dashed lines indicate thermal noise predictions from Eq. (\ref{['eq:Szz_dot']}), and solid lines represent experimental data measured using a laser Doppler vibrometer. The experimental system noise floor is shown in gray for reference.
• Figure 4: Minimum detectable sound limited by thermomechanical noise in slender microcantilevers.a, b, Square root of the single-sided pressure-referred thermomechanical noise, $G^{1/2}_{pp}(f)$, for microcantilevers with varying length ($b = 2~\mu\mathrm{m}$) and varying width ($L = 500~\mu\mathrm{m}$), respectively. Dashed lines indicate theoretical predictions from Eq. (\ref{['eq:Spp3']}), and solid lines represent experimental data.