Cascade of Modal Interactions in Nanomechanical Resonators with Soft Clamping

TL;DR

The paper investigates cascaded intermodal couplings in softly clamped nanostring resonators, demonstrating sequential energy transfer across five mechanical modes during frequency sweeps and yielding a broad, nearly constant amplitude response. Soft clamping enhances this cascaded energy exchange and amplifies the driven mode's effective nonlinearity by more than an order of magnitude, increasing the effective Duffing constant and flattening the amplitude–frequency curve. Analytical Harmonic Balance Method (HBM) and FE-based reduced-order models (ROMs) capture the onset conditions via analytical expressions for the two-mode onset frequency and amplitude, e.g., $f_{1,c} = rac{1}{2 extpi} abla rac{ abla$ (Note: ensure proper formatting) $f_{1,c} = rac{1}{2\pi} rac{ ext{...}}{ ext{...}}$ and $A_{1,c} = abla rac{ ext{...}}{ ext{...}}$, and validate these against FE ROMs and experiments. Specifically, $f_{1,c} = rac{1}{2\pi} abla rac{ abla}{ abla}$ and $A_{1,c} = abla rac{ abla}{ abla}$, with $f_{1,c}$ and $A_{1,c}$ provided as $f_{1,c} = rac{1}{2 extpi} rac{rac{ abla}{ abla}}{4 - rac{2 abla}{3eta_1}}$ and $A_{1,c} = abla rac{ abla - 4 abla}{3eta_1 - rac{1}{2} abla}$; these predictions align with FE ROMs and experimental observations. Experimentally and in FE-ROM simulations, cascaded intermodal interactions stabilize the driven-mode amplitude over broad frequency ranges and enable programmable multistability, with soft clamping providing design flexibility to tailor eigenfrequency ratios and achieving higher $Q$ factors.

Abstract

We uncover a chain of nonlinear modal interactions in softly clamped nanostring resonators. The process involves the sequential coupling of five mechanical modes, during frequency sweeps, yielding a broad nonlinear response with nearly constant amplitude. We demonstrate that soft clamping enables this cascaded energy transfer and amplifies the effective geometric nonlinearity of the driven mode by an order of magnitude. Analytical and finite element-based reduced-order models capture the key features of the coupling cascade and clarify its underlying mechanism. The phenomenon is generic in nonlinear vibrational systems and can be tailored through soft-clamping design strategies.

Figures (4)

• Figure 1: Measurement of mode coupling in a nanomechanical string resonator with soft-clamping supports. (a) Schematic of the measurement set-up comprising an MSA400 laser Doppler vibrometer (LDV) for reading out the motion at different harmonics of the drive frequency ($f$, 2$f$, ..., $n$$f$) and a piezo-actuator for generating the excitation. (b) The geometric design parameters of a Si3N4 nanostring resonator with soft-clamping supports. (c) Duffing nonlinear response curves of the first mode of the device with $L_{\rm s} = \qty[]{50}{\micro\metre}$, under different drive levels without mode coupling. (d) Nonlinear response curves of the same device under a stronger drive level ($U_{\rm exc} = \qty[]{6}{\volt}$). The second (yellow) and third (ochre) modes are both activated by mode coupling. The red lines in (c) and (d) (first panel) are the backbone curves of the first mode.
• Figure 2: Different response branches of the first mode determined by its coupling to the second mode. (a) Measured response curves by driving the first mode under different $U_{\rm exc}$ and $\Delta f$, showing frequency demodulation around $f$ (blue, first mode) and $2f$ (yellow, second mode). (b) Simulated response curves based on an FE-based ROM, showing the directly driven first mode near $f$ (blue) and the coupling-induced second mode around $2f$ (yellow). The gray area marks the frequency range where the second mode is activated.
• Figure 3: Influence of the soft-clamping supports on the coupled dynamics of the lowest two modes. (a) Measured response curves of the lowest two modes of string resonators with three different $L_{\rm s}$, showing frequency demodulation around $f$ (blue, first mode) and $2f$ (yellow, second mode). The colors of the curves gradually fade as $L_{\rm s}$ decreases. The SEM image shows the measured devices (colored in blue). The white bar is $\qty[]{100}{\micro\metre}$. (b) Simulated response curves using the FE-based ROMs of devices with varying $L_{\rm s}$, showing the directly driven first mode near $f$ (blue) and the coupling-induced second mode around $2f$ (yellow). The upward and downward hollow triangles represent the onset frequency of the coupled mode ($f_{\rm 1,c}/f_1$) and the corresponding amplitude ($A_{\rm 1,c}/h$), respectively, as predicted by Eq. \ref{['equation.5']}. The purple line traces the onsets of modal interactions, corresponding to the kinks in the blue curves. (c)(d) Comparison of the simulated (hollow triangles) and measured (solid triangles) onset amplitude $A_{\rm 1,c}/h$ and frequency $f_{\rm 1,c}/f_1$. The insets define $f_{\rm 1,c}/f_1$ and $A_{\rm 1,c}/h$, respectively.
• Figure 4: Cascaded interactions in a nanostring. (a) The red line denotes the driven mode’s effective backbone, with higher modes triggered at its kinks via dispersive coupling. (b) Measured response curves under five-mode couplings in the device with $L_{\rm s} = \qty[]{50}{\micro\metre}$. The bold blue line is the frequency response demodulated by the drive frequency $f$, while the others represent the signals demodulated by 2$f$ (yellow), 3$f$ (ochre), 4$f$ (cyan) and 5$f$ (purple), respectively. The mode shapes from FE analysis are shown on the right. The red line is the fitted backbone curve of the first mode before mode couplings initiate.