Nonlinear Modal Interactions in Clamped-Clamped Mechanical Resonators

A theoretical and experimental investigation is presented on the intermodal coupling between the flexural vibration modes of a single clamped-clamped beam. Nonlinear coupling allows an arbitrary flexural mode to be used as a self-detector for the amplitude of another mode, presenting a method to measure the energy stored in a specific resonance mode. The observed complex nonlinear dynamics are quantitatively captured by a model based on coupling of the modes via the beam extension; the same mechanism is responsible for the well-known Duffing nonlinearity in clamped-clamped beams.

Figure 1

(a) Setup, with a colored scanning electron micrograph of the resonator beam. DSP is the digital signal processor, $R$ is used to balance the bridge and set to $22\Omega$ and A is the amplifier. (b) Frequency responses of the third mode (amplitude $|A_3|$) for different drive amplitudes of the first mode $|A_1|$ on resonance. The drive current of mode 3 is $I_3=1.5\mathrm{mA}$. Inset: beam shapes of the first (solid) and third mode (dashed) for increasing amplitude of the first mode. (c) Resonance frequency of mode 3 ($f_{R,3}$) as a function of $|A_1|$: measurements (black squares) and model (red line). For small drive currents a quadratic dependence is observed (dashed blue line). Inset: schematic of the resonator showing the beam shape of the first (blue) and third (red) mode. (d) On a different device, the resonance frequency of the first mode is used to detect the second mode. Inset: beam shape of the first (blue) and second (red) mode. The error bars are within data markers.

Figure 2

Frequency-frequency response of the simultaneously driven and detected first and third mode. The drive frequency of the third mode is swept, while driving the first mode at a fixed frequency. After each sweep the frequency of the first mode is increased. The amplitudes $|A_1|$ (a) and $|A_3|$ (b) are recorded. The driving currents are $I_1=0.8\mathrm{mA}$ and $I_3=7.0\mathrm{mA}$. Red indicates a high amplitude and blue corresponds to a low-amplitude response. Simulations are shown in (c) and (d). The frequency response is plotted for the amplitude of mode 1 (e) and mode 3 (f) at $f_1=48\mathrm{kHz}$.

Figure 3

(a) Frequency-frequency response for the forward sweep of the third mode. Red indicates a high amplitude and blue corresponds to a low amplitude. On resonance of the first mode, the third mode is linear and off-resonance it is nonlinear. (b) Forward (black lines) and back (red lines) sweeps below resonance, on resonance and above resonance, with $f_1=45$, 52 and 55 kHz, respectively. Driving currents are $I_1=2\mathrm{mA}$ and $I_3=8\mathrm{mA}$. (c) Result of the simulation with parameters extracted from the experiment. (d) Simulated responses for the situation in (b).