Nonlinear behaviour of self-excited microcantilevers in viscous fluids

Abstract

Microcantilevers are increasingly being used to create sensitive sensors for rheology and mass sensing at the micro- and nano-scale. When operating in viscous liquids, the low quality factor of such resonant structures, translating to poor signal-to-noise ratio, is often manipulated by exploiting feedback strategies. However, the presence of feedback introduces poorly-understood dynamical behaviours that may severely degrade the sensor performance and reliability. In this paper, the dynamical behaviour of self-excited microcantilevers vibrating in viscous fluids is characterized experimentally and two complementary modelling approaches are proposed to explain and predict the behaviour of the closed-loop system. In particular, the delay introduced in the feedback loop is shown to cause surprising non-linear phenomena consisting of shifts and sudden-jumps of the oscillation frequency. The proposed dynamical models also suggest strategies for controlling such undesired phenomena.

Measuring viscosity with nonlinear self-excited microcantilevers

“…In particular, it has been highlighted that the microcantilever dynamic response is strongly affected by the delay present in the feedback loop. [12][13][14] In this work, it is shown how the nonlinear dynamics of a cantilever embedded in a feedback loop with an adjustable phase-shifter can be used as a high-sensitivity or threshold rheological sensor. The frequencies of the oscillations in the feedback loop are studied as a function of the viscosity of different water-glycerol solutions and the delay that is introduced in the loop by the phase-shifter.…”

“…These oscillations result from the balance between the feedback gain, which constantly amplifies the motion of the cantilever (inducing unstable exponentially growing trajectories), and the nonlinear saturation, which limits the system trajectories and stabilizes the system dynamics on stable self-sustained oscillations with a certain frequency and amplitude. 5,14 The frequency and amplitude of the oscillation guarantee that the gain of the loop is unitary and that the total phase shift around the feedback loop is an integer multiple of 2p radians. 14,15 The total delay s tot [indicated in Fig.…”

“…5,14 The frequency and amplitude of the oscillation guarantee that the gain of the loop is unitary and that the total phase shift around the feedback loop is an integer multiple of 2p radians. 14,15 The total delay s tot [indicated in Fig. 1(a)] accounts for the intrinsic delay due to the propagation of the electrical signals across the electronic components and to elastic waves in the cantilever and probe holder materials.…”

“…The total phase shift around the feedback loop must be an integer multiple of 2p radians for self-sustained oscillations to exist. 14,15 Therefore, the cantilever will naturally adjust its phase (and hence its oscillation frequency) to compensate for the overall phase imposed by the adjustable phase-shifter, the fixed polarity, and the intrinsic total delay of the setup.…”

“…In this case, the oscillation frequency is a noninjective function of fluid viscosity, as, for example, the purple and green lines show two possible oscillation frequencies for each value of viscosity (black and red circles, respectively). However, as discussed in the supplementary material, the Nyquist Stability Criterion states that the stable solution has the highest real part of the transfer function G jx ð Þ ¼ pCT jx ð ÞPS jx ð Þe Àjxs tot : 14,20 The real part of this transfer function is calculated numerically and plotted within the inset of Fig. 4(d), showing that for R 2 ¼ 9 kX (purple line, red circles), the stable solution occurs at 59 kHz, while for R 2 ¼ 10 kX (green line, black circles), the stable solution is at 33 kHz.…”