Nonlinear dynamics of a microelectromechanical oscillator with delayed feedback

Abstract

We study the dynamics of a nonlinear electromechanical oscillator with delayed feedback. Compared to their linear counterparts, we find that the dynamics is dramatically different. The well-known Barkhausen stability criterion ceases to exist, and two modes of operation emerge: one characterized by hysteresis in combination with a bistable frequency and amplitude; the other, by self-stabilization of the oscillation frequency and amplitude. The observed features are captured by a model based on a Duffing equation with delayed force feedback. Nonlinear oscillators with delayed force feedback are exemplary for a large class of dynamic systems. Oscillators are ubiquitous in nature and engineering, with implementations in physical, life, and social sciences. [1][2][3] Typical components of an oscillator are a resonant system and a positive feedback loop. Oscillators are self-sustained and produce an ac output signal from a dc input. Compared to a resonator, which is driven by an ac signal, an oscillator exhibits a reduced line width and improved phase-noise performance. These properties make them interesting as sensitive detectors and as timing references. A common example is the quartz crystal oscillator, with widespread application in electronic circuits.In current applications the oscillating element is typically linear, implying that the displacement is proportional to the driving force as in Hooke's law. In a linear system, displacement-proportional feedback modifies the spring constant, whereas velocity-proportional feedback modifies the damping. Stability for the latter systems is bounded by the well-known Barkhausen criterion. 4,5 It states that a selfsustained oscillation occurs when the phase shift around the loop is an integer multiple of 2π , and the loop gain exceeds the value k/Q, where k is the spring constant and Q the quality factor without feedback. Although the linear regime has been studied in great detail, much less is known about nonlinear oscillating elements. In a nonlinear oscillator, the oscillating frequency depends on the amplitude of the motion. Micro-and nanoscale electromechanical devices exhibit strong nonlinearity, which makes them ideal devices to study such systems.6-8 We demonstrate that in the presence of a delayed force feedback, the nonlinearity gives rise to intricate dynamic behavior near the oscillation threshold.To construct a strongly nonlinear oscillator, we use a doubly clamped micromechanical beam with a high aspect ratio as the oscillating element. Figure 1(a) shows the device, with dimensions L × w × h = 750 × 8 × 0.5 μm 3 , fabricated from silicon nitride using standard microfabrication processes.9 An integrated piezoelectric actuator enables periodic driving of the beam, as well as static tuning of its resonance frequency. The motion of the beam is probed with pm/ √ Hz sensitivity using an optical deflection technique, by, is shown in Fig. 1(b), reflecting a laser beam off the surface of the device onto a position-sensitive photodetector. 10 The experiments are condu...

Nonlinear behaviour of self-excited microcantilevers in viscous fluids

“…In this case, the oscillator will work in a bistable state [33]. This was also observed in figure 4, where, as the jump approached, the presence of the high-frequency motion started to be visible (green line).…”

“…The single-degree-of-freedom equation of motion that describes the vibration of the cantilever immersed in a viscous fluid in the feedback loop is therefore given by [33] (13) where the dot stands for time derivative and F(t) is the forcing term from the dither piezo. m CT = LWTρ c is the total mass of the beam, where T and ρ c are, respectively, the thickness of the beam and the density of the constituent material, while c i = 2πf 0i m CT /Q i , k i = (2πf 0i ) 2 m CT and y i are the intrinsic damping coefficient, stiffness and displacement associated to the i-th resonance mode, respectively.…”

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.…”

A Versatile Mass-Sensing Platform With Tunable Nonlinear Self-Excited Microcantilevers

“…One of the main drawbacks of using feedback loops to improve sensing performance is the presence of different sources of nonlinearities introduced, for example, by the nonlinear electronic components required to process the signals or even by the intrinsic mechanical nonlinearities of the resonator. The analysis of the dynamics of microresonators in presence of mechanical nonlinearities has shown interesting and surprising phenomena, such as stable operation of the resonators far beyond the critical vibration amplitude [24], [25] or bistable regimes [26], [27]. It was also shown that the frequency of oscillation is strongly dependent on the delay affecting the feedback signal [27]- [29].…”