Self-Sustained Micromechanical Oscillator with Linear Feedback

Abstract

Autonomous oscillators, such as clocks and lasers, produce periodic signals without any external frequency reference. In order to sustain stable periodic motions, there needs to be external energy supply as well as nonlinearity built into the oscillator to regulate the amplitude. Usually, nonlinearity is provided by the sustaining feedback mechanism, which also supplies energy, whereas the constituent resonator that determines the output frequency stays linear. Here we propose a new self-sustaining scheme that relies on the nonlinearity originating from the resonator itself to limit the oscillation amplitude, while the feedback remains linear. We introduce a model to describe the working principle of the self-sustained oscillations and validate it with experiments performed on a nonlinear microelectromechanical (MEMS) based oscillator.Autonomous oscillators are systems that can spontaneously commence and maintain stable periodic signals in a self-sustained manner without external frequency references. They are abundant both in Nature and in manmade devices. In Nature made systems, the self-sustained oscillators are the fundamental piece that describes systems as diverse as neurons, cardiac tissue, and predator-prey relationships [1]. In manmade devices, self-sustained autonomous oscillators are overwhelmingly used for communications, timing, computation, and sensing [2], with examples such as quartz watches [3] and laser sources [4]. A typical oscillator consists of a resonating component and a sustaining feedback element: the constituent resonator determines the oscillation frequency, whereas the feedback system draws power from an external source to compensate the energy loss due to damping during each oscillation of the resonator [5]. In order to initiate the oscillations, the initial gain of the feedback must be larger than unity, so that energy accumulates to build up oscillation amplitude [6]. However, to avoid ever increasing oscillations, some limiting mechanism must act to ensure that, eventually, the vibrational amplitude no longer grows.In the conventional designs of oscillators, the resonating element is operated in the linear regime, where its resonant frequency is independent of the excitation levels, and the necessary amplitude limiting mechanism is enacted in the feedback loop by introducing a nonlinear element (Fig. 1a). However, maintaining the resonating element in the linear regime has been challenging for a variety of applications requiring self-sustained oscillators made from micro-/nanoelectromechanical (M/NEMS) resonators [7][8][9], mostly because these resonators exhibit significantly reduced linear dynamic range. To limit the amplitude, common mechanisms include impulsive energy replenishment [5], saturated gain medium [4,10] or amplifiers [11,12], automatic level control [13,14], phase locked loops [15,16], nonlinear signal transduction [17], and dedicated nonlinear components [18]. These mechanisms to incorporate nonlinear elements into the electronic feedback circuitry introduce tec...

Self-sustained oscillations with delayed velocity feedback

“…The key ingredient that makes oscillations stable is a nonlinear dependence of damping with the oscillation amplitude, such that energy dissipation increases or decreases when the amplitude respectively grows or drops. This kind of nonlinearity, together with a cubic component in the restoring force, has been experimentally observed to occur in c-c beam micromechanical oscillators [6], which can therefore exhibit self-sustained motion under the action of linear velocity feedback. Our emphasis was put on the effect of a time delay in the feedback force, assumed to originate in the electric circuit that reads, conditions, and reinjects the oscillation signal.…”

“…The former is a Van der Pol-like nonlinearity, while the cubic contribution to the restoring force defines a Duffing oscillator [7]. Both kinds of nonlinearity, with α, β > 0, have been experimentally verified to occur in micromechanical oscillators formed by silica beams clamped at their two ends (clampedclamped, or c-c beams [4,6]). In the main oscillation mode, c-c beams vibrate much like a plucked string, so that x(t) can be associated with the displacement of the middle point of the beam with respect to its rest position.…”

“…In a series of recent experiments on micromechanical oscillators, it has been shown that selfsustained oscillations can be achieved with a feedback force proportional to the oscillation velocity [6]. Since a purely linear mechanical system cannot display stable periodic motion, this kind of feedback force must necessarily be compensated by some nonlinear contribution from the oscillator dynamics itself.…”

Energy exchange between coupled mechanical oscillators: linear regimes

“…On the other hand, devices such as pacemakers and certain kinds of sensors base their functioning on the maintenance of oscillatory motion for long periods at low power consumption, which demands small dissipation rates [5][6][7]. In the laboratory, observation of energy dissipation in a mechanical system is a standard tool to disclose the ingredients that shape its behavior, in particular, the occurrence of nonlinearity [8][9][10][11].…”

Resonance tracking in a micromechanical device using phononic frequency combs

“…Nonetheless, thermal noise shaping within the resonator feedback loop often dominates oscillator frequency stabilities over short-to-medium integration times with noise sources comprising both mechanical and electrical sources. In recent years, approaches to specifically engineer nonlinearities in a feedback oscillator or coupled oscillator configuration have yielded striking improvements in phase noise and frequency stability of micro/nanomechanical resonator oscillators [14][15][16][17][18][19].Here in this paper, using a recently established phononic frequency comb pathway, we demonstrate an alternative approach towards micro/nanomechanical resonant frequency tracking with benefits to frequency stabilities observed relative to the standard feedback configuration. Phononic frequency combs are produced via nonlinear interactions between a driven phonon mode and one or more additional parametrically excited modes [20][21].The Fermi-Pasta-Ulam (FPU) framework has been previously employed as a basis for describing phononic frequency comb formation [20][21].…”