Duffing revisited: phase-shift control and internal resonance in self-sustained oscillators

Arroyo, Sebastián I.; Zanette, Damián H.

doi:10.1140/epjb/e2015-60517-3

Cited by 24 publications

(26 citation statements)

References 20 publications

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“…1. This is the so-called backbone approximation to the resonance curve [7,9]. Within this approximation, the time delay as a function of the frequency along the backbone curve is…”

Section: Further Approximations I Backbone Approximationmentioning

confidence: 99%

Self-sustained oscillations with delayed velocity feedback

Zanette

2017

Pap. Phys.

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We study a model for a nonlinear mechanical oscillator, relevant to the dynamics of microand nanomechanical time-keeping devices, where periodic motion is sustained by a feedback force proportional to the oscillation velocity. Specifically, we focus our attention on the effect of a time delay in the feedback loop, assumed to originate in the electric circuit that creates and injects the self-sustaining force. Stationary oscillating solutions to the equation of motion, whose stability is insured by the crucial role of nonlinearity, are analytically obtained through suitable approximations. We show that a delay within the order of the oscillation period can suppress self-sustained oscillations. Numerical solutions are used to validate the analytical approximations.

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“…1. This is the so-called backbone approximation to the resonance curve [7,9]. Within this approximation, the time delay as a function of the frequency along the backbone curve is…”

Section: Further Approximations I Backbone Approximationmentioning

confidence: 99%

Self-sustained oscillations with delayed velocity feedback

Zanette

2017

Pap. Phys.

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“…(2) can be found through perturbation theory [24]. However, in contrast with previous treatments with weak nonlinearity [20], we do not assume that the cubic force is small as compared to the linear term [25]. The resulting zeroth-order equation is therefore the nonlinear Duffing equation without damping:ẍ 0 + x 0 + 4 3 βx 3 0 = 0.…”

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confidence: 99%

Self-Sustained Micromechanical Oscillator with Linear Feedback

et al. 2016

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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...

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“…(5). In the literature on nonlinear oscillating systems, the limit of weak damping/forcing is also known as the backbone approximation [10,8]. As in the previous section, the following results correspond to the case where oscillators 1 and 2 have, respectively, hardening and softening nonlinearity (β 2 < 0 < β 1 ), with ω 1 < ω 2 .…”

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confidence: 88%

Frequency stabilization by synchronization of Duffing oscillators

Zanette

2016

EPL

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We present analytical and numerical results on the joint dynamics of two coupled Duffing oscillators with nonlinearity of opposite signs (hardening and softening). In particular, we focus on the existence and stability of synchronized oscillations where the frequency is independent of the amplitude. In this regime, the amplitude-frequency interdependence (a-f effect) -a noxious consequence of nonlinearity, which jeopardizes the use of micromechanical oscillators in the design of time-keeping devices-is suppressed. By means of a multiple time scale formulation, we find approximate conditions under which frequency stabilization is achieved, characterize the stability of the resulting oscillations, and compare with numerical solutions to the equations of motion.

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Duffing revisited: phase-shift control and internal resonance in self-sustained oscillators

Cited by 24 publications

References 20 publications

Self-sustained oscillations with delayed velocity feedback

Self-sustained oscillations with delayed velocity feedback

Self-Sustained Micromechanical Oscillator with Linear Feedback

Frequency stabilization by synchronization of Duffing oscillators

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