Self-sustained oscillations with delayed velocity feedback

Abstract

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.

Effects of noise on the internal resonance of a nonlinear oscillator

“…We recognize the overall leaning shape of the Duffing resonance peak 11 . However, in contrast with the standard harmonically–forced Duffing oscillator, the present system does not have a low–amplitude solution for high frequencies 18 . This is a consequence of the linear dependence of the self–sustaining force on the velocity.…”

“…As a model for the interaction between two oscillation modes of a clamped–clamped (c–c) beam, we consider two linearly coupled one–dimensional oscillators, described by coordinates x 1 ( t ) and x 2 ( t ) 6 . The first oscillator is subjected to a self–sustaining force proportional to its own velocity at a delayed time, , and is affected by a cubic (Duffing 11 ) nonlinearity in the restoring force and a quadratic (van der Pol 17 ) nonlinearity in the damping force 18 , 19 . This oscillator represent the main oscillation mode of the c–c beam.…”

“…In our case, this force is proportional to the oscillation velocity delayed by a prescribed time. Such kind of feedback is able to generate self–sustained stationary oscillations if a van der Pol–like nonlinearity 17 is also present in the damping force 18 , 19 . The reinjection of the self–sustaining force allows the mechanical oscillator to attain stable periodic motion, characteristic of a wide class of dynamical systems spanning from plasma physics 20 to neurosciences 21 .…”

Self-Excited Dynamics of Discrete-Time Lur'e Systems

“…Lur'e systems have been widely studied in the classical literature on stability theory [3]. Within the context of self-excited systems, Lur'e systems under various assumptions are considered in [4,5,6,7,8,9,1,10,11,12]. Self-oscillating discrete-time systems are considered in [13,14].…”