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

We address two aspects of the dynamics of the forced Duffing oscillator which are relevant to the technology of micromechanical devices and, at the same time, have intrinsic significance to the field of nonlinear oscillating systems. First, we study the stability of periodic motion when the phase shift between the external force and the oscillation is controlled -contrary to the standard case, where the control parameter is the frequency of the force. Phase-shift control is the operational configuration under which self-sustained oscillators -and, in particular, micromechanical oscillators- provide a frequency reference useful for time keeping. We show that, contrary to the standard forced Duffing oscillator, under phase-shift control oscillations are stable over the whole resonance curve. Second, we analyze a model for the internal resonance between the main Duffing oscillation mode and a higher-harmonic mode of a vibrating solid bar clamped at its two ends. We focus on the stabilization of the oscillation frequency when the resonance takes place, and present preliminary experimental results that illustrate the phenomenon. This synchronization process has been proposed to counteract the undesirable frequency-amplitude interdependence in nonlinear time-keeping micromechanical devices

Wake states and frequency selection of a streamwise oscillating cylinder

This paper presents the results of an in-depth study of the flow past a streamwise oscillating cylinder, examining the impact of varying the amplitude and frequency of the oscillation, and the Reynolds number of the incoming flow. These findings are presented in a framework that shows that the relationship between the frequency of vortex shedding fs and the amplitude of oscillation A* is governed by two primary factors: the first is a reduction of fs proportional to a series in A*2 over a wide range of driving frequencies and Reynolds numbers; the second is nonlinear synchronization when this adjusted fs is in the vicinity of N = (1 - fs/fd)-1, where N is an integer. Typically, the influence of higher-order terms is small, and truncation to the first term of the series (A*2) well represents the overall trend of vortex shedding frequency as a function of amplitude. However, discontinuous steps are overlaid on this trend due to the nonlinear synchronization. When fs is normalized by the Strouhal frequency fSt (the frequency of vortex shedding from an unperturbed cylinder), the rate at which fs/fSt decreases with amplitude, at least for fd/fSt = 1, shows a linear dependence on the Reynolds number. For a fixed Re = 175, the truncated series shows that the rate of decrease of fs/fSt with amplitude varies as (2 - fd/fSt)-1/2 for 1 < or egal fd/fSt < or egal 2, but is essentially independent of fd/fSt for fd/fSt < 1. These trends of the rate of decrease of fs with respect to amplitude are also used to predict the amplitudes of oscillation around which synchronization occurs. These predicted amplitudes are shown to fall in regions of the parameter space where synchronized modes occur. Further, for the case of varying fd/fSt, a very reasonable prediction of the amplitude of oscillation required for the onset of synchronization to the mode where fs = 0.5fd is given. In a similar manner, amplitudes at which fs = 0 are calculated, predicting where the natural vortex shedding is completely supplanted by the forcing. These amplitudes are found to coincide approximately with those at which the onset of a symmetric vortex shedding mode is observed. This result is interpreted as meaning that the symmetric shedding mode occurs when the dynamics crosses over from being dominated by the vortex shedding to being dominated by the forcing

Numerical simulation of Faraday waves oscillated by two-frequency forcing

We perform a numerical simulation of Faraday waves forced with two-frequency oscillations using a level-set method with Lagrangian-particle corrections (particle level-set method). After validating the simulation with the linear stability analysis, we show that square, hexagonal and rhomboidal patterns are reproduced in agreement with the laboratory experiments [Arbell and Fineberg, Phys. Rev. Lett. 84, 654 (2000) and Phys. Rev. Lett. 85, 756 (2000)]. We also show that the particle level-set's high degree of conservation of volume is necessary in the simulations. The numerical results of the rhomboidal states are compared with weakly nonlinear analysis. Difficulty in simulating other patterns of the two-frequency forced Faraday waves is discussed.Comment: 20 pages, 12 figure

Euler flow predictions for an oscillating cascade using a high resolution wave-split scheme

A compressible flow code that can predict the nonlinear unsteady aerodynamics associated with transonic flows over oscillating cascades is developed and validated. The code solves the two dimensional, unsteady Euler equations using a time-marching, flux-difference splitting scheme. The unsteady pressures and forces can be determined for arbitrary input motions, although only harmonic pitching and plunging motions are addressed. The code solves the flow equations on a H-grid which is allowed to deform with the airfoil motion. Predictions are presented for both flat plate cascades and loaded airfoil cascades. Results are compared to flat plate theory and experimental data. Predictions are also presented for several oscillating cascades with strong normal shocks where the pitching amplitudes, cascade geometry and interblade phase angles are varied to investigate nonlinear behavior