Hysteresis suppression for primary and subharmonic 3:1 resonances using fast excitation

Nonlinear Dynamics - We analyze the effect of a fast harmonic excitation on hysteresis and on entrainment area in a forced van der Pol–Duffing oscillator near the primary and the 3:1...     

Quasi-Periodic Solutions and Stability for a Weakly Damped Nonlinear Quasi-Periodic Mathieu Equation

Quasi-Periodic (QP) solutions are investigated for a weakly dampednonlinear QP Mathieu equation. A double parametric primary resonance(1:2, 1:2) is considered. To approximate QP solutions, a double multiple-scales method is applied to transform the original QP oscillator to anautonomous system performing two successive reductions. In a first step,the multiple-scales method is applied to the original equation to derive afirst reduced differential amplitude-phase system having periodiccomponents. The stability of stationary solutions of this reduced systemis analyzed. In a second step, the multiple-scales method is applied again tothe first reduced system (RS) to obtain a second autonomous differentialamplitude-phase RS. The problem for approximating QP solutions of theoriginal system is then transformed to the study of stationary regimesof the induced autonomous second RS. Explicit analytical approximationsto QP solutions are obtained and comparisons to numerical integrationare provided.     

Effect of quasi-periodic gravitational modulation on the convective instability in Hele-Shaw cell

The aim of the present paper is to examine the effect of a quasi-periodic gravitational modulation on the onset of convective instability in Hele-Shaw cell. The quasi-periodic modulation considered here consists in a modulation having two incommensurate frequencies. This study is an extension of a previous work by Aniss et al. [Asymptotic study of the convective parametric instability in Hele-Shaw cell, Phys. Fluids 12 (2) (2000) 262-268] in which only a periodic gravitational modulation was considered. We have shown that for Pr=O(1) or Pr?1, the gravitational modulation has no effect on the convective threshold as expected. However, for Pr=O(ε²), it turns out that a modulation with two incommensurate frequencies has a stabilizing or a destabilizing effect strongly depending on the frequencies ratio.     

Quasi-periodic bursters and chaotic dynamics in a shallow arch subject to a fast–slow parametric excitation

Nonlinear Dynamics - In this paper, various nonlinear dynamics of a one-degree-of-freedom shallow arch model are investigated. The arch is subject to an imposed displacement of its support that is...     

Predicting Homoclinic Bifurcations in Planar Autonomous Systems

An analytical method to predict the homoclinic bifurcation in a planar autonomous self-excited weakly nonlinear oscillator is presented. The method is mainly based on the collision between the periodic orbit undergoing the homoclinic bifurcation and the saddle fixed point. To illustrate the analytical predictive criteria, two typical examples are investigated. The results obtained in this work are then compared to Melnikov's technique and to a previous criterion based on the vanishing of the frequency. Numerical simulations are also provided.     

Energy harvesting in a Mathieu–van der Pol–Duffing MEMS device using time delay

This paper investigates quasi-periodic vibration-based energy harvesting in a delayed nonlinear MEMS device consisting of a delayed Mathieu–van der Pol–Duffing type oscillator coupled to a delayed piezoelectric coupling mechanism. We use the multiple scales method to approximate the quasi-periodic response and the related power output near the principal parametric resonance. The effect of time delay on the energy harvesting performance is studied. It is shown that for appropriate combination of time delay parameters, there exists an optimum range of excitation frequency beyond the resonance where quasi-periodic vibration-based energy harvesting is maximum. Numerical simulations are performed to confirm the analytical predictions.     

Suppression of hysteresis in a forced van der Pol–Duffing oscillator

This paper examines the suppression of hysteresis in a forced nonlinear self-sustained oscillator near the fundamental resonance. The suppression is studied by applying a rapid forcing on the oscillator. Analytical treatment based on perturbation analysis is performed to capture the entrainment zone, the quasiperiodic modulation domain and the hysteresis area as well. The analysis leads to a strategy for the suppression of hysteresis occurring between 1:1 frequency-locked motion and quasiperiodic response. This hysteresis suppression causes the disappearance of nonlinear effects leading to a smooth transition between the quasiperiodic and the frequency-locked responses. Specifically, it appears that a rapid forcing introduces additional apparent nonlinear stiffness which can effectively suppress hysteresis in a certain range of the rapid excitation frequency. This work was motivated by the important issue of controlling and eliminating hysteresis often undesirable in mechanical systems, in general, and in application to microscale devices, especially.     

Periodic and quasiperiodic galloping of a wind-excited tower under external excitation

The galloping of tall structures excited by steady and unsteady wind may be periodic or quasiperiodic (QP) with amplitudes having the same order of magnitude. While the onset of periodic and QP galloping was studied, their control on the other hand has received less attention. In this paper, we conduct analytical study on the effect of a fast harmonic excitation on the onset of periodic and QP galloping in the presence of steady and unsteady wind. We consider the cases where the unsteady wind activates either external excitation, parametric one or both. A perturbation analysis is performed to obtain close expressions of QP solution and the corresponding modulation envelopes. We show that at various loading situations, the periodic and QP galloping onset is significantly influenced by the amplitude of the fast external excitation. In the case where the unsteady wind activates parametric excitation, the QP galloping occurs with higher frequency modulation compared to the case where the unsteady wind activates external excitation. In the case where external and parametric excitations are activated simultaneously, fast harmonic excitation eliminates bistability in the amplitude response and gives rise to a new small QP modulation envelope.     

Using delayed damping to minimize transmitted vibrations

In Mokni et al. [Mokni L, Belhaq M, Lakrad F. Effect of fast parametric viscous damping excitation on vibration isolation in sdof systems. Commun Nonlinear Sci Numer Simulat 2011;16:1720-1724], it was shown that in a single degree of freedom system a fast nonlinear parametric damping enhances vibration isolation with respect to the case where the nonlinear damping is time-independent. The present work proposes additional enhancement of vibration isolation using delayed nonlinear damping. Attention is focused on assessing the contribution of a delayed nonlinear damping over a fast parametric damping in terms of minimizing transmissibility. The results show that a nonlinear damping with delay greatly improves vibration isolation.