This study focuses on the bifurcation analysis of a Duffing-Van der Pol (DVDP) oscillator with rigid barriers, subjected to additive white noise excitations. This involves mapping the non-smooth vibro-impact system on a continuous phase plane through non-smooth variable transformations. The formulation considers the general case of asymmetrically positioned both sided rigid barriers. The probabilistic description of the system dynamics is subsequently defined through the corresponding Fokker-Planck (FP) equation. The stationary probability density function of the response is numerically obtained by solving the FP equation using finite element method. The stochastic stability of the vibro-impact DVPD oscillator is examined both in terms of D- and P-bifurcations. © 2016 Elsevier Ltd

Sayan Gupta

Department of Applied Mechanics

Bifurcation (mathematics)

Circuit oscillations

FOKKER PLANCK EQUATION

Lyapunov methods

Oscillators (mechanical)

Probability density function

Stochastic systems

Vibrators

White noise

DUFFING-VAN DER POL OSCILLATOR

FOKKER PLANCK

P-BIFURCATION

VARIABLE TRANSFORMATION

Vibro-impact

finite element method

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IIT Madras

International Journal of Mechanical Sciences

Wind tunnel tests on a NACA 0012 airfoil have been carried out to study the transition in aeroelastic response from an initial state characterised by low-amplitude aperiodic fluctuations to aeroelastic flutter when the system exhibits limit cycle oscillations. An analysis of the aeroelastic measurements reveals multi-fractal characteristics in the pre-flutter regime. This has not been studied in the literature. As the flow velocity approaches the flutter velocity from below, a gradual loss in multi-fractality is observed. Measures based on the generalised Hurst exponents are developed and are shown to have the potential to warn against impending aeroelastic flutter. The results of this study could be useful for health monitoring of aeroelastic structures. © 2016 Elsevier Ltd

Fulltext

Journal of Sound and Vibration

Multi-fractality in aeroelastic response as a precursor to flutter

This study carried out investigations on the bifurcation characteristics of a Duffing-Van der Pol (DVDP) oscillator subjected to white noise excitations. Dynamical (or D-) bifurcations are characterised by dramatic changes in the dynamical behaviour leading to topological changes in the phase portrait. An additional mode of bifurcation - phenomenological (or P-) bifurcation, is observed in stochastically excited systems when the stationary joint probability density function of the state variables undergo topological changes in the probability space. While D-bifurcation analysis is quantified in terms of the sign changes in the largest Lyapunov exponent, P-bifurcation analysis is usually qualitative and through visual inspection. In this study, a new quantitative measure for P-bifurcations based on the Shannon entropy is proposed. A comparison of the parameter regimes of the noisy DVDP oscillator identified by the three bifurcation methodologies have been presented. © 2016 Elsevier Ltd. All rights reserved.

Probabilistic Engineering Mechanics

Investigations on the bifurcation of a noisy Duffing-Van der Pol oscillator

The aeroelastic response of a NACA 0012 airfoil in the flow regimes prior to flutter is investigated in a wind tunnel. We observe intermittent bursts of periodic oscillations in the pitch and plunge response, that appear in an irregular manner from a background of relatively lower amplitude aperiodic fluctuations. As the flow speed is increased, the intermittent bursts last longer in time until eventually transitioning to a fully developed periodic response, indicating the onset of flutter. The repeating patterns in the measured response are visualized using recurrence plots. We show that statistics of the recurrence states extracted from these plots can be used to develop model-free precursors that forewarn an impending transition to flutter, well before its onset. © 2015 Elsevier Ltd.

Journal of Fluids and Structures

Precursors to flutter instability by an intermittency route: A model free approach

The stochastic bifurcations in a vibro-impact Duffing–Van der Pol oscillator, subjected to white noise excitations, are investigated. Bifurcations in noisy systems occur either due to topological changes in the phase space—known as D-bifurcations—or due to topological changes associated with the stochastic attractors—known as P-bifurcations. In either case, the singularities in the phase space near the grazing orbits due to impact lead to inherent difficulties in bifurcation analysis. Loss of dynamic stability—or D-bifurcations—is analyzed through computation of the largest Lyapunov exponent using the Nordmark–Poincare mapping that enables bypassing the problems associated with discontinuities. For P-bifurcation analysis, the steady-state solution of the Fokker–Planck equation is computed after applying suitable non-smooth coordinate transformations and mapping the problem into a continuous domain. A quantitative measure for P-bifurcations has been carried out using a newly developed measure based on Shannon entropy. A comparison of the stability domains obtained from P-bifurcation and D-bifurcation analyses is presented which reveals that these bifurcations need not occur in same regimes. © 2016, Springer Science+Business Media Dordrecht.

Nonlinear Dynamics

Stochastic bifurcations in a vibro-impact Duffing–Van der Pol oscillator

Nonlinear oscillators subjected to colored Gaussian/non-Gaussian excitations are modelled through a set of three coupled first-order stochastic differential equations by representing the excitation as a first-order filtered white noise. A 3-D finite element (FE) formulation is developed to solve the corresponding 3-D Fokker Planck (FP) equations. The joint probability density functions of the state variables, obtained as a solution of the FP equation, are typically non-Gaussian and are used for computing the crossing statistics of the response - an essential metric for time variant reliability analysis. The method is illustrated through a noisy Lorenz attractor and a Duffing oscillator subjected to additive colored noise. The increase in state-space dimension when the Duffing oscillator is additionally excited with a parametric Gaussian noise is effectively handled by using stochastic averaging to reduce the state-space dimension. Investigations are carried out to examine the accuracy of the FE method vis-a-vis Monte Carlo simulations. The proposed method is observed to be computationally significantly cheaper for these three problems. © 2014 Elsevier Ltd. All rights reserved.

Finite element solution of Fokker-Planck equation of nonlinear oscillators subjected to colored non-Gaussian noise

Bifurcation analysis of a stochastically excited vibro-impact Duffing-Van der Pol oscillator with bilateral rigid barriers

Journal	Data powered by TypesetInternational Journal of Mechanical Sciences
Publisher	Data powered by TypesetElsevier Ltd
ISSN	00207403
Open Access	No