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.

Sayan Gupta

Department of Applied Mechanics

Circuit oscillations

Excited states

Lyapunov methods

Oscillators (mechanical)

Probability density function

Topology

White noise

DUFFING-VAN DER POL OSCILLATOR

Gaussian white noise

JOINT PROBABILITY DENSITY FUNCTION

LARGEST LYAPUNOV EXPONENT

P-BIFURCATION

Quantitative measures

Shannon entropy

WHITE NOISE EXCITATION

Bifurcation (mathematics)

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IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Probabilistic Engineering Mechanics

Fluid–elastic systems nearing dynamic instabilities are known to be sensitive to fluctuations in fluid flow. A cantilever plate in axial flow with random temporal fluctuations, is examined numerically for its dynamical behaviour. The numerical model comprises of a nonlinear structural model for the flexible plate, coupled with unsteady lumped vortex model for the fluid forces. As the mean flow velocity is increased, the system transitions to limit cycle oscillations from a state of rest, through a regime of intermittent oscillations. The conditions for onset and disappearance of intermittency are discussed and are interpreted using stochastic bifurcation theories. While the onset of intermittency is found to be unaffected by the time scales of the flow fluctuations, they are observed to affect the length of the intermittency regime. The effect of plate flexibility on intermittency is also discussed. © 2019 Elsevier Ltd

Journal of Fluids and Structures

Intermittency in a cantilever plate in a randomly fluctuating fluid flow

The phenomenon of multiplicative noise induced intermittency (mNII) observed due to random temporal fluctuations in the system parameters of nonlinear dynamical systems is investigated using maps. Generic conditions for the onset and disappearance of mNII are derived. The role of correlation in the noise is investigated. The phenomenon of mNII is interpreted in terms of stochastic bifurcations as well. The developments are verified through numerical results for a set of simple1-d maps. Further, through a numerical example involving an aeroelastic system, it is shown that the proposed results are applicable for analysis of complicated engineering systems by considering Poincáre maps from the times histories of the response. © 2019 Elsevier Ltd

International Journal of Non-Linear Mechanics

Multiplicative noise induced intermittency in maps

In this study, we examine the effect of uniformly distributed noise superimposed on a steady, incompressible mean flow, past a circular cylinder undergoing transverse oscillations. We use a Duffing–Van der pol combined system for the present investigation in order to model the associated nonlinear behavior. We observe that as we vary the mean velocity as the bifurcation parameter, noise brings in major qualitative and quantitative changes on the structural response of the system compared to the deterministic cases. Noise induces new dynamical states in this vortex-induced vibration (VIV) system. The noise, depending on its coefficient of intensity, has a role in advancing the lock-in regime. The time histories of the response prior and post lock-in show different dynamics compared to the deterministic cases. Also, depending on the intensity of the noise coefficient, the probability density functions of the response amplitude also undergo qualitative and quantitative changes. We investigate the existence of P (phenomenological)-bifurcations in the VIV system by analyzing the joint probability density functions and quantify the response using Shannon entropy. © 2019, Springer Nature Singapore Pte Ltd.

Lecture Notes in Civil Engineering

Effect of noise on vortex-induced vibrations of circular cylinders

Aeroelastic systems with hardening nonlinearity exhibit supercritical Hopf bifurcation when the flow velocity exceeds a critical velocity leading to self-sustaining large amplitude limit cycle oscillations known as flutter. This study investigates the effects of irregular fluctuations in the flow on the dynamical stability characteristics of a two-degree-of-freedom pitch-plunge aeroelastic system with hardening nonlinearity. Dynamical or D-bifurcations are investigated through the computation of the largest Lyapunov exponent, while phenomenological or P-bifurcation analysis is carried out by examining the structure of the joint probability density function of the response quantities and their instantaneous time derivatives. The qualitative nature of P-bifurcation analysis makes it difficult to pinpoint the regimes of different response dynamics. In the light of this difficulty, a quantitative analysis using the Shannon entropy measure has been undertaken to quantify the P-bifurcation regime. This regime is shown to be coincident with the intermittency regime observed in the response time histories prior to flutter oscillations in fluctuating flows. © 2018, Springer Science+Business Media B.V., part of Springer Nature.

Nonlinear Dynamics

Intermittency in pitch-plunge aeroelastic systems explained through stochastic bifurcations

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

International Journal of Mechanical Sciences

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

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

Recent wind tunnel studies have shown that an aeroelastic system, prior to losing stability through flutter, goes through a regime where the response is characterized by intermittent bursts of periodic oscillations. The focus of this study is to investigate the reasons for this intermittent behavior through a numerical model. The studies indicate that the intermittency is observed only when the flow is accompanied by small random fluctuations. A stochastic bifurcation analysis is carried out to gain an understanding for this phenomenological interesting behavior. © 2016 The Authors.

Procedia Engineering

Numerical Investigations on Intermittency Route to Aeroelastic Flutter

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.

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.

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.

Journal	Data powered by TypesetProbabilistic Engineering Mechanics
Publisher	Data powered by TypesetElsevier Ltd
ISSN	02668920
Open Access	No