Chaos and bifurcation analysis of stochastically excited discontinuous nonlinear oscillators
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ORIGINAL PAPER
Chaos and bifurcation analysis of stochastically excited discontinuous nonlinear oscillators Pankaj Kumar · S. Narayanan
Received: 21 February 2020 / Accepted: 14 September 2020 © Springer Nature B.V. 2020
Abstract Dynamics of discontinuous nonlinear systems subjected to random excitation is studied. Such systems occur in many mechanical and aerospace applications involving impact, friction, clearance, backlash, freeplay etc. These systems are characterized by sharp switches in dynamical behaviour described by discontinuous stochastic differential equations. An adaptive time stepping approach is developed in combination with a bisection algorithm to locate precisely the discontinuity point in the numerical integration advanced by the Milstein method. The Brownian tree approach is used to direct the integration along the correct Brownian path. The examples of a Duffing oscillator with one- and two-sided impacts and a linear oscillator with a nonlinear discontinuous dry frictiontype damper (Coulomb damping) subjected to combined harmonic and white noise excitations are considered. Stable periodic motion, D-bifurcation and chaotic dynamics are exhibited in different parametric regimes. The path-wise numerical integration procedure demonstrates the accuracy and efficiency of the proposed scheme in the dynamic analysis of the noisy vibroimpact oscillator and the friction oscillator.
P. Kumar Dynamic Analysis Group, Bharat Heavy Electricals Limited, Nagpur 440001, India S. Narayanan (B) Indian Institute of Information Technology (Design and Manufacturing), Kancheepuram, Chennai 600 127, India e-mail: [email protected]
Keywords Vibro-impact oscillator · Coulomb friction · Largest Lyapunov exponent · Chaotic motion · Adaptive time integration · Bisection method · Brownian tree
1 Introduction Chaos and bifurcations in nonlinear dynamical systems have been an intense field of research in many areas of physics, engineering and other applications for the last five decades and more. Most of these studies are devoted to deterministic nonlinear systems. Though studies like stochastic bifurcations in stochastic and randomly excited nonlinear systems are also available in the literature, such studies are very much less. Also if the nonlinearities are of the discontinuous type with stochasticity included, the studies are even lesser. This paper addresses these two aspects of stochasticity and discontinuity in nonlinear dynamical systems. Discontinuous dynamic systems like vibro-impact systems and systems with dry friction damping frequently occur in many engineering problems, where effects resulting from nonsmooth phenomena have to be taken into account in the analysis of their dynamics [5,10,11,19,20,36,43]. The mechanical and structural systems that involve moving parts with amplitude constraints on motion such as that occur in rotor-stator rubs in gas turbines, gear rattle, aircraft flaps, automotive braking systems, tube interaction in heat exchangers, impact dampers, ship roll against ice berg are some
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