Chaos of the Rayleigh–Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation

2022 ◽  
Vol 192 ◽  
pp. 1-18
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen
Keyword(s):  
Duffing Oscillator ◽  
Harmonic Excitation ◽  
Periodic Perturbation

Superharmonic Resonance of Fractional-Order Mathieu–Duffing Oscillator

2019 ◽  
Vol 14 (7) ◽  
Author(s):  
Jiangchuan Niu ◽  
Xiaofeng Li ◽  
Haijun Xing
Keyword(s):  
Fractional Order ◽  
Order Term ◽  
Duffing Oscillator ◽  
Steady State Solution ◽  
Harmonic Excitation ◽  
Frequency Equation ◽  
Superharmonic Resonance ◽  
Duffing System ◽  
Resonance Response ◽  
Kbm Method

The superharmonic resonance of fractional-order Mathieu–Duffing oscillator subjected to external harmonic excitation is investigated. Based on the Krylov–Bogolubov–Mitropolsky (KBM) asymptotic method, the approximate analytical solution for the third superharmonic resonance under parametric-forced joint resonance is obtained, where the unified expressions of the fractional-order term with fractional order from 0 to 2 are gained. The amplitude–frequency equation for steady-state solution and corresponding stability condition are also presented. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional-order term, excitation amplitudes, and nonlinear stiffness coefficient on the superharmonic resonance response of the system are analyzed in detail. The results show that the KBM method is effective to analyze dynamic response in a fractional-order Mathieu–Duffing system.

scholarly journals Limiting Phase Trajectories and Resonance Energy Transfer in a System of Two Coupled Oscillators

2010 ◽  
Vol 2010 ◽  
pp. 1-24 ◽  
Author(s):  
L. I. Manevitch ◽  
A. S. Kovaleva ◽  
E. L. Manevitch
Keyword(s):  
Energy Transfer ◽  
Coupled Oscillators ◽  
Energy Exchange ◽  
Duffing Oscillator ◽  
Normal Modes ◽  
Resonance Energy ◽  
Harmonic Excitation ◽  
Conservation Of Energy ◽  
Phase Trajectories ◽  
Limiting Phase Trajectories

We study a problem of energy exchange in a system of two coupled oscillators subject to 1 : 1 resonance. Our results exploit the concept of limiting phase trajectories (LPTs). The LPT, associated with full energy transfer, is, in certain sense, an alternative to nonlinear normal modes characterized by conservation of energy. We consider two benchmark examples. As a first example, we construct an LPT and examine the convergence to stationary oscillations for a Duffing oscillator subjected to resonance harmonic excitation. As a second example, we treat resonance oscillations in a system of two nonlinearly coupled oscillators. We demonstrate the reduction of the equations of motion to an equation of a single oscillator. It is shown that the most intense energy exchange and beating arise when motion of the equivalent oscillator is close to an LPT. Damped beating and the convergence to rest in a system with dissipation are demonstrated.

scholarly journals Dynamic Analysis and Design of a Novel Ring-Based Vibratory Energy Harvester

Vibration ◽  
2019 ◽  
Vol 2 (3) ◽  
pp. 271-284
Author(s):  
Ibrahim F. Gebrel ◽  
Ligang Wang ◽  
Samuel F. Asokanthan
Keyword(s):  
Magnetic Force ◽  
Duffing Oscillator ◽  
Harmonic Excitation ◽  
Ring Structure ◽  
Restoring Force ◽  
Flexural Mode ◽  
Analysis And Design ◽  
Energy Harvesters ◽  
Numerical Predictions ◽  
Proposed Model

This paper aims to focus on the design and analysis of a novel ring-based mono-stable energy-harvesting device that is considered as an alternative to the beam and tube models used thus far. The highly sensitive ring second flexural mode, when combined with the nonlinear external magnetic force, results in an ideal combination that yields increased frequency range, and can be considered as novel in the field of vibration-based energy harvesters. A mathematical model for the ring structure, as well as a model to generate nonlinear magnetic force that acts on the ring structure, is formulated. The discretized form of the governing equations is shown to represent a Duffing oscillator in the presence of an external magnetic field. The forms of the system potential energy, as well as the restoring force, are examined to ensure that the mono-stable behavior exists in the proposed model. Numerical predictions of time response, frequency response, phase diagram, and bifurcations map when the system is subjected to ambient harmonic excitation, have been performed for the purposes of gaining an insight into the dynamics and power generation of this new class of harvesters.

Modified Path Integral Solution of Fokker–Planck Equation: Response and Bifurcation of Nonlinear Systems

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
Pankaj Kumar ◽  
S. Narayanan
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Path Integral ◽  
Duffing Oscillator ◽  
Planck Equation ◽  
Harmonic Excitation ◽  
Integration Scheme ◽  
Pi Method ◽  
Non Gaussian ◽  
Fokker Planck

Response of nonlinear systems subjected to harmonic, parametric, and random excitations is of importance in the field of structural dynamics. The transitional probability density function (PDF) of the random response of nonlinear systems under white or colored noise excitation (delta correlated) is governed by both the forward Fokker–Planck (FP) and the backward Kolmogorov equations. This paper presents a new approach for efficient numerical implementation of the path integral (PI) method in the solution of the FP equation for some nonlinear systems subjected to white noise, parametric, and combined harmonic and white noise excitations. The modified PI method is based on a non-Gaussian transition PDF and the Gauss–Legendre integration scheme. The effects of white noise intensity, amplitude, and frequency of harmonic excitation and the level of nonlinearity on stochastic jump and bifurcation behaviors of a hardening Duffing oscillator are also investigated.

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