Analytical and Numerical Studies of Hysteresis Observed in Higher Order Rayleigh- Duffing Oscillator

2021 ◽  
Vol 8 (3) ◽  
Keyword(s):  
Duffing Oscillator ◽  
Higher Order ◽  
Numerical Studies

Excitation-Induced Stability in a Bistable Duffing Oscillator: Analysis and Experiments

2014 ◽  
Vol 10 (1) ◽  
Author(s):  
Z. Wu ◽  
R. L. Harne ◽  
K. W. Wang
Keyword(s):  
Perturbation Theory ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Excitation Level ◽  
Frequency Range ◽  
Numerical Studies ◽  
Method Of Harmonic Balance ◽  
Mathieu’S Equation ◽  
Excitation Conditions ◽  
Good Agreement

The excitation-induced stability (EIS) phenomenon in a harmonically excited bistable Duffing oscillator is studied in this paper. Criteria to predict system and excitation conditions necessary to maintain EIS are derived through a combination of the method of harmonic balance, perturbation theory, and stability theory for Mathieu's equation. Accuracy of the criteria is verified by analytical and numerical studies. We demonstrate that damping primarily determines the likelihood of attaining EIS response when several dynamics coexist while excitation level governs both the existence and frequency range of the EIS region, providing comprehensive guidance for realizing or avoiding EIS dynamics. Experimental results are in good agreement regarding the comprehensive influence of excitation conditions on the inducement of EIS.

scholarly journals The influence of nonlinear terms in mechanical systems having two degrees of freedom

2006 ◽  
Vol 28 (3) ◽  
pp. 155-164
Author(s):  
Nguyen Duc Tinh
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Degrees Of Freedom ◽  
Averaging Method ◽  
Duffing Oscillator ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Higher Order ◽  
White Noise Excitation ◽  
Two Degree Of Freedom

For many years the higher order stochastic averaging method has been widely used for investigating nonlinear systems subject to white and coloured noises to predict approximately the response of the systems. In the paper the method is further developed for two-degree-of-freedom systems subjected to white noise excitation. Application to Duffing oscillator is considered.

Waveform Invariance in Nonlinear Periodic Systems Using Higher-Order Multiple Scales

2016 ◽  
Author(s):  
Matthew Fronk ◽  
Michael J. Leamy
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Higher Order ◽  
Linear Quadratic ◽  
Numerical Studies ◽  
Specific Order ◽  
Spectral Variance ◽  
Force Displacement

This paper carries-out a higher-order, multiple scales perturbation analysis on nonlinear monoatomic and diatomic chains with the intent of predicting invariant waveforms. The chains incorporate linear, quadratic, and cubic force-displacement relationships, and linear dampers. Multi-harmonic results for 1st and 2nd order expansions are reported in closed form, while results for the 3rd order are computed numerically on a case-by-case basis, thus avoiding difficulties associated with large symbolic expressions. Dimensionless parameters are introduced which characterize the amplitude-dependent nonlinear nature of a given chain. Interpretation of the perturbation solutions suggests that the nonlinear chains support certain waveforms which propagate invariantly; i.e., the spectral content does not change significantly over time and space. Numerical simulations confirm this finding using initial conditions corresponding to a specific order of the perturbation solution, and subsequent FFT’s of the response track the growth (or decay) of spatial harmonic content. A variance parameter computes mean fluctuation of the harmonics about their initial values. For a variety of parameter sets, the numerical studies confirm that spectral variance reduces when waves receive 2nd order initial conditions as compared to 1st order ones. Furthermore, chains given 3rd order initial conditions exhibit smaller variance when compared to those given 1st and 2nd order ones. The studies’ results suggest that introducing higher-order multiple scales perturbation analysis captures long-term, non-localized invariant waves (or cnoidal waves), which have the potential for propagating coherent information over long distances.

scholarly journals An Exact Solution to the Quadratic Damping Strong Nonlinearity Duffing Oscillator

2021 ◽  
Vol 2021 ◽  
pp. 1-8 ◽  
Author(s):  
Alvaro H. Salas ◽  
S. A. El-Tantawy ◽  
Noufe H. Aljahdaly
Keyword(s):  
Elliptic Function ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Duffing Oscillator ◽  
Higher Order ◽  
Strong Nonlinearity ◽  
Classical Dynamics ◽  
Elementary Functions ◽  
Weierstrass Elliptic Function ◽  
Quadratic Damping

The nonlinear equations of motion such as the Duffing oscillator equation and its family are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions before. Thus, in this work, the stability analysis of quadratic damping higher-order nonlinearity Duffing oscillator is investigated. Hereinafter, some new analytical solutions to the undamped higher-order nonlinearity Duffing oscillator in the form of Weierstrass elliptic function are obtained. Posteriorly, a novel exact analytical solution to the quadratic damping higher-order nonlinearity Duffing equation under a certain condition (not arbitrary initial conditions) and in the form of Weierstrass elliptic function is derived in detail for the first time. Furthermore, the obtained solutions are camped to the Runge–Kutta fourth-order (RK4) numerical solution.

Spectral Analyses of Nonlinear Interactions

1995 ◽  
Author(s):  
Balakumar Balachandran ◽  
Khalil A. Khan
Keyword(s):  
Coupled Oscillators ◽  
Higher Order ◽  
Nonlinear Interactions ◽  
Phase Coupling ◽  
Spectral Analyses ◽  
Weakly Nonlinear ◽  
Numerical Studies ◽  
Analytical Approximations ◽  
Higher Order Spectra

Abstract Signals pertaining to motions of nonlinearly coupled oscillators are studied using higher-order spectral analyses. The analyses is used to understand the role of phase coupling in nonlinear interactions between two or more Fourier components. For certain motions of weakly nonlinear systems, analytical approximations are obtained for relevant higher-order spectra and coherence functions. Numerical studies are conducted to verify analytical predictions and to illustrate the usefulness of spectral analyses for different cases.

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