Lagging Motion of Forced Nonlinear Oscillators

2003 ◽  
Author(s):  
Sand Woo Karng ◽  
Ki Young Kim ◽  
Ho-Young Kwak
Keyword(s):  
Nonlinear Systems ◽  
Applied Field ◽  
Nonlinear Oscillator ◽  
Phase Plane ◽  
Inverted Pendulum ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Expansion Ratio ◽  
Amplitude Response ◽  
Calculation Results

The lagging motion of nonlinear oscillators with respect to the externally driven field was treated analytically and the calculation results were compared with observed results. Such lagging problem may occur in nonlinear systems whose behavior crucially depends on the frequency of the applied force. The lagging motion of the nonlinear oscillator with respect to the harmonically driven field made the oscillator respond in a way that reduced the effect of the applied field. The calculation considering the lagging motion yielded proper results in the expansion ratio of the bubble under ultrasound and trajectories in phase plane, the frequency spectrum for a forced inverted pendulum, and the amplitude response of the Duffing oscillator.

Methods and Gaussian Criterion for Statistical Linearization of Stochastic Parametrically and Externally Excited Nonlinear Systems

1989 ◽  
Vol 56 (1) ◽  
pp. 179-185 ◽  
Author(s):  
R. J. Chang ◽  
G. E. Young
Keyword(s):  
Nonlinear Systems ◽  
Goodness Of Fit ◽  
Duffing Oscillator ◽  
A Priori ◽  
Nonlinear Oscillators ◽  
Statistical Linearization ◽  
Goodness Of Fit Test ◽  
Chi Square ◽  
Non Gaussian ◽  
Parametric And External Excitations

The methods of Gaussian linearization along with a new Gaussian Criterion used in the prediction of the stationary output variances of stable nonlinear oscillators subjected to both stochastic parametric and external excitations are presented. The techniques of Gaussian linearization are first derived and the accuracy in the prediction of the stationary output variances is illustrated. The justification of using Gaussian linearization a priori is further investigated by establishing a Gaussian Criterion. The non-Gaussian effects due to system nonlinearities and/or large noise intensities in a Duffing oscillator are also illustrated. The validity of employing the Gaussian Criterion test for assuring accuracy of Gaussian linearization is supported by performing the Chi-square Gaussian goodness-of-fit test.

Laplace–Pade Parametric Homotopy Perturbation Method for Uncertain Nonlinear Oscillators

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
S. Chakraverty ◽  
N. R. Mahato
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Nonlinear Oscillator ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Parametric Form ◽  
Homotopy Perturbation

Nonlinear oscillators have wide applicability in science and engineering problems. In this paper, nonlinear oscillator having initial conditions varying over fuzzy numbers has been initially taken into consideration. Here, the fuzziness in the uncertain nonlinear oscillators has been handled using parametric form. Using parametric form in terms of r-cut, the nonlinear uncertain differential equations are reduced to parametric differential equations. Then, based on classical homotopy perturbation method (HPM), a parametric homotopy perturbation method (PHPM) is proposed to compute solution enclosure of such uncertain nonlinear differential equations. A sufficient convergence condition of parametric solution obtained using PHPM is also proved. Further, a parametric Laplace–Pade approximation is incorporated in PHPM for retaining the periodic characteristic of nonlinear oscillators throughout the domain. The efficiency of Laplace–Pade PHPM has been verified for uncertain Duffing oscillator. Finally, Laplace–Pade PHPM is also applied to solve other uncertain nonlinear oscillator, viz., Rayleigh oscillator, with respect to fuzzy parameters.

ANALYTICAL ESTIMATES OF THE EFFECT OF NONLINEAR DAMPING IN SOME NONLINEAR OSCILLATORS

2000 ◽  
Vol 10 (09) ◽  
pp. 2257-2267 ◽  
Author(s):  
JOSÉ L. TRUEBA ◽  
JOAQUÍN RAMS ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Nonlinear Damping ◽  
Damping Coefficient ◽  
Critical Parameters ◽  
Nonlinear Dissipation ◽  
Simple Pendulum ◽  
Melnikov Theory ◽  
Damped Systems

This paper reports on the effect of nonlinear damping on certain nonlinear oscillators, where analytical estimates provided by the Melnikov theory are obtained. We assume general nonlinear damping terms proportional to the power of velocity. General and useful expressions for the nonlinearly damped Duffing oscillator and for the nonlinearly damped simple pendulum are computed. They provide the critical parameters in terms of the damping coefficient and damping exponent, that is, the power of the velocity, for which complicated behavior is expected. We also consider generalized nonlinear damped systems, which may contain several nonlinear damping terms. Using the idea of Melnikov equivalence, we show that the effect of nonlinear dissipation can be equivalent to a linearly damped nonlinear oscillator with a modified damping coefficient.

Parametric Identification of Nonlinear Systems Using Higher-Order Spectra

1997 ◽  
Author(s):  
Khalil A. Khan ◽  
B. Balachandran
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Oscillator ◽  
Transfer Functions ◽  
Nonlinear Oscillators ◽  
Parametric Identification ◽  
Higher Order ◽  
Numerical Results ◽  
Higher Order Spectra

Abstract A novel procedure to parametrically identify the nonlinearities in a system is presented. In this procedure, the relationships among higher-order transfer functions and higher-order spectra are utilized and expressions are derived for coefficients of non-linearities. Two different nonlinear oscillators are considered as examples to illustrate the procedure. Numerical results are also provided for a nonlinear oscillator.

Response Envelope Statistics for Nonlinear Oscillators With Random Excitation

1978 ◽  
Vol 45 (1) ◽  
pp. 170-174 ◽  
Author(s):  
W. D. Iwan ◽  
P.-T. Spanos
Keyword(s):  
Computer Simulation ◽  
Linear System ◽  
Analytical Method ◽  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Nonlinear Oscillators ◽  
Random Excitation ◽  
Weakly Nonlinear ◽  
Response Envelope

An approximate analytical method is presented for determining both the stationary and nonstationary amplitude or envelope response statistics of a lightly damped and weakly nonlinear oscillator subject to Gaussian white noise. The method is based on the solution of an equivalent linear system whose parameters are functions of the response itself. The solution derived by the approximate method is compared with that obtained by computer simulation for a Duffing oscillator.

Response Spectra for Nonlinear Oscillators

1975 ◽  
Vol 97 (4) ◽  
pp. 1223-1226 ◽  
Author(s):  
J. E. Manning
Keyword(s):  
Nonlinear Oscillator ◽  
Response Spectrum ◽  
Duffing Oscillator ◽  
Response Spectra ◽  
Nonlinear Oscillators ◽  
Random Excitation ◽  
Heuristic Approach ◽  
Perturbation Approach ◽  
Spectral Bandwidth ◽  
Nonlinear Spring

Methods are presented for calculating the response spectrum of a nonlinear oscillator with broadband random excitation. A perturbation approach is used for oscillators with slightly nonlinear spring elements to show that the effect of the nonlinearity is a slight shift in the spectrum peak. A new heuristic approach is used for oscillators with large nonlinearities and small damping to show that in addition to a shift in the peak an increase in spectral bandwidth also occurs. The Duffing oscillator is studied as a specific example.

Characteristics in Response Integration and Bifurcation of a Forced Duffing Oscillator

2007 ◽  
Author(s):  
Jang-Der Jeng ◽  
Yuan Kang ◽  
Yeon-Pun Chang ◽  
Shyh-Shyong Shyr
Keyword(s):  
Applied Sciences ◽  
Phase Plane ◽  
Duffing Oscillator ◽  
High Order ◽  
Chaotic Motions ◽  
Integration Algorithm ◽  
High Order Harmonic ◽  
Stiffness And Damping

The Duffing oscillator is well-known models of nonlinear system, with applications in many fields of applied sciences and engineering. In this paper, a response integration algorithm is proposed to analyze high-order harmonic and chaotic motions in this oscillator for modeling rotor excitations. This method numerically integrates the distance between state trajectory and the origin in the phase plane during a specific period and predicted intervals with excitation periods. It provides a quantitative characterization of system responses and can replace the role of the traditional stroboscopic technique (Poincare´ section method) to observe bifurcations and chaos of the nonlinear oscillators. Due to the signal response contamination of system, thus it is difficult to identify the high-order responses of the subharmonic motion because of the sampling points on Poincare´ map too near each other. Even the system responses will be made misjudgments. Combining the capability of precisely identifying period and constructing bifurcation diagrams, the advantages of the proposed response integration method are shown by case studies. Applying this method, the effects of the change in the stiffness and the damping coefficients on the vibration features of a Duffing oscillator are investigated in this paper. From simulation results, it is concluded that the stiffness and damping of the system can effectively suppress chaotic vibration and reduce vibration amplitude.

Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

2014 ◽  
Vol 24 (05) ◽  
pp. 1450075 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Fourier Series ◽  
Nonlinear Systems ◽  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Series Solution ◽  
Parametric Oscillator ◽  
Periodic Motions ◽  
Parametric Oscillators ◽  
Harmonic Term ◽  
Nonlinear Behaviors

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

Sequences of Global Bifurcations and the Related Outcomes after Crisis of the Resonant Attractor in a Nonlinear Oscillator

1997 ◽  
Vol 07 (11) ◽  
pp. 2437-2457 ◽  
Author(s):  
W. Szemplińska-Stupnicka ◽  
E. Tyrkiel
Keyword(s):  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Nonlinear Resonance ◽  
Basins Of Attraction ◽  
Global Bifurcations ◽  
System Behavior ◽  
Coexisting Attractors ◽  
System Parameters

The problem of the system behavior after annihilation of the resonant attractor in the region of the nonlinear resonance hysteresis is considered. The sequences of global bifurcations, in connection with the associated metamorphoses of basins of attraction of coexisting attractors, are examined. The study allows one to reveal the mechanism that governs the phenomenon of the post crisis ensuing transient trajectory to settle onto one or another remote attractor. The problem is studied in detail for the twin-well potential Duffing oscillator. The boundary which splits the considered region of system parameters into two subdomains, where the outcome is unique or the two outcomes are possible, is defined.

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