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.

A Practical Technique for Spectral Analysis of Nonlinear Systems Under Stochastic Parametric and External Excitations

1991 ◽  
Vol 113 (4) ◽  
pp. 516-522 ◽  
Author(s):  
R. J. Chang
Keyword(s):  
Nonlinear System ◽  
Linear System ◽  
Spectral Response ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Matching Condition ◽  
External Noise ◽  
Equivalent Linearization ◽  
Parametric Noise ◽  
Parametric And External Excitations

A practical approach is developed for analyzing the spectral response of a nonlinear system subjected to both parametric and external Gaussian white noise excitations. The technique is implemented through the combined methods of equivalent external excitation and equivalent linearization to derive an equivalent linear system under equivalent external noise excitation. The spectral response is then obtained through utilizing the input/output spectral relation and covariance matching condition. A parametric noise excited linear system, Duffing oscillator, and nonlinear system with hysteretic nonlinearity are selected for investigation. The validity of the proposed method for analyzing spectral response is further supported by some analytical solutions and FFT technique through Monte Carlo simulations.

On Asymptotic Approximations of First Integrals for Weakly Nonlinear Oscillators

2001 ◽  
Author(s):  
S. B. Waluya ◽  
W. T. van Horssen
Keyword(s):  
Perturbation Method ◽  
Periodic Solutions ◽  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
First Integrals ◽  
Asymptotic Approximations ◽  
Weakly Nonlinear ◽  
Time Periodic Solutions ◽  
Time Periodic

Abstract In this paper a nonlinear oscillator problem will be studied. It will be shown that the recently developed perturbation method based on integrating vectors can be used to approximate first in tegrals and periodic solutions. The existence, uniqueness, and stability of time-periodic solutions are obtained by using the approximations for the first integrals.

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.

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.

Calculation of Nonlinear Systems Under Narrow Band Excitation Using Equivalent Linearization and Path Continuation

2021 ◽  
Author(s):  
Alwin Förster ◽  
Lars Panning-von Scheidt
Keyword(s):  
Nonlinear Systems ◽  
Narrow Band ◽  
Gaussian White Noise ◽  
Random Excitation ◽  
Mean Value ◽  
Equivalent Linearization ◽  
Efficient Procedure ◽  
Stochastic Excitations ◽  
Wide Range ◽  
The One

Abstract Turbomachines experience a wide range of different types of excitation during operation. On the structural mechanics side, periodic or even harmonic excitations are usually assumed. For this type of excitation there are a variety of methods, both for linear and nonlinear systems. Stochastic excitation, whether in the form of Gaussian white noise or narrow band excitation, is rarely considered. As in the deterministic case, the calculations of the vibrational behavior due to stochastic excitations are even more complicated by nonlinearities, which can either be unintentionally present in the system or can be used intentionally for vibration mitigation. Regardless the origin of the nonlinearity, there are some methods in the literature, which are suitable for the calculation of the vibration response of nonlinear systems under random excitation. In this paper, the method of equivalent linearization is used to determine a linear equivalent system, whose response can be calculated instead of the one of the nonlinear system. The method is applied to different multi-degree of freedom nonlinear systems that experience narrow band random excitation, including an academic turbine blade model. In order to identify multiple and possibly ambiguous solutions, an efficient procedure is shown to integrate the mentioned method into a path continuation scheme. With this approach, it is possible to track jump phenomena or the influence of parameter variations even in case of narrow band excitation. The results of the performed calculations are the stochastic moments, i.e. mean value and variance.

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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