Chaos in a Weakly Nonlinear Oscillator With Parametric and External Resonances

1991 ◽  
Vol 58 (1) ◽  
pp. 244-250 ◽  
Author(s):  
K. Yagasaki
Keyword(s):  
Invariant Torus ◽  
Chaotic Dynamics ◽  
Averaging Method ◽  
Double Resonance ◽  
External Excitation ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
External Forces

This paper describes a study of the chaotic dynamics of a weakly nonlinear single degree-of-freedom system subjected to combined parametric and external excitation. We consider a case of double resonance in which primary resonances, with respect to parametric and external forces, exist simultaneously. By using the averaging method and Melnikov’s technique, it is shown that chaos may occur in certain parameter regions. These chaotic motions result from the existence of orbits homoclinic to a normally hyperbolic invariant torus which corresponds to a hyperbolic periodic orbit in the averaged system. The mechanism and structure of chaos in this situation are also described. Furthermore, the existence of steady-state chaos is demonstrated by numerical simulation.

Dynamics of a Weakly Nonlinear System Subjected to Combined Parametric and External Excitation

1990 ◽  
Vol 57 (1) ◽  
pp. 209-217 ◽  
Author(s):  
Kazuyuki Yagasaki ◽  
Masaru Sakata ◽  
Koji Kimura
Keyword(s):  
Chaotic Dynamics ◽  
Averaging Method ◽  
Invariant Tori ◽  
Homoclinic Orbits ◽  
External Excitation ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Average System ◽  
Theoretical Results

In this paper we study the dynamics of a weakly nonlinear single-degree-of-freedom system subjected to combined parametric and external excitation. The averaging method is used to establish the existence of invariant tori and analyze their stability. Furthermore, by applying the Melnikov technique to the average system it is shown that there exist transverse homoclinic orbits resulting in chaotic dynamics. Numerical simulation results are also given to demonstrate the theoretical results.

Chaos and Three-Dimensional Horseshoes in Slowly Varying Oscillators

1988 ◽  
Vol 55 (4) ◽  
pp. 959-968 ◽  
Author(s):  
Stephen Wiggins ◽  
Steven W. Shaw
Keyword(s):  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Equation Of Motion ◽  
Necessary Condition ◽  
Stable And Unstable Manifolds ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Unstable Manifolds ◽  
Additional Equation

We present general results pertaining to chaotic motions in a class of systems termed slowly varying oscillators which consist of weakly perturbed single-degree-of-freedom systems in which parameters vary slowly in time according to an additional equation of motion. Our results include an analytical method for detecting transversal intersections of stable and unstable manifolds (typically a necessary condition for chaotic motions to exist) and a detailed description of the chaotic dynamics that occur when this situation exists.

The Dynamics of a Nonharmonically Excited System Having Rigid Amplitude Constraints

1992 ◽  
Vol 59 (3) ◽  
pp. 693-695 ◽  
Author(s):  
Pi-Cheng Tung
Keyword(s):  
Dynamic Response ◽  
Bifurcation Analysis ◽  
Piecewise Linear ◽  
Coefficient Of Restitution ◽  
Degree Of Freedom ◽  
Linear Oscillator ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Excited System

We consider the dynamic response of a single-degree-of-freedom system having two-sided amplitude constraints. The model consists of a piecewise-linear oscillator subjected to nonharmonic excitation. A simple impact rule employing a coefficient of restitution is used to characterize the almost instantaneous behavior of impact at the constraints. In this paper periodic and chaotic motions are found. The amplitude and stability of the periodic responses are determined and bifurcation analysis for these motions is carried out. Chaotic motions are found to exist over ranges of forcing periods.

The Dynamic Response of a System With Preloaded Compliance

1988 ◽  
Vol 110 (3) ◽  
pp. 278-283 ◽  
Author(s):  
S. W. Shaw ◽  
P. C. Tung
Keyword(s):  
Dynamic Response ◽  
Piecewise Linear ◽  
Harmonic Excitation ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Linear Features ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Stable Periodic ◽  
Model Equations

We consider the dynamic response of a single degree of freedom system with preloaded, or “setup,” springs. This is a simple model for systems where preload is used to suppress vibrations. The springs are taken to be linear and harmonic excitation is applied; damping is assumed to be of linear viscous type. Using the piecewise linear features of the model equations we determine the amplitude and stability of the periodic responses and carry out a bifurcation analysis for these motions. Some parameter regions which contain no simple stable periodic motions are shown to possess chaotic motions.

Bifurcation and Chaos in Friction-Induced Vibration

2002 ◽  
Author(s):  
Yong Li ◽  
Z. C. Feng
Keyword(s):  
Friction Model ◽  
Degree Of Freedom ◽  
Stable Limit ◽  
Mechanical Oscillator ◽  
Chaotic Motions ◽  
Sliding Speed ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Friction Models ◽  
Friction Induced Vibration

Friction-induced vibration is a phenomenon that has received extensive study by the dynamics community. This is because of the important industrial relevance and the evere-volving development of new friction models. In this paper, we report the result of bifurcation study of a single-degree-of-freedom mechanical oscillator sliding over a surface. The friction model we use is that developed by Canudas de Wit et al, a model that is receiving increasing acceptance from the mechanics community. Using this model, we find a stable limit cycle at intermediate sliding speed for a single-degree-of-freedom mechanical oscillator. Moreover, the mechanical oscillator can exhibit chaotic motions. For certain parameters, numerical simulation suggests the existence of a Silnikov homoclinic orbit. This is not expected in a single-degree-of-freedom system. The occurrence of chaos becomes possible because the friction model contains one internal variable. This demonstrates a unique characteristic of the friction model. Unlike most friction models, the present model is capable of simultaneously modeling self-excitation and predicting stick-slip at very low sliding speed as well.

RESONANCE CONTROL FOR A FORCED SINGLE-DEGREE-OF-FREEDOM NONLINEAR SYSTEM

2004 ◽  
Vol 14 (04) ◽  
pp. 1423-1429 ◽  
Author(s):  
ANDREW Y. T. LEUNG ◽  
JIN CHEN JI ◽  
GUANRONG CHEN
Keyword(s):  
Feedback Control ◽  
Nonlinear System ◽  
Singularity Theory ◽  
Degree Of Freedom ◽  
Theory Approach ◽  
Nonlinear Feedback ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Resonance Control

The main characteristic of a forced single-degree-of-freedom weakly nonlinear system is determined by its primary, super- and sub-harmonic resonances. A nonlinear parametric feedback control is proposed to modify the steady-state resonance responses, thus to reduce the amplitude of the response and to eliminate the saddle-node bifurcations that take place in the resonance responses. The nonlinear gain of the feedback control is determined by analyzing the bifurcation diagrams associated with the corresponding frequency-response equation, from the singularity theory approach. It is shown by illustrative examples that the proposed nonlinear feedback is effective for controlling three kinds of resonance responses.

scholarly journals On the critical forcing amplitude of forced nonlinear oscillators

2013 ◽  
Vol 3 (4) ◽  
Author(s):  
Mariano Febbo ◽  
Jinchen Ji
Keyword(s):  
Primary Resonance ◽  
Nonlinear Oscillators ◽  
Analytical Form ◽  
Excitation Amplitude ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Saddle Node ◽  
Basis Method

AbstractThe steady-state response of forced single degree-of-freedom weakly nonlinear oscillators under primary resonance conditions can exhibit saddle-node bifurcations, jump and hysteresis phenomena, if the amplitude of the excitation exceeds a certain value. This critical value of excitation amplitude or critical forcing amplitude plays an important role in determining the occurrence of saddle-node bifurcations in the frequency-response curve. This work develops an alternative method to determine the critical forcing amplitude for single degree-of-freedom nonlinear oscillators. Based on Lagrange multipliers approach, the proposed method considers the calculation of the critical forcing amplitude as an optimization problem with constraints that are imposed by the existence of locations of vertical tangency. In comparison with the Gröbner basis method, the proposed approach is more straightforward and thus easy to apply for finding the critical forcing amplitude both analytically and numerically. Three examples are given to confirm the validity of the theoretical predictions. The first two present the analytical form for the critical forcing amplitude and the third one is an example of a numerically computed solution.

Chaotic Dynamics of a Quasi-Periodically Forced Beam

1992 ◽  
Vol 59 (1) ◽  
pp. 161-167 ◽  
Author(s):  
K. Yagasaki
Keyword(s):  
Chaotic Dynamics ◽  
Averaging Method ◽  
Single Mode ◽  
Homoclinic Orbits ◽  
Equation Of Motion ◽  
Degree Of Freedom ◽  
Finite Degree ◽  
Chaotic Motions ◽  
Higher Modes ◽  
Straight Beam

A straight beam with fixed ends, forced with two frequencies is considered. By using Galerkin’s method, the equation of motion of the beam is reduced to a finite degree-of-freedom system. The resulting equation is transformed into a multi-frequency system and the averaging method is applied. It is shown, by using Melnikov’s method, that there exist transverse homoclinic orbits in the averaged system associated with the first-mode equation. This implies that chaotic motions may occur in the single-mode equation. Furthermore, the effect of higher modes and the implications of this result for the full beam motions are described.

Ideal Elastic-Plastic Oscillators Subjected to Stochastic Input

1999 ◽  
Author(s):  
S. Caddemi ◽  
M. Di Paola
Keyword(s):  
External Noise ◽  
Spectral Density Function ◽  
Power Spectral Density Function ◽  
External Excitation ◽  
Homogeneous Poisson Process ◽  
Single Degree Of Freedom ◽  
Elastic Plastic ◽  
Single Degree ◽  
Power Spectral ◽  
Probabilistic Response

Abstract The paper deals with the evaluation of the probabilistic response of an ideal elastic-plastic single degree of freedom oscillator subjected to a normal white noise. The analysis has been conducted on the hypothesis that accumulated plastic displacements are a compound homogeneous Poisson process independent of the external excitation. In this case plastic displacements can be treated as an additional external noise, to be identified, acting on a linear system. In the paper a time domain approach to obtain the two variable non stationary correlation function is proposed. Hence the evolutionary power spectral density function is also obtained. A numerical example is presented in order to show the accuracy of the presented procedure in comparison to Monte Carlo simulations.

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