Grazing Bifurcation and Periodic Motion Switching in a Piecewise Linear, Impacting Oscillator Under a Periodical Excitation

2005 ◽  
Author(s):  
Albert C. J. Luo ◽  
Lidi Chen
Keyword(s):  
Nonlinear Dynamics ◽  
Periodic Motion ◽  
Local Stability ◽  
Piecewise Linear ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Grazing Bifurcation ◽  
Piecewise Linear System ◽  
Local Stability Analysis ◽  
Mapping Structures

The grazing bifurcation and periodic motion switching of the harmonically forced, piecewise linear system with impacting are investigated. The generic mappings based on the discontinuous boundaries are introduced. Furthermore, the mapping structures are developed for the analytical prediction of periodic motions in such a system. The local stability and bifurcation conditions for specified periodic motions are obtained. The regular and grazing, periodic motions are illustrated. The grazing is the origin of the periodic motion switching for this system. Such a grazing bifurcation cannot be estimated through the local stability analysis. This model is applicable to prediction of periodic motions in nonlinear dynamics of gear transmission systems.

Arbitrary Periodic Motions and Grazing Switching of a Forced Piecewise Linear, Impacting Oscillator

2006 ◽  
Vol 129 (3) ◽  
pp. 276-284 ◽  
Author(s):  
Albert C. J. Luo ◽  
Lidi Chen
Keyword(s):  
Nonlinear Dynamics ◽  
Periodic Motion ◽  
Local Stability ◽  
Piecewise Linear ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Grazing Bifurcation ◽  
Piecewise Linear System ◽  
Local Stability Analysis ◽  
Mapping Structures

The grazing bifurcation and periodic motion switching of the harmonically forced, piecewise linear system with impacting are investigated. The generic mappings relative to the discontinuous boundaries of this piecewise system are introduced. Based on such mappings, the corresponding grazing conditions are obtained. The mapping structures are developed for the analytical prediction of periodic motions in such a system. The local stability and bifurcation conditions for specified periodic motions are obtained. The regular and grazing, periodic motions are illustrated. The grazing is the origin of the periodic motion switching for this system. Such a grazing bifurcation cannot be estimated through the local stability analysis. This model is applicable to prediction of periodic motions in nonlinear dynamics of gear transmission systems.

Existence, Stability and Bifurcation of Periodic Motions in a Periodically Forced, Piecewise, Linear Oscillator With Impacts

2004 ◽  
Author(s):  
Albert C. J. Luo ◽  
Lidi Chen
Keyword(s):  
Nonlinear Dynamics ◽  
Periodic Motion ◽  
Piecewise Linear ◽  
Linear Oscillator ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Perfectly Plastic ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
The Stability

The nonlinear dynamics of a generalized, piecewise linear oscillator with perfectly plastic impacts is investigated. The generic mappings based on the discontinuous boundaries are constructed. Furthermore, the mapping structures are developed for the analytical prediction of periodic motions of such a system. The stability and bifurcation conditions for specified periodic motions are obtained. The periodic motions and grazing motion are demonstrated. This model is applicable to prediction of periodic motion in nonlinear dynamics of gear transmission systems.

DYNAMICS OF A HARMONICALLY EXCITED OSCILLATOR WITH DRY-FRICTION ON A SINUSOIDALLY TIME-VARYING, TRAVELING SURFACE

2006 ◽  
Vol 16 (12) ◽  
pp. 3539-3566 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
BRANDON C. GEGG
Keyword(s):  
Relative Motion ◽  
Efficient Method ◽  
Periodic Motion ◽  
Dry Friction ◽  
Local Stability ◽  
Analytical Prediction ◽  
Time Varying ◽  
Periodic Motions ◽  
Mapping Structures ◽  
Excited Oscillator

In this paper, periodic motion in an oscillator moving on the periodically traveling belts with dry friction is investigated. The conditions of stick and nonstick motions for such an oscillator are obtained in the relative motion frame, and the grazing and stick (or sliding) bifurcations are presented as well. The periodic motions of such an oscillator are predicted analytically and numerically, and the analytical prediction is based on the appropriate mapping structures. The local stability and bifurcation for such periodic motions are obtained. The periodic motions are illustrated through the displacement, velocity and force responses in absolute and relative frames. This investigation provides an efficient method to predict periodic motions of such an oscillator involving dry-friction. The significance of this investigation lies in controlling motion of such friction-induced oscillator in industry.

Periodic Motions in a Periodically Forced Oscillator Moving on the Oscillating Belt With Dry Friction

2005 ◽  
Author(s):  
Albert C. J. Luo ◽  
Brandon C. Gegg
Keyword(s):  
Relative Motion ◽  
Efficient Method ◽  
Periodic Motion ◽  
Dry Friction ◽  
Displacement Velocity ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Mapping Structures ◽  
Forced Oscillator

In this paper, periodic motion in an oscillator moving on a periodically vibrating belt with dry-friction is investigated. The conditions of stick and non-stick motions for such an oscillator are obtained in the relative motion frame, and the grazing and stick (or sliding) bifurcations are presented as well. The periodic motions are predicted analytically and numerically, and the analytical prediction is based on the appropriate mapping structures. The eigenvalue analysis of such periodic motions is carried out. The periodic motions are illustrated through the displacement, velocity and force responses in the absolute and relative frames. This investigation provides an efficient method to predict periodic motions of such an oscillator involving dry-friction. The significance of this investigation lies in controlling motion of such friction-induced oscillator in industry.

scholarly journals PERIODIC MOTIONS AND CHAOS WITH IMPACTING CHATTER AND STICK IN A GEAR TRANSMISSION SYSTEM

2009 ◽  
Vol 19 (06) ◽  
pp. 1975-1994 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
DENNIS O'CONNOR
Keyword(s):  
Stability Analysis ◽  
Bifurcation Analysis ◽  
Local Stability ◽  
Transmission System ◽  
Gear Transmission ◽  
Periodic Motions ◽  
Local Stability Analysis ◽  
Gear Transmission System ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

This paper focuses on periodic motions and chaos relative to the impacting chatter and stick in order to find the origin of noise and vibration in such a gear transmission system. Such periodic motions are predicted analytically through mapping structures, and the corresponding local stability and bifurcation analysis are carried out. The grazing and stick conditions presented in [Luo & O'Connor, 2007] are adopted to determine the existence of periodic motions, which cannot be achieved from the local stability analysis. Numerical simulations are performed to illustrate periodic motions and stick motion criteria. Such an investigation may provide some clues to reduce the noise in gear transmission systems.

scholarly journals Motion Switching and Chaos of a Particle in a Generalized Fermi-Acceleration Oscillator

2009 ◽  
Vol 2009 ◽  
pp. 1-40 ◽  
Author(s):  
A. C. J. Luo ◽  
Y. Guo
Keyword(s):  
Local Stability ◽  
Analytical Prediction ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Dynamical Behaviors ◽  
Global View ◽  
Mapping Structures ◽  
Stability And Bifurcations ◽  
First Time

Dynamic behaviors of a particle (or a bouncing ball) in a generalized Fermi-acceleration oscillator are investigated. The motion switching of a particle in the Fermi-oscillator causes the complexity and unpredictability of motion. Thus, the mechanism of motion switching of a particle in such a generalized Fermi-oscillator is studied through the theory of discontinuous dynamical systems, and the corresponding analytical conditions for the motion switching are developed. From solutions of linear systems in subdomains, four generic mappings are introduced, and mapping structures for periodic motions can be constructed. Thus, periodic motions in the Fermi-acceleration oscillator are predicted analytically, and the corresponding local stability and bifurcations are also discussed. From the analytical prediction, parameter maps of periodic and chaotic motions are achieved for a global view of motion behaviors in the Fermi-acceleration oscillator. Numerical simulations are carried out for illustrations of periodic and chaotic motions in such an oscillator. In existing results, motion switching in the Fermi-acceleration oscillator is not considered. The motion switching for many motion states of the Fermi-acceleration oscillator is presented for the first time. This methodology will provide a useful way to determine dynamical behaviors in the Fermi-acceleration oscillator.

An Anaytical Prediction of Period-1 to Chaos in a Periodically Forced, Damped, Hardening Duffing Oscillator

2015 ◽  
Author(s):  
Yu Guo ◽  
Albert Luo
Keyword(s):  
Differential Equation ◽  
Periodic Motion ◽  
Duffing Oscillator ◽  
Numerical Results ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Discrete Maps ◽  
Mapping Structures ◽  
Stability And Bifurcations

In this paper, periodic motions of a periodically forced, damped Duffing oscillator are analytically predicted by use of implicit discrete mappings. The implicit discrete maps are achieved by the discretization of the differential equation of the periodically forced, damped Duffing oscillator. Periodic motion is constructed by mapping structures, and bifurcation trees of periodic motions are developed analytically, and the corresponding stability and bifurcations of periodic motion are determined through eigenvalue analysis. Finally, from the analytical prediction, numerical results of periodic motions are presented to show the complexity of periodic motions.

Periodic Motions in a Periodically Forced Oscillator Moving on an Oscillating Belt With Dry Friction

2006 ◽  
Vol 1 (3) ◽  
pp. 212-220 ◽  
Author(s):  
Albert C.J. Luo ◽  
Brandon C. Gegg
Keyword(s):  
Relative Motion ◽  
Efficient Method ◽  
Periodic Motion ◽  
Dry Friction ◽  
Displacement Velocity ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Mapping Structures ◽  
Forced Oscillator

In this paper, periodic motion in an oscillator moving on a periodically oscillating belt with dry friction is investigated. The conditions of stick and nonstick motions for such an oscillator are obtained in the relative motion frame, and the grazing and stick (or sliding) bifurcations are presented as well. The periodic motions are predicted analytically and numerically, and the analytical prediction is based on the appropriate mapping structures. The eigenvalue analysis of such periodic motions is carried out. The periodic motions are illustrated through the displacement, velocity, and force responses in the absolute and relative frames. This investigation provides an efficient method to predict periodic motions of such an oscillator involving dry friction. The significance of this investigation lies in controlling motion of such a friction-induced oscillator in industry.

Nonlinear Dynamics of a Gear Transmission System: Part II — Periodic Impacting Chatter and Stick

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis O’Connor
Keyword(s):  
Local Stability ◽  
Transmission System ◽  
Gear Transmission ◽  
Periodic Motions ◽  
Motion Mechanism ◽  
Local Stability Analysis ◽  
Gear Transmission System ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
System Part

In Part I, the motion mechanism of impacting chatter and stick motion in the gear transmission dynamical system was discussed. This paper focuses on periodic motions relative to the impacting chatter and stick in order to find the origin of noise and vibration in such a gear transmission system. Such periodic motions are predicted analytically through mapping structures, and the corresponding local stability and bifurcation analysis are carried out. The grazing and stick conditions presented in Part I are adopted to determine the existence of periodic motions, which cannot be achieved from the local stability analysis. Numerical simulations are performed to illustrate periodic motions and stick motion criteria. Such an investigation may provide some clues to reduce the noise in gear transmission systems.

Bifurcation Tree of Period-1 Motion to Chaos in a Pendulum With Periodic Excitation

2016 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Harmonic Amplitude ◽  
Analytical Prediction ◽  
Algebraic Equations ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Discrete Maps ◽  
Periodically Driven ◽  
Mapping Structures ◽  
Nonlinear Algebraic Equations ◽  
Analytical Bifurcation Trees

In this paper, the bifurcation trees of period-1 motions to chaos are presented in a periodically driven pendulum. Discrete implicit maps are obtained through a mid-time scheme. Using these discrete maps, mapping structures are developed to describe different types of motions. Analytical bifurcation trees of periodic motions to chaos are obtained through the nonlinear algebraic equations of such implicit maps. Eigenvalue analysis is carried out for stability and bifurcation analysis of the periodic motions. Finally, numerical simulation results of various periodic motions are illustrated in verification to the analytical prediction. Harmonic amplitude characteristics are also be presented.

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