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.

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.

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.

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.

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.

On the Mechanism of Stick and Non-Stick, Periodic Motions in a Forced Linear Oscillator Including Dry Friction

2004 ◽  
Author(s):  
Albert C. J. Luo ◽  
Brandon C. Gegg
Keyword(s):  
Dry Friction ◽  
Local Stability ◽  
Displacement Velocity ◽  
Linear Oscillator ◽  
Periodic Motions ◽  
Friction Forces ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Linear Friction ◽  
Friction Oscillator

In this paper, the dynamics mechanism of stick and non-stick motion for a dry-friction oscillator is discussed. From the theory of Luo in 2004, the conditions for stick and non-stick motions are achieved. The stick and non-stick periodic motions are predicted analytically through the appropriate mapping structures. The local stability and bifurcation for such periodic motions are obtained. The stick motions are illustrated through the displacement, velocity and force responses. This investigation provides a better understanding of stick and nonstick motions of the linear oscillator with dry-friction. The methodology presented in this paper is applicable to oscillators with non-linear friction forces.

Periodic Motions in the Periodically Forced Oscillator With Multiple Discontinuities

2006 ◽  
Author(s):  
Albert C. J. Luo ◽  
Mehul T. Patel
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Forced Oscillator ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the stability and bifurcation of periodic motions in periodically forced oscillator with multiple discontinuities is investigated. The generic mappings are introduced for the analytical prediction of periodic motions. Owing to the multiple discontinuous boundaries, the mapping structures for periodic motions are very complicated, which causes more difficulty to obtain periodic motions in such a dynamical system. The analytical prediction of complex periodic motions is carried out and verified numerically, and the corresponding stability and bifurcation analysis are performed. Due to page limitation, grazing and stick motions and chaos in this system will be investigated further.

On the Mechanism of Stick and Nonstick, Periodic Motions in a Periodically Forced, Linear Oscillator With Dry Friction

2005 ◽  
Vol 128 (1) ◽  
pp. 97-105 ◽  
Author(s):  
Albert C. J. Luo ◽  
Brandon C. Gegg
Keyword(s):  
Dry Friction ◽  
Local Stability ◽  
Displacement Velocity ◽  
Linear Oscillator ◽  
Nonlinear Friction ◽  
Periodic Motions ◽  
Friction Forces ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Friction Oscillator

In this paper, the dynamics mechanism of stick and nonstick motion for a dry-friction oscillator is discussed. From the theory of Luo in 2005 [Commun. Nonlinear Sci. Numer. Simul., 10, pp. 1–55], the conditions for stick and nonstick motions are achieved. The stick and nonstick periodic motions are predicted analytically through the appropriate mapping structures. The local stability and bifurcation conditions for such periodic motions are obtained. The stick motions are illustrated through the displacement, velocity, and force responses. This investigation provides a better understanding of stick and nonstick motions of the linear oscillator with dry friction. The methodology presented in this paper is applicable to oscillators with nonlinear friction forces.

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.

Periodic Motions in a Coupled Van Der Pol-Duffing Oscillator

2017 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, periodic motions of a periodically forced, coupled van der Pol-Duffing oscillator are predicted analytically. The coupled van der Pol-Duffing oscillator is discretized for the discrete mapping. The periodic motions in such a coupled van der Pol-Duffing oscillator are obtained from specified mapping structures, and the corresponding stability and bifurcation analysis are carried out by eigenvalue analysis. Based on the analytical prediction, the initial conditions of periodic motions are used for numerical simulations.

Bifurcation Trees of Period-1 Motions to Chaos in a Quadratic Nonlinear Oscillator With Time-Delayed Displacement

2015 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyuan Xing
Keyword(s):  
Differential Equation ◽  
Bifurcation Analysis ◽  
Nonlinear Oscillator ◽  
Numerical Results ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Discrete Maps ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, periodic motions in a periodically forced, damped, quadratic nonlinear oscillator with time-delayed displacement are analytically predicted through implicit discrete mappings. The implicit discrete maps are obtained from discretization of differential equation of such a quadratic nonlinear oscillator. From mapping structures, bifurcation trees of periodic motions are achieved analytically, and the corresponding stability and bifurcation analysis are completed through eigenvalue analysis. From the analytical prediction, numerical results of periodic motions are illustrated to verify such an analytical prediction.

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