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.

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.

Dynamics of a Simplified van der Pol Oscillator Revisited

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arun Rajendran
Keyword(s):  
Bifurcation Analysis ◽  
Local Stability ◽  
Van Der Pol Oscillator ◽  
Nonsmooth Dynamics ◽  
Mapping Technique ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Chaotic Motions ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, the dynamic characteristics of a simplified van der Pol oscillator are investigated. From the theory of nonsmooth dynamics, the structures of periodic and chaotic motions for such an oscillator are developed via the mapping technique. The periodic motions with a certain mapping structures are predicted analytically for m-cycles with n-periods. Local stability and bifurcation analysis for such motions are carried out. The (m:n)-periodic motions are illustrated. The further investigation of the stable and unstable periodic motions in such a system should be completed. The chaotic motion based on the Levinson donuts should be further discussed.

Periodic Motions With Impacting Chatter and Stick in a Gear Transmission System

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis O’Connor
Keyword(s):  
Transmission System ◽  
Gear Transmission ◽  
Time Varying ◽  
Impact Model ◽  
Periodic Motions ◽  
Transmission Systems ◽  
New Model ◽  
Gear Transmission System ◽  
Mapping Structures ◽  
First Time

In this paper, an impact model with possible stick between the two gears is proposed for gear transmission systems, which includes the piecewise backlash model and the traditional impact model for the first time. The new model presented in this paper possesses a time-varying boundary for two dynamical systems either to switch or to impact. Such a model can catch impacting chatter and stick phenomena in gear transmission systems. Based on the new model, periodic impacting chatter and stick in a gear transmission system can be investigated. For doing so, switching sets on the time-varying boundaries are introduced to define basic mappings. Mapping structures based on basic mappings are developed for characterizing motions in gear transmission systems, and from such mapping structures, periodic motions with impacting chatter and stick in such a gear transmission system are predicted analytically. Numerical simulations are performed for illustration of periodic motions with impacting chatter and stick phenomena.

Impacting chatter and stick in a transmission system with two oscillators

2009 ◽  
Vol 223 (3) ◽  
pp. 159-188 ◽  
Author(s):  
A C J Luo ◽  
D O'Connor
Keyword(s):  
Local Stability ◽  
Transmission System ◽  
Gear Transmission ◽  
Periodic Motions ◽  
Transmission Systems ◽  
Motion Mechanism ◽  
System Parameters ◽  
Mapping Structures ◽  
Non Linear ◽  
The Relationship

In this article, an investigation on non-linear dynamical behaviours of a transmission system with a gear pair is conducted. The transmission system is described through an impact model with a possible stick between the two gears. From the theory of discontinuous dynamical systems, the motion mechanism of impacting chatter with stick is investigated. The onset and vanishing conditions for stick motions are developed, and the condition for maintaining stick motion is obtained as well. The corresponding physical interpretation is given for a better understanding of non-linear behaviours of gear transmission systems. A parameter map is presented to provide a global picture of the relationship between system parameters and corresponding motion. Grazing and stick conditions are utilized to determine the existence of periodic motions. Such periodic motions are predicted analytically through mapping structures, and the corresponding local stability and bifurcation analyses are carried out. Numerical simulations are performed to illustrate periodic motions and stick motion criteria. A better understanding of the motion mechanism between two gears may be helpful for improving the efficiency of gear transmission systems.

SINGULARITY, SWITCHABILITY AND BIFURCATIONS IN A 2-DOF, PERIODICALLY FORCED, FRICTIONAL OSCILLATOR

2013 ◽  
Vol 23 (03) ◽  
pp. 1330009 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
MOZHDEH S. FARAJI MOSADMAN
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Degrees Of Freedom ◽  
Eigenvalue Analysis ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the analytical dynamics for singularity, switchability, and bifurcations of a 2-DOF friction-induced oscillator is investigated. The analytical conditions of the domain flow switchability at the boundaries and edges are developed from the theory of discontinuous dynamical systems, and the switchability conditions of boundary flows from domain and edge flows are presented. From the singularity and switchability of flow to the boundary, grazing, sliding and edge bifurcations are obtained. For a better understanding of the motion complexity of such a frictional oscillator, switching sets and mappings are introduced, and mapping structures for periodic motions are adopted. Using an eigenvalue analysis, the stability and bifurcation analysis of periodic motions in the friction-induced system is carried out. Analytical predictions and parameter maps of periodic motions are performed. Illustrations of periodic motions and the analytical conditions are completed. The analytical conditions and methodology can be applied to the multi-degrees-of-freedom frictional oscillators in the same fashion.

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.

Stability and Bifurcation Analysis of a Delayed Discrete Predator–Prey Model

2018 ◽  
Vol 28 (09) ◽  
pp. 1850116 ◽  
Author(s):  
A. M. Yousef ◽  
S. M. Salman ◽  
A. A. Elsadany
Keyword(s):  
Stability Analysis ◽  
Local Stability ◽  
Chaotic Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Stability Analysis ◽  
Prey Model ◽  
Stability And Bifurcation Analysis ◽  
Different Types ◽  
Two Parameters

A discrete predator–prey model with delayed density dependence in the rate of growth of the prey is considered. In particular, we analyze the model presented by Kot [2005] which consists of three coupled difference equations and contains two parameters. Existence and local stability analysis of fixed points of the model are addressed. The normal form technique and perturbation method are applied to the different types of bifurcations that exist in the model being investigated. It is proved that the existence of transcritical and Neimark–Sacker bifurcations can occur in the model. In addition, the chaotic behavior of the model in the sense of Marotto is proved. To verify the results obtained analytically, we perform numerical simulations which also explore further the richer dynamics of the model.

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.

Period-1 Motions to Chaos in a Parametrically Excited Pendulum

2017 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Numerical Methods ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Periodic Motions ◽  
Parametrically Excited ◽  
Pendulum System ◽  
Discrete Maps ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
Simulation Results

In this paper, bifurcation trees of period-1 motions to chaos are investigated in a parametrically excited pendulum. To construct discrete mapping structures of periodic motions, implicit discrete maps are developed for such a pendulum system. The bifurcation trees from period-1 motions to chaos are predicted semi-analytically through period-1 to period-4 motions. The corresponding stability and bifurcation analysis are carried out through eigenvalue analysis. Finally, numerical simulations of periodic motions can be completed through numerical methods. Such simulation results are illustrated for verification of the analytical predictions.

Complete Bifurcation Trees of Independent Period-2 Motions to Chaos in a Parametrically Excited Pendulum

2019 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Bifurcation Analysis ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Periodic Motions ◽  
Parametrically Excited ◽  
Stability And Bifurcation ◽  
Period 2 ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

Abstract In this paper, bifurcation trees of independent period-2 motions to chaos are investigated in a parametrically excited pendulum. The implicit discrete mapping method is employed to obtain periodic motions in such a system. Analytical predictions of periodic motions are based on the mapping structures and peroidicity. The bifurcation trees of independent period-2 motions to chaos are studied, and the corresponding stability and bifurcation analysis are completed through eigenvalue analysis. Finally, sampled period-2 motions are simulated numerically in comparison to the analytical predictions. The infinite bifurcation trees of independent period-2 motions to chaos can be obtained.

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