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.

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.

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 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.

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.

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.

Periodic Motions of a Semi-Active Suspension System With MR Damping.

2006 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arun Rajendran
Keyword(s):  
Piecewise Linear ◽  
Active Suspension ◽  
Suspension System ◽  
Mapping Technique ◽  
Periodic Motions ◽  
Mapping Structures ◽  
The Stability ◽  
Magneto Rheological ◽  
Mr Damping ◽  
Full Nonlinearity

Periodic motions in a hysteretically damped, semi-active suspension system are investigated. The Magneto-Rheological damping varying with relative velocity is modeled through a piecewise-linear model. The theory for discontinuous dynamical systems is employed to determine the grazing motions in such a system, and the mapping technique is used to develop the mapping structures of periodic motions. The periodic motions are predicted analytically and verified numerically. The stability and bifurcation analyses of such periodic motions are performed, and the parameters for all possible motions are developed. This model is applicable for the semi-active suspension system with the Magneto-Rheological damper in automobiles. The further investigation on the Magneto-Rheological damping with full nonlinearity should be completed.

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.

Nonlinear Dynamic Analysis of a Flexible Rod Fastening Rotor Bearing System

2010 ◽  
Author(s):  
Heng Liu ◽  
Chen Li ◽  
Weimin Wang ◽  
Xiaobin Qi ◽  
Minqing Jing
Keyword(s):  
Periodic Motion ◽  
Great Influence ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mass Eccentricity ◽  
Rotor Bearing ◽  
Rod Fastening Rotor ◽  
The Stability ◽  
Tie Rods ◽  
Bearing System

This paper is concerned the stability and bifurcation of a flexible rod-fastening rotor bearing system (FRRBS). Here the shaft is considered as an integral or continuous structure and be modeled by using Timoshenko beam-shaft element which can take the effects of axial load into consideration. And using Hamilton’s principle, model tie rods distributed along the circumference as a constant stiffness matrix and an add-moment which caused by unbalanced pre-tightening forces. Then the model is reduced by a component mode synthesis method, which can conveniently account for nonlinear oil film forces of the bearing. This study focuses on the influence of nonlinearities on the stability and bifurcation of T periodic motion of the FRRBS subjected to the influence of mass eccentricity. The periodic motions and their stability margin are obtained by shooting method and path-following technique. The local stability and bifurcation behaviors of periodic motions are obtained by Floquet theory. The results indicate that mass eccentricity and unbalanced pre-tightening forces of tie rods have great influence on nonlinear stability and bifurcation of the T periodic motion of system, cause the spillover of system nonlinear dynamics and degradation of stability and bifurcation of T periodic motion.

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.

Effect of Thrust Magnetic Bearing on Stability and Bifurcation of a Flexible Rotor Active Magnetic Bearing System

2003 ◽  
Vol 125 (3) ◽  
pp. 307-316 ◽  
Author(s):  
Y. S. Ho ◽  
H. Liu ◽  
L. Yu
Keyword(s):  
Periodic Motion ◽  
Magnetic Bearing ◽  
Rotor System ◽  
Magnetic Bearings ◽  
Active Magnetic Bearing ◽  
Flexible Rotor ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mass Eccentricity ◽  
The Stability

This paper is concerned with the effect of a thrust active magnetic bearing (TAMB) on the stability and bifurcation of an active magnetic bearing rotor system (AMBRS). The shaft is flexible and modeled by using the finite element method that can take the effects of inertia and shear into consideration. The model is reduced by a component mode synthesis method, which can conveniently account for nonlinear magnetic forces and moments of the bearing. Then the system equations are obtained by combining the equations of the reduced mechanical system and the equations of the decentralized PID controllers. This study focuses on the influence of nonlinearities on the stability and bifurcation of T periodic motion of the AMBRS subjected to the influences of both journal and thrust active magnetic bearings and mass eccentricity simultaneously. In the stability analysis, only periodic motion is investigated. The periodic motions and their stability margins are obtained by using shooting method and path-following technique. The local stability and bifurcation behaviors of periodic motions are obtained by using Floquet theory. The results indicate that the TAMB and mass eccentricity have great influence on nonlinear stability and bifurcation of the T periodic motion of system, cause the spillover of system nonlinear dynamics and degradation of stability and bifurcation of T periodic motion. Therefore, sufficient attention should be paid to these factors in the analysis and design of a flexible rotor system equipped with both journal and thrust magnetic bearings in order to ensure system reliability.

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