Complex Motions in a Periodically Forced Oscillator With Multiple Discontinuities

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Mehul T. Patel
Keyword(s):  
Singularity Theory ◽  
Periodic Motions ◽  
Discontinuous System ◽  
Chaotic Motions ◽  
Discontinuous Boundary ◽  
Mapping Structures ◽  
Forced Oscillator ◽  
Local Singularity ◽  
Stability And Bifurcations ◽  
Motion Flow

The complex motions in periodically forced oscillator with multiple discontinuities are investigated in this paper. From the local singularity theory in discontinuous systems, the passability condition of motion flow from one domain to the adjacent one in phase space is presented, and the sliding and grazing conditions of motion flows to discontinuous boundaries are given to understand the mechanism of such complex motions. From the separation boundary, the generic mappings are defined for mapping structures. A generalized mapping structure is introduced for all complex periodic motions, and the local stability and bifurcations of periodic motions in such a discontinuous system are discussed. The passability of motion flow to the discontinuous boundary is used for motion continuity, and the grazing condition of flow to the discontinuous boundary is employed for the vanishing of complex motion. Complex periodic motions with sliding and grazing in such a discontinuous system are predicted analytically and chaotic motions are also illustrated.

Bifurcation and Stability of Periodic Motions in a Periodically Forced Oscillator With Multiple Discontinuities

2008 ◽  
Vol 4 (1) ◽  
Author(s):  
Albert C. J. Luo ◽  
Mehul T. Patel
Keyword(s):  
Local Stability ◽  
Singularity Theory ◽  
Periodic Motions ◽  
Sliding Motion ◽  
Discontinuous Boundary ◽  
Mapping Structures ◽  
Forced Oscillator ◽  
Local Singularity ◽  
Stability And Bifurcations ◽  
Bifurcation And Stability

The local stability and existence of periodic motions in a periodically forced oscillator with multiple discontinuities are investigated. The complexity of periodic motions and chaos in such a discontinuous system is often caused by the passability, sliding, and grazing of flows to discontinuous boundaries. Therefore, the corresponding analytical conditions for such singular phenomena to discontinuous boundary are presented from the local singularity theory of discontinuous systems. To develop the mapping structures of periodic motions, basic mappings are introduced, and the sliding motion on the discontinuous boundary is described by a sliding mapping. A generalized mapping structure is presented for all possible periodic motions, and the local stability and bifurcations of periodic motions are discussed. From mapping structures, the switching points of periodic motions on the boundaries are predicted analytically. Two periodic motions are presented for illustrations of the passability, sliding, and grazing of periodic motions on the boundary.

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.

On Period-1 and Period-2 Motions of a Jeffcott Rotor System

2019 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Solutions ◽  
Mapping Method ◽  
Rotor System ◽  
Periodic Motions ◽  
Period 2 ◽  
Jeffcott Rotor ◽  
Mapping Structures ◽  
Point Scheme ◽  
Stability And Bifurcations ◽  
Specific Mapping

Abstract In this paper, the semi-analytical solutions of period-1 and period-2 motions in a nonlinear Jeffcott rotor system are presented through the discrete mapping method. The periodic motions in the nonlinear Jeffcott rotor system are obtained through specific mapping structures with a certain accuracy. A bifurcation tree of period-1 to period-2 motion is achieved, and the corresponding stability and bifurcations of periodic motions are analyzed. For verification of semi-analytical solutions, numerical simulations are carried out by the mid-point scheme.

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.

Dynamic Mechanism of a Train Suspension System

2010 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis O’Connor
Keyword(s):  
Numerical Simulations ◽  
System Parameter ◽  
Suspension System ◽  
Dynamic Mechanism ◽  
Impact Model ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors ◽  
Mapping Structures

Nonlinear dynamical behaviors of a train suspension system with impacts are investigated. The suspension system is modelled through an impact model with possible stick between a bolster and two wedges. Based on the mapping structures, periodic motions of such a system are described. To understand the global dynamical behaviors of the train suspension system, system parameter maps are developed. Numerical simulations for periodic and chaotic motions are performed from the parameter maps.

Analytical Dynamics of a Ball Bouncing on a Vibrating Table

2012 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Numerical Simulations ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Theory Of Flow ◽  
Mapping Structures ◽  
Ball Bouncing ◽  
Vibrating Table

In this paper, the theory of flow switchability for discontinuous dynamical systems is applied. Domains and boundaries for such a discontinuous problem are defined and analytical conditions for motion switching are developed. The conditions explain the important role of switching phase on the motion switchability in such a system. To describe different motions, the generic mappings and mapping structures are introduced. Bifurcation scenarios for periodic and chaotic motions are presented for different motions and switchability. Numerical simulations are provided for periodic motions with impacts only and with impact chatter to stick in the system.

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.

On Symmetric Periodic Motions With Different Excitation Periods in a Discontinuous System With a Hyperbolic Boundary

2020 ◽  
Author(s):  
Albert C. J. Luo ◽  
Chuanping Liu
Keyword(s):  
Dynamic System ◽  
Numerical Simulations ◽  
Periodic Motions ◽  
Discontinuous System ◽  
Mapping Structures ◽  
Specific Mapping ◽  
Hyperbolic Boundary

Abstract In this paper, symmetric periodic motions with different excitation periods in a discontinuous dynamic system with a hyperbolic boundary are presented analytically. The switchability conditions of flows at the hyperbolic boundaries are developed. Periodic motions with specific mapping structures are predicted analytically, and numerical simulations of periodic motions are carried out. The corresponding G-functions are presented for illustration of motion switchability at the hyperbolic boundaries.

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.

Periodic Motions of the Machine Tools in Cutting Process

2007 ◽  
Author(s):  
Brandon C. Gegg ◽  
Steve S. Suh ◽  
Albert C. J. Luo
Keyword(s):  
Mechanical Model ◽  
Machine Tools ◽  
Degree Of Freedom ◽  
Cutting Process ◽  
Force Distribution ◽  
Periodic Motions ◽  
Numerical Predictions ◽  
Discontinuous Boundary ◽  
Mapping Structures ◽  
Two Degree Of Freedom

In this paper, a two-degree of freedom dynamical system with discontinuity is developed to describe the vibration in the cutting process. The analytical solutions for the switchability of motion on the discontinuous boundary are presented. the switching sets based on the discontinuous boundary is introduced and the basic mappings are introduced to investigate periodic motion in such a mechanical model. The mapping structures for the stick and non-stick motions are discussed. Numerical predictions of motions of the machine tool in the cutting process are presented through the two-degree of freedom system with discontinuity. The phase trajectory and velocity and force responses are presented and the switchability of motion on the discontinuous boundary is illustrated through force distribution and force product on the boundary.

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