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.

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 an Autonomous System With a Discontinuous Vector Field

2020 ◽  
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Autonomous System ◽  
Parameter Study ◽  
Spring Stiffness ◽  
Algebraic Equations ◽  
Periodic Motions ◽  
Mapping Structures ◽  
The Stability

Abstract In this paper, periodic motions in an autonomous system with a discontinuous vector field are discussed. The periodic motions are obtained by constructing a set of algebraic equations based on motion mapping structures. The stability of periodic motions is investigated through eigenvalue analysis. The grazing bifurcations are presented by varying the spring stiffness. Once the grazing bifurcation occurs, periodic motions switches from the old motion to a new one. Numerical simulations are conducted for motion illustrations. The parameter study helps one understand autonomous discontinuous dynamical systems.

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.

Periodic Motions in a Discontinuous Dynamical System With Two Circular Boundaries

2019 ◽  
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Periodic Motions ◽  
Discontinuous Dynamical System ◽  
Mapping Structures

Abstract In this paper, periodic motions in a discontinuous dynamical systems are studied. The discontinuous dynamical system has three domains partitioned through two circular boundaries. On the three domains, there are three distinct dynamical systems. From the G-functions, the switchability conditions of a flow from one domain to anther domain at the boundary are developed. The flow mappings from a boundary to a bounbary are developed for each domain and boundary. From the mapping structures, periodic motions in the discontinuous dynamical system are predicted. Numerical simulations of periodic motions and motion switchability at boundaries are presented in the discontinuous dynamical system.

Switching Mechanism and Complex Motions in an Extended Fermi-Acceleration Oscillator

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Extended Model ◽  
Time Varying ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Bouncing Ball ◽  
Mapping Structures ◽  
The Masses ◽  
First Time

In this paper, an extended model of the Fermi-acceleration oscillator is presented to describe impacting chatters, grazing, and sticking between the particle (or bouncing ball) and piston. The sticking phenomenon in such a system is investigated for the first time. Even in the traditional Fermi-oscillator, such a sticking phenomenon still exists but one often ignored it. In this paper, the analytical conditions for the grazing and sticking phenomena between the particle and piston in the Fermi-acceleration oscillator are developed from the theory of discontinuous dynamical systems. Compared with existing studies, the four exact mappings are used to analyze the motion behaviors of the Fermi-oscillator instead of one or two mappings. Mapping structures formed by generic mappings are adopted for the analytical predictions of periodic motions in the Fermi-acceleration oscillator. Periodic and chaotic motions in such an oscillator are illustrated to show motion complexity and grazing and sticking mechanism. Once the masses of the ball and primary mass are in the same quantity level, the model presented in this paper will be very useful and significant. This idea can apply to a system possessing two independent oscillators with impact, such as gear transmission systems, bearing systems, and time-varying billiard systems.

Analytical Dynamics of Complex Motions in a Train Suspension System

2011 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis O’Conner
Keyword(s):  
Numerical Simulations ◽  
Suspension System ◽  
Analytical Prediction ◽  
Impact Model ◽  
Analytical Dynamics ◽  
Complex Motion ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors ◽  
Mapping Structures

Nonlinear dynamical behaviors of a train suspension system with impacts are investigated. The suspension system is described through an impact model with possible stick between a bolster and two wedges. The analytical conditions that reflect the motion mechanisms for the complex motion are given. The mapping structures for periodic and chaotic motions of such a system can be described. The analytical prediction of the complex motions can be conducted from the mapping structure, and numerical simulations for periodic and chaotic motions can be carried out in sequel.

PARAMETRIC ANALYSIS OF BIFURCATION AND CHAOS IN A PERIODICALLY DRIVEN HORIZONTAL IMPACT PAIR

2012 ◽  
Vol 22 (11) ◽  
pp. 1250268 ◽  
Author(s):  
YU GUO ◽  
ALBERT C. J. LUO
Keyword(s):  
Dynamical Systems ◽  
Parametric Analysis ◽  
Periodic Motions ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
Periodically Driven ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
Different Types ◽  
The Impact

In this paper, complex motions and chaos in the periodically driven horizontal impact pair are investigated using the theory of switchability for discontinuous dynamical systems. Domains and boundaries are defined due to the discontinuity caused by impacts. Analytical conditions for switching of stick and grazing motions are derived in detail. Generic mappings are introduced to describe different periodic motions and to identify the mapping structures of chaos. The periodic motions in such impact pair are analytically predicted, and the corresponding stability and bifurcation analysis of periodic motions are carried out. Parameter maps with different types of motions are developed. Periodic and chaotic motions with different mapping structures are illustrated numerically for a better understanding of physics of ball motions in the impact pair.

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.

Bifurcation Analysis of a Fermi-Acceleration Oscillator Under Different Excitations

2011 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Chaotic Motions

In this paper, switchability and bifurcation of motions in a double excited Fermi acceleration oscillator is discussed using the theory of discontinuous dynamical systems. The two oscillators are chosen to have different excitation and parameters. The analytical conditions for motion switching in such a Fermi-oscillator are presented. Bifurcation scenario for periodic and chaotic motions is presented, and the analytical predictions of periodic motions are presented. Finally, different motions in such an oscillator are illustrated.

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.

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