Periodic Motions With Grazing in a Discontinuous Dynamical System With Two Circular Boundaries

2019 ◽  
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Periodic Motion ◽  
Necessary Conditions ◽  
Periodic Motions ◽  
Discontinuous System ◽  
Discontinuous Dynamical System ◽  
Sufficient And Necessary Conditions

Abstract In this paper, studied are periodic motions with grazing in a discontinuous dynamical system with two circular boundaries. The grazing motion is for a periodic motion switching to another periodic motions. Thus, the sufficient and necessary conditions of motion switching, grazing and sliding on the boundaries are discussed first. Periodic motions with grazing in the discontinuous system are presented for illustration of motions switching.

scholarly journals Analysis of Discontinuous Dynamical Behaviors of a Friction-Induced Oscillator with an Elliptic Control Law

2018 ◽  
Vol 2018 ◽  
pp. 1-33 ◽  
Author(s):  
Jinjun Fan ◽  
Ping Liu ◽  
Tianyi Liu ◽  
Shan Xue ◽  
Zhaoxia Yang
Keyword(s):  
Dynamical System ◽  
Vector Fields ◽  
Control Law ◽  
Periodic Motions ◽  
Separation Boundary ◽  
Discontinuous System ◽  
Dynamical Behaviors ◽  
Discontinuous Dynamical System ◽  
Discontinuous Boundary ◽  
Mass Spring

This paper develops the passability conditions of flow to the discontinuous boundary and the sticking or sliding and grazing conditions to the separation boundary in the discontinuous dynamical system of a friction-induced oscillator with an elliptic control law and the friction force acting on the mass M through the analysis of the corresponding vector fields and G-functions. The periodic motions of such a discontinuous system are predicted analytically through the mapping structure. Finally, the numerical simulations are given to illustrate the analytical results of motion for a better understanding of physics of motion in the mass-spring-damper oscillator.

Analytical Predication of Complex Motion of a Ball in a Periodically Shaken Horizontal Impact Pair

2011 ◽  
Vol 7 (2) ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Periodic Motion ◽  
Analytical Prediction ◽  
Periodic Excitation ◽  
Complex Motion ◽  
Chaotic Motions ◽  
Discontinuous Dynamical System ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, complex motions of a ball in the horizontal impact pair with a periodic excitation are studied analytically using the theory of discontinuous dynamical system. Analytical conditions for motion switching caused by impacts are developed, and generic mapping structures are introduced to describe different periodic and chaotic motions. Analytical prediction of complex periodic motion of the ball in the periodically shaken impact pair is completed, and the corresponding stability and bifurcation analysis are also carried out. Numerical illustrations of periodic and chaotic motions are given.

Switching Dynamics of a Periodically Forced Discontinuous System With an Inclined Boundary

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Brandon M. Rapp
Keyword(s):  
Dynamical System ◽  
Vector Fields ◽  
Normal Vector ◽  
Periodic Motions ◽  
Separation Boundary ◽  
Discontinuous System ◽  
Normal Vector Field ◽  
Periodically Driven ◽  
Switching Dynamics ◽  
Straight Line

This paper presents the switching dynamics of flow from one domain into another one in the periodically driven, discontinuous dynamical system. The simple inclined straight line boundary in phase space is considered as a control law for the dynamical system to switch. The normal vector-field product for flow switching on the separation boundary is introduced, and the passability condition of flow to the discontinuous boundary is presented. The sliding and grazing conditions to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the local stability and bifurcation analysis are carried out. Numerical illustrations of periodic motions with grazing to the boundary and/or sliding on the boundary are given, and the normal vector fields are illustrated to show the analytical criteria.

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.

Complex Dynamics of Bouncing Motions at Boundaries and Corners in a Discontinuous Dynamical System

2017 ◽  
Vol 12 (6) ◽  
Author(s):  
Jianzhe Huang ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Vector Fields ◽  
Complex Dynamics ◽  
Singularity Theory ◽  
Local Theory ◽  
Periodic Motions ◽  
Border Collision Bifurcations ◽  
Discontinuous Dynamical System ◽  
Stability And Bifurcations

In this paper, from the local theory of flow at the corner in discontinuous dynamical systems, obtained are analytical conditions for switching impact-alike chatter at corners. The objective of this investigation is to find the dynamics mechanism of border-collision bifurcations in discontinuous dynamical systems. Multivalued linear vector fields are employed, and generic mappings are defined among boundaries and corners. From mapping structures, periodic motions switching at the boundaries and corners are determined, and the corresponding stability and bifurcations of periodic motions are investigated by eigenvalue analysis. However, the grazing and sliding bifurcations are determined by the local singularity theory of discontinuous dynamical systems. From such analytical conditions, the corresponding parameter map is developed for periodic motions in such a multivalued dynamical system in the single domain with corners. Numerical simulations of periodic motions are presented for illustrations of motions complexity and catastrophe in such a discontinuous dynamical system.

Complex Motions in Horizontal Impact Pairs With a Periodic Excitation

2011 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Periodic Motion ◽  
Analytical Prediction ◽  
Periodic Excitation ◽  
Chaotic Motions ◽  
Discontinuous Dynamical System ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis

In this paper, the horizontal impact pair with a periodic excitation is studied from the theory of discontinuous dynamical system. Analytical conditions for motion switching are obtained. From generic mappings, analytical prediction of periodic motion is presented, and the corresponding stability and bifurcation analysis are carried out. Periodic and chaotic motions are illustrated numerically.

Grazing Bifurcations and Parametric Characterization of a Hysterically Damped Semi-Active Suspension System

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arun Rajendran
Keyword(s):  
Necessary Conditions ◽  
Active Suspension ◽  
Suspension System ◽  
Local Theory ◽  
Final States ◽  
Discontinuous System ◽  
Comprehensive Investigation ◽  
System Parameters ◽  
Sufficient And Necessary Conditions

The criterion for grazing motions in a hysterically damped semi-active suspension system is obtained from the local theory of non-smooth dynamical systems on the connectable and accessible domains. The generic mappings for such a suspension system are introduced. The sufficient and necessary conditions for grazing at the final states of mappings are expressed. The initial and final switching sets of grazing mapping, varying with system parameters, are illustrated for the grazing parametric characteristics. The initial and grazing, switching manifolds in the switching sets are defined through grazing mappings. Finally, numerical illustrations of grazing motions are very easily carried out with help of the analytical predictions. This paper provides a comprehensive investigation of grazing motions in the suspension system for a better understanding of the grazing mechanism of such a discontinuous system.

On Existence and Bifurcations of Periodic Motions in Discontinuous Dynamical Systems

2021 ◽  
Vol 31 (04) ◽  
pp. 2150063
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Mechanical Model ◽  
System Parameter ◽  
Period Doubling ◽  
Periodic Motions ◽  
Discontinuous System ◽  
Saddle Node Bifurcation ◽  
Discontinuous Dynamical System ◽  
Mapping Structures ◽  
Parametric Maps

In this paper, the existence and bifurcations of periodic motions in a discontinuous dynamical system is studied through a discontinuous mechanical model. One can follow the study presented herein to investigate other discontinuous dynamical systems. Such a sampled discontinuous system consists of two subsystems on boundaries and three subsystems in subdomains. From the theory of discontinuous dynamical systems, switchability conditions of a flow at and on the boundaries are developed. From such switchability conditions, grazing motions of a flow at boundaries are discussed, and sliding motions of a flow on boundaries are presented. Based on the motions in each domain and on each boundary, generic mappings are introduced. Using the generic mappings, mapping structures for specific periodic motions are developed. Based on the grazing conditions and appearance and vanishing conditions of sliding motions, parametric dynamics of the existences of the specific periodic motions are presented. In addition, the traditional saddle-node bifurcation, Neimark bifurcations and period-doubling bifurcation are used for parametric dynamics of periodic motions. Bifurcation trees of periodic motions varying with a system parameter are presented first, and phase trajectories of periodic motions are illustrated. The [Formula: see text]-functions are presented for the illustration of the motion switchability at the boundaries and sliding motions on the boundaries. Codimension-2 parametric dynamics of periodic motions are studied and how to develop the 2D parametric maps for specific periodic motions are presented. In the end, periodic motions with grazing are illustrated.

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