scholarly journals On Dynamical Behavior of a Friction-Induced Oscillator with 2-DOF on a Speed-Varying Traveling Belt

2017 ◽  
Vol 2017 ◽  
pp. 1-19 ◽  
Author(s):  
Jinjun Fan ◽  
Shuangshuang Li ◽  
Ge Chen
Keyword(s):  
Dynamical Systems ◽  
Mechanical Model ◽  
Dynamical Behavior ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
The Masses ◽  
Motion Complexity

The dynamical behavior of a friction-induced oscillator with 2-DOF on a speed-varying belt is investigated by using the flow switchability theory of discontinuous dynamical systems. The mechanical model consists of two masses and a speed-varying traveling belt. Both of the masses on the traveling belt are connected with three linear springs and three dampers and are harmonically excited. Different domains and boundaries for such system are defined according to the friction discontinuity. Based on the above domains and boundaries, the analytical conditions of the passable motions, stick motions, and grazing motions for the friction-induced oscillator are obtained mathematically. An analytical prediction of periodic motions is performed through the mapping dynamics. With appropriate mapping structure, the simulations of the stick and nonstick motions in the two-degree friction-induced oscillator are illustrated for a better understanding of the motion complexity.

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.

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.

scholarly journals On the periodic motions of dynamical systems

1927 ◽  
Vol 50 (0) ◽  
pp. 359-379 ◽  
Author(s):  
George D. Birkhoff
Keyword(s):  
Dynamical Systems ◽  
Periodic Motions

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.

In-Depth Analysis of Bocce Backhand Throw and Hitting Technology Based on Mechanics Mechanism

2012 ◽  
Vol 246-247 ◽  
pp. 89-93
Author(s):  
Qun Ying Shi
Keyword(s):  
Early Development ◽  
Experimental Analysis ◽  
Mechanical Model ◽  
Experimental Basis ◽  
Key Factors ◽  
Depth Analysis ◽  
The Masses

The bocce movement is a sport, loved by the masses and the sports field in the early development of the technology is more mature. But sports competition is growing as bocce, bocce technology more and more attentions, especially backhand throw to hit the technology. Backhand throw strike technology is the analysis of bocce on the basis of this paper, the establishment of a backhand throw to hit the mechanical model, and thus, by experimental analysis and summary of the key factors affect the backhand throw hit effect, and thus to improve backhand throw hit technology to provide the theoretical and experimental basis.

The Stick Motion of a Harmonically Excited, Friction-Induced Oscillator on a Sinusoidally Traveling Surface

2006 ◽  
Author(s):  
Albert C. J. Luo ◽  
Brandon C. Gegg ◽  
Steve S. Suh
Keyword(s):  
Dynamical Systems ◽  
Numerical Simulations ◽  
Dry Friction ◽  
The Other ◽  
Linear Oscillator ◽  
Analytical Prediction ◽  
Nonlinear Friction ◽  
Friction Forces

In this paper, the methodology is presented through investigation of a periodically, forced linear oscillator with dry friction, resting on a traveling surface varying with time. The switching conditions for stick motions in non-smooth dynamical systems are obtained. From defined generic mappings, the corresponding criteria for the stick motions are presented through the force product conditions. The analytical prediction of the onset and vanishing of the stick motions is illustrated. Finally, numerical simulations of stick motions are carried out to verify the analytical prediction. The achieved force criteria can be applied to the other dynamical systems with nonlinear friction forces possessing a CO - discontinuity.

Nonlinear Vibrations and Chaotic Dynamics of the Laminated Composite Piezoelectric Beam

2015 ◽  
Vol 137 (1) ◽  
Author(s):  
Minghui Yao ◽  
Wei Zhang ◽  
Zhigang Yao
Keyword(s):  
Numerical Simulation ◽  
Axial Load ◽  
Nonlinear Vibrations ◽  
Dynamical Behavior ◽  
Laminated Composite ◽  
Transverse Load ◽  
Nonlinear Phenomenon ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Piezoelectric Beam

This paper investigates the complicated dynamics behavior and the evolution law of the nonlinear vibrations of the simply supported laminated composite piezoelectric beam subjected to the axial load and the transverse load. Using the third-order shear deformation theory and the Hamilton's principle, the nonlinear governing equations of motion for the laminated composite piezoelectric beam are derived. Applying the method of multiple scales and Galerkin's approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of principal parametric resonance and 1:9 internal resonance. From the averaged equations obtained, numerical simulation is performed to study nonlinear vibrations of the laminated composite piezoelectric beam. The axial load, the transverse load, and the piezoelectric parameter are selected as the controlling parameters to analyze the law of complicated nonlinear dynamics of the laminated composite piezoelectric beam. Based on the results of numerical simulation, it is found that there exists the complex nonlinear phenomenon in motions of the laminated composite piezoelectric beam. In summary, numerical studies suggest that periodic motions and chaotic motions exist in nonlinear vibrations of the laminated composite piezoelectric beam. In addition, it is observed that the axial load, the transverse load and the piezoelectric parameter have significant influence on the nonlinear dynamical behavior of the beam. We can control the response of the system from chaotic motions to periodic motions by changing these parameters.

Bifurcation Tree of Period-1 Motion to Chaos in a Pendulum With Periodic Excitation

2016 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Harmonic Amplitude ◽  
Analytical Prediction ◽  
Algebraic Equations ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Discrete Maps ◽  
Periodically Driven ◽  
Mapping Structures ◽  
Nonlinear Algebraic Equations ◽  
Analytical Bifurcation Trees

In this paper, the bifurcation trees of period-1 motions to chaos are presented in a periodically driven pendulum. Discrete implicit maps are obtained through a mid-time scheme. Using these discrete maps, mapping structures are developed to describe different types of motions. Analytical bifurcation trees of periodic motions to chaos are obtained through the nonlinear algebraic equations of such implicit maps. Eigenvalue analysis is carried out for stability and bifurcation analysis of the periodic motions. Finally, numerical simulation results of various periodic motions are illustrated in verification to the analytical prediction. Harmonic amplitude characteristics are also be presented.

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.

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