Stability and Bifurcation for the Equispaced, Periodic Motion of a Horizontal Impact Damper

2001 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Periodic Motion ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Impact Damper ◽  
Stability And Bifurcation ◽  
Period 2 ◽  
Good Agreement

Abstract Stability and bifurcation conditions for the asymmetric, periodic motion of a horizontal impact damper under a periodic excitation are developed through four mappings for two switch-planes relative to discontinuities. Period-doubling bifurcation for equispaced motion does not occur, but the asymmetric period-1 motions change to the asymmetric, period-2 ones through a period doubling bifurcation. A numerical prediction for equispaced to chaotic motions is completed. The numerical and analytical predictions of the periodic motion are in very good agreement. The asymmetric, periodic motions are also simulated.

An Unsymmetrical Motion in a Horizontal Impact Oscillator

2002 ◽  
Vol 124 (3) ◽  
pp. 420-426 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Numerical Investigation ◽  
Dynamic Systems ◽  
Periodic Motion ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Discontinuous Systems ◽  
Impact Oscillator ◽  
Good Agreement

Stability and bifurcation for the unsymmetrical, periodic motion of a horizontal impact oscillator under a periodic excitation are investigated through four mappings based on two switch-planes relative to discontinuities. Period-doubling bifurcation for unsymmetrical period-1 motions instead of symmetrical period-1 motion is observed. A numerical investigation for symmetrical, period-1 motion to chaos is completed. The numerical and analytical results of periodic motions are in very good agreement. The methodology presented in this paper is applicable to other discontinuous dynamic systems. This investigation also provides a better understanding of such an unsymmetrical motion in symmetrical discontinuous systems.

Period-1 to Period-8 Motions in a Nonlinear Jeffcott Rotor System

2020 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C.J. Luo
Keyword(s):  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Mapping Method ◽  
Rotor System ◽  
Period Doubling Bifurcation ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Period 2 ◽  
Jeffcott Rotor

Abstract In this paper, a bifurcation tree of period-1 to period-8 motions in a nonlinear Jeffcott rotor system is obtained through the discrete mapping method. The bifurcations and stability of periodic motions on the bifurcation tree are discussed. The quasi-periodic motions on the bifurcation tree are caused by two (2) Neimark bifurcations of period-1 motions, one (1) Neimark bifurcation of period-2 motions and four (4) Neimark bifurcations of period-4 motions. The specific quasi-periodic motions are mainly based on the skeleton of the corresponding periodic motions. One stable and one unstable period-doubling bifurcations exist for the period-1, period-2 and period-4 motions. The unstable period-doubling bifurcation is from an unstable period-m motion to an unstable period-2m motion, and the unstable period-m motion becomes stable. Such an unstable period-doubling bifurcation is the 3rd source pitchfork bifurcation. Periodic motions on the bifurcation tree are simulated numerically, and the corresponding harmonic amplitudes and phases are presented for harmonic effects on periodic motions in the nonlinear Jeffcott rotor system. Such a study gives a complete picture of periodic and quasi-periodic motions in the nonlinear Jeffcott rotor system in the specific parameter range. One can follow the similar procedure to work out the other bifurcation trees in the nonlinear Jeffcott rotor systems.

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.

scholarly journals Bifurcation Analysis in Population Genetics Model with Partial Selfing

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yingying Jiang ◽  
Wendi Wang
Keyword(s):  
Center Manifold ◽  
Evolutionary Game ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Expected Payoff ◽  
Center Manifold Theorem ◽  
Population Evolution ◽  
Dynamical Behaviors ◽  
Period 2 ◽  
Dynamics Of Population

A new model which allows both the effect of partial selfing selection and an exponential function of the expected payoff is considered. This combines ideas from genetics and evolutionary game theory. The aim of this work is to study the effects of partial selfing selection on the discrete dynamics of population evolution. It is shown that the system undergoes period doubling bifurcation, saddle-node bifurcation, and Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-3, 6 orbits, cascade of period-doubling bifurcation in period-2, 4, 8, and the chaotic sets. These results reveal richer dynamics of the discrete model compared with the model in Tao et al., 1999. The analysis and results in this paper are interesting in mathematics and biology.

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.

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.

Chaos and Quasi-Periodic Motions on the Homoclinic Surface of Nonlinear Hamiltonian Systems With Two Degrees of Freedom

2005 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Energy Spectrum ◽  
Hamiltonian System ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Periodic Motion ◽  
Degrees Of Freedom ◽  
Weak Interactions ◽  
Kam Theorem ◽  
Periodic Motions ◽  
Chaotic Motions

The numerical prediction of chaos and quasi-periodic motion on the homoclinic surface of a 2-DOF nonlinear Hamiltonian system is presented through the energy spectrum method. For weak interactions, the analytical conditions for chaotic motion in such a Hamiltonian system are presented through the energy incremental energy approach. The Poincare mapping surfaces of chaotic motions for such nonlinear Hamiltonian systems are illustrated. The chaotic and quasiperiodic motions on the phase planes, displacement subspace (or potential domains), and the velocity subspace (or kinetic energy domains) are illustrated for a better understanding of motion behaviors on the homoclinic surface. Through this investigation, it is observed that the chaotic and quasi-periodic motions almost fill on the homoclinic surface of the 2-DOF nonlinear Hamiltonian systems. The resonant-periodic motions are theoretically countable but numerically inaccessible. Such conclusions are similar to the ones in the KAM theorem even though the KAM theorem is based on the small perturbation.

Complete Bifurcation Trees of Independent Period-2 Motions to Chaos in a Parametrically Excited Pendulum

2019 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Bifurcation Analysis ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Periodic Motions ◽  
Parametrically Excited ◽  
Stability And Bifurcation ◽  
Period 2 ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

Abstract In this paper, bifurcation trees of independent period-2 motions to chaos are investigated in a parametrically excited pendulum. The implicit discrete mapping method is employed to obtain periodic motions in such a system. Analytical predictions of periodic motions are based on the mapping structures and peroidicity. The bifurcation trees of independent period-2 motions to chaos are studied, and the corresponding stability and bifurcation analysis are completed through eigenvalue analysis. Finally, sampled period-2 motions are simulated numerically in comparison to the analytical predictions. The infinite bifurcation trees of independent period-2 motions to chaos can be obtained.

Switching Bifurcation and Chaos in an Extended Fermi-Acceleration Oscillator

2008 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Local Stability ◽  
Period Doubling ◽  
Time History ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
Numerical Predictions ◽  
Global View ◽  
Acceleration Model

The Fermi acceleration oscillator is extensively used to interpret many physical and mechanical phenomena. To understand dynamic behaviors of a particle (or a bouncing ball) in such a Fermi oscillator, a generalized Fermi acceleration model is developed. This model consists of a particle moving vertically between a fixed wall and the piston in a vibrating oscillator. The motion switching bifurcation of a particle in such a generalized Fermi oscillator is investigated through the theory of discontinuous dynamical systems. The analytical conditions for the motion switching are developed for numerical predictions. Thus, periodic motions in the Fermi-acceleration oscillator are given and the corresponding local stability and bifurcation are presented. Periodic and chaotic motions in such an oscillator are presented via the displacement time-history. From switching bifurcation and period-doubling bifurcation, parameter maps of periodic and chaotic motions will be developed for a global view of motions in the Fermi acceleration oscillator. To illustrate motion switching phenomena, the acceleration responses of the particle and base in the Fermi oscillator are presented. Poincare mapping sections are also used to illustrate chaos, and energy dissipation in chaotic motions can be evaluated.

Period-1 Motions in a Time-Delayed Duffing Oscillitor With Periodic Excitation

2014 ◽  
Author(s):  
Albert C. J. Luo ◽  
Hanxiang Jin
Keyword(s):  
Fourier Series ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
The Stability ◽  
Amplitude Characteristics

In this paper, analytical solutions of period-1 motions in a time-delayed Duffing oscillator with a periodic excitation are investigated through the Fourier series, and the stability and bifurcation of such periodic motions are discussed by eigenvalue analysis. The symmetric and asymmetric period-1 motions in such time-delayed Duffing oscillator are obtained analytically, and the frequency-amplitude characteristics of period-1 motions in such a time-delayed Duffing oscillator are investigated. Numerical illustrations of period-1 motions are given by numerical and analytical solutions.

Sign in / Sign up

Export Citation Format

Share Document