Analytical Solutions for Period-1 Motions in a Nonlinear Jeffcott Rotor System

2014 ◽  
Author(s):  
Jianzhe Huang ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Solutions ◽  
Periodic Motion ◽  
Rotor System ◽  
Eigenvalue Analysis ◽  
Hopf Bifurcations ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Jeffcott Rotor

In this paper, the analytical solutions of period-1 solutions are developed, and the corresponding stability and bifurcation are also analyzed by eigenvalue analysis. The Hopf bifurcations of periodic motions cause not only the bifurcation tree but quasi-periodic motions. The quasi-periodic motion can be stable or unstable. Displacement orbits of periodic motions in the nonlinear Jeffcott rotor systems are illustrated, and harmonic amplitude spectrums are presented for harmonic effects on periodic motions of the nonlinear rotor.

Periodic Motions and Bifurcation Trees in a Buckled, Nonlinear Jeffcott Rotor System

2015 ◽  
Vol 25 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Jianzhe Huang ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Solutions ◽  
Degrees Of Freedom ◽  
Rotor System ◽  
Hopf Bifurcations ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Jeffcott Rotor ◽  
Analytical Bifurcation Trees

In this paper, analytical solutions for period-m motions in a buckled, nonlinear Jeffcott rotor system are obtained. This nonlinear Jeffcott rotor system with two-degrees of freedom is excited periodically from the rotor eccentricity. The analytical solutions of period-m solutions are developed, and the corresponding stability and bifurcation are also analyzed by eigenvalue analysis. Analytical bifurcation trees of period-1 motions to chaos are presented. The Hopf bifurcations of periodic motions cause not only the bifurcation tree but quasi-periodic motions. The quasi-periodic motion can be stable or unstable. Displacement orbits of periodic motions in the buckled, nonlinear Jeffcott rotor systems are illustrated, and harmonic amplitude spectrums are presented for harmonic effects on periodic motions of the nonlinear rotor. Coexisting periodic motions exist in such a buckled nonlinear Jeffcott rotor.

Period-1 Motion to Chaos in a Nonlinear Flexible Rotor System

2020 ◽  
Vol 30 (05) ◽  
pp. 2050077 ◽  
Author(s):  
Yeyin Xu ◽  
Zhaobo Chen ◽  
Albert C. J. Luo
Keyword(s):  
Period Doubling ◽  
Rotor System ◽  
Harmonic Amplitude ◽  
Flexible Rotor ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Saddle Node ◽  
Jeffcott Rotor ◽  
And Control

In this paper, a bifurcation tree of period-1 motion to chaos in a flexible nonlinear rotor system is presented through period-1 to period-8 motions. Stable and unstable periodic motions on the bifurcation tree in the flexible rotor system are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are analyzed by eigenvalue analysis. On the bifurcation tree, the appearance and vanishing of jumping phenomena of periodic motions are generated by saddle-node bifurcations, and quasi-periodic motions are induced by Neimark bifurcations. Period-doubling bifurcations of periodic motions are for developing cascaded bifurcation trees, however, the birth of new periodic motions are based on the saddle-node bifurcation. For a better understanding of periodic motions on the bifurcation tree, nonlinear harmonic amplitude characteristics of periodic motions are presented. Numerical simulations of periodic motions are performed for the verification of semi-analytical predictions. From such a study, nonlinear Jeffcott rotor possesses complex periodic motions. Such results can help one detect and control complex motions in rotor systems for industry.

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.

Periodic Motion in a Nonlinear Vibration Isolator Under Harmonic Excitation

2017 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Nonlinear Vibration ◽  
Analytical Solutions ◽  
Periodic Motion ◽  
Harmonic Excitation ◽  
Eigenvalue Analysis ◽  
Harmonic Amplitude ◽  
Vibration Isolator ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Forced Oscillator

In this paper, periodic motions of a periodically forced oscillator with a nonlinear isolator are studied through generalized harmonic balanced method. Both symmetric and asymmetric period-1 motions are obtained. Stability and bifurcation of the periodic motions are determined through eigenvalue analysis. Numerical illustrations of both symmetric and asymmetric are given. From the harmonic amplitude spectrums, the harmonic effects on periodic motions are determined, and the corresponding accuracy of approximate analytical solutions can be observed.

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.

Analytical Solutions of Period-1 to Period-2 Motions in a Periodically Diffused Brusselator

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyu Guo
Keyword(s):  
Analytical Solutions ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Stable Period ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Amplitude Spectra ◽  
Unstable Solutions ◽  
Stability And Bifurcations

In this paper, the analytical solutions of periodic evolutions of the periodically diffused Brusselator are obtained through the generalized harmonic balanced method. Stable and unstable solutions of period-1 and period-2 evolutions in the Brusselator are presented. Stability and bifurcations of the periodic evolution are determined by the eigenvalue analysis, and the corresponding Hopf bifurcations are presented on the analytical bifurcation tree of the periodic motions. Numerical simulations of stable period-1 and period-2 motions of Brusselator are completed. The harmonic amplitude spectra show harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically.

Period-1 and Period-2 Motions in a Brusselator With a Harmonic Diffusion

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyu Guo
Keyword(s):  
Numerical Simulations ◽  
Analytical Solutions ◽  
Eigenvalue Analysis ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Stable Period ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcations

In this paper, the analytical solutions of periodic evolution of Brusselator are investigated through the general harmonic balanced method. Both stable and unstable, period-1 and period-2 solutions of the Brussellator are presented. Stability and bifurcations of the periodic evolution are determined by the eigenvalue analysis. Numerical simulations of stable period-1 and period-2 motions of Brusselator are completed. The harmonic amplitude spectrums show harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically.

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.

Internal Resonance of the Jeffcott Rotor With Nonlinear Spring Characteristics

2001 ◽  
Author(s):  
Yukio Ishida ◽  
Tsuyoshi Inoue
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Nonlinear Phenomena ◽  
Almost Periodic ◽  
Periodic Motions ◽  
Nonlinear Analyses ◽  
Rotor Systems ◽  
Gyroscopic Moment ◽  
Jeffcott Rotor ◽  
Two Degree Of Freedom

Abstract The Jeffcott rotor is a two-degree-of-freedom linear model with a disk at the midspan of a massless elastic shaft. This model executing lateral whirling motions has been widely used in the linear analyses of rotor vibrations. In the Jeffcott rotor, the natural frequency of a forward whirling mode pf and that of a backward whirling mode pb have the relation of internal resonance pf : pb = 1 : (−1). Recently, many researchers analyzed nonlinear phenomena by using the Jeffcott rotor with nonlinear elements. However, they did not take this internal resonance relationship into account. While, in many cases of the practical rotating machinery, such a relationship holds apprximately due to small gyroscopic moment. In this paper, nonlinear phenomena in the vicinity of the major critical speed and the rotational speeds of twice and three times the major critical speed are investigated in the Jeffcott rotor and rotor systems with small gyroscopic moment. Especially, the influences of internal resonance on the nonlinear resonances are studied in detail. The following were clarified theoretically and experimentally: (a) the shape of resonance curves becomes far more complex than that of a single resonance, (b) almost-periodic motions occur, (c) these phenomena are influenced remarkably by the asymmetrical nonlinearity and gyroscopic moment, and (d) the internal resonance phenomena are strongly influenced by the degree of the discrepancies among critical speeds. The results teach us the usage of the Jeffcott rotor in nonlinear analyses of rotor systems may induce incrrect results.

On Periodic Motions in a Periodically Driven van der Pol-Duffing Oscillator

2020 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Solutions ◽  
Analytical Method ◽  
Duffing Oscillator ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Bifurcation Tree ◽  
Periodically Driven ◽  
Implicit Mapping ◽  
Mapping Structures ◽  
Bifurcation And Stability

Abstract In this paper, the symmetric and asymmetric period-1 motions on the bifurcation tree are obtained for a periodically driven van der Pol-Duffing hardening oscillator through a semi-analytical method. Such a semi-analytical method develops an implicit mapping with prescribed accuracy. Based on the implicit mapping, the mapping structures are used to determine periodic motions in the van der Pol-Duffing oscillator. The symmetry breaks of period-1 motion are determined through saddle-node bifurcations, and the corresponding asymmetric period-1 motions are generated. The bifurcation and stability of period-1 motions are determined through eigenvalue analysis. To verify the semi-analytical solutions, numerical simulations are also carried out.

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