Numerical Method and Bifurcation Analysis of Jeffcott Rotor System Supported in Gas Journal Bearings

2008 ◽  
Vol 4 (1) ◽  
Author(s):  
Jiazhong Zhang ◽  
Wei Kang ◽  
Yan Liu
Keyword(s):  
Dynamical System ◽  
Numerical Method ◽  
Bifurcation Analysis ◽  
Reynolds Equation ◽  
Rotor System ◽  
Nonlinear Phenomena ◽  
Direct Integral ◽  
Stability And Bifurcation ◽  
Jeffcott Rotor ◽  
The Stability

From the viewpoint of nonlinear dynamics, the stability and bifurcation of the rotor dynamical system supported in gas bearings are investigated. First, the dynamical model of gas bearing-Jeffcott rotor system is given, and the finite element method is used to approach the unsteady Reynolds equation in order to obtain gas film forces. Then, the method for stability analysis of the unbalance response of the rotor system is developed in combination with the Newmark-based direct integral method and Floquet theory. Finally, a numerical example is presented, and the complex behaviors of the nonlinear dynamical system are simulated numerically, including the trajectory of the journal and phase portrait. In particular, the stabilities of the system’s equilibrium position and unbalance responses are studied via the orbit diagram, phase space, Poincaré mapping, bifurcation diagram, and power spectrum. The results show that the numerical method for solving the unsteady Reynolds equation is efficient, and there exist a rich variety of nonlinear phenomena in the system. The half-speed whirl encountered in practice is the result from Hopf bifurcation of equilibrium, and the numerical method presented is available for the stability and bifurcation analysis of the complicated gas film-rotor dynamic system.

Periodic Motions in the Periodically Forced Oscillator With Multiple Discontinuities

2006 ◽  
Author(s):  
Albert C. J. Luo ◽  
Mehul T. Patel
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Forced Oscillator ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the stability and bifurcation of periodic motions in periodically forced oscillator with multiple discontinuities is investigated. The generic mappings are introduced for the analytical prediction of periodic motions. Owing to the multiple discontinuous boundaries, the mapping structures for periodic motions are very complicated, which causes more difficulty to obtain periodic motions in such a dynamical system. The analytical prediction of complex periodic motions is carried out and verified numerically, and the corresponding stability and bifurcation analysis are performed. Due to page limitation, grazing and stick motions and chaos in this system will be investigated further.

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.

On an Independent Period-1 Motion in a Flexible Rotor System

2019 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Numerical Simulations ◽  
Internal Resonance ◽  
Mapping Method ◽  
Rotor System ◽  
Flexible Rotor ◽  
Periodic Motions ◽  
Phase Trajectories ◽  
Stability And Bifurcation ◽  
The Stability ◽  
Nonlinear Rotor System

Abstract This paper investigates stable and unstable period-1 motions in a rotor system through the discrete mapping method. The discrete mapping of a nonlinear rotor system is for stable and unstable period-1 motions. The stability and bifurcation of periodic motions are determined. Numerical simulations of periodic motions are completed and phase trajectories, displacement orbits and velocity plane are illustrated. The period-1 motion near the internal resonance is determined with large vibration in the nonlinear rotor system.

Stability and Bifurcation of Nonlinear Bearing-Flexible Rotor System with a Single Disk

2010 ◽  
Vol 148-149 ◽  
pp. 141-146
Author(s):  
Di Hei ◽  
Yong Fang Zhang ◽  
Mei Ru Zheng ◽  
Liang Jia ◽  
Yan Jun Lu
Keyword(s):  
Nonlinear Dynamic ◽  
Degrees Of Freedom ◽  
Rotor System ◽  
Dynamic Responses ◽  
Flexible Rotor ◽  
Runge Kutta Method ◽  
Stability And Bifurcation ◽  
Single Disk ◽  
The Stability ◽  
Predictor Corrector

Dynamic model and equation of a nonlinear flexible rotor-bearing system are established based on rotor dynamics. A local iteration method consisting of improved Wilson-θ method, predictor-corrector mechanism and Newton-Raphson method is proposed to calculate nonlinear dynamic responses. By the proposed method, the iterations are only executed on nonlinear degrees of freedom. The proposed method has higher efficiency than Runge-Kutta method, so the proposed method improves calculation efficiency and saves computing cost greatly. Taking the system parameter ‘s’ of flexible rotor as the control parameter, nonlinear dynamic responses of rotor system are obtained by the proposed method. The stability and bifurcation type of periodic responses are determined by Floquet theory and a Poincaré map. The numerical results reveal periodic, quasi-periodic, period-5, jump solutions of rich and complex nonlinear behaviors of the system.

Stability of the Jeffcott Rotor with Local Rubbing

2011 ◽  
Vol 55-57 ◽  
pp. 933-936
Author(s):  
He Li ◽  
Xiao Zhe Chen ◽  
Bang Chun Wen
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Stability Condition ◽  
Center Manifold ◽  
System Stability ◽  
Double Zero ◽  
Stability And Bifurcation ◽  
Jeffcott Rotor ◽  
The Stability ◽  
Bifurcation Behavior

The stability and bifurcation behavior of Jeffcott rotor with local rubbing are investigated in terms of Hartman-Grobman theorem in this paper. The case with double zero real part of eigenvalues is analyzed by means of the theory of center manifold and n-dimension Hopf bifurcation. Along with discussion for the effects of parameters on system stability and bifurcation behavior, numerical simulation of rotor locus is conducted and the stability condition is derived.

Stability and Bifurcation Analysis of a Prey–Predator Model

2021 ◽  
Vol 31 (04) ◽  
pp. 2150059
Author(s):  
T. N. Mishra ◽  
B. Tiwari
Keyword(s):  
Dynamical System ◽  
Linear Stability ◽  
Bifurcation Analysis ◽  
Finsler Space ◽  
Second Order ◽  
Critical Values ◽  
Numerical Examples ◽  
Stability And Bifurcation Analysis ◽  
Predator Model ◽  
The Stability

The purpose of the present paper is to study the stability of a prey–predator model using KCC theory. The KCC theory is based on the assumption that the second-order dynamical system and geodesics equation, in associated Finsler space, are topologically equivalent. The stability (Jacobi stability) based on KCC theory and linear stability of the model are discussed in detail. Further, the effect of parameters on stability and the presence of chaos in the model are investigated. The critical values of bifurcation parameters are found and their effects on the model are investigated. The numerical examples of particular interest are compared to the results of Jacobi stability and linear stability and it is found that Jacobi stability on the basis of KCC theory is global than the linear stability.

scholarly journals Analytical and numerical approach to defining the stability region of a dynamical system

2017 ◽  
Vol 13 (2) ◽  
pp. 55-62
Author(s):  
M. Neštický ◽  
O. Palumbíny ◽  
G. Michalčonok
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Computer Simulation ◽  
Differential Equations ◽  
Numerical Method ◽  
Stability Region ◽  
Lyapunov Method ◽  
Numerical Approach ◽  
Region Of Stability ◽  
The Stability

Abstract The paper presents two different approaches to estimating the region of stability of differential equation. Estimation of the region of stability is an essential practice in relation to control of the dynamical system. In this paper the objects of examination are differential equations with quasi-derivation. The equations have features that do not allow the application of classical methods for establishing stability. The goal is to compare the results of an analytic approach using Lyapunov method and computer simulation using a numerical method. The brief description of both methods are introduced and graphical results are presented and compared

Analytical Solutions of Period-1 Motions in a Periodically Forced van der Pol Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Eigenvalue Analysis ◽  
Van Der Pol Oscillator ◽  
Comprehensive Understanding ◽  
Van Der Pol ◽  
Stability And Bifurcation ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balanced method. The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis, and numerical illustrations of periodic-1 solutions are given to verify the approximate motion. This investigation provides more accurate solutions of period-1 motions in the van der pol oscillator for a better and comprehensive understanding of motions in such an oscillator.

Effect of Thrust Magnetic Bearing on Stability and Bifurcation of a Flexible Rotor Active Magnetic Bearing System

2003 ◽  
Vol 125 (3) ◽  
pp. 307-316 ◽  
Author(s):  
Y. S. Ho ◽  
H. Liu ◽  
L. Yu
Keyword(s):  
Periodic Motion ◽  
Magnetic Bearing ◽  
Rotor System ◽  
Magnetic Bearings ◽  
Active Magnetic Bearing ◽  
Flexible Rotor ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mass Eccentricity ◽  
The Stability

This paper is concerned with the effect of a thrust active magnetic bearing (TAMB) on the stability and bifurcation of an active magnetic bearing rotor system (AMBRS). The shaft is flexible and modeled by using the finite element method that can take the effects of inertia and shear into consideration. The model is reduced by a component mode synthesis method, which can conveniently account for nonlinear magnetic forces and moments of the bearing. Then the system equations are obtained by combining the equations of the reduced mechanical system and the equations of the decentralized PID controllers. This study focuses on the influence of nonlinearities on the stability and bifurcation of T periodic motion of the AMBRS subjected to the influences of both journal and thrust active magnetic bearings and mass eccentricity simultaneously. In the stability analysis, only periodic motion is investigated. The periodic motions and their stability margins are obtained by using shooting method and path-following technique. The local stability and bifurcation behaviors of periodic motions are obtained by using Floquet theory. The results indicate that the TAMB and mass eccentricity 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. Therefore, sufficient attention should be paid to these factors in the analysis and design of a flexible rotor system equipped with both journal and thrust magnetic bearings in order to ensure system reliability.

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