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

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.

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

Journal of Vibration and Acoustics ◽

10.1115/1.1570448 ◽

2003 ◽

Vol 125 (3) ◽

pp. 307-316 ◽

Cited By ~ 27

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 ◽

Mass Eccentricity ◽

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.

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

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.148-149.141 ◽

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 ◽

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.

Investigation on the Stability and Bifurcation of a 3D Rotor-Bearing System

Journal of Vibration and Acoustics ◽

10.1115/1.4023843 ◽

2013 ◽

Vol 135 (3) ◽

Cited By ~ 1

Author(s):

Yi Liu ◽

Heng Liu ◽

Jun Yi ◽

Minqing Jing

Keyword(s):

Path Following ◽

Rotor System ◽

Periodic Motions ◽

Stability Margins ◽

Rotating Speed ◽

Mass Eccentricity ◽

3D Elements ◽

Order Of Magnitude ◽

The stability and bifurcation of a flexible 3D rotor system are investigated in this paper. The rotor is discretized by 3D elements and reduced by using component mode synthesis. Periodic motions and stability margins are obtained by using the shooting method and path-following technique, and the local stability of the periodic motions is determined by using the Floquet theory. Comparisons indicate that 3D and 1D systems have a general resemblance in the bifurcation characteristics while mass eccentricity and rotating speed are changed. For both systems, the orbit size of the periodic motions has the same order of magnitude, and the vibration response has identical frequency components when typical bifurcations occur. The stress distribution and location of the maximum stress spot are determined by the bending mode of the rotor. The type of 3D element has a slight effect on the stability and bifurcation of the rotor system. Generally, this paper presents a feasible method for analyzing the stability and bifurcation of complex rotors without much structural simplification.

SINGULARITY, SWITCHABILITY AND BIFURCATIONS IN A 2-DOF, PERIODICALLY FORCED, FRICTIONAL OSCILLATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413300097 ◽

2013 ◽

Vol 23 (03) ◽

pp. 1330009 ◽

Cited By ~ 5

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 ◽

Mapping Structures ◽

Stability And Bifurcation Analysis ◽

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.

Periodic Motions in an Autonomous System With a Discontinuous Vector Field

Volume 7B: Dynamics, Vibration, and Control ◽

10.1115/imece2020-23633 ◽

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 ◽

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.

Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC) ◽

Period-3 Motions in a Parametrically Exited Inverted Pendulum

10.1115/detc2020-22176 ◽

2020 ◽

Author(s):

Albert C. J. Luo ◽

Chuan Guo

Keyword(s):

Numerical Simulations ◽

Inverted Pendulum ◽

Period Doubling ◽

Mapping Method ◽

Eigenvalue Analysis ◽

Symmetric Motion ◽

Bifurcation Tree ◽

Implicit Mapping ◽

Period Doubling Bifurcations

Abstract In this paper, period-3 motions in a parametrically exited inverted pendulum are analytically investigated through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the period-3 motions are predicted through eigenvalue analysis. The symmetric and asymmetric period-3 motions are obtained on the bifurcation tree, and the period-doubling bifurcations of the asymmetric period-3 motions are observed. The saddle-node and Neimark bifurcations for symmetric period-3 motions are obtained. The saddle-bifurcations of the symmetric period-3 motions are for symmetric motion appearance (or vanishing) and onsets of asymmetric period-3 motion. Numerical simulations of the period-3 motions in the inverted pendulum are completed from analytical predictions for illustration of motion complexity and characteristics.

On Period-1 and Period-2 Motions of a Jeffcott Rotor System

Volume 8: 31st Conference on Mechanical Vibration and Noise ◽

10.1115/detc2019-97259 ◽

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.

Stability and Bifurcation of Unbalanced Response of a Squeeze Film Damped Flexible Rotor

Journal of Tribology ◽

10.1115/1.2927236 ◽

1994 ◽

Vol 116 (2) ◽

pp. 361-368 ◽

Cited By ~ 37

Author(s):

J. Y. Zhao ◽

I. W. Linnett ◽

L. J. McLean

Keyword(s):

Power Spectra ◽

Squeeze Film ◽

Integration Scheme ◽

Flexible Rotor ◽

Jump Phenomenon ◽

Trigonometric Collocation ◽

Numerical Integration Scheme ◽

The Stability ◽

Nonsynchronous Vibrations

The stability and bifurcation of the unbalance response of a squeeze film damper-mounted flexible rotor are investigated based on the assumption of an incompressible lubricant together with the short bearing approximation and the “π” film cavitation model. The unbalanced rotor response is determined by the trigonometric collocation method and the stability of these solutions is then investigated using the Floquet transition matrix method. Numerical examples are given for both concentric and eccentric damper operations. Jump phenomenon, subharmonic, and quasi-periodic vibrations are predicted for a range of bearing and unbalance parameters. The predicted jump phenomenon, subharmonic and quasi-periodic vibrations are further examined by using a numerical integration scheme to predict damper trajectories, calculate Poincare´ maps and power spectra. It is concluded that the introduction of unpressurized squeeze film dampers may promote undesirable nonsynchronous vibrations.

An Experimental Investigation on the Response of a Flexible Rotor Mounted in Pressure Dam Bearings

Journal of Mechanical Design ◽

10.1115/1.3254831 ◽

1980 ◽

Vol 102 (4) ◽

pp. 842-850 ◽

Cited By ~ 15

Author(s):

R. D. Flack ◽

M. E. Leader ◽

E. J. Gunter

Keyword(s):

Experimental Investigation ◽

Rotor System ◽

Flexible Rotor ◽

The response of a flexible rotor mounted in six bearing sets has been experimentally determined. One set of axial groove bearings and five sets of pressure dam bearings were tested. Conventional synchronous tracking was used in the analysis and other techniques utilizing an FFT analyzer were developed. The stability of the system was seen to strongly depend on the design of the step bearings. The dam bearings were also noted to lock into subsynchronous whip during deceleration after the system went unstable. The response of the system with varying degrees of unbalance is also analyzed and several structural resonances of the rotor system are discussed.