Nonsmooth Bifurcations in a Cracked Rotor-Active Magnetic Bearings System

2014 ◽  
Vol 989-994 ◽  
pp. 2825-2828 ◽  
Author(s):  
Feng Hong Yang ◽  
Hong Zhi Tong
Keyword(s):  
Equations Of Motion ◽  
Magnetic Bearings ◽  
Active Magnetic Bearings ◽  
Periodic Motions ◽  
Cracked Rotor ◽  
Rotating Shaft ◽  
Chaotic Motions ◽  
Amb System ◽  
Nonlinear Equations Of Motion ◽  
Smooth System

A cracked rotor-active magnetic bearings (AMB) system with the time-varying stiffness is modeled by a piecewise smooth system due to the breath of crack in a rotating shaft. The governing nonlinear equations of motion for the nonsmooth system are established and solved with the numerical method. The simulation results show that a grazing bifurcation, period-double bifurcation and chaotic motions exist in the response. These nonsmooth bifurcations can give rise to jumps between periodic motions, quasi-periodic motions and chaos.

A Time-Varying Stiffness Rotor Active Magnetic Bearings Under Combined Resonance

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
U. H. Hegazy ◽  
M. H. Eissa ◽  
Y. A. Amer
Keyword(s):  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Multiple Scale ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Gyroscopic Effects ◽  
Nonlinear Equations Of Motion ◽  
Combined Resonance

This paper is concerned with the nonlinear oscillations and dynamic behavior of a rigid disk-rotor supported by active magnetic bearings (AMB), without gyroscopic effects. The nonlinear equations of motion are derived considering a periodically time-varying stiffness. The method of multiple scales is applied to obtain four first-order differential equations that describe the modulation of the amplitudes and the phases of the vibrations in the horizontal and vertical directions. The stability and the steady-state response of the system at a combination resonance for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical nonlinear behaviors, including multiple-valued solutions, jump phenomenon, hardening, and softening nonlinearity. A numerical simulation using a fourth-order Runge-Kutta algorithm is carried out, where different effects of the system parameters on the nonlinear response of the rotor are reported and compared to the results from the multiple scale analysis. Results are compared to available published work.

Nonlinear Analysis of a Rigid Rotor on Magnetic Bearings

1995 ◽  
Author(s):  
C. Nataraj
Keyword(s):  
Nonlinear Analysis ◽  
Equations Of Motion ◽  
Forced Response ◽  
Magnetic Bearings ◽  
Rigid Rotor ◽  
Stability Criteria ◽  
Safe Operation ◽  
Qualitative And Quantitative ◽  
Quantitative Results ◽  
Nonlinear Equations Of Motion

A simple model of a rigid rotor supported on magnetic bearings is considered. A proportional control architecture is assumed, the nonlinear equations of motion are derived and some essential nondimensional parameters are identified. The free and forced response of the system is analyzed using techniques of nonlinear analysis. Both qualitative and quantitative results are obtained and stability criteria are derived for safe operation of the system.

Dynamic Modeling and Analysis of Active Magnetic Bearings

2014 ◽  
Vol 494-495 ◽  
pp. 685-688
Author(s):  
Rong Gao ◽  
Gang Luo ◽  
Cong Xun Yan
Keyword(s):  
Dynamic Characteristic ◽  
Dynamic Modeling ◽  
Magnetic Bearing ◽  
Integrated System ◽  
Magnetic Bearings ◽  
Active Magnetic Bearing ◽  
Active Magnetic Bearings ◽  
Modeling And Analysis ◽  
Amb System ◽  
Mathematics Method

Active magnetic bearing (AMB) system is a complex integrated system including mechanics, electronic and magnetism. In order to research for the basic dynamic characteristic of rotor supported by AMB, it is necessary to present mathematics method. The dynamics formula of AMB is established using theory means of dynamics of rotator and mechanics of vibrations. At the same tine, the running stability of rotor is analyzed and the example is presented in detail.

A Numerical Study on the Effect of Unbalance and Misalignment Fault Parameters in a Rigid Rotor Levitated by Active Magnetic Bearings

2019 ◽  
Author(s):  
Prabhat Kumar ◽  
Rajiv Tiwari
Keyword(s):  
Numerical Study ◽  
Equations Of Motion ◽  
Rotor System ◽  
Magnetic Bearings ◽  
Inertia Force ◽  
Dimensional System ◽  
Rigid Rotor ◽  
Active Magnetic Bearings ◽  
Fault Parameters ◽  
Misalignment Fault

Abstract This paper focusses on analysing the vibration behaviour of a rigid rotor levitated by active magnetic bearings (AMB) under the influence of unbalance and misalignment parameters. Unbalance in rotor and misalignment between rotor and both supported AMBs are key fault parameters in the rotor system. To demonstrate this dynamic analysis, an unbalanced rigid rotor with a disc at the middle levitated by two misaligned active magnetic bearings has been mathematically modelled. One of the novel concepts is also described as how the force due to active magnetic bearings on the rigid rotor is modified when the rotor is parallel misaligned with AMBs. With inclusion of inertia force, unbalance force and force due to misaligned AMBs, the equations of motion of the rigid rotor system are derived and converted into dimensionless form in terms of various non-dimensional system and fault parameters. Numerical simulations have been performed to yield the dimensionless rotor displacement and controlling current responses at AMBs. The prime intention of the present paper is to study the effect on the displacement response of the rigid rotor system and the current consumption of AMBs for different ranges of disc eccentricities and rotor-AMB misalignments.

scholarly journals Control Performance, Stability Conditions, and Bifurcation Analysis of the Twelve-Pole Active Magnetic Bearings System

2021 ◽  
Vol 11 (22) ◽  
pp. 10839
Author(s):  
Sabry M. El-Shourbagy ◽  
Nasser A. Saeed ◽  
Magdi Kamel ◽  
Kamal R. Raslan ◽  
Mohamed K. Aboudaif ◽  
...  
Keyword(s):  
High Speed ◽  
Coupling Effect ◽  
Dimensional Space ◽  
Dominant Role ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Magnetic Bearings ◽  
Control Parameters ◽  
Active Magnetic Bearings ◽  
Pole System

The active magnetic bearings system plays a vital role in high-speed rotors technology, where many research articles have discussed the nonlinear dynamics of different categories of this system such as the four-pole, six-pole, eight-pole, and sixteen-pole systems. Although the twelve-pole system has many advantages over the eight-pole one (such as a negligible cross-coupling effect, low power consumption, better suspension behaviors, and high dynamic stiffness), the twelve-pole system oscillatory behaviors have not been studied before. Therefore, this article is assigned to explore the effect of the magneto-electro-mechanical nonlinearities on the oscillatory motion of the twelve-pole system controlled via a proportional derivative controller for the first time. The normalized equations of motion that govern the system vibrations are established by means of classical mechanics. Then, the averaging equations are extracted utilizing the asymptotic analysis. The influence of all system parameters on the steady-state oscillation amplitudes is explored. Stability charts in a two-dimensional space are constructed. The stable margin of both the system and control parameters is determined. The obtained investigations reveal that proportional gain plays a dominant role in reshaping the dynamics and motion bifurcation of the twelve-pole systems. In addition, it is found that stability charts of the system can be controlled by simply utilizing both the proportional and derivative gains. Moreover, the numerical simulations showed that the twelve-poles system can exhibit both quasiperiodic and chaotic oscillations besides the periodic motion depending on the control parameters’ magnitude.

Free and Forced Localization in a Nonlinear Periodic Lattice

1993 ◽  
Author(s):  
Alexander F. Vakakis ◽  
Melvin E. King ◽  
Arne J. Pearlstein
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equation ◽  
Structural Disorder ◽  
Periodic Lattice ◽  
Periodic Motions ◽  
Partial Differential ◽  
Chaotic Motions ◽  
Solitary Solutions

Abstract Free and forced localized periodic motions in an infinite nonlinear periodic lattice are analytically investigated. The lattice consists of weakly coupled identical masses, each connected to the ground by a nonlinear stiffness. In order to study the localized motions of the discrete system a continuoum approximation is assumed, and the ordinary differential equations of motion are replaced by a single nonlinear partial differential equation. The time-periodic solutions of this equation are then obtained by an averaging method, and their stability is examined using an analytic linearized method. It is shown that localized periodic motions of the lattice correspond to standing solitary solutions of the partial differential equation of the continuous approximation. For the free lattice, localized free motions occur when the coupling stiffnesses forces are much smaller than the nonlinear effects of the grounding stiffnesses. Moreover, these free localized motions are detected in the perfectly periodic nonlinear lattice, i.e., even in the absence of structural disorder (a feature which is an essential prerequisite for linear mode localization). When harmonic forcing is applied to the chain, localized, non-localized, and chaotic motions occur, depending on the spatial distribution and the magnitude of the applied loads. A variety of spatially distributed harmonic loads and analytic expressions for the resulting localized motions of the chain are derived.

Linear Dynamic Coupling in Geared Rotor Systems

1986 ◽  
Vol 108 (2) ◽  
pp. 171-176 ◽  
Author(s):  
J. W. David ◽  
L. D. Mitchell
Keyword(s):  
Gear Tooth ◽  
Equations Of Motion ◽  
Dynamic Coupling ◽  
Rotating Shaft ◽  
Rotor Systems ◽  
Linear Dynamic ◽  
Heavy Lift ◽  
Reaction Forces ◽  
Coupling Terms ◽  
Nonlinear Equations Of Motion

The ability to analyze accurately the torsional-axial-lateral coupled response of geared systems is the key to the prediction of dynamic gear forces, shaft moments and torques, dynamic reaction forces, and moments at all bearing points. These predictions can, in turn, be used to estimate gear-tooth lives, shaft lives, housing vibrational response, and noise generation. Typical applications would be the design and analysis of gear drives in heavy-lift helicopters, industrial speed reducers, Naval propulsion systems, and heavy, off-road equipment. In this paper, the importance of certain linear dynamic coupling terms on the predicted response of geared rotor systems is addressed. The coupling terms investigated are associated with those components of a geared system that can be modeled as rigid disks. First, the coupled, nonlinear equations of motion for a disk attached to a rotating shaft are presented. The conventional argument for ignoring these dynamic coupling terms is presented and the error in this argument is revealed. It is shown that in a geared system containing gears with more than about 50 teeth, the magnitude of some of the dynamic-coupling terms is potentially as large as the magnitude of the linear terms that are included in most rotor analyses. In addition, it is shown that the dynamic coupling terms produce the multi-frequency responses seen in geared systems. To quantitatively determine the effects of the linear dynamic-coupling terms on the predicted response of geared rotor systems, a trial problem is formulated in which these effects are included. The results of this trial problem shows that the inclusion of the linear dynamic-coupling terms changed the predicted response up to eight orders of magnitude, depending on the response frequency. In addition, these terms are shown to produce sideband responses greater than the unbalanced response of the system.

Control of a rotating shaft via active magnetic bearings

1997 ◽  
Author(s):  
Jean-Christophe Ponsart ◽  
Jean Levine ◽  
Jacques Lottin ◽  
Frangois Reverdy
Keyword(s):  
Magnetic Bearings ◽  
Active Magnetic Bearings ◽  
Rotating Shaft

Chaotic Vibration Analysis of a Coaxial Rotor System in Active Magnetic Bearings and Contact With Auxiliary Bearings

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Reza Ebrahimi ◽  
Mostafa Ghayour ◽  
Heshmatallah Mohammad Khanlo
Keyword(s):  
Dynamic Behavior ◽  
Rotor System ◽  
Magnetic Bearings ◽  
Effective Control ◽  
Speed Ratio ◽  
Maximum Lyapunov Exponent ◽  
Active Magnetic Bearings ◽  
Coaxial Rotor ◽  
Speed Parameter ◽  
Amb System

In many cases of rotating systems, such as jet engines, two or more coaxial shafts are used for power transmission between a high/low-pressure turbine and a compressor. The major purpose of this study is to predict the nonlinear dynamic behavior of a coaxial rotor system supported by two active magnetic bearings (AMBs) and contact with two auxiliary bearings. The model of the system is formulated by ten degrees-of-freedom in two different planes. This model includes gyroscopic moments of disks and geometric coupling of the magnetic actuators. The nonlinear equations of motion are developed by the Lagrange's equations and solved using the Runge–Kutta method. The effects of speed parameter, speed ratio of shafts, and gravity parameter on the dynamic behavior of the coaxial rotor–AMB system are investigated by the dynamic trajectories, power spectra analysis, Poincaré maps, bifurcation diagrams, and the maximum Lyapunov exponent. Also, the contact forces between the inner shaft and auxiliary bearings are studied. The results indicate that the speed parameter, speed ratio of shafts, and gravity parameter have significant effects on the dynamic responses and can be used as effective control parameters for the coaxial rotor–AMB system. Also, the results of analysis reveal a variety of nonlinear dynamical behaviors such as periodic, quasi-periodic, period-4, and chaotic vibrations, as well as jump phenomena. The obtained results of this research can give some insight to engineers and researchers in designing and studying the coaxial rotor–AMB systems or some turbomachinery in the future.

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