Dynamic Characteristics of Multi-degrees of Freedom System Rotor-bearing System with Coupling Faults of Rub-impact and Crack

Extensive studies on rotor systems with single or coupled multiple faults have been carried out. However these studies are limited to single-span rotor systems. A finite element model for a complex rotor-bearing system with coupled faults is presented. The dynamic responses of the rotor-bearing system are obtained by using the rotor dynamics theory and the modern nonlinear dynamics theory in connection with the continuation-shooting algorithm（commonly used for obtaining a periodic solution for a nonlinear system） for a range of rub-impact clearances and crack depths. The stability and Hopf instability of the periodic motion of the rotor-bearing system with coupled faults are analyzed by using the procedure described. The results indicate that the finite element method is an effective way for determining the dynamic responses of such complex rotor-bearing systems. Further for a rotor system with rub-impact and crack faults, the influences of the clearances are significantly different for different rub-impact stiffness. On the contrary, the influence of crack depths is rather small. The instability speeds of the rotor-bearing system increase due to the presence of the crack fault. The results obtained using the new finite element model, presented for computation and analysis of dynamic responses of the rotor-bearing systems with coupled faults, are in accordance with measurements in experiment. The formulations given can be used for diagnosis of faults, vibration control, and safe and stable operations of real rotor-bearing systems.

Experimental Investigation on Vibration Control of Rotor-bearing System with Active Magnetic Exciter

Vibration control is an efficient way to minimize a rotating machine’s vibration level so that its vibration fault-free can be realized.While,several factors,such as unbalance,misalignment and instability,contribute to the serious vibration of rotating machines.It is necessary that one apparatus can depress vibration caused by two or more reasons.The fault self-recovery(FSR) mechanism is introduced and investigated.Strategies of vibration control are investigated theoretically using numerical method firstly.Active magneticelectric exciter(AME) are selected as the actuator of a FSR device because it can provide suitable force by varying the control current in the exciters depending upon a proportional and derivative control law.By numerical study,it is indicate that only a small control force is needed to improve stability margins of the compressor and prevent subsynchronous vibration fault efficiently.About synchronous vibration,three control strategies,searching in whole circle,fast optimizing control(FOC),and none mistaking control,are investigated to show which of the control strategy can realize the fault self-recovery in the shortest time.Experimental study is conducted on a test rig with variable rotating speed.Results of the test indicate that the non-mistake control strategy can minimize synchronous vibration in less than three seconds.The proposed research can provide a new insight for subsynchronous and synchronous vibration restraining about centrifugal compressor.

Simulations and Experimental Investigation on Motion Stability of a Flexible Rotor-bearing System with a Transverse Crack

In the classical process for stability studies on the rotor-bearing system with crack faults, the simple discrete model is adopted for research on such problems, which neglect some needful dynamical influence factor, such as the material damping, shearing effect and gyroscopic effects, etc. Therefore, it is necessary to find a precise calculation model for simulation of the rotor-bearing system with cracks faults. In this paper, instead of the traditional simple discrete model, finite element （FE） model is adopted to investigate the motion stability of a nonlinear rotor system with crack fault. According to finite element theory, the FE model of the cracked rotor system is established firstly. It should be pointed out that the element where the crack occurs is modeled by a particular crack element and the supports at both ends are simulated by two nonlinear loads. Then, based on dimensionless and dimensionality reduction, the Newmark-[3 method and the shooting method are employed to study the effect of eccentricity and the depth of crack on instability speed and bifurcation feature. Furthermore, the simulation results are verified by some corresponding experiments. The simulation and experimental results show that instability speed does not change monotonically, but decreases firstly and then increases when the amount of eccentricity increases. Moreover, as the type of instability changes, the instability speed jumps concomitantly. Additionally, the presence of crack fault can disturb the oil whirl, as a result, instability speed tends to increase slightly, but it does not affect the type of instability and jumping phenomenon. This research presents an effective and convenient method which uses the finite element method （FEM） to research the motion stability of the nonlinear rotor-bearing system with cracked faults and other nonlinear force, and the proposed method can provide a theoretical reference for stability analysis and vibration control in more complex relevant rotor-bearing system.

Simulation Research on Oil Whirl Fault in a High-speed Rotor-bearing System

Under the sliding bearing support for the rotor-bearing system, the dynamic model of a rotor-bearing system is established under the action of non-linear oil film force. The rotor-bearing system has been studied by the application of nonlinear dynamics theory, and the system＇s response was obtained by the numerical integration approach. The effects of eccentricity, speed, lubricant viscosity, radius gap, bearing length and journal radius on the system＇s response have been studied by using an amplitude-frequency curve, three-dimensional spectrum and bifurcation, which provides a theoretical basis for the diagnosis of the oil whirl fault effectively in the rotor-bearing system supported by a sliding bearing.

Bearing loads of dual rotor-bearing system under stationary and transient conditions

Aimed at aeroengine vibration failure, bearing loads of dual rotor-bearing system caused by unbalance are calculated under stationary and transient conditions. The three-dimensional （3-D） finite element method （FEM） model of dual rotor-bearing system was established. Applying the ro- tor dynamics function of Ansys 12.0, bearing loads were calculated under various unbalance force in stationary condition, and the transient vibration characteristics and the effect of acceleration on bearing loads were discussed. On the basis of simulation results, the influence disciplines of unbal- ance on bearing loads and theoretic reference for reducing bearing loads during start-up were ob- tained.

Study of nonlinear dynamic characteristics of rotor-bearing systems

Based on the short-bearing model, the stability of a rigid Jeffcott rotor system is studied in a relatively wide parameter range by using Poincaré maps and the numerical integration method. The results of the calculation show that the period doubling bifurcation, quasi-periodic and chaotic motions may occur. In some typical parameter regions, the bifurcation diagrams, phase portrait, Poincaré maps and the frequency spectrums of the system are acquired with the numerical integration method. They demonstrate some motion state of the system. The fractal dimension concept is used to determine whether the system is in a state of chaotic motion. The analysis result of this paper provides the theoretical basis for qualitatively controlling the stable operating states of the rotors.

Finite Element Harmonic Solution of the Coupled Rotor-bearing System

Fluid-solid interaction problems have been studied q uite extensively in the past years. Rotor-bearing system is a typical example. Fluid field is changed under the exciting of rotor vibration. On the same ti me, a net force caused by fluid pressure exerts on rotor, which will change roto r vibration. So, the fluid-solid coupled analysis method must be used. Traditionally, numerical difference method was used to solve fluid problems. The coupled fluid-solid equation could not be set up based on the method. It is no t until finite element method was used in fluid dynamics area then can the coupl ed dynamics be researched. Recently many experimental, analytical and numerical studies have been used in the area . But in these investigations, it is a ssumed that the solid vibration could not be influenced by fluid. In the other w ords, the force exerted on solid from fluid was neglected in the papers. So, the models built were some kinds of semi-coupled model only. In this paper, the Galerkin finite-element method, two-dimension vibration equ ation of rigid body and Navier-Stokes equations are used to build a full-coupl ed fluid-solid model in rotor-bearing system. Some assumptions are taken: 1) In fluid equation, the nonlinear terms are relatively small and neglected. 2) The gravity takes no effect on this system. 3) The bearing and the rotor are long. Flow and leakage along the axis is neglec ted. 4) The fluid is a kind of Newtonian incondensable viscous fluid. 5) The rotor is considered to be a rigid body. Using the model established, we calculated all the examples given by paper , results show the error are less than 7%. So the full-coupled model is built c orrectly. Examples are given in the end of the paper. After analyzing the examples, we get some conclusions: 1) In rotor-bearing system, while being taken under two conditions that whether coupled method is taken or not, difference of pressure and vibration amplitude could reach 76% and 120%. Therefore coupled method must be taken to investigate fluid-solid system. 1) Amplitude of fluid pressure can be more or less influenced by rotor unbalance , gap, eccentricity and other factors. 2) By using coupling method, results show that the amplitudes of vibration and p ressure are greater than ignoring the method. It should be paid more attention t o.

New analysis model for rotor-bearing systems based on plate theory

The purpose of this work is to develop a new analysis model for angular-contact,ball-bearing systems on the basis of plate theory instead of commonly known approaches that utilize spring elements.Axial and radial stiffness on an annular plate are developed based on plate,Timoshenko beam,and plasticity theories.The model is developed using theoretical and inductive methods and validated through a numerical simulation with the finite element method.The new analysis model is suitable for static and modal analyses of rotor-bearing systems.Numerical examples are presented to reveal the effectiveness and applicability of the proposed approach.

An Improved Method of Detecting Chaotic Motion for Rotor-Bearing Systems

Based on reconstructing the phase space and calculating the largest Lyapunov exponent, an improved method of detecting chaotic motion is presented for rotor-bearing systems. The method is an improvement to the Wolf method and the Rosenstein algorithm. The improved method introduces the correlation integral function method to estimate the embedding dimension and the reconstruction delay simultaneously, and it makes tracks for the evolutions of every pair of the nearest neighbors to improve the utilization of the reconstructed phase space. Numerical calculation and experimental verification show that the improved method can estimate the proper reconstruction parameters and detect chaotic motion of rotor-bearing systems accurately. In addition, the analytical results show that the current approach is robust to variations of the embedding dimension and the reconstruction delay, and it is applicable to small data sets.

转速数对滑动轴承动力学系数影响研究

建立了滑动轴承的基于短轴承非线性油膜力模型,通过转速对轴承动力学特性的影响研究,得到了偏心率、最小油膜厚度、最大油膜压力、摩擦功耗、温升、临界轴颈质量、刚度系数、阻尼系数、二维和三维油膜压力的影响因素研究;在对三维油膜压力分析时发现转速存在一个临界值,当转速低于这个临界值时,转速对最大油膜压力影响较大,当转速高于这个临界值时,转速对最大油膜压力影响不大。

A comprehensive comparative investigation of frictional force models for dynamics of rotor−bearing systems

Vibrations of a rotor-bearing system(RBS)can be affected by the frictional forces between the components of the inherent bearings.Thus,an in-depth investigation of the influences of the frictional moments of the bearings on the vibrations of the RBS can be helpful for understanding the vibration mechanisms in the rotating machinery.In this study,an improved dynamic model of a RBS considering different frictional force models is presented.A comparative investigation on the influences of the empirical and analytical frictional force models on the vibration characteristics of the RBS is proposed.The empirical frictional force models include Palmgren’s and SKF’s models.The analytical frictional force model considers the rolling friction caused by the radial elastic material hysteresis,slipping friction between the ball and races,viscosity friction caused by the lubricating oil,and contact friction between the ball and cage.The influences of the external load and rotational speed on the vibrations of the RBS are analyzed.The comparative results show that the analytical frictional force model can give a more reasonable method for formulating the effects of the friction forces in the bearings on the vibrations of the RBS.The results also demonstrate that the friction forces in the bearings can significantly affect the vibrations of the RBSs.

ON-LINE ELIMINATION OF OIL WHIP

A new method to eliminate the oil whip online is put forward by use ofpassive electromagnetic damper. The damper works contactless and with DC current. Neither sensor norclosed loop control is needed. The dynamic equations of rotor-bearing system are built up bycombining d'Alemdert principle with Rize way, and the nonlinear oil film forces based on unsteadyshort bearing model are coupled to system. Such nonlinear equations are numerically solved byNewmark integration method. The calculated results show that the bifurcation behavior of the systemcan be. changed and the oil whip of the rotor may be well damped by external damping. Thebifurcation diagrams also show that the subharmonic vibration amplitude decreases in motion and thespeed at which the system losses its stability increases obviously by exerting external damping.Then experiments are carried out to demonstrate this phenomenon. It is observed that the complextrajectories of the journal motion are disappeared and the rotor-bearing system became stable whenthe power of passive electromagnetic damper is turned on. The experiments have good repeatability.

APPLICATION OF MECHANIZED MATHEMATICS TO ROTOR DYNAMICS

Based on the mechanized mathematics and WU Wen-tsun elimination method, using oil film forces of short-bearing model and Muszynska's dynamic model, the dynamical behavior of rotor-beating system and its stability of motion are investigated. As example, the concept of Wu characteristic set and Maple software, whirl parameters of short-bearing model, which is usually solved by the numerical method, are analyzed. At the same time, stability of zero solution of Jeftcott rotor whirl equation and stability of self-excited vibration are studied. The conditions of stable motion are obtained by using theory of nonlinear vibration.

NONLINEAR NUMERICAL ANALYSIS OF A FLEXIBLE ROTOR EQUIPPED WITH SQUEEZE COUPLE STRESS FLUID FILM JOURNAL BEARINGS

This study performs a dynamic analysis of a rotor supported by two squeeze couple stress fluid film journal bearings with nonlinear suspension. The numerical results show that the stability of the system varies with the non-dimensional speed ratios and the dimensionless parameter l＊. It is found that the system is more stable with higher dimensionless parameter l＊. Thus it can conclude that the rotor-bearing system lubricated with the couple stress fluid is more stable than that with the conventional Newtonian fluid. The modeling results thus obtained by using the method proposed in this paper can be used to predict the stability of the rotor-bearing system and the undesirable behavior of the rotor and bearing center can be avoided.

Dimensional Amplitude Response Analysis of Vibrations Produced by Defects in Rolling Contact Bearings

Usage of rolling contact bearings in variety of rotor-dynamic applications has put forth a need to develop a detailed and easy to implement techniques for the assessment of damage related features in these bearings so that before mechanical failure,maintenance actions can be planned well in advance.In accordance to this,a method based on dimensional amplitude response analysis and scaling laws is presented in this paper for the diagnosis of defects in different components of rolling contact bearings in a dimensionally scaled rotor-bearing system.Rotor,bearing,operating and defect parameters involved are detailed for dimensional analysis using frequency domain vibration data.A defect parameter for modeling all the three dimensions of the defect as well as the different shapes like square,circular,rectangular is put forth which takes into account the volume as well as the surface area of the defect.Experimental data set is generated for the‘model’bearing(designated as SKF30205J2/Q)using Box-Behnken design of response surface methodology for solution of the theoretical model by factorial regression approach.Obtained metamodel is then used for the prediction of the objective variable,i.e.,Vibration acceleration amplitude at the defect frequency component for other types of‘test’bearings(designated as SKF 30305C and SKF 22220 EK)using the developed scaling laws.Confirmation experiments showed that the computable relationship amongst objective variable and the dimensionless parameters can be forecast and correlated.

Dynamic Behaviors Analysis of Reduced Rotor Models with Looseness Based on the TPOD Method

The transient proper orthogonal decomposition(TPOD) method is used to study dynamic behaviors of the reduced rotor-bearing models,and the fault-free model is compared with the models with looseness fault.A 22 degree of freedoms(DOFs) rotor model supported by bearings is established.Both one end and two ends pedestal looseness of the liquid-film bearings are studied by analyzing the time history and the frequency-spectrum curves.The effects of the initial displacement and velocity values to frequency components of the original systems and the dimension reduction efficiency are discussed.Moreover,the effects of variation of initial conditions on the efficiency of the TPOD method are studied.Reduced models can provide guidance significance from the perspectives of the theory and numerical simplification to discuss the characteristics of pedestal looseness fault.

Finite Element Method Applied to the Eigenvalue Analysis of Flexible Rotors Supported by Journal Bearings

This work deals with a finite element procedure developed to perform the eigenvalue analysis of damped gyroscopic systems, represented by flexible rotors supported on fluid film journal bearings. The rotor finite element model is based on the Timoshenko beam theory, accounting for the shaft rotary inertia and gyroscopic moments. The governing equations for the hydrodynamic journal bearing are obtained through the Galerkin weighted residual method applied to the classical Reynolds equation. A perturbation scheme on the fluid film governing equation permits to obtain the zero-th and first order lubrication equations for the bearings, which allow the computation of the dynamic force coefficients associated with the bearing stiffness and damping. The rotor-bearing system equation, which consists of a case of damped gyroscopic equation, is rewritten on state form to compute the complex eigenvalues. The natural frequencies at several operating conditions are obtained and compared to the technical literature data. The influence of the effective damping on the eigenvalue real part sign is analyzed for some examples of rotor-bearing systems, showing how the stability conditions can be predicted by the eigenvalue analysis. The procedure implemented in this work can provide useful guidelines and technical data about the selection of the more appropriate set of bearing parameters for rotating machines operating at stringent conditions.

Dynamic Stability Study of Static Gas Bearing for Small Cryogenic Turbo-Expander

An experimental method is presented to analyze the dynamic stability of the gas bearing for small cryogenic turbo-expanders. The rotation imbalance response and the shape of the rotor orbit were obtained for different speeds up to 110,000 rpm, and the critical speed of the rotor-bearing system was determined by a Bode diagram. An FFT signal analytical method was applied to identify the resonance frequency, and the waterfall plot was presented. During the whole process of speeding up to the designed speed of 110,000 rpm, the rotor-bearing works stably with no whirl instability, which is validated in a waterfall plot. Also, the tested rotor-bearing model was analyzed theoretically. It was proved that the experimental results were highly consistent with those of theoretical calculations. Thus the experimental method proposed here to analyze the dynamic stability of the gas bearing is feasible.

Dynamics modeling and electromechanical coupling characteristics analysis of cage induction motors

Induction motors, as typical electromechanical energy conversion devices, have received limited attention in previous studies on electromechanical coupling vibrations, precise modeling, and the use of electromechanical coupling effects for fault diagnosis and condition assessment in motor drive systems. This study proposes a comprehensive model of cage induction motors that integrates the multiple coupled circuit model with a rotor-bearing dynamics model. The model accounts for the linear increase in the magnetomotive force across the slot and incorporates the skidding characteristics of bearings in the rotor-bearing system. By calculating the time-varying mutual inductance parameters based on the air-gap distribution in the vibration environment, the electromechanical coupling vibration of the cage motor is investigated. Furthermore, this study examines the electromechanical coupling vibration characteristics influenced by various factors, including bearing clearances, radial loads, and the vertical excitation frequencies of the stators. The results show that the proposed model improves the excitation inputs for the electrical and mechanical systems of the motor compared with conventional models. Increased bearing clearance and radial load affect the current and torque similarly but have opposite effects on the slip ratio. This study provides a deeper understanding of electromechanical coupling mechanisms and facilitates the use of such phenomena for fault diagnosis and condition assessment in motor-driven systems.

Interference torque of a three-floated gyroscope with gas-lubricated bearings subject to a sudden change of the specific force

A numerical study is carried out to investigate the influence of a sudden change of the specific force on a three-floated gyroscope with a rotor supported on gas-lubricated bearings.The interference torque is calculated to evaluate the influence by modeling the transient behavior of the rotor-bearing system. In combination with dynamic equations of the rotor displacement,the Reynolds equation is solved on the surface of spiral-grooved conical bearings. It is assumed that a steady state has been obtained with a constant specific force, and then the specific force is suddenly changed and maintained constant after that. Responses of the sudden change are obtained by solving the equations. Numerical results show that the direction of the sudden change of the specific force is the main factor which influences the interference torque curve. With a sudden change along the input direction, the interference torque fluctuation has a constant frequency and a decreasing amplitude. With a sudden change along the output direction, the interference torque fluctuates in a small range. With a sudden change along the spin direction, the change of the interference torque combines a fluctuation and a gradually-changing quasi-equilibrium value. This study provides a supplement to the steady-state error model of the three-floated gyroscope.