Rotor drop and following thermal growth simulations using detailed auxiliary bearing and damper models

Catcher bearings (CBs) or auxiliary bearings provide mechanical backup protection in the events of magnetic bearing failure. This paper presents numerical analysis for a rotor drop on CBs and following thermal growths due to their mechanical rub using detailed CB and damper models. The detailed CB model is determined based on its material, geometry, speed and preload using the nonlinear Hertzian load–deflection formula, and the thermal growths of bearing components during the rotor drop are estimated using a 1D thermal model. A finite-element squeeze film damper provides the pressure profile of an annular oil film and the resulting viscous damping force. Numerical simulations of an energy storage flywheel with magnetic suspensions failed reveal that an optimal CB design using the detailed simulation models stabilizes the rotor drop dynamics and lowers the thermal growths while preventing the high-speed backward whirl. Furthermore, CB design guides based on the simulation results are presented.     

Linear and non-linear control techniques applied to actively lubricated journal bearings

The main objectives of actively lubricated bearings are the simultaneous reduction of wear and vibration between rotating and stationary machinery parts. For reducing wear and dissipating vibration energy until certain limits, one can use the conventional hydrodynamic lubrication. For further reduction of shaft vibrations one can use the active lubrication action, which is based on injecting pressurized oil into the bearing gap through orifices machined in the bearing sliding surface. The design and efficiency of some linear (PD, PI and PID) and a non-linear controller, applied to a tilting-pad journal bearing, are analysed and discussed. Important conclusions about the application of integral controllers, responsible for changing the rotor-bearing equilibrium position and consequently the “passive” oil film damping coefficients, are achieved. Numerical results show an effective vibration reduction of unbalance response of a rigid rotor, where the PD and the non-linear P controllers show better performance for the frequency range of study (0-80 Hz). The feasibility of eliminating rotor-bearing instabilities (phenomena of whirl) by using active lubrication is also investigated, illustrating clearly one of its most promising applications.     

Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model

Linear models and synchronous response are generally adequate to describe and analyze rotors supported by hydrodynamic bearings. Hence, stiffness and damping coefficients can provide a good model for a wide range of situations. However, in some cases, this approach does not suffice to describe the dynamic behavior of the rotor-bearing system. Moreover, unstable motion occurs due to precessional orbits in the rotor-bearing system. This instability is called “oil whirl” or “oil whip”. The oil whirl phenomenon occurs when the journal bearings are lightly loaded and the shaft is whirling at a frequency close to one-half of rotor angular speed. When the angular speed of the rotor reaches approximately twice the natural frequency (first critical speed), the oil whip phenomenon occurs and remains even if the rotor angular speed increases. Its frequency and vibration mode correspond to the first critical speed. The main purpose of this paper is to validate a complete nonlinear solution to simulate the fluid-induced instability during run-up and run-down. A flexible rotor with a central disk under unbalanced excitation is modeled. A nonlinear hydrodynamic model is considered for short bearing and laminar flow. The effects of unbalance, journal-bearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold are verified. The model simulations are compared with measurements at a real vertical power plant and a horizontal test rig.     

A theoretical-experimental comparison of the synchronous response of a bowed rotor in five different sets of fluid film bearings

A comparison of theoretical and experimental synchronous unbalance responses of a bowed Jeffcott rotor in fluid film bearings has been completed. A transfer matrix method was used to predict theoretically the response of a 25·4 mm shaft in fluid film bearings and results were compared to data from a previous experimental study. Four bearing types were used: two axial groove, pressure dam, tilting pad and four-lobe. Very good agreement was found for all bearing types at the rotor critical speed (3000 rpm). Differences less than 15% in peak response were found and the theoretical and experimental peaks were found to occur within 200 rpm. Worst agreement was found for the preloaded four-lobe bearings and this disagreement was found for speeds other than the critical speed. Also, for equal bow and unbalance the tilting pad and four-lobe bearings were found to produce the least and most damping at the critical speed, respectively. Previous to this time a comparison of theoretical and experimental synchronous responses of a rotor system representing industrial turbomachines has not been published, nor has a comparative study of the different bearing types.     

IDENTIFICATION OF SPEED-DEPENDENT BEARING PARAMETERS

Bearing dynamic characteristics have been a major unknown in the modelling and analysis of large turbo-generators. An identification algorithm for bearing dynamic characterization by using unbalance response measurements is developed for multi-degree-of-freedom (m.d.o.f.) flexible rotor-bearing systems. The algorithm identifies the bearing dynamic parameters, consisting of four effective stiffness and four damping coefficients for each bearing, utilizing frequency domain synchronous unbalance response measurements from the accelerometers attached to the bearing housings in the horizontal and vertical directions, for a minimum two different unbalance configurations. The procedure of identifying bearing dynamic coefficients by using the proposed algorithm is presented and demonstrated through a numerical example. Adding noise to the simulated signal checks the robustness of the algorithm against measurement noise. Combinations of regularization and the generalized singular value decomposition (SVD) are used to tackle an ill-posed problem due to the nearly circular orbit of the rotor at the bearings, as a special case for nearly isotropic bearings. It is demonstrated that by measuring noisy bearing responses with the direction of rotation of the rotor both in the clockwise and anticlockwise directions, the bearing estimation problem for circular orbit becomes well-conditioned. The regularization algorithm is tested for an experimental rotor-bearing rig. The response reproduction capabilities are excellent even in the presence of measurement noise.     

Optimum support characteristics for rotor-shaft system with preloaded rolling element bearings

Preloading of rolling element bearings is often used to avoid clearance in the bearings and achieve precise dynamic requirement. Preloading gives rise to an expression of the restoring force, which is a non-linear function of the deformation of the rolling elements. In this paper, frequency-dependent optimum support characteristics have been found out by simultaneously minimizing the unbalance response (UBR) of the rotor and maximizing the stability limit speed (SLS) of a flexible horizontal rotor-shaft system comprising an unsymmetrically placed rotor disc placed on an elastic shaft mounted on preloaded rolling element bearings at the ends supported on viscoelastic polymeric supports. A sensitivity study of the UBR and SLS with respect to the support characteristics has been presented to have an idea about the permissible deviation of the support characteristics from the respective optimum, at any frequency. Thus, the sensitivity study helps the quality control man as well as the manufacturer of such supports to estimate the permissible deviation in the most sensitive frequency zones. The results presented in this work are in terms of non-dimensional parameters of the system and are, therefore, valid for any system under consideration.     

Dynamics and stability of rigid rotors levitated by passive cylinder-magnet bearings and driven/supported axially by pointwise contact clutch

A stable rotor—supported laterally by passive magnetic bearings and longitudinally by magnetic forces and a clutch—loses suddenly its contact to the clutch and executes abruptly longitudinal movements away from its original equilibrium position as a result of small increases in angular velocity. Such an abrupt unstable behaviour and its reasons are thoroughly theoretically as well as experimentally investigated in this work. In this context, this paper gives theoretical as well as experimental contributions to the problem of two dimensional passive magnetic levitation and one dimensional pointwise contact stability dictated by mechanical–magnetic interaction. Load capacity and stiffness of passive multicylinder magnetic bearings (MCMB) are thoroughly investigated using two theoretical approaches followed by experimental validation. The contact dynamics between the clutch and the rotor supported by MCMB using several configurations of magnet distribution are described based on an accurate nonlinear model able to reliably reproduce the rotor-bearing dynamic behaviour. Such investigations lead to: (a) clear physical explanation about the reasons for the rotor's unstable behaviour, losing its contact to the clutch and (b) an accurate prediction of the threshold of stability based on the nonlinear rotor-bearing model, i.e. maximum angular velocity before the rotor misses its contact to the clutch as a function of rotor, bearing and clutch design parameters.     

Magnetic Drag and Energy Losses in Noncontact Bearings Based on Superconducting Tapes

In the paper, the problem of magnetic drag and origination of energy losses in noncontact bearings based on high-temperature superconducting tapes is considered. The model configurations of bearings in which a superconducting tape is a stator and a set of permanent magnets is a rotor are investigated. It is shown that the magnetic friction can be neglected in the case where more than eight permanent magnets compose the rotor. This result indicates the possibility to create scaled magnetic bearings for the systems of long-term energy storage, for example, flywheel energy storage systems.     

A novel structure of permanent-magnet-biased radial hybrid magnetic bearing

The paper proposes a novel structure for a permanent-magnet-biased radial hybrid magnetic bearing. Based on the air gap between the rotor and stator of traditional radial hybrid magnetic bearings, a subsidiary air gap is first constructed between the permanent magnets and the inner magnetic parts. Radial magnetic bearing makes X and Y magnetic fields independent of each other with separate stator poles, and the subsidiary air gap makes control flux to a close loop. As a result, magnetic field coupling of the X and Y channels is decreased significantly by the radial hybrid magnetic bearing and makes it easier to design control systems. Then an external rotor structure is designed into the radial hybrid magnetic bearing. The working principle of the radial hybrid magnetic bearing and its mathematical model is discussed. Finally, a non-linear magnetic network method is proposed to analyze the radial hybrid magnetic bearing. Simulation results indicate that magnetic fields in the two channels of the proposed radial hybrid magnetic bearing decouple well from each other.     

Control of hysteretic instability in rotating machinery by elastic suspension systems subject to dry and viscous friction

Most of the undesired whirling motions of rotating machines can be efficiently reduced by supporting journal boxes elastically and controlling their movement by viscous dampers or by dry friction surfaces normal to the shaft axis, which rub against the frame. In the case of dry dampers, resonance ranges of the floating support configuration can be easily cut off by planning a motionless adhesive state of the friction surfaces. On the contrary, the dry friction contact must change automatically into sliding conditions when the fixed support resonances are to be feared. Moreover, the whirl amplitude can be restrained throughout the speed range by a proper choice of the suspension-to-shaft stiffness ratio and of the support-to-rotor mass ratio.This theoretical research deals firstly with the natural precession speeds and looks for Campbell plots in dependence on the shaft angular speed, for several rotor-suspension systems. Then, the steady response to unbalance is investigated, in terms of rotor and support orbits and of conical path of the rotor axis. In this search, the ranges of adhesive or sliding contact are identified in particular for system with dry friction damping. At last, the destabilizing influence of the shaft hysteresis in the supercritical regime is focalized and the counterbalancing effect of the other dissipative sources is verified. In the nonlinear case of dry friction dampers, the control of linear stability is fulfilled by a perturbation procedure, checking the magnitude of Floquet characteristic multipliers on the complex plane. Moreover, the nonlinear stability far from steady motion is tested by the direct numerical solution of the full motion equations. The comparison configuration of suspension systems with viscous dampers and no dry friction is examined through an analytical first approximation approach and closed-form results for stability thresholds are derived in particular for the symmetric case.     

Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings

The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.     

Dynamic instability of supercritical driveshafts mounted on dissipative supports—Effects of viscous and hysteretic internal damping

The case of a rotating shaft with internal damping mounted either on elastic dissipative bearings or on infinitely rigid bearings with viscoelastic suspensions is investigated in order to obtain the stability region. A Euler-Bernoulli shaft model is adopted, in which the transverse shear effects are neglected and the effects of translational and rotatory inertia, gyroscopic moments, and internal viscous or hysteretic damping are taken into account. The hysteretic damping is incorporated with an equivalent viscous damping coefficient. Free motion analysis yields critical speeds and threshold speeds for each damping model in analytical form. In the case of elastic dissipative bearings, the present results are compared with the results of previous studies on finite element models. In the case of infinitely rigid bearings with viscoelastic suspensions, it is established that viscoelastic supports increase the stability of long shafts, thus compensating for the loss of efficiency which occurs with classical bearings. The instability criteria also show that the effect of the coupling which occured between rigid modes introducing external damping and shaft modes are almost more important than damping factor. Lastly, comparisons between viscous and hysteretic damping conditions lead to the conclusion that an appropriate material damping model is essential to be able to assess these instabilities.     

Design sensitivity analysis of dynamic responses for a BLDC motor with mechanical and electromagnetic interactions

This paper presents a design sensitivity analysis of dynamic responses of a BLDC motor with mechanical and electromagnetic interactions. Based on the equations of motion which consider mechanical and electromagnetic interactions of the motor, the sensitivity equations for the dynamic responses were derived by applying the direct differential method. From the sensitivity equation along with the equations of motion, the time responses for the sensitivity analysis were obtained by using the Newmark time integration method. The sensitivities of the motor performances such as the electromagnetic torque, rotating speed, and vibration level were analyzed for the six design parameters of rotor mass, shaft/bearing stiffness, rotor eccentricity, winding resistance, coil turn number, and residual magnetic flux density. Furthermore, to achieve a higher torque, higher speed, and lower vibration level, a new BLDC motor was designed by applying the multi-objective function method. It was found that all three performances are sensitive to the design parameters in the order of the coil turn number, magnetic flux density, rotor mass, winding resistance, rotor eccentricity, and stiffness. It was also found that the torque and vibration level are more sensitive to the parameters than the rotating speed. Finally, by applying the sensitivity analysis results, a new optimized design of the motor resulted in better performances. The newly designed motor showed an improved torque, rotating speed, and vibration level.     

INFLUENCE OF A LEVELNESS DEFECT IN A THRUST BEARING ON THE DYNAMIC BEHAVIOUR OF AN ELASTIC SHAFT

This paper examines the non-linear dynamic behaviour of a flexible shaft. The shaft is mounted on two journal bearings and the axial load is supported by a defective hydrodynamic thrust bearing at one end. The defect is a levelness defect of the rotor. The thrust bearing behaviour must be considered to be non-linear because of the effects of the defect. The shaft is modelled with typical beam finite elements including effects such as the gyroscopic effects. A modal technique is used to reduce the number of degrees of freedom. Results show that the thrust bearing defects introduce supplementary critical speeds. The linear approach is unable to show the supplementary critical speeds which are obtained only by using non-linear analysis.     

Stability and transient dynamics of a propeller–shaft system as induced by nonlinear friction acting on bearing–shaft contact interface

This paper investigates the friction-induced instability and the resulting self-excited vibration of a propeller–shaft system supported by water-lubricated rubber bearing. The system under consideration is modeled with an analytical approach by involving the nonlinear interaction among torsional vibrations of the continuous shaft, tangential vibrations of the rubber bearing and the nonlinear friction acting on the bearing–shaft contact interface. A degenerative two-degree-of-freedom analytical model is also reasonably developed to characterize system dynamics. The stability and vibrational characteristics are then determined by the complex eigenvalues analysis together with the quantitative analysis based on the method of multiple scales. A parametric study is conducted to clarify the roles of friction parameters and different vibration modes on instabilities; both the graphic and analytical expressions of instability boundaries are obtained. To capture the nature of self-excited vibrations and validate the stability analysis, the nonlinear formulations are numerically solved to calculate the transient dynamics in time and frequency domains. Analytical and numerical results reveal that the nonlinear coupling significantly affects the system responses and the bearing vibration plays a dominant role in the dynamic behavior of the present system.     

Nonlinear dynamics of a rub-impact micro-rotor system with scale-dependent friction model

Scale effects in dry friction at microscale and the coefficients of friction due to adhesion force and two- and three-body deformations are considered. A rub-impact micro-rotor model with scaling nonlinear rub-impact force is presented and the nonlinear dynamic characteristics of the system in micro-electro-mechanical systems (MEMS) are investigated when the rotating speed, imbalance, damping coefficient, scale length and fractal dimension are regarded as the control parameters. Effects of scale length, fractal dimension, velocity-dependent impact factor and contact form on the coefficients of dry friction are investigated and discussed, and used to study the nonlinear behavior of rub-related vibrations with a large number of numerical simulations. The effects of rotating speed, imbalance, damping coefficient, and friction coefficient on the micro-rotor system responses are studied. It is indicated that the rub-impact micro-rotor system with the scale effects in friction alternates among the periodic, quasi-periodic and chaotic motions as the system parameters change. The results can be effectively used to diagnose the rub-impact fault, reduce the failure and improve the characteristics of a micro-rotor system, and optimize the design of micro-rotating machinery in MEMS.     

Linear parameter varying control design for rotating systems supported by journal bearings

Linear parameter varying (LPV) control is a model-based control technique that takes into account time-varying parameters of the plant. In the case of rotating systems supported by lubricated bearings, the dynamic characteristics of the bearings change in time as a function of the rotating speed. Hence, LPV control can tackle the problem of run-up and run-down operational conditions when dynamic characteristics of the rotating system change significantly in time due to the bearings and high vibration levels occur. In this work, the LPV control design for a flexible shaft supported by plain journal bearings is presented. The model used in the LPV control design is updated from unbalance response experimental results and dynamic coefficients for the entire range of rotating speeds are obtained by numerical optimization. Experimental implementation of the designed LPV control resulted in strong reduction of vibration amplitudes when crossing the critical speed, without affecting system behavior in sub- or super-critical speeds.     

Nonlinear dynamic analysis of a rotor-bearing-seal system under two loading conditions

The operating speed of the rotating machinery often exceeds the second or even higher order critical speeds to pursue higher efficiency. Thus, how to restrain the higher order mode instability caused by the nonlinear oil-film force and seal force at high speed as far as possible has become more and more important. In this study, a lumped mass model of a rotor-bearing-seal system considering the gyroscopic effect is established. The graphite self-lubricating bearing and the sliding bearing are simulated by a spring-damping model and a nonlinear oil-film force model based on the assumption of short bearings, respectively. The seal is simulated by Muszynska nonlinear seal force model. Effects of the seal force and oil-film force on the first and second mode instabilities are investigated under two loading conditions which are determined by API Standard 617 (Axial and Centrifugal Compressors and Expander-compressors for Petroleum, Chemical and Gas Industry Services, Seventh Edition). The research focuses on the effects of exciting force forms and their magnitudes on the first and second mode whips in a rotor-bearing-seal system by using the spectrum cascades, vibration waveforms, orbits and Poincaré maps. The first and second mode instability laws are compared by including and excluding the seal effect in a rotor system with single-diameter shaft and two same discs. Meanwhile, the instability laws are also verified in a rotor system with multi-diameter shaft and two different discs. The results show that the second loading condition (out-of-phase unbalances of two discs) and the nonlinear seal force can mainly restrain the first mode instability and have slight effects on the second mode instability. This study may contribute to a further understanding about the higher order mode instability of such a rotor system with fluid-induced forces from the oil-film bearings and seals.     

Modeling of a rotor speed transient response with radial rubbing

A rotor-stator model of a turbogenerator is introduced in order to investigate speed transients with rotor-to-stator rubbing caused by an accidental blade-off imbalance. In order to assess the angular deceleration of the rotor due to rubbing, the angular position of its cross-section is considered as an unknown of the problem. Displacement fields are discretized through a finite element formulation. The highly nonlinear equations due to contact conditions are solved through an explicit prediction-correction time-marching procedure combined with the Lagrange multiplier approach dealing with a node-to-line contact strategy. The developed numerical tool is suitable for analyzing rotor-stator interactions in turbomachines as the system passes through critical speeds during an accidental shutdown. The sensitivity of the system response to modeling, physical and numerical parameters is investigated. The results highlight the significant role of the friction coefficient together with the diaphragm modeling, from rigid to fully flexible, in the interaction phenomenon. Rigid models have the advantage of simplicity and provide reasonable estimations of the overall response of the turbine. A flexible model, however, may be more computationally intensive but is more appropriate in order to accurately capture quantities of interest such as shaft eccentricity and bearing loads.     

Rotor–stator contact dynamics using a non-ideal drive—Theoretical and experimental aspects

The possible contact between rotor and stator is considered a serious malfunction that may lead to catastrophic failure. Rotor rub is seen as a secondary phenomenon caused by a primary source, i.e. sudden mass unbalance, instabilities generated by aerodynamic and hydrodynamic forces in seals and bearings among others. The contact event gives rise to normal and friction forces exerted on the rotor at impact events. The friction force plays a significant role by transferring some rotational energy of the rotor to lateral motion. A mathematical model has been developed to capture this for a conventional backup annular guide setup. It is reasonable to superpose an impact condition to the rub, where the rotor spin energy can be fully transformed into rotor lateral movements. Using a nonideal drive, i.e. an electric motor without any kind of velocity feedback control, it is even possible to stop the rotor spin under rubbing conditions. All the rotational energy will be transformed in a kind of “self-excited” rotor lateral vibration with repeated impacts against the housing. This paper studies the impact motion of a rotor impacting a conventional backup annular guide for the case of dry and lubricated inner surface of the guide. For the dry surface case, the experimental and numerical analysis shows that the rotational energy is fully transformed into lateral motion and the rotor spin is stopped. Based on this study this paper proposes a new unconventional backup bearing design in order to reduce the rub related severity in friction and center the rotor at impact events. The analysis shows that the rotor at impacts is forced to the center of the backup bearing and the lateral motion is mitigated. As a result of this, the rotor spin is kept constant.