Development and Experimental Validation of Auxiliary Rolling Bearing Models for Active Magnetic Bearings ( AMBs ) Applications

Nowadays, the search for increasing performances in turbomachinery applications has led to a growing utilization of active magnetic bearings (AMBs), which can bring a series of advantages thanks to their features: AMBs allow the machine components to reach higher peripheral speeds; in fact there are no wear and lubrication problems as the contact between bearing surfaces is absent. Furthermore,AMBs characteristic parameters can be controlled via software, optimizing machine dynamics performances. However, active magnetic bearings present some peculiarities, as they have lower load capacity than the most commonly used rolling and hydrodynamic bearings, and they need an energy source; for these reasons, in case of AMBs overload or breakdown, an auxiliary bearing system is required to support the rotor during such landing events. During the turbomachine design process, it is fundamental to appropriately choose the auxiliary bearing type and characteristics, because such components have to resist to the rotor impact; so, a supporting design tool based on accurate and efficient models of auxiliary bearings is very useful for the design integration of the ActiveMagnetic Bearing System into themachine.This paper presents an innovativemodel to accuratelydescribe the mechanical behavior of a complete rotor-dynamic system composed of a rotor equipped with two auxiliary rolling bearings. Themodel, developed and experimentally validated in collaboration with Baker Hughes a GE company (providing the test case and the experimental data), is able to reproduce the key physical phenomena experimentally observed; in particular, the most critical phenomenon noted during repeated experimental combined landing tests is the rotor forward whirl, which occurs in case of high friction conditions and greatly influences the whole system behavior. In order to carefully study some special phenomena like rotor coast down on landing bearings (which requires long period of time to evolve and involves many bodies and degrees of freedom) or other particular events like impacts (which occur in a short period of time), a compromise between accuracy of the results and numerical efficiency has been pursued. Some of the elements of the proposed model have been previously introduced in literature; however the present work proposes some new features of interest. For example, the lateral and the axial models have been properly coupled in order to correctly reproduce the effects observed during the experimental tests and a very important system element, the landing bearing compliant suspension, has been properly modelled to more accurately describe its elastic and damping effects on the system. Furthermore, the model is also useful to characterize the frequencies related to the rotor forward whirl motion.


Introduction
Rotors equipped with AMBs technology have the advantage to allow the machine to reach high performances, but a secondary rolling bearings system is required, because magnetic bearings are not able to support the rotor in case of overload or failure of the energy supply system.Such auxiliary bearings system, whose main elements are catcher bearings, is directly engaged during rotor landings; thus, in order to safely support the rotor, it must have very high mechanical and dynamical properties in order to guarantee the desired impact, thermal, and wear resistance.The landing system shall also be able to protect the rotor from possible damage and guarantee long service life; for these reasons, the development of a tool able to 2 International Journal of Rotating Machinery represent such rotor-dynamic systems and based on efficient rolling bearing models is useful to provide an appropriate support during the design phase.
The current work presents an improved auxiliary bearing system model with a FEM rotor model able to represent real system dynamic behavior and to reproduce the key effects observed during experimental tests.The model is able to predict the behavior of the complete system during transient, steady and emergency conditions (e.g., delevitation and landing phenomena) and it is composed of different submodels: the rotor submodel, which includes a FEM model to represent the lateral dynamic of the rotor and two rigid body models to reproduce axial and torsional rotor dynamics, and the auxiliary rolling bearing submodel, characterized by a combination of multibody and contact models able to represent the dynamics and the interactions of the auxiliary system elements.
The model has been developed and experimentally validated in collaboration with Baker Hughes a GE company which provided the system layout data and the results of the experimental tests performed on a dedicated scale test rig (Section 2.5).
In this paper, a comparison between the most significant experimental and simulated results is presented; through the analysis of the results, it is possible to observe a satisfying reproduction of the real system behavior.Real rotor displacements, auxiliary bearing elements displacements, interaction forces and moments, whirling frequencies, rotor, and bearing speeds and acceleration are very well represented by the simulated results thanks to the optimal calibration of the model parameters.
Rotating system behavior has been studied extensively in the past by many researchers.Szcygielski [1] conducted one of the initial studies concerning a high-speed rotor touching a rigid body, modelling the rotor like a gyro pendulum; the preliminary analytical results showed a good representation of the experimental trajectories of the gyro axis.
Among many researchers, Muszynska [2], Beatty [3], and Ehrich [4] studied the rubbing phenomena, revealing that the rotating systems show nonlinear dynamical behavior such as sub-and super-synchronous responses and even chaotic motion.
Other researchers investigated theoretically and experimentally the dry friction backward whirl motion; Bartha [5,6] observed that, during rotor impact and rub phenomena, the rotor vibration shows backward orbiting due to the presence of a tangential friction force.
Wilkes et al. [7] investigated, through experimental and numerical methods, the nature of multimode dry friction whip and whirl for a variety of rub materials and clearances; their experimental results showed multiple whirl and whip regions.
In order to reduce rubbing and the related phenomena, Lahriri et al. [8] proposed new backup bearing design, using special pins to center the rotor during impacts and mitigate its lateral motion.Furthermore, they developed a mathematical model to capture phenomena arising from rotor impacts.
Schmied and Pradetto [9] studied the behavior of a rotor landing onto auxiliary bearings and developed a numerical predictive model: simulation results reproduced a fully backward whirl rotor orbit for a few rotations, until the auxiliary bearing reached the rotor speed.
Other researches focus on the steady-state responses of rotor-dynamic systems: H. Xie et al. [10] studied a complex rotor system supported by auxiliary bearings with clearance at each end of the rotor.The influence of rotor imbalance, clearance, support stiffness, and damping has been investigated using direct numerical integration of the governing equation of motion and the harmonic balance method.
L. Della Pietra et al. [11] studied and experimentally investigated the static and the dynamic behavior of a rigid balanced rotor on journal bearings; the conducted study underlined the importance of the static behavior of the rotor bearing system for a correct interpretation of some aspect of the dynamic behavior.In fact they observed that a small dynamic imbalance can significantly modify the rotordynamic behavior inducing a whirl with large orbits.
Other researchers designed innovative auxiliary bearings.Hawkins et al. [12] and McMullen et al. [13] have performed endurance tests, testing different variety of bearings and landing sleeves materials in order to improve the design of the auxiliary system.They observed excessive wearing of the thrust washer and concluded that a thrust surface with lower friction coefficient could provide better performance.
M. Dougdag et al. [14] carried out an experimental verification of a simplified model of a ball bearing in both static and dynamic modes and test its capabilities to simulate accurately fault effects; the developed model considers the balls scrolling in the cage and the effect of the load rotating vector.
Ransom et al. [15] and Masala et al. [16] performed experimental tests and developed specific numerical model on a scaled and a full scale AMB supported subsea compressor.R. G. Kirk [17] analysed AMB turbomachinery applications concerning the operation of the rotor drop onto the auxiliary bearings.His studies focused on the selection of specific auxiliary bearings able to withstand the contact forces following the speed rotor drop.The proper design of AMB machinery requires the calculation of the anticipated loading for the auxiliary bearings; so, Kirk presented results of transient response evaluation of a full-size compressor rotor to illustrate some of the important parameters in the design for rotor drop.
In another research E.E.Swanson et al. [18] studied in detail the dynamic behavior of the rotor/bearing/support systems in the case of total loss of AMB support.They presented test results for the initial rotor drop transient for several backup bearing configurations.The cases reported are rigidly mounted antifriction bearing, rigidly mounted solid bushings (lubricated and unlubricated), and two soft mounted solid bushing configurations.
Other researchers like J. Wilkes et al. [19] observed strong circular forward whirl following an AMB failure and developed specific models to reproduce such phenomenon.They developed an innovative model of auxiliary bearing system that represents a great step forward in the reproduction of rotor-dynamic system behavior; in fact the model allows reproducing the forward whirl rotor orbit observed during the landing tests.The model is based on a formulation where a friction force between the axial face of the rotor sleeve and the thrust face of the catcher bearing has been included.
The model proposed in this research work represents an efficient and flexible tool, able to reproduce the behavior of the whole system and the physical phenomena connected to the auxiliary rolling bearings system (e.g., forward whirl).A compromise between accuracy of the results and numerical efficiency is mandatory, in order to carefully study some interesting phenomena like the rotor full coast down on landing bearings, which requires long period of time to evolve and involves many bodies and degrees of freedom, and particular events like impulsive forces which occur in a very short period of time.
The present work proposes some innovative features of interest; the new formulation takes into account the reciprocal coupling between the lateral and the axial rotor-dynamic models, enabling the reproduction of the forward whirl observed in the radial dynamics and the oscillatory axial behavior of the system during the whirling phenomenon.The elastic and damping elements of the compliant axial and radial auxiliary bearing suspension have been modelled taking into account their features and their influence on the whole system behavior.
The developed model can simulate the rotor dynamics in presence of different unbalances and represent the behavior of the whole system under different operating conditions: rotor constant speed, rotor decelerated speed, and different axial load profiles; it also includes the estimation of power and energy losses, so it can be used as an effective supporting tool for the machine designer to foresee the main sources of power losses.Furthermore the model is also useful to characterize the frequencies related to some characteristic phenomena of the system, like the backward and forward oscillation frequencies of the machine components.

Materials and Methods
In this Section the architecture of the rotor-dynamic system (Figure 2) is described and the structure of the developed numerical model is presented; Section 2.3 is dedicated to the description of the auxiliary bearing system and the dynamic and contact models developed to represent the bearing elements behavior.
. .General Architecture.The rotor-dynamic system represented in the numerical model is composed of a rotor supported by two radial and one axial magnetic bearings and equipped with an auxiliary bearing system based on two rolling bearings (Figure 1).
Between the rotor sleeve and the catcher bearings a radial clearance is present: in emergency conditions, the rotor passes through the radial clearance, lands on the inner rings of the catcher bearings, and continues to rotate supported by the auxiliary system.
The real rotor-dynamic system is equipped with a force actuator which applies an external axial thrust load on the nondrive end of the system; at this side of the system there is also an axial clearance between the rotor sleeve and the auxiliary bearing; under the axial thrust, the rotor translates along its symmetry axis.
Between the outer race and the housing specific elastic elements (Figure 1) are placed: the axial support for the bearing and the elastic ribbon damper.The axial support acts on the outer race of the bearing, while the ribbon is placed between the outer ring of the auxiliary bearing and the housing; it creates both radial and tangential force components and has both elastic and damping characteristics.In the event of a full circular whirl mode, the ribbon characteristics greatly influence the frequency of the whirl motion.
The whole model is able to predict the behavior of the complete rotor-dynamic system during transient, steady and emergency conditions (e.g., emergency landing phenomena) and it is composed of two main submodels (see Figure 2): (1) Rotor model, which includes a FEM model representing the lateral dynamics of the rotor and two simplified models, where the rotor is represented as a rigid body, to reproduce axial and torsional rotor dynamics; the main purpose is to have a good compromise between accuracy of the results and computational efficiency, considering the long time duration of the events to be simulated.
(2) Auxiliary rolling bearing model, characterized by a combination of multibody and contact models, able to represent the dynamics of the elements of the auxiliary system.
In order to reproduce the forward whirl phenomenon observed during combined radial-axial landing experimental tests, the lateral and the axial model have been coupled (see Figure 2): the axial force calculated in the rotor axial model represents an input for the catcher bearing lateral model.
The effect produced by this coupling allows well reproducing the phenomena observed during the experimental tests and, especially, the forward whirling.
. .Rotor Model.The lateral dynamic of the rotor is described by a FEM model, while the axial and the torsional dynamics are described by two simplified models (Figure 2) where the rotor is represented as a rigid body.
The kinematic quantities calculated by the rotor model (e.g., rotor nodes radial and axial displacements y, z, , and , rotor nodes velocities ẏ , ż , α , and β , and rotor angular velocity Ω) represent the inputs of the auxiliary rolling bearing model; the interaction forces and moments (e.g., radial interaction forces between the rotor and the bearing   , interaction torques , and axial forces F ax ) between the bearing elements are calculated in the auxiliary bearing model and constitute the inputs of the rotor model.So, there is a reciprocal exchange of kinematic and dynamic quantities between the submodels at each time step of the simulation.
During the experimental tests carried out on a dedicated test rig, in particular during combined radial-axial landings, coupling effects between the lateral and the axial dynamics have been observed.Under an axial thrust load the experiments show a forward whirl rotor lateral orbit; in order to represent such effects, a specific coupled formulation between the two dynamic models has been developed.
The whole model provides a good representation of the dynamical behavior of the rotor with a low computational effort.
. . .Rotor Lateral Model.In order to describe the lateral dynamics, the rotor is represented through a finite element discretization, where each rotor element is considered as a 2D BEAM (Figure 3).Such FEM model simulates the lateral rotor behavior, according to the classical rotor-dynamics equations of motion and accounting for magnetic and auxiliary bearings actions and external loads [20]; each BEAM element is based on the Timoshenko Theory [20,21].Each node has 4 degrees of freedom: two radial translations and two rotations around the translational axes.
The inputs of the model are the forces and moments acting on the rotor in correspondence of the nodes where the magnetic and the auxiliary bearings are located and the external loads and torques (i.e., the unbalance load).The outputs of the rotor model are the kinematic quantities: positions x and velocities ẋ for each degree of freedom of the rotor nodes.
The equations of motion that represent the rotor lateral behavior are reported below [20]: where x contains the two translational and the two rotational degrees of freedom of the rotor nodes (, , , ), while M rot , C rot , G rot , and K rot are, respectively, the mass, damping, stiffness, and gyroscopic matrices of the rotor; the vector  contains the external forces and moments acting on the rotor nodes.Furthermore, a modal reduction for the rotor FEM lateral model has been implemented.The modal reduction used is based on the Craig-Bampton method and generates reduced rotor matrices (mass, stiffness, damping, and gyroscopic matrix) on the base of the number of nodes and modes chosen by the user to describe the considered system [22]; by using the modal reduction, it is possible to simulate long period of time in quite short simulation time, reaching a good compromise between results accuracy and computational efficiency.
. . .Rotor Axial Model.In order to reproduce the axial rotor dynamics, a simple model has been developed in which the rotor is represented as a rigid body able to translate along its symmetry axis.The model describes the axial vibration of the rotor and the axial force, which are input for the lateral model and must be known at each instant of the simulation in order to have a real coupling between the lateral and the axial dynamics.
Figure 4 shows a scheme of the system axial model; the   ,   parameters are the equivalent axial stiffness and damping of the system which have variable values depending on the rotor axial displacement;  ℎ is the thrust force applied by the actuator and is used to simulate the axial landing, while   is the rotor mass.In correspondence of the initial instant of the simulation, the rotor is placed at its initial position (  (0) = 0) where the axial clearance is equal to   ; when the axial thrust force is applied to the system, the rotor moves according to the following dynamic equations: where  ℎ is the axial thrust force acting on the rotor and   is the rotor mass.
By knowing   () and the equivalent stiffness   and damping   of the system (Figure 4), the axial reaction force   can be calculated as ( The axial force   () depends on the axial thrust load force  ℎ and acts on the lateral surface of the rotor sleeve and on the thrust surface of the inner race of the bearing.The axial contact between the rotor sleeve and the inner race of the rolling bearing produces a tangential force that is proportional to   () and the friction coefficient; for this reason, it acts on the lateral degrees of freedom of the rotor and of the bearing inner race and it represents the coupling between the lateral and the axial dynamic model: the axial force   , calculated through (3), is the main input of the rotor lateral model and of the rolling bearing inner race lateral model.
The equivalent axial stiffness   of the system can be calculated considering the stiffness elements (bearing stiffness   , support stiffness   , and housing stiffness  ℎ ) and taking into account the axial displacement of the rotor: when the axial system displacement is less than the axial clearance, no stiffness is considered to work; as the system translation exceeds the axial clearance   , the support stiffness   , and the bearing stiffness   work together; so, before the axial displacement of the shaft reaches the value   +   corresponding to the maximum deflection of the support, the equivalent axial stiffness can be obtained through the series of   and   and the axial force   () can be calculated as follows: where   represents the axial coordinate of the rotor.When the axial displacement of the shaft exceeds the value   +   (and the maximum deflection of the support is reached), the housing and the bearing stiffnesses  ℎ and   work in series and the support stiffness   does not play any role anymore: if . . .Rotor Spinning Model.In order to determine the rotor rotational velocity during the simulation, an approximate torsional model has been implemented, in which the rotor is described as a rigid body able to rotate around its symmetry axis.This simplified model allows obtaining a good representation of the torsional dynamics of the rotor, ensuring again a good compromise between accuracy and numerical efficiency.
The torques acting on the rotor are the auxiliary bearing moments and the windage torques (see ( 6)): where  represents the torsional moment of inertia and Ω is the rotor angular speed;  1 and  2 are the torques exerted by the two auxiliary bearings on the rotor while   is the windage torque.
The rotor angular speed decreases, during a simulation, because of friction effects (see Figure 5).
. .Auxiliary Bearing Model.The catcher bearing model describes the dynamics of the bearing components and allows the calculation of the interaction forces and moments between their contact surfaces.By solving the equations of motion of each element of the auxiliary bearing system, it is possible to calculate the position of the center of gravity of each element of the catcher bearing.
From a dynamical point of view, a simplifying assumption has been made: the catcher bearing elements are able to move in the plane, so the bearing inner race has 3 degrees of freedom (i.e., it can translate along the  and  directions and rotate around the  axis) while the outer race can only translate in the plane without rotating.
Equations (7) describe the translational dynamics of the rolling bearing inner race; the bearing inner race degrees of freedom are two because the inner race motion is considered to move in the radial plane of the bearing  (Figure 5).This simplifying hypothesis allows obtaining a good compromise between the accuracy computational efficiency: where x i contains the coordinates of the inner race center of gravity, M i is the inner race mass matrix, C i is the inner race damping matrix, P is the weight force of the inner race, and the vector F is the sum of the radial forces acting on the bearing inner race (see also Figure 7): F io is the interaction force on the inner race exerted by the rolling elements, F ir is the contact force exchanged between the inner race and the rotor during the contact (including a normal F rN component and a tangential one F fr ; see (15)), and F rub is the resultant force generated during the axial rub between the thrust surface of the bearing inner race and the lateral surface of the rotor sleeve (see (19)).Equation ( 8) describes the rotational dynamics of the bearing inner race: where   is the torsional moment of inertia of the bearing inner race and   is the moment acting on the inner race and due to the action of the radial friction force   ;   is the ball bearing drag torque due to the rolling friction of the rolling elements during the motion and   is the resultant torque due to the axial rub between the thrust surface of the bearing inner race and the lateral surface of the rotor sleeve.
The outer surface of the outer race interacts with the housing through the ribbon damper, which is an elastic-damping element; furthermore, there is another elastic element: the axial support on the outer race, placed between the external surface of the outer race and the statoric flange.
Equation ( 9) describes the translational dynamics of the bearing outer race, which can move into the radial plane  (Figure 5): where x o contains the coordinates of the outer race center of gravity, M o and C o are the bearing outer race mass and damping matrices, the vector F oi is the interaction force on the outer race exerted by the rolling elements, and F oh is the radial force exchanged between the outer race and the housing flange.
The auxiliary bearings have spherical rolling elements, which have been modelled as force elements, without representing their dynamic behavior.The elastic force exerted by the rolling elements allows representing the interactions between the outer and the inner race of the bearings.
. .Contact Models.The catcher bearing model simulates the plane motion of the auxiliary bearing elements and calculates the possible interaction forces and moments among them; the kinematic physical quantities are determined by solving the equations of motion of each elements (see Section 2.3), while the interaction forces/moments are calculated through appropriate contact models based on the Hertz theory.
In this section, the contact models implemented in the proposed bearing model will be explained, showing the formulation used to calculate the interaction forces and moments between the contact surfaces.
The surfaces which interact during the motion are (see Figure 7): (1) radial contact between the rotor landing sleeve and the inner surface of the bearing inner race: the contact force F ir exchanged between the inner race and the rotor during the contact includes a normal F rN component and a tangential one F fr ; furthermore, a friction moment T fr is generated; (2) axial contact between the thrust surface of the rolling bearing inner race and the lateral side of the landing rotor sleeve: during axial or combined axial-radial rotor landings, axial rubbing force and moment are generated F rub , T rub ; (3) elastic interaction between the bearing outer race and the bearing inner race through the rolling elements: the balls exert an elastic force F io on the inner race and F oi on the outer races; (4) the interaction force between the housing and the outer race is exerted through the ribbon damper element F oh .
. . .Rotor-Inner Race Radial Interaction.Concerning the development of this part of the model, see also the paper [19].During a radial landing, the inner surface of the bearing inner ring and the rotor landing sleeve get in contact; the rotor bounces, after the impact, on the inner ring and continues to rotate supported by the auxiliary bearings.As a result of the interaction with the rotor, the inner ring accelerates and tends to reach the rotor speed.
The contact forces F rN exchanged between the rotor and the bearing inner ring is made up of two components: F rN is the normal contact force and F fr is the friction force due to the radial rubbing.
The mathematical steps required to calculate such forces are briefly reported below.
It is very useful to define a radial-tangential (( 1 ), ( 1 )) reference system and express the quantities of interest (e.g., bodies centers of gravity coordinates; velocities) in this specific system (Figure 8).Furthermore, it is convenient to refer the analysis to the relative displacements and velocities between the rotor and the inner race in radial and tangential coordinates.The relative displacements d ri between the two bodies can be obtained from the difference between the coordinates of the rotor node O r and the coordinates of the inner race center of gravity O i : Through the use of the relative coordinates, it is possible to define a relative angle of rotation   and, consequently, to determine the rotational matrix R needed to convert the quantities expressed in the Cartesian reference system (( 0 ), ( 0 )) into the radial-tangential reference system (( 1 ), ( 1 )).
Equation (11) defines the relative angle of rotation   : while (12) defines the transformation matrix R: The relative displacements d ri and velocities k ri expressed in the Cartesian reference system can be expressed into the radial-tangential reference system using the rotational matrix R: The contact forces and moments are calculated through the contact model if there is a interpenetration   between the two bodies.By knowing the physical value of the radial clearance, the penetration between the two bodies along the radial direction can be calculated as follows: Using the interpenetration   , it is possible to calculate the normal contact force F rN and the friction force F fr due to the radial rubbing between the rotor sleeve radial surface and the inner surface of the inner ring: where  ℎ is the Hertzian contact constant,  V is the viscous contact constant,   is the radial friction coefficient, and  is the sign function of the tangential component of the relative velocity V  : The friction force F fr due to the radial rubbing between the rotor sleeve and the inner ring generates a torque T fr responsible for the acceleration of the inner race: During a pure radial landing, the rotor falls and bounces on to the inner ring of the bearing, which begins to rotate and quickly reaches the rotor angular speed; in case of high radial friction, the rotor performs a backward whirl orbit because of the friction effects between the rotor and the bearing inner ring.
On the other hand, during a combined axial-radial rotor landing due to AMBs failure (both radial and axial magnetic bearings), the rotor gets in contact with the inner surface of the inner ring and the thrust face of the catcher bearing.In case of high friction effects, a circular forward whirl rotor orbit has been observed in some experimental tests; in fact, as a consequence of repeated rotor landings, the bearing elements begin to be worn and the friction coefficient increases; the tangential actions due to that axial friction result to be stronger than the radial rubbing force.
. . .Rotor Sleeve-Inner Race Axial Interaction.As regards the development of this part of the model, see also the paper [19].Concerning the lateral dynamics, the rotor can perform small rotations around  and  axes; as a consequence of the rotor dynamics, the interpenetration between the lateral face of the rotor sleeve and the thrust surface of the inner ring cannot have a constant value; by knowing the inclination angles  and  of the rotor at any instant of time, it is possible to calculate the interpenetration value  ri at each point of the thrust surface and, consequently, calculate the axial pressure   at any point on the contact surface.
In order to calculate the resultant force and the resultant torque acting on the rotor due to the friction at the thrust surface, it is necessary to evaluate the relative slip velocity between the same point (, ) on the rotor and on the bearing.
The vector k A,r represents the velocity of the point (, ) on the rotor, while the vector k A,i represents the velocity of the point (, ) on the bearing inner ring: The relative velocity k A,ri between the point  on the rotor and the same point  on the bearing can be simply calculated by subtracting the inner ring velocity k A,r from the rotor velocity k A,i .By knowing the relative velocity between the rotor and the bearing, the resultant force F rub , acting on the rotor and due to friction at the contact interface, can be evaluated by integrating the pressure field over the contact area: where   is the friction coefficient between the lateral face of the rotor sleeve and the thrust face of the bearing.The torque due to the axial contact between the rotor and the bearing inner race can be determined as follows: ( The torque on the rotor and on the bearing is considered to be equal and opposite: . . .Outer Race-Inner Race Radial Interaction.The interaction force F io between the bearing inner and outer races is the radial reaction due to the relative motion of the races. The races can interact through the rolling elements, which are modelled as force elements; through the load-deflection curve of the catcher bearing it is possible to extrapolate the stiffness characteristics of the rolling elements ( 1 and  2 ) and define the interaction force F io with a second order polynomial function: where   is the relative displacement between the inner and the outer races centers of gravity.
The rolling elements exert also another important action on the catcher bearing: the rolling friction due to the balls generates a drag torque onto the inner ring, which decelerates under this effect.The approach to calculate the resistant torque T CB is based on the Harris-Palmgren formulation [23], but, instead of considering the applied static load, the International Journal of Rotating Machinery dynamic radial and axial forces acting on the bearing have been taken into account: where  1 is a factor depending on the bearing design and relative bearing load,   depends on the magnitude and direction of the applied load, and   is the pitch diameter of the bearing.For a ball bearing, Harris [23] defines  1 as where   represents the equivalent dynamic load and   is the basic static load rating usually provided in the manufacturer catalog, while  and  represent empirical coefficients.For a single row angular contact bearing the equivalent load F s can be defined as follows: where   = 0.5 and   varies with the bearing contact angle : For a radial ball bearing   is given by The numeric parameters  1 ,  2 ,  3 ,  4 , and  5 of ( 26) and ( 27) have been determined on the base of the bearing geometry and characteristics:  1 = 7.142910 −5 ,  2 = −1.244310−2 ,  3 = 0.64129,  4 = 0.9, and  5 = −0.1.

. . . Outer Race-Housing Interaction.
The ribbon damper is located between the bearing outer ring and the housing; it has both elastic and damping characteristics (i.e., structural and viscous damping) and generates radial and tangential force components.The characteristics of the ribbon greatly influence the frequency of the rotor whirl motion.
The interaction force between the outer race and the housing F oh can be expressed through the force exerted by the ribbon: where I is the identity matrix, x o is the vector of the coordinates of the outer race center of gravity,   is the ribbon viscous damping,   = 6.3 * 10 7 N/m is the ribbon stiffness, and Q is a matrix defined as follows: The parameter  is a cross coupled stiffness factor, while  V is the sign of the relative tangential velocity between the outer bearing and the housing.For combined axial-radial rotor landings, because of the axial thrust force applied to the bearing inner race, a reaction force exerted by the housing on the outer race through the axial support is present.The force contribution due to this interaction is the damping force F oh : where   is the Coulomb friction coefficient between the outer race and the housing and must be added to (28).
Figure 10 shows a scheme of the test rig that is equipped with a sensors system able to estimate components temperature, forces, acceleration, vibrations, and speeds: (1) load cells (); (2) proximity sensors (); (3) encoders (); The tests performed on the test rig allow the identification of the system behavior in landing conditions.In particular the tests carried out on the test rig belong to three different types (with different rotor speeds): radial landings (deactivating just radial AMBs), axial landings (deactivating just axial AMB), and combined radial-axial landings (deactivating both the AMBs).The experimental tests taken as a reference in this paper for the model validation (Section 3) are no whirling rotor radial landing, a pure radial landing characterized by a normal friction condition; whirling rotor radial landing, a pure radial landing with high friction coefficient; whirling rotor combining axial-radial landing with high axial thrust force value.

Results
The proposed model allows performing the simulation for all the classical operating conditions and to set up parameters like the rotor angular velocity, the friction coefficients, and the landing time.
In this section the most interesting results of the main simulations are reported and compared to the related experimental results.In particular, it is easy to observe the different dynamic behavior of the rotor in case of radial landing (with normal friction conditions and high friction conditions) and of combined axial-radial landing.During a pure radial landing characterized by normal friction conditions, the rotor bounces on to the bearing inner rings and, after the first bounces, the rotor and the inner ring rotate at the same angular speed.In case of a pure radial landing characterized by high radial friction coefficient, the rotor realizes a backward orbit; while in a combined axial-radial landing with a high axial thrust force value acting on the rotor, the tangential forces generated in the axial contact cause a rotor forward whirl.The backward and forward whirl phenomena occur in particular conditions of high axial or radial friction; in fact, after a sequence of radial or combined axial-radial landings, some bearing elements begin to be worn (e.g., bearing inner race) as a consequence of the high impact and friction forces and consequently, the friction coefficient increases causing the whirling phenomenon.
All the simulation results reported in the paper have been obtained by using both variable step ODE algorithms and fixed step ODE algorithms [24].The choice of the constant step for the fixed step algorithms and of the maximum step size for the variable step algorithms (together with the related tolerances) has been made to have a characteristic integration frequency (i.e., the inverse of the time step) always greater than the most significant frequencies of the considered system.
. .No Whirling Landing.In case of overloads or failure of the energy supply, the AMBs are disabled and the rotor falls and impacts on the inner rings of the auxiliary bearings.
The reported results show the displacements, forces, and orbits of specific rotor nodes; in particular, the nodes where the auxiliary bearings are located at the drive end and at the nondrive end of the machine (for brevity denoted DE node and NDE node) have been considered.The reported simulation and experimental test reproduces a rotor landing, in which the rotor has an angular speed equal to 7000 RPM and the AMBS are deactivated; so, as a consequence, the rotor lands on the auxiliary bearing system bouncing on to the bearing inner rings.During this first radial landing test (low radial friction without backward whirling), the following values were set for the speed Ω, the radial friction   , the axial friction   , and the axial load  ℎ : Ω =7000rpm,   =0.15 (low radial friction),   =0.3, and  ℎ =0N (no axial load).
In Figures 11 and 14 it is possible to observe the vertical and the horizontal displacements of the DE rotor node; by comparing the simulated plots with the experimental displacements reported in Figures 13 and 16, it is possible    to note a good representation of dynamic rotor behavior.The vertical displacements show some initial peak values, which correspond to the first rotor bounces on the inner rings characterized by a high energy content; after that, the rotor and the bearing inner ring rotates at the same angular speed.In Figures 12 and 15   Figure 17 shows the orbit described by the DE rotor node during the simulation; the blue circle is the nominal gap between the landing bearing bore and the landing sleeve, while in Figure 18 the experimental rotor orbit is reported.
. .Whirling Radial Landing.During a radial landing the rotor falls and impacts on the inner ring of the auxiliary bearing, it bounces on the ring, and in case of high friction and any conditions of wear, after some second tenths the rotor begins to whirl; in case of radial landing the contact force between the rotor sleeve and the inner ring of the bearing causes a backward whirl rotor orbit (Figure 27), considering the rotor angular speed convention shown in Figure 19.The reported results show the displacements, forces, and orbits of specific rotor nodes; in particular, the nodes where the auxiliary bearings are located at the drive end and at the nondrive end of the machine (for brevity denoted DE node and NDE node) have been considered.The rotor backward whirl is the predominant phenomenon in case of pure radial landings because of the radial friction effects.
The reference simulation reproduces a pure radial landing where the rotor rotates with an angular speed equal to 7000 RPM, the AMBS are deactivated, and the rotor lands on the auxiliary bearing system; between the inner race of the catcher bearing and the rotor landing sleeve surface, friction radial forces arise: such forces are responsible for the rotor backward whirl orbit.
In order to emphasize such behavior, the radial friction coefficient has been set to a quite high value with respect to the combined axial-radial landings, where the axial friction coefficient is predominant on the radial one.The high value of the friction coefficient simulates the wear condition of the bearing elements due to the repeated landings.
During this second radial landing test (high radial friction with backward whirling), the following values were set for the speed Ω, the radial friction   , the axial friction   , and the axial load  ℎ : Ω =7000rpm,   =0.3 (high radial friction),   =0.3, and  ℎ =0N (no axial load).
In   to better highlight amplitudes, phases and frequencies).By comparing the simulated plots with the experimental displacements reported in Figures 23 and 26, it is possible to note a good representation of the backward rotor orbit.The vertical displacements show some initial peak values, which represent the rotor bounces on the inner ring; after 0,5 s the vertical displacements switch from maximum negative values to maximum positive values: such oscillatory trend produces the backward whirl rotor orbit.Figure 27 show the orbits described by the DE rotor node during the simulation; the blue circle represents the nominal gap between the landing bearing bore and the landing sleeve.In Figures 22 and 25 the vertical and the horizontal forces on the rotor node in correspondence of the auxiliary bearing located at the DE side of the machine are reported; it is possible to observe the high dynamical values of such forces during the backward whirl.
In Table 1 the comparison between some significant results of the experimental tests and the related numerical    simulations is reported: there is a good alignment between the displacements, and the activation time of the backward whirl has the same values.
. .Whirling Combined Axial-Radial Landing.In this section the most significant results of the axial-radial combined    [Hz] 51 49 landing have been reported.The results reported in this paper are aimed at highlighting the forward whirl, the most important phenomenon observed during the combined landings; it occurs when the tangential forces between the lateral surface of the inner ring and the thrust surface of the rotor sleeve, caused by the axial friction, have a high value.In this condition such tangential forces prevail on the radial friction effects, which are responsible for the rotor backward whirl, and the forward whirl may occur.As in the previous case, the forward whirl is a particular phenomenon which may take place in case of high friction and wear conditions.The considered simulation corresponds to a test in which the rotor rotates with an angular speed equal to 7000 RPM supported by the AMBs; the AMBs are deactivated, the rotor falls, and lands on to the auxiliary bearings.At the same time also the axial AMB is deactivated and an axial thrust force acts on the system determining the axial displacement of the rotor and the contact between the inner race and the landing sleeve.
In case of a combined radial-axial landing the rotor performs a forward whirl orbit: in fact, during the axial landing the thrust surface of the inner ring gets in contact with the lateral surface of the rotor sleeve; because of this interaction, friction tangential forces are generated and the rotor begins to perform a forward whirl orbit.During the axial-radial landing test (with forward whirling), the following values were set for the speed Ω, the radial friction   , the axial friction   , and the axial load  ℎ : Ω =7000rpm,   =0.15 (low radial friction),   =0.3, and  ℎ =4000N.
In Figures 28, 30  By making a comparison between the resultant vertical displacements and forces of the simulation and the experimental results, it is possible to find a good agreement between experimental and numerical physical quantities.
The orbits of the DE and NDE rotor nodes resulting from the numerical simulation are a bit different from the experimental orbits: in the experimental test the rotor begins

Figure 1 :
Figure 1: General scheme of a magnetic bearings supported rotor with its auxiliary landing system.

Figure 5 :
Figure 5: Scheme of the rotor motion.

Figure 6 :
Figure 6: Schematic representation of the auxiliary system.

Figure 7 :
Figure 7: Schematic representation of the interactions between the elements of the auxiliary system.

Figure 8 :
Figure 8: Interaction normal and friction forces between the rotor and the bearing inner ring.

Figure 9 :
Figure 9: Experimental apparatus; from left to right: axial actuator, NDE radial magnetic bearing, speed sensor, DE radial magnetic bearing, magnetic thrust bearing, and power transmission coupling.

Figure 13 :Figure 14 :
Figure 13: Experimental rotor vertical displacements during a landing test at DE (blue line) and NDE (red line) rotor nodes.

Figure 15 :
Figure 15: Predicted rotor horizontal forces during a landing test (DE node).

Figure 16 :
Figure 16: Experimental rotor horizontal displacements during a landing test at DE (blue line) and NDE (red line) rotor nodes.

Figure 17 :
Figure 17: Predicted rotor orbit during a landing test (DE node).
Figures 20 and 24  it is possible to observe the vertical and the horizontal displacements of the DE rotor node (the time zooms of such quantities are reported in Figure21

Figure 23 :
Figure 23: Rotor vertical displacements during a radial landing experimental test at DE (blu line) and NDE (red line) rotor nodes.

Figure 24 :
Figure 24: Predicted rotor horizontal displacements during a radial landing test (DE node).

Figure 26 :
Figure 26: Rotor horizontal displacements during a radial landing experimental test at DE (blue line) and NDE (red line) rotor nodes.

Figure 27 :Figure 28 :
Figure 27: Predicted rotor backward whirl orbit at the DE rotor nodes.

Figure 29 :FYFigure 30 :
Figure 29: Time zoom of the predicted rotor vertical (in blue) and horizontal (in red) displacements at DE node during a combined landing test.