An analytical investigation of turbocharger rotor-bearing dynamics with rolling element bearings and squeeze film dampers A Ashtekar

Cummins Turbo Technologies, USA

L Tian, C Lancaster

Cummins Turbo Technologies, UK

ABSTRACT

The objective of this investigation is to examine the dynamics of a turbocharger supported by a deep groove or angular contact ball bearing and a squeeze film damper. In this novel approach a six degree of freedom 3D discrete element bearing model was interlaced with a first principle squeeze film damper model to determine the combined stiffness and damping of the turbocharger support. The combined model accounts for the current and the past dynamic states of the system to provide a more accurate support behavior than the current simplified 2-D bearing models used for rotor dynamic analysis. In addition, the Reynolds equation is iteratively solved for the squeeze film damper model to determine the damper behavior while accounting for side leakages. This allows for examining any shape or size of dampers. The combined model was then used to determine the dynamics response of the turbocharger by coupling it with a traditional quasi-static model as well as a time dependent rotor dynamic models. The effect of bearing component (inner race, outer race, cage and roller) defects on support stiffness and excitation will be examined. The damper will affect not only the turbocharger dynamics but also the bearing dynamics, affecting the bearing life.

1 INTRODUCTION

Turbochargers have commonly been equipped with journal bearings to support the turbines and rotor assembly. However, ball bearings have become popular as a replacement for journal bearings in turbochargers. Wang (1), in his review of ceramic bearing technology, points out that the hybrid ceramic bearing can provide better acceleration response, lower torque requirement, lower vibrations and lower temperature rise than journal bearings. Hybrid ceramic ball bearings contain steel inner and outer races, ceramic balls and usually a machined cage. Ceramic balls, as compared to their steel counter parts, are lighter, smoother, stiffer, harder, corrosion resistant, and electrically resistant. These fundamental characteristics allow for a wide range of performance enhancements in bearing rotor system. Ceramic balls are particularly well suited for use in harsh, high temperatures and/or corrosive environment. Therefore, hybrid ceramic bearings are ideal for turbocharger applications. Miyashita et al. (2), Keller et al. (3) and Tanimoto et al. (4) have employed ball bearings in small, automotive turbochargers. However, challenges still remain for high speed, high output turbochargers which demand large bore bearings operating at DN numbers over 2 million. As the bearing size increases, the dynamics of the bearing rotor system becomes critical for comprehensive design and satisfactory operation of the turbocharger.

Investigators have attempted to analytically analyze the dynamics of turbocharger rotor system. San Andrs et al. (5,6,7) has presented comprehensive models to predict turbocharger dynamics. Inclusion of a complete fluid-film bearing model provided an insight into the effects of bearing dynamics on the dynamics of a turbocharger. Bou-Said et al. (8) also investigated the rotor dynamics of a turbocharger with linear and non-linear aerodynamic bearing models. Pettinato et al. (9) demonstrated the advantages of such turbocharger rotor dynamic models by employing them to improve the design of bearings used in a turbocharger. Bonello (10) implemented non-linear model to study the dynamics of turbocharger on full floating and semi-floating ring bearings. However, most of the work in turbocharger rotor dynamic models has been concentrated on turbochargers with journal bearings. Therefore these models are unable to predict the rotor dynamics of turbochargers which use rolling element bearings. Nevertheless, investigators have attempted to develop analytical models to study the dynamics of simple rotor systems with rolling element bearings. Gupta (11-13) was among the first to present a three dimensional bearing dynamic model. The model developed was capable of analyzing motion of all bearing components. Meyer et al. (14) introduced the effects of defects on bearing and demonstrated the vibrations patterns associated with the defects. Saheta et al. (15) and Ghaisas et al. (16) presented a six degree of freedom, fully dynamic discrete element model. Their models consider bearing components as sections of spheres and cylinders, which significantly reduced the computational effort associated with bearing dynamic modeling. Sopanen et al. (17, 18) developed a bearing model which included the effects of inclusions. However in their analysis, cage dynamics and centrifugal loads were ignored. Ashtekar et al. (19, 20) developed a six degree of freedom bearing model which included the effects of bearing surface defects. In general, the previous investigators concentrated on the bearing dynamics and ignored the complicated interaction of the roller bearing with the shaft/rotor system. However, for a complete understanding and examination of high speed, high output turbochargers it is critical to combine the effects of the bearing and shaft/rotor dynamics. In high speed applications, the rotor undergoes various mode shapes resulting in complex motion of bearing rotor system. Lim et al. (21) and Hendrikx et al. (22) developed a bearing model including the effects of rotor flexibility; however they neglected the effect of bearing cage on the dynamics of the system. Tiwari (23, 24) considered the effects of imbalance and bearing preloading on the rotor dynamics, however, a simplified ideal bearing model was considered and rotor was assumed to be rigid. Prenger (25) presented a bearing model capable of modeling tapered roller bearings and angular contact bearings. Prengers model included the effect of flexible shafts; however, only simple shaft models were considered and the model was

unable to handle high speed applications. BEAST software developed by Stacke et al (26) is known to consider rotor flexibility; however, neither the model nor the results are available in public domain.

In this investigation a model was developed to represent the turbocharger bearing rotor system. The model combines a discrete element bearing model and a flexible rotor model to simulate the dynamics of the bearing rotor system. The model was then used to investigate the motion of each bearing components and determine the forces and deflection of the rotor as a function of various operating conditions. The results from the model were used to investigate the bearing performance at various preloads, rotor imbalances and operating speeds.

2 MODEL DESCRIPTION

A ball bearing consists of an inner race, outer race, rolling elements (balls) and a cage which separates the balls. These bearing components interact with each other directly or indirectly, affecting the motion and forces occurring between them. The turbocharger rotor is supported by the inner race and thus its motion and forces are also affected by the dynamics of the turbocharger rotor. As these motions and forces are eventually transmitted from the inner race to all other bearing components, the turbocharger rotor affects the dynamics of all bearing components. Similarly, any dynamic instability within the bearing is transmitted to the turbocharger rotor. In this study, the analytical investigation includes a bearing dynamic model which interacts with a flexible turbocharger rotor model to predict the dynamics of rotor system.

2.1 Dynamic bearing Model A key aspect of modeling the bearing dynamics with Discrete Element Method is obtaining the forces and moments acting on the bearing components. In the current model, the gravitational forces, contact forces and rotor interaction forces are considered as a part of the analysis.

Rolling element contact forces are considered when balls are in contact with other bearing components. Although bearing component surfaces deform to some degree when in contact, these deformations are typically very small in comparison to the balls characteristic length. Hence, in the contact model the detailed deformations of the contacting surfaces are ignored and instead the two contacting surfaces are allowed to overlap slightly. The degree of overlap is then used to determine the contact forces acting on the bearing components. To simplify the overlap calculations, bearing components are assumed to be made of simplified geometry consisting of sections of sphere and cylinders. The overlap, , between the elements is given by;

= (1 + 2) |2 1 | Where, 1 and 2 are the radius of the bodies and 1 and 2 are the position vectors of the respective bodies. It is possible to consider other shapes in the simulations; however, the contact detection schemes become more computationally intensive (Ting (27), Ting et al. (28), Matuttis et al. (29)). The normal contact force can be determined using the overlap and Hertzian force-deflection relationship

= 3 2 Where, K is the Hertzian stiffness coefficient. This approach of calculating normal contact force is much simpler and less computationally intensive than the method described by Gupta (30).

The Hertzian stiffness, K for two general, non-conformal contacting solids is given by Hamrock (31)

= 3 2

29 1 2

Where R is the curvature sum given by,

1

= 1

+ 1

; 1

= 1

+ 1

; 1

= 1

+ 1

The values of and are curvatures of the body in X and Y planes respectively. is the effective elastic modulus obtained from the elastic properties of bodies in contact and is given by,

= 2

1 2

+ 1 2 Where , , , are the modulus of elasticity and Poissons ratio for the two bodies. Parameters , , and require iterative calculations, however, current study uses the approximate solution provided by Hamrock (31)

= 2 ; = ; = 2 + ln; = 1 + ; = 2 1 In addition to a normal force, a tangential force exists at the point of contact between the ball and race. This tangential force is determined using a traction model, the relative tangential velocity at the point of contact, and the normal force at the contact. In this investigation, the Kragelskiis (32) model is used

= ( + | |)(| |) + , is related to slip velocity, , which is the difference in instantaneous velocities of bodies in contact.

Here, values of A, B, C, and D were calculated using the method used by Gupta (33). The tangential friction force is then given by

= | | Evaluation of the tangential friction forces at the contact can be quite involved because of the variations in local slip velocities from point to point in the contact ellipse. However, as pointed out by Gupta (30, 34), for most bearing applications the contact ellipse is sufficiently narrow along the direction of rolling so that the variations in the slip velocity and hence friction force along the semi minor axis can be neglected. Thus the total friction force can be evaluated by integrating the friction forces along the major axis of contact ellipse (30, 34). The resulting tangential force also creates a moment about the ball center and a moment about the center of contact ellipse. The moment about the ball center results in rolling motion while the moment about contact ellipse center causes the ball to spin.

The resulting contact forces and moments act equally but in the opposite directions on both of the bodies in contact. After calculating the total force and moments acting on the bodies, Newtons second law is used to calculate the linear and angular acceleration of the bodies. The accelerations are integrated with respect to time to obtain velocities and displacements in linear and angular directions. Each body has 6 DOF and thus each body is associated with 6

equations that are integrated. System presented in this paper has two bearing and each bearing has 15 such bodies. The above procedure is repeated at each time step using the new component states until some end condition, typically maximum time is reached.

2.2 Squeeze Film Damper Hamrocks (31) solution of Reynolds equation is used to calculate the reaction forces due to squeeze film damper on the outer race. For a long damper assumption, (l/d > 4), the side leakages can be neglected and the Reynolds equation reduces to

3

= 1220

This can be solved to get the relationship

= 120( + )33(1 2)3/2

The damping coefficient can be expressed as,

= 120( + )33(1 2)3/2

Where, is the absolute viscosity of the oil, is the radius of the bearing outer race, is the outer race velocity along Z axis, c is the clearance in the squeeze film damper and z is the position of the outer race CG.

The above relationship is suitable for any bearing with l/d ratio greater than four. For these bearings, the side leakages can be ignored so that the infinitely long bearing assumption holds true. However, for shorter bearings, consider the Reynolds equation with side leakages.

3

+

3

= 120

For a bearing,

= r and = (1 ) Therefore, equation reduces to

3

+ 23

= 620

= / Thus,

3

+ 24 3 = 12

This equation does not have an exact solution and thus needs to be solved iteratively. For a given and c, a solution is evaluated for a bearing position defined by h (or z, y in this case). Iterative solutions were obtained for a range of l and c values to generate a database of SFD. Table 1 shows the range of l and c values considered for the study.

Table 1: Range of l and c values

L (mm) 2 3 6 10 100

C (m) 20 30 50 100 200

Figure 1 shows the database plots. Intermediate values were obtained using linear interpolation. Figure 2 shows the implementation of database into DBM. Please note that the reaction force is opposite to the direction of OR velocity. Similarly, the reaction force along Y axis is also calculated. Both these reaction forces are added to the total forces acting on the OR discrete element. In addition, to include the effects of the anti-rotation pin, all outer race rotational degrees of freedom were constrained to be fixed. For each case, DBM was run and the motion of Inner Race, Outer Race, and Reaction Force at SFD and Damping Coefficient was recorded. Lower IR motion and reaction forces are primary benefits of well-tuned SFD.

From Figure 3 to Figure 5 it can be seen that the damping coefficient is sensitive to length of the bearing and the clearance. Higher damping reactions were observed at very small lengths and damping reactions reduced as length increased. However, after an optimum point the damping reactions shot up as length was increased to approach long SFD assumption. IR motion continued to reduce as the length of bearing increased. This is primarily due to the geometrical constraints due to larger contact surface between bearing and housing. Increased clearance had negative effect on reaction forces as well as IR motion.

Figure 1: Database for SFD

Figure 2: Short SFD model in DBM

Figure 3: Effect of length on reaction forces

Figure 4: Effect of length on damping

Figure 5: Effect of clearance on reaction forces

2.3 Dynamic Bearing Rotor Model (DBRM) Figure 6 depicts a schematic representation of the bearing rotor system as represented by the DBRM.

Figure 6: Dynamic Bearing Rotor Model

In this investigation, the DBM and damper model were used to determine the bearing response which was passed on to the flexible rotor dynamic model (FRM). The rotor model is an implicit solution and the ODE is solved for each steady state step.

Figure 7: 26 Node Dynamic Bearing Rotor Model

The shaft and the two wheels are modeled by the FRM and each angular contact bearing is modeled by a DBM. Interface nodes are established at points where the bearing supports on the rotor. These nodes are made coincident with the bearing inner race center of gravity. Thus the dynamic response is passed from one model to other. Figure 6 depicts these interface point interactions as two headed arrows indicating that the exchange of dynamic response occurs from both the sides, namely, DBM and the rotor model. The two bearings have a single piece outer race. Therefore, the outer races of the two DBMs, each representing one of the bearing, are rigidly linked to each other. Figure 6 illustrates these linkages shown as lines. Finally the single piece outer race of the DBMs is attached to the ground through a spring-damper arrangement representing the squeeze film damper.

Also, to include the effects of the anti-rotation pin, all outer race rotational degrees of freedom were constrained to be fixed. The two models, DBM and FRM, run parallel, communicating with each other at each time step. Any motion and/or forces due to turbocharger rotor flexibility affects the dynamics of all the bearing components and similarly dynamic response of bearing components affect the dynamics of the entire turbocharger bearing rotor system.

3 ADDITIONAL RESULTS AND OBSERVATIONS

3.1 Model Interaction Study To allow for the union of an explicit Bearing model with an implicit rotor model, three different methods were used.

In the first method the REB stiffness was evaluated using the DBM. The model was subjected to a varying IR motion and the reaction forces from the model were compiled to determine the stiffness matric of the bearing. This matrix can be used as support stiffness for any rotordynamic model of choice to investigate the turbocharger rotordynamics in presence of the REB. The table shows the matrix for a turbocharger bearing.

Table 2: Bearing Stiffness matrix

Bearing Direction Bearing Stiffness

Compressor

Fx 1.66E+06 -5.64E-07 2.86E-06 -9.62E-09 -1.23E+04

Fy -5.60E-07 1.66E+06 2.47E-06 1.23E+04 9.60E-09

Fz 2.86E-06 2.46E-06 1.43E+06 1.79E-08 -2.88E-08

Myz -9.43E-09 1.23E+04 1.79E-08 1.13E+02 1.03E-10

Mzx -1.23E+04 9.52E-09 -2.88E-08 1.03E-10 1.13E+02

Turbine

Fx 1.67E+06 2.38E-06 3.71E-06 1.82E-08 -1.24E+04

Fy 2.37E-06 1.67E+06 -1.05E-05 1.24E+04 -1.82E-08

Fz 3.70E-06 -1.05E-05 1.43E+06 -7.48E-08 -3.58E-08

Myz 1.82E-08 1.24E+04 -7.48E-08 1.13E+02 -1.69E-10

Mzx -1.24E+04 -1.82E-08 -3.59E-08 -1.70E-10 1.13E+02

This is a simple and efficient approach to be incorporated in any rotordynamic model. However, this method oversimplifies the REB and ignores the internal dynamics and instabilities of the REBs. The results from this method are nonetheless useful to analyze basic steady state dynamics. Comparison with other methods shows that the basic dynamics can be evaluated to acceptable accuracy.

In the second method, the model was used with a quasi-static approach model. The rotordynamic model is implicit and passes on the node state to the bearing model. The explicit DBM model is ramped up to the state and allowed to reach a steady state. The forces and displacements are passed back to the rotor model and the simulation continues. This method does not completely account for the past dynamics of the system but offers an REB solution that is analogues to journal bearing models. This method produces stable solutions and accounts for bearing internal dynamic response. However, the simulation resources and time required are significant.

In the third method the DBM was run in parallel to the rotordynamic model. Each time the rotor model passes on the states of the node, the past REB state is used as the starting point and the DBM explicit model is ramped on from the old node state to the new one. This allows for including the transient effects in the model. Rotor transients have a significant effect on bearings dynamics. However, these models do have the possibility of diverging solutions in rotordynamic models.

The results for each of these methods will be compared against each other and evaluated for a range of imbalances. The imbalance affects the rotordynamics as well as has a significant effect on the bearing dynamics.

3.2 Preloading Angular contact bearing are commonly preloaded, however, Hagiu (35) has demonstrated that wrong preloading will cause considerable reduction in bearing life. Figure 8 shows ball loads for two DBRM conditions, one which has preloading and one without preloading. Both of these cases are operating at the speed of 50000 rpm with 10 gm-mm imbalance. Please note the increased force fluctuation for the case of unloaded bearing. The results also demonstrate that occasionally the ball-race load becomes zero indicting loss of ball-race contact. The loss of load between the ball and inner race can cause ball sliding and skidding. The results demonstrate the significant effect of wrong bearing preloading in turbocharger. It is also to be noted that excessive preloading can lead to premature fatigue failure of the bearing. The effects of these bearing instability is examined on the turbocharger

Figure 8: Unloaded vs Preloaded Bearing Forces

3.3 Bearing Defects REB component configuration is of the planetary type with IR being the sun and the rolling elements being the planets. The rolling elements roll over the IR as well as the OR and drive the cage at the same time. Due to continuous but periodic nature of this interaction, a defect on any of the bearing components results in periodic excitation in the system.

Ball Defect Frequency = (Pd/(2*Bd)) * (N/60) * (1 (Bd/Pd*cos)^2)

Cage defect frequency = N/120 * (1 Bd/Pd*cos)

OR defect frequency = Nb/2 * (N/60) * (1 (Bd/Pd*cos))

IR defect frequency = Nb/2 * (N/60) * (1 + (Bd/Pd*cos))

Where, Pd = Pitch Diameter, Bd = Ball Diameter, Nb = Number of Balls, N = Speed, = contact angle of the angular contact bearing.

These equations provide a good guidance; however, they ignore the 3D nature of the bearings. i.e. there is no guarantee that the ball will pass over the defect each time. The ball track may be wide and might miss the defect in a periodic manner. Thus, a defect was introduced in the DBM using the defect models by Ashtekar et al. (19,20) and their effects on the rotordynamics were observed. This allows for a realistic simulation of the defects and their effects on turbocharger.

4 SUMMARY AND CONCLUSIONS

Investigation into replacing journal bearings of a high speed turbocharger with hybrid ceramic ball bearings requires a detailed analytical model of bearing rotor dynamics. For the analytical investigation rolling element bearing demands significant speed and accuracy of contact force calculations. An analytical model that meets these demands has been developed. A coupled dynamic model was developed for the ball bearing rotor systems. The model combines a discrete element bearing model and a flexible rotor model. A squeeze film iterative model was also developed to model the squeeze film dampers required to counter the high stiffness of the REB. The analytical model was used to investigate the different approaches to model the REB into the system. The model differences were highlighted under imbalance conditions to

demonstrate the dynamics ignored by simplified REB models. The model was also used to demonstrate the effects of preloading on the turbocharger dynamics. The analytical model was also used to gain knowledge of effects of the REB defects on turbocharger dynamics.

REFRERENCE LIST

(1) Wang L., Snidle R., and Gu L., 2000, "Rolling contact silicon nitride bearing

technology: a review of recent research," Wear, 246(1-2), pp. 159-173. (2) Miyashita, K., Kurasawa, M., Matsuoka, H., Ikeya, N., and Nakamura, F., 1987,

"Development of High Efficiency Ball-Bearing Turbocharger," SAE Paper No. 870354

(3) Keller, R., Scharrer, J., and Pelletti, J., 1996, "Alternative Performance Turbocharger

Bearing Design," SAE Paper No. 962500

(4) Tanimoto K., Kajihara K., and Yanai K., 2000, "Hybrid ceramic ball bearings for

turbochargers," SAE Paper 2000-01-1339, pp. 1-14.

(5) San Andres L., Rivadeneria J. C., Chinta M., Gjika K., La Rue G., 2007, Nonlinear

Rotordynamics of Automotive Turbochargers: Predictions and Comparisons to Test

Data, J. Eng. Gas Turbines Power, 129(2), pp. 488-494. (6) San Andres L., Rivadeneria J. C., Gjika K., Groves G., La Rue G., 2007, A Virtual Tool

for Prediction of Turbocharger Nonlinear Dynamic Response: Validation Against Test

Data, J. Eng. Gas Turbines Power, 129(4), pp. 1035-1047. (7) San Andres L., Rivadeneria J. C., Gjika K., Groves G., La Rue G., 2007,

Rotordynamics of Small Turbochargers Supported on Floating Ring Bearings---

Highlights in Bearing Analysis and Experimental Validation, J. Tribol., 129(2), pp. 391-398.

(8) Bou-Said B., Grau G., Iordanoff J., 2008, On Nonlinear Rotor Dynamic Effects of

Aerodynamic Bearings With Simple Flexible Rotors, J. Eng. Gas Turbines Power, 130, 012503.

(9) Pettinato B. C., DeChoudhury P., 2003, Rotordynamic and Bearing Upgrade of a

High-Speed Turbocharger, J. Eng. Gas Turbines Power, 125, pp. 95-102 (10) Bonello, P., 2009, Transient modal analysis of the non-linear dynamics of a

turbocharger on floating ring bearings, Proc. I. Mech. E. Part J: J. Engineering.

Tribology, 223, pp. 79-93. (11) Gupta, P. K., 1979, Dynamics of Rolling Element Bearings Part IV: Ball Bearing

Results, J. Lubrication Technology, 101, pp. 319-326. (12) Gupta, P. K., 1979, Dynamics of Rolling Element Bearings Part I: Cylindrical roller

bearing analysis, J. Lubrication Technology, 101, pp. 293-304. (13) Gupta, P. K., 1979, Dynamics of Rolling Element Bearings Part III: Ball Bearing

Analysis, J. Lubrication Technology, 101, pp. 312-318.

(14) Meyer L., Ahlgren F., and Weichbrodt B., 1980, "An analytic model for ball bearing

vibrations to predict vibration response to distributed defects," J. Mechanical Design,

102, pp. 205-210.

(15) Saheta, V.,2001, Dynamics of Rolling Element Bearings using Discrete Element

Method, M.S. thesis, Purdue University, West Lafayette, IN, USA

(16) Ghaisas, N., Wassgren, C., Sadeghi, F., 2004, Cage Instabilities in Cylindrical Roller

Bearings, J. Tribology, 126(4) pp. 681-689. (17) Sopanen J., and Mikkola A., 2003, "Dynamic model of a deep-groove ball bearing

including localized and distributed defects. Part 1: theory," Proc. I. Mech. E., Part K: J.

Multi-body Dynamics, 217(3), pp. 201-211. (18) Sopanen J., and Mikkola A., 2003, "Dynamic model of a deep-groove ball bearing

including localized and distributed defects. Part 2: implementation and results," Proc. I.

Mech. E., Part K: J. Multi-body Dynamics, 217(3), pp. 213-223. (19) Ashtekar A., Sadeghi F., and Stacke L., 2010, "Surface defects effects on bearing

dynamics," Proc. I. Mech. E., Part J: J. Engineering Tribology, 224(1), pp. 25-35. (20) Ashtekar A., Sadeghi F., and Stacke L. , 2008, "A New Approach to Modeling Surface

Defects in Bearing Dynamics Simulations," J. Tribology, 130(4), pp. 041103. (21) Lim T., and Singh R., 1990, "Vibration transmission through rolling element bearings,

part I: Bearing stiffness formulation," J. Sound and Vibration, 139(2), pp. 179199. (22) Hendrikx R., Van Nijen G., and Dietl P. , 1999, "Vibrations in household appliances

with rolling element bearings," Proc. International Seminar on Modal Analysis,

Katholieke Universiteit Leuven, 3, pp. 15371544. (23) Tiwari M., 2000, "Effect of Radial Internal Clearance of a Ball Bearing on the Dynamics

of a Balanced Horizontal Rotor," J. Sound and Vibration, 238(5), pp. 723-756. (24) Tiwari M., 2000, "Dynamic Response of an Unbalanced Rotor Supported on Ball

Bearings," J. Sound and Vibration, 238(5), pp. 757-779. (25) Prenger N., 2003, "Modeling the Dynamics of Rolling Element Bearings Using

ADAMS," M.S. thesis, Purdue University, West Lafayette, IN, USA.

(26) Stacke L., Fritzson D., and Nordling P., 1999, "BEASTa rolling bearing simulation

tool," Proc. I. Mech. E., Part K: J. Multi-body Dynamics, 213(2), p. 6371. (27) Ting J., 1992, "A robust algorithm for ellipse-based discrete element modelling of

granular materials," Computers and Geotechnics, 13(3), pp. 175-186. (28) Ting J., Khwaja M., and Meachum L., 1993, "An ellipse-based discrete element model

for granular materials," J. Numerical and Analytical methods in Geomechanics, 17(9), pp. 603-623.

(29) Matuttis H. G., Luding S., and Herrmann H. J., 2000, "Discrete element simulations of

dense packings and heaps made of spherical and non-spherical particles," Powder

Technology, 109(1-3), pp. 278-292. (30) Gupta P. K., 1975, "Generalized dynamic simulation of skid in ball bearings," J. of

Aircraft, 12, pp. 260265. (31) Hamrock B., Schmid S., and Jacobson B., 2004, Fundamentals of fluid film lubrication,

Marcel Dekker, Ohio.

(32) Kragelskii I.V., 1965, Friction and wear, Butterworth, London.

(33) Gupta P. K., 1984, Advanced Dynamics of Rolling Elements, Springer-Verlag, New

York.

(34) Gupta P. K., 1974, "Transient ball motion and skid in ball bearings," J. Lubrication

technology, 97, pp. 261-269. (35) Hagiu G., and Gafitanu M., 1994, "Preload-service life correlation for ball bearings on

machine tool main spindles," Wear, 172(1), pp. 79-83.

ABSTRACT1 INTRODUCTION2 MODEL DESCRIPTION2.1 Dynamic bearing Model2.2 Squeeze Film Damper2.3 Dynamic Bearing Rotor Model (DBRM)

3 ADDITIONAL RESULTS AND OBSERVATIONS3.1 Model Interaction Study3.2 Preloading3.3 Bearing Defects

4 SUMMARY AND CONCLUSIONSREFRERENCE LIST