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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.
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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
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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)
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= 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
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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.
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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
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Figure 2: Short SFD model in DBM
Figure 3: Effect of length on reaction forces
Figure 4: Effect of length on damping
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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
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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
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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
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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
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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.
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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