DESIGN OF HIGH SPEED, HIGH POWER SPINDLES
BASED ON ROLLER BEARINGS
By
ISMAEL A. HERNANDEZROSARIO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1989
ACKNOWLEDGMENTS
The author wants to extends his sincere gratitude to
Dr. Jiri Tlusty, Dr. Scott Smith and H. S. Chen. The deepest
of all gratitude goes to my loving wife Laura.
This research was funded under National Science
Foundation grant # MEA8401442 Unmanned Machining, High
Speed Milling.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ................................... ii
ABSTRACT ............. .................. .......... vi
CHAPTER
1. HIGH SPEED, HIGH POWER MILLING
Introduction .................................. 1
Development of High Speed Milling .............. 2
High Speed, High Power Machining ............... 4
Goals and Scope ................................ 6
2. LITERATURE SEARCH
Analytical Developments ........................ 8
High Speed Bearings: Experimental Results ..... 38
3.EXPERIMENTAL EQUIPMENT
High Speed, High Power Milling Machine ........ 43
Test Spindles ........................ ....45
Oil Supply to the Bearings .................... 52
Instrumentation ............................. ..... 55
Oil Circulating System ........................ 57
Evaluation of Cooling Capacity ................60
Seals ......................................... 64
4. THERMAL ANALYSIS
Thermal Analysis of the Spindle Housing .......68
Friction in Rolling Bearings ................. 68
Heat Generation ............................... 70
Heat Removal ............................ ...... 82
Steady State Temperature Fields ............... 86
Thermally Induced Loads ...................... 89
Computation of Thermal Loads .................. 91
iii
5. EXPERIMENTAL RESULTS AND DISCUSSION
Test Procedure ................................ 97
Curve Fitting of Experimental Data ............98
Temperature ................................... 99
Steady State Temperatures Versus
Spindle Speed ......................... 100
Steady State Temperatures Versus
Oil Flow Rate ......................... 102
Overall Temperature Equation ............ 120
Steady State Temperatures:
Comparison ................................ 121
Power Measurements ........................... 127
Motor Power Losses ...................... 127
Mechanical Power Losses ................. 127
Hydraulic Power Losses .................. 128
Configuration Power Losses ................... 129
Mechanical Power Losses ................. 129
Hydraulic Power Losses ..................129
Hydraulic Power Losses Versus
Spindle Speed ......................... 130
Hydraulic Power Losses Versus
Oil Flow Rate ......................... 131
Overall Hydraulic Power Losses
Equation .............................. 132
Power Losses: Comparison ................ 133
Bearing Loads ................................ 148
Externally Applied Load ................. 148
Bearing Thermal Loads ................... 149
Performance of the Seals ..................... 150
Bearing Failures ............................ 151
Radiax Bearing Failure .................. 152
High Speed Bearing Failure .............. 154
6. BEARING LOADS
Load Deflection Relationships ................ 157
Radial Loads ................................. 162
Axial Loads ................................. 165
Combined Loading ............................ 168
Bearing Life Calculation ..................... 172
Bearing Preload ............................. 173
Preloading Methods .......................... 174
Case 1: Variable Preload ............... 174
Case 2: Constant Preload ................ 180
High Speed Loads ............................ 183
Cylindrical Roller Bearings ................. 183
Tapered Roller Bearings ...................... 187
Centrifugal Forces ...................... 187
Gyroscopic Moment ....................... 189
Combined Loading ........................ 191
7. CONCLUSIONS
Spindle Configurations ....................... 193
Cylindrical Roller Bearings ............. 193
Tapered Roller Bearings ................. 194
Experimental Conclusions .................... 195
Empirical Equations ..................... 195
Bearing Preload ......................... 198
Recommendations .............................. 199
Design Modifications for
Configuration I ....................... 199
Design Modifications for
Configuration II ...................... 200
Final Comment ................................ 201
APPENDIX
RADIAL LOAD COMPUTATION PROGRAM .............. 204
COMBINED LOAD COMPUTATION PROGRAM ............ 205
LOAD DEFLECTION COMPUTATION PROGRAM .......... 206
HIGH SPEED CYLINDRICAL ROLLER BEARING
PROGRAM .................................... 207
BIBLIOGRAPHY ............................................. 208
BIOGRAPHICAL SKETCH .....................................212
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DESIGN OF HIGH SPEED, HIGH POWER SPINDLES
BASED ON ROLLER BEARINGS
By
Ismael A. HernandezRosario
May 1989
Chairman: Dr. Jiri Tlusty
Major Department: Mechanical Engineering Department
An experimental investigation was performed on two
spindle configurations based on roller bearings to determine
their potential for High Speed, High Power Machining
applications. The type of roller bearings considered were
super precision tapered roller bearings and double row
cylindrical roller bearings. The idleload performance of
each spindle was evaluated in terms of maximum operating
speed, operating temperatures, lubrication requirements and
required power to operate the spindle. The tapered bearing
spindle was provided with a constant preloading mechanism.
Neither spindle was operated at the target 1.0 million
DN (DN is the product of the bearing bore diameter in mm
times the spindle speed in rpm), although both spindles
exceeded the speed capabilities of current machine tools
with similar bearing arrangements. The spindle based on
tapered roller bearings is strongly recommended for High
Speed, High Power applications for its low power losses and
low operating temperatures at 9,000 rpm, a DN value of
900,000 (the maximum speed achieved) at an operating
temperature of 77 degrees Centigrade and 15 kW of power
losses. The configuration based on cylindrical roller
bearings is not recommended to operate above the speed of
6,000 rpm, DN value of 600,000 after which the operating
temperature and power losses are above the practical limit.
vii
CHAPTER I
HIGH SPEED, HIGH POWER MILLING
Introduction
The development of advanced cutting tools has
drastically reduced the time required to perform metal
removal operations. These new tool materials are capable of
operating at speeds up to an order of magnitude higher than
previously existing tools [1]. Thus, the use of these tools
to the maximum capabilities is called High Speed Machining
(HSM). These tools can be used for HSM of steel at
operating speeds of 200 m/min. using Coated Carbides, or
HSM of cast iron at 1000 m/min. using Silicon Nitrides or
HSM of aluminum at speeds between 10005000 m/min. using
High Speed Steels or Solid Carbides [2].
The main advantage of HSM is the capability to remove
metal faster. The increased metal removal rates (MRR) is
extremely attractive for such industries where machining
accounts for a considerable portion of the processing time
or the manufacturing cost, such as the aerospace industry,
e.g. aircraft frames and engines, or the manufacturing of
automotive engine blocks. Considering that by 1986 over 115
billion dollars were spent on metal removal operations [3],
any increase in productivity would have substantial
economic effects.
Development of High Speed Milling
The first person to investigate high speed metal
removal was Dr. Carl J. Salomon in Germany from 1924 to 1931
[4]. Dr. Salomon investigated the relationship between
cutting speed and cutting temperature. As a result of his
investigation, Dr. Salomon concluded that as cutting speed
increased, so did the cutting temperature, until a critical
maximum temperature was reached. Once at this critical
temperature, any further increase in the cutting speed would
produce a decrease in cutting temperature. As the cutting
speed was increased even further, the cutting temperature
would drop to usual operating levels. Thus, around the
critical temperature there is a range of very high
temperatures at which tools can not operate. Below this
range, usual metal removing operations were performed. Once
above this critical temperature range, increased metal
removal rates could be obtained if the necessary cutting
speeds were achieved. The benefit of the region above the
critical temperature is that the cutting speed could be
increased such that infinite metal removal rates were
theoretically possible. Unfortunately, Dr. Salomon's work
and experimental data were mostly destroyed during World
War Two, and limited information of his work is currently
available.
In 1958, R. L. Vaughn, working for Lockheed, started a
research program sponsored by the United States Air Force to
investigate the response of some high strength materials to
high cutting speeds (152,400 surface meters per minute, smm)
[4]. Some of the conclusions presented by Vaughn [4] which
are of particular interest to this dissertation are
1. High speed milling could be used for machining
high strength materials.
2. Productivity will increase with the use of HSM.
3. Surface finish is improved with HSM.
4. The amount of wear per unit volume of material
removed decreased with HSM.
5. An aluminum alloy, 7075T6, was machined at
36560 smm with no measurable tool wear.
6. The increase in cutting force was between 33 to
70%, over conventional machining forces.
7. At the time of investigation, the technology
available could not make maximum use of HSM.
During the 1960's and 1970's various companies, such as
Vought and Lockheed, experimented with HSM. In each case,
the investigators agreed on the potential increase in
productivity that HSM may yield [4]. Raj Aggarwal
summarizes the results from published data on HSM
investigations [5]. These investigations have shown that
an increase in cutting speed will produce a reduction in
power consumption per unit volume of metal removed (unit
horsepower). Although the effect of increased cutting speed
would depend on the chip load used during the investigation,
in general, lower unit horsepower was obtained for higher
speeds. On the effect of cutting speed on cutting forces,
results varied, while some researchers measured some
decrease in the forces; others found little or no change.
At this point, it is important to note that companies
involved in HSM are reluctant to publish their complete
results and test conditions, based on commercial
competitivity [ 6 ], which makes the comparison of results
quite difficult. An area of agreement is the application of
HSM to the end milling of thin aluminum ribs, where improved
surface finish was obtained [5].
High Speed, High Power Machining
T. Raj Aggarwal concludes that high speed capabilities
alone will not produce a relevant increase in productivity
[5]. To obtain significant improvements in productivity,
high speed milling must be coupled to high power machining.
High power machining refers to those machining operations
where the power requirements are above the capacity of
common machine tools (1020 kW). The combination of HSM and
high power milling is called High Speed, High Power (HSHP)
milling.
F.J. McGee [7] directed a HSM program for the Vought
Corporation. As part of the research program, he identified
the ideal HSHP machine tool for their investigation as
having a spindle rated at 20,000 rpm speed and 75 kW power;
unfortunately such a machine tool was not available. The
closest available spindle was a 20,000 rpm, 22 kW spindle
by Bryant. McGee [7] stresses the fact that the spindles
5
currently available in the market do not have the power
required to make optimum use of HSM. In an effort to
correct this lack of HSHP milling machines the trend has
been to retrofit existing machines with high speed spindles
with improved power capabilities [8]. Although this
procedure will improve the HSM capabilities of the existing
machine tools, there is still the need for a spindle capable
of achieving spindle speeds of 10,000 rpm with power
capabilities above 30 kW.
Tlusty [2] defines the requirements for a HSHP spindle
capable of high metal removal rates and without power
limitations. For the face milling of cast iron and steel,
Tlusty recommends [2] the use of a spindle based on 100 mm
diameter roller bearings, tapered roller bearings (TRB) or
double row cylindrical roller bearings (CRB), a 10,000 rpm
and 115 kW rating. Tlusty shows that such a spindle could
make optimum use of the new cutting tool materials. The
high stiffness values which are inherent to roller bearings
combined with the use of stability lobes would make the
maximum use of the new cutting tools.
A spindle based on 100 mm bore TRB or CRB which
operates at 10,000 rpm, would operate well above the
catalogue maximum for these types of bearings. Usually
these bearings are operated below 4000 rpm [9,10]. In order
to operate these spindles above such speeds, special
lubrication and cooling systems should be provided. The
consequences of thermal differential expansions must also be
determined. If HSM is to be ever fully implemented, then
HSHP spindles must be developed. For these spindles to be
developed, the performance of large diameter CRB and TRB in
machine tools operating at very high speeds must be
researched and understood.
Goals and Scope
This dissertation is an experimental investigation on
the HSHP performance of two types of large diameter (over
100 mm) bearings which are widely used in machine tools,
double row cylindrical roller bearings and tapered roller
bearings. The goal of this dissertation is to establish
which of these bearings could best be used in HSHP spindles
and what are their requirements for a successful spindle
design. For each spindle configuration its lubrication and
cooling requirements, its operating temperatures and its
maximum operating speed must be determined. Parameters
which characterize the performance of a spindle must also be
defined or identified.
The design and/or development of new bearing geometries
is beyond the scope of this dissertation. The high cost of
developing and producing a new, nonstandard bearing
geometry is above the economic capabilities of the machine
tool industry. However, redesigning of spindles is well
within the economic bounds of the machine tool industry.
The goal of this dissertation is to provide new and much
needed knowledge on the HSHP performance of large diameter
7
bearings in machine tools and to identify those parameters
which are essential for a successful HSHP spindle design
based on roller bearings.
CHAPTER II
LITERATURE SEARCH
Analytical Developments
In 1963, Harris presented the first paper [11] in which
an analytical method was used to predict the behavior of a
bearing assembly. In this paper, Harris presents a method to
estimate the operating temperature of rolling element
bearings assuming steady state operation and using a finite
difference scheme. The operating temperature at several
different nodal points of a bearing assembly could be
estimated since at each nodal point the net increase in
energy is zero at steady state. By definition, at steady
state, the amount of heat transferred into a nodal point
equals the amount of heat transferred out of the nodal
point.
According to Harris [11], the heat generated in the
bearings is due to a load torque which resist the rotation
of the rolling elements plus a viscous torque induced by the
lubricant surrounding the rolling elements. By comparing
this generated heat to the heat dissipation capacity of the
assembly, the operating temperature may be estimated. Since
the generated heat is the result of power losses, it's
computation is relevant to this dissertation.
As presented by Harris, the heat generation depends on
the type of bearing used (ball or roller bearing), the
bearing geometry (contact angle), loading conditions (radial
or thrust), bearing diameter and lubricant properties. The
load torque can be estimated using equation (2.1) [11].
M, = 0.782 fx Pe d, (Nmm) (2.1)
where
Mr: Load torque (Nmm).
f.: Load torque factor.
P,: Equivalent applied load (N).
d.: Bearing pitch diameter (mm).
The load torque factor is a function of bearing design
and the relative bearing load. Palmgren [11] experimentally
determined relations for estimating f1 for most bearing
types. For ball bearings the factor f. is given by
1 = z ( (2.2)
where
Po: Static load (N).
Co: Static Load Rating (N).
The coefficient z and the exponent y were determined
experimentally and are given below in Table 2.1 from [11].
For roller bearings, the value of f. was also determined
experimentally. The value of f, for several types of roller
bearings is given in Table 2.2, 4lso from [11].
Table 2.1 Coefficient z and Exponent y
for Ball Bearings
Bearing Type Contact Angle (0) z y
Deep Groove 0 0.0009 0.55
Angular Contact 30 0.001 0.33
Angular Contact 40 0.0013 0.33
Thrust 90 0.0012 0.33
Selfaligning 10 0.0003 0.40
Source: Harris, T.A., "How to Predict Temperature
Increases in Rolling Bearings," Product Engineering,
December 1963.
Table 2.2 Load Torque Factor Values f,
for Roller Bearings
Bearing Type f.
Cylindrical 0.00025 to 0.0003
Spherical 0.0004 to 0.0005
Tapered 0.0004 to 0.0005
Source: Harris, T.A., "How to Predict
Temperature Increases in Rolling
Bearings," Product Engineering, December
1963.
The equivalent load P. is a function of the type of
bearing, the geometry of the bearing, and the direction of
the load [11]. For ball bearings, the equivalent load is
given by either equation (2.3) or equation (2.4), whichever
yields the larger value of P.. For radial roller bearings,
P. is given by equation (2.5) or equation (2.6), whichever
is larger. Equation (2.7) estimates the value of P8 for
thrust bearings (ball or roller).
P. = 0.9 F. cot(ac) 0.1 F,. (N) (2.3)
P,= Fr (N) (2.4)
P. = 0.8 F. cot(a) (N) (2.5)
P, = Fr (N) (2.6)
Pe = F. (N) (2.7)
where
F.: Axial load (N).
a: Contact angle (o).
F,: Radial load (N).
The lubricant flowing inside the bearing cavity will
induce drag forces on the rollers. These drag forces oppose
the motion of the rollers, generating heat. The expressions
presented in [11] to determine the viscous torque are given
as equations (2.8) and (2.9). The viscous torque is a
function of the bearing diameter, the kinematic viscosity of
the lubricant, the lubrication method, and the rotational
speed.
M = 9.79x10fo(L*n)2f d.3 (Nmm) (2.8)
when u*n > 2000
M, = 1.59 x 105fcod3 (Nmm) (2.9)
when u*n 5 2000
where
M,: Viscous torque (Nmm).
fo: Viscous torque factor for circulating oil
lubrication [11]:
Angular Contact Ball Bearings (2 rows) fo=8.0
Tapered Roller Bearings fo= 8.0
Cylindrical Roller Bearings (1 row) fo = 6.0
4: Kinematic viscosity (cS).
n: Rotational speed (rpm).
The heat generation rate is then the sum of the two
torques Mr and M, times the rotational speed in rpm, times a
conversion factor. Thus, the heat generated at a given
rotational speed under an opposing torque M is given by
(2.10).
Qc = 1.05 x 104 n M (W) (2.10)
where
Q.: Heat generation rate (W).
M: Total opposing torque Mr + M,. (Nmm)
The next step in the development of bearing analysis
theory was to develop an understanding of the internal
behavior of the bearings. To achieve this understanding, a
large research effort was undertaken during the drive to
develop more efficient and reliable aircraft engines.
Faster and more powerful engines required the development
of more reliable bearings which could operate at higher
speeds for longer periods of time. To design bearings for
these operating conditions, complicated bearing analysis
methods and computer programs were developed. Some of the
papers which developed the understanding and modeling of
high speed bearings will now be discussed in chronological
order.
The next model was presented, also by Harris [12], in a
paper which introduced a method to predict the occurrence of
skidding in cylindrical roller bearings operating at high
speed. Skidding occurs when a rolling element slides
over the raceway surface instead of rolling over it. While
in this condition, the cage speed is below the rotational
speed of the bearing. Skidding is due to the fact that
during high speed operation, centrifugal forces eliminate
the normal load component acting between the rolling element
and the raceway, causing the sliding of the rollers over the
raceway surface. This deteriorates the roller and/or the
raceway surfaces decreasing considerably the fatigue life of
the bearing.
To predict skidding, it is necessary to estimate
bearing internal speeds and loads. The rotational speeds of
the rolling elements, cage and rollers, must be known if the
effects of centrifugal force on bearing behavior are to be
approximated. Since the speed of the rolling elements is
affected by the loading conditions we must solve
simultaneously for the loading conditions and internal
speeds. The equations needed to solve for bearing internal
loads and for the cage speed must first be presented.
The internal speeds of a roller are shown in Figure
2.1, from [12]. The rotational speed of the jI roller is
given by w,j. The rotational speed of the cage is given as
w,, while the speed of the inner ring is w. The model is
presented by Harris for the case when the inner race is
rotating while the outer race is static. This is just the
case for spindle bearing systems investigated in this
dissertation. The sliding velocities can be determined as
Vj = 0.5(d. D.)(ww4) 0.5Dlww (2.11)
Vj = 0.5(d,, + D,.,)w 0.5D.w, (2.12)
where
V.j: Sliding velocity at the inner contact of the
j* rolling element (m/sec).
Voj: Sliding velocity at the outer contact of the
jt0% rolling element (m/sec).
dn: Bearing pitch diameter (m).
D,: Roller diameter (m).
V/2( d4 Dv)Vc
VcO UTERiNS
L/2( d DV)
V2 2dvVvj
V/2( d,Dv( VVc)
2 ( dio
Figure 2.1 Internal Bearing Speeds, from
Harris, T.A., "An Analytical Method to Predict
Skidding in High Speed Roller Bearings," ASLE
Transactions, July 1966.
In Figure 2.2, the loads acting on a roller are shown
using the nomenclature used by Harris [12]. The i subscript
refers to the inner race contact, the o subscript refers to
the outer race contact, the j subscript refers to the j"
roller, while y and z subscripts indicate horizontal and
vertical components respectively. Thus, the load Qo,
indicates a vertical load, acting on the outer race contact
of the jl roller. The loads Q.o, and Q,., are the
reactions to external applied loads acting on the j"
roller. Load Fj is caused by the cage acting on the
roller. Loads Qyoj and Qyj are loads caused by the fluid
pressure acting on the rollers at each roller raceway
contact, while the drag forces acting at each contact are
given by Foj and Fj. The boldface version of the previous
are the dimensionless forms of the corresponding loads. The
effect of high speed operation on the roller, which induces
a centrifugally oriented force, is F,. The
elastohydrodynamic loads are introduced by Harris here.
During steady state operation, the summation of the
forces acting on each roller, in directions y and z, must
equal zero. In dimensionless terms the force balance is
given by [12] as
18
Ozoj L
/ ^"l
0zijj
Figure 2.2 Loads Acting on a High Speed Roller
Harris, T.A., "An Analytical Method to Predict
Skidding in High Speed Roller Bearings," ASLE
Transactions, July 1966.
___ (Q=ij + F.) Q.oj = 0
RL
%tj + FVj 
Ro
_ (Qy.oj Fo.j Faj) = 0
RL
(2.14)
where
Ro: Equivalent external radius of the cylinder
(mm).
R,: Equivalent internal radius of the cylinder
(mm).
Qyo.j =
Fj =
Q.IJ
lw,, E'R
Qzoj
Qma,j
lw E'Ro,
Qyrj
1. E'R
lw1 E'R.
lw, E'Rc,
F,, E'
iw E'Ro
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
i: Roller length (mm).
E
1 o2
(2.21)
(2.13)
QZ.o=
E: Modulus of elasticity (N/m2).
a: Poisson's Ratio.
The lubricant induced loads Fjj,Foj, Qyoj and Qa.j are
given next, in dimensionless form, as presented by Harris in
[12].
Fi = 9.2G03 UjO7 +
Foj = 9.2G03 Uoj0'7 +
Vc~j Ic:j
Q^ = 18.4 (1T)G3 U07
= 18.4 (1T)G3 U,10.7
(2.22)
(2.23)
(2.24)
(2.25)
where
G = aE'
a: Is the pressure coefficient of viscosity
(mm2IN).
Hj= 1.6
Hoj = 1.6
Go6 UijO7
0. 5 33 
Go06 UojO7
Q j0.2.13
4o VLj
Vij =
E' R
(2.26)
(2.27)
(2.28)
ILo Voj
Vo = _____ (2.29)
E' Ro
)o Uij
U 1j = (2.30)
2E' RL
ILoC Ucj
Uo= = (2.31)
2E' Ro
f^3 Gqij[l(y/4qij)2]^2
I1L = 2 e dy (2.32)
J0
f)= J Gqoj[l(y/4qoj)2] X/2
Ioj = 2 e" dy (2.33)
J0
As it can be seen from above, the elastohydrodynamic
loads are nonlinear functions of the roller speeds and
lubricant properties. Note that the operating temperature
is an input to the analysis and it is not corrected for each
iteration.
The model provides a method to solve for the cage speed
we. If there is skidding, the cage speed will be below the
expected value of
w. = 1/2 w(l Dw/d.) (2.34)
which is the cage speed during rolling motion.
To determine the cage speed equations (2.13) and (2.14)
are not enough. Torque balances must be performed at each
bearing location and for the complete bearing. This would
provide the necessary equations to solve for cage speed w.
and roller rotational speed wj, cage load on the roller Fd,
and outer race contact load Q,oj,. As it was noted before,
the other loads are nonlinear functions of lubricant
properties and roller speed. The inner race contact loads
Qzj, are computed from static load analysis of the complete
bearing.
The solution method would require the computation of an
initial cage speed from a known inner race speed. Using
this cage speed, the conditions at each roller location are
then computed. The loads are added up and they must balance
the externally applied loads. Harris does not solve the
model in this manner, due to the required computational
tools which were not available to him at the time. Instead,
he introduces some simplifying assumptions:
1. Since not all the rollers are loaded, Harris
only considers the loaded contacts.
2. At steady state, the speed and load conditions
at any loaded roller location is the same as in
the most heavily loaded roller. The drag force
acting on a loaded roller is determined by
dividing the computed drag force by the number
of loaded rollers.
These simplifications drastically reduce the number of
computations needed to solve for the roller speeds. Still,
Harris's analysis yields a sufficiently close prediction of
the occurrence of skidding. It shows that skidding does not
exist in preloaded bearings. As soon as the centrifugal
effects remove the preload, skidding starts. In those
applications where out of roundness bearings are used, the
geometry of the bearing improves skidding behavior.
The simplifications do take a toll of the accuracy of
the model. The simplified model can predict the occurrence
of skidding but would not quantify it. The model is also
limited for heavily loaded bearings. With the development
of the digital computer, the complete analysis developed by
Harris will later be used by other researchers in the
development of more accurate models, as it will be shown
below.
Boness [13] provides some experimental data which
corroborates the results obtained by Harris in his
simplified model. At the same time, the experimental
results aroused some doubt on the validity of Harris 's
simplifying assumptions. The results presented in [13],
show that for each roller, the rotational speed is
different. The oil film thickness is also different at each
roller location. This explains the limitations of Harris's
model. Boness also found that by decreasing the amount of
lubricant in the bearing cavity, the amount of skidding
could be reduced by 75 percent. To obtain this amount of
reduction in skidding, a very small amount of oil must be
used; which is not always possible since at high speed
applications oil provides the only reliable source of
cooling.
Poplawski [14] presents an analytical model which is
based on the model developed by Harris [12] and the
experimental results presented by Boness [13]. In his
model, Poplawski considers the rotational speed of each
roller as an independent variable which must be solved for
in order to compute the operational conditions of the
roller. Poplawski's model is quite similar to the complete
analysis presented by Harris [12], but no simplifications
are necessary thanks to the availability of powerful
computers. It also includes the computation of the drag
forces at each roller location.
In Figure 2.3 the loads acting on a high speed roller,
are shown according to Poplawski [14]. The similarity
between this model [14] and the one presented by Harris [12]
is obvious. In the model shown in Figure 2.3, there is an
extra load acting on the roller, which is a drag force
caused by the cage driving the roller and it is labeled
fFFj. Therefore, rewriting equations (2.13) and (2.14) to
include this term equations (2.35) and (2.36) are obtained.
Ro
__ (Q^j + F fvFm) Q.j = 0 (2.35)
R.
Ro
QOj + F:j (Qwoj Fon Fj) = 0 (2.36)
Ri
Another difference between this model and the original
is the computation of the deflection of the rollers. Harris
[12] uses a load deflection behavior which ignores the
lubricant film between the rollers and the raceways.
Poplawski's model [14], does includes the deformation of the
oil film between the rollers and the raceways. He modifies
the deflection equation to
8j = 8,sinoj + 86ycos4j (Gx/2)+hj+hoj (2.37)
8j = 86j +68j (2.38)
where
09 8hA
6j =) + P:j (2.39)
=8 + Po ___ (2.40)
(K'2 I6hEJ Pcij
where
h: Oil film thickness given by
8 C06(jO.u)0'7 E'0.3 R13 lW,0'3
h= _____________ (2.41)
3 po.3.3
6h 0.302 a'6(jiou)07 E'03 R013 1,.O.13
= (2.42)
6P p.3.3
The deflection behavior was used to determine the inner
race contact loads as
n
F,, = EPij sin4> (2.43)
j=l
n
Fy = EP1j cosoj (2.44)
j=l
which is the same method used by Harris.
Figure 2.3 Loads acting on a High Speed Roller
from Poplawski, J.V., "Slip and Cage Forces in a
High Speed Roller Bearing," ASME Journal of
Lubrication Technology, April 1972.
One major development of Poplawski's model is the
evaluation of the drag forces in more detail than in
previous models. The equations used to determine the drag
forces are as presented by Harris [12] but now evaluated at
each roller location. The drag force acting on an unloaded
roller with translator motion is given [14] as
Fa^wm = Focj FJ (N)
= 9.2(1+2t)G3Uou.'7 (2.45)
There is also considerable friction between the cage
and the guiding surfaces, either in the outer race or the
inner race, depending on which side is used for cage riding.
For inner race rotation and inner race guiding, Poplawski
suggests that the force is given by
Fp:xc = fs N (N) (2.46)
where N is the normal force acting on the pilot.
The last drag force component to be considered is due
to the churning of the oil by the rollers and Poplawski
introduces the following relationship
Fo1h ,= 1/2 ?C.S.V.2 (2.47)
where
: Effective density of the mixture=%oil oil
CD: Drag Coefficient
S.: Effective Drag Area (mm2)
V,: cage orbital velocity (m/sec.)
As before, the force balance equations are not enough
to solve for the unknown variables, namely F,, w., w,, and
Poj. Torque balances are performed, based on an initial
cage speed, for each loaded roller to estimate the
rotational roller speed. Once the speed is computed for all
rollers, the drag forces acting on the unloaded rollers and
on the cage are estimated. Then a second torque balance is
performed for the complete bearing assembly between the drag
forces and the cage loads. If equilibrium does not exist,
the cage speed is corrected and the roller conditions are
computed once again. This procedure is iteratively repeated
until equilibrium of the complete bearing is achieved.
Poplawski's model has very good correlation with the
experimental data presented by Boness [13]. It is a more
complete model in the fact that it includes the speed of
each roller as an independent variable. The incorporation
of the individual roller drag forces makes of it a more
realistic model. The work presented in [14] helped in the
further development of the bearing analysis methods.
In an effort to quantify the heat generation rates,
Witte [15] derived some theoretical equations, which were
later modified to accommodate experimental results for
tapered roller bearings. A heat generation potential factor
G was developed based on the geometry of a tapered roller
bearing under pure thrust load. The author called this
factor G; it is based strictly on the geometry of the
bearing and it is a constant for a particular bearing
series. The G factor can be obtained from a tapered bearing
catalog or computed according to the equation given by Witte
in [15] as
D35
G = ___________________ (2.48)
D0'7(nl)2/3 (sin a)X/3
The G factor is related to the resisting torque of the
bearing. The lower the G factor is, the lower the heat
generation for that bearing. The relationship between pure
axial load, the G factor, and resisting torque is given by
Witte [15] as
M = l.lxlO"'G F. (Sut)05 (F.O)"3 (2.49)
where
M: Resisting torque (lb.in.).
G: Bearing Geometry Factor.
F.: Axial Load (lb.).
S: Bearing Speed (rpm).
1: Lubricant viscosity at atmospheric pressure
(cP).
0: Lubricant pressureviscosity index (in2/ilb).
and it is limited for (SI) values larger than 3000 and
for axial loads which are less than twice the axial load
rating of the bearing.
For the case when radial loads are applied instead of
an axial load, equation (2.49) should be modified to
compensate for the different orientation of the load.
Equation (2.50) gives the relation between M, G, and radial
loads.
M = l.lxlO4'G (Sut)0 (fTF,/K)X/3 (2.50)
where
fT: Equivalent thrust load factor
K: Ratio of basic dynamic radial load rating to
basic dynamic thrust load rating.
and it is limited for (S.L) values larger than 3000 and
for radial loads which are less than twice the radial load
rating of the bearing. The fT and K factors can be obtained
either from [15] or from the manufacturer of the tapered
bearing.
Witte obtained good correlation between his equations
and experimental data. One shortcoming of his experiments
was that he used less than 1.9 liters per minute of
lubricating oil. This is quite low compared with what is
commonly used in high speed bearing applications.
Astridge and Smith [16] performed an experimental
investigation in an attempt to quantify the power losses,
and heat generation in high speed cylindrical roller
bearings. They used bearings with bore diameters of 300 and
311 mm, operating them at 1.1 million DN. The bearings were
operated with diametral clearance, simulating operating
conditions in aircraft engine bearing applications. From
their experimental results and other published data,
Astridge and Smith [16] suggested 10 sources of heat
generation:
1) Viscous dissipation between rollers and races.
2) Viscous dissipation between rollers ends and
guide lips.
3) Elastic hysteresis in rollers and races.
4) Dissipation in films separating cylindrical end
faces of rollers and cage.
5) Dissipation in films separating cage and
traces.
6) Dissipation in films separating cage side faces
and chambers wall.
7) Displacement of oil by rollers.
8) Flinging of oil from rotating surfaces.
9) Oil feed jet kinetic energy loss.
10) Abrasive wear and asperity removal.
As it can be seen from the list, most of the sources
are due to the drag forces acting between the rollers and
the lubricant. The lubricant is displaced by the rollers
as they move within the bearing cavity. According to
Astridge and Smith [16] the single most important source of
heat generation is due to the churning of the oil between
the rollers and the raceways. In the case considered in
[16] not all the rollers were loaded.
Performing a parametric study, Astridge and Smith
identified which of the parameters related to bearing
operation have a larger influence on heat generation. The
ones with a stronger effect on heat generation were found to
be speed, oil flow rate, oil viscosity and pitch diameter.
In [17] Rumbarger et al., presented a sophisticated
computer analysis for single row high speed cylindrical
roller bearings. The authors incorporate into a single
model the loaddeflection behavior, the kinematic and the
EHD behavior and the thermal behavior. Previous models did
not consider the effects on bearing behavior of the
interaction between these components of bearing performance.
In the model presented by Harris in [12] a single overall
bearing temperature is considered, while in [17] the
temperature at each contact is computed based on the
kinematics, the EHD conditions and the loads present at that
contact. In contrast with Poplawski's model, in the model
presented in [17] the drag forces are computed for each
roller element using the estimated EHD conditions for the
speed and temperature estimated for each contact. These
loads are then compared with previous speed and temperature
iterations, which if different are corrected. If the
kinematic conditions in a roller location change, the
overall load distribution may be affected, causing a change
in the elastohydrodynamic conditions and in thermal
performance. Due to the iterative solution method used, the
computation needs are enormous. It is then necessary to
limit the analysis to steady state operation, otherwise the
required computation capabilities would make the codes too
complex and expensive to use.
In this model, the elastic, kinematic, and thermal
analysis are similar to the ones used in the models
presented by Harris and Poplawski which were discussed
previously. The model presented by Rumbarger et al. in [17]
is relevant since it introduces the use of a complete fluid
analysis to evaluate the viscous effects of the lubricant on
the rolling elements. Therefore, the discussion of the
model would be concentrated into this new development
presented in [17].
The authors in [17] identified two main viscous drag
torque sources. The first source is the viscous drag caused
by the rolling elements moving through the lubricant. As
the rollers rotate within the bearing cavity, the lubricant
flows around them and between the rollers and the guiding
surfaces. The second source of viscous drags according to
Rumbarger et al. is caused by the motion of the cage within
the bearing. As the cage rotates, it is in contact with the
lubricant at the inner and outer surfaces, at the lands and
at the side surface. The total drag torque is the sum of
the drag torques acting on each rolling element plus the
drag torque acting on the cage.
The total drag torque acting on a roller is the sum of
the drag torque acting on the roller surface, plus the drag
torque acting on the roller end, plus the retarding torque
caused by the contact between the roller and the cage. The
torque acting on the roller surface is computed by the
authors of [17] as
T = Tw A r (Nm) (2.51)
where
T: Drag torque acting over the element surface
(Nm).
T,: Wall Shear Stress (N/m2).
A: Surface area of the roller (m2).
r: Reference radius from the center of rotation
(m).
The authors [17] recommend for the computation of the
shear stress acting on the rollers equation (2.52).
T = f(1/2 ?U2) (N/m2) (2.52)
where
f: Friction factor computed from the Reynolds
number assuming turbulent flow [17].
: Fluid mass density (oil and air mixture)
(Kg/m3)
U: Mass average velocity of the fluid (m/sec)
To compute the drag torque acting on the roller ends
the authors recommend equation (2.53).
T..u = 0.5 w' r' C, (Nm) (2.53)
where
T,,.: Drag torque acting on the end of the roller
(Nm)
w: Rotational speed of the roller (rad/sec.)
Cn: Correlation factor:
3.87/(NR.)5 for laminar flow
Nn.<300,000
O.15/(NR.)0'7 for turbulent flow
NR.>300,000
The last torque component acting on the rollers is due
to the contact between the rollers and the cage. To
estimate this torque the authors recommend the following
equation
Fj3 N Vt Vutil
T.Afta E ( E _________A__..r
A0 N=1 VR V.:j
S/2
2* E Avkrvk] (2.54)
k=l
where
Fj.: Contact force between the roller and the
guiding surface (N).
N: Number of horizontal lamina.
Vn1: Velocity of the race at the ith horizontal
lamina (m/sec.).
V.j.: Velocity of the roller at the ilh horizontal
lamina (m/sec.).
S: Number of vertical lamina.
Tma: Torque produced at the rollercage contact
(Nm).
c:: Friction coefficient between the roller and
the guiding shoulder.
AH.,: Area of the ith horizontal lamina (m2).
rj: Distance from the i"" horizontal lamina to
the center of the roller (m).
Avjk: Area of the kh vertical lamina (m2).
r.vj: Distance from the kt"h vertical lamina to
the center of the roller (m).
AQ: Total contact area between the rollers and
the guiding surfaces (m2).
which is obtained by dividing the contact area into
various vertical and horizontal lamina.
To compute the torque induced by the cage moving
through the lubricant, equation (2.51) is used for the inner
and outer surfaces of the cage and for the lands. For the
sides, equation (2.53) is used.
The main problem of the fluid model is it sensitivity
to the amount of oil inside the bearing cavity. The authors
used a volume percent of 15 to 20%; the percent of the total
bearing cavity volume which the oil occupies. These values
of volume percent yielded good correlation between the
experimental results and the model computations. The
accuracy of this procedure is questionable, since there is
no reliable way to measure the amount of oil inside the
bearing cavity. The density used for the mixture is
computed based on an amount of oil present in the cavity,
which is difficult to determined. The major contribution of
the model presented in [17] is the use of an
interdisciplinary approach to solve for the operating
conditions of a high speed bearing [18].
Since the model in [17] was presented, several advanced
computer codes have been developed for the analysis and
design of high speed bearings. The driving force for the
development of these codes have been the need for more
reliable bearings for combat aircraft mainshaft bearings
[19]. Two main types of bearing analysis codes have been
developed, for quasistatic or steady state analysis and for
dynamic or transient analysis. The first is represented by
programs such as SHABERTH for the analysis of shaftbearing
systems, and CYBEAN, for the analysis of cylindrical roller
bearings [20]. The dynamic analysis codes are represented
by the program DREB, which is used to analyze the transient
behavior of ball and roller bearings [20].
A major shortcoming of these computer codes is that
their results are seldom compared to experimental results
as pointed out by Parker in [19]. Another problem pointed
out by Parker [19], is that even if comparison to
experimental results is intended, there are some
computations which cannot be compared since there is no
experimental way to obtain experimental data to match the
computations. For example, some programs include in their
output roller skew angles and element temperatures which are
yet to be measured experimentally. Another problem with the
computer codes is the dependency on the volume percent of
oil in the bearing cavity to estimate the thermal behavior
of the assembly. Those researchers which have attempted a
comparison between the computer results and experimental
data are required to chose such a volume percent such that
their computations approximate the experimental results [17,
19,21,22,23].
Although the computer programs are still to be
improved, they have facilitated the development of advanced
bearing designs. The use of an interdisciplinary approach
to the analysis of the behavior of high speed bearing can
only be done using the computer. The problem is too complex
to be solved by a single individual without the assistance
of a high speed computer. The codes currently can only be
used in high speed supercomputers, which means there are not
available to most engineers involved in designs with bearing
applications.
High Speed Bearings: Experimental Results
In 1974, Signer et al. [24] presented experimental data
on high speed angular contact ball bearings. ACBB of 120 mm
diameter, 20 and 24 contact angles were tested to 3
million DN. The test conditions were made to simulate the
operating conditions in an aircraft turbine.
It was found in this investigation [24] that power
losses increased linearly with speed and with increased oil
flow rate through the inner race. Inner race lubrication
was more effective than other lubrication in reducing the
operating temperature, for the same oil flow rate. It was
interesting to find that when the oil flow rate was
increased over 3.8x103 cubic meters per minute (1.0 gpm),
the temperature increased, probably due to the increased
quantity of lubricant within the bearing cavity and to the
resultant churning.
Parker and Signer [25] present the results of their
investigation of high speed tapered bearings. The bearings
used had 120.65 mm bore with capability to use either jet
lubrication or conerib lubrication. The use of conerib
lubrication proved to be more efficient in limiting the
operating temperature. It was also found that the use of
conerib lubrication instead of jet lubrication reduced the
power consumption.
The experiment showed that the bearing temperatures and
power losses increased with spindle speed. The effect of
load on bearing temperature was insignificant.
In [25], Parker and Signer presented results of their
testing of TRB to DN values higher than one million. Since
TRBs have a better loaddeflection characteristic than ACBBs
or CRBs for the same envelope, they are preferred for some
applications where weight or space are critical. It was
also demonstrated that by providing the conerib/roller end
contact with sufficient lubrication, TRB can be operated to
very high speeds. The lubrication method recommended then
was the use of holes drilled through the cone, through which
oil was forced into the conerib area.
Parker and Signer used specially designed TRB to
investigate the high speed performance with conerib
lubrication versus the performance with oil jet lubrication.
The bearings used were of standard design but provided with
conerib lubrication to improve their high speed operating
performance.
The bearing tested had a bore diameter of 120.6 mm, an
outside diameter of 206.4 mm, a cup angle of 340, and it
contained 25 rollers. The test speeds were 6,000, 10,000,
12,500 and 15,000 rpm. The oil flow rates used were 1.9 x
103 to 15.1 x 103 m3/min.
The test results obtained in [25], showed that conerib
lubrication plus jet oil lubrication was a better
lubrication arrangement than oil jet lubrication alone.
In fact, the higher speeds could not be achieved safely with
oil jet lubrication alone. As for oil flow rate, by
increasing the oil flow rate, temperatures decreased while
power losses increased. It was also shown that for oil flow
rates over 11.4 x 103 m3/min, a further increase in oil
flow will not produce a significant temperature decrease.
Observing the power losses induced by the increased oil flow
rate, the use of oil flow rates larger than 11.4 x 103
m3/min do not seem justifiable.
Spindle speed also produced considerable increase in
temperature and power losses. The effects of load on
bearing temperature were insignificant compared to the
effects of the oil flow rate and spindle speeds tested. The
authors of [25] used the equation derived in [15] to
estimate the heat generation rates. The power losses
estimated using the equation from [15], had good correlation
with the experimental results.
Parker et al. [26] presented results of computer
optimized TRB bearings. These bearings were designed by
first optimizing the standard TRB design, as the ones used
in [25]. The optimized design was then presented to a
leading TRB manufacturer who suggested changes which would
allow the bearing to be economically manufactured. The
bearings used in [26] used 23 rollers, it had a cup angle of
310, 120.65 mm bore diameter and outer diameter of 190.5 mm.
The bearings were provided with conerib lubrication
and instead of oil jet lubrication, the front of the bearing
was lubricated through holes in the cone and through the
spindle. Oil was forced centrifugally through these holes
into the front of the bearing. Test speeds varied from
6,000 rpm to 20,000 rpm. Oil flow rates varied from 3.8 x
103 to 15.1 x 103 m3/min.
The computer optimized bearing operated at lower
temperatures, lower power losses and higher spindle speeds
than the standard bearing. Effects of oil flow rate,
spindle speed and load on bearing temperatures and losses
were similar for the optimized bearing and the standard
bearing.
Currently, aircraft engines operate at a maximum DN
value of 2.4 million [23,27]. The mean time between bearing
removal is up to 3000 hours from 300 hours ten years ago
[34]. Improvements in the lubrication methods have allowed
researchers to operate ACBB and CRB to 3.0 million DN, while
TRB have been operated to a 2.4 million DN [23,27]. The use
of AISI M50, a vacuuminduction melted, vacuum arc melted
alloy, has greatly improved the fatigue life of high speed
bearings.
CHAPTER III
EXPERIMENTAL EQUIPMENT
High Speed, High Power Milling Machine
The Machine Tool Laboratory at the University of
Florida is equipped with a HSHP milling machine, shown in
Figure 3.1. The spindle is driven by a 115 kW, 3000 rpm,
ASEA D.C. motor by means of a two stage flat belt
transmission. The first stage is a belt from the motor to
the intermediate shaft, located in the column of the milling
machine. The second stage, is from the intermediate shaft
to the the spindle. The speed ratio used for the high speed
test between the motor and the spindle was 0.26.
The spindle is mounted on the HSHP milling machine on
the front, bolted to a mounting bracket. Lubrication
connections and instrumentation are external to the HSHP
machine, making the change of spindles a simple task. To
change the spindle mounted on the machine, the current
spindle is unbolted and removed using a hoist. The next
spindle can then be mounted and bolted. The lubrication
system can easily be modified to accommodate several spindle
designs.
AXIS SERVO
Figure 3.1 HSHP Milling Machine
This HSHP milling machine permits a complete
investigation of the configurations under study. Each
configuration is tested not only for idle operation
performance, but also for cutting capabilities and chatter
stability.
Test Spindles
The two spindle bearing configurations shown in Figure
3.2 and Figure 3.3 were tested for HSHP performance. Their
operating temperatures, lubrication needs and power demands
were investigated at several speeds, during idle, no load
operation. Both spindles were equipped with circulating oil
lubrication. The amount of oil circulated was varied from
1.5 liters to 3.8 liters per minute, per bearing. The
spindles were tested for maximum operation speed.
Configuration I is based on double row cylindrical
roller bearings (CRB) NN 30K/SP manufactured by SKF. It has
one NN3019K/SP on the drive side and a NN3022K/SP on the
tool side. The CRBs support the radial loads while the
thrust load is supported entirely by a Radiax, a 234420
BMI/SP series angular contact thrust ball bearing (ACTBB) by
SKF, with a contact angle of 60. This configuration is
sometimes referred to as Standard Configuration I by SKF
researchers [28].
The preload in this configuration is provided,
individually for each bearing. The radial bearings are
preloaded radially by eliminating any clearance between the
outer race and the rollers. As it can be seen from Figure
3.2, tightening the nut A pushes on the inner race of the
lower CRB, moving the inner race and the rolling elements up
the tapered. As the rolling elements are driven up the
taper of the spindle, the diametral clearance between the
elements and the outer race is reduced. Tightening the nut
further, contact between all the rollers and the outer race
is produced, completely eliminating any clearance. If nut A
is tighten even more then interference is produced.
The ACTBB is preloaded by tightening the nut B to press
together the bearing assembly. As the nut B is tighten, any
gap between the races and the bearing spacer C is
eliminated. Once the nut B is completely tighten, the
preload between raceways and balls is achieved. The
preloading of this configuration is done during the assembly
of the spindle and cannot be released, unless the spindle is
completely disassembled.
The maximum speed achieved by this configuration was
8,000 rpm. The operation temperatures were above the
recommended for the type of oil used. The power losses were
almost 14 kW, which means that for a 20 kW milling machine
could only perform 5 kW of useful work at 8,000 rpm. When
the spindle was driven over 8,000 rpm, the ACTBB failed
within seconds of starting the test. This happened twice:
at 9,000 and 10,000 rpm. The failure was too fast for the
PROMESS sensor to detect any increase in the load of the
bearings. After discussing the failure with SKF
researchers, it was concluded that the cause of the failure
was the loss of preload. The loss of preload induced
skidding, which was the mode of failure of the bearing. To
correct the problem, the mounting preload must be increased
and a larger amount of lubricant must be provided to the
upper raceway. To achieve this increase in preload, the
spacer separating the two raceways, spacer C, must be
ground, bringing the two raceways closer together. This
increase in preload would also induce an increase in bearing
temperature, which could not be permitted, since operating
temperatures are already too high.
Configuration II is based on TRB. This configuration
operates under constant preload. A constant preload is
maintained by the bearing in the drive side, the HYDRA
RIB, by TIMKEN, Figure 3.4. The bearing is provided with a
hydraulic chamber and piston mechanism which provide a load
to the back of the rollers. As the chamber is pressurized,
the piston displaces forward, pushing on the rollers. This
forward displacement of the rollers produces the diametral
interference or preload. The preload force is proportional
to the hydraulic pressure in the chamber. If during the
operation of the spindle the loads acting on the rollers
increase, the piston would retract to a point where the load
on the rollers equals the preset value. If on the other
hand, the load on the bearing is reduced during the
operation of the spindle, the piston would move forward
until the preset load on the rollers is reestablished.
Beaing
NN3019 K
Being
234420 ll
Being
M3022K
Figure 3.2 Configuration I Test Spindle
Hydra Rib
Bearing
>Oil Distribution
Rings
 High Speed
Bearing
Figure 3.3 Configuration II Test Spindle
Oil Jet
Hydraulic Oil
Input \
Snap Ring ,
Rib Chamber
Figure 3.4 HYDRARIBT Bearing
Roller
cone
Piston
Outer Race
Inner Race
Tapered Roller
Cone Rib
Lubrication Ring
Cone Rib
Lubrication Hole
Figure 3.5 High Speed Bearing with ConeRib Lubrication
The High Speed (HS) bearing, 100 mm diameter, in the
tool side, Figure 3.5, is provided with conerib
lubrication. The cone is lubricated through holes drilled
from the back of the bearing to the conerib. At the back
of the bearing, there is a ring which entraps the oil
supplied by jets forcing it centrifugally into the holes.
This configuration operated successfully up to 10,000
rpm. The only failure experienced with this configuration
happened when lubrication to the conerib interface was
interrupted. The operation temperature was at all times
very acceptable with very low oil flow rates. The power
losses were lower than those for Configuration I.
Oil Supply to the Bearings
As mentioned before, the configurations are equipped
for circulating oil lubrication. Figures 3.2 and 3.3 show
the oil inlet and outlet points for each configuration.
Configuration I, is provided with two oil inlets per
bearing, one at each side of the spindle housing. Once
inside the housing the oil is forced around the bearings
through a groove in the outer surface of the outer race the
bearing. The oil enters the bearing through three holes in
the outer ring 120 degrees apart, provided for that purpose.
Through these holes the oil is forced into the bearing
cavity between the two rows of elements as shown in Figure
3.6. The oil is then forced out of the bearings, by the
rolling element motion and centrifugal forces. The oil is
then sucked out of the bearings through the exit ports.
Configuration II is provided with three oil inlet
points: two for the high speed bearing, and one for the
HYDRARIBT. Once the oil enters the housing it is directed
to the front of the bearings by the distribution ring. Both
bearings are provided with rings at the front (small end of
the rollers). The high speed bearing is provided with a
second distribution ring which feeds three oil jets. These
jets direct the flow to the back of the cone, which is
provided with a special ring. This ring entraps the oil
from the jets, which is then fed centrifugally into the
conerib interface through holes drilled for that purpose in
the cone.
Configuration II was designed for horizontal use. When
mounted in the vertical position, the upper bearing does not
receive the required lubrication due to gravitational
forces. Since the oil is sprayed up from the distribution
ring, in vertical applications, it does not have the
necessary pressure to force the oil through the bearing. To
correct this problem, a screw type pump was provided above
the HYDRARIBT. This pump supplied the necessary pressure
drop to overcome gravity and provide an efficient flow of
oil as long as a supply of 3.8 1pm is maintained to the top
bearing.
OIL\
Figure 3.6 Oil Supply to Double Row Bearings
Instrumentation
During the tests of configurations I and II, the
temperatures were monitored using type K thermocouples
placed at strategic positions in the test rig. The
thermocouples were connected to a digital display
thermometer. The thermocouples were located at the
following positions:
1. In the oil supply line.
2. In the oil return line.
3. At the oil exit point of each bearing.
4. At the outer race of the bearings.
5. On the surface of the housing.
The thermocouples at 1. and 2. measured the bulk oil
temperatures before and after passing through the housing.
The thermocouples used at 3. were in the suction line
removing oil from each bearing. These thermocouples
measured the exit temperature of the oil from each bearing,
while 2. measures the temperature of the mixture of the oil
from all bearings. Position 4. was measured for each
bearing through a hole in the housing. Position 5. was
measured at surface points above position 4. The
thermocouples used in 1., 2., and 3. were in direct contact
with the oil. The thermocouples used in 4. were
encapsulated in a bayonet type assembly. The thermocouples
used in 5. were in direct contact with the housing.
Sensor
Sensor
Figure 3.7 PROMESS Sensor
The load on the bearings was monitored using the
PROMESS sensor. As shown in Figure 3.7, strain gages are
located on the outer surface of the outer ring. As the
loaded elements pass over the strain gages, these will
provide an electrical signal proportional to the rolling
element load. The PROMESS sensor is especially useful when
monitoring the transient loads on the bearings.
The spindle speed was measured by using a magnetic
pickup and gear installed at the top of the spindle. The
speed was displayed on a electronic counter at all times.
This speed was compared against the speed measured using a
handheld tachometer. The speed was monitored throughout the
test.
The input power to the motor was monitored using a set
of current and voltage meters in the motor controller box.
These meters measured the current and voltage supplied to
the D.C. motor. The input power was computed from these
measurements.
Oil Circulating System
There are several lubrication methods used in machine
tools among them, grease lubrication, oil mist lubrication,
airoil lubrication ("OL"), and circulating oil
lubrication. Although the amount of oil required for
lubrication is small, for high speed applications large
amounts of oil must be used to provide the bearings with the
necessary cooling. The oil circulating through the bearing
cavity removes a large part of the heat generated. So far,
circulating oil is the only lubrication method which
provides the necessary cooling for high speed bearing
applications.
The oil used throughout the investigation was a SAE 10
equivalent oil, common in machine tools. A single type of
oil was used. The use of a heavier oil will increase the
hydraulic power losses and consequently, the operating
temperature of the bearings. The power available for useful
work (milling) will also be reduced due to an increase in
hydraulic power losses. If on the other hand a lighter oil
is used, the oil may exceed its operating range at high
speeds and degrade. The friction between the rolling
elements and the raceways would then increase, inducing an
even larger operating temperature.
Figure 3.8 shows the circulating loop for the cooling
and lubrication of the housing. Since circulating oil
lubrication is going to be used to cool the bearings, large
quantities of oil are necessary. The oil must be kept at
constant temperature, since the experimental investigation
would be affected by a variable supply oil temperature. The
oil is pumped from a 280 1 storage tank to the spindle by
the supply pump. The supply pump is a variable vane pump
with an operating range from 4 1pm to 53 1pm. Just before
reaching the spindle, the oil flow is distributed into three
streams. Each stream is controlled by a combination of
needle valve and a flow meter. Here, the amount of oil
going into each bearing is measured and controlled. If
configuration I, is being tested, each of the three streams
is then split in two, to supply the oil to the bearing from
both sides of the housing.
Once the oil has circulated through the bearings,
removing heat from the bearing cavity, it is sucked out of
the housing and returned into the storage tank by the
suction pump. Due to the amount of churning within the
bearings, the oil exiting the spindle is sucked out as foam.
In the storage tank the oil is defoamed and cooled. To
defoam the oil, it is passed through the screens, which
removes the entrapped air. The oil is then pumped from the
tank through the heat exchanger by the circulating pump.
The cooling fluid in the heat exchanger is chilled water,
from the laboratory's air conditioning system. After
passing through the heat exchanger, the cold oil is returned
back to the storage tank, near the warm oil return point,
refer to Figure 3.8.
It is a known fact that the larger the difference in
temperature between the two fluids in the heat exchanger,
the more efficient it works. The need to remove the foam
from the oil before it passes through the heat exchanger
limits the alternatives as where to locate the inlet to the
heat exchanger. If the suction point of the cooling circuit
is placed next to the warm oil return, all the foam coming
into the tank will be pumped into the heat exchanger,
reducing its efficiency. Therefore, the suction of the
cooling circuit must be placed on the proper side of the
screens, the closest possible to the warm oil return.
Evaluation of Cooling Capacity
In the initial stages of the investigation, it was
observed that the temperature of the supply oil increased
during the test. This increase in temperature significantly
affected the investigation since the bearing temperature
could not be related to a constant oil supply temperature.
Therefore, an evaluation of the cooling system was
performed. The question to be answered was if the
circulating system was capable of providing the necessary
cooling effect, removing from the warm oil all the heat it
acquired from the bearings.
The amount of heat removed by the oil, from the spindle
is given by
Qo = m c(To., T,) (kW) (3.1)
where
QoiB;: Heat removed by the oil from the bearings
(kW).
m: Oil flow rate (1pm).
: Oil density (g/ml).
c: Heat capacity of the oil (kJ/(kg C)).
T.: Supply oil temperature (C).
Tou.: Return oil temperature (C).
The heat removed from the oil in the heat exchanger is
given by
QoLxmx = m c(TM. Toue)
where
The
given by
(kW) (3.2)
QoCIHE: Heat removed from the oil (kW).
m: Oil flow rate (1pm).
z: Oil density (g/ml).
c: Heat capacity of the oil (kJ/(kg C)).
T,,: Oil temperature entering the heat
exchanger ("C).
Tout: Oil temperature exiting the heat
exchanger (C).
heat removed by the water in the heat exchanger is
Qw.t.H = m c(Tou, T..)
where
(kW)
(3.3)
Qwf.m.3: Heat removed by the water from the oil
(kW).
m: Cooling water flow rate (1pm).
n: Density of the water (g/ml).
c: Heat capacity of the water (kJ/(kg C)).
Tj,: Water temperature entering the heat
exchanger (C).
Tot: Water temperature exiting the heat
exchanger (C).
Experimental data was collected at steady state, it is
listed in Table 3.1. With this data, the amount of heat
removed from the oil in the heat exchanger, the amount of
heat acquired by the oil from the bearings and the amount of
heat acquired from the oil were computed. It was found that
the heat exchanger did have the necessary capacity to cool
the oil to the desired supply temperature. As it can be
observed from Table 3.1, the temperature of the oil entering
the heat exchanger is much lower than the temperature of the
returning oil. Therefore, the problem was not that the heat
exchanger could not supply the necessary cooling, but that
the warm oil was not getting to the heat exchanger until it
is too late. Upon inspection of the tank, it was found that
the oil inlet to the heat exchanger was too far from the
warm oil return point. This caused the warm oil to
concentrate on one side of the tank, heating that side of
the tank. This accumulation of warm oil increased until it
reached the heat exchanger oil inlet. By that time, the
amount of oil which needed to be cooled was above the
cooling capacity of the heat exchanger, which in the mean
time was circulating cool oil.
To solve the problem the oil inlet point into the heat
exchanger was moved closer to the oil return point. It
could not be moved close enough since it must be placed
after the screens, otherwise, the foam would make its way
into the heat exchanger, reducing its cooling capacity. The
final solution was to return the cold oil beside the warm
oil return. This kept the return side cold and there was no
chance for the warm oil to accumulate in that side.
Table 3.1 Heat Exchanger Temperatures
Water Oil
Speed Oil Flow In Out In Out Ret.Oil
3000 1.9 ipm 100C 160C 310C 13C 440C
3000 3.0 1pm 110C 17C 33C 14C 410C
3000 3.8 1pm 10C 180C 33C 15C 38C
4000 0.8 1pm 8C 12C 140C 12C 660C
4000 1.9 1pm 110C 180C 36C 21C 530C
4000 3.0 1pm 11C 20C 370C 230C 490C
4000 3.8 1pm 110C 21C 370C 22C 490C
5000 1.9 Ipm 100C 220C 260C 220C 660C
5000 3.0 ipm 100C 25C 31C 250C 62C
5000 3.8 1pm 9C 27WC 330C 27C 61C
6000 0.8 1pm 7C 110C 12C 10C 640C
6000 1.5 1pm 6C 12C 16C 12C 67C
Seals
As both configurations are lubricated using circulating
oil and mounted in the vertical position, proper sealing is
imperative. Any oil that leaks out of the housing, through
the bottom, will fall on the workpiece. This oil may affect
the life of the tool by exaggerating the thermal cycling of
the tool, causing the failure of the tool. Also, it
represents a hazard to the operator, since at high speeds,
the oil is sprinkled onto the surroundings, making the area
quite slippery.
Due to the high rotational speeds, noncontact seals
must be used. Noncontact seals have the extra advantage
that they do not contribute to the friction torque, thus
reducing the amount of heat generated. A similar
arrangement of labyrinth seal was used for both
configurations. A section view of the seal, for
Configuration I and Configuration II, is shown in Figure 3.9
and in Figure 3.10, respectively. Both configurations were
effectively sealed for most of our operating conditions.
Spindle
Suction Pump
...... Screens
Variable Output Pump
Figure 3.8 Circulating Oil System
Spindle
Housing
Oil Suction Points
Figure 3.9 Seal for Configuration I
Oil Jet
Housing
CneRib
Lubriction Inpuit
il i /ints
Oil Su.,,tion Poin
H50 Tper
Figure 3.10 Seal for Configuration II
CHAPTER IV
THERMAL ANALYSIS
Thermal Analysis of the Spindle Housing
To estimate the heat generation rates of bearings,
researchers and bearing manufacturers have developed several
empirical and theoretical equations. These equations relate
heat generation to bearing geometry, operating conditions
and lubrication parameters. In this chapter these
relationships will be presented and compared among
themselves and to experimental results.
Also in this chapter, thermal profiles are presented,
showing temperature distribution along the housing. The
presence of thermal gradients between the bearings and the
spindle housing may induce an increase of the original
preload, which in some instances may cause bearing seizure.
The thermal gradient is induced by the faster increase in
rolling element temperature compared to the housing during
the acceleration of the spindle to the operating speed.
Friction in Rolling Bearings
The heat generated in the bearings is the product of
frictional power losses. The sources of these frictional
losses as identified in [16,29,30] are:
1. Elastic hysteresis in rolling. As the bearing
rolls there are deformations in the raceways
and in the rolling elements. The energy
consumed in producing this deformation is
partly recovered when the element rolls to the
next position.
2. Sliding in rollingelement/raceway contacts due
to the geometry of the contacting surfaces.
3. Sliding due to deformation of contacting
elements.
4. Sliding between the cage and the rolling
elements, and between the cage and the guiding
surfaces.
5. Sliding between roller ends and inner and/or
outer ring flanges.
6. Viscous drag of the lubricant on the rolling
elements and cage. The viscous friction is
produced by the internal friction of the
lubricant between the working surfaces. Also
the churning of the oil between the cage and
the rolling elements, between the raceways and
the rolling elements and flanges. These losses
increase with speed and amount of lubricant in
the bearing cavity.
In the experimental investigation, the effect of the
above power losses were grouped into two measurable amounts,
Mechanical Power losses and Hydraulic Power losses. The
mechanical power losses are the consequence of mechanical
friction in the bearing cavity, without oil being circulated
through the bearing. The hydraulic power losses are the
results of viscous friction between the oil in the bearing
cavity and the rolling elements. These two main sources of
heat are discussed in Chapter V.
Heat Generation
The increase in temperature during the operation of the
bearings is the result of friction losses, which are
manifested as heat. The sources of friction in a bearing,
as mentioned above, include the friction at the contact
between rolling element and each raceway, friction between
the cage and the rolling elements and viscous drag between
the circulating oil and the rolling elements. Several
empirical relations have been developed to estimate the
amount of heat generated in a bearing.
The frictional power consumed by a bearing is given by
[9,11] as
Hf = 1.05x104 n M (W) (4.1)
where
Hf: heat generated (W)
n: spindle speed (rpm)
M: friction torque (Nmm)
Also from [9,11], the bearing manufacturer estimates
the friction torque as
M = 0.5 Vj, F d (Nmm) (4.2)
where
Vif: friction coefficient for the bearing
F: bearing load (N)
d: bore diameter of the bearing (mm)
The friction coefficient IL is given in [9,11] for
several types of bearings
for cylindrical roller bearings .f = 0.0011
for thrust ball bearings = 0.0013
for tapered roller bearings = 0.0018
These friction coefficients are for single row bearings
operating at average speed and at a load for a life of 1000
million revolutions.
The loads acting on the bearings are reactions to the
belt tension. The magnitude of this tension is computed
following the procedure suggested by the manufacturer in
[31] for the type of belt used. For Configuration I, the
tension load is 8600 N, while for Configuration II, the
tension load is 3600 N. With the tension load known and
using load equilibrium, the bearing reactions for
configuration were determined. For Configuration I, the load
on the lower bearing (NN3022 K) was estimated at 3000 N, for
the top bearing (NN3019 K) it was 12000 N and the center
bearing, the Radiax, 400 N which is the weight of the
spindle. For Configuration II, the load acting on the lower
bearing, the High Speed Bearing, was estimated as 1600 N,
while at the HydraRibT the belt tension component was
5200 N and an axial component of 400 N due to the weight of
the spindle.
Using equation (4.2) to compute the friction torque for
both configurations, using double the friction coefficient
for the double row bearings, the following estimates were
obtained:
for Configuration I
M30O22. = 360 Nmm
Mmo K = 1232 Nmm
M234420Moi = 46 Nmm
for Configuration II
Mm. 3xIB = 489 Nmm
M.s = 146 Nmm
The heat generated, computed using equation (4.1), at
the different test speeds, for each configuration are listed
next.
Configuration I
@ 3,000 rpm 516 Watts
@ 5,000 rpm 859 Watts
@ 7,000 rpm 1204 Watts
@ 8,000 rpm 1376 Watts
Configuration II
@ 3,000 rpm 781 Watts
@ 5,000 rpm 1312 Watts
@ 7,000 rpm 1837 Watts
@ 9,000 rpm 2362 Watts
A more accurate way to compute the friction moment is
by dividing it into two parts: an idling torque M, and a
load torque M.. The sum of the two is the friction torque.
The idling torque represents the friction torque during idle
operation of the bearing and is given by [9,11] as
Mo, = fxlO'8(vn)2'3d,3 vn>= 2000 (4.3)
Mo, = foX16OxlOd,3 vn< 2000 (4.4)
where
fo,: factor depending on bearing design and
lubrication method, for vertical spindles and
oil jet lubrication:
for double row ACBB............. 9
for CRB ...................... 46
for TRB ....................... 810
v: oil viscosity at working temperature (cS)
d,: mean diameter of the bearing (mm)
The friction torque due to the applied load can be
computed using an equation recommended by Palmgren, [11].
Mi = fx Fed. (Nmm) (4.5)
where
M.: friction torque due to the load (Nmm)
f,: factor dependent on the geometry of the
bearing and relative load.
Fe: equivalent force, as described below (N)
d,: mean bearing diameter (mm)
Recalling equation (2.2), for ball bearings, the factor
f. is given by
f = z( ) (4.6)
for angular contact ball bearings, z=0.0001 and y=0.33 [11].
For roller bearings, f. will be
for cylindrical roller bearings:
f1= 0.00020.0003
for tapered roller bearings:
f1=0.00030.0004
Fa for ball bearings is given by the following
equations, also from [11].
Fe = 0.9F. ctna 0.1F, (4.7)
or
Fe = F, (4.8)
whichever is larger, (4.7) or (4.8).
For radial roller bearings, F, is given below as
Fa = 0.8F. ctn a (4.9)
or
F. = F, (4.10)
whichever is larger, (4.9) or (4.10).
In Figures 4.1 and 4.2, the computed generated heat is
plotted at different test speeds and oil flow rates for
Configurations I and II, respectively. The generated heat
was computed by adding the idle friction torque and the
applied load friction torque and substituting the sum into
equation (4.1). In Figures 4.3 and 4.4, the power losses
determined experimentally for Configurations I and II,
respectively, are plotted. The experimental power losses
shown in the figure represent the sum of the Mechanical
Power Losses and the Hydraulic Power Losses, which are
defined in Chapter V. As it can be observed by comparing
Figures 4.1 and 4.2 against Figures 4.3 and 4.4, equations
(4.2) to (4.3) predicted a heat generation much lower than
the measured during the test. The supply oil temperature
and the return oil temperature were used to compute an
average oil temperature for the computation of the viscosity
of the oil inside the bearing cavity.
For their tapered bearings, TIMKEN recommends in [10]
the equations that follow to estimate the friction torque
and the heat generation.
M = kx G (SA)"5 (F.))"'3 (Nm) (4.11)
where
M: bearing operating torque (Nm)
k.: conversion factor = 7.56x 106 (metric units)
G: bearing geometry factor as given in the TIMKEN
bearing catalog [32,33].
for HYDRARIB. = 152.7
for High Speed Bearing = 129.5
S: spindle speed (rpm)
4: oil viscosity (Centipoise)
F.a: equivalent axial load (N)
if the bearing is under combined loading, the equivalent
load F.,q is determined as
K
if __ F., > 2.5 then F., = F,. (4.12)
F,
otherwise
1
F.a = f, F, (N) (4.13)
K
where
F.: axial load (N)
F,: radial load (N)
K: bearing K factor, from the TIMKEN bearing
catalog [32]:
for HYDRARIBT = 1.63
for High Speed Bearing = 1.23
f,: axial load factor, function of (KF.)/F, as
given in the bearing catalog [33].
To compute the heat generation rate for tapered roller
bearings, equation (4.14), which is recommended by TIMKEN
for their bearings was used. The computed generated heat is
plotted in Figure 4.5, versus spindle speed at constant oil
flow rate.
Q = k2 G S"5 A5 FaQ ./3 (4.14)
where
Q: heat generation (W)
k2: conversion factor (metric)= 7.9x107
As it can be observed by comparing Figure 4.5 against
the experimental measurements in Figure 4.4, equation (4.14)
predicted quite well the generated heat for Configuration
II.
o 1.5 LPM
* 3.8 LPM
A 2.3 LPM
1.2[
3o.8
I,
4J
L
C9
M A I
a 3.0 LPM
0 1000 2000 3000 4000 5000 8000 7000 8000 M000 10000
Spindle Speed RPM
Figure 4.1 Computed Generated Heat vs. Spindle Speed
Configuration I
 I 
o 0.8 LPM
* 2.3 LPM
S1.5 LPM
S3.0 LPM
1.6
1.2
.8
0.0
0
Figure 4.2 Computed Generated Heat vs. Spindle Speed
Configuration II
I 000 2000 3000 4000 5000 8000 7000 8000 9000 10000
Spindle Speed RPM
m .. .. ... if, S ... . m
I
o 1.5 LPM
3.8 LPM
A 2.3 LPM
a 3.0 LPM
I
0 1000 2000 3000 4000 5000 6000 7000 8000 0W00 10000
Spindle Speed RPM
Figure 4.3 Experimentally determined Heat Generation
Configuration I
20
v
a 15
01
I
"3
0
L. 10
a,
o 0.8 LPM
* 3.0 LPM
A 1.5 LPM
o 2.3 LPi
1000 2000 3000 400) 5000 000 7000 8000 8000 10000
Spindle Speed RPM
Figure 4.4 Experimentally Determined Heat Generation
Configuration II
15
10
4j
E 5
o 0.8 LPM
o 2.3 LPM
A 1.5 LPM
* 3.0 LPM
I000 2000 3000 4000 5000 8000 7000 8000 0000 10000
Spindle Speed RPM
Figure 4.5 Computed Heat Generation
Configuration II
12
I
8
03
I4
u
..... i
t
Heat Removal
During high speed operation of rolling elements, the
heat generated within the bearings is considerable, as it
will be shown in Chapter V. This heat must be removed to
avoid excessive thermal loads on the elements. If the
temperature rises too much, the lubricant may exceed its
operating range and the oil film between raceways and
elements could be eliminated.
Circulating oil lubrication has the largest heat
removal capacity, due to the amount of oil which is forced
through into the bearing cavity. The amount of heat removed
by the oil can be computed by multiplying the mass flow of
the circulating oil, by its heat capacity, times the change
in temperature.
Pol x= (m c)o1(TotT.jxy) (kW) (4.15)
where
Po1x: power removed by the oil (kW)
m: oil flow rate (1pm)
c: specific heat time the density of the oil,
1566 (KJ/(m3 C))
Tout,: oil temperature at the exit of the housing
(C)
T=uBP.y: oil temperature at the inlet of the
housing (C)
Table 4.1 Removed Heat/Generated Heat
~_______ Configuration I________
Speed Oil Flow Generated Removed Percentage
RPM Rate (LPM) Heat (kW) Heat (kW) Removed
3000 1.5 3.6 2.7 76
3000 2.3 4.9 4.0 81
3000 3.0 5.5 4.5 82
3000 3.8 5.6 5.0 90
5000 1.5 6.6 4.6 70
5000 2.3 8.1 6.3 78
5000 3.0 9.0 7.0 78
5000 3.8 11.1 9.2 83
6000 1.5 9.0 5.4 61
6000 2.3 10.8 7.4 68
6000 3.0 11.8 9.2 77
6000 3.8 12.9 10.7 83
7000 1.5 12.5 5.26 42
7000 2.3 14.3 8.3 58
8000 1.5 14.4 5.8 40
8000 2.3 18.3 9.4 51
8000 3.8 20.7 17.3 83
Table 4.2 Removed Heat/Generated Heat
________Configuration II_____
Speed Oil Flow Generated Removed Percentage
RPM Rate (LPM) Heat (kW) Heat (kW) Removed
3000 0.8 2.9 2.0 70
3000 1.5 2.9 2.4 82
3000 3.0 2.9 2.7 93
3000 3.8 2.9 2.6 89
5000 0.8 4.3 3.4 80
5000 2.3 4.6 3.6 79
5000 3.0 4.8 3.9 82
5000 3.8 4.8 4.0 82
7000 1.5 7.0 4.8 64
7000 2.3 7.0 4.9 70
7000 3.0 7.0 5.5 79
9000 0.4 10.6 5.5 52
9000 0.8 11.4 6.0 53
9000 1.5 12.4 6.3 54
9000 2.3 13.3 7.7 58
As the amount of oil increases, so does the cooling
capacity, removing more heat from the bearings. The ratio
of the heat removed to the heat generated increases with
increased oil flow rate. In Tables 4.1 and 4.2, the
percentages of generated heat removed by the oil are listed,
for each configuration, at each speed and oil flow rate. As
it can be seen from the table, as the oil flow rate
increases, the percentage of the generated heat which is
removed increases. Also from the table, as speed increases,
for the same flow rate, the percentage of the generated heat
removed by the oil decreases. This can be explained by the
fact that as the speed increases, so does the temperature of
the bearing, as it will be shown later. A higher bearing
temperature will produce a larger heat conduction rate
through the housing due to a larger temperature gradient
between the housing and the environment. Thus, less heat is
convected away by the oil. Also with an increase spindle
speed, the oil in the bearing cavity traps a larger amount
of air, changing itself into foam and hence reducing its
convection capacity.
An increase in oil flow rate will also produce an
increase in power losses, as it will be shown in Chapter V.
The increase in power losses is, in some cases, large enough
to nullify the increased cooling capacity that a larger oil
flow rate produces. Therefore, the net effect may be an
insignificant decrease in temperature and a significant
increase in power losses. From the experimental results,
such as power losses, operating temperature and oil flow
rate, design recommendations will be made for each
configuration.
Steady State Temperature Fields
The steady state thermal fields were computed for both
configurations. The analysis was performed using finite
difference methods. The housing was divided into ring
elements as shown in Figure 4.6 for Configuration II. The
initial temperature was taken as room temperature except for
those elements covered by the boundary conditions. The
program was stopped when the surface temperature of the
model approximated the experimentally measured surface
temperature. Forced convection at the housing surfaces was
assumed since the spindle rotation produces a considerable
flow of air around the spindle. The equations used to
estimate the amount of heat conducted radially from one
element to the next are given by [34] as
2nkl
Qa.L=_____ (TjT) (Watts) (4.16)
Ro
ln
R1.
where
Qr.ei.x: Heat transmitted in the radial direction
(Watts)
k: thermal conductivity of the housing material
(cast iron= 52 W/(m2C) [34]
1: axial length of the element (m)
Tj: temperature of the j^ element (C)
Tj.: temperature of the ijh element (C)
Ro: outer radius of the housing (m)
R.: inner radius of the housing (m)
The equation used to compute the heat conducted from
one element to the next in the axial direction is
2nkrdr
Q^.__.= (TjT.) (Watts) (4.17)
x
where
Q...L: Heat transmitted in the axial direction
(Watts)
r: radius of the i* element (m)
dr: radial width of the element (m)
x: axial distance between nodes (m)
The equation used to estimate the heat conducted away
by the air surrounding the spindle housing is given by [34]
as
Qoov o. = h A (T,, T.) (Watts) (4.18)
where
Qoov.oi.: Heat removed away by convection
(Watts)
h: convection coefficient = 9 W/m2 [34]
A: heat transfer area (m2)
T2: housing surface temperature (C)
T.: temperature of the surroundings (C)
Equation (4.19) was used to estimating the radiation
heat transfer.
Qrmimn = o 6 F A (Th4 T.4) (Watts) (4.19)
where
Qd.L.o: Heat removed away by radiation (Watts)
a: StefanBoltsman constant = 5.66961x 10"
(W/(m2K') [34]
e: emissivity (.8) [34]
F: shape factor = 1.0 [34]
The boundary conditions used for the analysis of each
housing were:
1. The bearings are represented as elements with
constant temperature. The temperature assigned is
the temperature of the bearing at steady state
measured in the test.
2. The temperature at the inside surface of the
housing is assumed to be equal to the average
between the surface temperature of the center and
the average bearing temperature, for the given speed
and oil flow rate.
3. At the outer surface the housing loses heat to the
environment through convection and radiation.
4. There is forced convection and radiation at the top
surface.
5. The temperature of the environment was assumed
constant at 220C.
The computed thermal profiles for Configuration II at
5,000 rpm, 7,000 rpm and 9,000 rpm, and an oil flow rate of
2.3 1pm are shown in Figures 4.7, 4.8 and 4.9. The
computed thermal profiles will be used to compute thermally
induced loads on the rolling elements.
Thermally Induced Loads
As heat is generated in the bearings, a temperature
gradient is developed between the bearings and the outer
surface of the housing. Since the bearings and the housing
are heating at different rates, their expansions occur at
different rates. These differential expansions induce
loads on the bearings. These loads will be proportional to
the difference in thermal expansions between the bearings
and the housings.
Let's assume that the inner race, the rolling elements
and the outer race are all at the same temperature. The
thermal expansions of the inner ring, the outer race and the
ith ring of the housing model are given respectively by
6. = F d n (T T=) (m) (4.20)
6io = F do Tx (To T.) (m) (4.21)
85k = F dh,, n (Thi T.) (m) (4.22)
where
68.: thermal expansion of the inner ring (m)
6,T,: thermal expansion of the outer ring (m)
68,H: thermal expansion of the ilh housing element
(m)
r: thermal expansion coefficient 10.6xl06 C'
[30]
d: inner ring diameter (m)
do,: outer ring diameter (m)
dh.: diameter of the il housing element (m)
T: temperature of the inner ring (C)
To: temperature of the outer ring (C)
Th,: temperature of the ill housing element (C)
T.: starting temperature (C)
The thermal expansion of the outer race is prevented by
the much slower expansion of the housing. It is at this
joint that the thermally induced interference happens,
increasing the bearing preload. To determine the induced
load, the expansion of the housing must first be computed.
Using the thermal fields computed above, the expansion of
each ring element in the housing can be computed. The
expansion of the element in contact with the bearing can
therefore be computed, and after computing the expansion of
the outer ring of the bearing, the increase in interference
could be determined. By using the loaddeflection
relationships developed in Chapter VI, the thermally induced
load could be computed.
Computation of Thermal Loads
Following the procedure described above for computing
the thermal loads, a sample calculation will now be provided
for the 7,000 rpm test of Configuration II. The thermal
deflection at each concentric ring surrounding the lower
bearing is first computed using the temperature distribution
as shown in Figure 4.8. The outer diameter of the bearing
element is 0.158 m. The next element is .012 m larger and
the rest are divided using 0.026 m increments. Using
equations (4.20) to (4.22), the thermal deflections are
computed next using T. as 295K.
For the outer race element, the thermally induced
deflection is
6,o = 10.6 x 106*(0.158)*n*(340295) (m)
6T0 = 2.37x 104 (m)
This 68, would be the deflection of the outer ring if
it was not constrained by the other ring elements. To
determine then the actual deflection, the deflections of all
the rings must be computed. Once the thermal deflection of
the outermost ring is estimated the deflection of the outer
race of the bearing is determined. The minimum deflection
computed for any of the rings surrounding the outer race was
of 2.26 x 10' m. Thus, the maximum deflection of the outer
race of the bearing is that of the ring which deflected the
less or 2.26 x 104 m.
To determine the increase in preload, the thermal
deflection of a roller must first be computed. It is given
by equation (4.22) using the diameter of the roller instead
of the element diameter.
68, = 10.6 x 106*(0.013)*n*(340295) (m)
6,0 = 2.37x 10 (m)
The increase in preload can now be estimated by
subtracting the roller thermal deflection from the outer
ring thermal deflection. This difference is multiplied by
the stiffness of the bearing to obtain the increase in load.
Thus, the difference in thermal deflections is 1.1 x 101 m.
Using equation (6.12) and a roller stiffness value of 1.00 x
10' N/m [30] the load was computed as 3.1 x 102 N. This
load is negligible for the type of bearing used. This
coincides with the PROMESS sensor measurements.
93
Figure 4.6 Thermal Model for Configuration II
