Design of high speed, high power spindles based on roller bearings


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Design of high speed, high power spindles based on roller bearings
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Hernandez-Rosario, Ismael A., 1959-
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Table of Contents
    Title Page
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        Page ii
    Table of Contents
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    Chapter 1. High speed, high power milling
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    Chapter 2. Literature search
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    Chapter 3. Experimental equipment
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    Chapter 4. Thermal analysis
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    Chapter 5. Experimental results and discussion
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    Chapter 6. Bearing loads
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    Chapter 7. Conclusions
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    Appendix. Computer program listings
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    Biographical sketch
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Full Text








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 # MEA-8401442 Unmanned Machining, High

Speed Milling.



ACKNOWLEDGMENTS ................................... ii

ABSTRACT ............. .................. .......... vi



Introduction .................................. 1
Development of High Speed Milling .............. 2
High Speed, High Power Machining ............... 4
Goals and Scope ................................ 6


Analytical Developments ........................ 8
High Speed Bearings: Experimental Results ..... 38


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


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



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


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


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


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



Ismael A. Hernandez-Rosario

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 idle-load 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.





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 1000-5000 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


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, 7075-T6, 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 (10-20 kW). The combination of HSM and

high power milling is called High Speed, High Power (HSHP)


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


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, non-standard 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


bearings in machine tools and to identify those parameters

which are essential for a successful HSHP spindle design

based on roller bearings.



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


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, (N-mm) (2.1)


Mr: Load torque (N-mm).

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)


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 f-rom [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

Self-aligning 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

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)


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


M = 9.79x10-fo(L*n)2f d.3 (N-mm) (2.8)

when u*n > 2000

M, = 1.59 x 10-5fcod3 (N-mm) (2.9)

when u*n 5 2000


M,: Viscous torque (N-mm).

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


Qc = 1.05 x 10-4 n M (W) (2.10)


Q.: Heat generation rate (W).

M: Total opposing torque Mr + M,. (N-mm)

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


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.)(w-w4) 0.5Dlww (2.11)

Vj = 0.5(d,, + D,.,)w 0.5D.w, (2.12)


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( d|4 Dv)Vc


L/2( d| DV)

V2 2dvVvj
V/2( d,-Dv( V-Vc)

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 j-l 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


Ozoj L

/ ^"l


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

%tj + FVj -

_- (Qy.oj Fo.j Faj) = 0



Ro: Equivalent external radius of the cylinder


R,: Equivalent internal radius of the cylinder


Qyo.j =

Fj =


lw,, E'R

lw E'Ro,


1. E'R

lw1 E'R.

lw, E'Rc,

F,, E'
iw E'Ro







i: Roller length (mm).


1 o2




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


Fi- = -9.2G-0-3 UjO-7 +

Foj = -9.2G-0-3 Uoj0'7 +

Vc~j Ic:j

Q^ = 18.4 (1-T)G-3 U0-7

= 18.4 (1-T)G-3 U,10.7






G = aE'

a: Is the pressure coefficient of viscosity


Hj= 1.6

Hoj = 1.6

Go-6 UijO-7
0.-- 5 3-3 --

Go06 UojO-7

Q j0.2.13

4o VLj
Vij =
E' R




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)

f)= J Gqoj[l-(y/4qoj)2] X/2
Ioj = 2 e" dy (2.33)

As it can be seen from above, the elastohydrodynamic

loads are non-linear 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


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 non-linear functions of lubricant

properties and roller speed. The inner race contact loads

Qzj, are computed from static load analysis of the complete


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


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


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.

__ (Q^j + F fvFm) Q.j = 0 (2.35)

QOj + F:j (Qwoj Fon Fj) = 0 (2.36)

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)


0-9 8hA
6j =) + P:j (2.39)

=8 + Po ___ (2.40)
(-K'2 I6hEJ Pcij


h: Oil film thickness given by
8 C06(jO.u)0-'7 E'0.3 R-13 lW,0-'3
h= _____________ (2.41)
3 po.3.3

6h -0.302 a'-6(jiou)0-7 E'0-3 R013 1,.O.13
= (2.42)
6P p.3.3

The deflection behavior was used to determine the inner

race contact loads as

F,, = EPij sin4> (2.43)

Fy = EP1j cosoj (2.44)

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)G--3Uou.-'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)


: 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

G = ___________________ (2.48)
D-0'-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)0-5 (F.O)"3 (2.49)


M: Resisting torque (lb.-in.).

G: Bearing Geometry Factor.

F.: Axial Load (lb.).

S: Bearing Speed (rpm).

1: Lubricant viscosity at atmospheric pressure


0: Lubricant pressure-viscosity 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


M = l.lxlO-4'G (Sut)0- (fTF,/K)X/3 (2.50)


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


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


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


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 load-deflection 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 (N-m) (2.51)


T: Drag torque acting over the element surface


T,: Wall Shear Stress (N/m2).

A: Surface area of the roller (m2).

r: Reference radius from the center of rotation


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)


f: Friction factor computed from the Reynolds

number assuming turbulent flow [17].

: Fluid mass density (oil and air mixture)


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, (N-m) (2.53)


T,,.: Drag torque acting on the end of the roller


w: Rotational speed of the roller (rad/sec.)

Cn: Correlation factor:

3.87/(NR.)-5 for laminar flow


O.15/(NR.)0'7 for turbulent flow


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


Fj3 N |Vt Vutil
T.Afta E ( E _________A__..r
A0 N=1 VR V.:j
2* E Avkrvk] (2.54)


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 roller-cage contact


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 shaft-bearing

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,


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


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 3.8x10-3 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 cone-rib lubrication. The use of cone-rib

lubrication proved to be more efficient in limiting the

operating temperature. It was also found that the use of

cone-rib 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 load-deflection 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 cone-rib/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 cone-rib area.

Parker and Signer used specially designed TRB to

investigate the high speed performance with cone-rib

lubrication versus the performance with oil jet lubrication.

The bearings used were of standard design but provided with

cone-rib lubrication to improve their high speed operating


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

10-3 to 15.1 x 10-3 m3/min.

The test results obtained in [25], showed that cone-rib

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 10-3 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 10-3

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 cone-rib 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

10-3 to 15.1 x 10-3 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


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 M-50, a vacuum-induction melted, vacuum arc melted

alloy, has greatly improved the fatigue life of high speed




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



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


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 re-established.

NN3019 K

234420 ll


Figure 3.2 Configuration I Test Spindle

Hydra Rib

>Oil Distribution

- High Speed

Figure 3.3 Configuration II Test Spindle

Oil Jet

Hydraulic Oil
Input \

Snap Ring ,

Rib Chamber

Figure 3.4 HYDRA-RIBT Bearing




Outer Race

Inner Race

Tapered Roller

Cone Rib
Lubrication Ring

Cone Rib
Lubrication Hole

Figure 3.5 High Speed Bearing with Cone-Rib Lubrication

The High Speed (HS) bearing, 100 mm diameter, in the

tool side, Figure 3.5, is provided with cone-rib

lubrication. The cone is lubricated through holes drilled

from the back of the bearing to the cone-rib. 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 cone-rib 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

HYDRA-RIBT. 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

cone-rib 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 HYDRA-RIBT. 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



Figure 3.6 Oil Supply to Double Row Bearings


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.



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


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


Oil Circulating System
There are several lubrication methods used in machine

tools among them, grease lubrication, oil mist lubrication,

air-oil lubrication ("O-L"), 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


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)


QoiB;: Heat removed by the oil from the bearings


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)


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..)




Qwf.m.3: Heat removed by the water from the oil


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 13-C 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


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, non-contact seals

must be used. Non-contact 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.


Suction Pump

...... Screens
Variable Output Pump

Figure 3.8 Circulating Oil System


Oil Suction Points

Figure 3.9 Seal for Configuration I

Oil Jet


Lubriction Inpuit

il i /ints
Oil Su.,,tion Poin

H50 Tper

Figure 3.10 Seal for Configuration II



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 rolling-element/raceway contacts due

to the geometry of the contacting surfaces.

3. Sliding due to deformation of contacting


4. Sliding between the cage and the rolling

elements, and between the cage and the guiding


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 = 1.05x10-4 n M (W) (4.1)


Hf: heat generated (W)

n: spindle speed (rpm)

M: friction torque (N-mm)

Also from [9,11], the bearing manufacturer estimates

the friction torque as

M = 0.5 Vj, F d (N-mm) (4.2)


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 Hydra-RibT 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


for Configuration I

M30O22. = 360 N-mm

Mmo K = 1232 N-mm

M234420Moi = 46 N-mm

for Configuration II

Mm.- 3xIB = 489 N-mm

M.s = 146 N-mm

The heat generated, computed using equation (4.1), at

the different test speeds, for each configuration are listed


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, = foX16OxlO-d,3 vn< 2000 (4.4)


fo,: factor depending on bearing design and

lubrication method, for vertical spindles and

oil jet lubrication:

for double row ACBB............. 9

for CRB ...................... 4-6

for TRB ....................... 8-10

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. (N-mm) (4.5)


M.: friction torque due to the load (N-mm)

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= 0.0002-0.0003

for tapered roller bearings:

Fa for ball bearings is given by the following

equations, also from [11].

Fe = 0.9F. ctna 0.1F, (4.7)


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)


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 (N-m) (4.11)


M: bearing operating torque (N-m)

k.: conversion factor = 7.56x 10-6 (metric units)

G: bearing geometry factor as given in the TIMKEN

bearing catalog [32,33].

for HYDRA-RIB. = 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

if __ F., > 2.5 then F., = F,. (4.12)

F.a = f, F, (N) (4.13)

F.: axial load (N)

F,: radial load (N)

K: bearing K factor, from the TIMKEN bearing

catalog [32]:

for HYDRA-RIBT = 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 A-5 FaQ ./3 (4.14)


Q: heat generation (W)

k2: conversion factor (metric)= 7.9x10-7

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


o 1.5 LPM

* 3.8 LPM

A 2.3 LPM






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





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


o 1.5 LPM

3.8 LPM

A 2.3 LPM

a 3.0 LPM


0 1000 2000 3000 4000 5000 6000 7000 8000 0W00 10000
Spindle Speed RPM

Figure 4.3 Experimentally determined Heat Generation
Configuration I


a 15

L. 10

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




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





..... i


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(Tot-T.jxy) (kW) (4.15)


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


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


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

Qa.L=_____ (Tj-T) (Watts) (4.16)


Qr.ei.x: Heat transmitted in the radial direction


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

Q^.__.= (Tj-T.) (Watts) (4.17)

Q...L: Heat transmitted in the axial direction


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]


Qoov o. = h A (T,, T.) (Watts) (4.18)


Qoov.oi.: Heat removed away by convection


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)


Qd.L.o: Heat removed away by radiation (Watts)

a: Stefan-Boltsman 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


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)


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

r: thermal expansion coefficient 10.6xl0-6 C-'


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 load-deflection

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 10-6*(0.158)*n*(340-295) (m)

6T0 = 2.37x 10-4 (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 10-4 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 10-6*(0.013)*n*(340-295) (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 10-1 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.


Figure 4.6 Thermal Model for Configuration II