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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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Subjects / Keywords:
Contact loads ( jstor )
Flow velocity ( jstor )
Housing ( jstor )
Hydraulics ( jstor )
Lubrication ( jstor )
Operating temperature ( jstor )
Power loss ( jstor )
Roller bearings ( jstor )
Structural deflection ( jstor )
Torque ( jstor )

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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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DESIGN OF HIGH SPEED, HIGH POWER SPINDLES
BASED ON ROLLER BEARINGS








By

ISMAEL A. HERNANDEZ-ROSARIO


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 # MEA-8401442 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. 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.


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 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

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, 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)

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, 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





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, (N-mm) (2.1)

where

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)

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 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
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 = 9.79x10-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

where

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

(2.10).

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

where

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

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.)(w-w4) 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( d|4 Dv)Vc


VcO UTERiNS


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





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- = -9.2G-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


(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


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


Go06 UojO-7

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 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

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 non-linear 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

0-9 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 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

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)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)

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

D3-5
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)

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 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

loads.

M = l.lxlO-4'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 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)

where

T: Drag torque acting over the element surface

(N-m).

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, (N-m) (2.53)

where

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

(N-m)








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 roller-cage contact

(N-m).

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,

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 3.8x10-3 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

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

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

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 M-50, a vacuum-induction 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 re-established.






















Bea-ing
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 HYDRA-RIBT 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 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

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,

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

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 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








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, 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.













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


Cne-Rib
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 rolling-element/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 = 1.05x10-4 n M (W) (4.1)

where

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)

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 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

obtained:

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

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, = foX16OxlO-d,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 ...................... 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)

where

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:

f1=0.0003-0.0004
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 (N-m) (4.11)

where

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

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 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)

where

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

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(Tot-T.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=_____ (Tj-T) (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^.__.= (Tj-T.) (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: 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

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 10.6xl0-6 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 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.





93




















































Figure 4.6 Thermal Model for Configuration II




Full Text
196
the greater amount of rolling elements in Configuration I.
The larger the number of rolling elements, the larger the
frictional losses, and thus, the higher temperature.
Increasing the oil flow rate decreases the bearing's
operating temperature. The larger the amount of lubricant
circulated through the bearing the larger the cooling
effect. Configuration I was more responsive to increases in
oil flow rate than Configuration II. As it was described
before, most of the oil forced through the cylindrical
roller bearings is splashed away, increasing the power
losses and having little cooling effect. On the other hand,
tapered bearings have an efficient flow of oil through the
cavity. Thus, increasing the oil flow rate has very little
effect on the operating temperature.
Two parameters were defined to evaluate the performance
of each spindle independently of the machine tool on which
it was installed. These parameters are the mechanical power
losses and the hydraulic power losses. The mechanical power
losses refers to the power required to drive the spindle to
the desired speed without oil circulation. The hydraulic
power losses refers to the power required to drive the
spindle through the circulating oil at the desired speed.
The total power required to drive a spindle to the desired
speed is the sum of the mechanical power losses, plus the
hydraulic power losses, and plus the motor/transmission
power losses. The last term, the motor/transmission power
losses is independent of the spindle.


95
Figure 4.8 Thermal Profile (7,000 RPM)
Configuration II


189
Fac = z C0 sina0 (N) (6.110)
where
Fac Centric axial load induced by the centrifugal
forces acting on the rollers (N)
Gyroscopic Moment
The gyroscopic moment acting on each roller induces a
redistribution of the loads on the cup race contact [37].
The magnitude of the gyroscopic moment is given by
Mw;J = 8.37x1o-12 D4 wm:)wR;, sin[0.5(ai+ao) ] (6.111)
Assuming a linear load distribution along the length of
the roller, the distributed load at the back of the roller
is qx and at the front is q2. In terms of the centrifugal
load C0, and gyroscopic moment Moj, qx and q2 can then be
determined by taking moments about the roller center of
gravity. From the figure describing the geometry of the
tapered bearing
C0(A + x) + = C£B (6.112)
solving for x
(C*B- Ma;t)
x = A (6.113)
C0
The center of gravity for a trapezoid load distribution
is given by
L L(q1+2q2)
x = (6.114)
2 3(qi+q2)
and
(qa.+q2)L = 2C0 (6.115)


166
The load on the outer race F0 and the load on the inner
race Fj. are given by
cosa£+ctnasina£
F0 = Fia (N) (6.34)
sin(a0 + a*)
Fj. = (N) (6.35)
sina
Once the loads F0 and F^ are determined, the deflection
at the roller can be determined as
where
(mm) (6.36)
(mm) (6.37)
60: normal approach at the outer race (mm)
Si.: normal approach at the inner race (mm)
The axial deflection at the outer 6oai and inner 5Xm.
contact are given by equations (6.38) and (6.39)
respectively
60
=
sincij.
6*
=
sincii.
(mm)
(6.38)
(mm)
(6.39)


Temperature
103
Figure 5.1 Transient Temperatures
Configuration I: 3,000 RPM


156
Figure 5.27 Bearing Thermal Loads


86
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
Qr. R0
In
Ri.
where
Qra (Watts)


70
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
H* = 1.05x10-* n M (W) (4.1)
where
heat generated (W)
n: spindle speed (rpm)
M: friction torque (N-mm)


Spindle
Figure 3.9 Seal for Configuration I


167
Figure 6.2 Axial Loads on a Tapered Roller


174
interference will be. The effect of preload on the load
deflection behavior is shown in Figure 6.4. The slope of
the load deflection curve is defined as the stiffness of the
bearing.
In the case of the ACBB or TRB, one of the races is
displaced axially more than the other, loading the elements
with a force proportional to the relative displacement and
contact angle.
Preloading Methods
In Figure 6.5, a and c, two different spindle bearing
configurations are shown. In Figure 6.5a, both the inner
and outer rings of a pair of bearings are kept at a
constant distance form each other. A preload is obtained by
means of an adjusting nut, which maintains a constant
distance between both bearings. This is denoted as CASE 1.
In Figure 6.5b, the direction of the azimuth angle 9 for
each bearing is indicated. Also in Figure 6.5b, the
positive direction of the forces and deflections at each
bearing is illustrated. The figure also illustrates the use
of a spring (or a hydraulic cylinder) to provide a constant
preload on the bearings. Let this be CASE 2.
Case 1: Variable Preload
In Figure 6.5a, the variable preload configuration is
shown. The nut maintains a constant distance between the
bearings. Thus, the axial deflection at each bearing should
be of the same magnitude and in the same direction.


192
play a more relevant role in the deflection of the bearings.
The observed loading on the bearings, using the PROMESS
sensor, confirms these results, since no change in bearing
load was detected during the test of Configuration II.


Table 5.8 Curve Fit Constants for Hydraulic
Power Losses Oil Flow Rate Relationships
for Configuration II
Spindle
2
Speed
a7
a8
R
R
3000 rpm
1.7
0.0


5000 rpm
2.4
0.13
0.93
0.96
6000 rpm




7000 rpm
4.2
0.0
0.00
0.00
8000 rpm




9000 rpm
8.2
0.18
0.99
0.99


191
Equating (6.113) and (6.114), substituting (6.115) into
it and solving for q2
C2 6M,
cr3
q2 =
q* is given by
qi =
6M,
(6.116)
(6.117)
Combined Loading
The load deflection equations for a TRB under combined
loading and operating at high speeds are similar to those
under static load. To the static load deflection equations
we add the effect of the centrifugal force acting on the
bearings, computed in the section on combined loading. The
equilibrium equations for the high speed case will be
= Fc;) +k6r;J 1 *11 (N) (6.118)
F.3 = Fac tkS.;,3--13- (N) (6.119)
These equations are then solved in the same manner as
presented above. For the operating speeds at which the tests
were performed, the computations show that the effect of
spindle speed on bearing deflections can be neglected.
Again, the effect of increased external loads and preload


154
increasing the oil flow rate above 2.3 1pm would not reduce
the operating temperature significantly. Thus, 2.3 1pm
could be provided by the current lubrication scheme and 1.5
lpm would be provided through the jets directly to the
rolling elements. This should decrease the bearing
temperature.
High Speed Bearing Failure
As mentioned above, the HS bearing in Configuration II
failed during the test of an air seal. The spindle was
operated in excess of 9,000 rpm and oil flow rates larger
than 2.3 lpm. In previous tests, without the air seal, the
bearing operated satisfactory at these conditions.
Examination of the failed bearing indicated that the cone-
rib contact area suffered from lubricant starvation. The
rear end of the rollers was considerably damaged and the
surface discoloration indicates that the temperature
exceeded the recommended maximum.
At 10,000 rpm, the DN value for this bearing is one
million. If the cone-rib contact is neglected and
lubrication is not provided for this critical area, the
usual DN limit is 0.5 million DN [21,22,23,25]. For the
high speed tapered bearing used, 3.0 million DN has been
experimentally obtained [26,27]. Thus, the bearing was
receiving some cone-rib lubrication, but not enough to allow
the bearing to exceed 1.0 million DN.


82
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.
Pox (ni (Tout-TBUPPly) (kW) (4.15)
where
Pon: power removed by the oil (kW)
m: oil flow rate (1pm)
l^c: specific heat time the density of the oil,
1566 (KJ/(m3 C))
Tout oil temperature at the exit of the housing
(C)
Tauppiy: oil temperature at the inlet of the
housing (C)


87
k: thermal conductivity of the housing material
(cast iron= 52 W/(m2oC) [34]
1: axial length of the element (m)
T-j: temperature of the element (C)
T: temperature of the itl1 element (C)
Ro: outer radius of the housing (m)
Ri.: 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
QXJ.-Lx= (Tj -Ti.) (Watts) (4.17)
x
where
Heat transmitted in the axial direction
(Watts)
r: radius of the itia 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
Q=onv.ction = h A (Th Ta) (Watts) (4.18)
where
Qconv.ction: Heat removed away by convection
(Watts)
h: convection coefficient = 9 W/m2 [34]
A: heat transfer area (m2)
Th: housing surface temperature (C)


26
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.


190
Fo
Figure 6.7 Loads on a High Speed Tapered Roller


130
configuration. The indicated flows are per bearing, not
total oil flow rate. For Configuration I, the three
bearings received the same oil flow rate at all times. For
Configuration II, the HYDRA-RIBtm was supplied with 3.8 1pm
at all times. The flow through the High Speed bearing was
varied from 0.8 1pm to 3.8 1pm. The graphs indicate the
total hydraulic power loss for the configuration, not for
individual bearings.
Hydraulic Power Losses Versus Spindle Speed
The effect of spindle speed, at constant oil flow rate,
on hydraulic power losses is shown in Figure 5.19 and
Figure 5.20, for Configuration I and Configuration II,
respectively. As before, the experimental data is
represented by the symbols and the curves represent
approximation functions of the data. The approximation
functions were derived as before, using a least square
scheme. The approximation functions are of the form
Ph = a5 na6 (C) (5.8)
where
Pn: Hydraulic Power Losses (kW)
n: Spindle Speed (rpm)
a5: Curve fit constant one (kW/rpma6)
a6: Curve fit constant two


BIBLIOGRAPHY
1. Komanduri, R., "High Speed Machining," Mech. Eng,
Vol. 107/12, December 1985, pp. 64-76.
2. Tlusty, J., "Dynamics of Production Machinery,"
EML6905 course notes, University of Florida,
Gainesville, Florida, Spring Semester 1985.
3. King, R.I.,ed.,"Preface," in Handbook of High
Speed Machining Technology, New York: Chapman and
Hall,1985.
4. King, R.I.,ed.,"Historical Background," in
Handbook of High Speed Machining Technology, New
York: Chapman and Hall,1985, pp. 3-26.
5. Aggarwal, T.J.,"General Theory and Its
Application in the High Speed Milling of
Aluminum," in Handbook of High Speed Machining
Technology, New York: Chapman and Hall,1985,
pp. 197-240.
6. King, R.I.,ed., "Milling," in Handbook of High
Speed Milling Technology, New York: Chapman and
Hall,1985, pp. 195-196.
7. McGee, F.J.,"Machine System Design and
Performance," in Handbook of High Speed Milling
Technology, New York: Chapman and Hall,1985,
pp. 241-258.
8. Field, M., Harvey, S., and Kahles, J., "High Speed
Machining: An Update," American Machinist, Vol.
127/2, February 1983, pp. 88-92.
9. Svenska Kullagerfabriken (SKF), Bearings in
Machine Tools, King of Prussia, PA: Svenska
Kullagerfabriken (SKF), 1965.
10. TIMKEN, TIMKEN Bearings in Machine Tools,
Canton, OH: The TIMKEN Co., 1981.
208


Table
3.1 Heat Exchanger Temperatures
Speed
Oil
Flow
Water
on
Ret.Oil
In
Out
In
Out
3000
1.9
1pm
10C
16 C
31C
13 C
44C
3000
3.0
1pm
11C
17C
33 C
14C
41C
3000
3.8
1pm
10C
18C
33 C
15C
38 C
4000
0.8
1pm
8 C
12C
14C
12C
66 C
4000
1.9
1pm
11C
18C
36C
21C
53 C
4000
3.0
1pm
11C
20C
37C
23 C
49 C
4000
3.8
1pm
11C
21C
37C
22C
49C
5000
1.9
1pm
10C
22C
26 C
22C
66C
5000
3.0
1pm
10C
25C
31C
25C
62C
5000
3.8
1pm
9C
27C
33 C
27C
61C
6000
0.8
1pm
7C
11C
12C
10C
64C
6000
1.5
1pm
6C
12C
16 C
12 C
67 C


144
20
5 10
o
Q.
3 5
x
0
0 1000 2000 3000 4000 5000 8000 7000 8000 9000 10000
Spindle Speed RPM
o1.5LPM ^ 2.3 LPM
o 3.0 LPM
* 3.8 LPM
*
A
S s' D
A
O
S '' -ti
O
1 1 1 i i i
1 1 J
Figure 5.23 Hydraulic Power Losses versus Spindle Speed
Configuration I (Overall)


Temperature
107
Figure 5.5 Transient Temperatures
Configuration I: 8,000 RPM


151
housing blocked the lubricant flow into the cone-rib contact
causing the seizure of the bearing. Thus, the use of an air
seal is not recommended for Configuration II.
To improve the removal of oil from the housing, rings
could be provided to fling the oil away from the spindle and
into grooves in the housing. From these grooves the oil
could be sucked away avoiding the excess oil at the bottom
of the spindle housing. For Configuration II, a more
efficient oil distribution system could avoid the
accumulation of oil at the bottom of the housing. If the
ring which supplies oil to the Hydra-Rib is redesigned
such that instead of eight supply holes, only three are
provided, the amount of oil getting to the lower end of the
housing would be reduced. As the ring was originally
designed, the oil is not forced high enough into the
bearing, causing it to fall back down into the housing,
eventually reaching the lower end.
Bearing Failures
Three bearings failed catastrophically during the
investigation. For Configuration I, the Radiax bearing
seized twice during the investigation. The failures
occurred when the spindle was driven to 10,000 rpra. The
other bearing failure occurred when an air seal was used in
Configuration II. As the air seal was tested, the spindle
was driven above 9,000 rpm, when the HS bearing failed.


209
11. Harris, T.A., "How to Predict Temperature
Increases in Rolling Bearings," Product
Engineering, December 1963, pp. 89-98.
12. Harris, T.A., "An Analytical Method to Predict
Skidding in High Speed Roller Bearings," ASLE
Transactions, Vol. 9, July 1966, pp. 229-241.
13. Boness, R.J., "The Effect of Oil Supply on Cage
and Roller Motion in a Lubricated Roller Bearing,"
ASME Journal of Lubrication Technology, Vol. 92/1,
January 1970, pp. 39-51.
14. Poplawski, J.V., "Slip and Cage Forces in a High
Speed Roller Bearing," ASME Journal of Lubrication
Technology, Vol. 94/2, April 1972, pp. 143-152.
15. Witte, D.C., "Operating Torque of Tapered Roller
Bearings," ASLE Transactions, Vol. 16, February
1973, pp. 61-67.
16. Astridge, D.G., and Smith, C.F., "Heat Generation
in High Speed Cylindrical Roller Bearings,"
Instn. Mech. Engrs., C14, April 1972.
17. Rumbarger, J.H., Filetti, E.G., and Gubernick, D.,
"Gas Turbine Engine Mainshaft Roller Bearing-
System Analysis," ASME Journal of Lubrication
Technology, Vol. 95, October 1973, pp. 401-416.
18. Pirvics, J., and Kleckner, R.J., "Prediction of
Ball and Roller Bearing Thermal Performance by
Computer Analysis," in Advanced Power Transmission
Technology, ed. by G.K. Fischer, NASA CP-2210,
1983, pp. 185-201.
19. Parker, R.J., "Present Technology of Rolling
Element Bearings," in Advanced Power Transmission
Technology, ed. by G.K. Fischer, NASA CP-2210,
1983, pp. 35-47.
20. Hamrock, B.J., and Anderson, W.J., "Rolling-
Element Bearings," NASA RP-1105, June 1983.
21. Coe, H.H., and Schuller, F.T., "Calculated and
Experimental Data for a 118-mm Bore Roller Bearing
to 3 million DN," ASME Journal of Lubrication
Technology, Vol. 103, April 1981, pp. 274-283.
22. Coe, H.H., "Predicted and Experimental Performance
of Large-Bore High Speed Ball and Roller
Bearings," in Advanced Power Transmission
Technology, ed. by G.K. Fischer, NASA CP-2210,
1983, pp. 203-220.


67
Figure 3.10 Seal for Configuration II


98
7. Spindle Speed.
8. Bearing Loads.
Curve Fitting of Experimental Data
The experimental data was used to develop empirical
equations which describe the relationship between the
independent variables spindle speed and oil flow rate, and
to the dependent variables operating temperatures and power
losses. It was found through the experimental investigation
that for a given configuration, any combination of spindle
speed and oil flow rate would determine a set of operating
temperatures and corresponding power requirements. Using a
public domain computer program developed by the IBM PC User
Group of Greater Kansas City, the data was analyzed. The
program uses the least square method to fit either a
straight line, or a geometric curve or an exponential curve
to a set of data inputed as x,y coordinates.
The results from the experimental investigation can be
represented as tridimensional matrices. Two axis
representing the independent variables spindle speed and oil
flow rate, the third representing either operating
temperatures or power losses. For each point where
temperature was measured, for each configuration, such a
matrix could be developed. As it will be demonstrated
below, each configuration has it own behavior which
characterizes its performance.


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


6
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


55
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.


Table 2.2 Load Torque Factor Values fx
for Roller Bearings
Bearing Type
fx
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.


182
z
EK(6ancosajocosa*. +6xAsinaA+.SP^)^ cosa*. cos9j* -
j=l
Fz (a+b)/b =0 (6.84)
z
EK(6a.Bcosa9jBcosaB +6Psina+. 5P^0)** cosOb cos9jB +
j=l
Fz (a/b) = 0 (6.85)
In Case 2, there are four equations and four unknowns.
The four unknowns are 6,**, 5XB, 6Z*, and 6ZB. Again we use
the Newton-Raphson method on equations (6.82) to (6.85) to
solve for the deflections, given a set of external loads.
The derivative of each equation with respect to the
corresponding deflection is given in equations (6.86) to
(6.87)
D( 6.82) z
= E K q (6Z* cosa9j* cosa*
j=l
+ 6x* sina* + 0.5P D ( 6.8 3 ) z
= E K q (6zB cosa9jB cosOb
6xb j=l
+ 6xB sinaB+ 0.5 P^)*3-1 sin2OB (6.87)
D( 6.84) z
= E K q (6Z* cosa6j* cosa* + &XAk sina*
6Z* j=l
+ 0.5 P^*)*3-1 (cosa* cos9j*) 2 (6.88)
D( 6.85) z
= E K q ( 6zb cosa9jB cosa,, +6XB sina^
6zb j=l
+ 0.5 P^)*3-3-
(COSOb cos9jB)2
(6.89)


152
During each of the test where a bearing failed, sufficient
amounts of oil were circulated through the bearings. Bearing
failures which are not caused by material fatigue are
considered to be premature [9]. If the bearing is properly
lubricated the bearing should not fail. Based on this fact
the bearing failures are discussed next.
Radiax Bearing Failure
The Radiax bearing failed while Configuration I was
driven to 10,000 rpm. The oil flow rate to the bearing at
the time of the test was 3.8 1pm, which was considered
sufficient based on previous test to 8,000 rpm. Once the
failed bearing was removed from the spindle, it was
inspected to determine the cause of the failure. The lower
of the two rows of balls was intact and in excellent working
conditions. The rolling elements in the upper row were, on
the other hand, welded to the races and cage. The cage
showed considerable deformation. The rollers exhibit
considerable skidding and high temperature exposure.
Comparing the bearing surface to the available literature on
bearing failures [9,10,30], it was concluded that it was not
a material fatigue induced failure. The skid marks
indicated that the elements were not sufficiently preloaded.
Also, when skidding occurs, the heat generation rates
increase considerably so that the lubrication needs are
greater. Thus, it was concluded that the preload acting on


173
For thrust roller bearings
C= fc(laff cos a0)7/s* tana0 z3/4 d29/27 (N) (6.62)
The load on the rolling element is given by the
following equations, first for radial roller bearings
P = 0.401 Fm z cosaQ (N) (6.63)
for thrust bearings, ball or rollers
P = Fm z sina0 (N) (6.64)
where
Fm: average element load
z F
Fm = E 1_ (N) (6.65)
i=l z
Bearing Preload
As it was mentioned before, bearings in machine tools
are installed with preload. Preload refers to the use of an
initial or a mounting load on the bearings to improve its
load deflection behavior. Preload eliminates the diametral
clearance of the bearings, so all elements are in contact
with both races. When a load is applied, all the elements
participate in the support of the load, the individual
rolling element load is then decreased, therefore, the
deflection is reduced.
Radial bearings are preloaded by producing a radial
interference. This interference is usually achieved by
driving the bearing onto a tapered shaft. The more the
bearing is driven onto the taper,
the larger the


100
At steady state, the temperatures are plotted versus
spindle speed and versus oil flow rate. An approximation
function was derived from the experimental data using least
squares regression analysis. The temperature of the
bearings was the dependent variable, while spindle speed and
oil flow rate were plotted as the independent variables.
Steady State Temperatures Versus Spindle Speed
Bearing temperatures for Configuration I and II are
plotted vs. spindle speed at constant oil flow rate in
Figure 5.10 and 5.11, respectively. Figure 5.10 shows the
change in temperature of the NN 3022K/SP bearing,
Configuration I, as the spindle speed is varied. The lower
bearing was selected since it is the largest of the three
bearings and the one with the highest operating
temperatures. In Figure 5.11, the temperature of the High
Speed Tapered Bearing, Configuration II, is shown. The oil
flow rate to the Hydra-Rib was kept constant at 3.8 1pm at
all times. In the figures and throughout this chapter the
experimental data is represented by symbols while the lines
represent approximation functions derived from the
experimental data. From Figures 5.10 and 5.11, a set of
approximation functions were derived using a least square
fit technique to estimate the temperature of the bearing for
a given spindle speed at a constant oil flow rate. The
functions are in the form


168
The total axial deflection 6produced by the
external axial force is then the sum of 6oa and 6ja.
+6j. (mm) (6.40)
The induced radial deflection at the outer 6ox. and
inner contact are given by equations (6.41) and (6.42)
respectively
6or = 60/cosa0 (mm) (6.41)
Sir = SiCOsaj. (mm) (6.42)
The total radial deflection produced by the
external axial force is then the sum of 6or. and 6ir.
6rtoti = 6ox. +6ir (mm) (6.43)
Combined Loading
If an ACBB or a TRB is under combined radial and axial
loading, without diametral clearance, both races will remain
parallel, as shown by Figure 6.3. There will be a
displacement in the axial direction 6 and a displacement in
the radial direction 6r. The total approach of the
raceways is
Sou = 6asin0j + 6rcos0jcosa 0.5 Pd (mm) (6.44)
As before, the load deflection relationship for any
element was given by equation (6.23) as
Fe-j K 6e-j
(N)
(6.23)


AXIS SERVO
Figure 3.1 HSHP Milling Machine


181
FzA = E FejA cosaAcos0jA = Fz(a+b)/b (N) (6.72)
j=l
z
FzB = E F0jB cosaB cos0jB = -Fz (a/b) (N) (6.73)
j=l
Substituting for the deflections we obtain
z
Fxa = E K(6zA cosa0jA cosaA + 6xA sina
j=l
+ 0.5 P^)*1 sinaA = FB (6.78)
z
F^b = E K(6zB cosa0jB cosas +6xB sinOe
j=l
+ 0.5 Pas)*1 sinctB = Fb (6.79)
z
E K(6ZAcosa0 jocosa*. +6XAsinaA+. 5PtaA)a cosa*, cos0jA =
j=l
Fz (a+b)/b (N) (6.80)
z
E K(6zBcosa0jBcosaB +6JtBsinaB+. 5P^)^ cosas cos0jB =
j=l
-Fz (a/b) (N) (6.81)
It is more convenient to rewrite equations (6.78) to
(6.81) as
z
EK(6ZAcosa0jocosa*. +6xAsinaA+. SP^)*3 sina*. FB=0 (6.82)
j=l
z
EK(6zBcosa0jbCoscLb +6JtBsinaB+ 5P<*b)sinos Ff=0 (6.83)
j=l


194
two rows of elements and then out each row, Figure 3.6.
This configuration operated at a maximum spindle speed of
8,000 rpm with very high operating temperatures, Figures
5.10 and 5.12. Twice the ACBB failed when attempts were
made to drive the spindle to 10,000 rpm. The power losses
for the configuration reached 20 kW for the maximum speed
and oil flow rate used during the test, Figures 5.18, 5.19,
and 5.21.
Tapered Roller Bearings
Configuration II is based on tapered roller bearings
and it is equipped with a mechanism which maintains a
constant preload during the operation of the spindle. The
bearing at the drive side is a Hydra-Ribi, Figure 3.4. The
bearing at the tool side is a tapered bearing provided with
cone-rib lubrication which makes it suitable for high speed
operations, Figure 3.5. This configuration is also provided
with jet lubrication to the front of the bearings.
Configuration II operated successfully to 9,000 rpm with an
oil flow rate of 2.3 1pm. At this speed, the operating
temperature was 67C, Figures 5.11 and 5.13, and 15 kW in
power losses, Figures 5.18, 5.20, and 5.22. The operating
temperatures and power losses for Configuration II were
always lower than those of Configuration I for the same
spindle speed and oil flow rate.


36
Ah-, i.: Area of the ittl horizontal lamina (m2).
rHJi: Distance from the i**1 horizontal lamina to
the center of the roller (m).
Av;Jlc: Area of the kth vertical lamina (m2).
rVJlc: Distance from the ktl* vertical lamina to
the center of the roller (m).
Ag: 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


197
The effect of spindle speed on hydraulic power losses
is much stronger than the effect of oil flow rate, as it can
be concluded from Figures 5.19 to 5.22 and equations (5.8)
to (5.11). For the same combination of spindle speed and
oil flow rate, the power losses for Configuration I are
higher than for Configuration II as it can be seen from
equations (5.8) and (5.11). This is caused by the
inefficient flow patterns in the cylindrical roller bearings
and the larger amount of rolling elements in Configuration
I. The higher power losses for Configuration I is the cause
for the configuration's higher operating temperatures. The
larger the power losses, the more heat is generated, and
thus, the higher the operating temperatures.
It can be observed from Figures 5.10 to 5.13 that there
is an oil flow rate after which any further increase in flow
has little effect on bearing temperature. By comparing the
increase in power losses, Figures 5.19 to 5.22, to the
decrease in temperature for a given oil flow rate an optimum
flow rate can be selected. For Configuration I, the optimum
oil flow rate selected was 2.3 1pm, while for Configuration
II the optimum flow rate was selected as 1.0 1pm, 0.5 1pm
provided by the oil jets and 0.5 1pm provided through the
cone-rib contact. These flow rates would provide the best
combination of operating temperature and power losses. The
maximum operating spindle speed recommended for each
configuration based on operating temperatures and power
losses are 6,000 rpm for Configuration I and 9,000 rpm for


Hydraulic Oil
Figure 3.4 HYDRA-RIBt*, Bearing


16
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.


177
Figure 6.5 Preloading Methods
a) Variable Preload Configuration;
b) Definition of Positive Sense For
Each Configuration;
c) Constant Preload Configuration.


122
Spindle speed is the parameter with the strongest
effect on bearing temperature. For Configuration I the
effect was stronger than for Configuration II, due to the
bearing geometry and number of rolling elements for each
configuration. The effect of oil flow rate on bearing
temperature was also stronger on Configuration I than for
Configuration II, due to the inherent oil flow rates
patterns for each bearing. In Configuration II, the oil
flows efficiently through the bearing while in the CRB used
in Configuration I, the bearing is splashed every which way
leaving sections of the bearing without appropiate cooling
[16]. Therefore, an increase in oil flow rate would be more
significant for Configuration I, since the excess oil would
now reach more of the bearing surface.
For both configurations the general approximations
provide a good agreement with the experimental data.
Therefore, these general equations (5.3) and (5.4) could be
used to estimate the operating temperatures of these types
of bearings within the test range.
It should be noted here that the curves describing the
relationship between operating temperature and oil flow
rate/spindle speed coincide in shape with those presented by
other researchers in [21,22,23,24,25 and 26].


127
Power Measurements
Three main types of power losses were identified from
the test. There is the motor power losses Pmotor, the
mechanical power losses Pm., and the hydraulic power losses
Ph. These power losses are described next. The
identification of these power losses allows the comparison
of different spindle configurations based on these losses.
As it will be shown next, these losses are independent of
the machine the spindle is installed on.
Motor Power Losses
The motor power losses Pmotoir are independent of the
spindle mounted on the machine. They are produced by the
inefficiencies of the D.C. motor and the mechanical losses
in the intermediate shaft of the transmission system. To
measure Pmot0r, the spindle is disengaged from the
transmission by removing the belt between the intermediate
shaft and the spindle. The motor is then accelerated to
test speed, driving with it the intermediate shaft. The
input power is recorded. The input power is in this case
the motor power losses.
Mechanical Power Losses
The mechanical power losses, Pm, are produced by
mechanical inefficiencies in the spindle configuration, such
as the friction between rolling elements and rings, see


42
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
bearings.


121
Tb = 0.30 V"-24 n*6* (C) (5.3)
for the range: 1.5 < v < 3.8 1pm
3,000 < n < 8,000 rpm
For Configuration II, the derived general equation is
Tb = 0.97 v-0-10 n-7 (C) (5.4)
for the range: 1.5 < v < 3.8 1pm
3,000 < n < 9,000 rpm
The general approximation functions, equations (5.3)
and (5.4) were used to generate Figures 5.14 and 5.15, for
Configuration I and Figures 5.16 and 5.17, for Configuration
II. Note that Figures 5.14 and 5.15 are similar to Figures
5.10 and 5.12, while Figures 5.16 and 5.17 are similar to
Figures 5.11 and 5.13. As it can be seen from the figures,
the general equations yield good approximations to the
experimental data.
Steady State Temperatures: Comparison
Using the steady state operating temperature as a
performance parameter, we can conclude from the experimental
data that Configuration I should not be operated at spindle
speeds above 6,000 rpm. Configuration II, on the other
hand, performs such that operating temperatures at 9,000 rpm
and 1.5 1pm are acceptable. For all combinations of oil flow
rate and speed Configuration II operated at significantly
lower temperatures than Configuration I. As it will be
shown later, power losses for Configuration II were always
lower than for Configuration I.


drcosl theta)
a)
Figure 6.3 Combined Load Geometry
a) Geometry and Nomenclature
b) Deflections.


211
35. Johnson, R.C., Optimum Design of Mechanical
Elements, Second Ed., New York: John Wiley &
Sons, Inc., 1979.
36. Hamrock, B., and Dowson, D., Ball Bearing
Lubrication: The Elastohydrodynamics of
Elliptical Contacts, New York: John Wiley & Sons,
Inc., 1981.
37. Cornish, R.F., Orvos, P., and Dressier, G.,
"Design, Development and Testing of High Speed
Tapered Roller Bearings", The TIMKEN Company,
Contract AFAPL-TR-75-26, July 1975.


Figure 5.15 Bearing Temperature versus Oil Flow Rate
Configuration I (Overall)


76
load F.q is determined as
K
if Fa >2.5 then F.q = F. (4.12)
Fr
otherwise
F. =
where
1
ft Fr _
K
(N)
(4.13)
Fa: axial load (N)
Fr: radial load (N)
K: bearing K factor, from the TIMKEN bearing
catalog [32]:
for HYDRA-RIB, =1.63
for High Speed Bearing = 1.23
ft: axial load factor, function of (KF.J/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 S1-5 u*5 Feq 1/3 (4.14)
where
Q: heat generation (W)
k2: conversion factor (metric)= 7.9x1o-"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
II.


CHAPTER VII
CONCLUSIONS
Large bore, 100 nun diameter, tapered roller bearings
and double row cylindrical roller bearings were investigated
for their High Speed, High Power capabilities. Two spindle
configurations were tested; each based on a different
bearing type, to investigate their performance in terms of
maximum speed, operating temperatures, and power
requirements. Based on the experimental results,
conclusions and recommendations are presented regarding the
suitability of these configurations for HSHP use.
Spindle Configurations
Cylindrical Roller Bearings
Configuration I is based on double row cylindrical
roller bearings of the NN30 K type. It has a very common
spindle-bearing arrangement in which the CRBs receive the
radial load while a double row angular contact ball bearing
(ACBB) sustains the axial load, Figure 3.2. Lubrication to
the bearings is provided via holes in the outer ring of the
bearings. The oil flows into the bearing cavity between the
193


45
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


155
As it was mentioned before, the cause of the failure
was that the pressurized air supplied to the seal affected
the circulation of the oil to the cone-rib contact.
Preliminary test showed that the spindle was capable of
operating at spindle speeds in excess of 10,000 rpm. Thus,
the air seal was eliminated as a solution for the oil mist
problem.


Temperature
108
100
90
80
u 70
60*-
50
. o supply oil a return oil o outer
* surface
40
30g?"
[ 1.5 LPM^3.0^k--0.8
0 10 20 30 40 50 80 70 80 90
Test Time Min.
Figure 5.6 Transient Temperatures
Configuration II: 3,000 RPM
race
loo no


7
bearings in machine tools and to identify those parameters
which are essential for a successful HSHP spindle design
based on roller 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.
43


207
High Speed Cylindrical Roller Bearing Program
10 REM CYLINDRICAL HIGH SPEED BEARING
20 LPRINT CHR$(12)
30 DIM DR(21), D ( 2 5,21),F(21)
40 INPUT "FR,PD,KN,Z,Q,N,FI";FR,PD,KN,Z,Q,N,FI
50 LPRINT "RADIAL FORCE (LB) ";FR:LPRINT "DIAMETRAL
CLEARANCE ";PD
60 LPRINT "STIFFNESS (LB/IN) ";KN:LPRINT "NUMBER OF
ELEMENTS ";Z
70 LPRINT "EXPONENT ";Q:LPRINT "SPINDLE SPEED (RPM) ";N
80 E=360/Z:PI=3.1412:DR(1)=.0002:FC=(N~2)*1.772E-14
90 FOR 13=1 TO Z:D(I3,1)=.0001:NEXT 13
100 FOR J=1 TO FI
110 C=0:FF=0
120 FOR 1=1 TO Z
130 B= COS((I-1)*E*PI/180)
140 AA=(DR(J)*B-PD/2-D(I,J)):IF AA<0 THEN AA=0
150 A= (AA)~Q-D(I,J)Q-FC/KN
160 AAD= (DR(J)*B-PD/2-D(I,J)):IF AAD<0 THEN AAD=0
170 AD= Q*(AAD)(Q-1)+Q*D(I,J)(Q-1)
180 D(I,J+1)=D(I,J)+A/AD:IF D(I,J+1)<0 THEN
D(I, J+l)=ABS(D(I,J+l))
190 F(I)=D(I,J+l)~Q*KN*B:FF=FF+F(I):C=Q*KN*B 2*D(I,J+l)(Q-
1)+C
200 NEXT I
210 AR=(FR-FF)/C
220 DR(J+l)=DR(J)+AR
230 NEXT J
240 LPRINT USING "RADIAL DEFLECTION = #####.######";DR(J)
250 LPRINT "ANGULAR POSITION DEFLECTION FORCE
":FOR 11=1 TO Z
260 X=(11-1)*E
270 LPRINT USING #####.######";X,D(I1,J), F(I1)
280 NEXT II
290 STOP


120
The approximation functions derived for the Bearing
Temperature-Spindle Speed relationship, equation (5.1) with
Table 5.1 and Table 5.2, and those for Bearing Temperature-
Oil Flow Rate relationship, equation (5.2) with Table 5.3
and Table 5.4, match very well the experimental data. In
Figure 5.12, the curve for 8,000 rpm does not follow the
pattern of all the other constant speed curves since it
never achieved steady state. During the test, the operating
temperature exceeded 90C before reaching steady state,
therefore the oil flow rate was increased to avoid damaging
the bearings.
Overall Temperature Equation
An approximation function for TB which includes both
spindle speed and oil flow rate as variables, can now be
derived using the method recommended by Johnson [35].
First, we obtain individual exponential relationships
between each of the independent variables and the dependent
variable, equations (5.1) and (5.2) with Tables 5.1 to Table
5.4. These equations are now combined by determining an
average value for the exponent for each variable. The value
of the constant for the overall equation is determined by
substituting the experimental values of TB, v and n, and
solving for the constant. For Configuration I the resultant
general approximation function is


49
Figure 3.3 Configuration II Test Spindle


142
18 -
a
m
a
u
S
o
Q.
O
3
o 3000 RPM
* 7000 RPM
a 5000 RPM
+ 8000 RPM
6000 RPM
o.o
0.5
1.5 2.0 2.5
OI I Flow Rate LPM
Figure 5.21 Hydraulic Power Losses versus Oil Flow Rate
Configuration I


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 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.
1


88
T: temperature of the surroundings (C)
Equation (4.19) was used to estimating the radiation
heat transfer.
Qr.dition = a e F A (Th4 Ta4) (Watts) (4.19)
where
Heat removed away by radiation (Watts)
a: Stefan-Boltsman constant = 5.66961x 10~a
(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.


46
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


23
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.


69
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
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


165
The equation to be used to solve by iteration for 6*. is
6*.(m+l)=6r(m)+A/B (mm) (6.30)
where
z
A: (Fr E K(6r cos 9j 0.5 P^^cos j)
j=l
z
B: EqK(6r cos 8j 0.5 P j=l
A list of the computer program developed to determine
6*. is included in the Appendix. Several example calculations
from reference [30] were performed and compared. The match
was good.
Axial Loads
Considering a TRB under centric axial load, each roller
contributes equally to support the load. The axial load Fa
an a roller is given by
F
Fa = (N) (6.31)
zsina
where a is the contact angle measured in degrees.
For static equilibrium to exist
~For =0 (6.32)
Fia. + F*a F0a = 0 (6.33)
For a single tapered roller, the forces acting on it
are shown in Figure 6.2.


80
Figure 4.4 Experimentally Determined Heat Generation
Configuration II


183
High Speed Loads
At high speeds the effects of the centrifugal force on
each roller may become significant. If the centrifugal
force increases enough, the rolling elements may lose
contact with the inner race. The lack of proper contact
will produce a decrease in the load zone, increasing the
load at each element, hence reducing the life of the
bearing. The equations for high speed analysis are taken
from reference [30]. The simpler case to consider is the
case of cylindrical roller bearings under a radial external
load.
Cylindrical Roller Bearings
The loads on a high speed CRB under an external radial
load are illustrated on Figure 6.6. Besides the load at each
roller/raceway contact, there is the centrifugal force
acting outward on the centroid of the roller.
The equilibrium of forces is satisfied by
Foj Fj_-j Fc = 0 (6.90)
where
F0j: Normal load at the roller/outer race contact
(N)
Fi;J: Normal load at the roller/inner race contact
(N)
Fc: Centrifugal Force on the roller (N)


160
6totl dinner t ^ oute r- (ITim) (6.13)
where
6totai: total approach for any element considering
both raceways (mm)
approach at the inner race contact (mm)
6outer: approach at the outer race contact (mm)
In the case of ball bearings, 6lnn.r and are
different from each other, due to the change in curvature at
each raceway. If we express the deflection in equation
(6.2) and substitute into equation (6.13)
fFY/3= ^FV/3+ (F
Vlcl/ \w
2/3
(6.14)
where
Kb: load deflection constant for a ball
considering both contacts (N/mmx-s)
Eliminating the force F from equation (6.14) and
expressing Kto:
Kfa =
(6.15)
Then the load deflection equation for a ball
considering both contacts is given by equation (6.16) below
F = Kb 63/2 (N) (6.16)


31
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.


210
23. Zaretsky, E.V., Schuller, F.T., and Coe, H.H.,
"Lubrication and Performance of High Speed
Rolling-Element Bearings," NASA TM-86958, May
1985.
24. Signer, H., Bamberger, E.N., and Zaretsky E.V.,
"Parametric Study of the Lubrication of Thrust
Loaded 120-mm Bore Ball Bearings to 3 million DN,"
ASME Journal of Lubrication Technology, Vol. 96/3,
July 1974, pp. 515-524.
25. Parker, R.J., and Signer, H.R., "Lubrication of
High Speed, Large Bore Tapered Roller Bearings,"
ASME Journal of Lubrication Technology, Vol. 100,
January 1978, pp. 31-38.
26. Parker, R.J., Pinel, S.I., and Signer, H.R.,
"Performance of Computer Optimized Tapered Roller
Bearings to 2.4 million DN," ASME Journal of
Lubrication Technology, Vol. 103, January 1981,
pp. 13-20.
27. Zaretsky, E.V., "Design and Lubrication of High
Speed Rolling Element Bearings," NASA TM-87107,
1985.
28. Lewinschal, L., "Machine Tool Spindle
Applications," King of Prussia, PA: Svenska
Kullagerfabriken (SKF), February 1983.
29. Brown, P.F., and Dobek, L.J., "High Speed
Cylindrical Roller Bearing Development," Pratt &
Whitney Aircraft Group Contract AFWAL-TR-80-2072,
February 1980.
30. Harris, T.A., Rolling Bearing Analysis, Second
Ed., New York: John Wiley & Sons, Inc., 1984.
31. HABASIT, Habasit High Duty Flat Belts,
Engineering Manual 1210, Atlanta, GA: HABASIT
Belting Inc., 1985.
32. TIMKEN, TIMKEN Spring Rate Manual, Canton,
OH: The TIMKEN Co., 1971.
33. TIMKEN, TIMKEN Bearing Handbook, Canton,
OH: The TIMKEN Co., 1983.
34. Kreith, F., and Bohn, M.S., Principles of Heat
Transfer, Fourth Ed., New York: Harper & Row,
1986.


52
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


Temperature
111
Figure 5.9 Transient Temperatures
Configuration II: 9,000 RPM


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
v


136
Figure 5.19 Hydraulic Power Losses versus Spindle Speed
Configuration I


18
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.


110
Figure 5.8 Transient Temperatures
Configuration II: 7,000 RPM


77
1.2
0.0
O 1.5 LPM
a 2.3 LPM o 3.0 LPM
* 3.8 LPM
y
S
1 1 L
1 1 1 1 1 1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Spindle Speed RPM
Figure 4.1 Computed Generated Heat vs. Spindle Speed
Configuration I


188
Cojradiai: Radial component of the outer race
reaction to the centrifugal force (N)
Cfjradiai: Radial component of the flange reaction
to the centrifugal force (N)
Cfjaxlal: Axial component of the flange reaction
to the centrifugal force (N)
Cojaxiai! Axial component of the outer race
reaction to the centrifugal force (N)
Rewriting equations (6.100) and (6.101) in terms of
Fcj r C0j and Cfj
Fcj CojCOSOo c^-jcosa* = 0 (6.102)
Casino* CojsinOo = 0 (6.103)
Solving in equation (6.103) for C*-, in terms of Co;J
C*-, = C0j (sina0/sina*) (6.104)
Substituting equation (6.104) into equation (6.102) and
solving for Co;J in terms of the centrifugal force
Co;)cosa0 + Co;J (sindo/sina^JcoscLf = 0 (6.105)
Fcj [cosdo + (sincXoCOsa^J/sincL,.] = 0 (6.106)
FCj ~ COJ [cosa0sinaLe + sinOoCosa^]/sina* =0 (6.107)
C0j = Fcj [sina*/ (sin(a^ + a0)] (6.108)
Substituting then equation (6.108) into equation
(6.104)
C£d = Fcj [sina0/ (sinfa* + aD)] (6.109)
When all the axial components of the centrifugal force
reaction on the outer race are added, they act as a thrust
load which tends to reduce the preload. The magnitude of
this induced thrust load is


3
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,


150
absolute load magnitude. The observed signal indicated that
the constant preload mechanism worked effectively, avoiding
drastic changes in load during transients.
The loads measured using the PROMESS sensor were
relatively low for the configurations considering the type
of bearing used. This leads into the conclusion that for
the configurations investigated and the lubrications schemes
used, thermal differential expansions are not a problem.
Performance of the Seals
As it was mentioned in Chapter III, each spindle
configuration was provided with non-contact labyrinth seals.
In general, the seals performed as designed when operated
below 8,000 rpm. Once over this speed, oil mist was
produced by small oil droplets which escaped the spindle
housing. These oil droplets where then changed into mist by
the spindle induced whirl surrounding the housing.
To avoid the mist, the oil flow rate could be reduced,
or an air seal could be added to the labyrinth seal, or the
oil could be removed from the housing more efficiently. At
the high spindle speeds at which the mist occurs, reduction
of the oil flow rate would produce an increase in bearing
temperature, this is not acceptable.
An air seal was tested to improve the labyrinth seal in
Configuration II. The air flow which was required to
produce effective sealing action affected the lubricant flow
through the high speed bearing. The air flowing into the


79
25
20
g 15
i
TJ
0)
+>
10
I
O 1.5 LPM
* 2.3 LPM d 3.0 LPM
. 3.8 LPM
*
.4
//
/Z/
-
b
\
\
i .-j
0^
i 1 1 1 1 i i
0 1000 2000 3000 4000 5000 8000 7000 8000 9000 10000
Spindle Speed RPM
Figure 4.3 Experimentally determined Heat Generation
Configuration I


149
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 at 1600 N, while at the
Hydra-RibTM the belt tension component was 5200 N and an
axial component of 400 N due to the weight of the spindle.
Bearing Thermal Loads
The loads on the bearings were monitored using the
PROMESS sensor. The PROMESS sensor provides a signal
proportional to the load on the bearing elements. By
monitoring this signal, the load on each bearing can be
measured during the operation of the spindle. Therefore,
the effects of speed and thermal expansions can be
determined during transients and during steady state
operation of the spindle.
The loads on Configuration I during one of the tests is
shown in Figure 5.27. The increase in load due to
differential thermal expansions can be clearly observed in
the figure. As the test progresses, the thermally induced
load increases, peaks and starts to decrease as the housing
starts to expand. Finally, only the speed induced loads
remain.
The PROMESS sensor control box for Configuration II was
damaged so the controller for Configuration I was used
instead. This provided a signal which increased with
bearing load although not reliably enough to talk about


13
function of the bearing diameter, the kinematic viscosity of
the lubricant, the lubrication method, and the rotational
speed.
= 9.79xl0_sfo(u*n)2/3dm3 (N-mm) (2.8)
when y*n > 2000
Mv = 1.59 x 10-5/odm3
where
(N-mm) (2.9)
when y*n < 2000
Mv: Viscous torque (N-mm).
f0: Viscous torque factor for circulating oil
lubrication [11]:
Angular Contact Ball Bearings (2 rows) fo=8.0
Tapered Roller Bearings f0= 8.0
Cylindrical Roller Bearings (1 row) f0 = 6.0
y.: Kinematic viscosity (cS).
n: Rotational speed (rpm).
The heat generation rate is then the sum of the two
torques Mr and Mv 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).
Q£ = 1.05 x 10"4 n M (W) (2.10)
where
Q£: Heat generation rate (W).
M: Total opposing torque + Mv (N-mm)


199
Recommendations
Design Modifications for Configuration I
If Configuration I is to be used above 6,000 rpm, the
lubrication/cooling system must be modified. The
modification must reduce the operating temperatures and
decrease the hydraulic power losses. It is important to
note that circulating oil lubrication provides an excess of
lubricant to the bearing. Thus, reducing the amount of
lubricant to improve the operating behavior does not affect
the basic lubrication needs of the bearing. As it was shown
earlier, increasing the oil flow rate produces both a
decrease in the bearing's operating temperature and an
increase in hydraulic power losses. An oil flow rate of 2.3
lpm was recommended above, since it would provide the best
combination of operating temperatures and power losses.
Supplying oil directly to the elements would improve
the cooling efficiency. Currently, all the oil is supplied
through the outer race of the bearings, between the rows of
elements. As the oil circulates through the bearing cavity,
its temperature increases and it induces hydraulic power
losses. If the oil is supplied directly from the source to
the rolling elements, the oil reaching them would be much
cooler than if it is first forced through the bearing.
Since this oil would be splashed away from the elements, the
hydraulic power losses would be reduced, decreasing the
temperature even further.


9
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].
Mr = 0.782 fx PB d (N-mm) (2.1)
where
Mr: Load torque (N-mm).
fx: Load torque factor.
Pa: 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 fx for most bearing
types. For ball bearings the factor fx is given by
where
P0: Static load (N).
C0: 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 fx was also determined
experimentally. The value of fx for several types of roller
bearings is given in Table 2.2, Iso from [11].


75
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 (Su)-5 (Fq)1/3 (N-m) (4.11)
where
M: bearing operating torque (N-m)
ka.: 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)
U: oil viscosity (Centipoise)
F.^: equivalent axial load (N)
if the bearing is under combined loading, the equivalent


21
' <=> j
U
i3
=
V^o voj
E' R0
Uo Uj-j
2E Ri
y-o uo3
2E' R0
(2.29)
(2.30)
(2.31)
4
I a. j 2
Gqij[1-(y/4qij)2]
SU/2
dy (2.32)
Ioj = 2
Gqoj [ 1- (y/4qoj ) 2 ]1/2
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
iteration.
The model provides a method to solve for the cage speed
wc. If there is skidding, the cage speed will be below the
expected value of
w0 = 1/2 w(l- Dw/dm) (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 we
and roller rotational speed ww;),
cage load on the roller F^,


125
Figure 5.16 Bearing Temperature versus Spindle Speed
Configuration II (Overall)


Temperature
106
Figure 5.4 Transient Temperatures
Configuration I: 7,000 RPM


178
Using equation (6.25), the axial reactions F** and FxB
can be expressed as
z
~ E F Bj A
j=l
z
sina*.
(N)
(6.70)
F XB
E F0jB
sinos
(N)
(6.71)
j=l
The radial loads on the bearings, given equations
(6.67) and (6.68) can be expressed using equation (6.48) as
z
Fza = E Fe:)A cosaA cos0jA = Fz (a+b)/b (N) (6.72)
j=l
z
Fxb = E Fe;)B cosOb cos0jB = -Fz (a/b) (N) (6.73)
j=l
Equations (6.70) to (6.73) can be rewritten in terras of
the external loads FxA, FXB, and Fz. The elements forces
can be substitutded for their deflections using equations
(112) and (134), such that we are left with a system of
three equations and three unknowns. The three unknown
variables are 6X, and 6a.B. The three equations are
z
E KiS^cosaOj^cosa.* +6xsinaA+. SP^)*3 sinaA -
j=l
z
EK(6zBcosa0jBcoso^ +6JtsinaB+.5?^)^ sino** = 0 (6.74)
j=l
z
EK(SucosajAcosaA +6xsinaA+.SP^)'3 cosaA cos0jA =
j=l
F* (a+b)/b (6.75)
z
EK( Sa-sCosaOjeCosctB +6xsinoLB+. SP^)*3 coscta cos0jB =
j=l
-Fz (a/b)
(6.76)


47
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-
RIBtm 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.


62
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.


158
or
F = kto 63/2
(N)
(6.2)
where
F: normal force between rolling element and
raceway (N)
6: deformation (mm)
6*: dimensionless contact deformation, function of
the curvature difference
curvature sum (mm-x)
kto: load deflection coefficient for a contact
between a ball and one ring (N/mm1*5)
The dimensionless deflection 6*, is given by Harris, in
a set of graphs, as functions of F( ). These graphs were
computed using elliptical integrals. Hamrock and Dowson
[36] provide a simplified method for the computation of the
elliptical integrals and the contact deflection. The method
was developed by Brewe and Hamrock in 1977 by using least
square regression analysis of the relationship between the
elliptical integrals and the curvature difference. Using
this method, the deflection is given by
(mm)
(6.3)
where
(6.4)
0.6360
k' = 1.0339
(6.5)


74
Recalling equation (2.2), for ball bearings, the factor
fx is given by
\ C0 /
for angular contact ball bearings, z=0.0001 and y=0.33 [11].
For roller bearings, fx will be
for cylindrical roller bearings:
fx= 0.0002-0.0003
for tapered roller bearings:
^=0.0003-0.0004
F a for ball bearings is given by the following
equations, also from [11].
Fb = 0.9Fa etna 0.1F*. (4.7)
or
Fe = Fr (4.8)
whichever is larger, (4.7) or (4.8).
For radial roller bearings, FB is given below as
Fa = 0.8Fa ctn a (4.9)
or
Fb = Fr (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


17
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
roller, while y and z subscripts indicate horizontal and
vertical components respectively. Thus, the load ,
indicates a vertical load, acting on the outer race contact
of the roller. The loads Q*OJ and Qzj are the
reactions to external applied loads acting on the
roller. Load is caused by the cage acting on the
roller. Loads QyoJ and 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 F;). 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 Fw. 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


Figure 6.1 Loads on a Radial Bearing


51
Figure 3.5 High Speed Bearing with Cone-Rib Lubrication


85
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,


64
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, 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.


184
Figure 6.6 High Speed Loads on a Cylindrical Roller


171
For any given set of external loads, the above two
equations (6.50) and (6.51) roust be solved for and 6r.
Using the Newton-Raphson method discussed above, both 6a and
6*. can be determined. In this case, we have two p variables
and two f functions. The functions are obtained from
equations (6.50) and (6.51).
z
fi(Pm)=Fa-E K(6asin8j +6rcos0jcosa 0.5 P^J^sina (6.52)
j=l
z
f2(Pm)=Fr-EK(6asinj+6ircos9jcosa-0.5P j=l
The derivatives of these equations, f^ with respect to
6a and f2 with respect to 6*. are
z
f'x(Pm) = q EK (6a sin9j + cos9jcosa
j=l
- 0.5 P^ J*3 1 sinasin8j (6.54)
z
f'2(Pm) = q E K(6aisin9j + cos9jcosa
j=l
- 0.5 Pa)3 1 (cosacos8j)2 (6.55)
The equations used to determine the deflections are
then
6a(m+l)=6a(m) + f*(6a(m))/f1x(6(m)) m=0,l,2... (6.56)
6ir(m+l)=6x.(m) + fxifi^m) )/f ,3.(6r(m) ) m=0,l,2... (6.57)
In the Appendix, there is also a list of the program
developed for computing the axial and radial deflection for
a given set of external loads. Again, the results of the
program were compared with examples from reference [30], the
agreement was good.


153
the upper row was eliminated due to the high speed rotation,
this produced the rolling elements to skid. Once the balls
started skidding, the surface was damaged and the
temperature increased to the point where the rolling
elements welded themselves to the raceways. The bearing
manufacturer (SKF) confirmed this conclusion upon inspection
of the bearing.
To limit skidding at high speeds, the preload must be
increased. To increase the preload, the separating ring
between the two rows of balls must be ground. The increase
in preload will be proportional to the reduction in distance
between the rows. On the other hand, an increase in bearing
load would increase the heat generation rates, as shown in
Chapter IV. Larger heat generation rates would limit the
operating speed to less than 8,000 rpm, since at that speed
the operating temperature was already too high. Therefore,
increasing the preload would eliminate skidding, but would
increase the operating temperature above the safe limit.
To reduce the operating temperature, the lubrication
system should be modified. Oil jets should be added to
provide lubrication to the raceways directly instead of
using exclusively the holes in the outer ring. It could be
argued that the lubrication value of this scheme is minimal,
since the oil would be splashed away from the balls. On the
other hand, the objective of these jets is to provide
cooling to the elements (balls, raceways and cage) not
lubrication. From the test data, it was concluded that


4
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)
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


12
The equivalent load Pa 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 Pa. For radial roller bearings,
Pa is given by equation (2.5) or equation (2.6), whichever
is larger. Equation (2.7) estimates the value of Pa for
thrust bearings (ball or roller).
Pa = 0.9 Fa cot(a) 0.1 Fr (N) (2.3)
PB = Fr (N) (2.4)
Pa = 0.8 Fa cot(a) (N) (2.5)
PB = Fr (N) (2.6)
PB = F. (N) (2.7)
where
Fa: Axial load (N).
a: Contact angle ().
Fri 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


CHAPTER V
EXPERIMENTAL RESULTS AND DISCUSSION
Test Procedure
The spindles were tested for performance, power losses,
operating temperatures and maximum operation speed. The
tests were run at idle, no load from a cutting process, on
the spindle. The spindle was started at low speeds (300
rpm), the desired oil flow rate was pumped into the spindle
and the initial bearing temperatures and loads were
recorded. The spindle was then accelerated to the
predetermined test speed. The test speed and oil flow rates
were selected based on the performance of the spindle during
previous tests.
During the transient stages, measurements were taken
every minute. As the system stabilized, the measurements
were taken at longer intervals, according to the changes in
the test parameters. The parameters recorded during the
test were:
1. Inlet Oil Bulk Temperature.
2. Outlet Oil Bulk Temperature.
3. Oil Temperature at the exit from each Bearing.
4. Outer Race Bearing Temperature.
5. Oil Flow Rate per Bearing.
6. Power imputed to the Spindle Drive System.
97


53
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-RIB-tm. 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-RIBtm. 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.


59
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,


APPENDIX
COMPUTER PROGRAM LISTINGS


135
Figure 5.18 Mechanical Power Losses
Configurations I and II


54
0IL\
1
BEARING
SI
7
Figure 3.6 Oil Supply to Double Row Bearings


40
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
performance.
The bearing tested had a bore diameter of 120.6 mm, an
outside diameter of 206.4 mm, a cup angle of 34, 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


Radial Load Computation Program
10 REM RADIAL LOAD COMPUTATION PROGRAM
20 DIM F(20),DR(50),DTH(20)
30 PRINT "USE CONSISTENT UNITS":INPUT "EXTERNAL RADIAL FORCE
(N/LB)";FR
40 INPUT "DIAMETRAL CLEARANCE (mm/in)";PD
50 INPUT"ELEMENT'S LOAD-DEFLE. CONST (n/mm or lb/in)";KN
60 INPUT "NUMBER OF ELEMENTS";Z:INPUT "EXPONENT Q ";Q
70 INPUT "RESULTS ON SCREEN OR PRINTER (S/P)";X$
80 IF X$="S" THEN GOTO 120
90 LPRINT "RADIAL FORCE";FR:LPRINT"DIAMETRAL CLEARANCE ";PD
100 LPRINT"LOAD DEFLECTION CONSTANT";KN:LPRINT"NUMBER OF
BALLS";Z
110 LPRINT "EXPONENT Q";Q:GOTO90
120 PRINT "RADIAL FORCE";FR:PRINT"DIAMETRAL CLEARANCE ";PD
130 PRINT"LOAD DEFLECTION CONSTANT";KN:PRINT"NUMBER OF
BALLS";Z
140 PRINT "EXPONENT Q";Q
150 J=-l:E=360/Z:DR(0)=.001
160 J=J+1:A=0:C=0
170 FOR 1=0 TO Z-l:B= COS(3.1415926#*I*E/180)
180 DTH(I)= (DR(J)*B-.5*PD):IF DTH(I)<0 THEN DTH(I}=0
190 F(I)=KN*(DTH(I)~Q):A=(F(I)*B)+A
200 C=(Q*KN*(DTH(I))*(Q-l))*B~2+C:NEXT I
210 T=(FR-A)/C :IF T=0 THEN 260
220 DR(J+l)=DR(J)+T:V=ABS((DR(J+l)-DR(J))/DR(J+l))
230 IF V<.001 THEN 250:GOTO 160
240 GOTO 160
250 IF X$="S" THEN GOTO 320
260 LPRINT"RESULTS":LPRINT" BALL DEFLECTION
FORCE "
270 FOR L=0 TO Z-1:M=L*E
280 LPRINT USING "########.#####";M,DTH(L),F(L)
290 NEXT L
300 LPRINT"RADIAL DEFLECTION ",USING "##.########";DR(J)
310 LPRINT "ITERATION ",USING "##.";J:GOTO 380
320 PRINT"RESULTS":PRINT" BALL DEFLECTION
FORCE "
330 FOR L=0 TO Z-1:M=L*E
340 PRINT USING "########.#####";M,DTH(L),F(L)
350 NEXT L
360 PRINT"RADIAL DEFLECTION ",USING "##.########";DR(J)
370 PRINT "ITERATION ",USING "##.";J
380 END
204


145
0.0 0.5 1.0 1.5 2.0 2.5
Oil Flow Rote LPM
3.0
3.5 4.0
Figure 5.24 Hydraulic Power Losses versus Oil Flow Rate
Configuration I (Overall)


185
Recalling equation (6.12), the load deflection
relationship for a line contact is
F = kr fi1-13- (N)
(6.12)
Substituting equation (6.12) into equation (6.90)
k^3-*13- k^3--3-3- Fe = 0
(6.91)
Dividing by kr
6o,1*13- 6^-^ Fc/kr = 0
(6.92)
The total radial 6*,-, approach is the
sum of the
approach at each one of the two contacts.
= 6o;) + (mm)
(6.93)
Expressing 6o;J and substituting equation
(6.93) into
equation (6.92)
(6 + 6*.-,)1-3-3- -6i*3-*3-1 F0/kr = 0
(6.94)
Recalling equation (6.22), the deflection
of a roller
at the angular position is
6Sj = 6rcos8j 0.5 P
(6.22)
Substituting equation (6.22) into equation (6.93), and
this into equation (6.94) we obtain
(6rcos0j P^/2 S^)3-*3-3- Si-,1-11 Fa/kr = 0 (6.95)
The sum of all the radial loads at each inner race must
equal the external applied radial load, assuming it is
applied through the shaft. Recalling equation (6.25)
z
Fr = £ kr ej3--3-3- cosQj (N)
3=1
(6.25)
For z rolling elements, we can now solve for 6i;J and 6r
since we have z+1 equations available. We obtain z equations


56
Sensor
Figure 3.7 PROMESS Sensor


138
Table 5.
Power Losses
5 Curve Fit Constants for Hydraulic
Speed Relationships for Configuration I
Oil
Flow
2
Rate
a5
a6
R
R
0.8
1pm




1.5
1pm
3.6
E-8
2.1
0.98
0.99
2.3
1pm
5.0
E-6
1.6
0.95
0.97
3.0
1pm
2.6
E-4
1.2
0.97
0.98
3.8
1pm
1.9
E-5
1.5
0.99
0.99


186
by expanding equation (6.94), letting j=l,2...z. Equation
(6.24) is the z+1 equation. The equations can be expressed
in terras of 6i;J
( (6rcos6j P^/2 Si-j)1*11 Fc/kr)*/:LO = 6i;) (6.96)
A vector of initial approximations of 6^ is assumed
with an initial value for 6,.. A new vector of 6i is
computed using equation (6.96). To compute a new
approximation for 6r, equation (6.26) is used in conjunction
with the Newton-Raphson method described above. First we
determine the difference between the external applied radial
load and the radial load computed using the deflections
z
D = Fr E kr cosej (6.97)
j=l
The partial derivative of (6.97) with respect to 6;) is
z
DP = 1.11 E kr6ej *lxcos0j (6.98)
j=l
The new approximation of 6r is given by
D
&rnew = rola + (6.99)
DP
By using this method, a solution for the high speed
loading computations for cylindrical roller bearings was
obtained. Once the computation of bearing loads at high
speeds can be achieved, the effect of preload, radial loads,
spindle speed and centrifugal loads on bearing deflections
can be investigated. Computations were performed for a
variety of operating conditions and the results indicate
that at the operating speeds used in these tests the effect


126
Figure 5.17 Bearing Temperature versus Oil Flow Rate
Configuration II (Overall)


Table 5.2 Curve Fit Constants for Temperature Speed
Re]
Lationships
for Confie
furation II
Oil Flow
~T~
Rate
ax
a2
R
R
0.8 1pm
0.67
0.52
0.91
0.95
1.5 1pm
0.78
0.49
0.98
0.99
2.3 1pm
0.01
0.76
1.00
1.00
3.0 1pm
1.28
0.42
0.90
0.95
3.8 1pm
0.90
0.15
1.00
- 1.00


Temperature
105
Figure 5.3 Transient Temperatures
Configuration I: 6,000 RPM


201
reaching the lower seals. As currently designed, the excess
oil from the top bearing plus all the oil from the lower
bearing falls into the lower seal straining its capacity.
The above two measures would lessen the amount of oil
reaching the lower bearing, reducing drastically the amount
of mist being produced.
To improve the survivability of the tapered bearings,
the rib supporting the back of the bearing should be
provided in the outer race. Currently, the rib is located
at the back of the inner race. Due to the inherent oil flow
patterns of the tapered bearings, the contact between the
back of the bearing and this rib does not receive the
necessary lubrication. Cone-rib lubrication was developed
to solve this particular problem. Changing the rib to the
outer race would use the inherent flow patterns to
lubricate the cone-rib contact. Tapered bearing
manufacturers have in stock a bearing with such a design.
This bearing is not manufactured with the precision levels
required for machine tools. Bearing manufacturers should
provide a sample of these bearings to researchers to
investigate their high speed performance.
Final Comment
Two spindle configurations based on different types of
roller bearings were tested for High Speed, High Power
performance. Although the target speeds were not met, their


14
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.


91
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
6To = 10.6 x 10-6*(0.158) *ti* ( 340-295 ) (m)
6To = 2.37x 10~4 (m)
This 6To 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 4 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


139
Power
Table 5.
Losses
6 Curve Fit Constants for Hydraulic
Speed Relationships for Configuration II
Oil
Flow
Rate
a5
a6
R2
R
0.8
1pm
1.8
E-5
1.4
0.92
0.96
1.5
1pm
2.3
E-5
1.4
0.94
0.97
2.3
1pm
1.4
E-8
2.2
0.94
0.97
3.0
1pm
3.3
E-4
1.1
1.00
1.00
3.8
1pm
5.6
E-4
1.0
1.00
1.00


27
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 translatory motion is given [14] as
= F0j Fi:j (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
= f 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
Fchurn= 1/2 ^>CDSV02 (2.47)
where
^ : Effective density of the mixture=%oil oil
CD: Drag Coefficient
S: Effective Drag Area (mm2)
Vc: cage orbital velocity (m/sec.)


131
The value of a5 and a6 for each configuration, for
constant oil flow rates, are tabulated in Table 5.5 and
Table 5.6. The approximation functions indicate that
hydraulic power losses increase as spindle speed increases.
This is shown graphically in Figures 5.19 and 5.20. As it
can be seen from the figures, at 8,000 rpm and 3.8 1pm the
hydraulic power losses reach 14 kW for Configuration I. If
the mechanical power losses are added, the total power
losses amount to 20 kW. This is too large to be acceptable.
Thus, Configuration I is not recommended for use at these
high spindle speeds.
Figures 5.19 and 5.20 show that the power losses of
Configuration II, are lower than those of Configuration I.
The steady state hydraulic power losses for Configuration II
at 9,000 rpm and 2.3 1pm are 9.7 kW. This is almost 2 kW
lower then the power losses for Configuration I at 8,000 rpm
and the same oil flow rate. In Configuration II the effect
of speed on the hydraulic power losses is quite strong, as
demonstrated in the figures.
Hydraulic Power Losses Versus Oil Flow Rate
The effect of oil flow rate on hydraulic power losses
is shown in Figures 5.21 and 5.22 on Configuration I and
Configuration II, respectively for constant spindle speed.
The approximation functions were derived as before and are
in the form



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32
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


89
5. The temperature of the environment was assumed
constant at 22C.
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
6Ti = r di it
(T T.)
(m)
(4.20)
6to = r dQ ti
(T0 T)
(m)
(4.21)
= r dhi
Tt (Thl T.)
(m)
(4.22)
where
6t: thermal expansion of the inner ring (m)


132
P* = a7 vaS (C) (5.9)
where
Ph: Hydraulic Power Losses (kW)
v: Oil Flow Rate (1pm)
a7: Curve fit constant one (kW/lpmaB)
a8: Curve fit constant two
The corresponding values for a7 and a8 are tabulated in
Table 5.7 and Table 5.8 for constant spindle speed. For
Configuration I and II, hydraulic power losses increase with
oil flow rate. This is shown graphically in Figures 5.21
and 5.22. The figures also show that the power losses of
Configuration II are lower than those of Configuration I.
Configuration II is very insensitive to increases in oil
flow rate, this is due to the pumping action inherent in
tapered roller bearings, which facilitate the flow of oil
through the bearings.
Overall Hydraulic Power Losses Equation
As before, a general approximation equation was derived
which combines the effects of spindle speed and oil flow
rate on hydraulic power losses. These general approximation
equations are given as
for Configuration I:
Ph = 0.0000030 v-82 n1-6 (kW) (5.10)
for 3,000 < n < 8,000 rpm


61
The heat removed from the oil in the heat exchanger is
given by
QollHE
= m nc(Tln Tout) (kW) (3.2)
where
Qol1HE
Heat removed from the oil (kW).
m:
V
Oil flow rate (1pm).
Oil density (g/ml).
c:
Heat capacity of the oil (kJ/(kg C)).
Tj.:
Oil temperature entering the heat
exchanger (C).
Tout
Oil temperature exiting the heat
exchanger (C).
The heat removed by the water in the heat exchanger is
given by
QwanarHE m ^ C ( Tout Tj_n ) ( KW) (3.3)
where
QWat.rHE: Heat removed by the water from the oil
(kW).
m:
r
Cooling water flow rate (1pm).
Density of the water (g/ml).
c:
Heat capacity of the water (kJ/(kg C)).
Tn:
Water temperature entering the heat
exchanger (C).
Tout
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


Table 5.3 Curve Fit Constants for Temperature Oil
Flow Rate Relationships for Configuration I
Spindle
2
Speed
a3
a*
R
R
3000 rpm
53.10
-0.24
0.99
0.99
5000 rpm
75.60
-0.26
0.87
0.93
6000 rpm
79.16
-0.23
0.97
0.98
7000 rpm
93.36
-0.23
1.00
1.00
8000 rpm
85.75
0.00
1.00
1.00


Figure 5.22 Hydraulic Power Losses versus Oil Flow Rate
Configuration II


38
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


Temperoture
109
Figure 5.7 Transient Temperatures
Configuration II: 5,000 RPM


206
Load Deflection Computation Program
10 REM RADIAL LOAD-DEFLECTION BEHAVIOR COMPUTATION PROGRAM
20 DIM F(20),DR(50),DTH(20),FF(50),DDR(50)
30 INPUT"INPUT BEARING SERIES";BEA$
40 PRINT "USE CONSISTENT UNITS"
50 INPUT "INITIAL EXTERNAL RADIAL FORCE (N/LB) AND
INCREMENT";FR,INC
60 INPUT "FINAL RADIAL FORCE (N/LB) ";FRF:FIN=INT((FRF-
FR)/INC)+1
70 INPUT"HOW MANY ROWS OF ELEMENTS";NROW
80 IF FIN>50 THEN GOTO 420: INPUT"HOW MANY ROWS OF
ELEMENTS";NROW
90 INPUT "DIAMETRAL CLEARANCE (mm/in)";PD
100 INPUT"ELEMENT'S LOAD-DEFLE. CONST (n/mm or lb/in)";KN
110 INPUT "NUMBER OF ELEMENTS/ROW";Z:INPUT "EXPONENT Q ";Q
120 INPUT "RESULTS ON SCREEN OR PRINTER (S/P)";X$
130 IF X$="S" THEN GOTO 180
140 LPRINT "BEARING TYPE ";BEA$
150 LPRINT "INITIAL RADIAL FORCE";FR:LPRINT"DIAMETRAL
CLEARANCE ";PD
160 LPRINT"LOAD DEFLECTION CONSTANT";KN:LPRINT"NUMBER OF
BALLS";Z
170 LPRINT "EXPONENT Q";Q:LPRINT "ROWS";NROW:Jl=0:GOTO 230
180 PRINT "BEARING TYPE ";BEA$
190 PRINT "INITIAL RADIAL FORCE";FR:PRINT"DIAMETRAL
CLEARANCE ";PD
200 PRINT"LOAD DEFLECTION CONSTANT";KN:PRINT"NUMBER OF
BALLS";Z
210 PRINT "EXPONENT Q";Q:J1=0 :PRINT "NUMBER OF ROWS ";NROW
220 PRINT :PRINT :PRINT:PRINT:PRINT
230 J=-l:E=360/Z:DR(0)=.001:FR=FR/NROW
240 J=J+1:A=0:C=0
250 FOR 1=0 TO Z-1:B= COS(3.1415926#*I*E/180)
260 DTH(I)= (DR(J)*B-.5*PD):IF DTH(I)<0 THEN DTH(I)=0
270 F(I)=KN*(DTH(I)*Q):A=(F(I)*B)+A
280 C=(Q*KN*(DTH(I))(Q-1))*B~2+C:NEXT I
290 T=(FR-A)/C :IF T=0 THEN 340
300 DR(J+l)=DR(J) +T
310 V=ABS((DR(J+l)-DR(J))/DR(J+l))
320 IF V<.001 THEN 340
330 GOTO 240
340 FF(J1)=FR:FR=FR+INC:DDR(J1)=DR(J)
350 J1=J1+1:IF J1 360 IF X$="S" THEN GOTO 390
370 FOR L=0 TO FIN-1:LPRINT USING"#####.######";FF(L),DDR(L)
380 NEXT L:GOTO 410
390 FOR L=0 TO FIN-1:PRINT USING"#####.######";FF(L),DDR(L)
400 NEXT L
410 END
420 PRINT "CHANGE THE DIMENSION OF FF AND DDR":STOP


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
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
in


E: Modulus of elasticity (N/m2)
a: Poisson's Ratio.
20
The lubricant induced loads Fi;J,F0jf Qyo;) and are
given next, in dimensionless form, as presented by Harris in
[12].
Fi_-j = -9.2G~ 3 Ui;10-7 + (2.22)
FOJ = -9.2G_ 3 UOJ'7 + (2.23)
Ho3
Qyoj = 18.4 (1-x)G 3 U3-7 (2.24)
= 18.4 (l-x)G-3 U;10-7 (2.25)
where
G = aE'
a: Is the pressure coefficient of viscosity
(mm2/N).
G.e Ui;)o.7
Hid = 1.6 (2.26)
n o-i3
Vzij
G 6 U0-,-7
Hoj = 1.6 (2.27)
Q.oj0*13
Uo
=
E' R
(2.28)


73
@ 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 Mo and a
load torque Mx. 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
M0 = foxl0_8(vn)2/3dm3 vn>= 2000 (4.3)
M0 = foXl60xl0_8dra3 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 4-6
for TRB 8-10
v: oil viscosity at working temperature (cS)
U,: mean diameter of the bearing (mm)
The friction torque due to the applied load can be
computed using an equation recommended by Palmgren, [11].
Mo. = fx Fed (N-mm) (4.5)
where
Mx: friction torque due to the load (N-mm)
fx: factor dependent on the geometry of the
bearing and relative load.
Fa: equivalent force, as described below (N)
d: mean bearing diameter (mm)


81
Figure 4.5 Computed Heat Generation
Configuration II


39
operating temperature, for the same oil flow rate. It was
interesting to find that when the oil flow rate was
increased over 3.8xl0~3 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.


34
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 = xw A r (N-m) (2.51)
where
T: Drag torque acting over the element surface
(N-m).
tw: 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).
xw = / (1/2 U2 ) (N/m2) (2.52)
where
fz 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).
TnA = 0.5 w2 rs Cn (N-m) (2.53)
where
Tend: Drag torque acting on the end of the roller
(N-m)


162
Now that the load deflection characteristics for a
single rolling element is known, the load deflection
behavior of a complete bearing will be consider.
Radial Loads
Let's consider now radial loads on a radial bearing.
Radial bearings are those bearings design to support only-
radial loads. Two of the most common radial bearings are
deep groove ball bearings and cylindrical roller bearings.
Analyzing a radial bearing as the one in Figure 6.1, the
deflection at the 8jtl1 angular position is given by
= 6*. cos 0j 0.5*Pd (mm) (6.22)
where
] 1,2,... z
z: number of rolling elements
Pd: diametral clearance (mm)
The force at angular position 0j is given by
Foj = K (N) (6.23)
For equilibrium to be satisfied, the magnitude of the
sum of the internal radial components must add to the same
magnitude as the applied external radial load or
z
Fr = E Fe;} cos 0j (N)
j=l
(6.24)


148
Bearing Loads
Externally Applied Load
The only externally applied load acting on the bearings
during the investigation was induced by the tension of the
driving belt on the spindle. The magnitude of the belt
tension may be computed using equations provided by the belt
manufacturer for each belt type and the dimension of the
pulleys. Two different belt types were used, HABASITtm
types A-5 (Configuration I) and S-5 (Configuration II).
Both are rated to operate at 150 hp. and 10,000 rpm. The S-
5 type belt is recommended to operate with less noise, while
type A-5 is recommended for heavy shock loads. Both belts
operated satisfactory under the idle test conditions. During
the cutting test, the belts slipped when heavy cuts were
attempted. Belt type S-5 was considerably quieter than type
A-5.
The load on the bearings was computed from data and
equations supplied by the belt manufacturer [31]. The
manufacturer uses a combination of empirical factors based
on the belt type, operating speed, and pulley diameter. For
each one of these parameters, an empirical factor is
provided in [31]. Using this publication [31], the belt
tension load induced on Configuration I was computed at 8600
N while for Configuration II it was computed at 3600 N. As
it was mentioned in Chapter IV, the individual bearing loads
are on the lower bearing (NN3022 K) 3000 N, for the top
bearing (NN3019 K) 12000 N and for the center bearing, the


24
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
fpFdj. Therefore, rewriting equations (2.13) and (2.14) to
include this term equations (2.35) and (2.36) are obtained.
R0
_ (Qz3 + Fw fjpF^-j) Qzoj =0 (2.35)
Rd.
Ro
Qvi-d + (0^3 F0j F^) =0 (2.36)
R
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


128
Friction in Bearings section in Chapter IV. To determine
pme, the spindle is driven to test speed without circulating
oil. The input power is recorded. To calculate Pmo, we
subtract Pmoto*- from the input power. The difference is
the mechanical power losses. In equation form
Pm= PniPUt (kW) (5.5)
where
Plnput: Power imputed to the D.C. motor at any
set of operating conditions (kW)
Pmotor* Motor power losses (kW)
Pm: Mechanical power losses (kW)
Hydraulic Power Losses
The hydraulic losses are caused by viscous friction on
the rolling elements by the circulating oil. These losses
will depend on the type of oil, the oil flow rate, spindle
speed and bearing geometry. To measure the hydraulic power
losses Ph, the spindle is driven to the test speed and oil
flow rate. The input power is recorded. From the input
power, the mechanical and motor power losses are subtracted.
The difference is the hydraulic power losses Ph, for the
given speed and oil flow rate.
Since PmB and Ph are characteristic for each
configuration, they can be used as comparison parameters.
For any configuration, the lower the power losses are the


113
Figure 5.11 Bearing Temperature versus Spindle Speed
Configuration II


series. The G factor can be obtained from a tapered bearing
catalog or computed according to the equation given by Witte
in [15] as
29
Dx *3
G = (2.48)
D~0-17(nl)-2/3 (sin a)l/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 = 1.1x104 G Fa (Su)-s ()17 3 (2.49)
where
M: Resisting torque (lb.-in.).
G: Bearing Geometry Factor.
Fa: Axial Load (lb.).
S: Bearing Speed (rpm).
4: Lubricant viscosity at atmospheric pressure
(cP).
4>: Lubricant pressure-viscosity index (in2/lb).
and it is limited for (S4) values larger than 3000 and
for axial loads which are less than twice the axial load
rating of the bearing.


DESIGN OF HIGH SPEED, HIGH POWER SPINDLES
BASED ON ROLLER BEARINGS
By
ISMAEL A. HERNANDEZ-ROSARIO
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


60
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
= m^ c(Tout Tin) (kW) (3.1)
Heat removed by the oil from the bearings
(kW).
Oil flow rate (1pm).
Oil density (g/ml).
Heat capacity of the oil (kJ/(kg C)).
Supply oil temperature (C).
Return oil temperature (C).
QoilB
where
QoilB'
m:
V
c:
Ti:
T
-* out


22
and outer race contact load Qzoj. As it was noted before,
the other loads are non-linear functions of lubricant
properties and roller speed. The inner race contact loads
Qzlj, 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


102
Steady State Temperatures Versus Oil Flow Rate
Figures 5.12 and 5.13 show the change in bearing
temperature for constant speed and as the oil flow rate
varies for Configurations I and II, respectively. Again, a
set of approximation functions were derived to describe the
effect of oil flow rate on bearing temperature. The
equations are in similar form as in equation (5.1)
Tb = a3 Va4 (C) (5.2)
where
Tb: Bearing temperature (C)
v: Oil flow rate (1pm)
a3: Curve fit constant one (C/lpm4)
a4: Curve fit constant two
The value of the constants a3 and a4 are given for each
configuration at each test speed in Table 5.3 and Table 5.4.
As it can be observed from Figure 5.12 and Figure 5.13, the
temperature of the bearing decreases with oil flow rate.
The decrease in bearing temperature with flow rate is to be
expected, since by increasing the oil flow rate, the cooling
capacity increases. Note that as the oil flow rate is
increased the drop in temperature is smaller than with the
previous increase in oil flow rate. This is due to the
increase in power losses that an increase oil flow rate
produces. As it was mentioned before, there is a compromise
oil flow rate at which the bearing temperature is reasonable
while the power losses are not that large.


CHAPTER VI
BEARING LOADS
Load Deflection Relationships
The loads exerted on a bearing are transmitted through
the contacts between the rolling elements and the raceways.
The geometry of the rolling elements and their material
properties determines how well the bearing can transmit the
loads during operation. Heat generation and power losses
are also a function of the contact geometry.
Let's consider a contact between a rolling element and
a raceway. Hertz derived the equations for two bodies in
contact in 1881, references [30,36]. The basic assumptions
made by Hertz in his analysis were:
1. The materials are homogeneous and the yield strength
is not exceeded.
2. No tangential forces are induced between the solids.
3. Contact is limited to a small portion of the
surface.
4. The solids are at steady state.
For a steel ball/raceway contact the load deflection
equation is given by
F = 2.14x10s 6* -2/3 63/2 (N) (6.1)
157


99
The computer program was used to develop relationships
between one of the independent variables, the other was kept
constant, and one of the dependent variables. The geometric
relationship was selected since it was consistently the best
fitting equation to the experimental data. The best fit was
determined by observing the coefficient of determination
(R2) and the correlation coefficient (R) which are part of
the program's output. The determination coefficient and the
correlation coefficient are included with the curve fit
constants in the discussion below.
Temperatures
The temperature of the bearings was recorded at
different oil flow rates and speeds during both steady
states and transient stages. A comparison can be made
between the two configurations based on their operating
temperatures given a spindle speed and and oil flow rate.
The temperatures of the supply oil, return oil and the lower
bearing are plotted for several test speeds versus time, in
Figures 5.1 to 5.9. These are the plots of the transient
behavior of the configurations under investigation. It can
be seen from these graphs the time it takes each
configuration to achieve steady state. The effect of a
different oil flow rate can also be observed from these
graphs.


133
for 1.5 < v < 3.8 1pm
for Configuration II:
= 0.000016 v-15 n1* 4 (kW) (5.11)
for 3,000 < n < 9,000 rpm
for 0.80 < v < 3.8 1pm
Using equations (5.10) Figures 5.23 and 5.24 were
produced for Configuration I. While Figures 5.25 and 5.26
were generated using equation (5.11), for Configuration II.
These figures are similar to Figures 5.18 to 5.22, for the
respective configuration. As it can be seen from the
figures, the agreement of the general equation for P** is not
as good as the general equation for Ts. The general
approximation function tends to predict higher power losses
than those actually measured.
Power Losses: Comparison
Configuration II operated at all times with lower power
losses than Configuration I. The lower power losses for
Configuration II are due in part to the fact that
Configuration II is based on two bearings while
Configuration I is based on three. The geometry of the
bearing has the strongest effect on bearing temperature.
The tapered bearings have an inherent pumping action which
facilitates the flow of oil through the bearings, providing
for better oil flow and heat removal. The bearings in
Configuration I are double row bearings which receive the


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Dr. Charles L. Proctor II
Associate Professor of
Mechanical Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Assistant Professor of
Mechanical Engineering
This dissertation was submitted to the Graduate Faculty
of the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
May 1989
/ 4JLtjjf Cl
Dean, College of Engineering
Dean, Graduate School


Bearing Temperature
Figure 5.12 Bearing Temperature versus Oil Flow Rate
Configuration I


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
IV


92
by equation (4.22) using the diameter of the roller instead
of the element diameter.
6to = 10.6 x 10~s*(0.013) *Tt* ( 340-295 ) (m)
6to = 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 10-5 m.
Using equation (6.12) and a roller stiffness value of 1.00 x
108 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.


161
For symmetrical roller bearings, the deflection at the
inner and outer races will be the same, then equation (6.13)
can be rewritten as
6r = 2*6 (N) (6.17)
where
6r: total approach for each roller considering
both contacts (mm)
Expressing 6 in equation (6.12) and substituting into
equation (6.17)
where
Kh: load deflection coefficient for a roller
considering both approaches (N/mm10/9)
Eliminating the force F from equation (6.18) and
solving for KR
^ \ 10/9
kr =
= 0.463kr (N/mmxo/s) (6.19)
The load deflection relation for rollers is then
F = Kr 6xo/9 (N) (6.20)
Therefore, we can rewrite a general load deflection
equation for either balls or roller elements
F = K (N) (6.21)
where
q: exponent characteristic of the rolling element
= 3/2 (1.5) for balls
= 10/9 (1.11) for rollers


15
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 roller is
given by ww;J. The rotational speed of the cage is given as
wc, 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
Vid = 0.5 (d Dw)(w-wc) 0.5Dwww;J (2.11)
V0j = O.SicL, + Dw)wc 0.5Dwww-j (2.12)
where
Vi;J: Sliding velocity at the inner contact of the
jth rolling element (m/sec).
Voj: Sliding velocity at the outer contact of the
jttl rolling element (m/sec),
d: Bearing pitch diameter (m).
Dw: Roller diameter (m).


114
Table 5.1 Curve Fit Constants for Temperature Speed
Relationships for Configuration I
Oil Flow
2
Rate
a2
R
R
0.8 1pm




1.5 1pm
0.33
0.62
0.96
0.98
2.3 1pm
0.21
0.66
0.98
0.99
3.0 1pm
0.39
0.58
1.00
1.00
3.8 1pm
0.13
0.70
0.98
0.99


Figure 3.8 Circulating Oil System


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 # MEA-8401442 Unmanned Machining, High
Speed Milling.
11


205
Combined Load Computation Program
5 REM COMBINED LOAD COMPUTATION PROGRAM
10 DIM F(4 0),DR(30),DTH(40),DA(30)
20 INPUT "FR,FA,PD,KN,Z, Q";FR,FA,PD,KN, Z ,Q
30 INPUT "CONTACT ANGLE";ALP
40 LPRINT "RADIAL FORCE";FR:LPRINT"AXIAL FORCE ";FA
50 LPRINT "DIAMETRAL CLEARANCE";PD
60 LPRINT"LOAD DEFLECTION CONSTANT";KN:LPRINT"NUMBER OF
BALLS";Z
70 LPRINT "EXPONENT Q";Q
80 LPRINT "CONTACT ANGLE ALPHA ";ALP
90 J=-l;G=COS(3.1412*ALP/180):H=SIN(3.1412*ALP/180)
100 E=360/Z:DR(0)=.001:DA(0)=.001
110 J=J+1
120 A=0:R=0:AD=0:RD=0
130 FOR 1=0 TO Z-l
140 B= COS(3.1412*1*E/180):D=SIN(3.1412*I*E/180)
150 DTH(I)= (DA(J)*D+DR(J)*B*G-.5*PD)
160 IF DTH(I)<0 THEN DTH(I)=0
170 F(I)=KN*(DTH(I)~Q)
180 A=(F(I)*H)+A:AD=KN*Q*(DTH(I)(Q-1))*H*D+AD
190 R=(F(I)*B*G)+R:RD=KN*Q*(DTH(I)(Q-l))*(B*G)2+RD
200 NEXT I
210 T=(FR-R)/RD:V=(FA-A)/AD
220 IF T=0 THEN 250
230 DA(J+l)=DA(J)+V:DR(J+l)=DR(J)+T
240 IF ABS((DR(J+l)-DR(J))/DR(J+l))>.001 THEN 110
250 LPRINT"RESULTS":LPRINT" BALL DEFLECTION
FORCE "
260 FOR L=0 TO Z-1:M=L*E
270 LPRINT USING "########.#####";M,DTH(L),F(L)
280 NEXT L
290 LPRINT"RADIAL DEFLECTION ",USING "##.########";DR(J)
300 LPRINT"AXIAL DEFLECTION ",USING "##.########";DA(J)
310 LPRINT "ITERATION ",USING J
320 STOP


2
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


187
of spindle speed on deflection is negligible, or none. The
effect of radial load and preload are significant as would
be expected. These results were expected based on the
readings from the PROMESS sensor, which showed that the
centrifugal forces on the bearings were quite low. For a
double row cylindrical roller bearing, the measured
centrifugal forces would not cause any significant
deflection.
High Speed Tapered Bearings
The centrifugal loads induced by high speed operation
of TRB are computed first. The gyroscopic moment effect
will then be evaluated with its effects on the distribution
of the load. These will then be combined with the effects
produced by a combined loading situation. It is assumed
that misalignment of the bearing does not occur.
Centrifugal Forces
The loads induced on a TRB by high speed operation are
shown in Figure 6.7. The centrifugal load Fc must be
supported by the outer race contact and the flange contact.
The sum of the axial and radial forces must equal 0.
(6.100)
(6.101)
where
Fcj: Centrifugal force acting on the roller
(N)


Bearing Temperature
112
Figure 5.10 Bearing Temperature versus Spindle Speed
Configuration I


172
Bearing Life Calculation
The life of the bearing can be estimated using the
standard equation [9,33]
10s revolutions
(6.58)
C: dynamic load rating of the bearing (N)
L10: Rated Life (rev.)
P: equivalent load on the bearing (N)
p: 3 for ball bearings
10/3 for roller bearings
The dynamic load rating can be obtain from the
manufacturers catalog [9,33] or can be computed as
C= fc(icos do)0-7 z2/3 D3--8 for D < 25.4 (6.59)
C= fc(icos a0)0-7 z2/3 3.647D1-4 for D > 25.4 (6.60)
where
fo: Dcosdo/dn,
i: number of rows
z: number of elements
D: rolling element diameter (mm)
For radial roller bearings, the equation to use is
C= f<2(i 1.** cos a0)7/9 z3/4 D29/2V (N) (6.61)
where
1B££: effective roller length (mm)


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.
Vll


170
Substituting equation (6.44) into equation (6.23) we
obtain the force for the element at the 0j*** position
Fej = = KlS^sinQj + 6rcos8jcosa l/2Pa)
The axial and the radial components of the force FOJ
are
Fejxiai= Foj sina (N) (6.46)
Fejradial = Fej COS COSO. (N) (6.47)
with the constraint that if 6e;J< 0, then = 0.
For equilibrium to exist, the sum of the magnitudes of
the element forces, axially and radially, must equal the
magnitude of the external axial and radial loads.
z z
Fa = EFojaijeiaLX = EFe-,sina (N) (6.48)
j=l j=l
z z
Fr = E Fe;)racaJ.al = E FejcosGcosa (N) (6.49)
j=l j=l
where
Fa: is the applied external axial load (N)
Fr: is the applied external radial load (N)
Substituting equation (6.45) into equations (6.48) and
(6.49) we obtain the equations needed to determine and
6r.
z
Fa = E K(6asin8j + 6rcos8jcosa 0.5 P^J^sina (6.50)
j=l
Fr E K(6asin8j + 6rCos0jcosa- 0.SP^)^cosacosj
j=l
(6.51)


137
Q 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Spindle Speed RPM
Figure 5.20 Hydraulic Power Losses versus Spindle Speed
Configuration II


195
Experimental Conclusions
Empirical Equations
Equations were derived using the experimental data to
relate operating temperature and power losses to the spindle
speed and the oil flow rate. The equations derived describe
the relationship between: bearing temperature and spindle
speed, (5.1), bearing temperature and oil flow rate, (5.2),
mechanical power losses and spindle speed, (5.6), hydraulic
power losses and spindle speed, (5.8), and hydraulic power
losses and oil flow rate, (5.9). These equations match the
experimental data very well and could be used to estimate
the behavior of the configuration within the test range,
Figures 5.10 to 5.13 and Figures 5.19 to 5.22.
The experimentally derived equations were combined to
obtain general equations which estimate either the operating
temperature or the hydraulic power losses in terms of both
spindle speed and oil flow rate, equations (5.3), (5.4) and
(5.10). The general equation relating spindle speed and oil
flow rate to bearing operating temperature matches
experimental data as good as the individual relationships,
Figures 5.14 to 5.17 and Figures 5.23 to 5.26.
Based on the experimentally developed equations, it can
be concluded that speed has a direct effect on bearing
operating temperature. As the spindle speed increases, so
does the operating temperature of the bearing. The effect
of spindle speed on temperature was much stronger for
Configuration I than for Configuration II. This is due to


129
more efficient is the configuration. The lower the power
losses, the lower the cooling requirements, since power
losses are the source of heat.
Configuration Power Losses
Mechanical Power Losses
The mechanical power losses for each configuration are
shown in Figure 5.18. The equations which describe the
mechanical power losses for each configuration are given
next.
for Configuration I:
Pms = 8.6 x 10"4 n (kW) (5.6)
for Configuration II:
Pme = 5.6 x 10~4 n (kW) (5.7)
where n is spindle speed (rpm).
Since there is no oil being circulated, there is no
dependence on oil flow rate. By comparing equations (5.6)
and (5.7) it should be clear that the mechanical power
losses for Configuration II are lower than for Configuration
I. This is also shown in Figure 5.18.
Hydraulic Power Losses
The effect of spindle speed and oil flow rate on
hydraulic power losses was investigated. The graphs which
will be discussed next represent the power losses for each


159
2
E' = (6.6)
0.5968
E = 1.0003 + Rx (6.7)
Ry
F = 1.5277+ 0.6023 ln(Ry/Rx) (6.8)
d^+Dcosa
Ry = (6.9)
2d
fD
Rx = (6.10)
2f-l
fQ: conformity, r/D, for outer race
fa.: conformity, r/D, for inner race
Ea: Young's Modulus for body a (N/mm2)
Eto: Young's Modulus for body b (N/mm2)
For a steel roller/raceway contacts, the load
deflection equation is given as
F = 7.86x10* 18/9 610/9 (N) (6.11)
or
F = kr 610/9 (N) (6.12)
where
1: roller effective length (mm)
kj.: load deflection coefficient for a contact
between a roller and one ring (N/mm:LO/9)
The total approach of the two raceways for any rolling
element under load is the sum of the approaches at each
element/raceway contact, or


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. Hernndez-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
vi


48
Figure 3.2 Configuration I Test Spindle


96
Figure 4.9 Thermal Profile (9,000 RPM)
Configuration II


164
Substituting equation (6.23) into equation (6.24)
z
Fr = E K So-,'3 cos 6j (N) (6.25)
j=l
Substituting equation (6.22) into equation (6.25)
z
Fr = E KtSr cos 0j 0.5 Palmeos Qj (N) (6.26)
j=l
with the constraint that if 60;1 < 0, then Fe;J = 0.
For any given external radial load F,., equation (6.26)
must be solved for 6,.. With 6r known, the deflection and
load for each rolling element can be computed using equation
(6.23). Rewriting equation (6.26) such that we can solve
for those values of 6*. which will make it equal to zero
Fr E K(6r cos 0j 0.5 P^cos 0j = 0 (6.27)
To solve for 6r, the Newton-Raphson method for
determining the roots of an equation is used. The method is
based on equation (6.28).
f(p*J
= Pm + m=l,2,... (6.28)
f(Pm)
where
pm: mth iteration of the variable p
f(Pm): function evaluated at the mth iteration
f'(Pm): derivative of the function
In the case under consideration, equation (6.27) is our
f function while 6r is our p variable. The derivative of
equation (6.27) with respect to S*. is given by equation
(6.29)
z
f'(Pm) = EqK(6r cos 0j 0.5 P^J^^cos2 0j (6.29)
j=l


I certify that I have read this study and that in my
opinion it conforms to acceptable standard:
presentation and is fully adequate, in scope
a dissertation for the degree of Doctor of PI
of scholarly
nd quality, as
losophy.
<1
*
Dr.
iri Tlusty, Chairman
Graduate Research Professor
of Mechanical Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Dariel C. Drucker
Graduate Research Professor
of Aerospace Engineering,
Mechanics, and Engineering
Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy. ,
Dr. Donald W. Dareing
Professor of Mechanical
Engineering


117
Table 5.4 Curve Fit Constants for Temperature
Oil Flow Rate Relationships for Configuration II
Spindle
Speed
a3 a4
2
R R
3000 rpm
5000 rpm
6000 rpm
7000 rpm
8000 rpm
9000 rpm
42.53
48.49
-0.11
-0.12
(1757
0.84
0.99
- 0.92
59.81
-0.08
1.00
1.00
73.51
-0.10
0.96
0.98


25
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
6-j = 6xsin(t>j + fincse})-, (Gi/2)+hi;J+ho;, (2.37)
6d = 6o:j +6i j (2.38)
where
=
where
(2.39)
(2.40)
h: Oil film thickness given by
8 a-6(u0u)-7 E'3 R*13 lw0-13
h= (2.41)
2 pO.13
6h -0.302 a6(y.0u)-7 E'-3 R-13 lw0'13
= (2.42)
6P P1-13
The deflection behavior was used to determine the inner
race contact loads as
n
Fx = EP-l-j sinj (2.43)
j=l
n
Fy = EPij cos-j (2.44)
j=l
which is the same method used by Harris.


33
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


147
20
16
O 3000 RPM
L 9000 RPM
n
12
o
*
£
u
3
<-
o.o
0.5
X
Q_
5000 RPM
-Ol

A
7000 RPM

A
-O-
.O 1.5 2.0 2.5
Oil Flow Rate LPM
3.0
A
-O
3.5
4.0
Figure 5.26 Hydraulic Power Losses versus Oil Flow Rate
Configuration II (Overall)


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.
8


Hydraulic Power Lasses KW
146
o 0.8 LPM
a i.5 LPM
n 2.3 LPM
* 3.0 LPM
+ 3.8 LPM
i
8 j-
4 r
i
0L
0
n
A
O
i 1 1 1 -i- 1 .--l 1 1 1
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Spindle Speed RPM
Figure 5.25 Hydraulic Power Losses versus Spindle Speed
Configuration II (Overall)


72
bearing, the High Speed Bearing, was estimated as 1600 N,
while at the Hydra-Rib^ 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
Nnn3022k 360 N-rom
^nn3oi9k 1232 N-mm
M234420BP1 = 46 Nmm
for Configuration II
^hydha~rib 489 N~mm
Mhs = 146 N-mm
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


84
Table 4.2 Removed Heat/Generated Heat
Con:
Eiguration 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


123
Figure 5.14 Bearing Temperature versus Spindle Speed
Configuration I (Overall)


90
6To: thermal expansion of the outer ring (m)
6Thi: thermal expansion of the ith housing element
(m)
T: thermal expansion coefficient 10.6xl0-6 C_1
[30]
d*: inner ring diameter (m)
d0: outer ring diameter (m)
d^i.: diameter of the itl1 housing element (m)
Tj.: temperature of the inner ring (C)
T0: temperature of the outer ring (C)
Th: temperature of the itl* 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.


41
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
31, 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 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.


30
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 = 1. lxlO-4 G (Sy.0) 5 (fT F^/K)1'3 (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 (Su) 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


140
Table 5.
Power Losses Oil
7 Curve Fit Constants for Hydraulic
Flow Rate Relationships for Configuration I
Spindle
2
Speed
a7
R
R
3000 rpm
0.91
1.0
0.92
0.96
5000 rpm
1.4
1.1
0.99
1.00
6000 rpm
2.9
0.76
0.99
1.00
7000 rpm
5.0
0.60
1.00
1.00
8000 rpm
6.1
0.64
0.94
0.97
9000 rpm






198
Configuration II. At these spindle speeds, the power losses
and the operating temperatures are acceptable for HSHP
machining operations.
Bearing Preload
The PROMESS sensor indicated that the preload in
Configuration I varied during the test. The change on
preload was due to thermal expansions and to high speed
centrifugal effects. The magnitude of the thermally induced
preload as measured by the PROMESS sensor, 200 N, is
negligible for the double row large diameter bearings used.
The effect of high speed centrifugal effects on bearing
preload is of special consequence to the Radiax bearing.
This was the bearing which failed twice during the test. As
it was mentioned in Chapter V, the preload on the upper row
was eliminated at high spindle speeds. Without preload the
balls started to skid, which increased the heat generation
rates. The lubrication system was unable to remove the
excess heat induced by skidding and the bearing seized.
Therefore, if Configuration I is to be used at speeds higher
than 8,000 rpm, the mounting preload in the Radiax bearing
must be increased.
The preloading mechanism used in Configuration II was
effective throughout the test. At all times during the
test, the PROMESS sensor indicated that the preload was kept
constant. Therefore, the use of a constant preloading
mechanism is recommended in HSHP spindles.


Figure 6.4 Effect of Preload on Bearing Stiffness


101
Tb = al na2 (C) (5.1)
where
Ts: Bearing temperature (C)
n: Spindle speed (rpm)
al: Curve fit constant one (C/rpm~2)
a2: Curve fit constant two
The value of al and a2 for Configurations I and II are
given in Table 5.1 and Table 5.2. As it can be observed
from Figure 5.10 and Figure 5.11, the temperature of the
bearing increases with spindle speed. As the spindle speed
increases, the heat generation also increases and
consequently the higher operating temperatures. The steady
state operating temperature of Configuration I reached 85C
at 7,000 rpm and 1.5 1pm. At 8,000 rpm and 1.5 1pm, the
temperature reached 84C without achieving steady state. To
prevent bearing damage due to high temperature operation,
the oil flow rate was increased.
The steady state operating temperature for
Configuration II, reached 80C at 9,000 rpm and 0.4 1pm. At
all times the operating temperatures for Configuration II
were much lower than for Configuration I, for the same speed
and oil flow rate.
The approximation functions derived for each flow rate
equation 5.1 with the constants in Table 5.1 and Table 5.2,
match the experimental data very well, as it can be
observed from the figures.


71
Also from [9,11], the bearing manufacturer estimates
the friction torque as
M = 0.5 u* F d (N-mm) (4.2)
where
)i£: friction coefficient for the bearing
F: bearing load (N)
d: bore diameter of the bearing (mm)
The friction coefficient p.* is given in [9,11] for
several types of bearings
for cylindrical roller bearings p.f = 0.0011
for thrust ball bearings p.* = 0.0013
for tapered roller bearings p.* = 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


10
Table 2.1
Coefficient z and Exponent
for Ball Bearings
y
Bearing Type
Contact Angle
() 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.


Speed
RPM
3000
3000
3000
3000
5000
5000
5000
5000
6000
6000
6000
6000
7000
7000
8000
8000
8000
83
Table 4.1 Removed Heat/Generated Heat
Configuration I
Oil Flow
Rate (LPM)
Generated
Heat (kW)
Removed
Heat (kW)
Percentage
Removed
1.5
3.6
2.7
76
2.3
4.9
4.0
81
3.0
5.5
4.5
82
3.8
5.6
5.0
90
1.5
6.6
4.6
70
2.3
8.1
6.3
78
3.0
9.0
7.0
78
3.8
11.1
9.2
83
1.5
9.0
5.4
61
2.3
10.8
7.4
68
3.0
11.8
9.2
77
3.8
12.9
10.7
83
1.5
12.5
5.26
42
2.3
14.3
8.3
58
1.5
14.4
5.8
40
2.3
18.3
9.4
51
3.8
20.7
17.3
83


200
To provide the element rows with lubrication the
housing would have to be redesigned to include more oil
input points. These additional oil inlet points could be
used for oil jets directed to the element rows. Some of the
locating rings currently used on the spindle would also have
to be modified to allow oil to go through them and onto the
rolling elements. These changes would improve the
performance of the configuration by decreasing the power
losses and the operating temperature. If the power losses
decrease, then more power would be available for machining.
The lower temperatures would improve the accuracy of the
configuration and the life of the elements.
Design Modifications for Configuration II
The oil distribution system for Configuration II should
be modified. The ring which supplies the lubricating oil to
the Hydra-Rib should be modified as to improve the oil
flow into the bearing. Currently, it has eight lubrication
holes. Since the oil is forced up the bearing, the pressure
drop caused by the large number of holes reduces the oil
flow into the bearing. If only three evenly spaced holes
are provided the oil would reach the bearing with enough
pressure to flow through. This would be a more efficient
design.
Rings should be added to the spindle to splash away the
excess oil from the bearings and into grooves in the
housing. This modification would limit the amount of oil


176
= -6xB = 6X (mm) (6.66)
where
6^: axial deflection of bearing A (mm)
6xb: axial deflection of bearing B (mm)
6X: axial deflection of the system (mm)
The axial force, acting on each bearing, is the
combination of the preload and the resulting component of
the external applied radial load. The axial load on each
bearing, is of the same magnitude, but opposite direction.
= Fxb (N) (6.67)
where
F^: axial reaction on bearing A (N)
FxB: axial reaction on bearing B (N)
The external radial load acts on each bearing in
proportion to the distance from the bearing to the point of
application.
(a+b)
FzA = F z (N) (6.68)
b
a
Fzb = -Fz (N) (6.69)
b
where
FzA: radial load on bearing A (N)
Fzb: radial load on bearing B (N)
Fz: external radial load (N)


202
performance guarantees the maximum use of the cutting tools
available for the milling of steel and cast iron.
Parameters were identified which can be used to compare the
performance of a configuration independently of the machine
tool in which they are installed. A set of empirical
equations were derived to estimate the power losses at
operating conditions different from those in the test.
Finally, recommendations were presented for the design of
spindle configurations based on roller bearings in terms of
lubrication arrangements, operating speeds, and oil flow
rates.


Temperature
100 r -
sor o supply
a return
a bearing
80 -
a 70 r
60 r sp#-*
50l &
40
30
vNs-A/)
'^..Wte/inA/r^
* ~ ~ \Ann-n
20 *
ni
*p>-a>-o1
I.5LPM
j i ~t t.
0 10 20 30 40 50 60 70 80 90
Test Time min
Figure 5.2 Transient Temperatures
Configuration I: 5,000 RPM


57
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,
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


134
lubrication between both rows, after the oil cools the outer
ring. The oil is then splashed towards the sides and
between the bearings. This causes the large increase in
hydraulic power losses with little effect on bearing
temperature.
By observing Figures 5.10, 5.12, 5.19 and 5.21, it can
be concluded that an oil flow rate of around 2.3 1pm per
bearing would provide the best combination of operating
temperature and power losses for Configuration I. The
reduction in operating temperature does not justifies the
increase in power losses which occur once the oil flow rate
exceeds 2.3 lpra.
It should be noted here that the curves describing the
relationship between power losses and oil flow rate/spindle
speed coincide in shape with those presented by other
researchers in [21,22,23,24,25 and 26].
For Configuration II, Figures 5.11, 5.13, 5.20 and 5.22
show that a minimum oil flow rate of 1.0 1pm to the High
Speed bearing would provide the best combination of
operating temperature and power losses. As Figures 5.14 and
5.15 illustrate, the decrease in temperature due to
increased flow is insignificant after 1.0 1pm. Note that
the Hydra-Rib bearing is receiving a flow of 3.8 1pm at
all times. The 1.0 1pm oil flow rate to the High Speed
bearing should be provided as 0.5 1pm to the front and 0.5
lpm to the back through the cone-rib lubrication holes.


93
Figure 4.6 Thermal Model for Configuration II


BIOGRAPHICAL SKETCH
The author was born December 4, 1959, in Mayagez,
Puerto Rico. He went to college at the University of Puerto
Rico, at Mayagez, obtaining a B.S.M.E. in June 1982, and
accepted a faculty position with the UPR. He began graduate
study at Clemson University in Clemson, South Carolina,
where he completed his M.S.M.E. in December 1983. He
married his beautiful and enchanting wife Laura M. on
January 2, 1984. After completing his Ph.D. from the
University of Florida, he expects to return to his position
at UPR.
212


28
As before, the force balance equations are not enough
to solve for the unknown variables, namely F^, wc, ww;J and
Po;3. 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


37
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


180
Case 2: Constant Preload
In the spindle configuration shown in Figure 6.5c a
constant preloading force is maintained on both bearings.
Therefore, the axial force acting on each bearing is known
at all times. The axial deflection will be different at
each bearing. The axial forces at each bearing are equal to
the preloading force.
Fxa = = Fp (N) (6.77)
In this configuration, if the sum of the axial
components at one bearing, exceeds the external opposite
axial load, the bearing will start moving in the direction
of the axial component of the radial load. At this point,
axial deflections become so large that further computation
of the radial deflections is extremely difficult. As in
Case 1, the radial force at each bearing is proportional to
the external radial load.
a+b
Fza = Fz (N) (6.68)
b
a
FzB = -Fz (N) (6.69)
b
As before the following equations hold
z
FxA = E FejA sinaA (N) (6.70)
j=l
z
F^cb = 2 FejB sinos (N) (6.71)
j=l
z


Bearing Temperature
Figure 5.13 Bearing Temperature versus Oil Flow Rate
Configuration II


58
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


35
w: Rotational speed of the roller (rad/sec.)
Cn: Correlation factor:
3.87/(Nr)*5 for laminar flow
Nr<300,000
0.15/(Nr*,)0-17 for turbulent flow
Nro>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
FjG N |VRi W^\
Ta<3d- Uc [ 2 AhuThIJ
Ag n=i vRi vi3
S/2
- 2* E Av;)krvj)c] (2.54)
k=l
where
FjG: Contact force between the roller and the
guiding surface (N).
N: Number of horizontal lamina.
VR1: Velocity of the race at the itn horizontal
lamina (m/sec.).
Velocity of the roller at the itl* horizontal
lamina (m/sec.).
S: Number of vertical lamina.
Tati (N-m).
Uc: Friction coefficient between the roller and
the guiding shoulder.


19
Ro
+ Fw) Q2oJ =0 (2.13)
R
R0
Qyij + (Qyoj Fo;J F^) =0 (2.14)
Rj.
where
R0: Equivalent external radius of the cylinder
(iran).
R*: Equivalent internal radius of the cylinder
(iran).
Qaid.3
Qzij =
lw ERi
Qzoj
QzoJ =
lw ER0
Qyl j
0yi3=
lw E'Ra.
Qyoj
QyoJ
lw E'R0
(2.18)
Fw=
Fw
(2.19)
lw E'R0
(2.20)
lw er0
lw z
Roller length
(mm) .
E' =
E
(2.21)
1 a2
(2.15)
(2.16)
(2.17)


179
Equations (6.74), (6.75) and (6.76) are solved by using
the Newton-Raphson method described before. The summation
terms are computed first, then subtracted from the resultant
force. The difference is divided by the corresponding
derivative. The result is then added to the variable being
iterated.
The Configuration I spindle is of the variable preload
type, since the preload is applied during installation and
there is no mechanism to maintain that preload.
Configuration I differs from the one described above in that
the radial support is produced exclusively by the radial
bearings, while the Radiax supports all the axial loads.
To determine the deflections of each bearing, the load
in the bearings must be determined. For the radial
bearings, it is the case of a beam supported at two points
under concentrated load. The reaction at each bearing is
computed as above. The deflection at the radial bearing is
computed using the technique discussed in section on radial
loads, taking into consideration that the bearings now have
two rows of elements.
The external axial load is entirely supported by the
Radiax bearing. To determine the deflections, the double
row bearing is treated as two, face to face, ACBBs under
centric axial load.


78
1000 2000 3000 4000 5000 8000 7000 8000 9000 10000
Spindle Speed RPM
Figure 4.2 Computed Generated Heat vs. Spindle Speed
Configuration II


94
Figure 4.7 Thermal Profile (5,000 RPM)
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:
68