Spindle Deflections in High-speed Machine Tools

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Int J Adv Manuf Technot (19 ) 11:232-239

© 1996 Springer-Verlag London Limited

The nternational Journal of

Rdvanced

manufacturing

Technolo lu

Spindle Deflections in High speed Machine Tools Modelling and

Simulation

Subhajit hatterjee

Industrial Engineering Department, The U niversity of Tennessee, K noxville, USA

This rese arch attempts to develop spindle deflection error

models for high-speed machining systems. A mode l for

determining total spindle deflection at the tool-end is presented.

The model incorporates spindle bearing characteristics shifts

in ball contact angles and centrifugal force and gyroscopic

moment effects at high speeds. It uses the transfer matrix

method to determine the total deflections at the tool-end based

upon the po int contact deformations at the individual balls of

an angular contact ball-bearing assembly. A simulator is also

developed for simulating spindle end deflections fo r various

spindle rotational speeds. The results of the simulation s how

contact angle variations and peak deflections at particular

spindle rotational speeds. Imp ortan t re searc h issu es are also

presented.

Keywords High speed machining; Spindle deflection; Spindle

rotation simulation

1 Introduction

High-speed machining (speeds ab ove 5000 r .p.m .) i s emerging

as a powerful tool for increasing productivity in finish

machining [1]. Industrial studies in this area have evolved

from early concep ts in the 1920s and rec ent adv ances in the

development of computer control sys tems have provided

the capabi l i ty for accurate ly control l ing high-performance

automat ic machines . The evolut ion of these high-speed

machines has been in parallel with progress in the field of

spindle and machine tool design. However, research in high-

speed machine tool des ign requires a t tent ion in many areas ;

part icular ly important are : the thermal growth problem,

positional errors due to deflections from high-speed gyroscopic

moments and centr i fugal forces , and the changes in bearing

stiffness characteristics at high speeds.

This paper is the resul t of prel iminary theoret ical error

modelling and simulation of high-speed spindle systems and

Correspondence and offprint requests o:

Subhajit Chatterjee, Industrial

Engineering Department, The U niversity of Tennessee, Knox ville,

TN 37996-1506, USA.

is organised as follows. The next section reviews pertinent

l i tera ture in the area of high-speed machining and machine

tool metrology. It is followed by a statement of the objectives

of this s tudy and presentat ion of a theoret ical model re la t ing

spindle deflection to spindle and operational parameters.

Initial results are then presented from simulation runs and

finally potential research areas discussed.

2 Literature Survey

Various types of machine tools and machining centres exist

and are classified for ease of specification. Some machining

centre classification examples are: horizontal spindle fixed

column; horizontal spindle moving column; vertical spindle

fixed column; vertical column moving column; vertical spindle

fixed bridge; vertical spindle travelling gantry; horizontal

spindle travelling column with ti l t rotary table, etc. The major

motio n directions are in the Cartesian directions and, ad ditional

degrees of freedom include table rota t ion and t i l t .

The literature survey will focus on research in four areas:

1. Spindle deflections at high speeds.

2. Error budget ing in machine tools .

3. Errors due to thermal effects.

4. New material possibili t ies in material structures.

2 1 Spindle Deflections at High Speeds

Preliminary work in bearing nonlinearity and spindle bearing

stiffness analysis has been reported by Shin et al . [2]. They

show analytic and expe rimen tal evidence of changes in stability

zones for angular contact bal l -bearing supported spindles a t

high speeds. According to their study, changes in bearing

stiffness at high speeds resulted in spindle instability. To

corrobo rate thei r theoret ical study, they perform ed experimen-

ta l machining tes ts a t a ma ximum spindle speed of 8000 r .p.m.

ANSI s tandard B5.54 refers to CNC machine tool perform-

ance evaluation (of which spindle error profile is one aspect).

Researchers have a lso worked in this area for qui te some

time and individual work with respect to machine tool

characterisation has been published [3,4].



Spindle Deflections in High speed Machine Tools 233

Researchers [5-7] point out the importance of the stiffness

of a machining system and the spindle in particular, to tool l ife

and proc ess stability. Particularly interesting is the o bserv ation

of improved surface finish with an intentional reduction of

tooling stiffness. Enhancement of chatter resistance through a

reduction of stiffness has also been reported [5].

I t should be noted that a majori ty of the reported tes t

resul ts are not in the high-speed regime. For high-speed

machines , some importan t requiremen ts are a very high degree

of damping, v ery high accuracy ( low radial and face runonts)

and cons iderat ion of spindle and tool ing s t ructure deform at ions

[6]. T he effectiveness of tapers at high speeds is also imp ortan t

as increase in taper diameters a t the front lead to changes in

axial position of the spindle, and consequently, the cutting

tool [6]. There is a need for formulating spindle error profiles

and quas i -s ta t ic deformat ion of the spindles by cons idering

centri fugal forces and gyroscopic moments of spindle m ounts .

Addi t ional ly, the use of wear-res is tant bearing types should

also be considered in high-speed spindles.

2 2 Error Budgeting in Machine Tools

Error budgeting is a systems analysis tool used for the

predict ion, control , and des ign of machine tools . An error

budget can be used to control the individual subsys tem errors

given the tota l acceptable sys tem error . The error budget ing

scheme can use error sources and coupl ing mechanisms in

conjunct ion with workpiece categories to re la te the error

source to workpiece errors [8] . In this s tudy the error source

is the deflection of the spindle and the coupling mechanism

could be spindle propert ies . I t i s a lso important to determine

the proport ion of the spindle error in re la t ion to other error

sources such as servo error and displacement error .

There is a significant body of research relating to quasi-

s ta t ic error formulat ion, detect ion and measurement , and

compensat ion for machine tools [9-13]. The three common

methods are:

1. Measurement of the twenty-one pos i t ional and angular

error terms independent ly and compensat ion through

interpolation.

2. Correla t ional models buil t on coordinate data and t r igono-

metric relationships.

3. Formulat ion of error envelopes us ing rigid body kinemat ics .

The rigid body kinematics technique uses rotation an d translation

of the links and joints of the machine tool to formulate the

error envelope. Compensation for each error source is then

accomplished by determining the rotational and translational

coefficients through measurement and then predicting errors

from kno wn positions. How ever, all of the reviewed work in

this area addresses the quasi-static errors; therefore, there is

stil l a need to investigate error envelopes for dynamic errors

due to spindle rotation and positioning at high speeds.

2 3 Errors due to Thermal Effects

The thermal error problem has been recognised for a long

time and experts view this error as the largest single source

of dimensional errors [14]. The re are six sources of therm al

errors:

1. Heat generated from the cut t ing process .

2. Heat generated by the machine.

3. Heat ing or cool ing provided by the cool ing sys tems.

4. Air-conditioning effects.

5. People effects.

6. Thermal memories from a previous environment .

I t has been shown that the heat produced by spindle bearing

friction in machine tools has a very serious effect and this

effect has been investigated by man y researchers [14]. Analytic

and f ini te-e lement techniques have been used to formulate

th er m al effects and opt imise machine tool s t ructures for

minimising thermal effects [15,16]. It is important to extend

these concepts to high-speed machining and to examine

temperature r ises and resul t ing deformat ions a t these con-

ditions. Additionally, the wear of bearings at these high

speeds and e levated temperatures should a lso be cons idered

in these studies.

2 4 New Material Poss ibilities in Material Structure s

Advances in materia l development have prompted research

efforts in examining their potential use in machine tools.

Rahm an et a l . [17] showed that cement i t ious composi tes when

used as lathe bed material exhibited significantly higher

damping ratios and higher first natural frequencies than their

convent ional counterparts . Another s tudy by Lee et al . [18]

demonstra ted the successful use of graphi te epoxy composi te

as a spindle material in machine tools. Their analysis of the

spindle bearing sys tem showed about a 20 increase in

damping of a graphi te spindle sys tem with the consequent

higher natural f requencies (and delayed chat ter) as compared

to a convent ional one. They also examined thermal s tabi l i ty

of the spindle sys tem and found that spindle expans ion was

almost negligible because of the almost zero coefficient of

thermal expans ion of the composi te materia l . This is a lso

advantageous because the bearing preload character is t ics can

be b et ter mainta ined when the the rmal expans ion is negl igible .

Howe ver, the experim ental work in both the abo ve s tudies was

conducted a t cut t ing speeds less than 1500 r .p.m. Therefo re , i t

is important to investigate new bearing spindle materials at

high speeds. Figure 1 shows the highly interrelated overall

l[ ERROR

GEOMETRIC MACHINE TOOL VIBRATION

STATIC LOADING CHATTER

THERMAL EFFECTS ~ SPINDLE IBRATION

8E.ARING SPINDLE DEFLECTION

THERMAL EFFECTS

Fig

1. Overall part error sources and error hierarchy.



234 S. Chatterjee

part error sources and the error hierarchy. The scope of this

work is l imited to dynamic error modell ing of spindle end

deflect ion aris ing from bearing deflect ions a t the bal l contact

points.

3 Modelling of the Spindle System

The s tudy was undertaken with two ini t ia l object ives :

1. Formulat ion of the error analys is and deflect ion model for

high speed spindle systems.

2. Exam inat ion through s imulat ion, of the effect of different

bearing preloads on high-speed spindle deflect ion.

The present s tudy a t tempts to model the spindle bearing

sys tem for deflect ions a t the bearing surfaces by taking into

account spindle and bearing mate ria l characteris t ics , rota t ional

speeds , bearing bal l centre migrat ion, centr i fugal forces and

gyroscopic moments a t high speeds . The deflect ions and

evolved forces are then cons idered as input condit ions in

determining, by the t ransfer matrix metho d TM M), the tota l

spindle displacement a t the spindle end tool end). The

spindle-bearing system will be analysed first and will be

fol lowed by the analys is of the TMM.

3 1 Analysis of the Spindle System

In this work, angular contact bal l -bearings are used in the

spindle-bearing sys tem. In the modell ing of the deflect ion a t

the bal l -spindle interface , centr i fugal forces and gyroscopic

moments are cons idered in determining the bal l centre

migrat ion, and consequently, the contact deformations . I t has

been shown that because o f bearing preloading and high

rota t ional speeds , a shif t occurs in the radius of curvature of

the inner race while the outer race is fixed [19,20]. Figure 2

shows the pos i t ions of the bal l centre and raceways for a

part icular angular pos i t ion of the bal l . From this f igure i t is

evident that a shif t occurs in the inner raceway groove centre

resulting in a change in the contact angles as shown in Fig.

3. Such changes in raceway contact angles lead to varia t ion

in load dis tr ibut ions on the bal ls . At high speeds , i t has been

shown that the outer contact angles are less than the nominal

contact angle and the inner raceway contact angle is greater

than the nominal contact angle [20]. Therefore , i t is necessary

to determine the ins tantaneous contact angles a t each bal l

pos i t ion for given rota t ional speeds . Once these angles are

known the t radi t ional force-deflect ion equat ions for point-

contact deformation can be used to determine the tota l

deflect ion deformation of the bal l -spindle interface and the

forces evolved.

The fol lowing equat ions , developed from Fig. 2 , are used

in determining the ins tantaneous contact angles and are based

on the works in [19,20].

W z + V 2 - F O -

0.5) D + ~o)2 = 0 1)

AF - W) 2 + RF - V) 2 - FI - 0.5) D + 802 = 0

2)

J s

Fig. 2. Angular position of rolling elements.

i A F

<

F i n a l

~ i f i o~

a n e r r ¢ e w a y

- ~q ~ g r o o v ~ cu r v a tu r e een lad r

t a.0.oo.

. / ,

B a l l ~ n t ~ r I . . . . . . . .

O u t e r r a e e w a r ~ e

Fig. 3. Positions of ball centre and raceway groo ve curvatures at

angular position ~, with and without applied load.

?to MV /D - Ko~aos W

FO - 0.5) D + 8o

KiSl s AF - W) - hi M RF - V)/ D

+

FI - 0.5) D + 8i

ko MV /D + KoS~o V

FO - 0.5) D + 8o

Ki~I 5 RF - V) + k~M AF - W)/D

FI -

0.5) D + 81

cos 13o) = V/ FO - 0.5) × D + 8i)

= 0 3)

CF= O 4)

5)



Spindle Deflections in High-speed Machine Tools 235

cos 13i) = R F - W FI - 0 .5) × D + ~i) 6)

sin 13o) = W/((FO - 0.5) × D + ~i) 7)

sin 13i) =

AF - W/((FI -

0.5) × D + ~i) 8)

CF = 2.095 D/2 ) 3 p PW2262/1000 9)

M = J(W1/o0 (W2Ao) 2 o~

s in F) 10)

The subscripts i and o refer to the outer and inner raceways .

The f i rs t two equat ions are Pythagorean re la t ionships between

the ins tantan eous bal~ centre pos i t ion and deflect ions of the

inne r and ou te r raceways . The th i rd and four th equa t ions

represent the sum of forces in two perpendicular direct ions

x and y). The solut ions of the f i rs t four equat ions are used

to p rov ide the inne r and ou te r con tac t ang le s found us ing

equa t ions 5 ) - 8 ) . I t i s to be no ted tha t these equa t ions a re

nonlinear and the values of gyroscopic M, and the centr i fugal

force CF depend on the pa r t i cu la r va lue o f the va r iab le s .

Also, A F and RF depe nd on axia l misal ignment o f the spindle

if any), and the re la t ive axia l and radia l displacement of the

two bearing races se t by bearing preload and assembly. These

three parameters , a long with ini t ia l guesses for the primary

variables w, v, ~i, ~o are used as inputs for solving the system.

This sys tem of equat ions is to be solved for each of the bal l

pos i t ions to determine the deflect ion a t the contact points

and deflect ion s tresses . Once this is done, force and moment

equat ions shown below are appl ied, in conjunct ion with the

contact deflect ions , to the ent i re spindle for determining more

accurate ly the axia l and radia l displacements ini t ia l ly input

as guesses) for the appl ied preloads :

M q

Fa X~ (Qq

s inai j - h 0 -~ cosaj) = 0

elas tic modulus , the length, a nd the mo men t of inert ia of the

s tructure and the rota t ional speed. The fol lowing is a

represen ta t ion of the t ransfer matrix for deflect ions in the y-

direct ions as used in this s tudy:

l l i i 1

l /E F/2EI

My 0 1 1

y o~2m loj2m FoJ2m/2EI (13o~3m/6EI) + 1

MY i 1

v

A similar matrix may be cons tructed for the deflect ions in

the z-direct ion.

Thus , by knowin g condit ions at the previous i - 1) segment

of the s t ructure , the con dit ions a t the next segment may be

determined. I t is assumed here that the spindle may deflect

bend) in the x, y and y, z planes leading to deflect ions a long

the y and z direct ions . The z-direct ion is vert ica l while the

y-direct ion is a long the axis of the spindle . The x-direct ion is

perpendicular to the y and z direct ions .

In sum m ary , the inne r and ou te r con tac t ang le s and the

resul t ing deflections determ ined a t each of the bal l pos i t ions

for the g iven m a te r ia l and ope ra t iona l cond i t ions fo rm inpu t

boundary cond i t ions to the TMM for de te rm in ing the to ta l

deflect ion a t the spindle end. The next sect ion presents ini t ia l

resul ts of the s imulat ion.

F~ - "Z~ (Qq cosetij + hq t~j) = 0

j

M - X u ((Qq s inaq - xi j -~-) cosa q)R i + hqMj)cost~j = 0

Here , the f i rs t two equat ions sum the forces for a l l the bal l

pos i t ions in the vert ical and horizontal direct ions for the

ent ire spindle whereas the third equ at ion is the sum o f the

m om ents .

The tota l deflect ion a t the bearing is then given by:

10-5 {5 __FR )

= 2.53 D-TCOS[3 + P

\ Z cosl

where P = K~ = contact deformation force and FR is the

bea r ing p re load and the de form a t ion va lues a re ob ta ined by

the so lu t ion o f the above equa t ions . These de f lec t ions fo rm

the inpu t boundary cond i t ions fo r the TMM m ethod .

3 2 Analysis of the Deflection at the Spindle End

The TM M techn ique was used to de te rm ine the to ta l de flec tion

at the spindle end [21]. The TMM technique divides a

s t ruc tu re in to s epa ra te s egm ents and f rom bo undary cond i t ions

of the forces and deflect ions , the net deflect ion and forces a t

the f ree end a re de te rm ined . The TMM techn ique requ i re s

as an input a matrix of coeffic ients whose e lements are the

4 Results

A com pute r p rogram , based on the Hooke-Jeeves S ea rch

Technique HJS T) [22] was used to de te rm ine the op t im um

inner and outer contact angle values for given input condit ions

and axia l deflect ions a t the bearing end. The input condit ions

were the ini t ia l guesses for the contact deformations a t the

inner and outer races , the f inal bal l centre pos i t ion, the

rela t ive axial and radia l deflect ion of the bearing, a nd the

bearing misal ignment referred to in F ig. 3) . The HJ ST

sea rches the func t ion space fo r the m in im um func t ion va lue

based on ini t ia l guesses for the decis ion variables . In this

case , the funct ion value was the squared sum of equat ions

1) - 4 ) and was m in im ised to ze ro to y ie ld cor re sponding

values of inner and outer rac e deformati ons and bal l centre

migrat ions . These values were then used to determine the

inner and outer race contact angles and the tota l tool end

deflect ions for a given spindle sys tem. Th e bearing and spindle

sys tem specif ications used for the s imulat ions are given below.

Nu mbe r of bal ls in bearing = 16

Nominal contact angle = 40 degrees

Ball diameter = 22.23 mm

Inner raceway d iam e te r = 102. 79 m m

Out er raceway diamete r = 147.73 mm

Inner g roove radius = 11 .63 m m



236 S Chatter jee

60 | ~ ; ~ ~ .+___..4___+

+

-

,~, : t

LLI

5°1

o

4o1--. . ~ ~ ~ ~

~ ~ } ~ ~ . ~ . - t ~ . . ~ ~ : : - ~ ~ ~ : ~ ....

d

~ ° i

..........................................................................................................................................................................................................................

2 0

22.5 67.5 112.5 . 157.5 202.5 247.5 292.5 337.5

4 5 9 0 1 8 5 1 80 2 2 5 2 7 0 3 1 5 8 6 0

0TA=0UTER R~,C£ ANGLE BA LL LO CA TI ON AN GL E, DE GR EE S

INA=Ii ~ER ACE ANGLE

P~£LOA~ IN KN I ~ OKN , OT A + OKN , INA ~ 10KN , OT A |

[] IOKN INA × 20KN OT A -~ - 20KN INA

CO 60-

J

b

O

50 . ~ _ _ _ ~ ~ ~ ~ ~ ,-~ . . . . . ~

..,

4 5 -

4 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~ 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .

25 --~-'-~--'~---~-.~-:~- -FE--~'-_E---~F-~ --~Z-- ~--~Z--~y -'.~-'-~F

2 0

15- ---~,,, × ~4 ~ :.4 :< ~. × × ~4 × × × ; . --- :~. -

~o~ z ~

....e75 ~k5 ~5_ _ __ __~ , 5 2 0 ~ . 5 ~ 4 k 5 ~ 0 ~ -~ ~ , ~

4 5 9 0 1 3 5 1 8 0 2 2 5 2 7 0 3 1 5 3 6 0

OTA = OUTER CONTAGT ANGLE B A L L ~,-,o,~,r.,,,.~r_~o,.,~, EG

R E E B

IN,k= NNERGONTA~TAN~LE

PRELOADINK-,N ~ 2 0 KN , OTA ~ 20 KN, tNA + 10KN, OTA t

- ~ -

10KN, INA ~ 5KN, OTA ~ 5KN, INA

Fig. 4. Contact angle variation at 5000 r.p.m.


Pitch diameter of ball circle = 125 mm

Bearing st iffness

=

375350 N/ m m ~~

Mass mom ent of bal l = 2 .21163 kg /mm e

Spindle length = 304.8 mm

Elastic ity modulus = 200000 N /m m 2

Moment of inert ia of shaf t = 39612.55 mm 4

Spindle diameter = 29 .27 mm

Assumed shaf t assembly misal ignment = 0 .001 radian

The results of this s imulation study are now presented. It

is seen from Figs 4-7 that the ball contact angle at a

circumferential posit ion varies with rotational speeds. The

inner and outer contact angles vary from the mean contact

angle of 40 degrees the inner race contact angle being higher

and the outer race contact angle be ing lower than the mean

contact angle. The patterns of ball excursions are very similar

at different preloads although at lower preloads the contact

angle variations at the races are larger than at higher preloads.

With increasing speeds the contact angles tend to exhibit

higher excurs ions from the nominal . This can be seen by

comparing Figs 4 and 7 for spindle speeds of 5000 and

30000 r .p .m.

O3

r r

<

O

60-

I

5 0 ~ ~ ~ ~

~ - ~ [] ~

= ~L....~.......~o..

45- ~- ~ -~ : ~ fL~-~-`L~`.--~ L- -L -~ .` L- ~ `---~e--~.L--~.~-~-

40' ~4 ~ ~- ~) ,( -. -~ )< ..'. .~I x . <~ ~. x ;-I x

2 5 ............................... ................................ ............................... ............................... ............................... ............................... ............................... .

20.

1 5 . ~ . ~ - ~ ` ~ : ~ . ~ - ~ - ~ - ~ ~ - ~ - ~ - ~ ~ - - ~ . ~ : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22.5 67.5 112.5 157.5 202.5 247.5 292.5 337.5

4 5 9 0 1 3 5 1 8 0 2 2 5 2 7 0 81 5 3 6 0

OTA==3UT~R ACE tNGLE

(NA=INNER ACE NGLE

BALL L O C A T I O N A N G L E , D E G R E E S

FRELOAO N KN

.m- - 0KN, OTA '~ OKN, INA :~ 10KN ,OTA

10KN , INA - -x - - 20KN ,OTA - -~1~-- 20KN , INA


CO 60 [3_ ~--.E ~--.-. E~,~ I:3 - u LJ ~ ~ -'~' [::3' ~ CZ'--'--~ ~

5 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 o , L ~ - - . ~ - . . . . ~ - - -

-

- -

-~-

ILl

c -', 4 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .

z 35,

<

F-- ~0 ........................................................................................... ..................................

o< 25 ~- -- --- - - ---- - - -- --- - - -'

' ' ' ' ' ' + ' ' ' -J~ - - - - - - - ' - - '~ -

20 ..................................................................................................................................................

o 152~.5 . . . . .75 1 1 2 .5 1 5~ '.5 ' 2 0~ .5 ' 24 ~ .5 ' 2 9 2. 5 ' 8 a7 .5 ~ -

4 5 9 0 1 3 5 1 8 0 2 2 5 2 7 0 3 1 5 3 6 0

B A L L P O S I T I O N , D E G R E E S

OTA= OUTER CONTACT ANGLE

INA=INNER CONTACT ANGLE

PRELOAD IN KN mE 0K N,O TA ~ 10KN ,OT A + 20 KN ,O TA

0 K N , I N A × I O K N , I N A . A 2 0 K N , I N A


z

g

LL

. I

LL

0

18o-r - -

-0.5

160 .................. ...~.__ ~ _;~ ...,l~=__ L._.~_.......~ ~ ~,- --> ~.- --~

.{i-4~ , _+___: '- :- - -+ .~ ; , ,, ; .t,. 4-- t,, t,, i ~ 0 .4

140

......................................................................................................................................................................................... -

12 o I . ............................ ............................. .............................. .............................. ............................. ................. =

1004. .................................................................................................................................................................................. 0 3 z

[

80 .................................................................................................................................................................................. z

5 0 ~....................................................................................................................................................................................2 ~O

40

20 ............................................................................................................................................................................................. 0.1

-20 _ , , - , , , . . . . . . . . . -0

22.5 67.5 112.5 157,5 202.5 247.5 292.5 337.5

M= MOMENT 45 90 185_ 180 225 270 315 860

CF=CENTRIFUGALFORCE BALL LOCATION, DEGREES

PRELOAD N KN [~ _ OF, 0KN ~ CF, 1oKN- -- ~- CF, 20KN 1

-E~-- M, 0KN --x- - M, 10KN ~ M, 20KN

Fig. 8. Centrifugal force and gyr osc opi c mom ent variations at

5000 r.p.m.

The variations of centrifugal forces and moments are shown

in Figs 8 and 9. Both the parameters vary with ball location

and gyroscopic moments are noted to be similar at 5000 and

30000 r .p .m.

The deflect ions at the bearings are shown in Figs 10-12. It

is seen that the deformat ion is maximum at certa in pre loads .



~o

L6

0 500 0 --0.2

0 4000 LO.4 5

L-o.6 Lu

2000

E:: -,.0.8 0

,~ 1000

Z

LU 0 = . . . . . ~r~, - . -1

0

- 1 0 0 0 - , . . . . . . T , i i 7 ~ ~ - 1 . 2

22.5 67,5 112.5 157.5 202,5 247.5 292.5 ,337.5

45 90 135 180 225 270 815 360

PRELOAD N KN BAL L LOCA TION, DEGRE ES

CF= CENTRIFUGALFORCE

M=MOMENT [- -~ CF, OKN + CF, 10KN --~4-- CF, 20KN

-~ - M, 0KN --x-- M, 10KN *- i- M, 20KN

Fig 9

Centrifugal force and gyroscopic moment variations at

30000 r.p.m.

0,06[

= °, °5 1 ..........................................................................

0 0 ,1 ............... .................. ................. ........... ..................................

0.o `31 ............................................. ......................

i ° °1 I .................. ..................................

0 02 1 ........................... ......................

0 5 10 20

PRELOAD, KN

I ~ Z DEFLECTION -+ - - Y DEFLECTION ]

Fig. 10.

D e f l e c t i o n a t b e a r i n g e n d a t 5 00 0

r .p .m.

&Of f

0.07-

0,06-

i 0.05

O

0.04-

0.08-

0.02.

0,01-

0 5000 10000 20000

PRELOAD, KN

m Z

DEFLECTION' -+-Y DEFLECTION

Fig. 11. Deflection at bearing end at 10000 r.p.m.

The de fo rma t ion i s fo r tw o d i r e c t ions pe rpe nd ic u la r t o t he

sha f t c e n t re - l ine . The r e a sons fo r t he pe a k ing o f t he de f l e c t ion

a t pa r t i c u l a r p re loa ds c a nno t ye t be de t e rmine d . The to t a l

de f le c t ions a t t he sp ind le e nd ob ta ine d by the TMM a re

shown in Figs 13 and 14 and a re s l ight ly h igher than those a t

t he be a r ing e nd . The de f l e c t ions i nc re a se w i th ro t a t i ona l

spe e ds fo r t he s imu la t e d c ond i t i ons .

In t h i s s tudy the be a r ing s t i f fne ss w a s a ssume d to be

c ons t a n t. Th i s how e ve r ma y no t be t he c a se as obse rve d by

others [2] ; changes in bear ing st i f fness can be taken in to

c ons ide ra t ion in t he p roc e ss .


0,11-

0.1.

,$

0 ~ I

0,08

0.07'

0 .06

a

0.05

0.04

lb b

PRELOAD, KN

+ z~DELFECTION -'+ -- Y-DEFLEC TION t

Fig. 12. Deflection at bearing end at 15000 r.p.m.

0.45

0.4 ̧

} 0 1 3 5

0,3

(3 0.25

0 0.2

uJ

o.15

LU

a 0.1

0,05

1.3-

PRELOAD, KN

+ Z-DEFLECTION -- + - Y-DEFLECTION

Fig. 13. Deflection at spindle end at 5000 r.p.m.

O

F-

O

uJ

¢

uJ

£3

1.2-

1,1

1-

0.9-

0 8-

0 7-

0.6-

0.5.

0.4

5 ib 2 0

PRELOAD, KN

+ Z-DEFLECTION ~ Y-DEFLECTION 1

Fig. 14. Deflection at spind le end at 15 000 r p.m

5 Discussion

This s imula t ion has shown var ia t ions in the contac t angles

ow ing to h igh ro t a t i ona l spe e ds r e su l t i ng in subse que n t

de f l e ct ions a t the be a r ing a nd the sp ind le e nds . A t l ow spe e ds

the c on ta c t a ng le s a re c lo se t o t he nomina l ; how e ve r t he

va r i a t i ons i nc re a se w i th ro t a t i ona l spe e ds . A n inc re a se i n t he

p re loa d se e ms to l im i t t he a ngu la r e xc u rs ions some w ha t bu t

var ia t ions a re s t i l l s ignif icant . Deflec t ions a t the sp indle end

a re more tha n those a t t he be a r ing e nd fo r t he sp ind le sy s te m

simu la t e d . I t i s t he re fo re ne c e ssa ry to s imu la t e d i f f e re n t



238 S. Chatterjee

sp in d le a n d b e a r in g sy s t e ms th a t imp a r t mo re r i g id i ty t o t h e

sy s t e m a n d e x a min e th e ro b u s tn e ss o f t h e s imu la to r .

A t t h i s s t a g e o n e o f t h e p r ima ry c o n c e rn s i s t o e s t a b l i sh

th e v a l id i t y o f t h i s mo d e l l i n g p ro c e ss a n d th e s imu la to r

w i th e x p e r ime n ta l d a ta . E f fo r t s a re p re se n t ly u n d e rw a y fo r

o b ta in in g in d u s t r i a l d a t a a n d /o r d e s ig n in g th e e x p e r ime n ta l

se t -u p . Sp e c i a ll y b u i l t mo to r - in t e g ra t e d sp in d le s a re n e c e ssa ry

a t spe e d s o f 1 0 0 0 0 r .p .m . a n d a b o v e . A d d i t i o n a l ly , th e

se n s i ti v i ty of t h e m o d e l o u tp u t w i th r e sp e c t t o t h e i n p u t

c o n d i t i on s i s a l so c u r re n t ly b e in g e x a min e d .

Th e v a r i a t i o n o f t h e c o n ta c t a n g le s a n d th e su b se q u e n t

c o n ta c t d e fo rma t io n s a n d d e f l e c t io n s a re imp o r t a n t fo r h ig h -

speed machin ing . Spec if ica l ly , the s p indle def lec t ion e rrors

c a n b e c o mp a re d w i th o th e r e r ro r so u rc e s ( se rv o e r ro r s ,

t h e rma l e r ro r s , e t c . ) a n d u se d in t h e d e v e lo p me n t o f ma c h in e

to o l e r ro r b u d g e t s .

A n o t h e r imp o r t a n t f a c to r i s t h e e f f ec t o f t h e v a r i a b i l i t y o f

the d imension of each ba l l on the overa l l sp indle def lec t ion

a t h igh speeds. This m ay have an e ffec t on the overa l l process

capabi l i ty . The e ffec t on to ta l def lec t ion , through s imula t ion ,

o f b a l l ma te r i a l d i f f e re n c e s o n c o n ta c t d e fo rma t io n a n d

a l t e rn a t iv e ma te r i a l s , su c h a s c o mp o s i t e s , b y in c o rp o ra t in g

th e ma te r i a l p ro p e r t i e s i n t h e s imu la t io n n e e d s t o b e e x a min e d .

Th e p re se n t mo d e l d o e s n o t i n c o rp o ra t e b e a r in g h e a t

e ffec ts . I t i s in tended in the fu ture to inc lude hea t e ffec ts in

th e mo d e l .

6 Conclusions

Th i s r e se a rc h h a s a t t e mp te d to mo d e l sp in d le e r ro r p ro f i l e s

fo r h ig h ro t a t i o n a l sp e e d s . Be a r in g c o n ta c t a n g le v a r i a t i o n s ,

v a r i a t i o n s i n c e n t r i fu g a l fo rc e s a n d mo me n t s , t o t a l t o o l e n d

d e f l ec t ion s h a v e b e e n mo d e l l e d . Ba se d o n th e r e se a rc h so me

imp o r t a n t fu r th e r r e se a rc h a re a s i n h ig h - sp e e d ma c h in in g a re :

1 . Co n s id e ra t io n o f a l t e rn a t iv e ma te r i a l s i n sp in d le s t ru c tu re s .

2 . Ex amin a t ion o f the e ffec t of var ia t ions in ba l l d imensions

on the def lec t ions.

3 . D e v e lo p m e n t o f e r ro r b u d g e t p ro c e sse s in ma c h in e to o l s .

4 . Examina t ion of the e ffec ts of var ious sp indle tapers on

def lec t ion a t h igh speeds.

5 . Ex a min a t io n o f e f f e c t s o f s t i f f e r b e a r in g ma te r i a l s o n

def lec t ions.

I t i s hoped tha t such invest iga t ions wi l l lead to a be t te r

u n d e r s t a n d in g o f t h e h ig h - sp e e d ma c h in in g p ro c e ss a n d b e t t e r

design of machine tools .

cknowledgement

Th e f in a n c i a l su p p o r t p ro v id e d b y th e O f f i c e o f Re se a rc h

A d min i s t r a t i o n , Th e U n iv e r s i t y o f Te n n e sse e , K n o x v i l l e , TN ,

is gra te fu l ly acknowledged.

References

1. R. I. King (ed. ),

Handbook of High-Speed Machining Technology

Chapman and Hall, New York, pp. 1-26, 1985.

2. Y. C. Shin, K. W. Wang and C. H. Ch en, Dynamic analysis of

a high speed spindle system , Transactions of the NA MRI/SM E

pp. 298-304, 1990.

3. Y. C. Shin, H. Chin and M. J. Brink, Characterization of CNC

machining centers , Journal of Manufacturing Systems 10(5),

pp. 407-421, 1991.

4. R. J. Hocken,

Technology of Machine Tools

vol. 5, Lawrence

Livermore Labo ratory CA, UCRL-52960-5, 1980.

5. E. Rivin, Trend s in tooling for CNC machine tools: machine

system stiffness ,

Manufacturing Review

4(4), pp. 257-263, 1991.

6. E. Rivin, Trends in tooling for CNC machine tools: tool-s pindle

interfaces ,

Manufacturing Review

4(4), pp. 264-274, 1991.

7. G. Chryssolouris, Effects of machine-t ool-work piece stiffness

on the wear behavior of superhard cutting materials ,

Annals of

the CIRP

31(1), pp. 65-69, 1982.

8. R. R. Don aldson, Erro r budgets ,

Technology of Machine

Tools vol. 5, Lawrence Livermore Laboratory CA, UCRL-

52960-5, pp . 9.14-1-9.14-14, 1980.

9. G. Zhang, R . Ouyang, B. Lu, R. Hocken, R. Veale and

A. Donmez, A displacement method for machine geometry

calibration , Annals of the C1RP 37(1), pp. 515-518, 1988.

10. R. Hocken, J. A. Simpson, B. Borchardt, L. Lazar, C. Reeve

and P. Stein, Three dimensional metrology ,

Annals of the

CIRF 26(2), pp. 403-408, 1977.

11. P. M. Ferreir a and C. R. L iu, A method for estimating and

compensating quasistatic errors of machine tools ,

Journal of

Engineering fo r .Industry

115, pp. 149-159, 1993.

12. R. Schultschik, The compo nents of the volumetr ic accuracy ,

Annals of the CIRP 25(1), pp. 223-228, 1977.

13. J. Mou and C. R. Liu, A metho d for enhancing the accuracy

of CNC machine tools for on-line inspection ,

Journal of

Manufacturing Systems

11(4), pp. 229-237, i992.

14. J. Bryan, International status of thermal research , Annals of

the CIRP

39(2), pp. 645-656, 1990.

15. J. Jedrzejewski, J. Kaczm arek and Z. Kowal, Numerical

optimization of thermal behavior of machine tools ,

Annals of

the CIRP 39(1), pp . 379-382, 1990.

16. R. Venugop al and M. Barash, Therm al effects on the accuracy

of numerically controlled machine tools ,

Annals of the CIRP

35(1), pp. 255-258, 1986.

17. M. Rahm an, M. A. Mansur and K. H. Chua, Evaluation of

advanced cementitious composities for machine-tool structures ,

Annals of the CIRP

37(1), pp . 373-376, 1988.

18. D. G. Le e, H.-C. Sin and N. P. Suh, Manufacturing of a

graphite epoxy composite spindle for a machine tool , Annals of

the CIRP

34(1), pp. 365-369, 1985.

19. A. B. Jones, A general theory for elastically constrained ball

and radial roller bearings under arbitrary load and speed

conditions , Journal of Basic Engineering pp. 309-320, June

1960.

20. T.

A.

Harris,

Rolling Bearing Analysis

3rd edn, John Wiley and

Sons, 1991.

21. A. Bhattacharya,

Principles of Machine Tools

2rid edn, New

Central Boo k Agency, Calcutta, India, pp. 605-609, 1975.

22. B. S. Gottfried and J. Weismann,

Introduction to O ptimization

Theory

Prent ice-Ha ll, Englew oods Cliffs, NJ, pp. 113-130, 1973.

omenclature

A F

final position, inner raceway groove centre

RF initial position, inner raceway groove centre

W final position of ball centre

V initial position of ball centre

D ball diameter, mm

ro inner raceway groove radius, mm

r~ inner raceway groove radius, mm

M gyroscopic moment, N-mm

FO ro/D

t71 ri/o

P bearing pitch diameter, mm

Ko oute r race load-de flection constan t, N/ram 1.5



~

CF

J

l

E

I

Y

M

V

m

P

o

inner race load-deflection constant, N/m m 1 5

centrifugal force, N

mass mom ent of inertia, N.mm 2

length of spindle, mm

modulus of elasticity, N/m m 2

moment of inertia of spindle, mm4

deflection of spindle along y-direction, mm

deflection of spindle along z-direction, mm

moment at spindle end, N.mm

shear force at spindle end, N

spindle mass, kg

material density

outer race contact angle

~o

F

W1

W2


inner race contact angle

nominal contact angle

inner race deformation

outer race deformation

angle between ball centre of rotation and the horizontal

mis-alignment (in degrees) of shaft assembly measured in a

plane perpendicular to shaft axis (x-direction)

ball and raceway angular raceway velocity ratio for ou ter

raceway control

ball orbital and angular raceway velocity ratio for rotating

inner raceway and outer raceway control

circumferential ball position

raceway control parameter

Spindle Deflections in High-speed Machine Tools

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