Technical Information

Accuracy of Feed Axes
Every year’s new machine tools show improvements in efficiency and power. Increasing feed rates and acceleration values reducemachining times. At the same time, increasing accuracy permits ever-closer tolerances. On the one hand, these developments enable
the machining of increasingly critical parts; on the other hand they simplify the manufacture of complex assemblies. Selective manual
assembly and bench work can often be reduced. Also, higher accuracy of parts normally results in improved function of component
assemblies. Dimensional accuracy in motor transmissions, for example, increases service life and reduces noise emission.
In the total error budget of a machine tool the positioning error values of the feed axes play a critical role. The following text discusses
these errors and compares them with other types of error. The primary problem involved with position measurement using rotary
encoder and ball screw is the thermal expansion of the ball screw. The resulting positioning error often outweighs the thermally induced
structural deformation and geometric error of machining centers. Several tests show the influence of the heating of the ball screw on the
results of machining – also with respect to the ball-screw bearing. When a linear encoder is used for position measurement, the thermal
expansion of the ball screw has no influence and the position drift is negligible. As requirements for machine tool accuracy and velocity
increase, the role of linear encoders for position measurement grows increasingly important.

The accuracy of modern machine tools is
measured with an increasing number of
new and revised inspection andacceptance tests. Where years ago purely
geometric acceptance tests predominated,today’s routine methods include dynamic
tests such as circular interpolation and freeform tests, thermal tests such as describedin ISO/DIS 230-3, and for production
machines, capability testing during
acceptance or regular inspection. The
influences of the cutting processes, thegeometric and thermal accuracy of themachine, its static and dynamic rigidity, andthe positioning response of the feed axeson the attainable accuracy of the workpiececan be more specifically analyzed. Machine
errors therefore become increasingly
transparent to the user.

Considering the increasing frequency ofchanging jobs and the concomitant
reduction in batch sizes, reducing thethermal or systematic error of a machine
tool through tedious optimization ofindividual production steps is seldom
feasible. “Accuracy of the first part” isgaining in importance. In particular thethermal error of machine tools is drawing
ever more interest.
The following text shows that thermal error
can be quite significant, especially for the
feed axes. Unlike structural deformation,
errors of the feed axes can be dramaticallyreduced through a choice of simple andreadily available measuring devices.

Feed drive mechanism of a millingmachining center

03/2006

Feed-drive system design

An exact error analysis of position
measurement via rotary encoder and feedscrew begins with a consideration ofprevalent mechanical feed-drive systems.
Although machine tool designs varyimmensely, the mechanical configurationof their feed drive is largely standardized(Fig. 1). In almost all cases, the recirculatingball screw has established itself as thesolution for converting the rotary motion of
the servo motor into linear slide motion. Itsbearing takes up all axial forces of the slide.
The servo motor and ball screw drive areusually directly coupled. Toothed-belt drives
are also widely used to achieve a compact
design and better adapt the speed.
For position measurement of feed axes on
NC machine tools it is possible to use
either linear encoders or recirculating ballscrews in conjunction with rotary encoders.
A position control loop via rotary encoderand ball screw includes only the servo
motor (Fig. 1 dashed line). In other words,
there is no actual position control of theslide, because only the position of theservomotor rotor is being controlled. To beable to extrapolate the slide position, themechanical system between the servo

xnom

Positioncontroller

motor and the slide must have a known
and, above all, reproducible mechanicaltransfer behavior. A position control loopwith a linear encoder, on the other hand,includes the entire mechanical feed-drive
system. Transfer errors from the mechanics
are detected by the linear encoder on theslide, and are corrected by the controller
electronics.
Differing terminology
Different terms are used to distinguishbetween these two methods of position
control. German-speaking and someEnglish-speaking communities generally
refer to them somewhat inaccurately as”direct and indirect measurement.”
However, these terms are rather poorly
chosen because, strictly speaking, bothmethods are direct. One method uses
the line grating on the linear scale as themeasuring standard, the other the pitch of
the ball screw. The rotary encoder simply
serves as an interpolating aid. Here theJapanese concepts of “Semi-Closed-Loopand Closed-Loop control” seemappropriate, since they more aptly describe
the actual problem.

Trend toward digitally driven axes
As a result of the trend toward digital axes
in drive technology, a large share of new
servo motors feature rotary encoders,which in principle can serve together withthe feed screw for position control. With
such a drive configuration the decisionmust be made as to whether to add alinear encoder or simply to use a ball screw
working in combination with the alreadyexisting motor encoder.
One should remember to consider the
problems discussed in the following text
regarding position measurement using arotary encoder/ball-screw system. They
can quickly increase the cost of an“economical” machine if the owner findsthat the accuracy does not suffice in
certain applications.

Velocity
controller

xactl

Fig. 1: 	Typical drive system of a numerically controlled machine tool with linear scale on the slide and rotary encoder on the motor.
Unlike in position feedback control with rotary encoder and ball screw, a linear encoder includes the feed drive mechanism in thecontrol loop.

2Positioning errors caused by mechanical influences

Kinematics error

Kinematics error that can be directly
attributed to position measurement using
feed screw and rotary encoder result fromball-screw pitch error, from play in the feedelements, and from pitch loss. Ball-screw
pitch errors directly influence the result of
measurement because the pitch of the ballscrew is used as a standard for linear
measurement. Play in the feed transfer
elements causes backlash. The pitch loss
results from a shift of the balls during thepositioning of ball-screw drives with twopoint preloading and can lead to reversal
error in the order of 1 µm to 10 µm[see Schröder, Wilhelm: Feinpositionierung
mit Kugelgewindetrieben].
Error compensation
Most controls are capable of compensatingsuch pitch error and reversal error.
However, to determine the compensation
values it is necessary to make elaborate
measurements with comparative
measuring devices such as interferometersand grid encoders. In addition, the reversal
error is often unstable over long periods oftime and must be regularly recalibrated(Fig. 2).

Strain in drive mechanisms

Forces leading to the deformation of feed
drive mechanisms cause a shift in theactual axis slide position relative to theposition measured with the ball screw androtary encoder. They are essentially inertia
forces resulting from acceleration of theslide, cutting process forces, and friction in
the guideways. The mean axial rigidity of afeed drive mechanism as shown in Fig. 1
lies in the range of 100 N/µm to 200 N/µm
(with a distance between ball nut and fixedbearing of 0.5 m and a ball-screw diameterof 40 mm).
Cutting force
The cutting force can quite possibly lie in
the kN range, but its effect is distributed
not only in the feed drive system, but also
over the entire structure of the machine
and therefore between the workpiece andthe tool. The deformation of the feed drive
system therefore normally has only a small
share in the total deformation of themachine. A linear encoder can recognize
and correct only this small portion of thetotal deformation. Critical component
dimensions, however, are normally finishedat low feed rates with correspondingly low
deformation of the feed drive system.

5 µm/div

5 µm/divY

Y

HEIDENHAIN

HEIDENHAIN
X

X

ACCOM

ACCOM

New

After 1 year

Fig. 2: 	Circular tests of a machining center without linear encoders in new condition andafter one year. The reversal error of the X axis has increased significantly.

5 µm/div

5 µm/div

f = 2, 4, 6 m/min
R = 10 mm
Y

Y

HEIDENHAIN

HEIDENHAIN

X

X

ACCOM

ACCOM

Position control with spindle/rotary encoder

Position control with linear encoder

Fig. 3: 	Circular tests of a machining center that has been retrofitted with linear encoders.
With position feedback control by rotary encoder and ball screw, the circles deviatesignificantly from the ideal path at high feed rates. With linear encoders, thecontour accuracy is considerably better.

Force of acceleration
A typical slide mass of 500 kg and amoderate acceleration of 4 m/s2 result in
deformations of 10 µm to 20 µm thatcannot be recognized by the rotary
encoder/ball-screw system. The present
industry trend toward accelerations insignificantly higher ranges will result inincreasingly large deformation values.
Force of friction
The force of friction in the guideways lies
between 1 % and 2 % of mass for roller
guideways and 3 % to 12 % of mass for
sliding guideways [see VDW-Bericht 0153:
Untersuchung von Wälzführungen zur
Verbesserung des statischen und
dynamischen Verhaltens von

Werkzeugmaschinen]. A mass exerting
5000 N therefore results in feed drive
deformation of only 0.25 µm to 6 µm.
Circular interpolation test for inspecting
machine tools
A typical example for errors dependent on
acceleration and velocity can be recorded
in a circular interpolation test on a verticalmachining center (Fig. 3). With position
feedback control by rotary encoder and ballscrew, the circles deviate significantly fromthe ideal path at high contour speeds. The
same machining center shows significantly
better contour accuracy when equippedwith linear encoders.

3Positioning error due to rising ball-screw temperature

Positioning error resulting from thermal
expansion of the ball screw presents thegreatest problem for position measurement
via rotary encoder and ball screw. This isbecause the ball-screw drive must serve adouble function: On the one hand it must
be as rigid as possible to convert the rotary
motion of the servo motor to linear feed
motion. On the other hand it must serve asa precision measuring standard. The twofold function therefore forces a compromise
because both the rigidity and the thermal
expansion depend on the preloading of theball nut and the fixed bearing. Both theaxial rigidity and the frictional moment are
roughly proportional to the preloading.
Friction in the ball nut
The largest portion of the friction in a feed
drive system is generated in the ball nut.
This is because of the complex kinematics
of a recirculating ball nut. Although at firstglance the balls may seem only to berolling, they are in fact subjected to a great
deal of friction. Besides the microslip
resulting from relative motion in thecompressed contact areas, the greatest
effect is from the macroslip due tokinematics exigencies. The balls are notcompletely held in the races and wobble
much like tennis balls rolling down a gutter.

The result is a continual pressing andpushing with occasional slipping of theballs. The friction among the balls isaggravated by high surface pressure dueto the absence of a retaining device toseparate them. As in every angular-contact
ball bearing a spinning friction results froma contact diameter that is not orthogonal tothe axis of ball rotation. Each ball therefore
rotates about its contact diameter. Studies
have also shown that the balls can move inthe thread only because of an additional
slip component brought on by the thread
pitch [see Weule, Hartmut /Rosum, Jens:
Optimization of the friction behaviour ofball screw drives through WC/C coatedroller bodies].
The recirculation system is a special
problem zone for ball screws. With every
entrance into the recirculation channel, justas with every exit, the movement of theball changes entirely. The rotational energyof the balls, which in rapid traverse typicallyrotate with 8000 rpm, must be respectively
started and stopped. In contrast to thepreloaded thread zone, in the recirculation
zone the balls are not under stress. The
play of energy causes the balls to collect in
the recirculation channel. Without elaborate
measures to reintroduce the balls into thethread at the end of the channel it tends tocongest, causing the familiar jamming of
the ball-screw drive.
The frictional moment of a ground
precision recirculating ball screw with
40 mm diameter and 10 mm pitch wasmeasured by Golz [see Golz, Hans Ulrich:Analyse, Modellbildung und Optimierungdes Betriebsverhaltens vonKugelgewindetrieben] for various preload
forces and rotational speeds (Fig. 4). The
Stribeck characteristic of frictional moment
is clearly recognizable. It confirms the highshare of solid-body friction and mixed
friction in ball-screw drives at low speeds.
Viscous friction dominates at high speeds.
It is interesting to note that for this typicalball screw the normal machining feed rates
lie far below the speeds at which thefrictional moment is at its minimum. The
rapid traverse feed rates, however, lie far
above it. The feed rates at which this ballscrew is at optimum efficiency therefore
seldom occur. The frictional moment is only
slightly dependent on the axis load of theball nut [see ibid.].

rpm

Spindle speed 

g
in
ad
o
l
e
Pr

e
rc
fo

Frictional moment

Frictional moment 

Nominal diameter
40.00 mmPitch
10.00 mmAxial force
0.00 N
Angle of contact
45.00 degrees
Ball diameter
6.35 mmInternal ball recirculation, no wiper

Mixedfriction

Viscous friction

Speed nFig. 4: 	Measured moment of friction of a two-point preloaded ball screw [see ibid.]. The Stribeck characteristic is plainly visible.

4Frictional heat generated in the ball nut
With a typical preload of 3 kN and allowing
for the missing wiper, this results in a noload or frictional moment of 0.5 Nm to
1 Nm. This means that in rapid traverse at aball screw speed of 2000 rpm approximately
100 W to 200 W of frictional heat is
generated in the ball nut.
More frictional heat to be expected
To increase rapid traverse velocity, either
the pitch or the rotational speed of therecirculating ball screw must increase. In
the last five years the maximum permissible
speed of recirculating ball screws has
doubled. Due to the continually increasing
requirement for acceleration, the preloading
and therefore the friction of the ball nut
could not be reduced. Recirculating ballscrew drives therefore generatesignificantly more heat than before and will
generate even more in the future.

Measurement of positioning accuracy
according to ISO 230-3
The influence of frictional heat on thepositioning response of the feed axis
becomes apparent when positioning tests
are conducted in accordance with theinternational standard ISO/DIS 230-3. This
standard contains proposals for making
uniform measurements of thermal shifts oflathes and milling machines as a result of
external and internal heat sources (Fig. 5).

the ends of the traverse range at an agreed
percentage of the rapid traverse velocity.
The change of the positions with respect tothe initial value is recorded. The test is tobe conducted until a satiation effect isclearly observable. Simpler test equipment
than a laser interferometer, such as dialgauges, can also be used for the axis test.
These tests enable any workshop to
conduct such inspections at a reasonable
cost.

Deformation in the machine structure
resulting from changes in ambientconditions or through heat generation in
the spindle drive are recorded with the aid
of five probes that measure against acylinder mounted in the tool holder. This
makes it possible to measure all five
relevant degrees of freedom. To test thefeed axes, it proposes a repeated positioning
to two points that lie as near as possible to

Structural deformation

Drift of positioning axis

– Ambient conditions– Heat generated in the spindle– Heat generated in the ball screw

Fig. 5:	Measurement of thermal displacement of a machining center in accordance with
ISO/DIS 230-3.

5

Influence of the ball-screw bearing on positioning accuracy

Differing types of behavior can be expected
depending on whether the ball screw can
expand freely. The various types of bearings
for recirculating ball screws are illustrated
in Fig. 6.

Figure 7 shows the result of a positioning
test as per ISO/DIS 230-3 on a verticalmachining center (built in 1998) retrofittedwith linear encoders. The X axis was
positioned to three points a total of 100
times at 10 m/min. Taking the standstill
periods for measured value acquisition into
account, the mean traversing speed during
the test was approx. 4 m/min. In addition
to the two positions at the ends of traverse
as recommended in the standard, a third
position at the midpoint of traverse wasmeasured. Figure 7 shows the position
values with respect to their initial values.
At first the ball screw / rotary encoder
system was used for position feedback
control. In a second test under otherwise
identical conditions, linear encoders wereused. The comparator system was a
VM 101 from HEIDENHAIN.
In spite of the moderate feet rate of
10 m/min (rapid traverse 24 m/min), theposition farthest from the fixed bearing of
the ball screw shifted by more than 110 µm
within 40 minutes. It is interesting to notethat the drift increases very quickly
immediately after switch-on. Any change inthe mean feed rate in a production process
therefore immediately affects positioning
accuracy. Similar results were published by
Schmitt [see Schmitt, Thomas: Modell derWärmeübertragungsvorgänge in dermechanischen Struktur von CNCgesteuerten Vorschubsystemen].

6

Fixed bearing at both ends (fixed/fixed), see Fig. 10

Fixed bearing at one end (fixed/preloaded), see Fig. 11
50 µm/mFig. 6: The various types of bearings for recirculating ball screws.

40 minDrift of position in [µm] Fixed bearing at one end
In the case of fixed/floating bearings, theball screw will expand freely away fromthe fixed bearing in accordance with its
temperature profile. The thermal zero pointof such a feed axis is at the fixed bearing.
This means that theoretically no thermal
shift would be found if the ball nut islocated at the fixed bearing. All otherpositions are affected by the thermal
expansion of the ball screw.

Fixed bearing at one end (fixed/floating), see Fig. 7

20

Linear encoder

0
-20

0 mm

-40
-60
360 mm-80

Ball screw / rotary
encoder-100-120

720 mmTime in [min] Fig. 7: 	Drift of three positions during positioning accuracy measurement in accordancewith ISO/DIS 230-3 on a machining center with ball screw in fixed/floating
bearings. Position measurements via rotary encoder and ball screw result in a
distinct drift of positions due to the thermal growth of the screw.

No drift in position values measured
with linear encoders
The measured positioning accuracy
depends directly on the number ofrepetitions, particularly if there are only
few repetitions. The measurements takenby the retrofitted linear encoders show nodrift.

Series production
To demonstrate the applicability of thisexperiment in actual production conditions,a small batch of aluminum workpieces
were machined on the same machine.
Eight 70 mm x 70 mm workpieces were
fixed on a vertical machining center. Fourpockets and two radii were machined using
four tools with an infeed of 1 mm in the
Z axis (Fig. 8).

After the 6-minute machining operation theeight parts were not exchanged. Rather,
the infeed in Z was increased by 1 mm andthe operation repeated. As a result of thethermal expansion of the ball screw, allworkpieces show a step pattern on the left
side. This pattern is particularly pronounced
on the workpiece farthest away from thefixed bearing. The right sides of the work­pieces are smooth because with each shift
in the positive X direction the previous step
was also removed. In principle the same
effect could be observed in the Y direction
as in the X direction, but because of thelesser amount of movement in the Y axis
the step pattern is significantly lesspronounced. In the X direction thecomparative measurement of the step
pattern shows a drift of approx. 90 µm
with a time constant of thermal expansion
slightly less than one hour (Fig. 9).

Fixed bearing at both ends
The situation is more complex in the caseof fixed/fixed bearings. Ideally rigid bearings
would prevent expansion of the ball screwat its end points. However, this would
require considerable force. To prevent
expansion of a ball screw with 40-mm
diameter, 2.6 kN must be applied per degree
Celsius of temperature increase. A typicalangular-contact ball bearing would quickly
fail under any large increase in temperature.
Under real conditions, the rigidity of the
purportedly fixed bearings with their seatslies in the area of 800 N/µm. This meansthat as the temperature of the ball screw
increases, the bearings deform significantly.
The end points of the ball screw do not
remain at their original position. The same
experiment as in Fig. 7 was conducted ona vertical machining center (built in 1998)
with fixed bearings at both ends. The
tested 1-m long feed axis was mechanicallydesigned to be very rigid. At each end of
the ball screw the same bearing was built
into seats that were machined directly into
the machine’s cast frame.

Tool 2Tool 3Tool 4

Time / 8 workpieces = 6 min

Fig. 8: 	Experimental setup for batch production with multiple workpieces. Four pockets
and two radii were machined using four tools to a depth of 1 mm each. To illustratedrift resulting from thermal expansion of the ball screw, the workpieces were notexchanged after machining. Instead, the part program was run repeatedly at
successively increasing depth.

2hDrift of machine datum in [µm] 

If additional work is to be done on
previously machined work­pieces withcritical dimensions, the machine datum
must be continually inspected and corrected.
The machine achieves thermal equilibrium
after one hour, but after an interruption inmachining it begins to drift in the reverse
direction. If the part program and with it
the mean feed rate are changed, it again
takes approx. one hour for the ball screw
to regain thermal equilibrium.

8 workpieces

Tool 1

Drift

100
80
60
40
20
0

Ball screw / rotary encoder
0

5

10

15

20

Linear encoder-20
Step/depth in [mm] 

Fig. 9: 	Result of the experiment in Fig. 8. The left pocket of the finished workpieces
plainly shows a step pattern resulting from thermal expansion of the ball screw.

7

Fixed/preloaded bearing
This type of bearing causes problems
for traversing programs with high mean
velocities because the bearing load isdetrimental to service life and the forces
to be withstood result in deformation of
the machine structure. A fixed/preloaded
bearing design is therefore often used asa sort of pressure valve (Fig. 6). With atypical preload of 50 µm/m, one would
expect that such a bearing configuration
would behave like a fixed/fixed combination
up to a temperature increase of approx. 5 K,
and beyond that, like a fixed/floating
combination.
Figure 11 shows the results of a positioning
test on a machining center with a ball screw
with fixed/preloaded bearings conductedalong the pattern of the previously described
experiments. Surprisingly, in spite of the
fixed/preloaded bearing configuration, a
position drift similar to that in Fig. 7
becomes apparent. This means that thefeed axis behaves roughly like one with
fixed/floating bearings. The thermal zero
point seems to lie near the fixed bearing.
Unlike the axes in the two previous
experiments, this axis had a travel of only
500 mm instead of one meter. The
magnitude of the drift is therefore notcomparable.
This experiment shows that the simple
model of the ball screw with fixed/
preloaded bearings does not stand up to
reality. As a rule, the end with the movablebearing is much less rigid than the end
with the fixed bearing. The cause lies in the
difference in the bearing designs. While thefirst end with a genuine, inherently
preloaded fixed bearing must continue toremain rigid when the second end has
begun to move, with increasing
temperature the second end loses its
preload and therefore also its rigidity.

8

Position drift in [µm]

30
0 mm

20
10
0
0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

450 mm-10-20
900 mm-30-40

Time in [min]

Fig. 10: 	Drift of three positions during positioning accuracy measurement in accordancewith ISO/DIS 230-3 on a feed axis with a ball screw in fixed bearings at both ends.
Position measurements via rotary encoder and ball screw result in a distinct driftof positions due to the thermal growth of the ball screw.

Position drift in [µm] The results of measurement in Fig. 10
show curves as theoretically expected. The
end points of the ball screw cannot be kept
in their original positions. They each moveby 20 µm to 30 µm in the direction of theforce generated by heat. The total expansionof the ball screw is about 50 % less thanthat shown in Fig. 7. This means that bydesigning fixed bearings at both ends, theexpansion could be halved. The thermal
zero point of the feed axis seems to lie at
the midpoint of the traverse range. This isalso expected because the bearings haveapproximately equal rigidity and the ballscrew was heated evenly over its entire
length.

50
40

500 mm

30
250 mm

20
10

0 mm
0
0

6

12

18

24

30

36

42

48

54

-10
Time in [min] Fig. 11: 	Drift of three positions during positioning accuracy measurement in accordancewith ISO/DIS 230-3 on a feed axis with a ball screw in fixed/preloaded bearings.
The results show a distinct drift due to position measurement by ball screw androtary encoder.

Influence of the temperature distribution along the ball screw

Apart from the ratio of bearing stiffness, theposition of the thermal zero point depends
particularly on the distribution of temperature
along the ball screw. Figure 12 shows athermographic snapshot of a ball-screw
drive after several hours of reciprocating
traverse between two points separated
by 150 mm. As the thermograph shows,even after several hours the temperature
increase remains almost exclusively in thearea of the ball nut traverse. The temperatureof the ball screw and therefore the thermal
expansion is very local.
Because the bearings of the ball screw canprovide at best only an evenly distributed
mechanical tension and ensure constant
expansion along the ball screw, they cannot
compensate the expansion resulting fromlocal temperature changes.
Fig. 12:	Local heating of a ball screw in the traverse range of the ball nut after 6 hours ofreciprocating traverse at 24 m/min between two positions separated by 150 mm[see A. Frank / F. Ruech: Position measurement in CNC Machines]. For thisthermographic snapshot, the machine table was moved aside at the end of thetraverse program. The illustration shows the higher temperatures of the belt drive,
locating bearing, and ball screw.

+10 K

 40 (35)mm
1m
700 N/µm

700 N/µm
0 N/µmPositioning error in [µm]

A simple calculation shows this clearly(Fig. 13). On a 1-m long ball screw witha fixed bearing at one end, a local
temperature increase of 10 K as indicatedby the red curve would result in a positioning
error as indicated by the green curve. A
fixed/fixed bearing configuration with arigidity of 700 N/µm results in an error
curve as indicated by the blue curve. As aresult of the forces exercised by the bearing,
the ball screw is compressed at its ends
where the temperature is not increased.
The area of the ball screw near its midpoint
expands due to the temperature increaseat almost the same rate as with the fixed/
floating configuration.
At 22 µm, the maximum positioning error
from the fixed/fixed configuration isroughly 2/3 of the error that occurs fromthe fixed/floating configuration.

40

20

0
0.5

20

1

Position in [m] Fig. 13:	Positioning error in a Semi-Closed Loop as a result of local temperature rise in therecirculating ball screw.
9Countermeasures

The test results show that the thermal
expansion of the ball screw as a result offriction in the bearings and particularly inthe ball nut results in significant positioning
errors if the axis is controlled in a SemiClosed Loop. Besides the use of linear
encoders, countermeasures aimed at
avoiding this error include coolantconducting hollow ball screws and purely
electronic compensation in the control
software.

Position error in [µm] 

Cooled ball screws
The circulation of the coolant requires ahole in the ball screw and rotating bushings
near the screw bearings. Apart from thesealing problems, the hole reduces the ballscrew’s mechanical rigidity in its alreadyweak axial direction. The greatest problem,
however, is a sufficiently accurate
temperature control of the coolant. A 1° Cchange in temperature changes the lengthof a 1-m long ball screw by 11 µm. In lightof the considerable amount of heat to be
removed it is not an easy task to maintaina temperature stability of < 1 K. This isparticularly so when the spindle or its
bearings are cooled with the same system.
In such a case, the required cooling capacity
can easily lie in the kilowatt range. The
temperature constancy of existing ballscrew chillers is usually significantly worse
than 1 K. It is therefore often not possible
to use them to control the temperature of
the ball screw. Switching controllers areoften used in the chillers to reduce costs.
Since each switching operation is triggeredby a violation of temperature limits, theindividual switching operation can beconsidered to be an expansion of thecooled ball screw and therefore an axis

positioning error. Figure 14 shows theresult of a positioning test on a verticalmachining center with liquid-cooled ballscrews in fixed/floating bearings. During
the test, the axis was moved slowly at
2.5 m/min between two points separated
by 500 mm. The maximum traverse range
was 800 mm. The position drift of theposition farther away from the fixed bearing
was recorded. The switching of the chiller
is plainly visible. Its hysteresis was 1 K.
Compared with the noncooled SemiClosed Loop design the position drift wassignificantly reduced. However, the switch
operations produce relatively quick changes
in position, which have a stronger effectduring the machining of workpieces withshort machining times than the slow
position drift evident in the noncooled
Semi-Closed Loop design.
Software compensation
Research is underway on compensation ofthermal deformation with the aid of analytic
models, neural networks, and empiricalequations. However, the main focus ofthese studies is on the deformation of themachine tool structure as a result of internaland external sources of heat. There is little
interest in investigation into compensationof axis drift.
As a whole, the possibilities of such
software compensation are frequently
overestimated in today’s generalatmosphere of enthusiasm for software
capabilities. Successful compensation in
the laboratory is usually achieved only after
elaborate special adjustments on the test
machine. It is usually not possible to apply
such methods to machines from series

10
Linear encoder

0
-10

0

10

20

30

40

Cooled ball screw / rotary encoder

-20
-30
-40
-50

Noncooled ball screw / rotary encoder

Time in [min]

Fig. 14:	X axis of a vertical machining center with liquid-cooled ball screw in fixed/floating
bearings. The diagram shows the drift of the position farthest away from the fixedbearing during reciprocating traverse over 500 mm (800-mm traverse range) at
2.5 m/min. The axis was also equipped with a linear encoder for test purposes.
10

production without time-consuming
adjustment of the individual machines.
The example of the feed axis shows thevariations of the input parameters to beconsidered.
To compensate the expansion of the ballscrew, its temperature must be knownwith respect to its position, since the local
temperature depends on the traversing
program. Direct temperature measurementof the rotating ball screw, however, is very
difficult. Machine tool builders therefore
often attempt to calculate the temperature
distribution. This is theoretically possible if
a heat analysis can be prepared for individual
sections of the ball screw. The heat in sucha section is generated by friction in the ballnut through thermal conductance along theball screw, and through heat exchange withthe environment. The friction of the ball nut
depends almost proportionately on the
preload of the ball nut and, in a complex
manner, from the type, quantity, andtemperature of the lubricant. The preload of
the ball nut normally changes by ±10 % to±20 % over its traverse range in an mannerdepending on the individual ball screw. In
the course of the first six months, the mean
preload typically decreases to 50 % of its
original value. Due to the complex interactionof static forces at play on the ball screw,certain jamming effects and an associatedincrease in friction are unavoidable. Eventhese few examples show that thecalculation of the actual frictional heat
presents formidable problems. Calculating
the heat dissipation is similarly difficult
because is depends strongly on largely
unknown ambient conditions. Even thetemperature of the air surrounding the ballscrew is normally unknown, although it
plays a decisive role in any calculation ofheat dissipation.
On the whole it seems certain that, evenin the relatively simple case of a fixed/
floating bearing, software compensationof ball-screw expansion without additionaltemperature sensors has little chance of
success. In the case of fixed/fixed andfixed/preloaded bearing one must
additionally take into account the bearing
rigidity and the preload-dependent frictionin the bearings. These factors makecompensation even more difficult.

A comparison with the usual geometricerror leads to similar results. If one observes
the pitch, roll, and yaw error of the feed
axes of 16 different NC machines one seesthat these types of error usually lie in therange of 10 µm/m to 50 µm/m (Fig. 16). The
positioning error is found by multiplying
these values by the respective Abbe
distance. The error does not attain thevalues of the feed axis until over 1 meter
traverse.

10

X

0
-10
-20
Y

-30

-50

Z

Spindle ON

-60
-70

0

1

2

3

4

5

Spindle OFF

6

7

8

9

10 11 12 13 14 15 16Time in [h] Rotational deformation in [µm/m] 

The deformation of the second machining
center is shown in Fig. 15. Under thesame conditions, also with 6000 rpm, itshows a maximum linear deformation of{X: 5 µm, Y: 45 µm, Z: 55 µm}. The rotational
deformation reached a maximum of{A: 25 µm/m, B: 10 µm/m}. The third machine
was equipped with a high-speed spindle
and jacket cooling. At 12000 rpm it showedlinear deformations of {X: 5 µm, Y: 5 µm,
Z: 40 µm} and rotational deformations ofmax. {A: 20 µm/m, B: 30 µm/m}. The
measured axis drift values attain at least
the same magnitude as the structuraldeformation. Particularly on spindles with
fixed/floating bearings, or machines witheffective cooling of the spindle, thepositioning error of the feed axes driven ina Semi-Closed Loop is significantly greaterthan the measured structural deformation.

20

-40

40
30
20

Rotation about X

10
Rotation about Y

0
-10
-20
-30
Spindle ON

-40
-50

0

1

2

3

4

5

Spindle OFF

6

7

8

9

10 11 12 13 14 15 16Time in [h] Fig. 15:	Structural deformation of a vertical machining center as a result of heat generated
in the spindle drive at 6000 rpm without load.

Deviation in [µm/m]

After this discussion of temperaturedependent positioning error of feed drives,
it remains to classify these types of error
with the other types of static and quasistatic
error in the total error budget of the tested
machining centers. The frame deformation
resulting from the heat generated by thespindle drive was examined on all threemachines in accordance with ISO/DIS 230-3.
After several hours of operation at a
maximum spindle speed of 6000 rpm,
the first machining center showed a linear
deformation of {X: 5 µm, Y: 60 µm,
Z: 15 µm}. The rotational deformation
reached a maximum of {A: 40 µm/m,
B: 70 µm/m}.

Linear deformation in [µm] Comparison of positioning error with other types of error

160
140
120

 Roll Pitch
 Yaw

100
80
60
40
20
0

500 550 600 650 700 750 900 950 1000 1200 1900 2000 2600 3200

Travel range of X axis in [mm] 

Fig. 16:	Pitch, roll, and yaw of feed axes of 16 different NC machines11Conclusion

The primary problem involved with position
measurement using rotary encoder andball screw is the thermal expansion of theball screw. With typical time constants of
1 to 2 hours, thermal expansion causes
positioning error in the magnitude of
0.1 mm, depending on the nature of thepart program. This positioning error
therefore outweighs the thermally induced
structural deformation and geometric errorof machining centers.
After every new part program the ballscrew requires approx. 1 hour to attain athermally stable condition. This also applies
to interruptions in machining. A rule of thumbfor thermal expansion is that, over the entire
length of a cold ball screw 1 meter in length,
the ball screw grows by approx. 0.5 µm to
1 µm after every double stroke. This
expansion accumulates within the time
constant.
As requirements for machine tool accuracy
and velocity increase, the role of linear
encoders for position measurement grows
increasingly important. This should betaken into consideration when deciding on
the proper feedback system design.

Literature
Schröder, Wilhelm: Feinpositionierung
mit Kugelgewindetrieben,
Fortschrittsbericht VDI Reihe 1 Nr. 277,
Düsseldorf; VDI Verlag 1997
VDW-Bericht 0153: “Untersuchung vonWälzführungen zur Verbesserung desstatischen und dynamischen Verhaltens
von Werkzeugmaschinen”
Weule, Hartmut / Rosum, Jens:
Optimization of the friction behaviour ofball screw drives through WC/C coatedroller bodies, Production Engineering
Vol. 1/1 (1993)
Golz, Hans Ulrich: Analyse,
Modellbildung und Optimierung desBetriebsverhaltens vonKugelgewindetrieben, Dissertation Uni
Karlsruhe, 1990
Schmitt, Thomas: Modell derWärmeübertragungsvorgänge in dermechanischen Struktur von CNCgesteuerten Vorschubsystemen,
Verlag Shaker, 1996
A. Frank / F. Ruech: Position
measurement in CNC Machines ...
Lamdamap conference, Newcastle 1999

More information:          
 

394843 · 01 · A · 02 · 10/2018 · PDF• Brochure: Linear Encoders for
Numerically Controlled Machine Tools• Brochure: Rotary Encoders• Brochure: Position Encoders for Servo
Drives