Please cite this article in press as: Jin C, et al. Thermal
characteristics of a CNC feed systemunder varying operating
conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 14Precision Engineering xxx
(2015) xxxxxxContents lists available at
ScienceDirectPrecisionEngineeringj our nal homepage: www. el sevi
er . com/ l ocat e/ pr eci si
onThermalcharacteristicsofaCNCfeedsystemundervaryingoperatingconditionsChaoJina,b,BoWua,YouminHua,PengxingYia,,YaoChengaaState
Key Lab of Digital Manufacturing Equipment and Technology, School
of Mechanical Science and Engineering, Huazhong University of
Science andTechnology, Wuhan 430074, Hubei, PR ChinabResearch
Institute of Zhejiang UniversityTaizhou, Taizhou 318000, Zhejiang,
PR ChinaarticleinfoArticle history:Received 28 February
2012Received in revised form11 July 2014Accepted 20 April
2015Available online xxxKeywords:Thermal errorWavelet neural
networkOperating conditionFeed systemabstractIn
high-speedandhigh-precisionfeedsystems,thermalpositioningerrorsaremainlycausedbythenon-uniformtemperaturevariationsand
resultingtime-varyingthermaldeformationsunderdifferentoperatingconditions.Theresearchpresentedhereultimatelyaimsto
developagenericmethodcapableofevaluatingthethermalcharacteristics(suchastemperatureriseofheatsources,thermalpositioningerror)ofthefeedsysteminducedby
varyingoperatingconditions(feedspeed,cuttingloadand preloadofball
screw).Thethermalcontactresistancebetweentheballsandtheinnerand
outerringsofsuppor-tingbearingis
calculatedusingtheHertziantheoryandJHMmethod.Experimentswerecarriedouton
ahigh-speedfeedsystemexperimentalbench,andtheinuencesofoperatingconditionson
temperaturerisesofsupportingbearingsandball screwnut
wereanalyzed.Basedon a
WNN-NARMAL2model,therelationshipbetweentemperatureriseofsupportingbearingsandoperatingconditionswas
established.Furthermore,withthetemperatureoftheballscrewnut
settobea
movingheatsourceload,thetem-peratureandthermaldeformationdistributionsoftheballscrewweresimulated.Theworkdescribedlaysa
solidfoundationforthermalerrorpredictionandcompensationofa
feedsystemundervaryingoperatingconditions. 2015
ElsevierInc.Allrightsreserved.1. IntroductionPositioning systems
with high speed, high accuracy and longstroke become more important
in precision machining. A high-speed precision feed system reduces
non-cutting operating timeand tool replacement time, making
production more economical.However, due to the friction at the ball
screw bearing and nut, ahigh-speed feed system generates a lot of
heat, causing thermalexpansion which adversely affects machining
accuracy. Therefore,the thermal positioning error of a ball screw
is one of the mostimportant objects to consider for high-speed and
high-precisionmachine tools.Researchers have considered many ways
of reducing thermalerrors, including the thermally symmetric design
of a structure,separation of the heat sources from the main body of
a machinetool, active cooling [1]. However, the costs associated
with theseCorresponding author. Tel.: +86 27 87557415; fax: +86 27
87557415.E-mail addresses: [email protected], [email protected]
(P. Yi).approaches are usually very high. In addition, there are
manyphysical limitations, which cannot be overcome solely by
designtechniques. As a result, error compensation techniques to
improvemachine accuracy cost-effectively have received signicant
atten-tion [13]. Some machine tool manufacturers have
implementedsoftware in their open-architecture CNC controller to
compen-sate the thermal error in real time [4]. Position feedback
systemsthat do not rely on the ball screw, such as scales,
represent awell-establishedandsuccessful approachtoreducingthermal
posi-tioning errors of feed system. However, the thermal errors of
aball screw system are caused by the non-uniform
temperaturevariations and time-varying thermal deformations in the
machinestructure. Therefore, accurate modeling of thermal errors
remainsa key challenge of error compensation.Most current research
is focused on the thermal error compen-sation of the whole machine
tools. Thermally induced error is atime-dependent nonlinear process
caused by nonuniformtemper-ature variation in the machine
structure. The interaction betweenthe heat source location, its
intensity, thermal expansion coef-cient and machine system
conguration creates complex
thermalhttp://dx.doi.org/10.1016/j.precisioneng.2015.04.0100141-6359/
2015 Elsevier Inc. All rights reserved.Please cite this article in
press as: Jin C, et al. Thermal characteristics of a CNC feed
systemunder varying operating conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 142 C. Jin et al. / Precision
Engineering xxx (2015) xxxxxxbehavior. Researchers have employed
various techniques, namely,nite element methods, coordinate
transformation methods, neu-ral networks, etc, in modeling the
thermal characteristics [59,3].Venugopal et al. [10] carried out
different types of experimentsunder no-load conditions. The thermal
error was predicted basedon the temperature of the lead screw nut
by using a basic linearexpansion model. Veldhuis et al. [11]
conducted ve different testsover a period of 10h. The thermal error
generated as a result ofthese tests was mapped against the
temperatures measured by 17thermocouples mountedonthe machine by
using a neural networkmodel. A nite element approach to simulate
the thermal behaviorof the ball screwtransmission was presented in
[12]. When deter-mining thermal errors arising in ball screws, it
is very important toaccurately identify the temperature
distribution along the screwand on this basis determine its axial
thermal elongation. Heiselet al. [13] used an infrared camera to
measure temperatures alongthe screw. Anexperimentally
determinedtemperature distributionand measured positioning errors
for 4000 cycles were obtained.In a high-speed feed system, bearings
are considered to be themain heat sources, and the thermal
properties of the bearingsneed to be carefully studied. For a
bearing, the thermal resistancesfor conduction through the bearing
elements themselves and forradiation can be calculated using the
dimensions, the thermal con-ductivities, the thermal-optical
properties, and the temperaturesof the elements. However, it ca be
said that the thermal contactresistances between the balls and the
rings, which are most closelyrelated to the temperature differences
across the bearings, are dif-cult to predict because few useful
calculation method have beenproposed yet. Fromthe above, it can be
seen that most of the workcarriedout thus far is basedonthe
principle of directly mapping thethermal error against the
temperature of critical machine elementsirrespective of the
operating conditions. The different operatingparameters used were
just a means to generate varying thermalstates on the machine. But
researchers [1,14] discovered the pointthat the thermal error of a
machine tool was strongly dependentupon the specic operating
parameters and conditions that themachine was put through.
Accordingly, research on changing ruleand dynamics characteristics
of heat sources, temperature eld,and structure thermal deformation
must be taken into account toreduce the error of thermal
deformation and improve the workingaccuracy of a machine
tool.Theresearchpresentedhereultimatelyaims at thedevelopmentof a
comprehensive model that can predict the temperature risesof heat
sources and thermal characteristics in a ball screwCNC
feedsystemunder different operating conditions (feed speed, load
andpreload of the ball screw). The thermal contact resistance
betweenthe balls and the inner and outer rings of supporting
bearing is cal-culated in Section 2. The experimental setup is
described in Section3. Based on an orthogonal experimental design,
the relationshipbetween temperature rise of bearings and operating
conditions isestablished based on a WNN-NARMAL2 neural network in
Section4. Furthermore, with the temperature of the ball screw nut
as amoving heat source load, the temperature and thermal
deforma-tion distributions of the ball screw under bearings and
ball screwnut heat sources were simulated in Section 5.2.
Expressions for thermal contact resistance2.1. Contact
resistanceThe contact resistances between the balls and the inner
andouter rings maybe treated in the same manner as
constrictionresistance since both resistances result from the
restriction of theheat owdue to small contact arrears. In the
ellipsoidal coordinatesystemthe Laplaces equation is:2T =u__f
(u)Tu_(1)where_f (u) =_(a2+u)(b2+u)u (2)And , b are the semi-major
and semi-minor axes of the ellipticcontact area, respectively;
whileis the variable along an axisnormal to the contact plane.The
boundary conditions are:u = 0, T = T0(3)u ,T = 0 (4)With Eqs. (1),
(3) and (4), the temperature distribution can beobtained: [6]T
=Q4k_udu_f (u)(5)where Q is all the heat leaving the elliptic
contact area. And by thedenition of the thermal contact
resistanceR=T0TuQ=14k_0du_f (u)=14k_0du_(a2+u)(b2+u)u(6)Using the
complete elliptic integral of the rst kind, Eq. (6) canbe written
in the following formasR =(a/b)4ka, (a/b) =2
F_e, 2_(7)Then, the contact thermal resistance between the ball
and theinner or outer ring can be determined by using Eq. (7). For
mostbearing, whose ball and both rings are made fromthe same
mate-rial, wecan write the contact thermal resistance per ball as:R
=12k_(ai/bi)ai+(ao/bo)ao_(8)These expressions permit us to predict
the total contact resis-tance resulting from the contact of an
arbitrary number of ballswith both the inner and outer rings by
connecting the thermalresistances in parallel.2.2. Contact areas in
a ball bearingWhen two elastic bodies having smooth round surface
are pressagainst each other, the contact area becomes elliptic.
Assumed thatthe angle between the two planes containing the
principal radiiof curvature of the bodies are perpendicular as in
the case of ballscontactingthe inner or outer ringof a bearing, the
followingexpres-sions can be derived:a = a_34PA +B_1 v21E1+1
v22E2__1/3b = b_34PA +B_1 v21E1+1 v22E2__1/3(9)A =12_1r1+1r2_, B
=12_1r
1+1r
2_(10)In which r1, r
1 are the radius of curvature for inner or outer raceand groove,
respectively. And r2, r
2 are the radii of rolling ball. Con-sidering the bearing model
shown in Fig. 1, for the contact at innerring side, the radius of
curvature r
1of the inner groove must bePlease cite this article in press
as: Jin C, et al. Thermal characteristics of a CNC feed systemunder
varying operating conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 14C. Jin et al. / Precision
Engineering xxx (2015) xxxxxx 3Fig. 1. Schematic of ball
bearing.treated as negative in Eq. (10); while at the outer ring
side contact,r1, r
1 must be treated as negative.The values of a* and b* are
calculated as follows:___I =2e2_F_e, 2_E_e, 2__J =2e2__E_e, 2_1
e2F_e, 2___JI =AB(11)InwhichF(e, /2) andE(e, /2) are the complete
elliptic integralsof the rst and second, respectively.F_e,
2_=_/20(1 e2sin2)12 dE_e, 2_=_/20(1 e2sin2)12 d(12)Eq. (11) can be
solved numerically by the NewtonDownhillmethod, and then e can be
determined, the value of I and J canbe calculated. Finally,a =_I
+J
_1/3, b = a(1 e2)1/2(13)2.3. Distribution of internal loading
under centric thrust loadIn ball bearing, depending on the contact
angles, ball gyroscopicmoments and ball centrifugal forces can be
of signicant magni-tude such that inner raceway contact angles tend
to increase andout raceway contact angles tend to decrease. Under
zero load thecenters of the raceway groove curvature radii are
separated by adistance BD dened by (as shown in Fig. 1)BD = (r
i +r
o2 rb) = (fi+fo1)D(14)Angular contact ball bearings subjected to
a centric thrust loadhave the load distributed equally among the
rolling elements.HenceQ =FaZ sin(15)where a is the contact angle
that occurs in the loaded bearings, andcan be determined as
follows. In the unloaded condition, the initialcontact angle is
dened bycos = 1 Pd2BD(16)Fig. 2. Angular contact ball bearing under
thrust load.InwhichBis thetotal curvature, andPdis
themounteddiametralclearance.Pd = dodi2D (17)A thrust load
Faapplied to the inner ring as shown in Fig. 2causes an axial
deection a. This axial deection is a componentof a normal deection
along the line of contact such that fromFig. 2n = BD_cos ocos
1_(18)Since Q = K1.5n, where K is the load-deection factor.
Substitut-ing Eq. (18) into (15), we get,FaZK(BD)1.5 = sin_cos ocos
1_1.5(19)Eq. (19) maybe solved numerically by the
NewtonRaphsonmethod, the equation to be satised iteratively is,
= +FaZK(BD)1.5 sin_cos ocos 1_1.5cos _cos ocos 1_1.5+1.5tan2cos
o_cos ocos 1_0.5(20)With 7204AC ball bearing as an example, under
w=5000rpmand Fa=0200N, the contact resistance is shown in Fig.
3.FromFig. 3, we can see with the thrust load increasing, the
ther-malcontact resistance decreases. That is because when the
thrustload increases, the normal load of each ball increases, then
the con-tact area extends. Withconsiderationof thermal contact
resistance,under w=5000rpmand Fa=200N, the temperature distribution
of7204AC ball bearing is shown in Fig. 4. The temperature
compari-sonof outer ring withandwithout considerationof thermal
contactresistance is shown in Fig. 5.FromFig. 4 the temperature
distribution of inner and outer ringwas inconsistent because
existence of thermal contact resistance.Please cite this article in
press as: Jin C, et al. Thermal characteristics of a CNC feed
systemunder varying operating conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 144 C. Jin et al. / Precision
Engineering xxx (2015) xxxxxxFig. 3. Thermal contact resistance
under thrust load.Fig. 4. Temperature distribution of ball
bearing.In Fig. 5, there is obvious difference between simulation
and mea-sured results when thermal contact resistance is not
considered.3. Experimental setupThe experimental apparatus is shown
in Fig. 6.The experimental bench is mainly a ball screwsystem. The
solidball screwhas anominal diameter of 32mm,leadof 20mm, travel
of700mm,thermal expansion coefcient of 12.3103mm/C, andis xed to
substructure by a back-to-back arrangement of angularcontact ball
bearings. An AC servomotor with maximum rotatingspeed of 3000rpmis
connected to the ball screw with elastic cou-plings. Theoretically,
the maximumfeed speed of the experimentalbench can reach 60m/min.
Furthermore, by adjusting the reliefvalves at both ends of the load
cylinder, axial load is implementedwithin the range of 04kN. The
preload of the ball screw can beadjusted by rotating the nuts near
the bearing housing using adigital-display torque wrench.A VM182
linear grating scale with a resolution of 0.5mis usedto measure
positioning errors at different positions of the feed sys-tem(as
seen in Fig. 7a). Pt100 resistance temperature sensors witha
measuring range of 0150C are used to measure the outer
ringtemperatures of the ball bearings (as seen in Fig. 7c and d)
andthe temperatures of the ball screw nut. The ball screw
tempera-tures at measuring points x =60mmand x =540mmwere
acquiredby infrared radiation thermometers with a resolution of
0.2C (asseen in Fig. 7e). The axial thermal displacement at the
right end ofthe ball screw was measured by a laser displacement
sensor LK-G30 with a resolution of 5 micrometer (as seen in Fig.
7f). The axialload is acquired by the pressure sensor installed
with the reliefvalves. According to the experiences of selecting
heat sensitivepoints [15,16], temperature sensors are mainly set at
the nearestplaces to heat sources. In the experiments, temperature
sensors aremainly placed at drive motor housings, left/right
bearings and ballscrew nut. The measuring points of temperature are
depicted indetail in Fig. 8.According to ISO230-2 Determination of
accuracy and repeat-ability of positioning of numerically
controlled axes and ISO230-3Determination of thermal effects
Determination of thermaleffects, the drive systemmakes
reciprocating movements at a cer-tain speed within the travel as
depicted in Fig. 9. In-between setsof back-and-forth movements, the
positional deviations at eightmeasuring points are measured by the
VM182 linear grating scale.4. Relationship between temperature
rises of heat sourcesand operating conditions4.1. Inuence of
varying operating conditions on temperature riseThe temperature
rises of a heat sources depends on the varyingoperating conditions,
such as runtime, travel, feed speed, cuttingload, ball
screwpreloadandsoon. Inorder toanalyze the inuencesof operating
conditions on temperature rise, an orthogonal exper-iment was
carried out on the HUST-FS-001 experimental benchFig. 5.
Temperature comparison of outer ring.Please cite this article in
press as: Jin C, et al. Thermal characteristics of a CNC feed
systemunder varying operating conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 14C. Jin et al. / Precision
Engineering xxx (2015) xxxxxx 5Fig. 6. High-speed feed
systemHUST-FS-001.Fig. 7. Photographs of the experimental setup.
(a) Linear grating. (b) Pressure sensor. (c) Temperature measuring
point of right bearing. (d) Temperature measuring point
ofleftbearing. (e) Temperature measuring point at x =540mmon the
ball screw. (f) Thermal displacement sensor at the right end of the
ball screw.Fig. 8. Temperature measuring points and data
acquisition system. (A) Ball screwnut. (C) Right bearing. (D) Left
bearing. (E) Motor housing.Please cite this article in press as:
Jin C, et al. Thermal characteristics of a CNC feed systemunder
varying operating conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 146 C. Jin et al. / Precision
Engineering xxx (2015) xxxxxxFig. 9. Standard test cycle for
thermal evaluation.considering feed speed, cutting load, ball screw
preload as con-trollable factors. Each operating condition has ve
different levels,so the orthogonal experiment was designed using
two orthogonaltables L25 (56) (up to 6 factors at 5 levels each)
[17]. The two tablesrepresent a total of 10 different levels for
the feed speed. Each ofthe 50 tests started from a cold state and
lasted for 40minwhenthe experimental bench can reach a thermal
steady state. Becausethe preload of the ball screw is difcult to
adjust, the orthogonalexperiment was rearranged as shown in Table
1.Using an analysis of variance [17], the inuences of
operatingconditions (feed speed, preload of ball screwand cutting
loads) onthe temperature rise of the right bearing can be evaluated
as shownin Tables 2 and 3.In Tables 2 and 3, ST is the total
corrected sumof squares of theexperiment, and ST has fT degrees of
freedom, fT=251=24. SV, SPand SFare the sums of squares due to
different operating condi-tions, and they each have 4 degrees of
freedom. Seis the sum ofsquares of the residual errors, and Sehas
fe=fT34=12 degreesof freedom. VV, VPand VFare the average sum of
squares of theoperating conditions at deferent levels, and Fi is
the F check value[17]. r is the number of different levels, r =5.
T1, T3, . . .,T25 are thesums of the temperature rises at different
feed speed levels. T20,T40, . . .,T100 are the sums of the
temperature rises at different ballscrew preload levels. T0.5,
T0.75, . . .,T1.5are the sums of the tem-perature rises at
different cutting load levels. With increasing feedspeed, ball
screwpreload and cutting load, the frictional heat in thebearings
becomes larger, resulting in higher bearing temperatures.FromTables
2 and 3, it can be seen that all the three operating con-ditions
affect the temperature rise of the right bearing, with thepreload
of the ball screwas the most signicant factor.The inuences of
operating conditions (feed speed, preload ofball screw and cutting
loads) on the temperature rise of the ballscrewnut are shown in
Tables 4 and 5. It can be seen that the tem-perature rise of the
ball screw nut is signicantly affected by boththe feed speed and
cutting load. As expected, the ball screwpreloadshows no signicant
effect onthe temperature rise of the ball screwnut.4.2. Prediction
of temperature rise under varying operatingconditions4.2.1. Wavelet
neural network based on the NARMA-L2 modelIn the past few decades,
modeling and identication tech-niques for nonlinear systems have
been extensively studied inapproximation, prediction, and control.
Several nonlinear mod-els have been proposed in the literature,
including the NARMA-L2model representation proposed by Narendra et
al. [18], which wasintroduced as an approximation of the nonlinear
autoregressivemoving averaging (NARMA) model. The NARMA-L2 model
takesthe formof the following nonlinear difference equation:y (k+1)
= f [y (k) ,y (k1) , . . .,y (kn +1) , U (k 1) , . . .,U (k n +1)]
+g [y (k) , y (k1) , . . .,y (k n +1) , U (k 1) , U (k 2) , . . .,U
(k n +1)] U (k)(21)where the input U(k) =[V(k), P(k), F(k)], and
output y(k) =T(k). V(k),P(k) and F(k) denote the feed speed, ball
screwpreload and cuttingload respectively at time step k. (n1)
represents the number ofhistory terms considered in the model. The
output of the networky(k) is the temperature rise T(k) of either
the bearing or the ballscrew nut. The NARMA-L2 model is nonlinear
with respect to theinput and output, but linear with respect to the
current input U(k).Because of the good function approximation
ability of theWavelet Neural Network (WNN) [19,20], a modied
wavelet neu-ral network (WNN-NARMAL2) based on the NARMA-L2 model
isintroduced, with the topological structure shown in Fig. 10.
Theupper part of the whole network is an approximation of
g[y(k),y(k 1), . . .,y(k n+1), U(k 1), U(k 2), . . .,U(k n+1)]U(k)
inEq. (21), and the lower part is an approximation of f[y(k), y(k
1),. . .,y(k n+1), U(k 1), U(k 2), . . .,U(k n+1)]. In both parts,
allthe units in each layer are fully connected to the nodes in the
nextlayer except for the input U(k). In the output layer, a
log-sigmoidtransfer function 1/(1+ex) is used to limit the output
range.To train the WNN-NARMAL2 network, the weights and thewavelet
coefcients are updated by the Particle SwarmOptimiza-tion (PSO)
algorithm [21]. The parallel search strategy of the PSOreduces the
possibility of premature convergence at local optima.The connection
weights and the wavelet coefcients of the WNN-NARMAL2 are
considered to be the particles of a population.According to Fig.
10, the ith particle is represented as follows:Xi=_Wih1i, aih1,
bih1, Wijh1, Wih2i, aih2, bih2, Wih2_(22)where Wh1i and Wh2i are
the connectionweights betweenthe inputand hidden layers of the
upper and lower part, respectively. Wjh1andWh2
aretheconnectionweights betweenthehiddenandoutputlayers of the
upper and lower part, respectively. ah1, ah2, bh1 and bh2are
dilation and translation parameters of wavelet functions.To
validate the predictive capability of the presented WNN-NARMAL2
model, the sequence of operations listed in Table 6 werecarried out
on the HUST-FS-001 experimental bench. The total run-time of the
experiment was 120min.The temperature rises of bearings and ball
screwnut are shownin Fig. 11.Please cite this article in press as:
Jin C, et al. Thermal characteristics of a CNC feed systemunder
varying operating conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 14C. Jin et al. / Precision
Engineering xxx (2015) xxxxxx 7Table1Orthogonal experiment results
for the temperature rise of bearings and ball screwnut.No Operating
conditions Temperature-riseof screwnut (C)Temperature-riseof left
bearing (C)Temperature-riseof right bearing (C)Feed speed V (m/min)
Ball-screwpreload P (Nm)Cutting load F (kN)1 1 20 0.67 2.3 0.3
0.423 20 1 2.6 1.1 1.235 20 1.33 3.2 1.5 1.748 20 1.67 4.0 1.8
1.9510 20 2 5.1 1.9 2.061 40 1 2.4 1.1 1.173 40 1.33 2.9 1.8 1.985
40 1.67 3.6 2.6 2.798 40 2 4.7 2.9 3.01010 40 0.67 3.2 1.9 2.1111
60 1.33 2.6 2.2 2.4123 60 1.67 3.3 2.3 2.5135 60 2 4.1 3.2 3.2148
60 0.67 3.1 2.3 2.31510 60 1 3.7 2.9 3.1161 80 1.67 3.3 2.7 2.8173
80 2 3.7 3.1 3.2185 80 0.67 2.4 1.9 2.0198 80 1 3.4 3.0 3.02010 80
1.33 3.9 3.5 3.6211 100 2 3.6 3.3 3.5223 100 0.67 2.2 2.2 2.2235
100 1 2.8 2.8 2.9248 100 1.33 3.7 3.7 3.72510 100 1.67 4.4 4.4
4.42612 20 0.67 3.4 1.6 1.72715 20 1 4.3 2.2 2.32818 20 1.33 5.0
2.7 2.82921 20 1.67 5.6 3.5 3.53025 20 2 6.3 4.1 4.33112 40 1 4.0
2.6 2.73215 40 1.33 4.6 3.4 3.63318 40 1.67 5.4 4.2 4.33421 40 2
6.0 5.1 5.23525 40 0.67 4.7 3.0 3.13612 60 1.33 4.3 3.3 3.43715 60
1.67 5.1 3.9 3.93818 60 2 5.8 5.2 5.33921 60 0.67 4.4 3.6 3.84025
60 1 5.1 4.1 4.14112 80 1.67 4.6 3.9 4.14215 80 2 5.5 4.7 4.74318
80 0.67 4.1 3.5 3.74421 80 1 4.9 4.5 4.64525 80 1.33 5.4 4.9
5.14612 100 2 5.0 4.6 4.74715 100 0.67 3.9 3.9 4.04818 100 1 4.7
4.7 4.74921 100 1.33 5.2 5.2 5.35025 100 1.67 5.8 5.9 6.0Table
2Variance analysis results of the temperature rise of the right
bearing (feed speed 1m/minto10m/min).Operating conditions
SifiViFiFeed speed (V) SP =T202+T402+T602+T802+T1002r1n_n
i=1Ti_2= 11.43 4 VP =Sp4= 2.86VPVe= 57.2Ball-screwpreload (P) SP
=T202+T402+T602+T802+T1002r1n_n
i=1Ti_2= 11.05 4 VF =SF4= 1.92VPVe= 43.13Cutting load (F) SF
=T0.672+T12+T1.332+T1.672+T22r1n_n
i=1Ti_2= 4.51 4 Ve =Se12 = 0.05 ST =n
i=1Ti21n(n
i=1Ti)2= 24.98Residuals (e) Se =ST SV SP SF =0.77 12 Ve =Se12 =
0.064ST ST =n
i=1Ti21n_n
i=1Ti_2= 19.81 24Please cite this article in press as: Jin C, et
al. Thermal characteristics of a CNC feed systemunder varying
operating conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 148 C. Jin et al. / Precision
Engineering xxx (2015) xxxxxxTable 3Variance analysis results of
the temperature rise of the right bearing (feed speed 12m/minto
25m/min).Operating conditions SifiViFiFeed speed (V) SV
=T122+T152+T182+T212+T252r1n_n
i=1Ti_2= 5.27 4 VV =SV4= 1.32VVVe= 26.4Ball-screwpreload (P) SP
=T202+T402+T602+T802+T1002r1n_n
i=1Ti_2= 11.43 4 VP =Sp4= 2.86VPVe= 57.2Cuttingload (F) SF
=T0.672+T12+T1.332+T1.672+T22r1n_n
i=1Ti_2= 7.68 4 VF =SF4= 1.92VFVe= 38.4Residuals (e) Se =ST SV
SP SF =0.60 12 Ve =Se12 = 0.05ST ST =n
i=1Ti21n_n
i=1Ti_2= 24.98 24Table 4Variance analysis results of the
temperature rise of the ball screwnut (feed speed 1m/min
to10m/min).Operating conditions SifiViFiFeed speed (V) SV
=T12+T32+T52+T82+T102r1n_n
i=1Ti_2= 5.77 4 VV =SV4= 1.44VVVe= 144Ball-screwpreload (P) SP
=T202+T402+T602+T802+T1002r1n_n
i=1Ti_2= 0.05) 4 VP =SP4= 0.0125VPVe= 1.25Cutting load (F) SF
=T0.672+T12+T1.332+T1.672+T22r1n_n
i=1Ti_2= 7.70 4 VF =SF4= 1.925VFVe= 192.5Residuals (e) Se =ST SV
SP SF =0.12 12 Ve =Se12 = 0.01ST ST =n
i=1Ti21n_n
i=1Ti_2= 13.64 24Table 5Variance analysis results of the
temperature rise of the ball screwnut (feed speed 12m/minto
25m/min).Operating conditions SifiViFiFeed speed (V) SV
=T122+T152+T182+T212+T252r1n_n
i=1Ti_2= 4.39 4 VV =SV4= 1.10VVVe= 366.7Ball-screwpreload (P) SP
=T202+T402+T602+T802+T1002r1n_n
i=1Ti_2= 0.004 4 VP =SP4= 0.001VPVe= 0.333Cutting load (F) SF
=T0.672+T12+T1.332+T1.672+T22r1n_n
i=1Ti_2= 7.88 4 VF =SF4= 1.97VFVe= 656.7Residuals (e) Se =ST SV
SP SF =0.036 12 Ve =Se12 = 0.003ST ST =n
i=1Ti21n(n
i=1Ti)2= 12.31 24Table 6Sequence of operating conditions for
validation of the WNN-NARMAL2 model.Operating conditions Experiment
step1 2 3 4 5 6 7Feed speed V (m/min) 20 8 10 15 13 25 20Preload of
ballscrewP (Nm)60 60 60 60 60 60 60Cutting load F (kN) 1.33 1.67 2
2 2 1.67 1.33Cycles*120 40 100 50 40 100 30Runtime t (min) 26 16 27
11 12 20 8*During every back-and-forth test cycle, there are time
intervals (0.5s) for linear grating scale to measure the positional
deviations at each measuring points andtimeintervals (0.5s) for the
experimental bench to return to the original testing point.
Considering 16 measuring points, there is totally (160.5+0.52) =9s
used formeasurement and original return in each cycle.Please cite
this article in press as: Jin C, et al. Thermal characteristics of
a CNC feed systemunder varying operating conditions. PrecisEng
(2015), http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE
IN PRESSG ModelPRE-6229; No. of Pages 14C. Jin et al. / Precision
Engineering xxx (2015) xxxxxx 9Fig. 10. Topological structure of
the WNN-NARMAL2 neural network.Fig. 11. Temperature rises of the
bearings and ball screw nut under varying oper-atingconditions.As
known, system identication is the premise of prediction,so the rst
2/3 of the data were used to train the network. Theestimated
network wasthen use to predict the temperature riseof the remaining
data, using the sequence of operating conditionsas input. In this
paper, the temperature rise of the right bearingis selected for
prediction. The training and prediction results areshown in Fig.
12. In Fig. 12, the solid lines show the experimentaldata, while
the dashed lines and dashed lines with markers denotethe training
and prediction results respectively. It can be seenthat the
training and prediction errors of temperature rise werebetween 0.4C
and 0.4C. Furthermore, this model describes theeffects of changes
in feed speed and cutting load very well, whichshows this model is
robust to variations in feed speed and cuttingload. Wecompared the
temperature rise predicted by a traditionalback propagation (BP)
neural network with that of the proposednetwork, and the results
showthat the WNN-NARMAL2 model hasbetter prediction accuracy.
Moreover, because of the lack of tracingability for small input
changes, it seems that the BP network is notcapable of prediction
in this situation.4.2.2. Prediction of temperature rise under
varying operatingconditionsIn order to select training sets closest
to operating conditions, adistance parameter can be dened based on
the results of varianceanalysis:distance =__SV V/25_2+_SF F/2_2+_SP
P/100_210(23)Fig. 12. Training and prediction of temperature rise
under varying operating conditions.Fig. 13. Relationship of the
distance parameter and the right bearing temperature rise.Please
cite this article in press as: Jin C, et al. Thermal
characteristics of a CNC feed systemunder varying operating
conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 1410 C. Jin et al. / Precision
Engineering xxx (2015) xxxxxxTable 7Training data selection for
unknown operating conditions.Feed speed V(m/min)Ball-screwpreload
P(Nm)Cuttingload F (kN)DistanceTraining data 1 3 100 0.67
1.1434Training data 2 25 80 1.33 1.0796Unknown conditions 6 80 2
1.0915Fig. 14. Predictionof temperature rise of the right bearing
under different operatingconditions.The relationship of the
distance parameter and right bearingtemperature rise is shown in
Fig. 13. It can be seen the distanceparameter is an effective
indicator which can used to identify thetemperature rise of heat
sources with high positive correlation.Based on the distance
parameter, the temperature rises of rightbearing in the orthogonal
experiment were classied into ve cat-egories using a dynamic
clustering algorithm [22] (as shown inFig. 13 with different
markers). In order to predict the temperaturerise under unknown
operating conditions, the two optimal clos-est sets of training
data can be obtained from the ve categorieswith the distance
parameter as indicator. For operating conditionsV=6m/min,
P=80NmandF =2kN, thethus identiedtrainingdataare shown in Table 7.
Although the sets V=3m/min, P=80Nm,F =2kN (with distance =1.0877)
and V=15m/min, P=80Nm,F =2kN (with distance =1.1178) are closest to
the given operatingconditions, there have not been chosen to be the
training data inorder to keep dispersity of the ve categories
[22].The WNN-NARMAL2 model describes the relationship
betweenoperating conditions and temperature rise. With the neural
net-work trained by data from the orthogonal experiments
identiedinTable 7, vericationis showninFig.
14withcomparisonbetweenthe measured and predicted temperature
rise.In Fig. 14, the solid lines with different markers showthe
exper-imental data, while the dashed lines with different markers
denotethe training and prediction results. After training, the
tempera-ture rise under the operating conditions of V=6m/min,
P=80Nmand F =2kN can be predicted as shown by the dashed line with
Fig. 15. The nite element model of the ball screwsystem.Fig. 16.
Temperature distribution of the ball screwunder bearings and ball
screwnut heat sources.Please cite this article in press as: Jin C,
et al. Thermal characteristics of a CNC feed systemunder varying
operating conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 14C. Jin et al. / Precision
Engineering xxx (2015) xxxxxx 11Fig. 17. Comparison of the
temperature rise between nite element simulation and measurement
results at x =60mm.Fig. 18. Comparison of the temperature
distribution between the nite element simulation and measurement at
60min.markers in Fig. 14. Furthermore, it can be seen that the
predictionerrors of temperature rise were between 0.2C and 0.2C. As
thepreloads of the three data sets are different in Table 7, the
predic-tion results in Fig. 14 veried that the model is robust to
variationsin the preload of the ball screw.5. Temperature and
thermal deformation distribution ofthe ball screwunder heat
sourcesIn this paper, a nite element model for the ball screw
systemwas established using the ANSYS software package as shown
inFig. 19. Comparison of temperature distribution between the nite
element simulation and measurement at 120min.Please cite this
article in press as: Jin C, et al. Thermal characteristics of a CNC
feed systemunder varying operating conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 1412 C. Jin et al. / Precision
Engineering xxx (2015) xxxxxxFig. 20. Thermal deformation
distribution of the ball screwwith the nite element simulation.Fig.
21. Comparison of the thermal deformation between nite element
simulation and experiment at x =580mm.Fig. 15. The temperature and
thermal deformation distributions ofthe ball screwunder bearing and
ball screwnut heat sources weresimulated.In the nite element model,
the X-axis is the axis of the ballscrew. The origin of the
coordinate systemis at the left end bearing.The SOLID90 element was
used to simulate the temperature elddistribution. To obtain good
calculating precision and speed simul-taneously, the elements of
the bearings are meshed much morerened than other ball
screwsystemcomponents. There are a totalof 30,355 solid elements in
the nite element model. With the dataas shown in Fig. 7, the
temperature of the ball screwnut wasset tobe a moving heat source
load within the travel [23].The predicted temperature distribution
of the ball screwunderbearings and ball screw nut heat sources is
shown in Fig. 16. Thecomparison of the temperature rise at 60mm in
the x directionbetween measurement and nite element simulation are
shown inFig. 22. Comparison of the deformation distribution between
nite element simulation and experiment at 60min.Please cite this
article in press as: Jin C, et al. Thermal characteristics of a CNC
feed systemunder varying operating conditions. PrecisEng (2015),
http://dx.doi.org/10.1016/j.precisioneng.2015.04.010ARTICLE IN
PRESSG ModelPRE-6229; No. of Pages 14C. Jin et al. / Precision
Engineering xxx (2015) xxxxxx 13Fig. 23. Comparison of the
deformation distribution between nite element simu-lation and
experiment at 120min.Fig. 17. The comparisons of temperature
distribution along the ballscrew between experiment measurement and
the nite elementsimulation are shown in Figs. 18 and 19.Having
obtained the temperature distribution of the ballscrew, the thermal
element can be switched to structure ele-ment, and the thermal
deformation distribution of the ball screwcan be calculated.
Considering that in the experimental benchthe ball-screw is xed at
one end and preloaded at the other,the displacement along x
direction at the right bearing endis xed. The modeled thermal
deformation distribution of theball screw is shown in Fig. 20 under
the operating conditionslisted in Table 6. The comparison of the
thermal deformation at580mm in the x direction between the
experiment measurementand nite element simulation is shown in Fig.
21. The compar-isons of thermal deformation distribution along the
ball screwbetween experiment measurement and nite element
simula-tion are shown in Figs. 22 and 23. Considering both bearing
andball screw heat sources, the thermal deformation distribution
ofthe nite element simulation was consistent with
experimentresults.6. ConclusionThermal error compensation is
challenging for high-speed feedsystems. A generic model is
presented in this paper, which iscapable of predicting the
temperature rise of sensitive points andthermal positioning error
of the feed system induced by varyingoperating conditions (feed
speed, cutting load and preload of theball screw). First, based on
an orthogonal experiment carried outon the custom-built HUST-FS-001
experimental setup, the inu-ences of varying operating conditions
on the temperature rise ofbearings and ball screw nut were
analyzed. The results show thatpreload of the ball screw is the
most signicant factor affectingthe temperature rise of the
bearings, while feed speed and cut-ting load are the most signicant
factors affecting the temperaturerise of the ball screw nut.
Second, the relationship between thetemperature rise of sensitive
points (bearings, ball screw nut) andoperating conditions was
established with the WNN-NARMAL2model. The WNN-NARMAL2 model can be
used in twoways: (1)During machining, it can predict the
temperature rise of heatsources under varying operating conditions,
so on-line operatingcondition optimization can be achieved. (2)
Before machining, itcanpredict thetemperatureriseof heat sources
under various oper-ating conditions, so the most suitable operating
conditions can bechosen. Third, fromthe predicted temperature rise
of the bearingsand ball screwnut, the temperature and thermal
deformation dis-tribution of the ball screw were acquired by the
Group Explicit(GE) nite difference method and nite element
simulation. Theresults show that the temperature distribution near
supportingbearings was mostly decided by the bearing heat source,
whilethe temperature of the ball screw within the travel was
affectedby the moving ball screw nut heat source. Finally, the
relation-ship between thermal deformation of the ball screw and
thermalpositioning error of the feed system wasestablished by a
NARMAmodel.With knowledge of the operating conditions and history
data,the proposed method can be trained off-line, and the thermal
char-acteristics of a feed systemcan be identied. The trained model
canbe used for on-line prediction of the temperature distribution
andthe thermal positioning error of a feed system using the
operat-ing conditions as the inputs. Furthermore, the thermal
positioningerror can be feed back to the NC systemto implement a
closed-loopcontrol. To summarize, the method presented in this
paper lays asolid foundation for thermal error compensation in
high-precisionfeed systems.Future research will be focused on
applying the presentedmethod into anactual agile manufacturing
environment withvary-ing operating conditions, where axial loads
are dominated byinertial loads, and repeated motion of the axis
over a small rangewill occur. For preload of ball screw, it can be
tested ofine dur-ing machining intervals or obtained by other
advanced methodsonline [24]. Considering of prediction errors in
thermal deforma-tion, more results will be usedtoassess andimprove
the robustnessof the WNN-NARMAL2 model for predicting the
temperature riseof critical machine elements and the robustness of
the models forpredicting the resulting thermal positioning
errors.AcknowledgementsThe work here is supported by the National
Natural ScienceFoundation of China (No. 51175208, No. 51075161),
the State KeyBasic Research Program of China (NO.2011CB706803), the
ChinaPostdoctoral Science Foundation(No. 2012M511609) andthe
Tech-nology Innovation Foundation of Huazhong University of
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