Research Article
A study of cooling system in a
grease-lubricated precision spindle
Advances in Mechanical Engineering
2016, Vol. 8(8) 1–15
Ó The Author(s) 2016
DOI: 10.1177/1687814016665296
aime.sagepub.com
Li Wu and Qingchang Tan
Abstract
In this article, an integrated thermal model is numerically analyzed to calculate the heat generation of spindle bearings
and temperature distribution of the spindle system, with consideration of the operating conditions and lubrication conditions, such as rotation speed, preload, and oil film thickness. Surface roughness is an important parameter of elastohydrodynamic lubrication, which is measured by an optical topographer and characterized by statistical parameters. The
experimental measurement system of the precision spindle is composed of a set of laser triangulation sensors and thermal infrared imagers. After the thermal characteristics, mathematical models are experimentally verified, and the effects
of cooling oil temperature, cooling oil flow rate, and characteristic dimensions of the cooling channel on keeping the
spindle gradient are analyzed in detail. The results indicated that temperature gradient across the spindle system can be
significantly controlled by optimizing the technical parameters of the oil-cooling system.
Keywords
Precision spindle, heat generation, elastohydrodynamic lubrication, oil film thickness, surface roughness, temperature
gradient, oil-cooling system
Date received: 2 April 2016; accepted: 29 July 2016
Academic Editor: Ramoshweu Lebelo
Introduction
Research on crucial functional components of
machine tools, especially high-precision, high-power,
and high-speed spindles, is an effective method
to improve modernized manufacturing industries.
Spindles must be assembled with high rotation precision and dynamics performance under the influence of
rotational speeds, applied loads, preload, lubrication
methods, and cooling conditions.1–3 In the machining
process, tool wear degree is important, which influences the spindle dynamics and thermal characteristics.4 Friction heat generation in the bearings is the
main heat source in the spindles. Through heat transfer, the heat transfers to inner ring, balls, outer ring,
spindle shaft, and housing with oil-cooling channel.
Then, the spindle system reaches the corresponding
temperature distribution.
For thermal characteristics of the spindle system,
many studies have been carried out through
experimental techniques and theoretical approaches.
Stein and Tu5 proposed a state-space mathematical
model for estimating the preload thermally induced
in high-speed spindles. The model is derived from
physical laws of heat transfer and thermoelasticity
and is suitable for various bearing configurations and
lubrication conditions.6–10 Bossmanns and Tu11,12
presented a finite difference thermal model to characterize the power distribution, heat transfer, and heat
sinks of a high-speed motorized spindle. Holkup
et al.13 presented a finite element-based thermomechanical model of the spindles with rolling
College of Mechanical Science & Engineering, Jilin University, Changchun,
China
Corresponding author:
Qingchang Tan, College of Mechanical Science & Engineering, Jilin
University, Changchun 130022, China.
Email: [email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License
(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
2
bearings. Transient changes of the temperatures may
alter the stiffness and contact forces in the bearings
under specified operating conditions, causing seizure
and damage of the spindles. Mizuta et al.14 investigated the heat transfer between the inner and the
outer rings of an angular ball bearing under the different rotational speeds and thrust forces. And, the thermal conductivity is calculated under different thermal
conductance resistances. Chien and Jang15 developed
a three-dimensional fluid motion mathematical model
of the built-in motorized spindle with a helical rectangular water-cooling channel. The effects of cooling
water flow rate on the spindle temperature distributions were numerically analyzed and verified by
experiments under different values of the heat sources.
The above works are significant to the development
of modern high-performance spindles. However, heat
generation rate of most spindles with rolling bearings
was calculated by empirical formula without considering the internal speeds, dynamic loads, and lubrication
states of bearings. And the heat transfer coefficients
were calculated by ignoring changes of lubricant temperatures, they did not consider the effect of cooling
parameters on the temperature distributions of the spindle system.
This work develops a thermal model of the multiplebearing spindle. In the model, rotational speed, centrifugal forces, gyroscopic moments, and lubricant oil film
are taken into account to predict friction heat of the
rolling bearings, and temperature distribution of the
spindle system is calculated by analyzing the heat conduction and convection heat transfer numerically and
experimentally. Then, from the model, oil-cooling system of the spindle is studied by considering initial temperature and oil flow rate of cooling oil and the cooling
channel dimensions to maintain the temperature gradient in a suitable range.
Figure 1. Spindle-bearing system.
Advances in Mechanical Engineering
Structure model of the spindle system
A high-precision spindle system with the helical oilcooling channel is shown in Figure 1, which is used in
the high-power large horizontal machining center rated
30/37 kW with a maximum speed of 4500 r/min. The
spindle bearings are assembled by tandem and back-toback arrangements, which are rigidly preloaded by two
sleeves, a set of adjusting rings, and a locknut. The
cooling system is used for controlling temperature of
the spindle bearing.
A proper initial preload can achieve an appropriate
bearing stiffness, increase bearing fatigue life, decrease
bearing noise, prevent skidding, and reduce the difference of the contact angle between the inner and outer
raceways of the spindle bearings at high speed. The
spindle bearings are assembled on the shaft by locking
device which provides the preload Fp, as shown in
Figure 2. The preload can be calculated accurately
through a series of empirical correction factors, and the
thrust force Pa0 carried by each bearing decides the
internal load distribution in the individual bearing.
Bearing contact angular ap and pre-deformation dp
can be solved by the following equation16
1:5
Fp
cos a0
1
=
sin
a
p
cos ap
ZDw 2 K
BDw sin (ap a0 )
dp =
cos ap
ð1Þ
Fp = z z1 z2 z3 FA
ð2Þ
where
where FA is the preload of the uninstalled bearings, and
z is the bearing factor, z1 is the correction factor of the
bearing contact angle, z2 is the correction factor of the
preload grade, and z3 is the correction factor of
Wu and Tan
3
D1
z(x) = G
‘
X
cos (2p
g n x)
nl
(2D)n
g
,
.1
1\D\2,
g
ð4Þ
is the characteristic
is the fractal dimension, G
where D
is an arbitrary parameter, and the
scale coefficient, g
value is usually 1.5. nl is the minimum cutoff frequency
of contour structure.
The mean square height, slope, and second derivative of a profile in an arbitrary direction are explained
as follows19
Figure 2. Combination bearings with preload and thrust load.
" #
" 2 #
dz 2
d2z
m0 = E(z ) = s , m2 = E
ð5Þ
, m4 = E
dx
dx2
2
the bearing rolling element, and these parameters can
be looked up in the bearing sample. Under the preload
Fp, the front bearings generate axial displacement dpA,
the rear bearing generates axial displacement dpB, and
the total displacement is dp = dpA + dpB.
When the thrust force Pa0 is applied on the preloaded combination bearings, entirety axial displacement da0 is generated. With the pre-deformation dp, the
relationship between the contact angle of the front
bearing a1 and the rear bearing a2 can be expressed as
follows16
1:5
1:5
Pa0
cosa0
cosa0
=2 sina1
1
sina2
1
cosa1
cosa2
ZDw 2 K
sinða1 a0 Þ sinða2 a0 Þ dpA + dpB
+
=
ð3Þ
BDw
cosa1
cosa2
Frictional heat generation of the
spindle-bearing system
Frictional heat generation of bearings is the dominant
heat source in a spindle-bearing system, and it is mainly
related to bearing internal speeds, dynamic loads, and
lubrication state in the bearing. These factors will be
discussed fully in the following sections.
Contact friction in the elastohydrodynamic
lubrication of grease
Rolling bearings are generally operated with oil lubrication. Rolling element–raceway contact friction depends
on contact geometry and load, rolling speed, and lubricant properties. A lubricant is a substance that is used
to reduce friction and provide smooth running. Based
on the Hertz elastic contact theory and Reynolds
hydrodynamic lubrication theory, oil film thickness can
be approximately calculated by considering the surface
roughness.17–25
For the contact of real surfaces, Weierstrass–
Mandelbrot model can be used for the lubricated contact. An optical topographer is used to measure a profile in an arbitrary direction x21
2
where m0, m2, and m4 are known as the zeroth, second,
and fourth spectral moments of a profile, respectively.
The parameter L was established during the 1960s to
indicate the degree to which a lubricant film separates
the surfaces in rolling ‘‘contact’’20
L= hmin
s2m + s2R
12
ð6Þ
where sm is the root mean square (rms) roughness of
the raceway, and sR is the rms roughness of the rolling
element. Full-film separation can be assumed for L 3.
When the lubricant film is insufficient to completely
separate the surfaces in rolling contact, that is, for L
3, it is possible that some of the surface peaks, also
called asperities, break through the lubricant film and
contact each other when the lubricant temperature in
the contact increases causing viscosity to decrease.
The ratio of contact to apparent area Ac/A0 is19
Ac
d
= pRs Ss DSUM F1
A0
Ss
ð7Þ
where Rs is the assumed constant radius of the spherical summits, Ss is the standard deviation of the summit
height distribution, DSUM is the area density of summits, and d is the distance between the summit height
and the surface mean plane. F1(t) is the integral19
F 1 ðt Þ =
ð‘
(x t)f(x)dx
ð8Þ
t
Other variables can be expressed as follows19
"
! #12
1
3 p 2
0:8968
m0
Rs =
, Ss =
1
8 m4
m0 m4 m22
h
4
1 1
m20
m4
d
ðpm0 m4 =m22 Þ2
p
ffiffi
ffi
, =
DSUM =
12
6pm2 3 Ss
0:8968
1 m m m2
ð 0 4= 2Þ
ð9Þ
4
Advances in Mechanical Engineering
Hamrock and Dowson proposed the calculation formula of dimensionless oil film thickness26
0:67
H0 =
G0:53 2:69U
1 0:61e0:73k
0:067
Qz
ð10Þ
normal force Q is distributed over an elliptical surface
defined by the projected major and minor axes 2an, 2bn,
respectively, as shown in Figure 3. The radius of a contact point (xn, yn) is defined as follows
82
912
!12 32
2
< =
1 1
Dw
rn0 = 4 R2n x2n 2 R2n a2n 2 +
a2n 5 + x2n
:
;
2
where
z = Q , E0 = E
= h0 U , Q
G = lE0 , U
0
E Rx
E0 R2x
1 j2
ð16Þ
In the expression for Ho, the equivalent radius in the
direction of rolling is given by
Rx =
Dw
ð17gÞ
2
ð11Þ
Sliding of the raceway over the ball in the direction
of rolling is determined by the difference between the
linear velocities of raceway and ball. Hence, the sliding
velocities in the y# (rolling motion) and x# (gyroscopic
motion) directions are determined as follows26
dm
vn + r0 n ½ðcn vn vx0 Þ
2
cosðan + un Þ vz0 sinðan + un Þ
The velocities U, with grease swept into the rolling
element–raceway contacts, for the inner and outer raceway contacts are given by26
dm
Ui =
½ð1 g Þðv vm Þ + gvR 2
dm
½ð1 + g Þvm + gvR Uo =
2
ð12Þ
ð13Þ
Point-contact oil film thickness in elastohydrodynamic lubrication (EHL) can be expressed as follows
h(x, y) = h0 +
x2
y2
+
+ dðx, yÞ
2Rx
2Ry
ð14Þ
where h0 is the oil film thickness of contact center; Rx
and Ry are the equivalent curvature radii along the x, y
direction; and d(x, y) is the deformation displacement.
At high bearing operating speeds, some of the frictional heat generated in each concentrated contact is dissipated in the lubricant momentarily residing in the inlet
zone of the contact. This effect tends to increase the temperature of the lubricant in the contact. It is clear that
the lubricant dynamic viscosity will be changed as a
result of temperature increase in the contact. Roelands
established an equation which defined the relationship
of dynamic viscosity, temperature, and pressure26
h(T , s) = h0 exp
(
"
0:68
ðln h0 + 9:67Þ 1 + 5:1 3 109 s
3
ð17Þ
vx0 n = r0 n vy0
Thickness of oil film center can be expressed as
h0 = H0 Rx
vy0 n =
#)
T 138 1:1
1
T0 138
ð15Þ
where
xn
un = arc sin
rn
In equation (16), co = 1 and ci = 21. The friction
shear stress is described as given by Harries and Barnsby
1
Ac
Ac v1
1
+ tlim
h
t = cv ma s + 1 A0
A0
h
ð18Þ
where Ac is the area associated with the asperity–
asperity contact, A0 is the total contact area, and sliding coefficient cv = 1 or 21 depending on the direction
of sliding velocity.
As shown in Figure 4, the relative axial displacement
of the inner rings is da, and the relative angular displacements is u; the axial distance between the loci of inner and
outer raceway curvature centers at the ball position is A1.
In accordance with a relative radial displacement of the
ring centers dr, the radial displacement between the loci of
the curvature centers at each ball position is A2; changes
of the two positions can be expressed as follows26
A1 = BDw sin a0 + da
A2 = BDw cos a0 + dr
ð19Þ
The position variables X1, X2, A1, A2 are related to
contact angles ai and ao and contact deformations di
and do as follows
X2
X1
, sin ao =
ðfo 0:5ÞDw + do
ðfo 0:5ÞDw + do
A2 X2
A1 X1
cos ai =
, sin ai =
ðfo 0:5ÞDw + di
ðfo 0:5ÞDw + di
cos ao =
Sliding motions and dynamic loads in ball bearings
Based on the Hertzian contact theory, the ball contacts
the outer-ring and inner-ring raceways, such that the
ð20Þ
Wu and Tan
5
Figure 3. Model of ball and raceway contact.
The remaining two equations pertaining to the position of the ball center can be obtained
ðA1 X1 Þ2 + ðA2 X2 Þ2 ½ðfi 0:5ÞDw + di 2 = 0
X12 + X22 ½ðfo 0:5ÞDw + do 2 = 0
ð21Þ
Friction shear stresses can be calculated by a given
lubricating fluid and a given condition of rolling contact surface separation.27 Figure 5 shows the force and
moment loads of a ball in loaded, grease-lubricated
angular-contact ball bearing.
From Figure 5, it can be seen that the following conditions of force equilibrium must be satisfied for steadystate operation of the bearing26
Figure 4. Position of the ball center and raceway groove
curvature center.
Figure 5. Ball model with forces and moments.
6
Advances in Mechanical Engineering
Qo cos ao Qi cos ai Fx0 o sin ao + Fx0 i sin ai Fz0 = 0
Qo sin ao Qi sin ai + F
Fy0 o Fy0 i + Fv = 0
x0 o
cos ao F cos ai = 0
x0 i
Qo sin ao + Fx0 o cos ao Fa =Z = 0
mv prD2w ðdm vm Þ1:95
32g
ð1
pffiffiffiffiffiffiffi2ffi
1q
ð
1 Dw
2
ð1
Dw
2
ð1
1 ao bo t y0 o cosðao + uo Þ + ai bi ty0 i cosðai + ui Þ dtdq = 0
1q
½ao bo t x0 o + ai bi tx0 i dtdq Mgy0 = 0
pffiffiffiffiffiffiffi2ffi
1q
pffiffiffiffiffiffiffi2ffi
1q
ð
ao bo t y0 o sinðao + uo Þ + ai bi t y0 i sinðai + ui Þ dtdq Mgz0 = 0
pffiffiffiffiffiffiffi2ffi
1q
ð23Þ
According to static analysis results, variables in
equations (22) and (23) are given initial values; the
unknown variables have been solved by the iterative
method.
Friction heat generation
In the ball–raceway contact, rate of the friction heat
generation is given by26
ð
1
a b
E_ nyj =
t nyj ynyj dAnj = njJ nj
J
pffiffiffiffiffiffiffi2ffi
1q
ð1
ð
tnyj ynyj dtdq n = i, o; j = 1, 2, . . . , Z
p
ffiffiffiffiffiffiffi
ffi
1 1q2
ð
_Enxj = 1 t nxj ynxj dAnj = anj bnj
J
J
pffiffiffiffiffiffiffi2ffi
1q
ð1
ð
tnxj ynxj dtdq n = i, o; j = 1, 2, . . . , Z
p
ffiffiffiffiffiffiffi
ffi
1
2
nX
=o X
Z E_ nyj + E_ nxj + E_ fdrag
ð26Þ
n=i j=1
pffiffiffiffiffiffiffi2ffi
pffiffiffiffiffiffiffi2ffi
1q
ð
1 ð25Þ
Rate of the total friction heat generation is obtained
by summation of the component heat generations
E_ tot =
where r is the weight of the lubricant in the bearing free
space divided by the volume of the free space, and mv is
the drag coefficient.
The moment equilibrium conditions are
Dw
2
dm vm Fv Z
E_ fdrag =
2J
ð22Þ
Viscous drag force Fv in equation (22) can be
expressed as
Fv =
friction heat is also generated due to the balls passing
through the lubricant and is given by
Heat transfer model and temperature distribution
Three fundamental modes for the transfer of the friction heat among the shaft, bearings, and housing are
the heat conduction, heat convection, and heat radiation. Heat radiation effect is minor and usually be
neglected.
The heat conduction includes the inner-ring shaft,
outer-ring housing, and inner and outer raceway balls.
The heat convection includes the grease lubrication, oil
cooling, and ambient air.11 Figure 6 shows the thermal
resistances and temperature nodes model along the
radial direction of the front bearing. Nine axial temperature nodes and eight radial temperature nodes are
defined. All temperature nodes and corresponding heat
transfer equations are calculated by Takabi and
Khonsari.28
Burton and Steph30 proposed that half of the friction heat is transferred to balls, and the other half is
transferred to the rings. It can be calculated by standard finite difference methods. The spindle is decomposed into a certain amount of nodes for analyzing. At
each node, heat influx equals heat efflux; therefore, the
sum of all heat flowing toward a temperature node is
equal to zero, as shown in Figure 7.
From the figure, the heat flows in a node can be
expressed as
E_ f 0 + E_ 10, c + E_ 20, v + E_ 30, c + E_ 40, c = 0
ð24Þ
1q
where J is a constant converting to W; the surface friction shear stress and the sliding velocity are obtained
from the above section.
For a grease-lubricated bearing, in addition to the
friction heat generated in the ball–raceway contacts,
ð27Þ
where E_ f 0 is the heat generation; E_ 10, c , E_ 20, v , E_ 30, c ,
E_ 40, c are the heat flows by the heat conduction; and
E_ 20, v is the heat flow by the convection. Based on
Newton’s law of cooling and Fourier’s law, equation
(27) can be expressed as
1
ðT1 T0 Þ + P2 S2 ðT2 T0 Þ
E_ fo +
R1
1
1
ðT3 T0 Þ +
ðT4 T0 Þ = 0
+
R3
R4
ð28Þ
For all nodes of the spindle system, thermal contact
resistance and heat transfer coefficients can be obtained
from previous works.27–30 Conductive and convective
resistance formulas for the spindle-bearing system
Wu and Tan
7
Figure 6. Temperature and thermal resistance model along the radial direction of the spindle system.
Table 1. Conductive and convective resistance formulas for
the spindle-bearing system.
Thermal
resistance
Conductive and convective resistance formulas
Rob , Rib
R=
c(a=b)
2
, c(a=b) =
4ka
p
p=2
ð
0
Rb
Rl
Rf
Figure 7. Two-dimensional temperature node system.
2
pkDw
.
1 1
1=ðPl SÞ = 1 0:332k Pr3 unxm 2 S
R=
k
1 Pf S = 1
0:0225 Re0:8 Pr0:3 S
hgap
,
1 !
hgap 3
k
or 1 Pf S = 1
1:86 Re Pr S
hgap
L
where Re =
presented in Table 1 with the corresponding Re and Pr
also are presented in this table. Equations similar to
equation (28) can be written and solved by the
Newton–Raphson method.
Experimental setup and data reduction
Rh , Ro , Ri , Rs
Rlb
Roh
Rair
du
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
i
1 (1 b2 =a2 )2 sin2 u
u hgap
ch
, Pr =
k
n
1 lnðDo =Di Þ
k 2pb 0:5
0:4
k
1=ðPv SÞ = 1 0:33
. D ReD Pr S
hring
hgap
R=
+
S
kring
kgap
c2
R = 1=ððc0 + c1 u Þ SÞ
R=
Experimental setup
The experimental apparatus is shown in Figure 8. The
spindle system is tested by five sets of Micro-Epsilon
laser triangulation sensors measurement instrument
and FLIR’s thermal infrared imager. The spindle is
operated in an air-conditioned environment, and the
temperature is 17.6°C. The spindle is cooled by Habor
Oil Cooler, the model is HBO-3RPTSB-BY-10, and the
oil temperature can be controlled between 10°C and
40°C. In the experiment, the initial temperature of cooling oil is 15°C. There are five temperature measurement
points SP01–SP05: SP01–SP03 measured the front
bearings, SP04 measured the middle part between the
front bearing and the rear bearing, and SP05 measured
the rear bearing. All data signals were collected and
converted by data acquisition system. Figure 9 shows
the measured temperature curve. The test is continued
until the steady-state condition is attained. Minimum of
3.5 h was required for the temperature measurements to
become steady. Test is stopped upon attainment of the
steady state by shutting down the spindle motor. When
the spindle is reaching the thermal balance, the temperature is changed very small; we consider this steady
state as the thermal balance. It means that the rate of
heat generation inside and the rate of heat dissipation
from the bearing reach a state of equilibrium. The time
to reach thermal balance is different for each combination of rotational speed and load, but we did hundreds
of tests, the longest time was almost 4 h except of thermal failure which happened before it reached the steady
state.
8
Advances in Mechanical Engineering
Figure 8. Measurement system of the spindle.
Figure 9. Measured temperature for test condition of 3000 r/min and 1750 N thrust load.
Data reduction
The experimental spindle of the horizontal machining
center is operated at eight different speeds from 0 to
4500 r/min, and the interval is 500 r/min. When the
measured points have small temperature changes at the
minimum speed, the speed adjusts to the next value,
until reaching the maximum speed. Acquisition and
pickup temperature data and the experimental temperatures are shown in Figure 10.
Results and discussions
In order to verify validity of numerical analysis models
presented in sections ‘‘Frictional heat generation of the
spindle-bearing system’’ and ‘‘Experimental setup and
Figure 10. Measured temperature during step test from 0 to
4500 r/min.
Wu and Tan
9
Figure 11. Viscosity of lubricating grease versus temperature.
Figure 12. Oil film thickness of the grease in contact center.
data reduction,’’ different spindle running conditions
are set, experiments and simulations can be compared
under operating conditions. Then the mathematical
model provides theory analysis for optimum design and
dynamic compensation of machine tool spindle system.
Numerical results
In this article, Kluber/NBU 15 grease is used to lubricate the spindle bearing. It has a kinematic viscosity of
21 mm2/s at 40°C and 4.7 mm2/s at 100°C, and the density of grease is 0.99 g/cm3. The dynamic viscosity–
temperature curve is shown in Figure 11.
Wilson developed thermal reduction factors for lubricant film thickness from numerical solutions of the thermal EHL problem for rolling–sliding contacts. The film
thickness reduction factor is taken into account,26 the
oil film thickness of contact center in the front bearings
can be calculated, and the result is shown in Figure 12.
Figure 12 shows comparison of central oil film thickness in the isothermal state and thermal effect on the
grease. As seen, in the isothermal state, center oil film
would be thicker at higher rotational speed. Since when
the grease temperature is increasing, grease viscosity
will decrease. Center oil film thickness will increase
more slowly with considering thermal effect, especially
at high speed, oil film thickness even decrease, and then
the friction shear stress will be changed greatly, leading
to an non-ignorable effect on friction heat generation.
In this article, the spindle bearing (71928CD/
P4ATBTA) is a triple angular-contact ball bearing; the
suffix TBT represents tandem duplex and back-to-back
combination mode. Corresponding to 1.1 million DN
with a 165 mm bore diameter of the spindle bearings,
the ball diameter is 13.37 mm. The supporting bearing
(71924CD/P4ADBA) is a back-to-back combination
bearing; the suffix DB represents back-to-back combination mode. The spindle system was operated under
1137.2 N preload, which is provided by the adjustable
ring. Variables of ball–raceway contacts under different
Figure 13. Axial load and axial displacement versus thrust
force: (a) axial load calculated based on the constant preload of
1137.2 N and (b) axial displacement calculated based on the
constant preload of 1137.2 N.
speeds are calculated using non-isothermal lubricant,
and the calculated results are shown in Figures 13–15.
From the figures, it is clear that axial load, centrifugal
forces, gyroscopic moments, and lubrication conditions
can significantly influence bearing contact deflections,
contact angles, and contact loads.
According to equation (6) and the experimental
results of rough surface, the simulation results of
10
Advances in Mechanical Engineering
Figure 14. Calculated variables of rolling elements in the spindle bearings: (a) ball–inner raceway and ball–outer raceway normal
contact deformations dij and doj, (b) ball–inner raceway and ball–outer raceway normal contact angles aij and aoj, and (c) semi-major
and semi-minor axes of the elliptic contact in inner and outer raceway.
parameter L at the 0° position are shown in Figure 16.
Area associated with the asperity–asperity contact can
be negligible, the friction shear stress is governed by the
grease in the contact, and the friction heat generations
of the spindle bearings are calculated by equations (24)
and (25). The calculated results are obtained as shown
in Figure 17. All temperature nodes and corresponding
heat transfer equations are calculated, and the calculated results are shown in Figure 18. From Figure 18, it
can be seen that temperature gradient is high, and rate
of the heat dissipation from the bearing outer ring is
higher than that from the inner ring.
Wu and Tan
Figure 15. Ball speeds of the spindle bearings.
Figure 16. Lambda parameters at the 0° position of the
spindle bearing.
11
Figure 18. Temperature distribution of spindle.
Figure 19. Comparison of temperatures.
due to the difficulty in predicting thermally induced
preload and lubrication condition in these speeds. But,
on the whole, prediction temperatures are in good
agreement with the experimental results. Therefore, it
can be said that the numerical model can be applied to
analyze the thermal characteristics and improve spindle
design.
Optimization and analysis of the cooling system
Figure 17. Frictional heat generation.
Comparison with experimental results
Acquisition and pickup temperature data are compared
with the corresponding simulation data. The experimental temperatures and numerical ones are shown in
Figure 19. From the figure, the temperature predictions
for 2000 and 2500 r/min are less accurate; this is mainly
From Figure 18, it can be seen that a large temperature
difference occurs, and the thermal balance is not stable.
If the spindle works for a long time, the instability
increases the ball–raceway contact load, friction, and
temperature, which results in the bearing seizure. So
the cooling system is used to balance the temperature
gradient of the spindle system.
For the oil-cooling system, physical model of the
spindle system with a helical rectangular oil-cooling
channel is shown in Figure 20. The convective heat
transfer of cooling oil changes with the flow type. Type
of the flow around the housing surface is divided into
turbulent and laminar.
12
Advances in Mechanical Engineering
Figure 20. Physical model of cooling system with the cylindrical channel.
The cooling oil used is ISO VG32; its physical properties are functions of temperature, from physical parameters of the cooling oil, and the equations are
obtained by the curve-fitting method as
k = 0:1381 1:012 5:5018 3 104 t
r = 869:1 1 7:9744 3 104 ðt 21Þ
h = 0:06673e0:051ðt21Þ
ð29Þ
cp = 4:428ðt 21Þ + 1884:2
As shown in Figure 21, the simulation results and
the experimental measurements are in good agreement
in both trend and magnitude. So the simulation results
can be used to calculate the convective heat transfer
coefficients of the fluid.
According to the Nusselt criterion, when the cooling
oil is under the condition of turbulent flow state, the
convective heat transfer coefficient can be calculated as
follows
0:7 0:8 0:3 0:5 r c h
Pf = 0:0225 u0:8 h0:2
gap k
ð30Þ
Under the condition of laminar flow state, the convective heat transfer coefficient can be calculated
according to the following formula
1 1
1
Pf = 1:86 u3 hgap L 3 k 2 r c 3
ð31Þ
From equations (30) and (31), it can be seen that the
temperature of the cooling oil and sectional dimensions
of the cooling channel are the important variables for
calculating coefficient of convection heat transfer. The
heat transfer coefficient equation of the cooling oil is
transformed to a function with temperature and hgap
variables. According to the structure finite element
analysis, value range of hgap is from 0.005 to 0.03 m, the
step is 0.005 m; value range of oil temperature is from
10°C to 40°C. Relationship between the heat transfer
coefficients, temperature, and gap characteristic dimension hgap is shown in Figure 22.
When gap characteristic dimension hgap is less than
0.01 m, the temperature has a minimal effect on the
convection heat transfer coefficient of the cooling fluid,
and the fluid is obvious in laminar flow. When hgap is
greater than 0.01 m, the state of the cooling fluid
changes from laminar flow to turbulent flow with the
increase in temperature, and the broken line is the turning point of the fluid state. In the turbulent state, the
convection heat transfer increases very quickly when
the temperature increases.
Based on cooling channel machining condition, and
the cooing capability of oil cooler, according to the calculated results of equations (30) and (31), the optimal
value of coefficient of convective heat transfer can be
obtained after restudy of the physical parameters, and
the optical physical parameters of cooling channel and
oil cooler also can be obtained; optimal value hgap is
19 mm, and the optimal temperature is 31°C.Figure 22
is obtained with cooling oil of 31°C and hgap of 19 mm.
Figure 23 shows that the maximum temperature is
higher than that in Figure 18, but the increasing value is
less than 3°C, and it has little effect on temperature distribution of the spindle system. The temperature difference across the bearing is lesser than that in Figure 18,
and the decreasing value is about 15°C. This optimized
cooling system can maintain the temperature gradient
across the bearing.
In this article, a computer program is developed to
implement analysis and calculation for a given spindle
configuration and operational conditions, the thermal
information and optimized parameters can be obtained,
and this is useful to provide dynamic compensation for
working machining center and improve the machining
center structure in the R&D.
Wu and Tan
13
Figure 21. Physical characteristics of cooling fluid: (a) heat conductivity coefficient, (b) density, (c) dynamic viscosity, and (d)
specific heat capacity.
Figure 23. Temperature along the radial direction after
optimization.
Figure 22. Coefficients versus temperatures of cooling fluid.
Conclusion
An analysis and modeling procedure for simulating
friction heat generation of the spindle system has been
developed. Thrust force, rotational speed, lubrication
condition, and surface asperities are calculated in detail.
The model provides a practical model for the thermal
characteristics of the spindle-bearing system due to the
finite number of thermal nodes in the heat transfer
model of the spindle system in the radial direction.
The heat transfer capability of the cooling system is
related to the cooling oil temperature, the cooling oil
flow, and the equivalent diameter of the cooling channel. The cooling oil flow is selected the maximum flow,
14
and then the relationship of the heat transfer coefficient
of the cooling system, the cooling oil temperature, and
the equivalent diameter of the cooling channel is
analyzed.
The system temperature gradient is mainly determined by the cooling system, as well as the applied
force and speed. Cooling channel sectional dimensions
and cooling oil temperature are the main parameters to
optimize temperature gradient of the spindle system.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
References
1. Min X, Shuyun J and Ying C. An improved thermal
model for machine tool bearings. Int J Mach Tool Manu
2007; 47: 53–62.
2. Lin C-W, Tu JF and Kamman J. An integrated thermomechanical-dynamic model to characterize motorized
machine tool spindles during very high speed rotation.
Int J Mach Tool Manu 2003; 43: 1035–1050.
3. Choi JK and Lee DG. Thermal characteristics of the
spindle bearing system with a gear located on the bearing
span. Int J Mach Tool Manu 1998; 38: 1017–1030.
4. Twardowski P, Legutko S, Krolczyk GM, et al. Investigation of wear and tool life of coated carbide and cubic
boron nitride cutting tools in high speed milling. Adv
Mech Eng 2015; 7: 1–9
5. Stein JL and Tu JF. A state-space model for monitoring
thermally induced preload in anti-friction spindle bearings of high-speed machine tools. J Dyn Syst: T ASME
1994; 116: 372–386.
6. Tu JF and Stein JL. On-line preload monitoring for antifriction spindle bearings of high-speed machine tools.
J Dyn Syst: T ASME 1995; 117: 43–53.
7. Tu JF and Stein JL. Active thermal preload regulation
for machine tool spindles with rolling element bearings.
J Manuf Sci E: T ASME 1996; 118: 499–505.
8. Tu JF. Thermoelastic instability monitoring for preventing spindle bearing seizure. Tribol T 1995; 38: 11–18.
9. Tu JF and Katter JG. Bearing force monitoring in a
three-shift production environment. Tribol T 1996; 39:
201–207.
10. Katter JG and Tu JF. Bearing condition monitoring for
preventive maintenance in a production environment.
Tribol T 1996; 39: 936–942.
11. Bossmanns B and Tu JF. A thermal model for high speed
motorized spindles. Int J Mach Tool Manu 1999; 39:
1345–1366.
Advances in Mechanical Engineering
12. Bossmanns B and Tu JF. A power flow model for high
speed motorized spindles—heat generation characterization. J Manuf Sci E: T ASME 2001; 123: 494–505.
13. Holkup T, Cao H, Kolář P, et al. Thermo-mechanical
model of spindles. CIRP Ann: Manuf Techn 2010; 59: 365–368.
14. Mizuta K, Inoue T, Takahashi Y, et al. Heat transfer
characteristics between inner and outer rings of an angular ball bearing. Heat Transf: Asian Res 2003; 32: 42–57.
15. Chien CH and Jang JY. 3-D numerical and experimental
analysis of a built-in motorized high-speed spindle with
helical water cooling channel. Appl Therm Eng 2008; 28:
2327–2336.
16. Harris TA and Kotzalas MN. Essential concepts of bearing technology: rolling bearing analysis. 5th ed. Boca
Raton, FL: CRC Press, 2006.
17. Hamrock BJ and Dowson D. Ball bearing lubrication:
elastohydrodynamics of elliptical contact. New York:
Wiley, 1981, pp.1–386.
18. Jang YH, Cho H and Barber JR. The thermoelastic Hertzian contact problem. Int J Solids Struct 2009; 46:
4073–4078.
19. McCool JI. Comparison of models for the contact of
rough surfaces. Wear 1986; 107: 37–60.
20. Dowson D. Elastohydrodynamic and micro-elastohydrodynamic lubrication. Wear 1995; 190: 125–138.
21. Greewood JA and Wu JJ. Surface roughness and contact:
an apology. Meccanica 2001; 36: 17–630.
22. Masjedi M and Khonsari MM. Theoretical and experimental investigation of traction coefficient in line-contact
EHL of rough surfaces. Tribol Int 2014; 70: 79–189.
23. Takabi J and Khonsari MM. On the dynamic performance of roller bearings operating under low rotational
speeds with consideration of surface roughness. Tribol
Int 2015; 86: 2–71.
24. Lugt PM. Grease lubrication in rolling bearings. New
York: John Wiley & Sons, 2013.
25. Lugt PM. A Review on grease lubrication in rolling bearings. Tribol T 2009; 52: 70–480.
26. Harris TA and Kotzalas MN. Advanced concepts of bearing technology: rolling bearing analysis. 5th ed. Boca
Raton, FL: CRC Press, 2006.
27. Katsuhiko N. Thermal contact resistance between balls
and rings of a bearing under axial, radial, and combined
loads. J Thermophys Heat Tr 1995; 9: 88–95.
28. Takabi J and Khonsari MM. Experimental testing and
thermal analysis of ball bearings. Tribol Int 2013; 60:
3–103.
29. Rohsenow WM, Hartnett JP, Cho YI, et al. Handbook of
heat transfer. New York: McGraw-Hill, 1998.
30. Burton RA and Staph HE. Thermally activated seizure of
angular contact bearings. ASLE Trans 1967; 10: 408–417.
Appendix 1
Notation
a
A
semi-major axis of the contact area
distance between raceway groove
curvature centers
Wu and Tan
b
B
c
d
dm
D
DSUM
Dw
E
E_
f
Fa
Fp
Fr
Fv
Fz#
G
G
h0
h
hgap
hmin
H0
k
K
Mgy#
Mgz#
Pa0
Pr
Q
z
Q
r
R
Re
Rs
Rx
Ry
sm
15
semi-minor axis of the contact area
fi + fo 2 1, total curvature
specific heat
distance between the summit height and
the surface mean plane
diameter of pitch circle
fractal dimension
area density of summits
ball diameter
Young’s modulus
heat generation rate
r/D
axial force
preload
bearing radial load
viscous drag force
centrifugal force
material parameter
characteristic scale coefficient
center thickness of oil film
lubricant film thickness
stereotype dimension of gap geometrical
characteristic
minimum lubricant film thickness
dimensionless oil film thickness
conductivity coefficient
load-deflection factor
gyroscopic moment in y# direction
gyroscopic moment in z# direction
applied load
Prandtl number
ball–raceway normal load
load parameter
raceway groove curvature radius
heat resistance
Reynolds number
assumed constant radius of the spherical
summits
equivalent curvature radii along the x
direction
equivalent curvature radii along the y
direction
rms roughness of the raceway
T
u
U
v
Z
rms roughness of the ball
area
standard deviation of the summit height
distribution
temperature
fluid velocity
speed parameter
lubricant velocity
number of balls per bearing
a
a°
g
g
d
h
k
L
m
n
j
P
r
s
t
v
vm
vR
mounted contact angle
free contact angle
(D cos a)=dm
arbitrary parameter
displacement of the bearing
dynamic viscosity
eccentricity
lubricant film parameter
drag coefficient
kinematic viscosity
Poisson’s ratio
heat transfer coefficient
density
normal contact stress or pressure
friction shear stress
rotational speed
orbital motion rotational speed
ball rotational speed
sR
S
Ss
Subscripts
f
h
i
j
n
o
R
s
x, x#
y, y#
fluid
housing
inner raceway
rolling element position
raceway
outer raceway
rolling element
shaft
x direction, transverse to rolling direction
y direction, rolling direction