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Mech. Sci., 12, 109–122, 2021
https://doi.org/10.5194/ms-12-109-2021
© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.
Research on the influence of air-gap eccentricity
on the temperature field of a motorized spindle
Xiaohu Li1,2, Jinyu Liu1,2 , Cui Li1,2, Jun Hong1,2, and Dongfeng Wang3,4
1Key Laboratory of Education Ministry for Modern Design & Rotor-Bearing System,
Xi’an Jiaotong University, Xi’an, 710049, China
2School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, 710049, China
3Henan Key Laboratory of high-performance bearing technology, Luoyang, 471039, China
4Luoyang Bearing Research Institute Co., Ltd., Luoyang, 471039, China
Correspondence: Xiaohu Li (li.xiaohu@mail.xjtu.edu.cn)
Received: 30 September 2020 – Revised: 18 December 2020 – Accepted: 22 December 2020 – Published: 9 February 2021
Abstract. The air-gap state between the stator and rotor is an important indicator to measure the performance
of a motorized spindle. It affects the temperature field distribution of the motorized spindle and the machining
accuracy of the mechanical parts. Since the accurate thermal model is the basis of the research on the temper-
ature field distribution of the motorized spindle, in this paper, firstly, the mechanical loss, electrical loss and
magnetic loss of the motor under different air-gap eccentricities are calculated and the heat-generating power
of an angular-contact ball bearing is obtained based on Harries contact theory. Secondly, the thermal model of
the motorized spindle is established and the steady-state temperature field of the motorized spindle is simulated
by using ANSYS, and the influence of air-gap eccentricity on the temperature field of the motorized spindle is
discussed. Finally, the circumferential temperature field distribution of the motorized spindle with the air-gap
eccentricity is verified by experiment. The results show that the air-gap eccentricity has a significant influence
on the non-uniform temperature field of the motorized spindle.
1 Introduction
A motorized spindle is the key component of the Computer
Numerical Control (CNC) machine tool. Its performance has
an important impact on the quality of parts processing, ma-
chine performance and production efficiency. After the motor
is assembled, the air gap is formed between the stator and ro-
tor. Ideally, the fixed air gap is evenly distributed, and thus
the magnetic field is generated. However, the air gap is al-
ways eccentric due to the assembly errors, parts manufactur-
ing errors and workload. The air-gap eccentricity will cause
unbalanced magnetic pull, which makes the uneven temper-
ature rise. Most studies show that up to 75 % of the total ge-
ometric errors of the workpiece are caused by the thermal
characteristics of the motorized spindle (Mayr et al., 2012).
Therefore, it is of great theoretical significance and practi-
cal value to study the influence of air-gap eccentricity on the
temperature field distribution of the motorized spindle.
Many scholars have studied the influence of air-gap ec-
centricity on the motor. As early as in the 20th century,
Smith (1911) and Smith and Johnson (1912) preliminarily
studied the influence of air-gap eccentricity on unbalanced
magnetic tension and loss. The results showed that the iron
loss and unbalanced magnetic tension increased with the in-
crease in eccentricity, which laid a foundation for the re-
search on the influence of air-gap eccentricity on the motor
performance. Ellison and Yang (1971) analyzed the flux den-
sity and permeability wave in the air gap of the induction
motor. It was found that the eccentricity of the rotor would
cause the induction motor to produce a series of pole-pair
force waves. The predicted noise composition was verified
by experiments. Li et al. (2007) analyzed the vibration of the
three-phase induction motor with static eccentricity. The re-
sults showed that the eccentric fault could be detected by a
vibration signal. Guo et al. (2003) expressed the air gap in the
form of Fourier series. The unbalanced magnetic pull force
Published by Copernicus Publications.
110 X. Li et al.: Research on the temperature field of a motorized spindle
caused by eccentricity was calculated, and the influence of
eccentric force on vibration was analyzed. Kim et al. (2014)
studied the influence of the rotor static eccentricity on the
magnetic force on the stator surface and magnetic harmonic
spectrum. The static eccentricity of the rotor will increase
the first and 31st harmonic components. Abdi et al. (2015)
used the finite-element method to calculate the unbalanced
magnetic tension of a brushless doubly-fed motor with dif-
ferent rotor eccentricities. This would increase the vibration
and noise of a brushless doubly fed motor (BDFM). Han
and Palazzolo (2016) adopted the magnetic equivalent cir-
cuit method to model the motor. It had been parameterized.
The radial and tangential forces increased with the increase
in eccentricity. He et al. (2016, 2017a, b) obtained a detailed
formula of unbalanced magnetic tension by studying the un-
balanced magnetic tension under different forms of static ec-
centricity. Oumaamar et al. (2017) predicted the influence of
air-gap static eccentricity on the related frequency of the neu-
tral line voltage spectrum through analysis. It indicated that
the voltage spectrum of the neutral point was more sensitive
to the air-gap static eccentricity. Bindu and Thomas (2018)
proved the applicability of the vibration signal and stator
current signal in diagnosing static eccentricity through ex-
periments on three-phase squirrel-cage asynchronous motors
with static eccentricity. Ding et al. (2015) studied the influ-
ence of different air-gap static eccentricity on magnetic field
strength and core loss based on electromagnetic theory and
the finite-element method. However, there are relatively few
studies on the air-gap eccentricity to the various losses of the
motorized spindle motor.
In the research of temperature field simulation of the mo-
torized spindle, Bossmanns and Tu (1999, 2001) analyzed
the energy flow distribution of the motorized spindle based
on the finite-difference method. The calculation methods of
heat transfer coefficients at different heat dissipation bound-
aries of shafting were presented. The steady and transient
temperature fields were analyzed, and they were verified
by experiments. Yan et al. (2016) established the network
method of spindle transient analysis on the basis of consid-
ering the heat-structure interaction. The improved bearing
and system model were studied. The advantage of the tran-
sient model was verified by experiments. Chen et al. (2013)
used the finite-element method to conduct thermal simulation
based on the power flow model. Experimental verification
was carried out. It can be seen that the above studies estab-
lished the thermal model of a high-speed motorized spindle
and simulated the temperature field. However, the tempera-
ture field distribution of the motorized spindle was analyzed
symmetrically, and the influence of air-gap eccentricity on
the motorized spindle heating and temperature field distribu-
tion is not considered. The temperature fields obtained were
uniformly distributed.
Therefore, in this paper, the thermal model of the mo-
torized spindle is established based on the consideration of
the influence of air-gap eccentricity on motor loss. Secondly,
based on the established thermal model, the influence of air-
gap eccentricity on the temperature field distribution of the
motorized spindle is studied by the finite-element method,
and the temperature field distribution law of the motorized
spindle in the circumferential direction is obtained and veri-
fied by experiments.
2 Motorized spindle heating model considering
air-gap eccentricity
In our research, since the influence of air-gap eccentricity on
the temperature field of the motorized spindle is mainly stud-
ied, the influence of other factors is ignored, and the variation
law of heat generation of the motor and bearing under differ-
ent eccentricity is only calculated in this chapter.
2.1 The power loss of the motor considering air-gap
eccentricity
As shown in Fig. 1, the geometric center of the stator and ro-
tor and rotating center of the rotor are coincident when there
is no eccentricity. However, when the air gap is eccentric,
they do not coincide. Considering dynamic eccentricity, it is
assumed that α1=0 at the smallest air gap of the motor. Be-
cause the inner diameter of the stator is much larger than the
air-gap length, the air-gap length δat α1is calculated by the
following formula:
δ(α1,t)=r1−r0−ecosα1=h−ecos α1,(1)
where his the uniform length of the air gap, and eis the
eccentricity of the rotor.
Motor loss is generally divided into four categories: me-
chanical loss, electrical loss, magnetic loss and additional
loss. The proportion of additional loss in the total loss is very
small, about 1 %–5 % of the rated power. So, the additional
loss is ignored, and the first three losses are mainly calcu-
lated.
Mechanical loss is mainly the friction loss between the ro-
tor and air during high-speed rotation. It is mainly generated
in the air gap between stator and rotor. Therefore, there will
be different mechanical loss at a different air-gap position.
The shear moment required for rotor operation at a certain
speed is
T=r0·ZτdA=
2π
Z
0
µr3
0lω1
δdα1.(2)
So, the power of mechanical loss is calculated by the follow-
ing formula:
Pw=ω1·T=
2π
Z
0
µlω2
1r3
0
δdα1,(3)
Mech. Sci., 12, 109–122, 2021 https://doi.org/10.5194/ms-12-109-2021
X. Li et al.: Research on the temperature field of a motorized spindle 111
Figure 1. Air-gap eccentric state of the motor.
where r0is the outer diameter of the rotor; τis the shear
stress of air, and µis the dynamic viscosity of air; Ais the
surface area of the rotor, and lis the rotor length.
The electrical losses of the stator and rotor windings are
different with different air gaps. The current at the uneven air
gap is k1times that at the average air gap by analyzing the
uneven air gap and the rate of current change. Then the power
of electrical loss with air-gap eccentricity can be calculated
by the following formula:
Pec =Z1+k1·(h−δ)/h2Pe
2πdα1.(4)
The motorized spindle uses a three-phase asynchronous mo-
tor as the drive. Alternating magnetic fields are generated
between the air gap, and that generates the magnetomotive
force. The fundamental wave magnetomotive force of the sta-
tor winding is as follows:
F1=4
π×I1N1sinωftcos pα,
F1=4
π×I1N1sin(ωft+2π
3)cos(pα +2π
3),
F1=4
π×I1N1sin(ωft−2π
3)cos(pα −2π
3).
(5)
The synthesis of three-phase fundamental wave magnetomo-
tive force is the synthesis of a fundamental wave of the stator
coil. Considering the fundamental wave distribution factor
and pitch factor, it is as follows:
Fd(α,t)=3√2
π×I1N1
pkd1kp1 cos(ωft−pα),(6)
where I1is the effective value of the stator-phase current; N1
is series turns of stator single-phase winding; pis the polar
logarithm; kd1 is the fundamental wave distribution factor of
the stator; kp1 is the stator fundamental wave pitch factor;
and ωfis the phase current angular frequency and αis an
electrical angle.
The rotor’s fundamental wave synthesis emf is as follows:
FZ(α,t)=3√2
π×I2N2
pkd2kp2
·coshωft−pα −π
2+ϕ1+ϕ2i,(7)
where I2is the effective value of the rotor phase current; N2
is the series turns of single-phase rotor winding; kd2 is the
fundamental wave distribution factor of the rotor; kp2 is the
fundamental pitch factor of the rotor; ϕ1is the angle of the
stator current hysteretic stator voltage; and ϕ2is the angle of
stator voltage with the rotor current hysteretic.
ϕ1=tan−1X1σ
R1,
ϕ2=tan−1sX2σ
R2,(8)
where X1σis the stator leakage reactance; X2σis the leakage
reactance of the rotor; R1is the resistance of stator winding;
R2is the resistance of rotor winding and sis the slip.
Then, the air-gap fundamental wave synthesis emf is as
follows:
F(α,t)=Fd(α, t)+Fz(α, t),(9)
(Fd1 =3√2
π
I1N1
pkd1kp1 ,
Fz2=3√2
π
I2N2
pkd2kp2 .(10)
Because the stator and rotor fundamental wave synthesis
magnetomotive force is relatively static, then
F(α,t)=Fcos(ωft−pα −β).(11)
Among them,
F=qF2
d1 +F2
z2+2Fd1Fz2cos θ ,
θ=π
2+ϕ1+ϕ2,
tanβ=Fz2cos(ϕ1+ϕ2)
Fd1−Fz2sin(ϕ1+ϕ2).
(12)
In the motorized spindle, the air-gap length is small, and the
eccentricity is also small. We only need to consider the first
third magnetic harmonic. Therefore, the air-gap permeability
is as follows:
∧(α1,t)≈µ0
h1+εcos(α1)+ε2cos2(α1)
+ε3cos3(α1).(13)
https://doi.org/10.5194/ms-12-109-2021 Mech. Sci., 12, 109–122, 2021
112 X. Li et al.: Research on the temperature field of a motorized spindle
The air-gap magnetic density is as follows:
B=µ0
F(α1,t)
δ(α1,t)= ∧(α1, t )F(α1, t ),(14)
where Fis the air-gap fundamental wave synthesis magne-
tomotive force.
Magnetic loss is caused by hysteresis and eddy current in
cores. When circulating magnetization, hysteresis power loss
per unit mass can be written in the following formula:
Pt=Cf B2
m,(15)
where Ptis hysteresis power loss; Cis the constant related to
the grade of electrical steel; fis the magnetization frequency
and Bmis the maximum magnetic induction intensity.
When the magnetic field in the iron core changes, the loss
caused by the induced current is called eddy current loss. The
eddy current loss of the stator and rotor of the motor is ex-
pressed as follows:
P=π2t2
1(f Bm)2
6ρrc
,(16)
where Pis the eddy current loss power, t1is the iron core
thickness, ρis the density of the iron core and rcis the resis-
tivity of the iron core.
2.2 Heating calculation of angular contact ball bearings
Bearing heating will lead to changes in grease and fitting
state, thermal deformation and other phenomena. It has an
important influence on the machining characteristics of the
motorized spindle. Its heat is mainly generated by the friction
of the bearing. The friction torque of the bearing is divided
into components of inner and outer rings. Then it becomes
the local component of the contact area (Harris and Kotza-
las, 2007), which can be written as follows:
Mij=Dw0.225f0(γ1ω1)2
3d3
m
+0.5f1P0i
C01
3
P1idmzd0,(17)
Moj=Dw0.225f0(γ1ω1)2
3d3
m
+0.5f1P0o
C01
3
P1odmzdi,(18)
where Dwis ball diameter; f0is the coefficient that depends
on bearing design and lubrication mode; γ1is the kinematic
viscosity associated with lubricants; dmis the pitch circle di-
ameter of ball bearing; f1is the coefficient related to bearing
type and load; P0is the equivalent static load of the bearing;
C0is the rated static load of the bearing; P1is the equiva-
lent load determining the friction moment; Zis the number
of balls; diis the diameter of the contact point of the inner
channel and dois the diameter of the contact point of the
outer channel.
Another important cause of bearing heating is the spin fric-
tion moment between rolling body and raceway contact area.
The spin friction torque in the contact area of the inner and
outer rings is as follows:
Msi =3KsiQiaiεi/8,(19)
Mso =3KsoQoaoεo/8,(20)
where Ksi,Kso are the friction coefficient between the ball
and the contact area of the raceway; Qi,Qoare normal con-
tact loads between the ball and the raceway. ai,aoare the
semi-axis length of the Hertz contact ellipse. εi,εoare ellip-
tic integrals of the second kind.
Therefore, the power of bearing heating is as follows:
Hij=ωrolliMij +ωsi Msi,(21)
Hoj=ωrolloMoj +ωso Mso,(22)
where ωrolli = −ωidm/Db,ωrollo = −ωodm/Db.Hij,Hoj
are the heating power generated by the contact area between
the rolling body and the inner and outer raceway. ωi,ωoare
the rolling angular velocity of the raceway with respect to
the inner and outer rings of the rolling body. ωsi,ωso are the
spin angular velocity of the rolling body with respect to the
contact area of the inner and outer circles.
As shown in Fig. 2, the motorized spindle heating model
with different rotation speeds is constructed from the collec-
tion of each motor loss and bearing heating under different
eccentricities.
2.3 Simulation parameters of the motorized spindle
heating considering air-gap eccentricity
According to the above calculation model, the temperature
field simulation parameters of the motorized spindle under
different air-gap eccentricity are calculated. Since the theo-
retical average air gap between the stator and rotor of the mo-
torized spindle is 300 µm, the mechanical power loss, electri-
cal loss, and magnetic loss varying with rotating speed under
the eccentricity of 0, 25, 50, 75 and 100 µm are calculated,
respectively.
The calculation results of mechanical loss are shown in
Fig. 3. The mechanical loss increases with the increase in ro-
tating speed and eccentricity, and it increases more obviously
with the increase in eccentricity at the same speed.
The electrical loss of the stator and rotor under differ-
ent eccentricities varies with the rotating speed as shown in
Figs. 4 and 5. The electrical loss increases with the increase
in rotating speed and eccentricity. It can be seen from the fig-
ures that the greater the air-gap eccentricity is at the same
speed, the more obvious the variation of the electric loss of
the stator and rotor is. Therefore, the influence of air-gap ec-
centricity on the electrical loss is not negligible.
Mech. Sci., 12, 109–122, 2021 https://doi.org/10.5194/ms-12-109-2021
X. Li et al.: Research on the temperature field of a motorized spindle 113
Figure 2. Motorized spindle heating model considering air-gap eccentricity.
Figure 3. Mechanical loss power under different air-gap eccentric-
ities.
Figure 4. The electrical loss of the stator with different eccentrici-
ties.
The maximum magnetic field strength under different ec-
centricities obtained by calculation is shown in Fig. 6. At the
same rotation speed, the maximum magnetic field strength
increases with the increase in eccentricity compared with
the situation without eccentricity. The maximum magnetic
field intensity increases by 9.09 %, 19.91 %, 32.81 % and
48.15 %, respectively, when air-gap eccentricity is 25, 50, 75
and 100 µm.
Figure 5. The electrical loss of the rotor with different eccentrici-
ties.
Figure 6. Maximum magnetic field strength with different rotating
speed and eccentricity.
The hysteresis loss of the motor stator is proportional to
the square of the maximum magnetic field strength. There-
fore, the stator hysteresis loss varies with the variation of air-
gap eccentricity, which is shown in Fig. 7. The stator hystere-
sis loss increases with the increase in rotating speed under
the same eccentricity, and it changes significantly with the
increase in rotating speed.
https://doi.org/10.5194/ms-12-109-2021 Mech. Sci., 12, 109–122, 2021
114 X. Li et al.: Research on the temperature field of a motorized spindle
Figure 7. Stator hysteresis loss power with different rotating speed
and eccentricity.
Figure 8. The stator eddy current loss power with different rotating
speed and eccentricities.
The eddy current loss of the stator and rotor is also closely
related to the maximum magnetic field strength. It varies with
the air-gap eccentricity too. The calculated eddy current loss
of the stator and rotor with different eccentricities varies with
the rotation speed, as shown in Fig. 8 and 9. The greater the
eccentricity is, the greater the eddy current loss is, and the
eddy current loss increases faster and faster with the increase
in eccentricity.
According to the motor loss power under different eccen-
tricity and bearing heating power, the overall heating power
of the motorized spindle with different eccentricities at each
speed is obtained. As shown in Fig. 10, the overall heating
power of the motorized spindle increases with the increase
in eccentricity, and the higher the rotating speed, the more
obvious the increase in heating power with eccentricity.
Figure 9. The rotor eddy current loss power with different rotating
speed and eccentricities.
Figure 10. Thermal power of the motorized spindle under different
eccentricities of each speed.
3 Study on temperature field simulation of the
motorized spindle considering air-gap eccentricity
3.1 Temperature field simulation of the motorized
spindle without eccentricity
The 3D model of the motorized spindle is established by
SolidWorks, and the minor components and small structure
of the motorized spindle are simplified to simplify calcula-
tion. Based on the specific structure of the model and the
solution requirements, a three-node triangular element is se-
lected to mesh the bearings, the stator and rotor, rounded cor-
ners and mating surfaces. The divided model has a total of
147 511 nodes and 65 428 units, as shown in Fig. 11. The
initial conditions are as follows: the ambient temperature is
16 ◦C; the coolant temperature is 16 ◦C; the coolant flow is
10 L min−1.
According to the theory of heat transfer, there are three ba-
sic modes of heat transfer, namely heat conduction, heat con-
vection and heat radiation (Minkowycz et al., 2015). Because
in the actual working conditions the radiation heat trans-
fer is very little, only thermal conduction and convection of
the motorized spindle are considered here. Taking 2000 rpm
as an example, the relevant thermal model parameters and
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X. Li et al.: Research on the temperature field of a motorized spindle 115
Figure 11. Gridding of the motorized spindle.
boundary conditions are shown in Table 1. In Table 1, λis the
thermal conductivity. ωδis the average air-gap velocity. utis
the circumferential velocity of the rotor end. c0,c1and c2
are constants measured in the experiment, 9.7, 5.33 and 0.8,
respectively. Based on the finite-element method, the steady-
state temperature field is simulated at 2000, 4000, 6000, 8000
and 10 000 rpm, respectively.
According to the structure and heat-transfer characteris-
tics of the motorized spindle, the circle corresponding to the
front-end cover is selected as circle 1 to analyze the distri-
bution of the circumferential temperature field at different
rotating speeds. In order to facilitate the analysis, the center
of circle 1 is taken as the origin, and the rectangular coordi-
nate system is established. The positions of eight measuring
points are shown in Fig. 12.
Because the influence of air-gap eccentricity is not consid-
ered in the simulation, the circumferential steady-state tem-
perature field at each speed obtained by simulation is rela-
tively uniform and in a symmetrical state.
The steady-state temperature field simulations at 2000,
4000, 6000, 8000 and 10 000 rpm are obtained. The circum-
ferential steady-state temperature field of each measuring
point of circle 1 at different rotating speeds is shown in
Fig. 13. Because the influence of air-gap eccentricity is not
considered in the simulation, the circumferential steady-state
temperature field at each speed obtained by simulation is rel-
atively uniform and in a symmetrical state.
3.2 Temperature field simulation of the motor
considering eccentricity
It is assumed that the lowest part of the motor is the mini-
mum air-gap position. The end surface temperature field of
the motor under 0, 25, 50, 75 and 100 µm air-gap eccentricity
is obtained. The positions of the minimum air-gap eccentric-
ity and maximum air-gap eccentricity are named the lower
half and upper half, respectively. The simulation boundary
conditions of 25 µm eccentricity are shown in Table 2.
3.3 Analysis of simulation results considering air-gap
eccentricity
Firstly, the overall temperature field distribution of the stator
and rotor without air-gap eccentricity and with different ec-
centricity values is analyzed, and Fig. 14 shows part of the
results. At 2000 rpm, the maximum temperature is 25.67 and
32.409 ◦C, respectively, when the air-gap eccentricity of the
motor is 25 and 100 µm. Compared with the non-eccentric
state, the temperature increases by 5.33 % and 32.99 %, re-
spectively, which is consistent with the calculation in the pre-
vious chapter. With the increase in air-gap eccentricity, the
loss of the motor increases gradually, which leads to the in-
crease in temperature rise of the stator and rotor. Further, the
circumferential temperature field of the stator and rotor also
presents a non-uniform state.
The circumferential temperature of the stator and rotor
in the different air-gap position is shown in Fig. 15. At the
smallest air-gap position, the motor generates more heat.
Therefore, the temperature of the stator and rotor is highest in
this position, which is 31.658 ◦C of the stator and 31.215 ◦C
of the rotor. The temperature of the motor decreases with
the increase in air-gap length. This will result in asymmet-
ric distribution of the circumferential temperature field be-
tween the stator and rotor, and there is a large temperature
difference between the stator and rotor, 0.703 and 0.585◦C,
respectively.
The maximum circumferential temperature difference of
the stator end surface under different eccentricities is shown
in Table 3. When the air gap is evenly distributed, the circum-
ferential temperature field of the stator is evenly distributed.
When the air-gap eccentricity is 25, 50, 75 and 100 µm, the
maximum temperature difference of the stator circumferen-
tial temperature field is 0.173, 0.342, 0.519 and 0.703 ◦C, re-
spectively. With the increase in air-gap eccentricity, the max-
imum temperature difference in the circumferential direction
of the stator increases gradually; that is, the degree of non-
uniformity in the circumferential temperature field increases.
4 Experimental study
4.1 Design of experimental scheme
In order to verify the accuracy of the temperature field simu-
lation of the motorized spindle and explore the actual distri-
bution law of the temperature field of the motorized spindle,
a non-contact motorized spindle temperature test bed with
static eccentricity is built. The motorized spindle is produced
by the Luoyang Bearing Research Institute. The air-gap ec-
centricity in the initial assembly of the motorized spindle is
ensured by the top wire at the stator housing position (as
shown in Fig. 16). Since the air gap of the motorized spin-
dle used in our experiment is required to be 300 µm, the
static eccentricity is set to about 100 µm, and the static eccen-
tricity measured by the air-gap gauge is 90 µm. The experi-
https://doi.org/10.5194/ms-12-109-2021 Mech. Sci., 12, 109–122, 2021
116 X. Li et al.: Research on the temperature field of a motorized spindle
Table 1. Thermal boundary conditions and thermal model parameter.
The boundary conditions Heat transfer coefficients Value
Heat transfer between front bearing and cooling water jacket h=Nufλ/d262.14
Heat transfer between stator and cooling water jacket h=Nufλ/d2102.93
Heat transfer between the air gap αδ=281+ω0.5
δ81.53
Heat transfer at the rotor end at=281+√0.45ut78.78
Heat transfer of the moving part of the spindle end αz=c0+c1u−c227.46
Heat transfer between shell and environment 9.7
Heating rate of the stator/W m−357801
Heating rate of the rotor/W m−336240
Heat generation rate of outer ring of front bearing/W m−3167 840
Heat generation rate of inner ring of front bearing/W m−325 8711
The rate of heat production of the front bearing ball/W m−31 308 943
Heat generation rate of outer ring of rear bearing/W m−3154 248
Heat generation rate of inner ring of rear bearing/W m−3257 093
Heat generation rate of rear bearing ball/W m−31 290 573
Figure 12. End circumference and measuring point distribution.
Figure 13. Circumferential steady-state temperature field distribu-
tion of circle 1 at different rotating speeds.
mental system includes infrared thermography, a frequency
converter, a water-cooling machine and a control system. A
frequency converter is used to control spindle-rotating speed
and display frequency and current. The water-cooling ma-
chine is used to provide a fixed flow of circulating coolant
and display the coolant-setting temperature, real-time tem-
Table 2. Thermal model parameters considering eccentricity.
Thermal load and boundary conditions Value
Heating rate of the upper part of the stator/W m−363165
Heating rate of the lower part of the stator/W m−368 827
Heat generation rate of the upper part of the rotor/W m−337222
Heat generation rate of the lower part of the rotor/W m−343 354
Heat transfer coefficient between the air gap/W (m2◦C)−181.53
Heat transfer coefficient of the rotor end/W (m2◦C)−178.78
Heat transfer coefficient of the stator cooling jacket/W (m2◦C)−162.14
perature and ambient temperature. Each device is shown in
Table 4.
The physical diagram of the motorized spindle tempera-
ture test bed and the overall structure layout are shown in
Fig. 17.
The vibration signal measured by the experiment can be
used to detect the eccentric state of the motor (Li et al.,
2007). In order to determine whether there is eccentricity
in the initial assembly of the motorized spindle, the vibra-
tion tests of the motorized spindle at 4800 and 13 200 rpm
were carried out by using the B&K data acquisition instru-
ment and a three-way acceleration sensor. Its vibration signal
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X. Li et al.: Research on the temperature field of a motorized spindle 117
Figure 14. The influence of air-gap eccentricity at 2000 rpm on the temperature field of the stator rotor.
Table 3. The circumferential maximum temperature difference of the stator under different air-gap eccentricity.
Eccentricity/µm 0 25 50 75 100
The maximum circumferential temperature difference/◦C 0 0.173 0.342 0.519 0.703
and spectrum diagram after fast Fourier transform are shown
in Figs. 18 and 19. The air-gap eccentricity of the motorized
spindle will cause unbalanced magnetic tension and thus pro-
duce vibration of a different frequency. When its static ec-
centricity exists, the unbalanced magnetic tension will cause
the vibration of the motorized spindle with a frequency of
4 times for the motorized spindle with two pole pairs. As can
be seen from the figure, the corresponding 4-fold frequency
conversion amplitude is large at different speeds, while other
frequency conversion is relatively small. It shows that the ini-
tial assembly of the motorized spindle has a certain static ec-
centricity.
Taking the temperature field of the motorized spindle un-
der no load as the main part, the motorized spindle tempera-
ture test bed is used to comprehensively explore the steady-
state temperature field distribution of the motorized spindle
end surface at different speeds (2000, 4000, 6000, 8000 and
10 000 rpm). The setting parameters of the infrared thermal
imager are 1 m away from the object, 50 % relative humidity,
and 0.95 sensitivity. The initial conditions of this group of
experiments are consistent with those of simulation experi-
ments.
4.2 Analysis of experimental results
The temperature field cloud diagram of the motorized spin-
dle end surface at 2000 rpm and different times is shown in
Fig. 20. The end surface temperature field presents a non-
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118 X. Li et al.: Research on the temperature field of a motorized spindle
Table 4. Non-contact temperature test bed equipment parameters.
Part Model Note
Computer LenovoZ480 4 GB of memory
Inverter VFD32 380 V, 15 kW
Infrared thermal imager FLIRT630 The temperature range is −40–150 ◦C, the accuracy is ±0.01 ◦C
Water-cooling machine MCW-35C Cooling capacity is 3.5 kW, the input power is 1.9kW
Motorized spindle A type Rated power is 10 kW, rated current is 24A
Figure 15. The temperature of the stator and rotor with 100 µm
eccentricity at different air-gap positions.
Figure 16. Schematic diagram of the motor spindle jacking wire.
uniform phenomenon at 10 min through analysis. With the
increase in time, the non-uniform state gradually deepens un-
til the initial thermal equilibrium state is reached at 40 min.
In the corresponding part of the front-end cover, circle 1 is
selected to study the distribution rule of the circumferential
temperature field at different rotating speeds. The circumfer-
ential steady temperature fields of each measuring point on
circle 1 at each rotating speed are shown in Fig. 21. The max-
imum temperature appears at point g at each rotating speed
(except 10 000 rpm). For example, at 2000 rpm, the maxi-
mum temperature is 29.623 ◦C at point g and the minimum
temperature is 28.158 ◦C at point c. Overall, the circumfer-
ential temperature field of circle 1 in operation is not uni-
form, and the non-uniform state of each rotating speed is that
the temperature of the lower part is higher than that of the
upper part. This is consistent with the simulation of temper-
ature distribution considering air-gap eccentricity. There are
Figure 17. Experimental system principle and object diagram.
(a) Schematic diagram of experimental design. (b) Water-cooling
machine. (c) The inverter. (d) Experimental system structure dia-
gram.
two reasons for this analysis. Firstly, in terms of heat dissipa-
tion, the heat-dissipating capacity of each component is un-
even. So, the thermal expansion is not uniform. Further, with
the working clearance changing, the circumferential thermal
resistance also changes. These will lead to the spindle cir-
cumferential temperature field in an asymmetric state. Sec-
ondly, the existence of air-gap eccentricity in the motor leads
to non-uniform heating power in terms of heat generation.
High heat generated at small air gaps leads to higher temper-
ature. As a result, the temperature of the lower part of the
stator and rotor is higher and the circumferential temperature
field presents a non-uniform state. Heat from the stator and
rotor is transferred to the end cover along the axial direction,
which leads to the non-uniform temperature field in the cir-
cumferential direction of the front-end cover. In addition, the
air-gap eccentricity of the motor will also make the tempera-
ture of the stator and rotor increased.
Suppose the maximum temperature of the circumferential
temperature field is Tmax and the minimum temperature is
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X. Li et al.: Research on the temperature field of a motorized spindle 119
Figure 18. Vibration signal of the motorized spindle at 4800 rpm.
Figure 19. Vibration signal of the motorized spindle at 13 200 rpm.
Tmin. Then the circumferential maximum temperature differ-
ence is calculated by 1T =Tmax −Tmin.
The degree of non-uniformity is expressed by the circum-
ferential maximum temperature difference. The maximum
temperature difference at each rotation speed on circle 1 is
shown in Table 5. The maximum temperature difference in-
creases from 1.465 to 3.468 ◦C with the increase in rotating
speed. It indicates that the non-uniformity at circle 1 gradu-
ally deepens with the increase in rotating speed. The reason
is the same as the calculated power loss trend of the motor.
With the same eccentricity, each power loss of the motor in-
creases significantly with the increase in rotating speed, and
the loss difference at different circular positions is more obvi-
ous. Therefore, its circumferential non-uniformity deepens.
5 Conclusions
In this paper, based on the established thermal model, the in-
fluence of air-gap eccentricity on the temperature field distri-
bution of a motorized spindle was studied and the circumfer-
ential temperature field distribution of the motorized spindle
was revealed. The conclusions are as follows.
In the study of thermal model, mechanical loss, electri-
cal loss and magnetic loss of the motor all increase with the
increase in air-gap eccentricity. The mechanical loss power
of 100 µm eccentricity increased by 6.31 % compared with
that without eccentricity at 2000 rpm. The electric loss of the
stator and rotor increased by 18.03 % and 18.04 %, respec-
tively. Stator hysteresis loss increased by 1.17 times. The
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120 X. Li et al.: Research on the temperature field of a motorized spindle
Figure 20. Transient temperature field distribution at different times of the motorized spindle at 2000rpm.
Figure 21. Circumferential temperature field of circle 1 at different rotating speeds.
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X. Li et al.: Research on the temperature field of a motorized spindle 121
Table 5. The maximum temperature difference of circle 1 at different rotating speeds.
Speed/rpm 2000 4000 6000 8000 10 000
Circumferential maximum temperature difference/◦C 1.465 1.653 2.451 3.195 3.468
eddy current loss power of the stator and rotor increased by
1.19 times. It indicates that air-gap eccentricity has a great in-
fluence on motor loss. Therefore, air-gap eccentricity should
not be neglected in the establishment of the thermal model.
Based on the finite-element method, the steady-state tem-
perature field simulation study of the motorized spindle was
conducted without considering air-gap eccentricity. The re-
sults show that the circumferential temperature field distribu-
tion is symmetrical, and the results of the temperature field
after considering air-gap eccentricity show that the increase
in air-gap eccentricity leads to the increase in temperature
rise of the stator and rotor. Further, the temperature is higher
at smaller air gap, and the circumferential temperature field
is in a state of asymmetric distribution, and with the increase
in eccentricity, the degree of non-uniformity increases up to
0.703 ◦C.
The steady-state temperature field experimental results of
the motorized spindle at various rotating speeds show that
the circumferential temperature field of the end cover is non-
uniform, and the non-uniform state at each rotating speed is
that the temperature of the lower half is higher than that of the
upper half. It is consistent with the simulation results of tem-
perature field considering air-gap eccentricity. The degree of
non-uniformity gradually deepens with the increase in rotat-
ing speed.
This paper provides a theoretical basis for the accurate
thermal characteristic analysis and the improvement of ma-
chined surface quality of the motorized spindle.
Data availability. All the data used in this paper can be obtained
from the corresponding author upon request.
Author contributions. XL proposed the research route and com-
pleted the first draft; JL modified and improved the paper and sup-
plemented relevant data; CL completed part of the experiment and
data analysis; JH provided guidance for the experiment and paper
framework; DW provided guidance and analysis for replenishing
experimental data.
Competing interests. The authors declare that they have no con-
flict of interest.
Special issue statement. This article is part of the special issue
“Robotics and advanced manufacturing”. It is not associated with a
conference.
Acknowledgements. The authors express their gratitude for the
financial support mentioned below.
Financial support. This work was supported by the National
Natural Science Foundation of China (grant nos. 52075428 and
51575434), the Major Science and Technology projects of Shaanxi
Province of China (grant no. 2018zdzx01-02-01HZ01) and the
Open Fund funded project of Henan Key Laboratory of High-
Performance Bearing Technology (grant no. 2020ZCKF04).
Review statement. This paper was edited by Peng Li and re-
viewed by Wei Wang and one anonymous referee.
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