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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

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 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-

https://doi.org/10.5194/ms-12-109-2021 Mech. Sci., 12, 109–122, 2021

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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