heat transfer

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International Journal of Machine Tools & Manufacture 47 (2007) 53–62

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An improved thermal model for machine tool bearings

Xu Mina, Jiang Shuyuna,�, Cai Yingb

aDepartment of Mechanical Engineering, Southeast University, Nanjing, 210096, P.R. ChinabWuxi Machine Tool Corporation, Wuxi, 214061, P.R. China

Received 28 November 2005; received in revised form 24 February 2006; accepted 27 February 2006

Available online 19 April 2006

Abstract

Thermal model for machine tool spindle is of great importance to machine tool design. Traditionally, the thermal contact resistance

between solid joints and the change of the heat generation power with the bearing temperature are often ignored when thermal

characteristics of a machine tool spindle are analyzed. This has caused inaccuracies in the thermal model. With the heat source models

and the heat transfer models from Bossmanns and Tu [Journal of Manufacturing Science and Engineering 123 (2001) 495–501,

International Journal of Machine Tools & Manufacture 39 (1995) 1345–1366], a model including the thermal contact resistance at solid

joints based on a fractal model and the change of the heat generation power, viz. the amount of the heat generation per second, with the

bearing temperature increases is developed. The complete thermal model is used to simulate the temperature distribution in grinding

machine housing with a conventional spindle bearing. Compared with experiment, it is shown that the completed model is much more

accurate than the traditional model which ignores the two important factors above. The thermal expansion of the housing system is

analyzed.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Machine tool spindle; Thermal characteristics; Rolling bearing; Joint; Heat generation power

1. Introduction

In recent years, with the development of the high speedmachining, the thermal characteristics of machine toolshave been held of much account by many researchers. It isone of important factors that affect the performance ofmachine tool systems. There have been a number ofthermal or thermo-mechanical models to investigate thethermal and dynamic responses of machine tool spindles.Bossmanns and Tu [1,2] developed a finite difference modelto characterize the heat generation, heat transfer and heatsinks of a high-speed motorized spindle. Lin et al., [3]presented an integrated model with experimental validationand sensitivity analysis for studying various thermo-mechanical-dynamic spindle behaviors at high speeds. Liand Shin [4] developed a more comprehensive integratedthermo-dynamic model for high-speed spindles using finiteelement method, which is coupled with the spindle dynamic

e front matter r 2006 Elsevier Ltd. All rights reserved.

achtools.2006.02.018

ing author. Tel.: +8625 83794920; fax: +86 25 83791414.

ess: [email protected] (J. Shuyun).

model through bearing heat generation and thermalexpansion of the whole system. They [5] also investigatedthe effects of bearing configuration on the thermo-dynamicbehavior of spindles using the model. In the modelsmentioned above, heat transfer and heat conduction,which are the important boundary conditions for thermalanalysis, are taken into account. But all of them ignoredthe thermal contact resistance at the solid joints and thechange of the heat generation power of the bearings withtemperature increases. They assumed that the temperaturesof two contact surfaces were coupled or set an experientialconstant value on the resistance for all kinds of joints andthe heat generation power was constant.There are many joints existing in a machine tool spindle

system, such as the interfaces between the bearing and theshaft, the bearing and the bearing support, the housing andthe covers and so on. When two surfaces are in contact, thepresence of surface roughness produces imperfect contactat the joint, no matter how much the pressure between thesurfaces is. The imperfect contact results in a sharptemperature drop across the joint. Such a temperature

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Fig. 2. The spindle system.

Table 1

The viscosity-temperature characteristic of the grease

Temperature (1C) Kinematic viscosity (mm2/s)

X. Min et al. / International Journal of Machine Tools & Manufacture 47 (2007) 53–6254

jump plays a significant role in the thermal characteristicsof the system. On the other hand, as the heat generationpower of bearings is in proportion to the viscosity of thebase oil of the grease and the viscosity changes greatly withtemperature, the change of the heat generation power isnotable. Sometimes, the decrease in heat generation powercan reach 50% when the temperature increases. Ignoringthese two facets, the previously developed models fail toprovide a precise analysis result for the system.

In this paper, the thermal characteristics of a givengrinding machine housing with a conventional spindlebearing was investigated experimentally and numerically,using finite element method. With the heat source modelsand the heat transfer models from Bossmanns and Tu [1,2],this paper develops the models for the thermal contactresistance at the solid joints based on a fractal model andthe heat generation power as the bearing temperaturevaries to complete the thermal model.

25 40

30 23

40 20

2. Grinding machine housing

The setup to be modeled is a grinding machine, housingof a centerless grinder, as shown in Fig. 1. Its spindlebearing system is a conventional spindle with a grindingwheel located on the bearing span (Fig. 2). The spindleworks at 1100 rpm and its maximum speed is 1500 rpm.Thematerial of the shaft is steel 40Cr and that of the frame iscast iron.

Two pairs of bearings were mounted on the spindle. Thebearing span is about 600mm.A double-row short cylind-rical roller bearing NN3018K and a pair of angularcontact ball bearings 7018 are on the left and a double-rowshort cylindrical roller bearing is on the right. From left toright, the bearings are marked 1, 2, 3 and 4, respectively.The bearing mounting area of the shaft was ground.

The angular contact ball bearings are mounted withclearance tolerance to avoid the preload variation of thebearings due to temperature variation and arranged in Oarrangement (back to back arrangement) and preloaded by

Fig. 1. Model of the grinding machine housing.

two sleeves and a locknut. The double-row short cylind-rical roller bearings are mounted with interference toler-ance. The preload of the bearings is 600N.The Asnic HQ72-102 grease is used to lubricate the

bearings. The viscosity-temperature characteristic of thebase oil of the grease is shown in Table 1.

3. Thermal model

3.1. Finite element model

The finite element model for the grinding machinehousing is established using the ANSYS software package,as shown in Fig. 3. Z-axis is on the axis of the shaft. Theorigin of coordinate is on the mid of the abrasion wheel.The SOLID87 element is used to simulate the temperaturefield distribution. And the structural element SOLID92 isadopted to calculate the displacement of the grindingmachine housing due to temperature variation.To get good calculating precision and speed simulta-

neously, the elements near the bearings are meshed muchmore refinedly than in other regions. There are a total of51,355 solid elements for the FEA model.The contact elements CONTA174 and TARGE170 are

used to simulate the joints in the system. Because the jointsbetween the housing and the covers are far from thebearings, the influence of the thermal resistances of thesejoints on the thermal characteristics of the whole machinetool system can be neglected. So only the thermal contactresistances between the bearing outer rings and the bearingsupport, and the inner rings and the shaft neck have beenconsidered. There are eight joints in the grinding carriage,

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Fig. 3. The finite element model.

Fig. 4. Joints between bearings and frame, bearings and shaft.

X. Min et al. / International Journal of Machine Tools & Manufacture 47 (2007) 53–62 55

shown as Fig. 4. To simulate the joints, create contact pairsat the joints and define the real constant TCC, i.e. thermalcontact conductance, of each contact element as thecontact conductance coefficient of relative joint, which isto be explained below.

3.2. Thermal contact resistance

The imperfect contact produced by surface roughnessindicates that it isn’t a full contact at the joint. There aremany contact spots and cavities between the surfaces. Inthe cavities, it is filled with filling material. Generally, thethermal conductivity of the filling is much lower than thatof the parts, so it gives to the thermal resistance and thetemperature jumps [6].

Conventional methods to study the thermal contactresistance are by experiment [7,8] and the models based onthe statistical characterization of the rough surfaces [9]. Butboth the experiments and the parameters of the modelsdepend strongly on the resolution and the precision of theroughness measuring instruments. They fail to provide ananalysis model that is suitable for all occasions. Toestablish a scale-independent model, some researchers haveintroduced fractal theory into the contact resistances [10],but they ignored the upper limit of the sample length Lu ofthe contact surface, which is important to the regularengineering surfaces [11,12]. A new fractal model for

thermal contact resistance is founded based on M–B fractalmodel to overcome the shortages of the common thermalcontact resistance models.

3.2.1. Thermal contact resistance.

The thermal contact resistance R can be expressed as [6]

R ¼1

Ahc

, (1)

where A is the apparent contact area of a contact region,hc is the contact conductance coefficient. Ignoring theradiation heat transfer, the contact conductance coefficientfollows the relation

hc ¼1

Lg

Ac

A

2k1k2

k1 þ k2þ

Av

Akf

� �, (2)

where Lg is the thickness of the void space between twocontact surfaces, Ac is the real contact area of the joint, Av

is the void area of the joint, k1, k2, kf are the thermalconductivities of the materials of the two parts and themedium, respectively. If Lg, Ac and Av are known, thethermal contact resistance can be given.

3.2.2. Dimensionless fractional contact area

A unique property of rough surfaces is that if a surface isrepeatedly magnified, increasing details of roughness areobserved right down to nanoscales. In addition, the profilesat all magnification appear quite similar in structure. Sucha behavior can be characterized by fractal geometry. Incontrast with the statistical parameters of the roughsurfaces, the fractal parameters do not depend on theresolution of the roughness measuring instruments or thelength scale of the sample and are scale-independent.Majumdar and Bhushan [13] developed a fractal

contact model based on the fractal theory to describe thecharacterization of the rough surfaces. However, as anengineering surface cannot be fully characterized by a purefractal because the surface contains a deterministic part inits shape, Wang and Komvopoulos proposed the conceptof a fractal-regular surface [11]. Take an engineeringplane for example, though the surface profile appearsrandom, multiscale and disordered on a microscale, as it isintentionally made flat, it is regular in macroscale anddifferent from the naturally formed fractal surfaces, such asthe surface of the Earth. Therefore, the engineeringsurfaces exhibit a fractal behavior only within a finiterange of length scales. To overcome the difficulty, theupper limit of the sample length Lu is proposed. The fractalsample length L in W–M function should be less than orequal to Lu. For the sample whose length is larger than Lu,it should be divided into a number of fractal domains withareas equal to L2

u. As the pressure distributes evenly, thedimensionless fractional contact area of the sampleA�ð¼ Ac=AÞ is equal to the dimensionless fractional contactarea of the fractal domain A�r ð¼ Arf =Lu2Þ, where Arf is thereal contact area of the fractal domain.

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Table 2

Contact conductance coefficients at the joints (m2�K/W)

No. of bearings #1 #2 #3 #4

Inner ring/shaft neck 7680 7680 10122 10122

Outer ring/bearing support 500 500 2000 2000


The improved M–B fractal model has three parameters:the fractal dimension of the surface D (1oDo2), thefractal roughness parameter G and the upper limit of thesample length Lu. The three characterization parameterscan be gotten from the power spectrum of the W–M

function [11].According to the contact theory [13], Ar

* is related to thedeformation style of the asperities. When an island is incontact with plane and brought deformation, if the contactarea of the island a is less than the critical microcontactarea ac, its deformation is plastic. If a4ac, it is elastic. ac isgiven by

ac ¼G2

ðH=2EÞ2=ðD�1Þ, (3)

where H is the hardness of the softer material between twoparts, E is the equivalent elastic modulus, defined asE ¼ [(1�v1

2)/E1+(1�v22)/E2]

�1, E1, E2, v1 and v2 are theelastic moduli and Poisson’s ratios of the two materials,respectively.

Therefore, if the largest contact area aL of the fractaldomain is less than the critical area, aLo ac, only plasticdeformation will take place at the joint. Ar

* follows therelation

A�r ¼p

H. (4)

If aL4ac, part of asperities are plastic contact and theother asperities are elastic contact. Calculating the integralsof the elastic and plastic forces at the joint and neglectingsmall term, Ar

* follows the relation [12]When Do1.5

A�r ¼3ffiffiffiffiffiffi2pp

4

!2=ð3�DÞLu

G

� �ð2D�2Þ=ð3�DÞ

g4ðDÞp

E

� �2=ð3�DÞ

,

(5)

when D41.5

A�r ¼ ½g3ðDÞ��2=D p

E

� �2=D 4

3ffiffiffiffiffiffi2pp

G

Lu

� �D�1

�g1ðDÞ� �

a�c ð3�2DÞ=2

(

þH

Eg2ðDÞ a�c

ð2�DÞ=2��2=D

ð6Þ

where p is the apparent pressure, a�c is the normalizedcritical microcontact area, a�c ¼ 2ac=L2

u;c is the domainextension factor for microcontact size distribution.

g1ðDÞ ¼ ½ð2�DÞ=D�D=2D=ð3� 2DÞ,

g2ðDÞ ¼ ½D=ð2�DÞ�ð2�DÞ=2,

g3ðDÞ ¼ 2D=2cð4�4DþD2Þ=4,

g4ðDÞ ¼ ½g1ðDÞg3ðDÞcðD�2Þð3�2DÞ=4

�2=ðD�3Þ½D=ð4� 2DÞ�ð3�2DÞ=ð3�DÞ.

3.2.3. Thickness of the void space between two contact

surfaces

For two contact planes, the thickness of the void space atthe joint Lg can be expressed as

Lg ¼ z1 þ z2 � d1 � d2, (7)

where z1, z2 are the height of the asperities of two planesrespectively,d1,d2 are the largest deformation of theasperities of the planes respectively. Many of the engineer-ing contact surfaces have the same roughness andmachining process. So Eq. (7) can be written as

Lg ¼ 2ðz� dLÞ, (8)

where z can be obtained by the roughness of the surfaces.On the other hand, according to the fractal theory, the

largest deformation of the asperities on the plane is

dL ¼ GD�1ða0LÞð2�D=2Þ (9)

where aL’ is the largest truncated area of the asperities,aL’ ¼ 2aL.The statistical distribution of the truncated microcontact

area a’ is given by

nða0Þ ¼D

2cð2�DÞ=2

ða0LÞD=2ða0Þ�ðDþ2Þ=2, (10)

where a’ ¼ 2a, then, the integrals in Eq. (10) can beevaluated as

Arf ¼

Z a0L

0

nða0Þa da0 ¼ cð2�D=2Þ D

4� 2Da0L (11)

From Eqs. (8)–(11), and A�r ¼ Arf =L2u; it is can be

obtained that

Lg ¼ 2 z� cðD�2ÞðD�2Þ=4GD�1 4� 2D

D

� �ð2�DÞ=2"

ðA�r Þð2�DÞ=2Lu2�D

#ð12Þ

3.2.4. Thermal contact resistance of the joint

In Eq. (2), Av/A ¼ 1-Ac/A. So the thermal contactresistant of the joint R is

R ¼Lg

A½A�r k þ ð1� A�r Þkf �, (13)

where k ¼ 2k1k2/(k1+k2). Ar* and Lg are determined from

Eqs. (4)–(6) and (12).

ARTICLE IN PRESSX. Min et al. / International Journal of Machine Tools & Manufacture 47 (2007) 53–62 57

The values of the contact conductance coefficients at thejoints are shown in Table 2.

3.3. Heat generation and heat transfer

3.3.1. Heat generation

The major heat generation of the system is caused by thecutting process and the friction between the balls and racesof the bearings [14]. Assumed that the majority of cuttingheat is taken away by coolant and chips, the heat generatedby bearings is the dominant cause of temperature change.The heat generated by a bearing can be computed as

Hf ¼ 1:047� 10�4nM, (14)

where Hf is the heat generated power (W), n is the rotatingspeed of the bearing (rpm), M is the total frictional torqueof the bearing (Nmm). The total frictional torque M

consists of two parts, one is the torque M1 due to appliedload and the other one is the torque M2 due to viscosity oflubricant.

That is

M ¼M1 þM2, (15)

where

M1 ¼ f 1p1dm, (16)

where f1 is a factor related to the bearing type and load, p1is the bearing preload (N), dm is the mean diameter of thebearing (mm).

M2 ¼ 10�7f 0ðn0nÞ2=3d3

m if n0n � 2000, (17)

M2 ¼ 160� 10�7f 0d3m; if n0no2000, (18)

where f0 is a factor related to bearing type and lubricationmethod, v0 is the kinematic viscosity of the lubricant (mm2/s).

From above equations, it is can be seen that the heatgeneration power is dependent on the viscosity of the baseoil of the grease. As the temperature rises, the viscosity willdecrease and therefore the heat generation power willdecrease, too. According to the viscosity-temperaturecharacteristic of the base oil of the grease shown in Table1, the heat generation power of each bearing is shown inTable 3.

Taking the heat generation powers with respect to thebearing working temperatures as body load table andapplying it to the bearing volumes as the thermal boundary

Table 3

The heat generation power of each bearing (W)

No. of bearings #1 #2 #3 #4

251C 44.3 44.3 60.8 60.8

301C 30.8 30.8 44 44

401C 28 28 40.6 40.6

condition, this paper complete the thermal analysis for thegrinding machine housing.

3.3.2. Heat transfer coefficient

The heat transfer coefficient for convection h isinvestigated in [2,4]. It is defined as

h ¼Nukfluid

d, (19)

where kfluid is the thermal conductivity of the ambient air,Nu is the Nusselt number. When the convection occurs atouter surfaces of a long cylinder, such as a shaft, d is thediameter of the cylinder.The Nusselt number Nu is computed from the Reynolds

number, Re, and the Prandtl number, Pr, based ondifferent convection conditions. For this research, thefollowing equation is used [15]:

Nu ¼ 0:133Re2=3Pr1=3, (20)

where

Re ¼ufluidd

nfluid, (21)

Pr ¼cfluidmfluid

kfluid

, (22)

where ufluid is the velocity, vfluid is the kinematic viscosity,cfluid is the specific heat capacitance and mfluid is the dynamicviscosity of the air.This equation is valid for Reo4.3� 105, 0.7oPro670.

For free convection around stationary surfaces, h ¼

9:7W=ðm2KÞ is used [2].

3.4. Bearing stiffness

When calculating the displacement of the grindingmachine housing due to temperature variation, the radialand the axial stiffness of the bearing must be known. Thebearing stiffness of the two pair of bearings can beobtained by the aid of the Rolling Bearing AnalysisSoftware developed by Xu and Jiang [16]. The values ofthe stiffness are listed in Table 4. To calculate the stiffness,the contact pairs discussed in 3.1 are used, too. Evaluatethe real constants of the contact element FKN and FKT,i.e. normal stiffness and tangent stiffness, with relativestiffness of the bearing.

Table 4

The stiffness of the bearings (N/m)

Bearing NN3018K 7018

Radial stiffness 5.3277e9 1.817e8

Axial Stiffness 1.7759e8 6.057e6

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Fig. 5. Steady-state temperature distribution of the entire grinding machine housing.

0 5000 20000 2500024

26

28

30

32

34

36

38

40

42

Tem

pera

ture

(°C

)

300001500010000Time (sec)

Fig. 6. Temperature history of the bearing #2.


4. Thermal analysis

The analysis includes the steady state and transient statetemperatures and the thermal expansion. For steady stateanalysis, the temperature distribution of the whole grindingmachine housing is presented. For transient state analysis,temperatures of six locations are measured with infraredthermoscope to compare with simulation.

4.1. Temperature analysis

Fig. 5 shows the steady-state temperature distribution ofthe entire grinding machine at a rotational speed of1100 rpm and the reference temperature is 25 1C. Thetemperature at the left of the housing is much higher thanthat at the right. The maximum temperature occurs at theinner ring of the bearing #2. This is mainly because the heatgeneration powers of the left bearings are much greaterthan that of the right bearings and the thickness of the leftbearing support is smaller than that of the right bearingsupport. This causes very different thermal expansionsbetween the left and right bearing supports, which will bediscussed in Section 4.2.

Fig. 6 shows the temperature variation of the bearing #2with respect to time. It rises sharply at the beginning andgently after a short time.

Fig. 7(a)–(f) are comparisons between the measured andpredicted temperature histories of six locations. Thelocations are on the side surface of the grinding carriageas shown in Fig. 1.

From Fig. 7, it can be seen that the temperaturepredictions for the locations match with the measuredvalues very well despite the elements far from the bearingswere coarser. In contrast to bearing #2, the temperatures ofthe six locations increase slowly at the very start andsharply at the early stage, and gradually saturate to thefinal temperature when the amount of heat generationbalances with the heat dissipation into the atmosphere.Their rising times are much longer than those of thebearings.

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0 5000 10000 1500024

26

28

30

32

34

36SimulationsMeasurements

SimulationsMeasurements





Tem

pera

ture

(°C

)

24

26

28

30

32

34

36

Tem

pera

ture

(°C

)

24

26

28

30

32

34

36

Tem

pera

ture

(°C

)

24

26

28

30

32

34

36

Tem

pera

ture

(°C

)24

26

28

30

32

34

36

Tem

pera

ture

(°C

)

24

26

28

30

32

34

36

Tem

pera

ture

(°C

)

Time (sec)

20000 0 5000 10000 15000

Time (sec)

20000

0 5000 10000 15000

Time (sec)

20000 0 5000 10000 15000

Time (sec)

20000

0 5000 10000 15000

Time (sec)

20000 0 5000 10000 15000

Time (sec)

20000

(a) (b)

(c) (d)

(e) (f)

Fig. 7. Comparison of the temperature histories, simulated, experimental: (a) Locations 1; (b) Locations 2; (c) Locations 3; (d) Locations 4; (e) Locations

5; and (f) Locations 6.


As shown in Fig. 8, the temperatures of the two partsare not continuous and there are temperature jumpsat the bearing and shaft joints and also at the bearingand bearing support joints. This is due to the thermalcontact resistance at the joints. As the heat flow has beendammed up by the resistance, the temperature of theparts in which heat is generated is higher than those of theother parts.

Fig. 9 is a comparison of temperature historiesmeasured, the calculated either with thermal contactresistance and the change of the heat generationconsidered or without thermal contact resistance andthe change of the heat generation considered. As shownin Fig. 9, the temperature without considering thethermal contact resistance and the change of the heatgeneration power is about 1.5 1C higher than the measuredvalues.

4.2. Thermal displacement analysis

Fig. 10 shows the thermal displacements of the grindingmachine housing, which takes the ANSYS’ coordinatesystem as datum and are magnified 1000 times, after heatbalancing. For clarity, the covers on the shaft are notshown in the figure. The black lines are the original shapeof the housing. As shown in Fig. 5, the temperature at theleft of the housing is much higher than that at the right.The displacement at the left of the housing is larger thanthat at the right, and the maximum displacement appearsat the left top corner of the coping, the X-, Y- andZ-component displacements of which are �14.776, 51.982,and 6.962e�3mm, respectively. The fact that the total heatgeneration power of the left bearings is much larger thanthat of the right bearings is the reason for the thermaldisplacement distribution.

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0 5000 10000 1500024

26

28

30

32

34

36

Tem

pera

ture

(°C

)

With contact resistanceWithout contact resistanceMeasurements

20000

Time (sec)

Fig. 9. Comparison of temperature histories among the three results of location 2.

Fig. 8. Simulated temperature distribution of the grinding machine housing.


To investigate the effect of the thermal expansion on themachining precision, the displacements of six points alongthe shaft are listed in Table 5. The locations of the sixpoints are shown in Fig. 2. Point A and F are at the mid ofthe two double-row short cylindrical roller bearingsrespectively. B, C, D and E are between A and F, sharingthe bearing, span approximately.

The results of both models, one considering the thermalcontact resistance and the change of the heat power, and

the other neglecting them, are listed in Table 5. As thethermal contact resistance and the variation of the heatgeneration power are been considered, the values ofdisplacements in result I are much less than the corre-sponding ones in result II.From Table 5, it can be seen that the X- and

Y-component displacements of the six points are significantmainly due to the displacement of the housing, meaningthat the X- and Y-component displacements of the

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Fig. 10. The thermal displacement of the grinding machine housing: (a) front view; (b) side view.


abrasion wheel are considerable. These displacementsmay affect the machining precision seriously. At thesame time, the Z-component displacements of the sixpoints are also large, as the shaft will expand along thebearing house for the intrinsic structure design whenthe thermal expansion occurs, but they have less effect onthe machining precision.

5. Conclusions

In this paper, a thermal model based on the Bossmannsand Tu’s model [1,2] has been developed to characterize theheat distribution of a grinding machine housing, inparticular the change of the heat generation power andthe thermal contact resistance at the solid joints based on

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Table 5

Displacements of six points (� 10�3mm)

Point A B C D E F

Result I: Considering the thermal contact

resistance and the change of the heat

power

X-component

displacement

6.0793 5.2543 4.0029 2.5072 1.6402 0.9971

Y-component

displacement

19.2 17.724 14.654 11.917 11.4 9.1554

Z-component

displacement

�2.785 �13.417 �19.574 �20.065 �22.095 �32.086

Compound displacement

of X- and Y-

20.139 18.486 15.191 12.178 11.517 9.21

Total displacement 20.331 22.842 24.777 23.471 24.916 33.381

Result II: Without considering the

thermal contact resistance and the change

of the heat power

X-component

displacement

9.2974 8.1949 6.0795 3.8323 2.5264 1.2326

Y-component

displacement

29.577 27.409 22.698 18.472 17.441 13.766

Z-component

displacement

�0.3385 �12.74 �19.833 �20.45 �22.734 �33.563

Compound displacement

of X- and Y-

31.004 28.608 23.498 18.865 17.623 13.821

Total displacement 31.006 31.316 30.749 27.823 28.765 36.297


fractal model. Compared with experimental results, it hasbeen shown that the model has much better accuracy thanthose without taking the heat generation change and thethermal resistance into account. Although this model wasdeveloped for analysis of a given grinding machinehousing, it can be used for thermal analysis of variousmachining systems without loss of generality, and theequations of the thermal contact resistance are suitable forall joints.

Acknowledgements

The authors gratefully wish to acknowledge thesupports of National Science Foundation throughgrant No.50475073 and Jiangsu Province Science andTechnology Plan through grant BK2002059, BE2003071,BE2004025, BA2005015.

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