Abstract: With the improvement of machining accuracy and efficiency, heat generation of guideway becomes the crucial

problem which will cause thermal convex deformation. Thermal convex deformation of guideway which accounts for about 22.7% of

the total errors is one of the main factors affecting the machining accuracy of machine tool. A conservation of energy-based

optimization method is proposed to balance the temperature field and reduce the thermal convex deformation. The main strategy

of this method is to transfer the heat from high temperature regions to the lower temperature regions through a heat transfer

system. The structure and contact area of the heat transfer system are determined according to the output and input heat ratio of

the guideway. The simulation and experimental results show that the temperature field of the guideway is more uniform after

optimized, the thermal convex deformation is reduced about 50%, it is of great significance to reduce the thermal convex

deformation of the guideway and improve the machining accuracy of machine tool.

Keywords: Conservation of Energy, Thermal Optimization, Temperature Distribution, Thermal Convex, Guideway

1. Introduction

The guideway is the benchmark used to determine the

relative position of the worktable and spindle in a machine

tool. In high-speed and high-precision machining, the

high-speed rotation of spindle and movement of feed system

will generate a large amount of friction heat and produce the

thermal deformation which seriously affect the machining

accuracy of machine tool. Research shows that machining

errors caused by thermal deformation account for 40% to 70%

of the total errors in precision machining [1]. Yun [2] studied

the thermal characteristics of the guideway and ball screw of

CNC machine tool based on the finite element method, it was

found that the thermal error of guideway accounts for about

22.7% of the total errors of machine tool. Therefore, studying

and optimizing the thermal deformation of machine tool’s

guideway is of great significance for improving the

machining accuracy of machine tool.

In the existing research, there are mainly two methods to

reduce the thermal error, one is the thermal error

compensation method [3, 4], and the other is thermal error

prevention method. The error compensation method is that it

uses the software to real-timely compensate the thermal

errors without changing the machine tool’s structure [5].

Barakat et al. [6] proposed an error compensation method for

kinematics and geometric errors in a coordinate measuring

machine which can effectively improve the accuracy of the

coordinate measuring instrument. Chen [7] developed a

kinematics modeling and a post-processing method to

implement the real-time error compensation for the

kinematics errors on a five-axis CNC machine tool.

The thermal error prevention method is to control the

thermal deformation or temperature rise directly through

changing the machine tool’s structure or cooling the internal

heat sources and its assemblies. Thermal optimization design

is one of the mainly used thermal error prevention method.

The main strategy is to design a symmetrical structure to

reduce the radial thermal deformation, or using the constraint 34 Ruoda Wang et al.: An Optimization Method for Thermal Convex Deformation of Machine

Tool’s Guideway Based on the Conservation of Energy

method to limit the thermal deformation, or transfer the

thermal deformation to the direction which does not

influence the machining accuracy [8-10]. The other thermal

error prevention method is to control the temperature rise of

assemblies of machine tool, and the cooling system becomes

the best choice. Xia et al. [11] designed a fractal tree-like

channel network radiator in the spindle cooling system based

on fractal theory, and verified the heat dissipation of the

device through experiments. Li et al. [12] used a

single-circuit thermosyphon to cool the spindle, the

temperature rise is controlled at 28.5°C.

In the above-mentioned research, numerical simulation

technology is usually used to discuss the main factors

affecting the temperature field and thermal deformation of

machine tool to provide a basis for thermal error

compensation and prevention. Guo et al. [13] analyzed the

main factors affecting the deformation of guideway using the

finite element method. Zeng et al. [14] discussed the

influence of thermal deformation of guideway in the axial

direction on the machining accuracy of workpiece through

thermal-structure coupling analysis.

In addition to the above-mentioned researches, some other

studies are carried out related to the thermal deformation

control. Kang et al. [15] built a thermal error model using the

neural network method and studied the relevant factors

affecting the thermal error. Hiroshi et al. [16] used the

approximate thermal deformation method to improve the

machining accuracy of machine tool. Li et al. [17] established

a finite element objective function for thermal error of

guideway, and optimized the boundary conditions of finite

element model using the response surface approximation

model method to improve the machining accuracy of machine

tool. Zhou et al. [18] used the fast heat conduction method to

balance the temperature field of spindle housing, and the

thermal tilt of the spindle was greatly reduced.

All the thermal error control method can effectively reduce

the thermal deformation of machine tool. However, there is

no research involving the prevention of thermal deformation

of guideway. In this paper, the thermal convex deformation

of guideway is studied and optimized using the principle of

conservation of energy. The main idea of the proposed

method is to transfer the heat from higher temperature

regions to lower temperature regions to balance the

temperature field of guideway and to reduce the thermal

convex deformation. Section 2 introduces the optimization

mechanism and design strategy of heat transfer system.

Section 3 is the simulation and optimization strategy of the

guideway. Section 4 verifies the optimization effect through

experiments, and the experimental results show that the

thermal convex deformation of the guideway is reduced

about 50% using the proposed thermal error optimization

method. Finally, some conclusions are given in section 5

based on the simulation and experimental results.

2. Optimization Mechanism

The main idea of the proposed thermal optimization method

is to transfer the heat from higher temperature regions to the

lower temperature regions to balance the temperature

distribution of guideway according to the principle of

conservation of energy. The key of this method is to determine

the heat need to be transferred and to design the heat transfer

system.

2.1. Calculation the Output and Input heat

As shown in Figure 1, a part of the frication heat generated

by guideway transfers to the ambient through convection, a

part of it transfers to the worktable and machine bed through

conduction which causes the temperature rises of the

components. Due to the influence of the location of heat

sources and the heat transfer process, the temperature

distribution of the guideway is uneven as shown in Figure 2

which will cause the convex deformation and affect the

machining accuracy of machine tool. In order to balance the

temperature distribution of the guideway and to reduce the

convex deformation, the heat in the higher temperature

regions should transfer to the lower temperature regions. The

total heat in the guideway can be calculated as,

c

0

9.8 ( )

l

g g g g

Q c S f x

*ρ *

=

∫

(1)

Figure 1. Heat transfer diagram of guideway.

where, Qg is the heat generated by the guideway, Qh is the

heat convection, cg and ρg are the specific heating capacity in

J/(kg°C) and density in kg/m3 of the guideway, respectively, Sg

is the cross-sectional area of the guideway in m2 , fc(x) is the

temperature rise function of the guideway, l is the length of

the guideway in meter.

The balance of the temperature distribution is to obtain a

uniform temperature rise along the axial direction of the

guideway, therefore, the linear fitting of the temperature rise

should be established as,

l ( ) 1 2

f x p x p

= + (2)

where, p1 and p2 are the coefficients.

As shown in Figure 2, the temperature distribution of the

guideway along the axial direction is separated into 4 regions

by the linear fitting curve and marked as Q1, Q2, Q3, and Q4.

The total heat in the guideway is Qg= Q1+ Q4. In order to

balance the heat along the axial direction of the guideway, the International Journal of Mechanical Engineering and Applications 2021; 9(2): 33-41 35

heat in the higher temperature region Q1 needs to be quickly

transferred to the lower temperature regions Q2 and Q3

through the high thermal conductivity material. That is to say,

the transferred heat Q1 should equal to Q2 and Q3,

1 2 3

Q Q Q

= +

(3)

Figure 2. Temperature distribution of guideway.

Figure 2. fc(x) is the temperature rise of the guideway, fl(x)

is the linear fitting of temperature rise, Q1, Q2, Q3, and Q4 are

4 heat regions, A and D are two ends of the guideway, B and

C are the intersections of fc(x) and fl(x).

To transfer the heat from higher temperature regions to the

lower temperature regions, a heat transfer system with high

thermal conductivity is designed in section 2.2. Since the heat

transfer system is connected with the guideway through

thermal silica gel, the thermal contact resistance is very small

and can be ignored.

Assuming that the heat dissipation rate of the system is

unchanged after using the heat transfer system, then, Eq. 3 can

be rewritten as Eq. 4 according to the conservation of energy.

( ) ( ) ( ) =

( ) ( ) ( ) ( )

C

g g c l

B

B D

g g c l g g c l

A C

C m f x f x dx

C m f x f x dx C m f x f x dx

−

− + −

∫

∫ ∫

(4)

In Eq. 4, the increase of internal energy of the heat transfer

system is ignored due to it is only a heat transfer medium

when the system is in thermal equilibrium. For a given

guideway, the temperature rise is related to the heat

generation, but the temperature distribution trend is basically

the same when the working stroke of the machine tool

remains unchanged especially in batch machining. Therefore,

the heat ratio of output and input along the axial direction of

the guideway can be calculated using the following

calculation process.

1) The first step is to obtain the temperature rise curve

using the temperature sensors to measure the temperature rise

along the axial direction or using the finite element method to

simulate the temperature distribution of the guideway, the

obtained temperature rise curve is marked as fc(x) as shown

in Figure 2.

2) The second step is to establish the linear fitting model

using the least squares method as shown in Eq. 2, and then

using the following formula to revise the parameter p2, the

revised linear fitting function is marked as fl(x) as shown in

Figure 2.

2

c 1 2

0

1

( ) 0

2

l

f x p l p l

− − =

∫

(5)

where, l is the length of guideway in meter.

3) The third step is to calculate the coordinates of

intersections of the temperature rise curve and the new linear

fitting line, which is marked as B and C as shown in Figure 2.

4) The fourth step is using Eq. 4 to calculate the output

heat Q1 and the input heat Q2 and Q3. And the calculated

results are used to calculate the heat ratio by the next step.

5) The fifth step is to calculate the heat ratio using the

following formula,

( )

2 ( ) ( )

i

i

C

g g c l

B

Q

C m f x f x dx

*η *=

−

∫

(6)

where,

1

( ) ( )

i

i

l

i g g c l

l

Q C m f x f x dx

+

= −

∫

is the subdivision

of the transferred heat as shown in Figure 3, li and li+1 are the

subdivision interval which can be determined through the

required heat transfer accuracy, for example, high heat

transfer accuracy requires a smaller subdivision interval, and

vice versa.

Using the above-mentioned method, the subdivided heat

ratio of output and input can be calculated. After determining

the heat ratio of output and input, next is using the heat ratio

to design the heat transfer system.

2.2. Design the Heat Transfer System

In order to transfer the heat quickly, the copper alloy with

high thermal conductivity is selected as the heat transfer

material in this study. Since the heat outputted from the higher

temperature region Q1 should be transferred to the heat

transfer system firstly, and then transferred to the lower

temperature regions Q2 and Q3, the heat transferred to the heat

transfer system is equal to the heat outputted from the

guideway when the system is in thermal equilibrium. Then,

the required quality and volume of the high thermal

conductivity material can be calculated as,

( )

0

( ) ( )

l C

h h g g c l

B

h h h

C m t C m f x f x dx

m V

*ρ *

∆ = −

=

∫ ∫

(7)

where, Ch is the specific heat capacity of the high thermal

conductivity material in J/(kg°C), mh and mg denote the

quality of the high thermal conductivity material and the

guideway in kg, respectively, ρh is density of the copper alloy

in kg/m3 .

Since Ch, Cg, mg, and ρh are all known, the quality and

volume of the high thermal conductivity material can be

calculated through Eq. 7. Next is to calculate the contact area

between the heat transfer system and the guideway using Eq. 8. 36 Ruoda Wang et al.: An Optimization Method for Thermal Convex Deformation of Machine

Tool’s Guideway Based on the Conservation of Energy

upper

i i

A A

*η*

= ×

(8)

where, Aupper is the area of the upper surface of the guideway

in m2 .

Figure 3. The output and input heat in guideway.

Figure 3. Qi is subdivision of the transferred heat, li and li+1

are the subdivision interval.

Figure 4. The heat transfer system.

Using Eq. 8, the required contact area between the heat

transfer system and the guideway at heat transfer regions A1,

A2, and A3 can be determined as shown in Figure 4. The heat

in the guideway at region Q1 is transferred to the copper alloy

through the contact area A1, then it conducts quickly in the

copper alloy and form a uniform temperature field. Thereby

the temperature in the copper alloy at the regions Q2 and Q3

will higher than the guideway at the same regions, then the

heat in the copper alloy will transfer to the guideway through

the contact area A2 and A3 to form a uniform temperature

field.

Using the aboved-mentioned thermal optimization method,

the heat ratio needs to be outputted and inputted can be

determained according to the principle of conservation of

energy, and the heat transfer system can be designed using

heat ratio subdivision method. In order to verify this method,

the thermal behavior of a guideway is simulated and

optimized in section 3.

3. Simulation and Optimization

3.1. Simulation

In this paper, a guideway with dimensions of 1180×23×22

mm is used to study the thermal behavior and its optimization

as shown in Figure 5. The material of the guideway is carbon

steel, and its material properties are shown in Table 1. The

guideway is fixed on the machine bed by bolts. In order to

simulate the thermal behavior of the guideway, the boundary

conditions should be calculated firstly. The heat of the

guideway is usually generated by the frication between the

slider and the guideway, which can be calculated as,

Q Fv

J

= *µ *(9)

where, µ=0.05 is the friction coefficient between the slider and

the guideway, F=2000 N is the pressure applied on the

guideway, v=0.5 m/s is the relative moving speed between the

slider and the guideway, J=4.2 J/cal is the thermal power

equivalent. Therefore, the heat generation of the guideway can

be calculated as Q=11.9 W, and the heat flux of the contact

surface between the slider and the guideway is 6159.4 W/m2 .

Figure 5. The cross-section of the guideway.

When the air flows through the surface of the guideway, the

process of heat transfer between the two is called convection.

This process includes the convection of fluid macroscopic

displacement and the heat conduction between fluid

molecules, which is the result of the combined action of

convection and heat conduction. This is the main heat

dissipation form of the guideway, and the heat convection

coefficient can be calculated as,

h Nu

L

= ⋅*λ *(10)

where, Nu is the Nusselt number, λ is the thermal conductivity

of the fluid in W/(m·k), L is the geometric feature size in

meter.

Since the heat transfer at the surface of the guideway

belongs to natural convection, the Nusselt number can be

calculated as,

3

2

Nu C Gr ( Pr)nm

Gr g tL

v

*β *

= ⋅

∆

=

(11) International Journal of Mechanical Engineering and Applications 2021; 9(2): 33-41 37

where, Gr is the Grashof number, Pr is the Prandtl number, g

is the acceleration of gravity in m/s2 , β is the volume

expansion coefficient in 1/k, ∆t is the temperature difference

between the fluid and the wall in °C, v is the kinematic

viscosity, the subscript m represents the qualitative

temperature, c and n are constants. Then, the heat convection

coefficient of the guideway is calculated as h=7.6 W/(m2 ·k).

Table 1. Material properties of the guideway.

Physical property

Value

Thermal Conductivity in w/m·K

55

Elastic Modulus in Gpa

12

Poisson’s ratio

0.3

Thermal expansion coefficient in m/K

1.2×10-5

Specific heat capacity in J/kg·K

470

Density in kg/m3

7.8×103

Figure 6. Finite element model of the guideway.

Figure 6 shows the finite element model of the guideway,

the chamfers and holes on the guideway are removed. Path1

and path2 represent the edges at the top and bottom of the

guideway. Because the heat source of guideway is motive, the

moving heat source is applied to the finite element model. The

initial and reference temperature is set as 20 °C, the heat

convection is applied on the surface of the guideway, the

simulation time is 14400 seconds. Using the Ansys software,

the temperature field and the thermal deformation of the

guideway can be simulated.

Figure 7 shows the temperature field of the guideway at

14400 seconds. It can be seen from Figure 7 that the higher

temperature region is occurred in the middle of the guideway,

the maximum temperature is 26.051°C. The lower

temperature regions are occurred in both ends of the guideway,

and the minimum temperature is 24.29°C. The temperature

distribution is uneven.

Figure 8 shows the temperature rise at the midpoint of the

guideway. It can be seen from Figure 8 that the temperature at

the midpoint rises sharply first and then stabilizes at 14400s.

At this time, the temperature field of the guideway reaches a

steady-state. The trends of the temperature rise curves at the

other points are similar to the midpoint, but the temperature

rises are different.

Applying the transient temperature field at 14400 seconds

and the displacement constraint to the finite element model,

the thermal deformation of the guideway can be simulated

through thermal-structure coupling analysis and the

simulation result is shown in Figure 9. The maximum thermal

deformation of the guideway in the Y-direction is 2.2525 µm.

The thermal deformation trend of the guideway is convex.

Figure 7. Temperature field of the guideway.

Figure 8. The temperature rises at the midpoint.

Using the numerical simulation method, the temperature

field and thermal deformation of a single guideway can be

obtained. However, the guideway is fixed on the machine bed

in actual situation, the heat conduction between the guideway

and machine bed as well as the influence of thermal

deformation of machine bed on the guideway should be

considered.

Because the heat transfer between the guideway and

machine bed is heat conduction, the thermal contact resistance

should be calculated firstly to simulate the thermal behavior

when the guideway is fixed on the machine bed. The thermal

contact resistance can be calculated as [19],

0.618

( )

105.6

c

H

R

k P

*σ *

= ×

×

(12)

where, σ is the root-mean-square value of surface roughness

which can be calculated through Eq. 13, k is the tuned average

thermal conductivity which can be calculated by Eq. 14, H is

the minimum hardness of the contact surfaces in HB, P is the

contact pressure in MPa, which can be calculated by Eq. 15.

2 2

1 2

*σ σ σ *

= +

(13)

where, σ1 and σ2 are the surface roughness of the two contact

parts in macron. 38 Ruoda Wang et al.: An Optimization Method for Thermal Convex Deformation of Machine

Tool’s Guideway Based on the Conservation of Energy

Figure 9. Thermal deformation of the guideway.

Figure 10. Temperature field of the guideway and machine bed.

1 2

1 2

k 2k k

k k

=

+

(14)

where, k1 and k2 are the thermal conductivities of the two

contact parts in W/(m·K).

/

p F S

= (15)

Applying the boundary conditions to the finite element

model when the guideway is fixed on the machine bed, the

temperature field and thermal deformation can be simulated

using the Ansys. The temperature field is shown in Figure 10,

the maximum temperature is 26.031°C which is basically

consistent with the temperature field when the guideway is

single. The maximum thermal deformation along the

Y-direction of the guideway is 10.923 µm when the guideway

is fixed on the machine bed as show in Figure 11, which is

much higher than the maximum thermal deformation when the

guideway is single. This is because the thermal deformation of

the machine bed is superimposed on the guideway. Therefore,

it is very important to control and optimize the thermal

deformation of the guideway.

3.2. Optimization of the Temperature Field

Figure 12 shows the temperature rises curves of path 1 and

path 2 along the axial direction. The temperature trends of

these two paths are almost the same, and the maximum

temperature difference between the two is only 0.015°C. It

represents that the transient temperature of the guideway has

reached the steady-state.

Figure 11. Thermal deformation along the Y-direction.

Figure 12. Axial temperature distribution of path1 and path2.

The temperature distribution along the axial direction of the

guideway can be obtained through calculating the average

temperature rise of path 1 and path 2 as shown in Figure 13

(curve T(X)). Using the least square fitting method, a

sixth-order polynomial can be obtained as shown in Eq. 16.

( )

6 5 4 3

31.22 -110.5 146.1 -88.37

2

20.33 1.81 24.85

T x x x x x

x x

= +

+ + +

(16)

In Eq. 16, the fitting error of the temperature rise of the

guideway is 0.02373, the root mean square error is 0.009427,

the coefficient of determination of the equation is 0.9993. That

is to say, the sixth-order polynomial can appropriately express

the temperature distribution along the axial direction of the

guideway. The linear fitting of the temperature rise of the

guideway can be obtained using Eqs. (2, 5) as shown in Figure

13 (curve G(X)). The intersections of the linear fitting and the

temperature rise curve is marked as J and K as shown in

Figure 13, and the coordinates of intersections J and K can be

calculated as J (0.23, 25.61), K (0.94, 25.6) using Eq. 17.

( )

6 5 4 3 2

31.22 -110.5 +146.1 -88.37 +20.33

1.81 24.85 =-0.014 25.62 0 1.18

x x x x x

x x x

+

+ + < <

(17)

In order to transfer the heat from the higher temperature

region Q1 to the lower temperature regions Q2 and Q3 quickly,

the material with high thermal conductivity should be selected.

Currently, materials with high thermal conductivity include International Journal of Mechanical Engineering and Applications 2021; 9(2): 33-41 39

the diamond, carbon fiber, copper alloys, aluminum alloys, etc.

Considering the factors of processability and price, the copper

alloy with thermal conductivity of λ=399 W/(m·K) is selected

as the heat transfer medium in this study. Substituting the

material properties of the copper alloy and the guideway as

shown in Table 1 into Eq. 4, the output and input heat can be

calculated as shown in Eq. 18.

Figure 13. Temperature distribution and its linear fitting.

Figure 13. T(X) is the temperature rise of the guideway, G(X)

is the linear fitting of the temperature rise, Q1 is the output heat

region, Q2 and Q3 are the input heat region, J, K is the

intersections of T(X) and G(X).

[ ]

[ ]

[ ]

0.93

3

1

0.23

0.23

3

2

0

1.18

3

3

0.93

0.46 10 3.59 ( ) ( )

=306.69 J

0.46 10 3.59 ( ) ( )

151.92 J

0.46 10 3.59 ( ) ( )

151.78 J

Q T x G x dx

Q G x T x dx

Q G x T x dx

= × × × −

= × × × −

=

= × × × −

=

∫

∫

∫

(18)

Substituting Q1, Q2 and Q3 into Eq. 6, the heat ratio of

output and input of the guideway can be calculated as ƞ1=50%,

ƞ2=24.76%, and ƞ3=25.24%. Substituting ƞ1, ƞ2, and ƞ3 into Eq.

8, the contact area between the guideway and the copper alloy

at regions E, F, and G as shown in Figure 13 can be calculated

as SE=13.1 cm2 , SF=81.95 cm2 , and SG =19.93 cm2 . The

subdivision of the contact area of E, F, and G can be

calculated using Eq. 6. Figure 14 shows the simplified

structure of the heat transfer system. In Figure 14, SE, SF, and

SG express the contact areas between the guideway and the

copper alloy which can be connected by thermal conductive

silicone.

Figure 14. The shape of the heat transfer system.

Figure 14. SE, SF, and SG are the contact areas between the

guideway and the copper alloy.

Substituting the properties of the copper alloy into Eq. 7 the

weight of the required copper alloy can be calculated as

mh=4.23 kg, the volume can be calculated as Vh=4.72×10-4 m3

as shown in Eq. 19.

[ ]

[ ]

0.93

3

0.23

0.93

3

0.23

3

4 3

0.39 10 ( ) ( )

0.46 10 3.59 ( ) ( )

8.96 10 4.23 4.72 10 m

h

h h

m T x G x dx

T x G x dx

V V

−

× × − =

× × × −

× × = → = ×

∫

∫

(19)

Using the optimization method for thermal convex

deformation of guideway, a heat transfer system can be

obtained as shown in Figure 14. In order to verify the

effectiveness of the prosed optimization method, the

experiments is carried out in section 4.

4. Experimental Verification

4.1. Experimental Setup

Because the slider moves when the guideway works, it is

difficult to arrange the sensors on the guideway to measure the

temperature and thermal deformation. We designed a thermal

behavior testing platform as shown in Figure 15 (1). The

testing platform includes a machine bed, a single guideway,

four adjustable heaters, a heat transfer system, an

d a data

acquisition system. The guideway is fixed on the machine bed

through bolts, the adjustable heaters are attached on the

surface of the guideway to simulate the friction heat. Six

temperature sensors and five eddy current displacement

sensors are used to measure the temperature and thermal

deformation of the guideway before and after optimized. The

installation sequence of the temperature sensors from left to

right is T1, T2, T3, T4, T5, and T6 as shown in Figure 15 (2)

and (4). The installation sequence of the eddy current

displacement sensors from left to right is D1, D2, D3, D4, and

D5 as shown in Figure 15 (2) and (4).

The heaters are evenly arranged in the stroke of the sliders. The

power of each heater is adjusted to a suitable value according to

the calculation result of the friction heat of the guideway. The

room temperature is around 29°C. The experiment time is 7200

seconds. Figure 16 shows the experimental results of the

guideway before and after optimized. 40 Ruoda Wang et al.: An Optimization Method for Thermal Convex Deformation of Machine

Tool’s Guideway Based on the Conservation of Energy

4.2. Experimental Results

Figure 16(a) shows the temperature rises of the guideway

before optimized, the temperature rises marked as Tem1 to

Tem6 are measured by the temperature sensors T1 to T6.

Because both ends of the guideway hang in the air, its

convective heat transfer is much larger than the middle part.

Therefore, the temperature rise at both ends of the guideway is

not considered in this study. After heating for 2 hours, the

maximum temperature rise is about 8°C which occurred in the

middle of the guideway, the maximum temperature difference

is about 4°C.

Figure 15. Experimental setup. (1) Thermal behavior testing platform of

guideway, (2) Experimental setup before optimization, (3) Data acquisition

device, (4) Experimental setup with heat transfer system, (5) Temperature and

displacement sensors arrangement.

Figure 15. T1, T2, T3, T4, T5, and T6 are the temperature

sensors, D1, D2, D3, D4, and D5 are the eddy current

displacement sensors.

Figure 16(b) shows the temperature rises of the guideway

with optimized, after heating for 2 hours, the maximum

temperature rise is about 8°C which occurred in the middle of

the guideway and is similarly to the maximum temperature

rise of the guideway before optimized. Except for the

temperature rises of both ends of the guideway, the maximum

temperature difference is about 2°C, which is reduced about

50% compared without optimization of the guideway.

Comparing with Figure 16(a) and Figure 16(b), the

temperature field of the guideway with heat transfer system is

more uniform than that without optimized.

Figure 16(c) shows the thermal deformation of the

guideway before optimized, ∆1 to ∆5 are the thermal

deformations measured by eddy current displacement sensors

D1 to D5. The maximum thermal deformation along the

Y-direction is 14.8 µm which occurred in the middle of the

guideway. Except for the thermal deformations of both ends of

the guideway, the thermal convex deformation of the

guideway is 7.6 µm.

Figure 16(d) shows thermal deformations of the guideway

with heat transfer system, the maximum thermal deformation

along the Y-direction is 15.8 µm which occurred in the middle

of the guideway. Except for the thermal deformations of both

ends of the guideway, the thermal convex deformation of the

guideway is 3.7 µm, which is reduced about 50% compared

without optimization of the guideway. Comparing with Figure

16(c) and Figure 16(d), the thermal equilibrium time is also

reduced significantly when the guideway is optimized using

the heat transfer system. The experimental results show that

the conservation of energy-based thermal optimization

method can effectively reduce the thermal convex

deformation of the guideway. It is of significance for

improving the machining accuracy of machine tool.

5. Conclusions

An optimization method for thermal convex deformation

of machine tools’ guideway is proposed in this paper. The

output and input heat are determined according to the

conservation of energy, the structure and contact area of the

heat transfer system is designed. Based on the numerical

simulation and the experimental results, the following

conclusions can be drawn.

(1) A heat transfer system is designed based on the principle

of conservation of energy, which can balance the

temperature field and reduce the thermal convex

deformation of the guideway significantly.

(2) The structure and contact area of the heat transfer

system are designed according to the subdivision of

output and input heat ratio.

(3) The temperature distribution along the axial direction of

the guideway and the linear fitting of the temperature

rise is used to determine the output and input heat ratio

of the guideway.

(4) A thermal behavior testing platform of guideway is

designed which can measure the temperature and

thermal deformation of the guideway conveniently.

(5) The experimental results show that the temperature

difference and the thermal convex deformation of the

guideway is reduced about 50% using the thermal

optimization method. It is of significance for improving

the machining accuracy of machine tool.

Figure 16. Experimental results, (a) Temperature rise of guideway before

optimized, (b) Temperature rise of guideway with heat transfer system, (c)

Thermal deformation along Y-direction before optimized, (d) Thermal

deformation along Y-direction with heat transfer system. International Journal of Mechanical Engineering and Applications 2021; 9(2): 33-41 41

Figure 16. Tem1 to Tem6 are the temperature rises of the

guideway, ∆1 to ∆5 are the thermal deformations of the

guideway.

6. Recommendations for Future Work

In this study, the temperature field of the guideway is

optimized based on the conservation of energy, and a fast heat

conduction system is designed to reduce thermal convex

deformation. With the development of additive manufacturing

technology, the complex structures of the heat transfer system

can be machined. Therefore, the proposed method is of great

significance to improve the accuracy of machine tools.

Acknowledgements

This paper is sponsored by the “Technology of on-line

monitoring system for thermal characteristics of NC machine

tools” (No. H2019304021); the “Project funded of Shanghai

science committee- Precision technology and its application

for five-axis machine tool based on the real-time

compensation” (NO. J16022).