analytical solution to transient temperature field based ... · ellipsoidal mobile heat sources, an...

Introduction

Since Rosenthal (Refs. 1, 2) produced an analytical treat-ment of the temperature field in welding, several modifica-tions to his approach and some new solutions for the tem-perature field in welding have been published. He derivedthe temperature distribution equations for point, line, andplane heat sources utilizing the heat conduction equationfor the quasistationary state. While this revealed the effectsof the welding parameters on weld pool shape, the solutionincorporates rather large errors in the temperature distribu-tion because of its many assumptions. Since then, many researchers (Refs. 3–13) have tried todefine temperature field analytically to obtain a temperaturefield map as realistic as possible. Initially, a two-dimensionalsurface Gaussian heat source with effective arc radius wasadopted to find the temperature distribution on weldedplates and weld pool geometry. Eagar and Tsai (Ref. 3) modi-

fied Rosenthal’s theory to include a two-dimensional surfaceGaussian distributed heat source with a constant distribu-tion parameter (which can be considered as an effective arcradius) and found an analytical solution for the temperatureof a semiinfinite body subjected to this moving heat source.Their solution was a significant step for the improvement oftemperature prediction in the near heat source regions.Goldak et al. (Ref. 4) first introduced the three-dimensional(3D) double-ellipsoidal moving heat source. Finite elementmodeling (FEM) was used to calculate the temperature fieldof a bead on plate and showed that this 3D heat source couldovercome the shortcomings of the previous 2D Gaussianmodel to predict the temperature of the welded joints withmuch deeper penetration. Nguyen et al. (Ref. 5) presentedan analytical solution of transient temperature distributionof a semiinfinite body subjected to three-dimensional heatdensity of a semiellipsoidal and double-ellipsoidal mobileheat source. Bo and Cho (Ref. 6) derived an analytical solu-tion to the transient temperature field in the finite thick-ness plate with Gaussian heat distribution during single-pass arc welding. Fachinotti and Cardona (Ref. 7) proposed asemianalytical solution to the thermal field induced by amoving double-ellipsoidal welding heat source in a semiinfi-nite body. Nguyen et al. (Ref. 8) again described an approxi-mate analytical solution for the double-ellipsoidal heatsource in finite thick plates. However, with the variation of temperature, the materialis nonhomogenous and nonisotropic, physics thermal prop-erties depend on temperature, and phase transformationsare taken into consideration to some degree. Grosh et al.(Ref. 9) derived an analytical solution for one-, two-, andthree-dimensional heat conduction from a moving energysource. Assuming the thermal properties of the solid are thefunctions of temperature, which is the coefficient of thermalconductivity; the volume and specific heat are the same lin-ear function of temperature; and the analytical results wereoutlined. Excellent agreement was noted between experi-ment and theory. Miettinen (Ref. 14) developed special algorithms to cal-culate important solidification-related thermophysical prop-erties: enthalpy and enthalpy-related data, density, andthermal conductivity for low-alloy and stainless steels at any

WELDING RESEARCH

APRIL 2017 / WELDING JOURNAL 113-s

SUPPLEMENT TO THE WELDING JOURNAL, APRIL 2017Sponsored by the American Welding Society and the Welding Research Council

Analytical Solution to Transient Temperature FieldBased on Coefficient of Thermal ConductivityBased on Gaussian and ellipsoidal mobile heat sources, an analyzing model of

temperature field was obtained in arc welding of finite thickness plates

BY K. GUO AND S. DAI

ABSTRACT Assuming that the material object of heat conduction is infinite, the coefficient of thermal conductivity, volume, and specificheat are the same function of temperature. By means of theKirchhoff transformation on thermal conductivity, a partial differential equation was obtained, and the approximate analyzing solution was obtained by Fourier integral. Based on Gaussian andellipsoidal mobile heat sources, an analyzing model of temperature field was obtained in arc welding of finite thickness plates.The model was applied to gas tungsten arc welding through anexperiment on carbon steel. The actual isotherms of the specimens’ cross sections at various times from the arc start pointwere compared with those of the simulation result, whichshowed this kind of temperature field model is feasible; thus the material physical properties are promoted under thetemperature field model.

KEYWORDS • Coefficient of Thermal Conductivity • Heat Source • Specific Heat • Gas Tungsten Arc Welding (GTAW)

Guo Supplement April 2017.qxp_Layout 1 3/10/17 2:59 PM 13

temperature from 1600° to 25°C, which means the algo-rithms can be used to predict thermophysical properties ofsteels. To reduce the errors, this paper considered that the ther-mal properties of the solid are the same exponential func-tion of temperature, and an analytical approximate solutionwas obtained in infinite plate. Applying the solution to a fi-nite thick plate, an analytical solution for the Gaussian andellipsoidal mobile heat source was derived and calibratedwith the experimental data.

Temperature Field Based on MaterialVariational Coefficient Assuming the material objects of the welding process tobe infinite, the influence of boundary conditions can be neg-lected. In general, the governing equation of heat conduc-tion for the three-dimensional transient temperature is rep-resented as

where (T), (T), and c(T) denote the thermal conductioncoefficient, density, and specific heat of the material, respec-tively. 0, 0, and c0 denote the thermal conduction coeffi-cient, density, and specific heat of the material on referencetemperature, respectively, assuming they are constants. Tand t denote the three-dimensional transient temperatureand time of welding, respectively. Equation 1 can be linearized using Kirchhoff’s transfor-mation, as

where (t) is the temperature-dependent thermal conductivity. Substituting Equation 2 into Equation 1, a concept formof governing equation with the new variable, U, can be ob-tained.

Considering the variety of temperatures during welding,the thermal conductivity, density, and specific heat of thematerial are the functions of temperature. Defining varia-tion coefficient , the property variation coefficients arevery small for most materials so that convergence can be en-sured. If the variety of temperatures aren’t sharply different,the variation coefficient is zero. To improve the model’s precision, we consider the coeffi-cient of thermal conductivity, volume, and specific heat tobe the same exponential functions of temperature.

Substituting Equations 4–6 into Equation 3, then

For convenience, let T0 = 0 substituting Equation 4 intoEquation 2, then

Let T = 1°C, at time t = 0. From Equation 8 we get

Substituting Equations 8 and 9 into Equation 7, then

Substituting Equation 8 into Equation 10, then

��x

�(T )�T�x

��

��+ ��y

�(T )�T�y

�

��

�

�

+ ��z

�(T )�T�z

��

��=(T )c(T )�T

�t1( )

U = �(�)�0T0

T� d� 2( )

�2U

�x2 + �2U

�y2 + �2U

�z2 = �(T )c(T )�(T )

� �U�t

3( )

�(T )c(T )=�0c0e�T 4( )

�(T )= �0e�T 5( )

�(T )�(T )c(T )

=� 6( )

�2U

�x2 + �2U

�y2 + �2U

�z2 = 1��U�t

7( )

U = exp �T �1( ) /� 8( )

U0 = exp ��1( ) /� 9( )

U(x ,y,z,t )= 1�

exp ��1( )� 1

(4��t )3 2

�exp � x2 + y2 + z2

4�t�

��

��10( )

WELDING RESEARCH

WELDING JOURNAL / APRIL 2017, VOL. 96114-s

Fig. 1 — Comparisons among the experiment and two calculationsat time t = 6.25 s with = 0.000032 and = 0 based on a single ellipsoidal heat source.

Fig. 2 — Comparisons among the experiment and two calculationsat time t = 12.5 s with = 0.000032 and = 0 based on a single ellipsoidal heat source.

Guo Supplement April 2017.qxp_Layout 1 3/10/17 2:59 PM Page 114

is very small for most of the materials, when

so Equation 11 can be expanded by the Taylor integral, andtemperature T of the plate can be obtained

and the convergence of series expansion can be assured. Expanding by the Taylor integral again, temperature T ofthe plate can be obtained.

Assuming there is a concentrated source, the source Qcan be obtained at reference temperature Q = exp(T) · Q0,where Q0 denotes the source at the reference temperature.

The temperature produced by the source will be

Substituting Equation 14 into Equation 13, we have

Letting T0 be the reference temperature of welding, the tem-perature of the plate can be solved

Assuming n = 1, m = 3, we finally receive the analyticalsolution to the temperature field of the plate

Let n = 1, m = 1, then

T = 1ln

exp 1

(4 t )3 2 exp = x2 + y2 + z2

4 t+1 11( )

where exp = x2 + y2 + z2

4 t1,

exp 1

4 t( )3 2 <1

thenexp 1

4 t( )3 2 exp = x2 + y2 + z2

4 t<1

T = 1 1( )n+1

nn=1exp 1

(4 t )3 2 exp = x2 + y2 + z2

4 t

n

12( )

T ==1( )n+1

nn=11m!m=1

mn

1

(4 t )3 2 exp = x2 + y2 + z2

4 t

n

=1( )n+1

nn=1 1+2!

+2

3!+ ...+

m 1

m!+ ...

n

1

(4 t )3 2 exp = x2 + y2 + z2

4 t

n

13( )

QT( )c T( ) =

Q0

0c014( )

T x ,y,z,t( )= Q0

0c0

=1( )n+1

nn=1 m=1

m 1

m!

n

1

4 t( )3 2 exp = x2 + y2 + z2

4 t

n

15( )

T x ,y,z,t( )=T0 =Q0

0c0

=1( )n+1

nn=1 m=1

m 1

m!

n

1

4 t( )3 2 exp = x2 + y2 + z2

4 t

n

16( )

T x ,y,z,t( )=T0 =Q0

4 t( )3 20c0

1+2+

2

6

exp = x2 + y2 + z2

4 t17( )

T x ,y,z,t( )=T0 =Q0

4 t( )3 20c0

exp = x2 + y2 + z2

4 t18( )

WELDING RESEARCH


Fig. 3 — Comparisons among the experiment and two calculationsat time t = 18.75 s with = 0.000032 and = 0 based on a singleellipsoidal heat source.

Fig. 4 — Comparisons among the experiment and two calculationsat time t = 6.25 s with = 0.000032 and = 0 based on a Gaussianheat source.


T x ,y,z,t( ) T0 = d0

t

0

D Q m

0c0 4 t( ) 3 2B y

B y

A x

A x

exp =x= x'( )2 + y= y'( )2 + z= z'( )2

4 t( )

exp = x'2 + y'2

2 2 1+2

dx'dy'dz'

=Qm 1+ 2+ 2 6( )

c 4 t( ) 3 20

td

A x

A xexp =

x= x'( )24 t( )=

x'2

2 2 dx'

which means the variation coefficient is zero, = 0.

Analytical Solution to the TemperatureField Subjected to a Gaussian Heat Sourcein Finite Thickness Plates The assumption is the effective heat source should pro-duce the same amount of heat input into the finite body asthe original heat source would in an infinite body. Considering a Gaussian heat source in a finite plate ofwidth 2B, length 2A, and thickness D, the local coordinate ofthe heat source O'(x',y',z') was constructed so that its axeswere parallel to the fixed coordinate of the finite plate O(x, y, z), where origin O' was located at O(x, y, z). In general,in quest of a description of temperature field, the solutionto basic Equation 17 for an infinite body with temporaryheat source Q0 applied at optional point of body with coordi-nates (x',y',z') was used:

Considering the Gaussian heat source

where denotes the parameter of Gaussian heat distribu-tion, and denotes maximum heat density magnitude. Based on a Gaussian heat source in an infinite body, theGaussian heat source in a finite plate was assumed to have asimilar form but a different maximum heat density magni-tude, assuming that heat convection and radiation were ig-norable due to the short welding time.

then

The temperature field caused by the total Gaussian heatsource equaled the following:

T x ,y,z,t( )=T0 =Q0

4 t( )3 20c0

1+2+

2

6

exp =x= x'( )2 + y= y'( )2 + z= z'( )2

4 t19( )

Q x' ,y'( )=Qm exp = x'2 + y'2

2 2

Q x ,y( )=Qm B y

B y

A x

A xexp = x'2 + y'2

2 2 dx'dy'

=2

2Qm erf

A= x2

+ erfA+ x

2

erfB y

2+ erf

B+ y2

20( )

where erf z( )= 2exp x2 2( )0

zdx 21( )

Qm =2 2( )=1Q erf

A x2

+ erfA+ x

2

1

erfB y

2+ erf

B+ y2

1

22( )

WELDING RESEARCH


Fig. 5 — Comparisons among the experiment and two calculations at time t = 12.5 s with = 0.000032 and = 0 based on aGaussian heat source.

Fig. 6 — Comparisons among the experiment and two calculationsat time t = 18.75 s with = 0.000032 and = 0 based on a Gaussian heat source.


where

Substituting Equation 22 into Equation 23, we get

Assuming the moving heat source with constant speed vin the x-direction and substituting x = x – v into Equation26, the final approximate solution can be obtained.

where

Analytical Solution to the TemperatureField Subjected to Ellipsoidal Heat Sourcein Finite Thickness Plates

SingleEllipsoidal Heat Source

Let us consider a finite plate of width 2B, length 2A, andthickness D as in a Gaussian heat source, assuming the ef-fective heat source should produce the same amount of heatinput into the finite thickness plates with a single-ellipsoidalheat source.

where ah, bh, and ch are single-ellipsoidal heat source param-eters, and Qm denotes maximum heat density magnitude.

M = erf4 t( )+2 2( ) A= x( )=2 2x

4 t( ) 2 t( )+ 2

+erf4 t( )+2 2( ) A+ x( )=2 2x

4 t( ) 2 t( )+ 224( )

N = erf4 t( )+2 2( ) B= y( )=2 2y

4 t( ) 2 t( )+ 2

+erf4 t( )+2 2( ) B+ y( )=2 2y

4 t( ) 2 t( )+ 225( )

T x ,y,z,t( )=T0 =Q 1+ 2+ 2 6)( )

2 0c0

1

2 2 +4 t( )0

t

expx2 + y2

2 2 +4 t( )M N P

erfD z

4 t( )erf

z

4 t( )d 26( )

where P= erfA x

2+ erf A+ x

2

1

erfB y

2+ erf B+ y

2

1

27( )

T x ,y,z,t( )=T0 =Q 1+ 2+ 2 6( )

2 0c0

1

2 2 +4 t( )0

t

exp( x v )2 + y2

2 2 +4 t( )Mv N Pv

erfD z

4 t( )erf

z

4 t( )d 28( )

Mv = erf4 t( )+2 2( ) A x+ v( )=2 2 x v( )

4 t( ) 2 t( )+ 2

+erf4 t( )+2 2( ) A x v( )=2 2 x v( )

4 t( ) 2 t( )+ 229( )

Pv = erfA x+ v

2+ erf A+ x+ v

2

1

erfB y

2+ erf B+ y

2

1

30( )

expB y

B y y y'( )24 t( )=

y'2

2 2 dy' exp0

D z z'( )24 t( ) dz'

=Qm 1+ 2+ 2 6( )0c0 4 t( ) 3 20

t

2

3

erfD z

4 t( )

erfz

4 t( )exp = x2 + y2

2 2 +4 t( )2 4 t( ) 4 t( )

2 +2 t( )M Nd 23( )

Q x' ,y' ,z'( )=Qm exp3x'2

ch2 = 3y'2

ah2 = 3z'2

bh2 31( )

Q x,y,z( )=Qm exp0

D

B y

B y

A x

A x x'2

2ch2

exp3y'2

ah2 = 3z'2

bh2 dx'dy'dz' 32( )

WELDING RESEARCH



The temperature field caused by the single-ellipsoidalheat source equals

Assuming the moving heat source with constant speed vin the x-direction and substituting x = x – v to Equation 36,the final approximate solution can be obtained as follows:

where

DoubleEllipsoidal Heat Source

Let us consider that the heat source consisted of twoquarters of different ellipsoids as shown in a single-ellipsoidheat source, following a similar procedure as described inthe previous section for the single-ellipsoidal heat source,density. Equations were obtained for the front and back half ofthe double-ellipsoidal heat source, respectively, as follows

T x ,y,z,t( ) T0 = d0

t

0

D 1+ 2+ 2 6( )Qm

0c0 4 t( ) 3 2B y

B y

A x

A x

expx x'( )2 + y= y'( )2 + z= z'( )2

4 t( ) exp3x'2

ch2 = 3y

'2

ah2 = 3z

'2

bh2 dx'dy'dz'

=Qm 1+ 2+ 2 6( )c 4 t( ) 3 20

td exp

A x

A x x x'( )24 t( )=

3x'2

ch2 dx'

expB y

B y y y'( )24 t( )=

3y'2

ah2 dy' exp

0

D z z'( )24 t( )=

3z'2

bh2 dz'

= 3 3Q(1+ 2+ 2 6)c

Rx Ry Rzerf 3D bh )( )0

t

1

12 t( )+bh2 12 t( )+ah2 12 t( )+ ch2

exp3x2

12 t( )+ ch2= 3y2

12 t( )+ah2exp

z2

12 t( )+bh2d 36( )

where Rx = erf12 t( )+ ch2 A= x( )= ch2x

ch 4 t( ) 12 t( )+ ch2

+erf12 t( )+ ch2 A+ x( )= ch2x

ch 4 t( ) 12 t( )+ ch2Sx=1 37( )

Ry = erf12 t( )+ah2 B y( )=ah2y

ah 4 t( ) 12 t( )+ah2

+erf12 t( )+ah2 B+ y( )=ah2y

ah 4 t( ) 12 t( )+ah2Sy

1 38( )

= α − τ −

α − τ α − τ⎡⎣

⎤⎦

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

+α − τ α − τ⎡

⎣⎤⎦

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

R erft D b D z

b t t b

erfb z

t t b

zh

h h

h

h

12 ( ) + ( )

4 ( ) 12 ( )+

4 ( ) 12 ( )+(39)

2

2

2

T x ,y,z,t( )=T0 =3 3Q 1+ 2+ 2 6( )

c

R Ry Rzerf 3D bh( )0

t

1

12 t( )+bh2 12 t( )+ah2 12 t( )+ ch2

exp3( x v )2

12 (t )+ ch2 = 3y2

12 (t )+ah2

expz2

12 t( )+bh2d 40( )

R= erf12 t( )+ ch2 A x+ v[ ]

ch 4 t( ) 12 t( )+ ch2ch

2 x v( )ch 4 t( ) 12 t( )+ ch2

+erf12 (t )+ ch

2 A+ x v( )

ch 4 t( ) 12 t( )+ ch2+

ch2 x v( )

ch 4 t( ) 12 t( )+ ch2

i erf3 A x+ v( )

ch)+ erf (

3 A+ x v( )ch

1

(41)

Qm = 24 3Q

+ +ahbhcherf 3D ch( ) Sx1 Sy

1 33( )

where Sx = erf3( A x )ch

+ erf 3( A+ x )ch

34( )

Sy = erf3 B= y( )ah

+ erf3 B+ y( )ah

35( )

Qmf =24 3Qrf

+ +ahbhchf erf 3D chf( ) Sxf1 Sy

1 42( )

Qmb =24 3Qrb

+ +ahbhchberf 3D chb( ) Sxb1 Sy

1 43( )

WELDING RESEARCH



rf + rb = 2 (46)

The temperature field caused by the double-ellipsoidalheat source equals

where

Although the mathematical formulations qualitatively re-vealed the effects of the welding parameters on the weldpool shape, including complicated integration, error func-tions, exponential functions, and square roots, we can getthe approximate solutions from Equations 28, 41, and 48.

Results and Discussion Miettinen (Ref. 14) developed special algorithms that canbe used to predict thermophysical properties of steels. Con-sider variational coefficient :

= 0 · eT

Assuming thermal conductivity 0 = 29 J/m/s/°C, accord-ing to the result of Ref. 14, by means of fitting the equation,we can get = 0.000032. Considering gas tungsten arc welding with medium-carbon steel Q235, with a width of 40 mm, length 80 mm,and thickness 3 mm; voltage, U = 9 V; current, I = 95 A; arcefficiency, = 0.8; welding speed, v = 1.52 mm/s; referencedensity, 0 = 7820 kg/m3; reference heat capacity, c0 = 600J/kg/°C. The parameter of the Gaussian heat source was =0.816 mm3. The parameter of the ellipsoidal heat source wasah = 3 mm; bh =1.7 mm; ch = 2.1 mm; T0 = 25°C. The tempera-ture of the melting point of the medium carbon steel was1475°C. With the help of Equations 28 and 41, variational coeffi-cient = 0.000032 and = 0, the temperature was calculat-ed using MATLAB software. Compared with measured dataof position cross section in the experiment and calculationat times of 6.25, 12.5, and 18.75 s, the error among experi-ment and calculation occurs, which are shown in Figs. 1–6. The figures show the sizes of the melting zones in the areawithin the thermal cycle. The points in the curve have a tem-perature of 1475°C, and denote the point of melting in theweld pool, which was obtained by both analytical solution andexperiment. From the analytic computation, the length,width, and depth of the melting zone were observed as time t= 6.25, 12.5, and 18.75 s, respectively. At time t = 6.25 s, therewas a weld pool in the steel. With respect to the increase inwelding time, the width of the weld pool increased at time t =12.5 s, and penetration of the weld pool occurred. At time t=18.75 s, the weld pool expanded sequentially. The figures show the errors for various measured loca-tions among the experiment, calculation 1 and calculation 2.It was observed that the relations vary with the change ofthe measurement locations and depend on the welding time.Considering the variational coefficient, the errors can be re-duced, which shows the maximum error will be 15%. Cause of the error may be the ignorable heat convectionand radiation, the serial in Equations 28 and 41, or othercauses. The feasibility of this kind of temperature field mod-el is validated through the experiment, thus promoting thematerial physical properties changes under the model of thetemperature field.

Conclusion

Considering the thermal properties of the solid are cer-tain functions of temperature and assuming they are thesame exponential function of temperature, an analytical ap-proximate solution was obtained in infinite plate. Applyingthis solution to a finite thickness plate, an analytical solu-tion under Gaussian and ellipsoidal mobile heat sources was

T x ,y,z,t( ) T0 =3 3Q 1+ 2+ 2 6( )

c

Ry Rzerf 3D bh( )0

t

expz2

12 t( )+bh2= 3y2

12 t( )+ah21

12 t( )+bh2

1

12 t( )+ah2rf Rxf

12 t( )+ chf2exp =

3 x v( )12 t( )+ chf2

+ rbRxb12 t( )+ chb2

exp =3 x v( )2

12 t( )+ chb2 d 47( )

Rxf = erf12 t( )+ chf2 A x+ v[ ]

chf 4 (t ) 12 (t )+ chf2

chf2 x v( )

chf 4 (t ) 12 (t )+ chf2

+erf12 t( )+ chf2 A+ x v( )

chf 4 (t ) 12 (t )+ chf2

+chf

2 x v( )chf 4 (t ) 12 (t )+ chf

2

i erf3 A x+ v( )

chf+ erf

3 A+ x v( )chf

1

48( )

Rxb = erf12 t( )+ chb2 A x+ v[ ]

chb 4 t( ) 12 t( )+ chb2

chb2 ( x v )

chb 4 t( ) 12 t( )+ chb2

+erf12 t( )+ chb2 A+ x v( )

chb 4 t( ) 12 t( )+ chb2+

chb2 x v( )

chb 4 (t ) 12 t( )+ chb2

i erf3 A x+ v( )

chb+ erf

3 A+ x v( )chb

1

49( )

Sxb = erf3 A= x( )chb

+ erf3 A+ x( )chb

45( )

where Sxf = erf3 A= x( )chf

+ erf3 A+ x( )chf

44( )

WELDING RESEARCH



iment, which shows the feasibility of an analytical approxi-mate solution, pure heat conduction was sufficiently accu-rate to simulate temperature fields and fusion zone shape tosome degree. The series expansion of temperature field wasconsidered one term and three terms of variational coeffi-cient, we will calculate terms of the series in our next workin order to reduce the errors.

This work was funded by the Natural Science Foundationof China under grant 11301377.

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WELDING RESEARCH


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References

Authors: Submit Research Papers Online

Peer review of research papers is now managed through an online system using Editorial Manager soft-ware. Papers can be submitted into the system directly from the Welding Journal page on the AWS website(aws.org) by clicking on “submit papers.” You can also access the new site directly ateditorialmanager.com/wj/. Follow the instructions to register or log in. This online system streamlines thereview process, and makes it easier to submit papers and track their progress. By publishing in the WeldingJournal, more than 70,000 members will receive the results of your research.

Additionally, your full paper is posted on the American Welding Society website for FREE access aroundthe globe. There are no page charges, and articles are published in full color. By far, the most people, at theleast cost, will recognize your research when you publish in the world-respected Welding Journal.

KAIWEN GUO ([email protected]) and SHAOJUN DAI are with School of Science, Tianjin Polytechnic University, Tianjin, China. GUO isalso with the Tianjin Key Laboratory of Modern Mechatronics Equipment Technology, Tianjin Polytechnic University.


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