THERMAL ERROR MODELING IN MACHINE TOOL

THERMAL ERROR MODELING IN

MACHINE TOOL

Poonam S. Patil*

Department of Electronics, Shivaji University, Kolhapur, Maharashtra, India-4160041_,

[email protected]

R. R. Mudholkar

Department of Electronics, Shivaji University, Kolhapur, Maharashtra, India-416004,

[email protected] Abstract:

Thermal error is the domineering factor contributing to machine tool accuracy. In order to reduce thermal errors different compensation methods are used. The result of thermal error compensation depends on effectiveness of thermal error modeling. Thermal errors are engendered due to abnormal heat change in temperature and stray internal and external heat sources. This article presents overview of various investigations about thermal error modeling methods. Different types of thermal error models are used depending on analysis, numerical calculation, experimental and observation result. We have elaborated the analytical and numerical principle based thermal error models and static and dynamic empirical based thermal error models. This paper aims at offering investigational information about thermal error modeling methods practiced in machine tool.

Keywords

:

Heat; Empirical based model; Principle based model; Thermal deformation; Thermal error.

1. Introduction

In machining process the accuracy and productivity are vital phenomena. Machining process have different types of errors such as geometric, kinematic, thermal, Cutting-force induced errors in addition to tool wear, and the errors induced by assembling and chattering [Yanget al. (2015)]. Among all these errors 40 to 70 % errors

are the result of thermal error [Bryan,(1990)] [Krulewich, (1998)]. The drift in properties of machine component with change in temperature is called thermal effect which results in thermal deformation [Liet al.(2009)].

Thermal errors are generated due to heat change in temperature, due to internal and external heat sources. Internal heat source includes friction in spindle bearings, heat generated by spindle and axis drive motor ,viscous shearing of fluids in hydraulic and coolant system, heat generated in transmission system such as gear boxes and belts, heat generated by friction in seals and heat generated or removed by cutting coolant. External heat source includes change in environmental temperature and heating due to radiation [Postlethwaite (1999)]. Internal heat sources in a machine tool results in temperature distribution in structures and resulting heat flow causes the thermal deformation. As the spindle system plays important role in a machine tool structure, thermal deformation in the spindle system degrades the machining accuracy and productivity [Hayatoet al.(2006)]. In

order to reduce thermal deformation, prediction and compensation of thermal error is essential [Duet al. (2015)].

For the detection thermal error several modeling methods are being used. These methods were classified in different groups as principle based model and empirical model. Principle based model includes analytical model and numerical model, whereas empirical based model includes static and dynamic model [Liet al.(2009)]. In this

paper principle based and empirical based thermal error modeling methods for machine tool are reviewed.

2. Thermal Error Models

2.1. Principle Based Model

which is followed by thermal deformation. The principle used is modeled by “Eq. (1)”.

L

α

L T

Δ = Δ (1)

Where, ‘α’ is thermal expansion coefficient, ‘L’ is sample length and ‘∆T’ is a change in temperature. In a principle based model relation between thermal error and heat generated is described by non-linear differential equation which can be simplified by numerical and analytical method [Liet al.(2009)].

2.1.1. Analytical Model

Analytical thermal error modeling is used to obtain the relation between heat generated and temperature field [Liet.al, (2009)].

Karpat et al. (2006) proposed predictive and analytical, thermal modeling for orthogonal cutting process. This

method is used to predict stress, forces and temperature distribution of tool chip. In thermal error modeling oblique moving band heat source theory is used and it is analytically combined with modified parallel shear zone theory. In machine tool chip non-uniform heat intensity is predicted from the stress distribution in secondary shear zone.

Kuo et al. (2006) projected analytical modeling to find out solution for the problem of rise in temperature for

moving plane heat source in surface grinding. In grinding operation large amount of frictional heat generated which flows through work piece. It results in rise in temperature at the interface between work piece and the grinding wheel. In this modeling variable separation method is applied to three dimensional heat conduction equations. In order to seek the solution for rise in temperature the general solution for three directions is presented in an integral form as a function of the initial conditions, which is multiplied by particular solution of the rise in temperature. When this theoretical solution was compared with experimental solution, it was seen that difference between them is less than 4%.

Analytical modeling provides high precision in the estimation of thermal deformation, but this model takes large time in the development [Dementjevet al.(2014)].

2.1.2. Numerical Model

As the heat flows in all directions and if the temperature inside the system gets decreased in that case, it is difficult to get analytical solution. Therefore numerical models are looked upon as alternative for thermal error modeling. The accuracy of the system depends on how the boundary conditions such as power of heat sources and heat transfer coefficient are well defined [Babu and Dhamotharan (2014)]. The numerical model includes Finite Element Analysis (FEA) method and Finite Difference element Method (FDM). FEA method found to be sturdy and strong, but it is very challenging in the problems involving boundary conditions and it also intricate to find out accurate heat transfer characteristics [Abdulshahedet al. (2015)].

• Heat generation and heat transfer

Heat in machine tool may be generated due to friction between rollers and races of bearings. Heat generated in the bearings is given by “Eq. (2)”.

4

1.047 10 . .

f

H

=

×

−

n M

(2)

Where, ‘n’ is the spindle rotational speed in rpm and ‘M’ is frictional torque of bearing in Nm (Newton Meter). Torques ‘M’ consists of two parts: first the torque due to applied load (M1) and second the torque due to viscosity of lubricants (M2).

1 2

M

=

M

+

M

(3)

Where ‘M1’is the mechanical friction torque in M⋅m and‘M2’viscous friction torque in N⋅m. Mechanical friction torque is related to the structure parameters of the bearing and the load that it bears by “Eq. (4)”.

1 1 1 m

M

=

f p d

(4)

Where ‘f1’is mechanical friction torque coefficient and ‘p1’is bearing load ‘N’, and ‘dm’ is the bearing pitch diameter in m. Viscous friction torque is related to the structure parameter and its lubrication method as given in “Eq. (5)” and “Eq. (6)”.

2

7 ₃ 3

2

10

0

( )

0 m

0

2 0

0 0

M

=

−

×

f V

d

v

≥

(5)

7 3

2

160 10 f d

0 m 0

2000

M

=

×

−

v

<

(6)

Where ‘f0’refers to the coefficient that is related to bearing type and lubrication condition, ‘Vo’ refers to the viscosity of lubricant under working temperature in mm2_{/s (millimeter square per second) [Yang et al.} (2014)][Chenet al.(2014)].

While studying thermal characteristics of machine tool environmental and radiation effects are not considered because machine tools are generally kept at constant temperature [Yang et al. (2015)]. Coefficient of convection

of heat transfer is as modeled by “Eq. (7)”.

N

fluid u

K

h

d

=

(7)

Where ‘Nu’ is Nusselt number and it was obtained from Reynolds number ‘Re’ and the Prandtl number ‘Pr’. For air ‘Nu’ is calculated by the formula of “Eq. (8)”.

1 1

3 2

0.664 R P

u e r

N

=

×

×

(8)

Nu of coolant in roller is calculated by the following formula 1 1

3 2

0.332 R P

u e r

N

=

×

×

(9)

Where,

d

=

fluid e

u

R

v

(10)

‘ufluid’ is the airflow velocity and‘Kfluid’is the kinematic viscosity of air[Chenet al.(2014)].

2.1.2.1. Finite Element Method

Finite element method includes dynamic stiffness of the bearings and contact forces, thermal expansion and temperature distribution [Brecheret al.(2015)]. A thermal characteristic of a spindle system is simulated by finite

element method. By using finite element method temperature and thermal deformation at finite number of points can be obtained. This method provides relatively accurate data [Babu and Dhamotharan(2014)].

Udup et al. (2013) designed a simulation model to find out the effects of temperature distribution in the spindle

and roller bearings. They used ANSYS program to employ the finite element method. Dynamic and thermal characteristics of rotating spindle driven by a synchronous motor and a transmission belt were used to develop testing configuration. Temperature sensor thermocouple was used for detection of the thermal behavior of each bearing. Dynamic behavior of bearings was studied by tri-axial accelerometer fixed on the front bearing and on the rear bearing one axial accelerometer. The rotation speed was obtained with a Laser-Tachometer.

2.1.2.2. Finite Difference Method

Heat flows from one element to adjacent element by conduction, while the heat transfer between surface area and cooling fluid is through convection. The governing heat equation for the spindle model consists of coaxial, cylindrical pipe element is as given by “Eq. (11)”.

surface surface surface face

q

outer inner

q

front

+q

end

+q

node

= Ct +

dT

q

dt

+

+

(11)

Where 'q' is the heat generated. The transient state and steady state characteristics of motorized spindle have been observed through simulation. The programming was done using MATLAB [Bossmanns and Tu(1999)]. Numerical methods provide accurate result, but it is very complicated to design finite element model involving boundary conditions and difficult to obtain characteristics of heat transfer. Besides this method suffers from large development time [Dementjevet al. (2014)]. The standard software normally do not incorporate the models

for numerical estimation of heat generation in the bearings and requires lots of preparatory work [Brecheret al.(2015)].

2.2. Empirical based thermal model

Empirical relation between temperature and thermal deformation is obtained by measuring temperature at few points in machine tool [Wanget al. (1998)]. Empirical model based on thermal error modeling are further

classified into static and dynamic models. Designing of static model is based on the information at a particular time, whereas dynamic method is based on present as well as past time information [Li et al.(2009)].

\2.2.1. Static Model

The statistical methods are used for thermal error modeling when the data is in irregular form and forming a mathematical equation is an intricate task [Wanget al. (1998)]. All the rage the static models are Multivariable

Regression Analysis (MRA), Artificial Neural Network (ANN) and Adaptive Network Fuzzy Inference (ANFIS) [Dementjevet al.(2014)]. In the proposed article MRA and ANFIS static models are discussed.

2.2.1.1. Multivariable regression analysis (MRA)

Multivariable regression analysis is used in the application including one dependent variable and more than one independent variable. The value of dependent variable is obtained from the known value of independent variable [Sathish Kumar and Ravindra (2012)].

Yu et al. (2016) have used multiple regression analysis to obtain the relation between the heat source of work

piece of rotating spindle and thermal error.

0 1 1 2 2 3 3

= + x + x

x

...

x +

i i i i i i i i in ni i

y

β β

β

+

β

+

+

β

ε

(12)

Where

β

i₀ is constant and

β

i1,

β

i2………,

β

inare regression coefficient and ε is the error. i=1, 2. . , n and x1, x2…,

xn are random variables representing temperature difference data sources, while yi is the reaction variable that gives thermal error. Thermal error model cannot be explained by simple first order linear model shown by “Eq. (12)”. The polynomial regression model and interactive regression is used to increase the accuracy of thermal error model, which is represented by “Eq. (13)” and “Eq. (14)”. respectively.

2 3

0 1 1 2 2 3 3

= + x + x

x ...

x +

m

i i i i i i i i in ni i

y

β β

β

+

β

+

+

β

ε

(13)

0 1 1 2 2 3 1 2

= + x + x

x x

...

x

x +

i i i i i i i i i ip ni ji i

y

β β

β

+

β

+

+

β

ε

wher n

e

≠

j

(14)

Lin et al. (2007) have proposed multivariable regression analysis method to design thermal displacement model

for spindle. Multivariable regression analysis method was used as the spindle system shows diverse displacement properties under different rotational speed. The thermal displacement analysis was done on the structural elements of tool containing many temperature sensors to sense the temperature. The general linear regression model is expressed in matrix form as shown in “Eq. (15)”.

11 12 1 ₀

0 1

21 22 2 1 2

1

1 2

1 x x

x

1 x x

x

1

1 x x

x

p

p

p n

n n n np

y

y

y

β

β

β



_{  }

 

 



_{  }

 

 



_{  }

 

 



_{  }

 

 

_{ }

 





 











   



ε

ε

=

+

ε

(15)

y

=

x

β ε

+

(16)

Regression method is the simple method to determine change in temperature and relative change in displacement. It is reliable but there is a change in thermal displacement with machining process and environmental conditions [Abdulshahedet al. (2015)]. The model based on multiple regression analysis

sometimes does not fulfill the requirements mainly on the accuracy [Dementjevet al. (2014)].

2.2.1.2. Adaptive Network Fuzzy Inference (ANFIS)

ANFIS includes combination of fuzzy inference system with adaptive networks. ANFIS provides mapping between input and output [Abdulshahedet al. (2015)]. Abdulshahed et al. (2015) have published an article on

adaptive neuro fuzzy inference system to design thermal error model. In ANFIS the data space is divided into rectangular sub-spaces known as ANFIS-Grid model and the fuzzy clustering model (ANFIS-FCM model). ANFIS combines learning laws of ANN and Fuzzy Logic. Basic architecture of ANFIS is shown in Fig. 2.

Fig. 2. Architecture of ANFIS [Abdulshahedet al. (2015)]

In ANFIS the first layer is the fuzzy layer that converts the input into fuzzy value using membership functions. Each node in the second layer is fixed and represented by circle with π value. Output value of second layer is the multiplication of node function and input variable. In the third layer each node is fixed and represented by circle with 'N'. The output of the third layer represents ratio of particular node firing strength to the sum of all nodes firing strength. In the fourth layer all nodes are adjustable nodes and output is square of node functions. The fifth layer indicates the output and denoted by summation of inputs.

Feng et al. (2016) have presented an article on thermal error modeling of the spindle using Neuro-Fuzzy system.

They used Adaptive Neuro Fuzzy System (ANFIS) along with Grey system to predict thermal deformation of spindle in the z-axis. The combined model of ANFIS and Grey system rendered decreased thermal error from 45 µm to less than 7 µm. ANFIS was compared with Back Propagation network (BP) and the comparison indicated that the ANFIS system is transparent as if-then rules were used. The if-then rules are better to understand and interpretation. In BP network it is difficult to find out the relation between input and output.

ANFIS can be used for prediction [Sargolzaei and Kianifar(2010)], as knowledge based discovery model [Huang et al. (2006)] and in control application [Marichalet al. (2007)].

2.2.2. Dynamic model

2.2.2.1. Common Dynamic Model

Common dynamic neural network model is used for dynamics of non-linear system. On the whole dynamic neural network is classified in two types Feed Forward Dynamic Network (FDNN) and Recurrent Neural Network (RNN). FDNN encompasses current and past variables. It requires large input neurons, as a result computing time increases. Recurrent Neural Network (RNN) is used for stationary system while Integrated Recurrent Neural Network (IRNN) is used for non-stationary nature of thermo-elastic system. IRNN is used for thermal error modeling to of spindle. RNN have two connections: feed forward and feedback. Feed forward recurrent neural network is shown in Fig.3. The feed forward recurrent neural network contains the previous output of context layer which is fed back to the input layer. RNN is represented by “Eq. (17)”.

{

}

( )

XC C

( ),

xT

( )

X t

=

F W

X

t W T t

(17)

( )

X ( 1)

( 1)

c C

X t

=

α

t

− +

X t

−

(18)

2

( )=X(t-1)+

( 2)

( 2) ...

c

X t

α

X t

− +

α

X t

− +

(19)

( )

Ex

( )

E t

=

W

X t

(20)

Where 'T (t)' is the input temperature measurement, 'E(t)' is the output thermal deformation, 'X (t)' is hidden units, 'Xc_{(t)' is context units , 'α (0< α<1)' is the coefficient for self connection, 'W}xc_{’, ‘Xc (t)', ‘W}xT_{’ are} weight matrices and 'F(.)' is a nonlinear vector function.

The RNN is implemented with the use of subsequent steps.

1. From the current, previous input and output time series the first order difference equation is obtained. 2. RNN model is constructed for non-linear relation between input difference and output difference data. 3. Output difference is integrated to obtain integrated model for original thermo-elastic system.

Dynamic models have advantage over static models in robustness of operating under different working conditions. The IRNN capture non-stationary conditions, while RNN capture stationary conditions [Yang and Ni(2005b)].

Fig.3. Feed forward recurrent neural network [Yang and Ni (2005b)].

Wang et al. (1998) have used Grey system theory for the modeling of thermal drift of the spindle. Grey system forms dynamic model that represented by differential “Eq. (21)”.

n (ξ) 1 (ξ)

(ξ) ( ) ( ) ( )

1 1

1 1 1 2 2 3 1

n 1

d x (k) _{+ a} d x (k) _{+...+ a x (k) = b x ( ) + b x ( ) ... x ( )}

dt

n

n m m

n k k b k

dt

ξ ξ ξ

−

−

− + + (21)

In this equation 'n' is the order of differential equation, 'm' is the number of the data types and 'ξ' is the transformation number. “Eq. (21)” is called as grey dynamic model and commonly it is represented by GM (n, m). The coefficients of the model are obtained by the least square method. In the present case, 'X1' is preferred as the thermal drift data series and 'X2', 'X3', …'Xm' selected as the temperature monitoring points. In a Grey theory irregular data set is converted into new data set by using a data generation method known as Accumulated Generating Operation (AGO). The AGO non-negative series is given by “Eq. (22)”.

(0)

“Eq. (22)”.can be transformed into “Eq. (23)”.

( ) ( 1)

1

( )

n

( )

k

x

ξ

k

x

ξ−

k

=

=



(23)

An off-line and on-line method is designed to use Grey system theory for the modeling of thermal error. In Dynamic Grey theory thermal error modeling minimum transducers are required for the temperature monitoring [Wang et al. (1998)].

2.2.2.2 Adaptive Dynamic Model

In empirical modeling the real time conditions are different from the experimental conditions on account of complex manufacturing process, which makes difficult to predict thermal error in small batch production. In long term applications, process is slow, thus Non-Adaptive Dynamic Models are not suitable for small batch productions and long term applications [Yang and Ni (2005a)]. In small batch and long term applications Adaptive Dynamic modeling is used.Yang et al. (2015) have used an Adaptive Recursive Dynamic modeling for a three-axes CNC machine tool. In the Recursive model, adaption is based on parameter estimation by Kalman-filter and multiple sampling of horizons. The horizons are integrated for Recursive modification of model coefficients in real time by means of a process-intermittent probing. They concentrated on the development of Dynamic model intended for on-line thermal error estimation. A Recursive model adaptation with multi-sample horizon is shown in Fig.4. In on-line application of thermal error estimation the sampling horizon corresponds to the number of sampling intervals, where a thermal-error measurement is not accessible. The recursive model gives more real mapping between temperature input and thermal deformation outputs based on on-line application. In on-line adaption the off-line trained integrated time series model is used with matching sampling rate. The Recursive dynamic model is used with multiple horizons for easy measurements and to reduce interrupts in production.

Fig.4. Recursive model adaptations with multi-sample horizon [Yang and Ni(2005a)].

The Dynamic model with multiple sampling-horizons is recursively updated with time instant‘t’shown by “Eq. (24)”.

. .

t= N Ts K (24)

Where 'N' is the Sampling horizon, 'Ts' is the sampling interval and 'k' = 1, 2, 3, . If the past information of thermal error is not available at time instant shown in “Eq. (25)”, then thermal error estimation at the previous instant is used as input to the dynamic model.

t≠N Ts k (25)

3. Conclusion

Thermal error is one of the major source for machine tool inaccuracy. The first step in analyzing the thermal error is development of appropriate thermal error models. In this paper various thermal error modeling methods are reviewed. From this we conclude the following-

1. The principle based model provides high precision in the estimation of thermal deformation, but these models consume large time in the development. It is noticed that analytical modeling becomes difficult if the temperature inside the system gets lowered. Finite Element Analysis method (FEA) and Finite Difference Method (FDM) numerical methods offers relatively accurate data but the temperature and thermal deformation at finite number of points is obtained.

Accuracy of the numerical methods depends on amount of precision employed in defining the boundary conditions.

2. In empirical modeling static method is used when data is in irregular form and difficult to form mathematical equation. In Multivariable Regression Analysis (MRA) the thermal displacement changes with machining process and environmental conditions. Artificial Adaptive Fuzzy Inference System (ANFIS) is mainly used as knowledge based discovery model. Empirical Dynamic model have advantage over static models in robustness of operating under different working conditions. Common dynamic method and adaptive dynamic method are technically reviewed. Common Dynamic neural network is used for stationary as well as non-stationary conditions. Non-adaptive dynamic models are not robust for small batch and long term applications. Therefore Adaptive dynamic modeling method is used for small batch and long term application.

4

.

Acknowledgement

The authors would like to thank UGC, New Delhi for funding the research project entitled as “Studies on Fuzzy Logic Thermal Compensation Technique for High Speed Precision Machine”.

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