experiments and simulation of thermal behaviors of the

CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 28,aNo. 1,a2015

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DOI: 10.3901/CJME.2014.1031.162, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn

Experiments and Simulation of Thermal Behaviors of the Dual-drive Servo Feed System

YANG Jun, MEI Xuesong*, FENG Bin, ZHAO Liang, MA Chi, and SHI Hu

State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Received April 29, 2014; revised October 11, 2014; accepted October 31, 2014

Abstract: The machine tool equipped with the dual-drive servo feed system could realize high feed speed as well as sharp precision.

Currently, there is no report about the thermal behaviors of the dual-drive machine, and the current research of the thermal

characteristics of machines mainly focuses on steady simulation. To explore the influence of thermal characterizations on the precision

of a jib boring machine assembled dual-drive feed system, the thermal equilibrium tests and the research on thermal-mechanical

transient behaviors are carried out. A laser interferometer, infrared thermography and a temperature-displacement acquisition system are

applied to measure the temperature distribution and thermal deformation at different feed speeds. Subsequently, the finite element

method (FEM) is used to analyze the transient thermal behaviors of the boring machine. The complex boundary conditions, such as heat

sources and convective heat transfer coefficient, are calculated. Finally, transient variances in temperatures and deformations are

compared with the measured values, and the errors between the measurement and the simulation of the temperature and the thermal

error are 2 ℃ and 2.5 μm, respectively. The researching results demonstrate that the FEM model can predict the thermal error and

temperature distribution very well under specified operating condition. Moreover, the uneven temperature gradient is due to the

asynchronous dual-drive structure that results in thermal deformation. Additionally, the positioning accuracy decreases as the measured

point became further away from the motor, and the thermal error and equilibrium period both increase with feed speeds. The research

proposes a systematical method to measure and simulate the boring machine transient thermal behaviors. Keywords: dual-drive feed system, thermal deformation, temperature distribution, thermal characteristics measurement, FEM

1 Introduction

A dual-drive feed system could induce thermal error, which is a key factor that constrains the precision and maintenance of computerized numerical control (CNC) machines. Dual-drive servo systems with such characteristics as large processing loads, high-speed feeds, excellent acceleration, and deceleration control are widely used in large-scale precision boring machine designs worldwide. However, the dual-drive feed system structure is complex and causes many problems. For example, the friction between two screws and a nut exhibits different characteristics. Moreover, master-slave motor control and other issues are not synchronized, which causes the machine to release a large amount of thermal energy; the development of thermal errors ultimately affects machining accuracy. But there was little literature to research the thermal error of boring machine equipped with the dual-drive servo system.

Nowadays, the basic geometric accuracy of currently used Chinese precision boring is close to that of the

* Corresponding author. E-mail: [email protected] Supported by National Hi-tech Research and Development Program of

China (863 Program, Grant No. 2012AA040701) © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2015

international level, with rectilinear coordinate positioning accuracy reaching 3 μm and repeat positioning accuracy reaching up to 1.5 μm. But the accuracy decreases and becomes far lower than the initial design value after the machine is used for a long period of time. This decreased accuracy over time primarily results from inadequate maintenance and accuracy stability, and the thermal error is the main factor for the inadequate accuracy, accounting for 70% of the total number of errors arising from various error sources[1]. Thermal error would account for a larger proportion of total error as the machine tools become more sophisticated. Furthermore, the dynamical characteristic of a spindle also affects the thermal error. ZHANG, et al[2], proposed a holospectrum-based balancing method to improve the machining accuracy.

A non-uniform temperature distribution causes thermal errors in CNC machine tools; this distribution becomes non-linear and non-stationary and varies with time. The mutual coupling of location and strength of the heat source, coefficient of expansion, and machine structure create complex thermal characteristics[3].

Of course, the first step is the measurement of the thermal error, POSTLETHWAITE, et al[4], made extensive use of thermal imaging for rapid assessment of machine tool thermal behavior and off-line development of the compensation models. WANG, et al[5], presented a new

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concept-in addition to the widely recognized error avoidance and error compensation approaches-of controlling machining thermal effects by monitoring of machine thermal status.

The second step is to establish thermal error model including experiment modeling and simulation. CHEN, et al[6–10], used artificial neural networks (ANNs) to establish a relationship between temperature and the thermal error of a machine tool; although the model was useful for making generalizations, its physical explanation was weak and its predictive ability was heavily dependent on typical learning samples. RAMESH, et al[11], developed a model using support vector machines with the effective prediction of thermal errors. In recent years, the finite element method (FEM), which is used to analyze temperature fields and thermal deformation of machine tools, has become a topic of increasing interest, where MIN, et al[12], established a variety of thermal boundary conditions for a thermal state model based on the Fourier thermodynamic equation and analyzed the gradient distribution of the screw temperature field under different heat fluxes. KIMA, et al[13–14], investigated a linear motor feed system and discussed how positioning accuracy is affected by machine guideway and the thermal deformation of linear encoders. Combining both FEM simulations and experiments, HSIEN, et al[15], analyzed the relationship between preload and screw feed speed and the temperature field as well as the thermal deformation correlation. MIAN, et al[16], also analyzed thermal error using finite element method, and established an efficient prediction model.

Furthermore, thermal error compensation is an essential part of a machine tool, there are many scholars researched this issue and made a lot of good scientific achievements, such as PAHK, et al[17–21].

However, precision CNC machine tool error is a mutual coupling of many complex factors that are affected by many variables, and therefore, it is extremely difficult to establish a theoretical equation from the perspective of elasticity and heat transfer. In addition, only a few studies have investigated thermal error and dual-drive servo feed systems.

The current study focuses on a box-type precision CNC coordinate boring machine equipped with the dual-drive servo system. Thermal balance experiments were performed using a laser interferometer, infrared thermography, and a temperature displacement acquisition system to measure the distribution of the temperature field and thermal deformation at different feed rates for a high-precision CNC boring machine. Then, the study analyzed how dual-drive structures and different feed speeds affect thermal characteristics of the precision boring machine. Subsequently, this paper presented a thermal model, based on the finite element analysis, to simulate the thermal behavior of the boring machine. The model considered the complex boundary conditions such as heat sources and convective heat transfer. Transient variances in

temperatures and deformations were allowed in the solution. The comparison of the experiments show that this model can predict the temperature distribution and positioning error under specified operating conditions very well.

2 Measurement Principle and Equipment

2.1 Experimental system

The experimental system is shown in Fig. 1(a), which focuses on prismatic precision CNC coordinate boring machine tools. The system analyzes the change in the temperature field and the thermal distortion of the feed system. The positioning accuracy of the boring machine is 3 μm, and the repeat positioning accuracy is 1.5 μm. The three axes of the linear synchronous dual-drive architecture are X, Y, and Z; this architecture has a travel range of 1200 mm´1000 mm´1000 mm and a maximum point feed rate of F=64 m/min; the actual processing maximum feed rate is F=45 m/min, and the screw is a hollow cooling structure.

The measurement equipment and functions are as follows: a Renishaw laser interferometer XL80 is used to obtain the position-dependent thermal error of the feed system; a FLIR SC7000 infrared camera is used to measure the XY plane temperature field; and a synchronous acquisition system is used to determine the temperature and thermal deformation. This system uses PT100 precision temperature sensors to measure the temperature for the feed system motors, bearings, nut seat, guideway, and environment. Pt100 is a thermistor sensor which is used to measure the temperature of the machine, and the temperature sensor is Platinum resistance with a magnetic case. In the experiments, the specific parameters are as follows: temperature range is –20 °C to 150 °C, and the accuracy can reach up Class “A”(0.15 °C+0.002 t) IEC-751 standard. In addition, a high-precision eddy-current sensor is applied to measure the screw terminal thermal expansion. The High-precision eddy-current sensors made in Germany are come from Micro-Epsilon. The sensor model is DT3010-M S2. The measuring range is 2 mm. Its resolution is 0.1 μm (0.005% FSO), and the frequency response is 25 kHz (–3 dB), temperature range operation: –50 °C to 150 °C/–60 °F to 300 °F. The three systems collect data synchronically, and the measurement system is shown in Fig. 1(b). The temperature-displacement synchronous acquisition system with bus control mode is constructed based on high-performance card SCXI made by NI company. Temperature current signals are input to the current terminal, and thermal error voltage signals are input to voltage terminal. The two terminals are connected to the signal adjusting module which filters and amplifies the signals, and the control module is responsible for the final A/D conversion and acquisition to achieve simultaneous measurement of temperatures and displacements.

Y YANG Jun, et al: Experiments and Simulation of Thermal Behaviors of the Dual-drive Servo Feed System

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Fig. 1. Measurement of thermal behaviors

Fig. 2 and Table 1 present and describe the machine

positions of the magnetic temperature sensors (PT100), denoted T1, , T9, and the eddy-current displacement sensor S1.

Fig. 2. Mounted sensor positions on the machine tools

Table 1. Temperature sensor and displacement sensor mounting position

Temperature sensor and displacement sensor

Installation location

T1

T2

T3

T4

T5

T6

T7

T8

T9

Above X-axis motor

Below X-axis nut

Ambient temperature

Below X-axis motor

X-axis screw

Front bearing above the X-axis

Rear bearing above the X-axis

Rear bearing below the X-axis

Front bearing below the X-axis

S1 Screw terminal heat elongation (X +)

below the X-axis

The measurement accuracy of the temperature by infrared thermography is related to emissivity, when the emissivity parameter is accurate, the measured temperature generally is more accurate. Since different material surface has different emissivity, the emissivity can be obtained by reverse correction method. The sensor Pt100 was used to collect accurate temperature T1 of the measured point, and the temperature T2 of the same point was obtained by the infrared thermography, then the emissivity was calculated with T1 and T2 by applying the equation embedded in the infrared thermography.

2.2 Measuring principle In a laboratory at a constant temperature of 20 °C, four

feed rates on X-axis were used as the experimental variables using the test conditions shown in Table 2. The feed axis measuring point distribution is shown in Fig. 3, where the measuring point range was [–50 mm, –1150 mm], and the distance between each point was 100 mm; there were a total of 12 measurement points, and the coordinate of 0 denotes the laser origin of the laser interferometer.

Table 2. X-axis feed system: experimental conditions and measurement parameters

Feed rate F/(m • min–1)

Measurement feed Fm/(m • min–1)

Single measurement cycle Total

measurement time t/min

Measurements number N

Run time tr/min

Measurement time tm/min

Reciprocating frequency

measurement n

6 0.5 30 10 3 590 14 12 0.5 30 10 3 630 15 18 0.5 30 10 3 590 14 24 0.5 30 10 3 525 16

Fig. 3. Distribution diagram of the measured points of the feed axis

The errors at each point in the cold state were measured

before the feed system was operated continually; these

errors were marked as geometry errors. The feed system axes were run back and forth continuously for 30 min at each feed rate speed, and the error value minus the geometric error was collected as the result data and recorded as the thermal error [21]. The feed rate was reduced to F=0.5 m/min when heat caused by the feed movement affected the measurement results. According to the standards of VDI and ISO, each measurement underwent


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three reciprocating cycles, where the measurement of the laser interferometer for each measured point had duration of 2 s. At each measurement point, the machine was suspended for 5 s, and the three cyclical measurements consumed a total time of 10 min. The reverse overrun was set at 2 mm to prevent the measurement error caused by the backlash effect.

Ekij denotes thermal error of the No. i measurement point

of the screw at the jth measurement measured by the three reciprocating cycles of the kth measurement value. Measured by the laser interferometer, the measurement points are denoted as i=1, 2, ,12, the number of measurements is denoted by j=1, 2, , N, and the number of feed axis reciprocating cycles in a single measurement is k=1, 2, , 6; Ei0 is the geometric error of the ith point of screw under the cold state, and thus,

6

0 01

1 .6

ki i

k

E E=

= å (1)

The ith measurement point of the jth measured thermal error is

6

01

1 ( ).6

k kij ij i

k

E E E=

= -å (2)

Assuming that the ith measurement point of the jth

measured temperature, Tij, is obtained by the infrared camera, then Tj is the screw characteristic temperature at the jth measurement:

12

1

1 .12j ij

i

T T=

= å (3)

The thermal deformation temperature acquisition system

was used to collect real-time synchronous data once per second for the top and bottom motor temperatures, front and rear bearing temperatures, nut seat temperature, ambient temperature, and the terminal thermal elongation of the lower screw of the dual-drive configuration of the X-axis.

3 Experimental Results and Analysis

Precision CNC machine tool thermal error is a result of many complex factors and is affected by many variables, such as the processing methods, material type, processing path, ambient temperature, cooling systems, cutting fluids, cutting parameters, screw preload, and, in particular, feed rate. In this paper, the thermal characteristics of a precision coordinate boring machine were analyzed based on experiments at different feed rates. In addition, because thermal characteristics are temperature sensitive, the test was performed in an enthalpy chamber at a temperature of 20 °C to avoid interference.

3.1 Temperature field characteristics of a feed system

3.1.1 Temperature distribution at a constant feed rate Fig. 4 illustrates that the temperature of the feed system

motors, bearings, and nut seat exhibits a periodic variation related to run time, where a single run time was 30 min, the measurement time was 10 min, and the feed rate decreased at a rate of F=0.5 m/min during the measurements. The movement generates less heat than is dissipated, thus causing the temperature to decrease; when the feed rate is once again high, the temperature has gradually increased, and therefore, the general increase in temperature exhibits cyclical changes with a cycle time of approximately 40 min, which is consistent with the measurement cycle.

Fig. 4. Temperature increase of screw over time

The temperature of the upper screw motor of the X-axis dual-drive structure is significantly higher than that of the lower screw motor, whereas the upper front and rear bearing temperatures are also higher than those of the lower front and rear bearings due to the active control of the upper motor. The lower motor is the driven end that controls the active motor, where the driven motor cannot be accurately synchronized due to the friction between the screw and nut. In addition, the power and load of the active end are larger and thus generate a larger amount of heat, which increases the temperature. The temperature of the upper motor can reach 39.2 °C at thermal equilibrium, which is higher than the front bearings (30.2 °C) and the rear bearings (23.3 °C). The temperature of the lower motor is also higher than that of the front and rear bearings, and the thermal equilibrium temperature is 35.3 °C. In contrast, the seat temperatures of the two nuts are similar, and the thermal equilibrium nut seat temperature is 28.6 °C. Therefore, the screw, motor, and bearing are at different temperatures at the top and bottom of the dual-drive coordinate boring machine, which causes the uneven temperature gradient distribution of the feed system and could easily lead to thermal deformation and affect the positioning and repetitive-positioning accuracy. This conclusion can be used as a reference for the thermal balance design of a precision coordinate boring machine.


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Fig. 5 presents a schematic diagram of the dual-drive structure of the X-axis.

Fig. 5. Dual-drive schematic diagram for the X-axis

The measurements taken from the points in Fig. 6

indicate that the temperature trends are consistent and gradually increase over time, eventually reaching thermal equilibrium. The time until equilibrium is reached is approximately 320 min, with a maximum temperature of 24.5 °C.

Fig. 6. Temperature at the screw measurement points

The temperature at the measurement points on the screw

fluctuated between 0 and 1.5 °C, and the temperatures of the measurement points near the motor side were lower than those at the measurement points far from the motor. Because the infrared emission rate is affected by light and other environmental factors, the mechanical structure near the motor end easily forms a block surface; this block surface is dark and may result in a proximal temperature that is lower than that in the distal motor, thus yielding a measurement system error. Thus, the entire screw temperature field is approximately an isothermal surface, as shown in Figs. 7(a) and 7(b). The screw temperature varies and is nonlinear, non-stationary, and slowly changing, which is consistent with Ref. [2].

3.1.2 Comparison of the temperature field at different feed rates

The temperature variations for a set point of 1050 mm on the screw at different feed rates were analyzed. The measurement point reached the maximum temperature at thermal equilibrium, and the temperature increased with increases in the feed rate. Figs. 8 and 9 illustrate that when F=6 m/min, the measurement point was at approximately

240 min when equilibrium is reached at its highest temperature of 21 °C, which is approximately the ambient temperature. Because the feed rate is relatively small at this cutting condition, the friction between the screw and nut and between the gateway and slider is small, generating a smaller amount of heat, while at the same time, the hollow cooling mechanism is applied to the screw; thus, under such a circumstance, the screw temperature is close to the ambient temperature. When F=24 m/min, the thermal equilibrium time was approximately 380 min, and the temperature at the 1050 mm measurement point was as high as 26 °C at the thermal stationary state. Figs. 8 and 9 indicate that the measurement point temperature and feed rate have a nonlinear relationship; within the same time interval, the temperature increases to a higher degree and more quickly for higher feed rates. The time needed to reach thermal equilibrium increases with increases in the feed rate. In an actual machining process, the thermal equilibrium time can be used as a reference for the warm-up time for a precision coordinate boring machine.

Fig. 7. Measurement of temperature at F=18 m/min

Fig. 8. Temperature over time at different feed rates


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Fig. 9. Three-dimensional temperature map

3.2 Thermal deformation of the feed system The thermal error of the feed system is closely related

with the location of the coordinates, and thus, the error is a position-dependent thermal error. In this section, the relationship between thermal deformation and run time and coordinate position is analyzed using the experimental data at a feed rate of F=18 m/min as the sample space. The X-coordinate of the experiment ranged from –50 mm to –1150 mm, the spacing between the measurement points was 100 mm, and there were a total number of 12 measurement points. The position error was measured by the laser interferometer, and the thermal expansion of the terminal screw was tested by an eddy displacement sensor.

3.2.1 Time characteristic of the position-dependent

thermal error The end near the motor is defined as the positive

direction of the X-axis, whereas the free end that is far from the motor side is defined as the negative direction of the X-axis. When F=18 m/min, the thermal error Ei,j of the 12 measurement points corresponding to the coordinate position can be determined using Eq. (2), where Fig. 10 presents the thermal error over time.

Fig. 10. Thermal deformation at the measurement points on the screw when F=18 m/min

Overall, the thermal errors at all measurement points

exhibited the same trend, i.e., a gradual increase over time. When the thermal error reached steady state, the equilibrium time was approximately 530 min, and the

thermal error was –3.3 μm at the measurement point at –50 mm, which is the minimum of all measurement points. In contrast, the thermal error at the measurement point at –1150 mm is the maximum (–19.7 μm). Thermal drift direction is toward the negative direction of the X-axis, i.e., the screw is expanding to the free end of the X-axis during processing.

According to Fig. 10, the thermal error increases and the positioning accuracy of the precision coordinate boring machine decreases with increasing distance from the motor position.

3.2.2 Thermal expansion of the screw

As shown by the time series of the screw below the X-axis in Fig. 11, when F=18 m/min, the screw terminal in the X-axis expanded and elongated in the negative direction, and the thermal expansion rate changed rapidly with sharp increases over time and reached thermal equilibrium after approximately 85 min when the elongation of the axial expansion of the end reached 20.3 μm, followed by cyclical fluctuations. The fluctuations are similar to the variation of the temperature field caused by the measurement cycle.

Fig. 11. Thermal expansion at the free end of the screw at F=18 m/min

When the screw terminal expansion reached thermal

equilibrium, the maximum thermal error was only 7.5 μm at the –1150 mm measurement point, which is far less than at the elongation of the end (20.3 μm). A comparison of Figs. 10 and 11 illustrates that the screw terminal variation of the thermal expansion and thermal error trends of the feed axis coordinate position are inconsistent; if the precision coordinate boring machine CNC system is under open-loop control, then these trends should be consistent. The reason for the observed inconsistency is because this experiment is under closed-loop control, and the CNC system can automatically compensate for the screw pitch error caused by thermal expansion; the thermal expansion and thermal error trends do not follow a simple equivalence relation, and thus, the two curve trends are inconsistent. Therefore, the following conclusions can be drawn: thermal elongation of the screw terminal is the result of the cumulative effect of the entire screw expansion, and the entire closed-loop control mode cannot completely eliminate the effect of screw expansion on the position


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accuracy. Some thermal error cannot be eliminated, and thus, such error must be addressed via other techniques, such as thermal error compensation or thermal equilibrium design.

In addition, when the feed rate was 6, 12, and 24 m/min, the thermal error at equilibrium at –1150 mm was still the largest of all measurement points, with corresponding errors of 8.6 μm, 10.3 μm, and 26.9 μm, respectively. As the feed rate increases, the motor power increases, and therefore, the amount of friction between the motor and bearings and screw and nuts increase; this increased friction generates more heat, leading to increased screw expansion and increases in the thermal error.

3.2.3 Thermal error changing with position

When the feed rate was F=18 m/min, the temperature field of the screw was an approximately isothermic surface, and thus, it can be assumed that Tj was the typical characteristic temperature of a single measurement, the value of which is the average temperature of all measurement points calculated using Eq. (3); the result is shown in Fig. 12.

Fig. 12. Temperature and position-dependent thermal errors at F=18 m/min

The measured value at the cold state is the geometric

error, which is the thermal error at measurement points 0 at 0 min. Fig. 12 illustrates that the thermal errors are position dependent and follow an approximately linear relationship; moreover, the thermal error increases with increases in the position coordinates.

This result indicates that the positioning accuracy is lower and the thermal error is greater as the coordinate position is farther away from the motor. Furthermore, the screw temperature field reached thermal equilibrium after 320 min, whereas the thermal errors increased and they did not reach thermal equilibrium until 530 min, which further demonstrates that the thermal error has a relative larger time lag than the change in temperature field. Fig. 13 presents the location and duration of the thermal error on a three-dimensional map. At the same measurement point, the thermal deformation exhibits a nonlinear relation with time, thus verifying that thermal errors are associated with the coordinate position and are nonlinear and non-stationary.

Fig. 13. Thermal distortion in the X-direction when F=18 m/min

4 Thermal Behavior Simulation

The FEA of the coordinate boring machine is conducted in ANSYS workbench module. And transient thermal - structural coupling is analyzed by putting the results of transient thermal analysis as the heat input of a quasi-transient structural. The model has a total of 401 004 nodes and 164 758 elements, in which 3D 4-node tetrahedral and 8-node quadratic hexahedral heat transfer elements that are compatible with load-deformation analysis are used. Element type is the SOLID90, which is a higher order version of the 3D eight node thermal element. The element has 20 nodes with a single degree of freedom, temperature, at each node. The 20-node elements have compatible temperature shapes and are well suited to model curved boundaries. The mesh is shown in Fig. 14.

Fig. 14. Mesh of the boring machine

The model of transient heat transfer is based on the heat

conduction equation [22]:

T/ ,pQ C T t q= ¶ ¶ + (4)

where Q is the power density, is the mass density, Cp is


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the specific heat capacity, T is the temperature distribution in space and time, is the gradient operator, and q is the heat conductivity.

The load-deformation analysis calculates the boring machine deformations induced by thermal expansion. The strain vector can be determined by Hookes’s law:

,T = (5)

where is the thermal expansion coefficient, T is the value of temperature change.

The main heat source of a ballscrew system is the friction caused by a moving nut and the support bearings. Some assumptions are made perform thermal analysis by FEM.

(1) Machining is not performed, thus the chip effect is not considered.

(2) The screw shaft is a solid cylinder. (3) Heat radiation to the surrounding is ignored. (4) The temperature-dependent non-linear properties of

the material (thermal conductivity, heat capacity, thermal expansion, etc) are not considered, since the ambient temperature is 20 ℃.

4.1 Calculation of the power heat sources

4.1.1 Motor heat generation model

The motor power losses can be obtained by calculating the motor efficiency [23]:

motor el in motor windge(1 ) ,Q P P = - + (6)

where motorQ is the motor generated heat (W), el inP is the motor input power, motor is motor efficiency, windgePis motor wind losses.

Motor power loss allocation between the rotor and the stator is determined by the motor slip and synchronous frequency, and the analytic expression is as follows:

sliprotor motor

ync

stator motor rotor

,

,

s

fQ Q

f

Q Q Q

ìïï =ïïíïïï = -ïî

(7)

where rotorQ is the rotor generated heat, statorQ is the stator generated heat, fslip is the slip frequency and fsync synchronous frequency (Hz).

4.1.2 Bearing thermal generation model

Harris pointed out that the heat generation of the bearing is mainly due to the external loads and the viscous friction of the lubricant. In addition, the rolling body spin motion in the raceway also generated heat by frictional torque. The heat generated by bearings is the dominant heat causing thermal deformations. The heat can be calculated by the

following equation [24–26]:

4total 1.047 10 ,Q n M-= ´ ´ ´ (8)

where Qtotal is the bearing generated heat (W), n is the rotating speed of the bearing (r/min), M the total frictional torque of the bearing (N • mm). The frictional torque M consists of two components: one is caused by the applied load M1 and the other one by the viscosity of lubricant M2:

1 2.M M M= + (9)

The applied load M1 can be given by the following

equation[27]:

1 1 ,mM f F d = (10)

max(0.9 tan 0.1 , ),a r rF F F F = - (11)

where 1f is a quotient related to the bearing type, F is the bearing load (N), and dm is the mean diameter of the bearing (mm). ,aF rF is axial and radial forces, respectively. The viscosity of lubricant M2 can be computed by Eq. (12):

7 2 3 3

2

7 32

10 ( ) , 2000,

160 10 , 2000,

o o m o

o m o

M f v n d v n

M f d v n

-

-

ìï =ïïíï = ´ <ïïî

≥ (12)

where fo is a factor related to bearing type and lubrication method and vo is the kinematic viscosity of the lubricant (mm2/s). 4.1.3 Computation of the screw thermal generation

We assumed that friction heat generation between the moving nut and the screw shaft is uniform at contacting surface and is proportional to contacting time, and heat generation at support bearings is also constant per unit area and unit time.

The screw shaft generated heat Q could be defined as[25]:

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nMQ = (13)

where n is the rotating speed of the nut, M is the total frictional torque.

When the axial load of the screw is Fa, the torque M1 required to drive the nut is:

1 ,2π

a hF PM

= (14)

where hP is the screw input power, is the nut efficiency of the ball screw.

The ball screw resistance moment M2 is obtained by the


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following:

22 (1 ).

2πa hF P

M

= - (15)

While the driving torque M1 and the resistance moment

M2 changes, the total friction torque M can be approximated as:

2 10.94 .M M M= + (16)

4.2 Heat transfer coefficient In the FEM model, a complete description of convection

requires a definition of the heat transfer coefficient , which can be given by FRANK, et al[28]:

,g

Nu

h

= (17)

where Nu is the Nusselt number, is the fluid thermal conductivity, and hg is the characteristic length. For free convection, Nu is a function of the Rayleigh number Ra and Prandtl Pr number. And

,gu hRe

v

= (18)

where u is the average velocity of fluid, v is the fluids kinematic viscosity.

The Prandtl Pr is determined by the materials of the boring machine:

,c

Pr

= (19)

where c is the fluid capacity, is the fluid dynamic viscosity.

Nu can be induced by applying Eqs. (18)–(19):

0.80.022 5 ,NNu Re Pr = (20)

where N is a constant. 4.3 Transient simulation analysis

Because the thermal balance test was performed in an enthalpy chamber at 20 °C, the simulation reference temperature was set 20 °C. The simulated temperature and thermal deformation distribution are shown as Fig. 15 and Fig. 16. The simulated temperature of the upper motor can reach 40.2 °C on thermal equilibrium, which match the measured value 39.2 °C. The screw temperatures in the range [–50 mm, –1150 mm] are significantly higher than the other ends, consistent with the experimental results. The screw shaft and scale deformed under running condition, this is one of the important factors leading to the decline positioning accuracy. The simulation results shown that the

X-axis scale has a larger thermal deformation, maximum thermal expansion of the screw reached 23.6 μm, while the measured value was 20.3 μm.

Fig. 15. Temperature distribution of the boring machine

Fig. 16. Deformation distribution of the boring machine

4.3.1 Comparison of the simulated and the measured temperature

In order to verify the thermal FEM model, key points’ temperature and thermal deformation was extracted to compare with the measured value. Fig. 17 shows the screw transient simulated temperature distribution at t=83 min, and the overall trends is similar to experimental Fig. 6. At thermal equilibrium, the simulated maximum temperature is 24.6 °C matched well with the measured 24 °C. But heat balance time of the simulation 83 min is much less than the measured value 320 min, which is due to temperature declined caused by the measuring, needing a longer time to reach thermal equilibrium. In addition, the simplified structure model leads to the heat transfer process shorter reducing the thermal equilibrium time also. Fig. 18 shows the comparison between the simulated and the measured


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temperatures of the screw at 12 measuring points.

Fig. 17. Simulation temperatures of the upper screw

Fig. 18. Comparison of the measured and simulated screw temperatures at steady state

Fig. 19 presents the simulated and the measured

temperatures such as motors and bearings at the steady state. The simulation error is at the range 2 °C.

Fig. 19. The measured and simulated temperatures of keypoints at steady state

4.3.2 Comparison of the simulated and the measured thermal error

Fig. 20 describes that the heat distortion values of the scale with 12 measuring points, which are extracted from the simulation, increases with time similar to the

experimental Fig. 10. The maximum thermal deformation of the simulation is –23.9 μm at X=–1050 mm, while the maximum measured position error is –19.7 μm at X=–1150 mm. This Shown that thermal deformation of the scale is the main factor to reduce positioning accuracy. At thermal equilibrium, steady-state value comparison is shown as Fig. 21, simulation error is about 2.5 μm. It can be seen clearly that calculated temperature and deformation variations from FEM are very close to the measured data of both tests.

Fig. 20. The simulated deformation of the scale in X-direction

Fig. 21. Comparison of the measured position error and the simulated scale deformation at steady state

5 Conclusions

This paper proposed a systematic method to analyze the thermal behavior of the boring machine equipped with the dual-drive servo system. The temperatures and the thermal deformation were measured under moving the X-axis long-term. A simplified FEM model for the machine was developed, and transient temperature distribution and the thermal deformation of the feed system in the machine tool were analyzed. The comparison between measured and simulated results showed very good agreement. From the results, the following conclusions can be drawn.


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(1) The temperature field and thermal deformation vary with time and are nonlinear and non-stationary. The temperature equilibrium time was 320 min, which is less than the thermal drift equilibrium time of 530 min, indicating that the screw temperature field is a slow-changing process and that the thermal deformation has relative lag characteristics.

(2) Three linear axes of a boring machine are synchronized in a dual-drive configuration. The screw shaft of each load, which is an uneven preload, and the dual drive servo motor control cannot be precisely synchronized, which can easily lead to uneven temperature gradient distributions. Such uneven distributions can result in linear screw expansion or bending and twisting of the flat heat in the three-dimensional space, eventually leading to thermal errors that decrease machining accuracy.

(3) Screw terminal thermal elongation is the cumulative effect of expansion of the entire screw, where CNC closed-loop control can eliminate part of the thermal drift but cannot completely eliminate the position-dependent thermal error caused by screw expansion. In addition, the positioning accuracy decreases and the thermal error increases as the X-axis position becomes farther away from the motor.

(4) As the feed rate increases, friction generates more heat, which results in more intense screw expansion and increased position-dependent thermal error. The temperatures at the measurement points on the screw, motor, bearings, and other key components have a non-linear relation with the feed rate that increases as the feed speed increases in the equilibrium state, which also increases the time needed to reach thermal equilibrium. During an actual machining process, the thermal equilibrium time can be used as a reference for the warm-up time of a precision coordinate boring machine.

References

[1] BRYAN J B. International status of thermal error research[J]. Annals of CIRP, 1990, 39(2): 645–656.

[2] ZHANG Yun, MEI Xuesong, SHAO Mingping, et al. An improved holospectrum-based balancing method for rotor systems with anisotropic stiffness[J]. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2013, 227 (2): 246–260.

[3] MOU J. A method of using neural networks and inverse kinematics for machine tool error estimation and correction[J]. ASME Trans Journal of Manufacturing Science and Engineering, 1997, 119: 247–254.

[4] POSTLETHWAITE S R, ALLEN J P, FORD D G. Use of thermal imaging, temperature and distortion models for machine tool thermal error reduction[J]. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 1998, 212(B8): 671–679.

[5] WANG Hui, HUANG Qiang, YANG Hong. In-line statistical monitoring of machine tool thermal error through latent variable modeling[J]. Journal of Manufacturing Systems, 2006, 25(4): 279–292.

[6] CHEN J S, CHIOU G. Quick testing and modeling of thermally-induced errors of CNC machine tools[J]. International

Journal of Machine Tools and Manufacture, 1995, 35(7): 1063–1074.

[7] YANG S, YUAN J, NI J. The improvement of thermal error modeling and compensation on machine tools by CMAC neural network[J]. International Journal of Machine Tools and Manufacture, 1996, 36 (4): 527–537.

[8] CHEN J S. Fast calibration and modeling of thermally-induced machine tool errors in real machining[J]. International Journal of Machine Tools and Manufacture, 1997, 35(2): 159–169.

[9] WANG K C, TSENG P C, LIN Kuoming. Thermal error modeling of a machining center using grey system theory and adaptive network-based fuzzy inference system[J]. JSME International Journal, Series C: Mechanical Systems, Machine Elements and Manufacturing, 2007, 49(4): 1179–1187.

[10] MIZE C D, ZIEGERT J C. Neural network thermal error compensation of a machining center[J]. Precision Engineering, 2000, 24(4): 338–346.

[11] RAMESH R, MANNAN M A, POO A N. Support vector machines model for classification of thermal error in machine tools[J]. International Journal of Advanced Manufacturing Technology, 2002, 20(2): 114–120.

[12] MIN X, JIANG S A. Thermal model of a ball screw drive system for a machine tool[J]. Journal of Mechanical Engineering Science, 2011, 1: 186–225.

[13] KIMA J, JEONG Y H, CHO D W. Thermal behavior of a machine tool equipped with linear motors[J]. International Journal of Machine Tools and Manufacture, 2004, 44: 749–758.

[14] UHLMANN E, HU J M. Thermal modeling of an HSC machining centre to predict thermal error of the feed system[J]. German Academic Society for Production Engineering (WGP), 2012, 6: 603–610.

[15] WU C H, KUNG Y T. Thermal analysis for the feed drive system of a CNC machine center[J]. International Journal of Machine Tools and Manufacture, 2003, l43: 1521–1528.

[16] MIAN N S, FLETCHER S, LONGSTAFF A P, et al. Efficient thermal error prediction in a machine tool using finite element analysis[J]. Measurement Science and Technology, 2011, 22(8): 365–372.

[17] PAHK H, LEE S. Thermal error measurement and real time compensation system for the CNC machine tools incorporating the spindle thermal error and the feed axis thermal error[J]. International Journal of Advanced Manufacturing Technology, 2002, 20(7): 487–494.

[18] POSTLETHWAITE S R, ALLEN J P, FORD D G. Machine tool thermal error reduction-an appraisal[J]. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 1999, 213(1): 1–9.

[19] YANG Hong, NI Jun. Dynamic modeling for machine tool thermal error compensation[J]. Journal of Manufacturing Science and Engineering, Transactions of the ASME, 2003, 125(2): 245–254.

[20] DU Z C, YANG J G, YAO Z Q, et al. Modeling approach of regression orthogonal experiment design for the thermal error compensation of a CNC turning center[J]. Journal of Materials Processing Technology, 2002, 129(1–3): 619–623.

[21] Li Yang, ZHAO Wanhua. Axial thermal error compensation method for the spindle of a precision horizontal machining center[C]//2012 IEEE International Conference on Mechatronics and Automation (ICMA 2012), Chengdu, China, August 5–8, 2012: 2319–2323.

[22] BAEHR D, STEPHAN K. Heat and mass transfer[M]. Berlin: Springer, 2009.

[23] EUGENE A, THEODORE B. Marks’ standard handbook for mechanical engineers[M]. 10th ed. New York: McGraw Hill Professional, 2007.

[24] HARRIS T A, HE S Q. Rolling bearing application[M]. Beijing: China Machine Press, 2014.

[25] HARRIS T A. Rolling bearing analysis[M]. New York: John Wiley


·87·

and Sons, 1991. [26] ZHAO Haitao, YANG Jianguo, SHEN Jinhua. Simulation of

thermal behavior of a CNC machine tool spindle[J]. International Journal of Machine Tools and Manufacture, 2007, 47(6): 1003–1010.

[27] JONES A B. A general theory for elastically constrained ball and radial roller bearings under arbitrary load and speed conditions[J]. Journal of Fluids Engineering, 1960, 82(2): 309–320.

[28] FRANK P, INCROPERA D P. Fundamentals of heat and mass transfer[M]. New York: SOS Free Stock, 2001.

Biographical notes YANG Jun, born in 1985, is currently a PhD candidate at State Key Laboratory for Manufacturing Systems Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, China. His research interests focus on precision engineering and thermal error compensation. E-mail: [email protected] MEI Xuesong, born in 1963, is currently a professor at State Key Laboratory for Manufacturing Systems Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, China. His research interests focus on precision engineering and nanotechnology. E-mail: [email protected]

FENG Bin, born in 1980, is currently a PhD candidate at State Key Laboratory for Manufacturing Systems Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, China. His research interests focus on precision engineering. E-mail: [email protected] ZHAO Liang, born in 1987, is currently a PhD candidate at State Key Laboratory for Manufacturing Systems Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, China. His research interests focus on precision engineering. E-mail: [email protected] MA Chi, born in 1989, is currently a PhD candidate at State Key Laboratory for Manufacturing Systems Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, China. His research interests focus on precision engineering. E-mail: [email protected] SHI Hu, born in 1984, is currently associate professor at State Key Laboratory for Manufacturing Systems Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, China. His research interests focus on precision engineering. E-mail: [email protected]

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