The thermal drift modeling of spindle system based on a physical driven deformation methodology

: T hermal error is one of the main factors that leads to the decline of CNC machine tool’s accuracy stability. The location of the typical thermal key points is determined by the finite element analysis model of the spindle system. The thermal experiments were conducted on a spindle of the milling machine tool. Furthermore, the distribution law of the spindle temperature field and the mechanism of its deformation was elaborated. On this basis, the temperature field model of each region of a spindle system was established based on the generation, conduction, and convection theory of heat, spindle speed and motor load. A physical driven deformation modeling (PDDM) method was put forward for building the relationship between the axial thermal drift error and the temperatures of the key points of spindle system. Then, with parameters identified using data of one speed, the influence of structural size uncertainty on prediction results was analyzed by the first-order second-moment (FOSM) method. The prediction residual errors of the suggested model with multiple size parameter fluctuation were provided. Finally, the effectiveness and robustness of the time-varying error model was verified by experiments and compensation.


Introduction
Machining errors that deteriorate the quality of productions are related largely to errors caused by heat, geometric error, and deformation error induced by cutting force.It has been observed that the time-varying thermal displacement accounts for approximately 50-80% of the total errors of the machined parts in precision machining [1][2].The spindle is the key functional component, and also the largest heat source of CNC machine tool.Its thermally induced deformation has a significant adverse effect on accuracy stability of the machine tool [3].The thermal deformation is a time-dependent nonlinear process caused by no uniform temperature variation in the mechanical structure.The interaction among thermal resistance coefficient, heat source intensity, the system structure, and heat source location create complicated thermal behavior in the machine tools [4].Thus, this paper conducted in-depth study of the time-varying thermal error in the spindle system, and the influence of thermal effect on the machining accuracy of the machine tool was researched.
In general, there are two primary strategies to minimize the thermal error of machine tool: thermal error avoidance and thermal error compensation [5,6].The purpose of former method is supposed to reduce the heat generation or the thermal deformation.Some samples of this strategy are adopting thermal symmetric structure (radial symmetry of spindle box), employing cool system, and thermally insensitive materials.These approaches mentioned above have a limit to achieve machining accuracy, however, manufacturing costs are relatively high.Error compensation technique is trying to produce an artificial error for adjusting the position of the tool to workpiece [7].Thermal error compensation is considered as be a"soft technique"that has a wide field of application at less-cost.
In the past decades, researchers have employed various techniques, namely, multiple regression, neural network, gray theory, support vector machine, time series etc., in modeling the thermal characteristics of machine tools [8][9][10][11][12].However, these models are essentially mapping thermal displacements against temperature variation of the thermal-sensitive point, but they don't reflect any physical or geometric rules.
Furthermore, the establishment of the accurate models require a large amount of input and output data, requires more temperature sensors, and the robustness of these models is poor [13].Thus, this type of model is more suitable for use in cases where physical or geometric relationships cannot be established.
The time-varying thermal error of spindle includes axial thermal growth error (ATGE) and radial thermal drift error (RTDE).In the present study, the reason of the spindle nonlinear behavior is explained from the theory and experiment of thermal bending.
Then, the RTDE of a spindle is compensated by the suggested model.The modeling methods of ATGE have been extensively studied, mainly divided into two categories:(a) ATGE is a function of the rotating speed whose advantage is that no additional sensors are required, so the error compensation is low cost and easy to implement [14,15].However, there exits poor robustness and applicability.For instance, when the spindle speed varies frequently or the ambient temperature varies greatly, the forecast effect of the model becomes worse.Moreover, the spindle with cooling devices is not suitable.
(b) ATGE is a function of the key point temperature, with the measured temperature as input, which has stronger robustness [16,17].However, the predicted results become poor if moving state of the spindle during actual machining process differs from those of thermal tests.
In addition, numerous other methods are applied to model the ATGEs of the spindle.
Kang et al. [18] presented the thermal deformation model of spindle based on the heattransfer mechanism and the rotating speed.Liu et al. [19] built the spindle thermal growth error based on the velocity and acceleration of temperature variation (MBOTV).
Lei et al. [20] put forward a compensation method of the spindle axial time-varying thermal error based on edge computing.Ma et al. [21] suggested a three-dimensional FEA model of a motorized spindle, and deduced its thermally induced error based on heat flow.Most of the above literatures pay attention to the thermal expansion of the spindle, ignoring thermal inclination.However, the axial thermal drift error (ATDE)affects the consistency of machining accuracy of high-performance parts.In order to solve this problem, this article intends to establish model for studying the thermal field of a spindle system, and the thermal error-temperature model of spindle was derived based on the constitutive model.Due to lacking of design, geometric sizes of the spindle system were difficult to measure accurately in the test site.Therefore, the influence of structural size uncertainty on prediction results was analyzed by the FOSM method.
The strong robustness of the error model was verified using both the experiment and compensation.

ANSYS thermal behavior simulation
During the operation of the spindle, the major heat source is the rotating friction of bearings, motor of spindle and cutting heat.Since most of the cutting heat is carried away by the coolant and the cutting, the heat intensity of the bearing and motor affect the temperature field distribution of the spindle unit, which results in thermal distortion of a spindle system.In this chapter, ANSYS was utilized to carry out steady-state thermal analysis of spindle system, and the temperature field distribution and thermal deformation of spindle system were obtained.
The heat generation power and the surface convection heat transfer coefficients of different components in a spindle system were computed in Table 1, and were employed as thermal load and boundary conditions of the thermal analysis in the ANSYS software.
The thermal behavior of spindle system was simulated according to the experimental procedure, and the temperature and thermal deformation distribution and variation rule of the spindle system were obtained.The simulation results were presented in Fig. 1.
Fig. 1(a)shows the spindle and the headstock concentrated the majority of the heat produced by the bearings and motor, with only a small amount of heat transmitted to the column through the spindle box.Fig. 1(b) shows the temperature of the front bearing of the spindle was significantly higher than that of the rear bearing.This is due to the heat generation rate of the front bearing group of the spindle unit was much bigger than that of the rear bearing one, and there was thermal resistance between the bearings and the shaft chamber, which prevented the bearing frictional heat from being dissipated timely.Moreover, there was temperature gradient in the process of heat entering and leaving the material.Fig. 1(c)-(d) show the Z-axis thermal deformation of the drilling machining center was mainly induced by the thermal elongation of the spindle and the column.Based on the steady-state thermal analysis mentioned above and the analysis of the heat source and the structure of the CNC machine tool, five thermal key points were selected to be arranged on the vertical milling center to detect the temperature variant of the spindle system.The specific distribution of measuring points was described in  The axial thermal growth data of a spindle was collected by the spindle error analyzer.
See Fig. 4 for the schematic layout of the spindle error analyzer.The displacement capacitive sensors were mounted around two precision balls, and fixed to the worktable via a bracket, to measure the test bar motion (simulated cutter).When the spindle deformed along the Z-direction, the lowest displacement sensor could acquire the error data .

Analysis of experimental data
In the experiment, a high-speed mechanical spindle with three groups of rotation speed of 6000, 10000, and 14000 rpm were chosen for thermal characteristics experiments.
The temperatures and ATDEs under various rotating speeds were shown in Fig. 5 and Fig. 6.It can be observed from Fig. 5 and Fig. 6 that as the temperature increased, the errors became large, which is interpreted that the test bar moved downward.Moreover, the rotating speed of the spindle was positively correlated with thermally induced deformation.It indicates that the increase of rotational speed caused the increase of the spindle internal heat accumulation, and the greater the thermal deformation.
On the base of the finite element simulation and designed experiment in this section, the thermally induces deformation process of the spindle system at 6000 rpm can be divided into the following three stages: Phase 1: When the spindle begins to rotate, heat is generated and transferred to the column through the spindle housing, resulting in an increase in the temperature near the heat source side of the column.In contrast, the other side of the column doesn't significantly increase in temperature.The thermal expansion of the column causes the thermal tilt of a spindle, and the test bar is far away from the z-axis displacement sensor, as depicted in Fig. 5.
Phase 2: After the spindle rotates for a period of time, the temperature of the left and right sides of the column still both increase, which leads to the thermal bending angle of the spindle becomes bigger.When the heat production and the heat dissipation reach dynamic thermal equilibrium, increase of the main shaft elongation slows down.
Phase 3: When the machine tool stops to cool down, it starts to contract by giving off heat.The temperature gradient of left and right sides is stabilized, so the temperature difference of the column changes slowly during Phase 3 (Fig. 5).As a result, the column continued to elongate, the axis of the spindle deflects counter clockwise in the x-z plane.

Thermal error modeling using PDDM
The mechanism and causes of the thermal behavior of CNC machining center spindle in Section 2 was analyzed, and the thermal expansion of spindle system was elaborated.
Based on the analysis in Section 2, the modeling of the time-varying thermal drift error was studied in this chapter.At present, the application of PBM method in the thermal error prediction has become a main research direction of precision control.In contrast to PBM's constitutive model-based design model, DDM requires sufficient data to map the temperature of key mechanical elements to thermal errors.Thus, this paper utilized Physics driven deformation modeling (PDDM) method to describe the thermal field of a spindle system, and the spindle thermal drift error was designed based on geometrical structure, so this model is robust with low prediction bias.

Thermal field of spindle system
The spindle system consists of the spindle, the spindle box and the column, as shown in Fig. 7. From Fig. 7, in the machine tool running, the heat produced by the bearing and the motor flows into the spindle, resulting in the temperature increase of the spindle and its housing.Meantime, the generated heat is conducted to the headstock and the column, and also dissipated into the surrounding air.The spindle system is simplified into five areas, namely, the spindle area, the upper and lower parts of the headstock and the left and right segments of the column.The critical temperature point for each region is selected to describe its thermal state.
According to the temperature rise mechanism of the spindle, the thermal change of the spindle system is a dynamic change process with time.The temperature variant of each region at any time can be expressed by Eq. ( 1) According to Ref. [22] and Fig. 6, the spindle speed has significant influence on its thermal characteristics.The temperature change caused by friction heat of bearings can be obtained by using the spindle speed as an input variable: ( The temperature field of spindle system varies with the motor heat applied to it.The temperature change induced by the heat production of spindle motor can be estimated by the load of spindle motor: ( In order to eliminate the effect of variation with time in the motor load, the average load during can be calculated as follows: (4) where is the real-time motor load collected by the computer with Visual C++ developing FOCUS program at a sampling period of 500 millisecond.
For certain region, the spindle heat is transferred to the surrounding air causing the temperature change (5) Because the temperature gradient exits in the various areas of a spindle system, which results in heat conduction occurring between the two adjacent components.For certain segment, the temperature variation caused by heat transfer to adjacent areas can be calculated as follows:

Calculation of ATDE
The real time temperature rise of every region in a spindle system can be derived from the above formula, and the expansion and contraction of these areas can be calculated according to the temperature field of spindle system.The thermal deformation diagram of spindle system is presented in Fig. 8. From Figure 8, (1) shows the initial thermal equilibrium state, (2) shows the thermal state.Assuming that is thermal expansion of the spindle, and are thermal expansion of the left and right sides of the column.
The values of , and are positive during thermal expansion and negative during contraction.
Take thermal deformation (2) in Fig. 8 consideration as an example, the relationship between the temperature field and the thermal elongation of the spindle system is built.
When machine running, the spindle will elongate due to the increase of internal temperature.After the spindle and the column are heated, thermal deformation occurs along their length direction.At the same time, one side of the column near the heat source experiences a significantly higher temperature rise rate than the other side (see Fig. 5), which leads to thermal tilting due to different expansion amounts on the column left and right sides, making the tool deviate from the original position, causing thermal drift deformation of the spindle.As shown in Fig. 6, the spindle temperature follows a similar trend to the axial thermal displacement.That is to say, a linear relationship between a special point of spindle and the thermal elongation can be established, which be obtained by adjusting the thermal expansion coefficient.Therefore, the thermal elongation of the spindle at time can be expressed as follows: Similarly, the column undergoes the deformation due to the thermal effect of machine tool.The temperature rise on the left side of the column is deferent from that on the right side, though, the temperature field is continuous and linear.This indicates that the thermal expansion of the column is linearly related to the temperature.According to Equations ( 8) and ( 9), the thermal expansion of the left and right sides of the column can be calculated as follows: The slope of the column can be obtained from the triangular proportion relation: (10) According to Fig. 1 (d) and Figure 3 system structure, the column can be divided into the upper and lower parts.If the deformation values and are taken from the left and right sides of the interface between the two parts respectively, then 、 move the tool up and causes the tool to move downwards.Because of the spindle system inclination deformation, the influence of the thermal deformation of the spindle on the machining precision varies with the tilt angle, that is (11)

Model parameters identification
For the model proposed here, it is necessary to identify some unknown parameters.
Among them, the heating coefficient, the heat dissipation coefficient and heat conductivity coefficient of each region of spindle system can be determined by fitting the respective temperature increase curves.The thermal expansion , ,and can be derived from the reverse derivation of the tested values .Therefore, regarding Eqs.( 7)-( 10), the corresponding independent variables and dependent variables and are all obtained.The unknown parameters of the proposed model are identified by the least square method, and the optimal objective function can be determined (12) and Eq. ( 13): The parameters identification results were presented in Table 2 using the data of 6000 rpm.

Analysis of geometric size uncertainty impact on prediction results
During the predicting the spindle thermally induced errors, due to the irregular structure of CNC machine tool (see Fig. 3), it is difficult to accurately measure the geometric size of the spindle system required for the time-varying drift error prediction.Therefore, it is essential to analyze the influence of measurement deviation of the geometric size on the predicted results of the error model.The first order second moment (FOSM) is one of the methods for calculating reliability [23].The FOSM need explicit the time-varying drift error model's function for reliability analysis.The FOSM is used to calculate the reliability of measuring deviation fluctuation values within a certain allowable deviation range.Its principle is to approximate the limit state function linearly at the design point.
According to Eqs. ( 1) and ( 11), the performance function of the axial thermal drift error prediction can be defined as: is a basic random vector whose components are independent of each other: Set the true value of the variable is Eq. ( 14) can be expanded at the point based on Taylor series, and the approximate expression of performance function is expressed below: The mean value and standard deviation of is as follows: The reliability index for the fluctuation values of measured deviation belonging to a certain allowable deviation range or not can be calculated using Eq. ( 18): The reliability can be calculated by Eq. ( 21): After multiple measurement of the actual structural dimensions of the spindle system, the mean and standard deviation of were presented in Table 3.By using the identification parameters of Table 2, the reliability indicator and the reliability was calculated by the FOSM method (Fig. 9):  The above result shows that when the geometric size of the spindle system varied in a certain range (see Table 3), the impact on the axial error prediction was negligible.In order to evaluate the effect of geometric size error on the thermal error prediction results, five groups of data were selected within their variance range based on the nominal dimensions of structure in Table 3.By using the parameters given in Table 2, the variant of residuals over time for five structural cases was depicted in Fig. 10.As can be seen from Fig. 10, for the five groups of geometric sizes, their residuals followed the same pattern over time, but the residuals difference between groups was very small.The maximum residuals were not more than 8.9m.From Fig. 10, the predicted residuals can be divided into two parts: one is caused by the differences between the thermal deformation of the spindle system and the time-varying drift error model; The other one is induced by the geometric size error.In Fig. 10, the variant of the residual curves over time was caused by the error of the proposed model.If the predicted error corresponding to actual measured dimension of the spindle system was taken as the basis, the difference between the other four structural sizes and it were indued by dimensional deviation.The prediction error caused by dimensional deviation can be express as follow: (23) where is residual value of case ; is the number of residual values at a certain structural case over a period of time.
According to Fig. 10 and Eq. ( 23), 4N residual data caused by dimensional error can be obtained.The histogram and error distribution law of predicted error statistics caused by the geometric parameters is shown in Fig. 11.It can be observed from Fig. 12 that the predicted thermal error change with time, that is, in Eq. ( 18) varies with time.Therefore, the reliability calculated according to the statistical results of the distribution of each random variable changes with time.Here, the average value of the predicted axial thermal error in Fig. 12 is 29.88 .Combined with the mean and standard deviation of 0. 45 and 2.18 of axial residuals, the calculated reliability index of the axial error prediction is 13.7 and the reliability is 1.

Simulations
The forecast effect of the suggested model was simulated in 1STOPT software.
Figure12-14 show the simulation results of the spindle at the different rotating speeds.
The results indicate that the estimated temperature is in good agreement with that measured one, which demonstrate that the PDMM can reasonably describe the temperature field distribution of the spindle system.
To evaluate the prediction precision of the error model, the evaluation parameters are introduced for the predicting thermal error [24].As can be seen from Fig. 15, the RSME, R and F of the three groups of experimental axial thermal error prediction models were

Conclusion
This paper proposes an effective technology to improve the axial positioning accuracy of a vertical drilling center.The thermal-structural coupling model of the spindle system is established by using the ANSYS simulation software.The distribution law of the temperature field and thermal deformation of a spindle system is obtained by numerical analysis to determine the location of the typical thermal key points.The test system for the spindle thermal characteristics is built, and the temperature and the thermal error experiments are carried out.Based on the structure of the spindle system and the results of thermal tests and numerical simulation, the spindle system is divided into five areas.
The correlation model of the temperature of the key points in every region is built based on the heat-transfer theory, spindle speed and motor load.According to the geometric relationship among different regions, the modeling method for the ATDEs of the spindle is put forward, and verified through simulation and experiments.The influence of geometric size uncertainty on prediction results is analyzed.The salient points of this article are outlined as follows: (1) PDMM method can record the dynamic process of the thermal field of the spindle system, and describe the tilt error at probable axial-deformations.Experimental results show that the forecast precision of the proposed model still reaches over 94%.
(2) When the geometric size varies within a given precision range, the maximum residual error of axial prediction is not more than 9 .The predicted residuals can be divided into two parts.One part comes from the presented model error, which causes the predicted residuals to fluctuate with time.The other part is caused by the error of structure size.It turns out that the mean of axial residual caused by dimensional uncertainty is 0. 45 ,and the variance is 4.74 .

Fig. 1 .Fig. 2 .
Fig. 1.Steady state temperature analysis and thermal deformation distribution of the spindle system FEM simulation at the rotating speed of 6000 r/min2.2Establish experimental platformTo explored the thermal characteristics of the mechanical spindle, the ATGs of the spindle was conducted on a vertical drilling center TC500R.The entire experimental device was composed of the vertical drilling center TC500R, the spindle error analyzer CPL290, the temperature sensor PT100, and the microcomputer, the flow of the spindle thermal test system was depicted in Fig.2.According to Fig.2, the capacitive displacement sensor transmitted the precision ball position signal to the spindle error analyzer SEA software through a cable, then the experimental outcomes were displayed in the numerical and graphical forms.The temperature sensor collected the temperature signal and sent it to the temperature acquisition module through the signal cable.The module automatically identified the temperature signal and then passed it to the computer via a communication module, and used LabView programming to display the result.

Fig. 3 .
Fig.3.From Fig.3, Temperature sensor T1 was fixed on the outer surface of the spindle sleeve and was used to indicate changes in the spindle temperature caused by the heating of the bearing and the motor.Temperature sensors T2 and T3 were influenced by the heating of the motor and bearings, and were utilized to measure the temperature changes of the headstock.Temperature sensors T4 and T5 were placed on the two sides of the column respectively, representing the tilt deformation of the column.

Fig. 3 .
Fig.3.Distribution of temperature sensors for the spindle system

Fig. 10 .
Fig. 10.Contrast of predicted results for different structural cases

Fig. 11 .
Fig. 11.Statistic results of predicted errors stemmed from the structural size deviations

Fig. 16 .
Fig. 16.Experimental results of axial thermal error compensation at variable speed

Table 2 .
Results of error model parameters identification at 6000rpm c 