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Prediction of Thermal Deformationfor a Ball Screw System Under CompositeOperating ConditionsA. S. Yang, S. Z. Chai, H. H. Hsu, T. C. Kuo, W. T. Wu, W. H. Hsiehand Y. C. HwangAbstract The position error of a feed drive system is mostly caused by thermaldeformation of a ball screw shaft. A high-speed ball screw system can generatemassive heat with greater thermal expansion produced, and consequently have anegative effect on the positioning accuracy. In this study, we applied the com-putational approach using the finite element method (FEM) to simulate the thermalexpansion process for estimating the deformation of the ball screw system. In thenumerical analysis, the deformation of the ball screw shaft and nut was modeledvia a linear elasticity approach along with the assumption that the material waselastic, homogeneous, and isotropic. To emulate the reciprocating movements ofthe nut at the speeds of 20, 40 and 60 m/min corresponding to the screw shaft, wealso employed a three-dimensional unsteady heat conduction equation with theheat generation from the main sources including the ball screw shaft, nut andbearings as the heat transfer model to solve the temperature distributions fordetermining the temperature rises and axial thermal deformations in a ball screwA. S. Yang C1 S. Z. Chai C1 H. H. HsuDepartment of Energy and Refrigerating Air-Conditioning Engineering, National TaipeiUniversity of Technology, Taipei 106, Taiwane-mail: .twT. C. Kuo C1 W. H. Hsieh (&)Department of Mechanical Engineering, and Advanced Institute of Manufacturing withHigh-tech Innovations, National Chung-Cheng University, Chiayi 621, Taiwane-mail: .twW. T. WuDepartment of Biomechatronics Engineering Nation Pingtung University of Scienceand Technology, Pingtung 912, Taiwane-mail: Y. C. HwangHIWIN Technologies Corp, Taichung 408, Taiwane-mail: .twG.-C. Yang et al. (eds.), Transactions on Engineering Technologies,DOI: 10.1007/978-94-017-8832-8_2,C211 Springer Science+Business Media Dordrecht 201417shaft under composite operating conditions. The simulated results demonstratedthat the countermeasures must be taken to thermally compensate great deteriora-tion of the positioning accuracy due to vast heat production at high rotating speedsof shaft for a ball screw system.Keywords Ball screwsC1FEMC1Heat transfer modelC1Machine toolC1PositioningaccuracyC1Thermal deformation1 IntroductionThe performance of a ball screw feed drive system in terms of speed, positioningaccuracy and machine efficiency plays a very important role in product quality andyield in manufacturing industries primarily including machine tools, semi-con-ductors, optoelectronics, and so on. Considering a high-speed precision ball screwsystem, the occurrence of contact surfaces (such as the interfaces between the balland nut grooves, the ball and screw grooves, and the bearing and shaft) producescontact friction at these junctions. The friction of the nut and ball bearings entails asudden and violent heating of balls, and in turn results in the temperature rises ofthe ball screw, leading to mechanical micro- deformations and an overheating ofthe coolant. Such a temperature heating of ball screw could also cause significantthermal deformations deteriorating the ball screw system accuracy in mechatronicstools or instruments 1.The development of fabrication technology for a variety of applicationsnecessitates high-precision apparatuses for achieving remarkably delicate goodswith high output 2. As indicated by Bryan 3, the thermal induced error inprecision parts has still been the key setback in the industry. Substantial effortswere done on the machine tools, thermal behavior and thermal error compensationon the spindles, bearings and ball screws, respectively. Ramesh et al. 4 and Chen5 carried out the air-cutting experiments to reproduce the loads under actual-cutting situations. To adjust the thermal conditions of the machine tool, Li et al. 6conducted the tests of varying spindle speed for controlling the loads. The per-formance of a twin-spindle rotary center was experimentally evaluated by Lo et al.7 for particular operating settings. Afterward, Xu et al. 8 incorporated thecontact resistance effect into a thermal model for simulation of machine toolbearings. Koda et al. 9 produced an automatic ball screw thermal error com-pensation system for enhancement of position accuracy.The frictional process from a high-speed ball screw system essentially releasedtremendous amounts of heat and results in the continuing temperature increase andthermal expansion, leading to deterioration of the positioning accuracy. In thisinvestigation, we considered the heat generation from two bearings and the nut asthe thermal loads with the prescribed heat flux values imposed on the inner sur-faces of grooves between the bearings and nut of the ball screw system.18 A. S. Yang et al.The convection boundary conditions were also treated for solid surfaces exposedto the ambient air. A FEM-based thermal model was developed to resolve thetemperature rise distribution and in turn to predict the thermal deformation of theball screw. In addition, simulations were conducted to appraise the influence ofcomposite operating conditions in terms of different speeds (1,000, 2,000 and3,000 rpm) as well as moving spans (500 and 900 mm) of the nut on the tem-perature increases and thermal deformations of a ball screw shaft.2 Description of Ball Screw SystemFigure 1 presents a schematic diagram of a ball screw feed drive system,encompassing a ball screw and driving unit. A continuous advance and returnmovement of the ball screw takes place in the range of 900 mm. It has 20-mmlead, 41.4-mm ball center diameter (BCD) and 1,715-mm total length. The outerand inner diameters of the screw shaft are 40 and 12.7 mm, respectively. Table 1presents the parameters of main components for the ball screw drive system, whichreally contains the ball screw shaft, ball screw nut and bearings.Figure 2 illustrates the moving velocity of the screw nut. This investigationconsiders the reciprocating movements of the nut at a maximum speed of 40 m/minpertaining to the screw shaft with a time period of 3.43 s and acceleration/deceleration of 2.1 m/s2as the baseline study case.3 Computational AnalysisThe physical model in this study investigates the thermal expansion process in a ballscrew system. Essentially, heat is generated mainly from the friction between theball and nut grooves as well as the ball and screw grooves. In view of the fact that astring of balls filled the grooves between the screw and nut are rotating very fast, theheat has been distributed evenly over the inner surface of raceways. The nut and twobearings are modeled as the fixed thermal loads imposed on the ball screw shaft. Thethermal resistance resulting from the lubrication oil film between the balls andraceways is assumed to be ignored here attributable to a very thin layer of oil film,and the effect of heat conduction by means of the lubricant and thermal deteriorationis negligible. Numerical calculations were performed by the FEM softwareANSYSC210to investigate the thermal behavior of a ball screw 10. The theoreticalformulation was based upon the time-dependent three-dimensional heat conductionequation for a ball screw system. The governing equations are stated as follows:ko2Tox2o2Toy2o2Toz2C18C19 qcoTot: 1Prediction of Thermal Deformation for a Ball Screw System 19The symbols q, c, k and T mean the density, specific heat, thermal conductivity,and temperature of the ball screw shaft and nut, respectively. Here the temperatureT is a function of the spatial coordinates (x, y, z) and time. The q, c, k values forcomputations are 7,750 kg/m3, 480 J/kg-oC and 15.1 W/m-oC. Figure 3 exhibitsthe heat generation by the nut and bearing for a period of 3.43 s. The friction effectFig. 1 Schematic of a ballscrew feed drive systemTable 1 Main component parameters of the ball screw feed drive systemBall screw shaft Ball screw nutTotal Length (mm) 1,715 Type FDCThread Length (mm) 1,295 Length (mm) 143.4LEAD (mm) 20 Diameter (mm) 70BCD (mm) 41.4Outer diameter(mm) 40Inner diameter (mm) 12.7 BearingLine number 2 Type TACContact type 4 points OD (mm) 30Ball diameter (mm) 6.35 ID (mm) 12.7Fig. 2 Moving velocity ofthe screw nut with respect tothe screw shaft20 A. S. Yang et al.between the balls and raceways of the nut and bearings is the most important causefor temperature increase.Given that the load of the nut contains two parts: the preload and dynamic load,_Gnut, the heat generation by the nut (in W), can be described as 11, 12:_Gnut 0:12pf0v0nM: 2Here f0is a factor determined by the nut type and lubrication method; v0is thekinematic viscosity of the lubricant (in m2/s); n is the screw rotating speed (inrpm); M is the total frictional torque of the nut (in N-mm). In this research,_Gbearingis the heat generated by a bearing (in W), defined as below 13._Gbearing 1:047 C2 10C04nM: 3The variables n is the rotating speed of a bearing and M is the total frictionaltorque of bearings, including the frictional torque due to the applied load and thefrictional torque due to lubricant viscosity.The convective heat transfer coefficient h (in W/m2-oC) is computed 14byh Nukfluid=d: 4Here Nusselt number Nu = 0.133Re2/3Pr1/3, while the variables Re and Prrepresent Reynolds number and the Prandtl number. The sign kfluidis the thermalconductivity of the surrounding air and d is the outer or inner diameter of the screwshaft (mm). More detailed information can be found in Ref. 15.Fig. 3 Heat generation by a nut and b bearing for a period of 3.43 sPrediction of Thermal Deformation for a Ball Screw System 21In this study, the ball screw is modeled using a linear elasticity approach andassumed as the elastic, homogeneous, and isotropic material. The governingequations for the ball screw deformation are as follows:qo2vxot2orxoxosyxoyoszxoz qgx: 5qo2vyot2oryoyosxyoxoszyoz qgy: 6qo2vzot2orzozosxzoxosyzoy qgz: 7The variables v, r and e symbolize the displacement, stress and strain vectors(tensors).r De C0 eth8The symbol ethis the thermal strain. Given the assumption of a linear elasticresponse, the stress-strain relationship is given by r De, where D has the formD k 2G kk000kk 2G k 000 2G 000000G 000G 0000 0G0BBBBBB1CCCCCCA9k vE1 v1 C0 2v; G E21 v10eth DTa; a; a; 0; 0; 0;T11D is the elasticity matrix consisting of the material properties, whereas theproperties of E, m and a are the Youngs modulus, the Poissons ratio and thecoefficient of thermal expansion (CTE). In this investigation, the values ofYoungs modulus, Poissons ratio and CTE of the ball screw are set to be1.93 9 1011Pa, 0.31 and 1.16 9 10-5m/m-oC. The term DT = T - Tref, Tref,isthe initial temperature of 27 C176C. A finite element method was used to solve the ballscrew model in accordance with the principal of virtual work. For each element,displacements were defined at the nodes and the associated displacements withinthe elements were subsequently obtained by means of interpolation of the nodalvalues by the shape functions. The strain-displacement and stress-strain equationsfor structure were solved with the Gaussian elimination method for sparsematrices 10.22 A. S. Yang et al.4 Experimental MeasurementsExperimental measurements were conducted to determine the time-dependentdistributions of temperature, temperature rise and thermal deformation for vali-dation of the thermal model by FEM and evaluate the performance of the coolingsystem. Figure 4 illustrates a schematic diagram of the experimental set up,containing the ball screw, driving unit, LNC controller, thermal/laser detectionsystem for measuring temperature/deformation and linked data acquisition system.A continual forward and backward movement of ball screw ensued in a 900-mmrange, having 20 mm lead, 41.4 mm BCD and 1,715 mm total length. The dataprocessing module consisted of four thermal couples as well as a high-precisionRenishaw XL-80 laser system with the test data recorded for every 600 s.5 Results and DiscussionSimulations were attained by the FEM software ANSYSC210to predict the thermaland deformation characteristics of a ball screw. In the analysis, we modeled astring of balls as a coil-like circular band fully filled the interior surface of race-ways between the screw and nut. Figure 5 displays the numerical grids for systemsimulations. The mesh setup had three unstructured sections, such as the bearings,shaft and coil-like circular band. Finer grids were paced in the areas near thegrooves and the solid surfaces. The average cell length was approximately0.0015 m with the least spacing of 0.0006 m to resolve the steep variations ofthermal properties near heat sources. The predictions of transient temperaturedistributions on the surface of the ball screw with different grids and time stepssuggested that grid independence could be attained using a mesh setup of1,367,867 grids with a time step of 5 s.In simulations, the reciprocating movements of the nut was at a speed of40 m/min respecting to the screw shaft with a 3.43-s period. The analysis wasbased on the rotational speed of 3,000 rpm and the initial temperature of 27 C176C.During the continual operations, Fig. 6 illustrates the temperature and thermaldeformation distributions along the axial distance of hollow ball screw shafts at t= 3,600 s. The frictional heat produces the temperature rise with high temperaturesoccurred in the core areas of the ball screws for hollow ball screws.In the structure analysis, the left side of the ball screw was treated as the freeend with the right side maintained as the fixed-end condition. The thermal loadsfrom the temperature distribution predictions were then input to solve the thermaldeformation. The predicted results indicated relatively larger local thermaldeformations near the high temperature regions occurred at the center of the shaftin general. The thermal deformation distributions showed a great reduction inthermal expansion for the hollow shaft at the left end of the screw.Prediction of Thermal Deformation for a Ball Screw System 23Fig. 4 Schematic diagram of the experimental set upFig. 5 Numerical grids of ball screw shaftFig. 6 Temperature and thermal deformation distribution of ball screw shafts at t = 3,600 s24 A. S. Yang et al.Figure 7 illustrates a comparison of the prediction with the measured temper-ature rise data along the axial axis of hollow ball screw at t = 3,600 s. Thecalculated temperature rise distribution was compared with the measurements ofdifferent positions. Overall, the results clearly indicated that the FEM model rel-atively over-predicted the temperature increase (i.e. 17.7 C176C in the central area ofthe shaft) with the discrepancy within approximately 18.7 % in the axial distanceof 0.15 to 0.4 m, showing that the simulation software may reasonably simulatethe thermal expansion phenomenon.As far as the transient developments of temperature rises are concerned, Fig. 8illustrates a comparison of time history of the predictions with measured tem-perature rises at three locations for the hollow ball screw. At the pre-specified axialdistances of 0.05, 0.68 and 1.62 m, all three temperature rise profiles grew quicklyin the early stage and tended to level off toward their steady-state values of 2.73,17.2 and 2.66 C176C for the test points of 1, 2 and 3, respectively, at t = 3,600 s. Dueto massive heat generated in continuing operations, both the predicted measuredresults of the test point 2 show a higher temperature rise with the difference under1.93 %, revealing the accuracy of the FEM simulations.In order to verify the thermal and structural models by FEM, Fig. 9 illustrates acomparison of the prediction with measured thermal deformation data along theaxial distance of hollow ball screw at t = 3,600 s. The thermal deformation alongthe axial direction was measured using a laser interferometer, for comparison withthe calculated results. It can be clearly seen that the deformations from the FEMpredictions and the experimental results were fairly close, deviating below 9.1 %for the axial distance from 0.2 to 0.9 m. Nevertheless, as compared to the test data,the over-predicted deformation was noted with a large error appeared at the axialdistance of 0.1 m owing to the over-estimation of temperature rise in this asso-ciated area.Fig. 7 Comparison ofprediction with measuredtemperature rise data alongthe axial distance of ballscrew at t = 3,600 sPrediction of Thermal Deformation for a Ball Screw System 25The comparisons of predictions with the experimental data indicated that theFEM method could reasonably predict the thermal expansion process for deter-mining the deformation of a ball screw system. Thus, calculations were extendedto consider the influences of different spindle speeds and traveling distance of thenut on the temperature rises and thermal deformations of a ball screw shaft.Table 2 illustrates the numerical test details in terms of the rotating speed and#3 #2 #1Fig. 8 Comparison of thetime history of the predictionswith measured temperaturerises at three locations for ballscrewFig. 9 Comparison of theprediction with measuredthermal deformation dataalong the axial distance ofball screw at t = 3,600 s26 A. S. Yang et al.moving extent of the ball screw under the composite operating conditions. Insimulations, the shaft starts the operation at a speed of 2,000 rpm with a 900-mmmoving distance of the nut for 300 s and next with a moving distance of 500 mmfor 300 s, then changes to the speed of 3,000 rpm with a 900-mm moving distancefor 300 s, and decreases the speed to 1,000 rpm with a 500-mm moving distance inthe last 300 s. The axial spans, defined as the moving distance of the nut, are

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