机床进给驱动电流测量倾向和数学模型的估计.pdf

International Journal of Machine Tools received in revised form 17 October 2005 accepted 21 October 2005 Available online 5 December 2005 Abstract Leveling is an important part of a machine tool installation process because it signifi cantly infl uences the product quality machine tool accuracy and machine lifetime Conventional leveling procedures are performed by skilled engineers using leveling instruments such as spirit or electronic levels It is diffi cult to monitor the level of a machine tool because an accurate leveling instrument is expensive and diffi cult to install Therefore a novel methodology for estimating the inclination angle of a machine tool feed drive is proposed in this paper to overcome the diffi culties of leveling The proposed methodology utilizes motor current measurements and a new mathematical model of the machine tool feed drive that considers the inclination Experimental results showed that the proposed method successfully estimates the inclination angle and enhances the accuracy of the machine tool feed drive model by considering the inclination effects r 2005 Elsevier Ltd All rights reserved Keywords Current measurement Machine tool feed drive Machine tool leveling Dynamic simulation 1 Introduction When leveling a machine tool its inclination is adjusted so that the feed drive is positioned at a planned orientation 1 Imperfect leveling results in structural deformation of the machine tool leading to premature wear of the moving parts and poor product and machine tool accuracy 2 3 As machining industries require improved levels of accuracy as well as high speed capacity machine tool leveling will become more important to achieve the maximum perfor mance of the machine tool The leveling procedure during installation and scheduled maintenance periods is a diffi cult task Leveling is one of the most time consuming processes required for a machine tool installation It requires repetitive work during the adjustment phase and the measurements must be repeated at various table positions to check the twist and inclination of the guideway Furthermore current industry leveling practices depend largely on the skills of experts because the method requires a great deal of empirical knowledge Therefore it has been diffi cult to automate the leveling procedure In addition because an accurate leveling instrument is expensive and diffi cult to install on an operating machine tool regular monitoring of the leveling problem is diffi cult We present a novel methodology for estimating the inclination angle of a machine tool feed drive in this paper The methodology utilizes measurements of the feed motor current and a mathematical model of the machine tool feed drive The feed drive inclination infl uences the motor torque because a gravitational force corresponding to the table weight is generated in the feed direction by the inclination Thus the amount of inclination can be estimated using the measured current and a mathematical feed drive model that considers the inclination The proposed method isolates the infl uence of the inclination from other mechanical effects such as guideway friction ARTICLE IN PRESS 0890 6955 see front matter r 2005 Elsevier Ltd All rights reserved doi 10 1016 j ijmachtools 2005 10 015 Corresponding author Tel 221235813 fax 23122159 E mail address bkmin yonsei ac kr B K Min A substantial amount of research on inclination mea surement methods and types of apparatus permissible leveling errors of a machine tool and various test methods has been conducted by Schlesinger 1 According to Schlesinger 0 00231 is the maximum permissible error in the leveling of horizontal and vertical milling machines More recent research results of machine tool leveling are diffi cult to fi nd In the fi eld of measuring machine tool inclinations some research on the geometric error of machine tools can be found in Refs 4 5 However most studies have dealt with the relative inclination between machine tool parts rather than the absolute inclination which is related to earth level Many researchers have used the feed motor current as a measure of a disturbance force such as a cutting force Altintas 6 estimated the cutting force using the feed motor current to detect tool breakage while Kim and Chu 7 investigated the frictional behavior of a machine tool using the feed motor current Extensive research works have elaborated the modeling of machine tool feed drives Younkin 8 presented a dynamic model of a feed drive system possessing backlash and stiction and Ebrahimi and Whalley 9 simulated the effect of para meters related to the system stiffness on the transient response of a feed drive system In the next section of this paper we present a dynamic mathematical model of a machine tool feed drive and derive the infl uence of the feed drive inclination on the feed motor current In Section 3 experiment results are used to verify the proposed method The fi ndings of this study are summarized in Section 4 2 Mathematical model of a feed drive In this section we introduce models of the feed drive components These include a servo motor coupling ball bearing leadscrew guideway and table In particular the table model was expanded to consider the inclination of the machine tool which results in an inclination of the table Consequently a mathematical model of a machine tool feed drive was constructed by combining the component models The inclination model which defi nes the relation ship between the feed motor current behavior and the feed drive inclination angle was also derived from a combina tion of the component models 2 1 Modeling of the feed drive components 2 1 1 Servo motor model Recently most feed drive actuators of machining centers are AC servo motors Because an AC servo motor model is complex the motor is frequently modeled as an equivalent DC motor using vector transformation or root mean squares A well known DC motor equation is Tm Kt i 1 Jm dom dt Tl Tm 2 where the variables Tm i om and Tlare the generated motor toque motor current angular velocity of the motor shaft and torque required to move the linear stage respectively The linear stage in our study consisted of the mechanical parts of the feed drive except for the servo motor A linear stage typically consists of a coupling ball bearing leadscrew guideway and table The parameters Kt and Jmare the torque constant and moment of inertia of the motor respectively 2 1 2 Coupling model In machining centers the motor shaft is typically connected to a leadscrew shaft by a mechanical coupling In our model the coupling was assumed to be suffi ciently stiff so that Tl Jc dom dt Tls 3 where Jcand Tlsare the moment of inertia of the coupling and the torque required to rotate the ball bearing lead screw respectively 2 1 3 Ball bearing leadscrew model The preload and friction must be considered to model a leadscrewprecisely Theball bearingleadscrewina machine tool feed drive is generally preloaded to increase the positioning accuracy and rigidity and to decrease the thermal deformation 10 The torque caused by the preload is similar to a frictional torque and the behavior can be modeled as 3 Tp sgn om 0 05 tan b 0 5 Fa0 P 2p 4 where Tp b Fa0 and P are the torque caused by the preload lead angle preload force and pitch of the ball bearing leadscrew respectively The mechanical effi ciency is used to model the friction of the ball bearing leadscrew because most leadscrew manu facturers provide this value instead of the frictional coeffi cient The mechanical effi ciency is defi ned as 11 Z l p dm m l p dm l p m dm 1 m tan b 1 m cot b 5 where the parameters dm m and Z are the mean diameter frictional coeffi cient and mechanical effi ciency of the ball bearing leadscrew respectively In this study only the Coulomb friction of the leadscrew was considered the viscous friction and Stribeck effect 12 were neglected Therefore the mechanical effi ciency was modeled as a constant value When an axial force Ft is applied to a preloaded ball bearing leadscrew the torque required to rotate the leadscrew shaft is given by Tls Jls dom dt Tp P 2 p Z Ft 6 ARTICLE IN PRESS Y Hun Jeong et al International Journal of Machine Tools Hyundai andavertical machining center ACE V30 Daewoo were used to demonstrate the proposed approach by comparing the current difference defi ned in the previous section with the measured machine angle Since the weight of the CNC tapping center was small it was convenient to use to verify the model with respect to various inclination values by ARTICLE IN PRESS Ft vt M Mg Guideway Ffriction Ground Fexternal Table Fig 1 Simplifi ed model of the inclined table and guideway Y Hun Jeong et al International Journal of Machine Tools Taylor Hobson Ltd was 0 2s 0 0000561 and the analog reading of Level 2 Niveltronic TESA had a grid scale of 2s 0 000561 Levels 1 and 2 were attached to the machine tool table to measure the pitch and roll angles of the inclination respectively Fig 2 shows the arrangement of the levels on the machine table during the tests 3 2 Verifi cation of the inclination model The y axis of the tapping center was used to verify the linear relationship Eq 12 between the feed drive inclination and the current difference The pitch angle was changed by adjusting the six leveling bolts in the bottom plane of the machine tool while the roll angle was maintained close to zero The inclination was changed from 600 to 600s at intervals of 150s Fig 3 displays the current differences with respect to various inclination angles of the y axis of the CNC tapping center The current difference was proportional to the inclination angle of the feed drive A small offset at zero inclination was observed which may have been caused by the current measurement noise outputoffsetsoftheHallsensorandlevel directional dependency of Coulomb friction or local waviness of the machine tool table However the amount of the offset was suffi ciently small to be negligible The maximum estimation error was 0 01481 Consequently the proportional relationship between current difference and inclination angle given by Eq 12 was verifi ed In addition this test also demonstrated that the sensitivity of themachinetoolwas6 571 10 5A s whichis equivalent to 13 5791A rad 3 3 Mechanical effi ciency of the ball bearing leadscrew According to the proposed model the proportional coeffi cient of the relationship between the current differ ence and inclination angle of the feed drive is affected by four parameters the motor torque constant table mass leadscrew pitch and mechanical effi ciency Once these parameters have been identifi ed Eq 12 can be easily used to estimate the feed drive inclination angle of an actual machine tool Typically the leadscrew pitch motor torque constant and table mass can be determined from the component datasheets and design information However the mechanical effi ciency of the ball bearing leadscrew is diffi cult to identify because of the combined frictional characteristics of the leadscrew and guideway ARTICLE IN PRESS Fig 2 Arrangement of the levels on the table 0 05 0 04 0 03 0 02 0 01 0 0 01 0 02 0 03 0 04 0 05 750 600 450 300 1500150300450600750 Inclination Angle sec Current Difference A y 6 571E 5 x SD 1 450E Fig 3 Measured current differences with respect to various feed drive inclination angles Y Hun Jeong et al International Journal of Machine Tools the resulting mechanical effi ciency of the ball bearing leadscrew was 0 5468 When applying the second method based on the static ratio of the feed motor current to the external force the relationship between the external force and motor current under constant velocity no friction and no preload conditions was derived from Eqs 1 3 6 and 7 i P 2pKtZ Fext 13 Eq 13 gives the relationship between the static external force and motor current since the characteristics of the servo drive were not considered In a previous study by the authors 16 using the same CNC machining center described in Section 3 1 the frequency response of the motor current with respect to the cutting force in an x axis feed drive was obtained experimentally This provided ratios of the motor current to the external force at various frequencies The value of the term P 2pKtZ in Eq 13 can be determined from the frequency response at exactly 0Hz However because it is diffi cult to obtain the frequency response at 0Hz in an actual machine tool feed drive the static ratio was assumed to be similar to the frequency response close to 0Hz In this study the static ratio 0 0029A N was obtained at a frequency response of 10Hz As a result the mechanical effi ciency calculated from Eq 13 was 0 5434 The mechanical effi ciency value was checked using the transfer function between the current and the external force obtained using the mathematical model introduced in Section 2 and the motor drive information provided by the CNC manufacturer The effi ciency determined using the transfer function at 10Hz was 0 565 which is close to the experimental result 3 4 Estimation of the inclination angle A test to estimate the inclination angle of a CNC machiningcenterinstalledinamachineshopwas conducted using the proposed method The measured current difference of the machine tool x axis was 0 234A TheparametersofEq 12 exceptthemechanical effi ciency were predetermined from the component data sheets and CAD drawings Fig 4 a and b show CAD models of the saddle and assembled machine tool The mechanical effi ciency was determined using the second method explained in the previous section The parameter values and sensitivity for estimating the inclination angle are summarized in Table 2 The calculated inclination angle was 0 2251 To verify the proposed estimation method the actual inclination angle of the feed drive was measured using Level 1 The measured inclination angle was 0 2321 which was within 3 0 of the estimated value Furthermore the mechanical effi ciency of the ball bearing leadscrew identi fi ed using the actual inclination angle and the measured current difference was 0 56 Therefore the proposed technique used to determine the mechanical effi ciency of the ball bearing leadscrew from the static ratio of the current to the external force is appropriate ARTICLE IN PRESS Table 1 Parameter values related to the mechanical effi ciency of the ball bearing leadscrew of the CNC tapping center ParameterValue unit Pitch P 0 01 m Torque constant Kt 1 2627 Nm A Mass of table Mt 300 0 kg Sensitivity PMtg KtpZ 13 5537 A rad Fig 4 CAD models of the a saddle and b assembled CNC machining center Table 2 Parameter values used to estimate the inclination angle of the x axis feed drive of the CNC machining center ParameterValue unit Current difference ji j ji j 0 234 A Pitch P 0 01 m Torque constant Kt 1 01 Nm A Mass of table Mt 1050 0 kg Mechanical effi ciency Z 0 5434 Sensitivity PMtg KtpZ 59 6794 A rad Y Hun Jeong et al International Journal of Machine Tools the difference between the simulated and the measured current was less than 4 0 The current differences calculated from the simulated and actual current measurement are shown in Fig 6 b The current difference maintained the same value for the whole velocity range except in the low velocity region The fl uctuation of the current difference at low speeds was caused by the instability of the friction in the region of boundary lubrication over which the friction force was dominated by the surface conditions 4 Conclusions In this study we proposed a novel indirect method to estimate the inclination angle of a machine tool feed drive The proposed model based approach detected the differ ence in the feed motor current of a machine tool feed drive operating in the backward and forward directions which resulted from differences in the relative direction of gravity during motion when the feed drive was inclined A precise mathematical model of a machine tool feed drive that considers the effect of gravity was derived and used to estimate the inclination angle The proposed method was evaluated with experiments using two real machine tools Comparisons between the current difference and the machine tool inclination angle measured by an electronic level confi rmed that the current difference was proportional to the inclination angle of the feed drive When the proposed method was applied to measure the inclination angle of a CNC machining center x axis feed drive with an inclination angle of 0 2321 the estimated inclination angle was 0 2251 representing an error of less than 3 0 ARTICLE IN PRESS Fig 5 Simulation model of a machine tool feed drive developed in the Dymola environment Table 3 Simulation parameters of the x axis feed drive of the CNC machining center ParameterValue unit Torque constant Kt 1 01 Nm A Moment of inertia of the motor Jm 0 0037 kgm2 Moment of inertia of the coupling and ball bearing leadscrew Jc Jls 0 003 kgm2 Lead angle of the ball bearing leadscrew b 41210 Preload of the ball bearing leadscrew Fa0 2450 N Pitch of the ball bearing leadscrew P 0 01 m Mechanical effi ciency of the ball bearing leadscrew Z 0 5434 Mass of the table Mt 1050 kg Inclination angle y 0 225 deg Coeffi cient of Coulomb friction mCoulomb 0 0729 Coeffi cient of viscous friction mviscous 1 2060 s m Coeffi cient of Stribeck effect friction mStribeck 0 0237 Characteristic velocity of the Stribeck effect s 444 6 s m Y Hun Jeong et al International Journal of Machine Tools Manufacture 46 2006 1343 13491348 The experiment result demonstrated that the current difference was unaffected by the table velocity Further more the current differences could be predicted with less than a 4 0 error from simulations using the proposed feed drive model In addition the mechanical effi ciency of the ball bearing leadscrew a factor of the relationship between the current difference and feed drive inclination angle could be accurately identifi ed by two methods based on experiment results of the relationship between the current d