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Assessment of Force-Induced Errors in CNC Turning
Assessment of Force-Induced Errors in CNC Turning S Hinduja (l), D Mladenov and M Burdekin Department of Mechanical Engineering, UMIST, Manchester, UK
Abstract This paper describes a procedure to evaluate the force-induced errors occurring in cyiindriml turned mrrponents. This procedure is based on a model which represents the realtime stiffness of the spindlebearing system and the rotational damping stiffness of the mrrponent held in a chuck. The stiffness values are determined from a single cutting test in which the defledion of a test bar is measured. The model also the takes into acmunt the stiffness of the mrrponent and toolpost. The model is validated by -ring predided error in the cyiindricityof a machined bar with the measured value.
kywwds: CNC turning. force-induced errors. spindle-bearing sydern
1 INTRODUCTION
2
Preasion manufacture relies to a significant extent on the wallty and accuracy of a machine tool. The geometric errors in the machine tool are based on 'mlaad' instrumentabbased tests origindly f m l a t e d by into national and Schlesinger and later in-ated international standards. Other irqmtant fadors which affed the accuracy of a machined m p e n t are forceinduced errors in the machine tml and workpwce. thermal errors due to the heat generated in the machine tool and by the cutting process. and tool wear.
2.1 Model
Several researchers have studied. charaderised and mrrpensated for one or m e of the above errors using various strategies. Donmez et al. [ I ] developed a mrrprehensive model to predict the g m t r i c and thermally induced errors in a machine tool and they mrrpensated for these in real time. On the other hand, Sata et al. [2] focused on the error caused by the thermal gowth of the tool which they were able to represent using an expnential function. Yang et al. [3] and Ni et al. [4]. mncentrated on m d e l l i r q the force-induced errors due to the spindle bearing system, tool post and mrrponent. the flexibilities of which were determined qmrimentally. The strategy is to mrrpensate for the above errors either off-line by mdlrylng the CNC tool path or in real time which usually requires on4me mitoring of the process parameters (cutting force. terrperatures etc.) [3-61. The latter is not usually possible Ibr all machines an the shop floor. More recently, Liu and Venwinod suggested a s h o p k friendly off-line strategy for mrrpensating for the forceinduced errors. However. to mrrpensate for a part. their strategy requires information relating to a simlar previously mchined part to exist in the database. Instead of mnsidering one or m e of the above errors. some researchers such as Asao et al. [8]. charaderised the total machining error by an expnential fundion. the mnstants for which were determined for various tools and cutting mnditions. But the disadvantage of this approach is that it is machinespecific and offers m nsght into the magnitudes of the various error mrrponents. This paper focuses on the force-hduced errors and it desaibes the development of a spring model to represent the spindlebearingchuck system of a lathe. This model also takes into acmunt the clamping stiffness d a mrrponent held in a chuck. The error predided by this spring model is then synthesised with the forceinduced errors due to the toolpost and mrrponent.
SPINDL~BEARINGCHUCKSYSTEM
The cutting loop mnsists of the spindle-bearing system, chuck, lathe bed, cutting tool. tml post and the workpiece. Whilst an accurate representation of fie system can be obtained by using finite elements [9]. the resulting model w w l d be tm m p l e x and the analysis timeoonsumng. Since the main p u p e of the model is to faalitate the determination of the defledion of the mrrponent at the cutting tool position. it was decided to use a mmbination of linear and rotational springs to represent the system. Two irrportant mnsiderations limed the eventual m p l e l o t y of the model. The first of these was that it should be possible to calculate the spnng m t a n t s of the model from a single udlirq test. The second mnsideration was that measuring the defledion of the rotating mrrponent diredly should be avoided because fadors such as eccentricity and surface roughness will affect the accuracy of the measwed data.
-
Defledion measurement
Preasion bar
Figure 1: Cutting loop In addition to the above. the real-time stiffness of the system should be measured rather than the static as has been done by most researchers [5. 91. Hence, as shown in Figure 1. it was decided that the measuring positims should be lomted on a bar outside the cutting loop. The portron of the system to the nght of the cutting tool. induding the bar. will not bend. This is bemuse the defledion curve for this part of the system is a straight Cne and tangential to the deRedion m e Ibr the left portron of the cutting loop. Sinm a straight line is determined by two points. the maxirmm number of measuring positions is two.Wth two defledion values, the r m s t mmplex model mnsists of one linear and two tdational springs [Figure 2). The first two springs (K1and Kz) represent the Overall bending and rotational stiffness of the spindle-bearh-huck system. The third spring ( K 3 ) represents the darrplng rotational bending stiffness of the m p e n t held in the muck. The two rotational springs are m n e d e d by a rigid link, m e length (Lc) is measured from the back face of the chuck to the centre ofthe damped portion of the mrrponent in the chudc.
hydraulic chudr
L&
-
tiers
Figure 3: Schematic representation of the experimental rig F r m the model shown in Figure 2, let B represent the defledion measured by either of the two transducers Wich are placed at distances L1 and L Zfrom the spring K3 respedively. If L is equal to either L1 or L2 and a i s the distance from the centre of the d a m portron of the mrrponent to the mtting tool position. then SL = S K ,
+SK2
+SK3
F +F.(L+f&+L,)+F.La
--
Kl = A+B.a
K2
K3
(11
where A and are m s t a n t s and F is the tangential or radial mrrponent of the cutting force. The rotation k)of the system to the nght of the cutting tool position can be calculated f r m t h e two LVDT readings. 8 L =-
62 -61
2.3 Experlmental detemlnatlon of Kl, IG and IG
As mentioned earlier. a single cut is sufficient to determine the sprhg m s t a n t s . In this cutting test, a cylindrical bar made of low-mrbm steel (Ens) was machined using a depth of cut (d) of 0.5 mn and a feedrate (s) of 0.079 lTmlrev (0.0031 inlrev). A SECO M e r of type PCLNR2020K12 with an insert of either type CNMG 120404MF2TP200 (i.e. nose radius 0.4 mn) or CNMG1204012MF2TP200 (nose radius 1.2 mn)was used. The diameter of the test wmkpmce was chosen to be large (89 mn in this case) so that its stiffness is an order of magnitude greater than other elements in the cutting loop. Moreover. the cylindrical shape enabled the stiffness of the m p e n t to be accurately calmlated A spindle speed of 1000 rpm was used. Since the intention was not to measure the forces during the above test, the following equations were used to calculate the three corrponents of the force.
M
F r m Figure 2.
The constants in e q s . (1) and (3) can be solved if & and & are measured for two different values of a i.e. two different positions at which the cutting force is applied. Once these oonstants are known. K1. KZ and k can be easity determined. 2.2 Experlmentalset-lp The experimental rig m s i s t e d of a test vmkpwce. setting device. displacement transducers, a sensor holder and a data acquisition system. The workpwce was a simple cylindrical bar with a setting device near its free end (Figure 3). The other end was held in a hydraulic chudr of a MHP slanLbed CNC lathe. The setting device had a blind hole and a precision extension bar of diameter 8 m and length 160 mm. was held inside it by eight m e w s . These saews. when adjusted. reduced the eccentriuty of the extension bar to as low as 1 pm (Figure 4) thus substantially reducing the noise in the displacement signals obtained during cutting tests. The other end of the extension bar rotated inside a sensor holder which was m n t e d in the tailstodc. This holder carried two sets of dsplacement transducers of the LVDT type. so that the defledion mld be measured not only in the vertical plane but also in the horizontal. although only one pair of sensors was used at any one time.
Figure 2: Model for spinde-bearing-chudrsystem
settinadevice
lJ
'
'
spindleaxis
Fgure 4: Adjusting the precision bar
FY -- 1901do 9 3 7 8 . ~ 08489 F -5636’’ 1175 3 0 5665
z-
The constants in the above equations were obtained expwimentally with the above tool and wmkpwce material using artting mnditions mrrespnding to finishing operations. The signal obtained from the first displacement transducer (LVDTI) is shown in Figure 5. Data was mptured at a frequency of 1 kHz resulting in 43500 data points for each channel for a 150 mlength of cut. A regression analysis ofthe data resulted in a linear variation as shown in Figure 5. The constants shown in the figure are in fact, the values ofA and in equation (2). W e t h a t in Figure 5. and in the subsequent figures, tool position = 0 mrresponds to a = 3 O m to a-nt for the distance from the centre d the damping position to the ends of the jaws and the small flange at the start of the m m p e n t . Similarly. a tool position o f 1 5 0 m mrresponds to a=180 m. The difference between the two signals was calculated (see eq. 3) and a subsequent regression anatysis of the difference also resulted in a linear variation for R. €k = 2.903E-5 + 6.137E-4.a =C+D.a From these f a r mnstants the values for the spnng mnstants were calculated as: 6 = 8 6 . 8 0 ~ 1 0Nlm; ~ IG = 1 . 7 3 ~ 1 0Nmlrad ~ and IG = 0 . 5 6 8 ~ 1 0NrWrad. ~ S m and Stone [IO] obtained a value ot 1x10’ ~ h n for the headstodc stiffness and this m n p a r e s favourably with the value of K1.
The above stiffness values were obtained with a damping length (CL) of 31 mn. To evaluate its influence. twoother lengths (43 and 37 mn) were investrgated. These values were arrived at by using an integer n u m e r of serrations an the jaws to damp the mrrponent. In all cases.the tdal darrplng pressure exerted by the hydraulic chuck was maintained at its maximm value of 66 kb. The results showed that K1 and Kt remained unaffeded but Ka inaeased from 0.568~10 Nmlrad at 31 mto 0.723~10 at 37 mm and 1.O7Ox1O6at 43 m. 2.4 Radlal dellectlon at the cuttlng edge In the X Z plane, b t h FX and FZ (Fig. 6) mntribute to the defledion of the mrrponent in the radial diredion. This defledion. at the p n t of cutting. is given by (see Fig. 2):
6, = 6K, + 6& +6&
L
m m
.-C
Tool position [mn] Figure 5: Variation ofdefledion with tool position
7) Wth the mrrponent rotating at a low speed, a LVDT s e n s a was w e d at a slow feedrate from diameter D1 to Dz. The resulting signal (Figure 7) has two distind parts. the first refers to and the semnd to D. Since the machined portion of the mrrponent (Dz) is sufficiently dose to the chuck. it will mntain only two errors i.e. & and b t . Portion D 1. of rrmrse. is assumed to be error free. If &tort=
(D1-D2)/2 - d
then &tort represents the error due to the spindle bearing system and the toolpost i.e. &tort=
6s+
Hence,t ,K
b t
=bdFx
Several tests with different depths of aR were mnduded and the mrrespnding values of &tort noted. Figure 7 shows the signal obtained with d = 0.5 mn. For exarrple. for this depth of cut. B t a was measured as 9.8 pm and & =5.4 Hence b5t= 4.4pm.Since Fxwas 165Nforthis depth of cut. the stiffness of the tmlpost was 38x1O6 Nlm.
v.
4 COMPONENT DEFLECTION The mrrponent deflection b m due to the radial and axial m q m e n t s of the cutting force was mlculated. w h r e possble. analyWAty or. in the mse of m m p e n t s with m e x geometry. by the finite element method. In the latter, the wodqnece was modelled with 8noded hexahedron elements. For the m q m e n t s investrgated. the defledion was less than 1-2 pn and therefore the error due to mrrponent flexibilw was not as significant as that due to the spindleharing systemor toolpost. 5
EXAMPLE
To verify the model developed herein. a cyiindriml bar of diameter 86 mn and length 150 mn was machined with a 0.5 m depth of aR. The profile was then measured on a CMM machine and this is shown in Figure 8 fromwhich it 141 K3
K3
3 TOOLPOST DEFLECTION The radial mrrponent of the cutting force. F i will also muse the toolpost to defled. This defledion wil not alter the profile of the mrrponent but will onty resdt in a mnstant inaease in the mrrponent diameter. A simple test was designed to evaluate the magnitude of this defledion. hrtA. short wdqnece was held in the chuck and p r m c h i n e d with a spring pass to ensure that it mntaned no f o r c e - M u d errors. Then the test metewas machined exceptforthe last5 rrm(Figure
Fx
f
Figure 6: Schematic representation in the X-Z plane
I I
.-
! I
-5000
0
-
I
-
1
-
r
-
I
-
I
-
r
-
moo imo ism m o 25000 am
1 35000
1
-20
60 80 100 120 140 160 Tool psition [mn] om = 25; od = 9.9; oa = 17.4; ob = 0.3; oc = 7 5
data points
Fgure 7: Determination of the t d p o s t stiffness is dear that the error in cyiindriaty is 50 p (i-e. twice m . the values shown in Figure 8 are radius values). Curve A is the delledion of the mrrponent at the cutting tool position due to the flexibihty of the spindle4maring-dwck system. This curve was calculated using q . 4 . Note that the starting value (approximately 9.8 pm) for this curve is given by the sum of b t and the value obtained for S, ( q . 4 ) when a = 0.Curve represents the deflection of the mrrponent at the cutting tool position due to its own flexibility. the maxirmm defledion (&) being 0.3 p m If one mnsidered only the force-kduced error. Wich is given by the sum of curves A and B. then, acmrding to the model. it w w l d be possible to reduce the cyiindriaty error by 2(0w&) i.e. 35.4 p.But in pradice this would not happen bemuse every lathe has a spindle msalignment error (a)which, in the case of the MHP lathe, was 8.2'. Since this angle is negative i.e. the axis is indined towards the tool post. it will reduce the cyikdriaty error. Curve C shows the linear variation of this error which i s given by L tan a. L being the length of the art. Hence, for the testplece (L = 150 mn).the redudion in the cyiindnuty error is twice oc i.e. 15 p.This means that be net predided error in cyiindriaty i s only 20.4 pm as -red to the measured value of 50 prn Further investigations revealed that the thermal growth of the t d muld be sgnificant. To substantiate this, a groove (3 mn wide and 3 m deep) was machined near the chuck end of the cyiindriml bar. This bar was machined again with the same cutting m i t i o n s as shown in Fgure 8. When the tool reached the centre of the groove. the feed was stopped and the tool allowed to mol down to ambient mnditions. The tool then mntinued to machine the small p b o n of the test bar between the groove and the chuck. The difference in the diameters immediately on either side of the groove gave the thermal growth of the tool. This difference was measured as 20 pm and a t h s a m n t is added to the predided value of 20.4 pm, the error in cylindriatymnbereducedfrom50 t o l 0 p r r . 6
1.
2.
CONCLUSIONS
A simple test has been developed which enables one to determine the effedive stiffness of the spindle bearingchuck system. This stiffness can be subsequently used to predid the forceinduced error h the mrrponent. The stiffness mnstarts mn also be used to -re the relative performancelaccuracy d different CNC machines. The model has diagnostic signifimnce in that the mntribution of the individual mrrponents can be assessed.
0
20
40
K1= 8 2 . 3 ~ 10"Nlrn K2 = 1.65 x l o 6Nmlrad k = 0.53 x l o 6Nmlrad s = 0 -203 rrmlrev VC= 270 Wmn
Figure 8: w 3.
7 [I]
[2]
[3]
[4]
[5] [6]
[8]
[9]
e
Px= 165 N Pz=89 N D=86mn d = 0.5 rrm C L = 3 1 rrm
d and predided defledions
For the machine and mrrponent used in the experiment. the magnitudes of the f o r M u c e d and tool-thermal growth errors were w a r a b l e and they were able to reduce the error in the cyiindriaty by 80%. REFERENCES
Donmez. MA., Blomquist. D.S., W e n . R.J., Liu. C.R. and Barash. M.M.. 1986. A general m e for machine tool accuracy enhancement by error mrrpensation. Precision Engineering. 814:187-I 96. !Ma. T., Takeuchi. Y. and We&. M., 1981, Irqxovement of Working Accuracy on NC Lathe by Corrpensatim for the Thermal E!xpansion of Tool. Annals of the CIRP. 3011:445448. Yang. S.. Yuan. J. and Ni. J.. 1997. ReaCtiTle cutting force induced error mrrpensation on a turning center, Int J Mach Tools Manufad. 37111:1597-1610. Ni. J.. 1997. CNC Machine Accuracy Enhancement thrwgh Real-Time Corrpensation. J Manu S a and Engg. Trans ASME. 119:717-725. Jednejewski. J. and Modrzy&i. W.. 1997. Intelligent supeMsion of thermal deformations in h g h preasion machine tools. MTDR Conf. 39:379-382. El Oufai. A.. Guillot. M.. and Bedrwni. A.. 2000. Accuracy enhancement of rmlti-axis CNC machines thrwgh on-line neuromrrpensation. J Intelligent Manufaduring. 11:53!-545. Liu. Z.Q. and Venwinod. P.K.. 1999, Error Corrpensation in CNC Turning Solely from Dimensional Measurements of Previously Machined Parts, 1999.Annals of CIRP. 4811:429432. hsao. T.. MPugaki. Y. and S a k a m b . M.. 1992. Preasion turning by means of a simplified predidive fundion of machining error. Annals of CIRP. 4111 447450. K a y . Y.. Chmg. Y.-P.. Tsai. J.-W.. Chen. S C . and Yang L.-Y.. 2001, Integrated 'CAE' strategies for the d e s y of machine tool spindle-bearing system. Finite Elements in Analysis and Design. 37:485-511.
[ l o ] Soon. M.P. and Stone. B.J..1998. The stiffness of statically indeterminate spindle s y s t e m with n m linear bearings. Int J Adv Manuf Tech, 14:787-794.
Abstract This paper describes a procedure to evaluate the force-induced errors occurring in cyiindriml turned mrrponents. This procedure is based on a model which represents the realtime stiffness of the spindlebearing system and the rotational damping stiffness of the mrrponent held in a chuck. The stiffness values are determined from a single cutting test in which the defledion of a test bar is measured. The model also the takes into acmunt the stiffness of the mrrponent and toolpost. The model is validated by -ring predided error in the cyiindricityof a machined bar with the measured value.
kywwds: CNC turning. force-induced errors. spindle-bearing sydern
1 INTRODUCTION
2
Preasion manufacture relies to a significant extent on the wallty and accuracy of a machine tool. The geometric errors in the machine tool are based on 'mlaad' instrumentabbased tests origindly f m l a t e d by into national and Schlesinger and later in-ated international standards. Other irqmtant fadors which affed the accuracy of a machined m p e n t are forceinduced errors in the machine tml and workpwce. thermal errors due to the heat generated in the machine tool and by the cutting process. and tool wear.
2.1 Model
Several researchers have studied. charaderised and mrrpensated for one or m e of the above errors using various strategies. Donmez et al. [ I ] developed a mrrprehensive model to predict the g m t r i c and thermally induced errors in a machine tool and they mrrpensated for these in real time. On the other hand, Sata et al. [2] focused on the error caused by the thermal gowth of the tool which they were able to represent using an expnential function. Yang et al. [3] and Ni et al. [4]. mncentrated on m d e l l i r q the force-induced errors due to the spindle bearing system, tool post and mrrponent. the flexibilities of which were determined qmrimentally. The strategy is to mrrpensate for the above errors either off-line by mdlrylng the CNC tool path or in real time which usually requires on4me mitoring of the process parameters (cutting force. terrperatures etc.) [3-61. The latter is not usually possible Ibr all machines an the shop floor. More recently, Liu and Venwinod suggested a s h o p k friendly off-line strategy for mrrpensating for the forceinduced errors. However. to mrrpensate for a part. their strategy requires information relating to a simlar previously mchined part to exist in the database. Instead of mnsidering one or m e of the above errors. some researchers such as Asao et al. [8]. charaderised the total machining error by an expnential fundion. the mnstants for which were determined for various tools and cutting mnditions. But the disadvantage of this approach is that it is machinespecific and offers m nsght into the magnitudes of the various error mrrponents. This paper focuses on the force-hduced errors and it desaibes the development of a spring model to represent the spindlebearingchuck system of a lathe. This model also takes into acmunt the clamping stiffness d a mrrponent held in a chuck. The error predided by this spring model is then synthesised with the forceinduced errors due to the toolpost and mrrponent.
SPINDL~BEARINGCHUCKSYSTEM
The cutting loop mnsists of the spindle-bearing system, chuck, lathe bed, cutting tool. tml post and the workpiece. Whilst an accurate representation of fie system can be obtained by using finite elements [9]. the resulting model w w l d be tm m p l e x and the analysis timeoonsumng. Since the main p u p e of the model is to faalitate the determination of the defledion of the mrrponent at the cutting tool position. it was decided to use a mmbination of linear and rotational springs to represent the system. Two irrportant mnsiderations limed the eventual m p l e l o t y of the model. The first of these was that it should be possible to calculate the spnng m t a n t s of the model from a single udlirq test. The second mnsideration was that measuring the defledion of the rotating mrrponent diredly should be avoided because fadors such as eccentricity and surface roughness will affect the accuracy of the measwed data.
-
Defledion measurement
Preasion bar
Figure 1: Cutting loop In addition to the above. the real-time stiffness of the system should be measured rather than the static as has been done by most researchers [5. 91. Hence, as shown in Figure 1. it was decided that the measuring positims should be lomted on a bar outside the cutting loop. The portron of the system to the nght of the cutting tool. induding the bar. will not bend. This is bemuse the defledion curve for this part of the system is a straight Cne and tangential to the deRedion m e Ibr the left portron of the cutting loop. Sinm a straight line is determined by two points. the maxirmm number of measuring positions is two.Wth two defledion values, the r m s t mmplex model mnsists of one linear and two tdational springs [Figure 2). The first two springs (K1and Kz) represent the Overall bending and rotational stiffness of the spindle-bearh-huck system. The third spring ( K 3 ) represents the darrplng rotational bending stiffness of the m p e n t held in the muck. The two rotational springs are m n e d e d by a rigid link, m e length (Lc) is measured from the back face of the chuck to the centre ofthe damped portion of the mrrponent in the chudc.
hydraulic chudr
L&
-
tiers
Figure 3: Schematic representation of the experimental rig F r m the model shown in Figure 2, let B represent the defledion measured by either of the two transducers Wich are placed at distances L1 and L Zfrom the spring K3 respedively. If L is equal to either L1 or L2 and a i s the distance from the centre of the d a m portron of the mrrponent to the mtting tool position. then SL = S K ,
+SK2
+SK3
F +F.(L+f&+L,)+F.La
--
Kl = A+B.a
K2
K3
(11
where A and are m s t a n t s and F is the tangential or radial mrrponent of the cutting force. The rotation k)of the system to the nght of the cutting tool position can be calculated f r m t h e two LVDT readings. 8 L =-
62 -61
2.3 Experlmental detemlnatlon of Kl, IG and IG
As mentioned earlier. a single cut is sufficient to determine the sprhg m s t a n t s . In this cutting test, a cylindrical bar made of low-mrbm steel (Ens) was machined using a depth of cut (d) of 0.5 mn and a feedrate (s) of 0.079 lTmlrev (0.0031 inlrev). A SECO M e r of type PCLNR2020K12 with an insert of either type CNMG 120404MF2TP200 (i.e. nose radius 0.4 mn) or CNMG1204012MF2TP200 (nose radius 1.2 mn)was used. The diameter of the test wmkpmce was chosen to be large (89 mn in this case) so that its stiffness is an order of magnitude greater than other elements in the cutting loop. Moreover. the cylindrical shape enabled the stiffness of the m p e n t to be accurately calmlated A spindle speed of 1000 rpm was used. Since the intention was not to measure the forces during the above test, the following equations were used to calculate the three corrponents of the force.
M
F r m Figure 2.
The constants in e q s . (1) and (3) can be solved if & and & are measured for two different values of a i.e. two different positions at which the cutting force is applied. Once these oonstants are known. K1. KZ and k can be easity determined. 2.2 Experlmentalset-lp The experimental rig m s i s t e d of a test vmkpwce. setting device. displacement transducers, a sensor holder and a data acquisition system. The workpwce was a simple cylindrical bar with a setting device near its free end (Figure 3). The other end was held in a hydraulic chudr of a MHP slanLbed CNC lathe. The setting device had a blind hole and a precision extension bar of diameter 8 m and length 160 mm. was held inside it by eight m e w s . These saews. when adjusted. reduced the eccentriuty of the extension bar to as low as 1 pm (Figure 4) thus substantially reducing the noise in the displacement signals obtained during cutting tests. The other end of the extension bar rotated inside a sensor holder which was m n t e d in the tailstodc. This holder carried two sets of dsplacement transducers of the LVDT type. so that the defledion mld be measured not only in the vertical plane but also in the horizontal. although only one pair of sensors was used at any one time.
Figure 2: Model for spinde-bearing-chudrsystem
settinadevice
lJ
'
'
spindleaxis
Fgure 4: Adjusting the precision bar
FY -- 1901do 9 3 7 8 . ~ 08489 F -5636’’ 1175 3 0 5665
z-
The constants in the above equations were obtained expwimentally with the above tool and wmkpwce material using artting mnditions mrrespnding to finishing operations. The signal obtained from the first displacement transducer (LVDTI) is shown in Figure 5. Data was mptured at a frequency of 1 kHz resulting in 43500 data points for each channel for a 150 mlength of cut. A regression analysis ofthe data resulted in a linear variation as shown in Figure 5. The constants shown in the figure are in fact, the values ofA and in equation (2). W e t h a t in Figure 5. and in the subsequent figures, tool position = 0 mrresponds to a = 3 O m to a-nt for the distance from the centre d the damping position to the ends of the jaws and the small flange at the start of the m m p e n t . Similarly. a tool position o f 1 5 0 m mrresponds to a=180 m. The difference between the two signals was calculated (see eq. 3) and a subsequent regression anatysis of the difference also resulted in a linear variation for R. €k = 2.903E-5 + 6.137E-4.a =C+D.a From these f a r mnstants the values for the spnng mnstants were calculated as: 6 = 8 6 . 8 0 ~ 1 0Nlm; ~ IG = 1 . 7 3 ~ 1 0Nmlrad ~ and IG = 0 . 5 6 8 ~ 1 0NrWrad. ~ S m and Stone [IO] obtained a value ot 1x10’ ~ h n for the headstodc stiffness and this m n p a r e s favourably with the value of K1.
The above stiffness values were obtained with a damping length (CL) of 31 mn. To evaluate its influence. twoother lengths (43 and 37 mn) were investrgated. These values were arrived at by using an integer n u m e r of serrations an the jaws to damp the mrrponent. In all cases.the tdal darrplng pressure exerted by the hydraulic chuck was maintained at its maximm value of 66 kb. The results showed that K1 and Kt remained unaffeded but Ka inaeased from 0.568~10 Nmlrad at 31 mto 0.723~10 at 37 mm and 1.O7Ox1O6at 43 m. 2.4 Radlal dellectlon at the cuttlng edge In the X Z plane, b t h FX and FZ (Fig. 6) mntribute to the defledion of the mrrponent in the radial diredion. This defledion. at the p n t of cutting. is given by (see Fig. 2):
6, = 6K, + 6& +6&
L
m m
.-C
Tool position [mn] Figure 5: Variation ofdefledion with tool position
7) Wth the mrrponent rotating at a low speed, a LVDT s e n s a was w e d at a slow feedrate from diameter D1 to Dz. The resulting signal (Figure 7) has two distind parts. the first refers to and the semnd to D. Since the machined portion of the mrrponent (Dz) is sufficiently dose to the chuck. it will mntain only two errors i.e. & and b t . Portion D 1. of rrmrse. is assumed to be error free. If &tort=
(D1-D2)/2 - d
then &tort represents the error due to the spindle bearing system and the toolpost i.e. &tort=
6s+
Hence,t ,K
b t
=bdFx
Several tests with different depths of aR were mnduded and the mrrespnding values of &tort noted. Figure 7 shows the signal obtained with d = 0.5 mn. For exarrple. for this depth of cut. B t a was measured as 9.8 pm and & =5.4 Hence b5t= 4.4pm.Since Fxwas 165Nforthis depth of cut. the stiffness of the tmlpost was 38x1O6 Nlm.
v.
4 COMPONENT DEFLECTION The mrrponent deflection b m due to the radial and axial m q m e n t s of the cutting force was mlculated. w h r e possble. analyWAty or. in the mse of m m p e n t s with m e x geometry. by the finite element method. In the latter, the wodqnece was modelled with 8noded hexahedron elements. For the m q m e n t s investrgated. the defledion was less than 1-2 pn and therefore the error due to mrrponent flexibilw was not as significant as that due to the spindleharing systemor toolpost. 5
EXAMPLE
To verify the model developed herein. a cyiindriml bar of diameter 86 mn and length 150 mn was machined with a 0.5 m depth of aR. The profile was then measured on a CMM machine and this is shown in Figure 8 fromwhich it 141 K3
K3
3 TOOLPOST DEFLECTION The radial mrrponent of the cutting force. F i will also muse the toolpost to defled. This defledion wil not alter the profile of the mrrponent but will onty resdt in a mnstant inaease in the mrrponent diameter. A simple test was designed to evaluate the magnitude of this defledion. hrtA. short wdqnece was held in the chuck and p r m c h i n e d with a spring pass to ensure that it mntaned no f o r c e - M u d errors. Then the test metewas machined exceptforthe last5 rrm(Figure
Fx
f
Figure 6: Schematic representation in the X-Z plane
I I
.-
! I
-5000
0
-
I
-
1
-
r
-
I
-
I
-
r
-
moo imo ism m o 25000 am
1 35000
1
-20
60 80 100 120 140 160 Tool psition [mn] om = 25; od = 9.9; oa = 17.4; ob = 0.3; oc = 7 5
data points
Fgure 7: Determination of the t d p o s t stiffness is dear that the error in cyiindriaty is 50 p (i-e. twice m . the values shown in Figure 8 are radius values). Curve A is the delledion of the mrrponent at the cutting tool position due to the flexibihty of the spindle4maring-dwck system. This curve was calculated using q . 4 . Note that the starting value (approximately 9.8 pm) for this curve is given by the sum of b t and the value obtained for S, ( q . 4 ) when a = 0.Curve represents the deflection of the mrrponent at the cutting tool position due to its own flexibility. the maxirmm defledion (&) being 0.3 p m If one mnsidered only the force-kduced error. Wich is given by the sum of curves A and B. then, acmrding to the model. it w w l d be possible to reduce the cyiindriaty error by 2(0w&) i.e. 35.4 p.But in pradice this would not happen bemuse every lathe has a spindle msalignment error (a)which, in the case of the MHP lathe, was 8.2'. Since this angle is negative i.e. the axis is indined towards the tool post. it will reduce the cyikdriaty error. Curve C shows the linear variation of this error which i s given by L tan a. L being the length of the art. Hence, for the testplece (L = 150 mn).the redudion in the cyiindnuty error is twice oc i.e. 15 p.This means that be net predided error in cyiindriaty i s only 20.4 pm as -red to the measured value of 50 prn Further investigations revealed that the thermal growth of the t d muld be sgnificant. To substantiate this, a groove (3 mn wide and 3 m deep) was machined near the chuck end of the cyiindriml bar. This bar was machined again with the same cutting m i t i o n s as shown in Fgure 8. When the tool reached the centre of the groove. the feed was stopped and the tool allowed to mol down to ambient mnditions. The tool then mntinued to machine the small p b o n of the test bar between the groove and the chuck. The difference in the diameters immediately on either side of the groove gave the thermal growth of the tool. This difference was measured as 20 pm and a t h s a m n t is added to the predided value of 20.4 pm, the error in cylindriatymnbereducedfrom50 t o l 0 p r r . 6
1.
2.
CONCLUSIONS
A simple test has been developed which enables one to determine the effedive stiffness of the spindle bearingchuck system. This stiffness can be subsequently used to predid the forceinduced error h the mrrponent. The stiffness mnstarts mn also be used to -re the relative performancelaccuracy d different CNC machines. The model has diagnostic signifimnce in that the mntribution of the individual mrrponents can be assessed.
0
20
40
K1= 8 2 . 3 ~ 10"Nlrn K2 = 1.65 x l o 6Nmlrad k = 0.53 x l o 6Nmlrad s = 0 -203 rrmlrev VC= 270 Wmn
Figure 8: w 3.
7 [I]
[2]
[3]
[4]
[5] [6]
[8]
[9]
e
Px= 165 N Pz=89 N D=86mn d = 0.5 rrm C L = 3 1 rrm
d and predided defledions
For the machine and mrrponent used in the experiment. the magnitudes of the f o r M u c e d and tool-thermal growth errors were w a r a b l e and they were able to reduce the error in the cyiindriaty by 80%. REFERENCES
Donmez. MA., Blomquist. D.S., W e n . R.J., Liu. C.R. and Barash. M.M.. 1986. A general m e for machine tool accuracy enhancement by error mrrpensation. Precision Engineering. 814:187-I 96. !Ma. T., Takeuchi. Y. and We&. M., 1981, Irqxovement of Working Accuracy on NC Lathe by Corrpensatim for the Thermal E!xpansion of Tool. Annals of the CIRP. 3011:445448. Yang. S.. Yuan. J. and Ni. J.. 1997. ReaCtiTle cutting force induced error mrrpensation on a turning center, Int J Mach Tools Manufad. 37111:1597-1610. Ni. J.. 1997. CNC Machine Accuracy Enhancement thrwgh Real-Time Corrpensation. J Manu S a and Engg. Trans ASME. 119:717-725. Jednejewski. J. and Modrzy&i. W.. 1997. Intelligent supeMsion of thermal deformations in h g h preasion machine tools. MTDR Conf. 39:379-382. El Oufai. A.. Guillot. M.. and Bedrwni. A.. 2000. Accuracy enhancement of rmlti-axis CNC machines thrwgh on-line neuromrrpensation. J Intelligent Manufaduring. 11:53!-545. Liu. Z.Q. and Venwinod. P.K.. 1999, Error Corrpensation in CNC Turning Solely from Dimensional Measurements of Previously Machined Parts, 1999.Annals of CIRP. 4811:429432. hsao. T.. MPugaki. Y. and S a k a m b . M.. 1992. Preasion turning by means of a simplified predidive fundion of machining error. Annals of CIRP. 4111 447450. K a y . Y.. Chmg. Y.-P.. Tsai. J.-W.. Chen. S C . and Yang L.-Y.. 2001, Integrated 'CAE' strategies for the d e s y of machine tool spindle-bearing system. Finite Elements in Analysis and Design. 37:485-511.
[ l o ] Soon. M.P. and Stone. B.J..1998. The stiffness of statically indeterminate spindle s y s t e m with n m linear bearings. Int J Adv Manuf Tech, 14:787-794.