- Email: [email protected]
Path Compensation with Respect to Manufacturing Tolerances
Copyright 0 IFAC Intelligcnt Manufacruring Systems, Seoul, Korea, 1997
PATH COMPENSATION WITH RESPECT TO MANUFACTURING TOLERANCES Tae-Il Seo
Philippe Depmce
Jean-Yves Hascoet
Institut de Recherche en Cybernttique de Nantes UMR CNRS 6597 - Ecole Centrale de Nantes 1, rue de la Noe, BP 92101, 44321 Nantes Cedex 3, FRANCE Tel. , (33],2.,0.37.16.71 F.. , (99) 2.,0.7,.7,.06 e-mail: [email protected]
Abstract: In this paper, we present our research dealing with the choice of the reference of compensation with respect to a given tolerance and the compensation tool trajectory in the milling process. The general purpose of our research is to compensate the tool deflection that occurs during the milling process. We consider it under two aspects : surface prediction generated by the tool deflection and surface prediction generated by the contact points between the cutting tool flute and the workpiece. These two aspects imply two different approaches to carry out the compensation method , called miTTOT method, which can generate a new tool trajectory. In order to compare these two approaches, we present some practical examples. Keywords: CAD/ CAM, Machine tool, Path Compensation, End Milling, Manufacturing Tolerance
L INTRODUCTION
pensated tool trajectory. This path generator reQuires three steps, cutting force model, tool deflection model and path compensation .
The end milling operation is very useful in the fields of industry. Despite the multiplication of NC machines-tools and the increase of CAD/CAM software performances, the dispersions due to the tool deflection make an inaccuracy of milling operation. Therefore, we cannot produce what we hope, in spite of the generation of a nominal tool trajectory by the CAD/CAM system. In order to increase the accuracy of milling operation, we have to take into account an approach to determine a new tool trajectory, which allows to compensate the errors of the tool deflection. Our ultimate purpose is to develop an approach to compensate the tool deflection by the generation of a new tool trajectory without reducing the productivity.
--
Cuttio Process
PIli!, Gtota1or:
_ Cutting Fol'I:C Model • Tool Deflection Model • PlIIth Compcnsllion
Fig. L Milling process with compensation We have used a cutting force model (Kline et aL , 1982), a tool deflection model (Armarego and Deshpande, 1994; Suh et al., 1995) and a path compensation method , called mirror method (Hascoet et al. , 1997). For the path compensation, the mirror method requires a reference to search a compensated tool trajectory. This refer-
To accomplish our purpose, firstly, we consider a general milling process shown in figure 1. This process schematizes a cutting process with an offline compensation, which can determine a com-
363
ence of compensation can be considered according to two different aspects of the milled surface generation. We have firstly considered the surface generation by the average tool deflection (Suh et al., 1995; Hasco~t et al., 1997). This approach allows us to simplify the process of the milled surface prediction. We have also considered the surface generation by the contact points between the tool flute and the workpiece (Fujii et al., 1979; Tlusty et al., 1991). This approach allows us to predict more exactly the milled surface. For these two different approaches, we present our considerations of the tool path compensation with respect to a given manufacturing tolerance.
2. CUTTING FORCE MODEL
forces along the tool flute. With a view to improving inaccuracy of the computed tool inertia, we have applied an equivalent tool diameter DE of a cantilever beam, which is determined in order to make its inertia the same as that of the real tool diameter D (Kops and Vo, 1990). We have calculated the tool deflection as a function of the tool angular position because the cutting force evolves according to the tool orientation in the workpiece. We have also calculated the avera.ge tool deflection for several tool angular positions under the same cutting conditions, radial depth, axial depth and feed rate per tooth. In fact, these two types of tool deflection are required to predict the milled surface.
4. COMPENSATION METHOD
Among the different cutting force models, we have chosen the model of Kline and DeVor, which allows to determine oot only cutting forces but also force centers (Kline et al., 1982). The force model is based on the determination of two specific coefficients KT and KR which can make the proportional relation between cutting forces and an unit chip section area. From experimental results, we characterize a tool-matter couple by these coefficients KT and KR, modeled by a function of the three cutting parameters, radial depth of cut, axial depth of cut and feed rate per tooth. From this model, not only the cutting forces along any direction (distributed along the tool flute) can be computed but also a cutting force center is applied to simplify the tool deflection model. The cutting force center is a point where a concentrated cutting force applied for the cantilever beam model. So, the bending moment produced by a concentrated cutting force and his center is equivalent to that produced by distributed cutting forces. We have characterized a tool-matter couple. We have chosen an end milling tool, 4 flutes, diameter 6 mm, 30o-helix angle, 30 mm-used length and a steel workpiece (middle carboo steel). The spindle speed is fixed by 1250 RPM. A set of experiments has been realized in order to determine the models of KT and K R. All simulation and experimental results presented in this paper are carried out with this specification of cutting tool and workpiece.
3. TOOL DEFLECTION MODEL To avoid the deflection model becoming sophisticated, the calculation of the deflection is then based on a cantilever beam model (Armarego and Deshpande, 1994; Suh et al., 1995). A concentrated cutting force is loaded at a force center instead of considering the distributed cutting . 364
The path compensation consists in correcting a theoretical path by integrating errors due to the deflection. Workpitu (Profi~)WI
.DT
------'(Profile)",
(Pro ItJo
(a) without com~ruation
.D(b) with compelUotwn
Fig. 2. Path compensation Figure 2 shows two cases of the milling opera.tions : with and without compensation. In the case of the figure 2-(a), (profile)w is the nominal profile (wished profile) which has to be obtained, and TN is the nominal trajectory computed to obtain (profile}w. In fact, TN is provided by a CAM system without taking into account the effects of the tool deflection. We obtain the deflected profile (profile)D which is deviated as compared to (profile)w because of the tool deflection. In the case of the figure 2-(b), our aim is to determine a new trajectory Tc, that associated to the defects of the tool defiection will allow to generate the profile Pc equal to the wished profile (profile}w. To obtain the compensated trajectory Tc, we use a method, called mirror method. The mirror method is an iterative process that allows to compute a new path which compensates the tool deflection ( Hasco~t et al., 1997). The mirror method is an iterative algorithm to generate a new tool trajectory which allows to
compensate tool detl.ection errors. In each step of iteration , we need a reference to generate a temporary compensated trajectory which compensates previous tool detl.ection errors. With a view to determine this reference of compensation , we have considered the process of milled surface prediction .
consequence, we have shown that the two milled surfaces are different (Depin~ et aI., 1997). We present the consideration of a reference of compensation to compare these two methods of surface generations. When a certain tolerance is given, we must respect this tolerance to manufacture a workpiece. So, we also consider the choice of the reference of compensation regarding to the given tolerance.
Tooi"rt1fiI~-i
5.1 Compemation when milled surfa~ is generated by tool profile
In this section, we present a compensation when milled surface is generated by tool profile. The profile of the deformed tool is a curve and its detl.ection is variable .
(aJ surftlCtl gtlluaud by,~ a ...uag~ loci tkflutiOll
.
" Tool "roJIk~
,,",f-, , ' " ,," ,,
3
Fig. 3. Surface generation
5. REFERENCE FOR THE COMPENSATION In order to apply the mirror method, the question of how to predict the milled surface takes precedence over all others. So, we have taken into account the two following aspectS: (1) prediction of surface generated by the profile of detl.ected tool, and (2) prediction of surface generated by the trace of the contact point between the tool flute and the workpiece. In the first case, we have made an assumption that the milled surface corresponds to the deflected tool profile. We have taken into account a deflected tool profile to generate a milled surface. In the view of calculation time, this assumption gives us some advantages to accomplish the tool trajectory compensation. But , if we want to predict a precise milled surface, we must consider the variation of the tool deflection with respect to the tool angular position. In the second case, we have made a different assumption that the milled surface is generated by contact points between a cutting tool and a workpiece. In this case, we must take into account the variation of the cutting force as a function of the tool angular position. Figure 3 shows the process of the surface generation by the profile of detl.ected tool and by the trace of the contact points. We have two different ways to predict the milled surface, as a
Fig. 4. Levels of compensation The error due to the tool deflection differs along the tool axis, when cutting tool mills a workpiece by a nominal trajectory. Figure 4-(a) shows a case of milling process without compensation. In this case, we consider maximum error Emu and minimum error Emin. The errors Emu and Emin are distances measured from the wished profile. If Emu (or E'min) produces an overcut error (or undercut error), the value of Emu (or E,.,;n ) is negative (or positive). For the calculation of the compensation, we have to choose a reference along the cutting tool. Three examples of compensation are presented on figure 4 ; (b) compensation with respect to the top of the part, which entails an undercut error compared to the theoretical profile, (c) compensation with respect to the bottom of the part , which entails an overcut error, (d) some intermediate situation between (b) and (c) (Sub et al., 1995 ). In the fields of industry, the dimension to obtain is toleranced : one has to produce a surface in a 365
valid domain . In our case, the errors due to the deflection have to belong to this area. We propose to determine the level of compensation (see figure 4 (b) , (c) and (d)) according to the toleranced dimension .
domain {(Prel )mu ; (Prr ! )min} assure the respect of the dimension
N::.:·.
We have tested this choice of compensation reference on an example of cylindrical workpiece (Dl!pincl! et al. , 1997). In this case, we have succeeded in reducing the maximum error by approximately 80%.
I'ffJ/Ilrll)
5.2 Compensation when milled surface is generated by the only one moving contact point E_{:c..... ) (a) T"lt~fPlCt and trrors
E,.., (:c .... )
As mentioned in the previous section , this case is different, but can be considered in the same way. In this case, we choose three points for the reference of compensation.
(b) Domain "f~eftrtnct
Fig. 5. Reference of compensation taking into account a tolerance The first step consists in verifying the possibility to respect a given tolerance N;;:,:x. When we manufacture a workpiece by the nominal tool trajectory, if a tolerance N; ...n is given , we can define a following discrimina;t.a.lol to distinguish between a possibility and an impossibility for respecting the tolerance.
(b) CtwrlpeTUlUUJI1 fo r
W"'"I1,"W>! uro'
M_ _
Ct.tol
== (cmu - Emu ) . (Cm;n - Em;n)
(1)
-
M_
We can verify possibility to respect N;m._x by following definitions. ....n
.6. 101 ~ 0, impossible to respect N;;:;:x. The compensation is necessary. • If .6. 101 < 0, possible to respect N;;;:;:x. The compensation is not necessary.
• [f
(t:) COIPIpenslJ1llJll/o' IMmAXIJIIIDII
u ror
rd} ComJH.flS(Jlion lo~ IM (lVerog~ \·,II.. ~ Mm·u n IM mlUlIIIlDII and nu"""",,,
Fig. 6. References of compensation
we require the compensation , we try to determine a reference of compensation p~e! (see figure 5-(a)) . After the milling process with the compensated trajectory for the reference Pr e !, if .6./(11 < 0, the compensation is successful. In order to consider a domain of the reference of compensation , we take into account two limits of the reference. In figure 5- (b) , two different profiles of the deformed tool are shown. Profile (1) is determined so that the maximum error arrives at the point N + Cmu . In this case , (Pnl )mu is the reference of compensation . So, in order to determine (Prt! )mu, we continue to carry out the mirror method until the maximum error will be equal to N + Cmu. By the same way, Profile (2) is determined so that the minimum error arrives at the point N + Cmin. In this case, (Pnt )min is the reference of compensation. So, in order to determine (Pret )min, we continue to carry out the mirror method until the minimum error is equal to N + Cmin. Therefore, if (P~tt )mu > (P ret )min , the two limits of reference (Prr ! )mu and ( P~d )min make a domain of reference of compensation with respect to N;:::.,:·. All the references of compensation taken inside this
[f
Figure 6 shows uncompensated profile and three compensated profiles. Figure 6-(a) shows the milled profile without compensating. Here, Emu and Emin are the minimum and maximum errors whkh move as function of the cutting conditions. So, we cannot easily predetermine positions of these errors. But , we can compensate with respect to three points as the previous case. Three examples of compensation are presented in figure 6 ; (b) compensation for the minimum error, (c) compensation for the maximum error and (d) compensation for the average value between maximum and minimum errors. Firstly, we can verify a possibility to respect tbe given tolerance N:;;:;:z. In figure 7-(a), the milled profile generated by contact points and a given reference of compensation are shown. This milled profile is obtained after compensation with respect to a given reference. As compared to a tolerance N;::,:~ , if both Emu and E min are in the domain of tolerance (or if .a.tol ? 0), the compensation is not necessary. But, if not , the compensation must be carried out to respect the given tolerance.
N:::,:z
366
\"\"r~". . . . "
Pndicud milkd profik
T"
~
•, , ,,
Wis~d
profile I
Fig. 8. Bandwidth allowing to guarantee the tolerance "f--N+c_ _
;~e _ _~'·~~:::,N N+c_ (b) Limiu wirl!
rts~ct
Both (Tc)mu and (TC)min constitute a bandwidth. So, we can make the following conclusion
. •
• If an arbitrary compensated tool trajectory Tc exists in a bandwidth consisting of two
10 rht rokranct
Fig. 7. Limits of compensation with respect to the tolerance
limits compensated trajectories (Tc)mu and (Tc )minl Tc can compensate the errors of the tool deflection to guarantee a given tolerance
We can consider the domain of reference of compensation in the way as in the previous section. In figure 7, two profiles generated by contact points are shown. They are different pattern profiles due to the variation of the tool de8ection as a function of the tool angular position . To obtain profile (1), we continue to carry out the iteration of the mirror method until the maximum error Emp. is tangent to the limit (1). After that we have determined the profile (1) by iterations of the mirror method , we can know a compensated position indicated by (Pc}mu. By the same way, for a limit (2), we can determine a profile (2) and a compensated position indicated by (PC)min ' As a matter of fact , (Pc )mu and (Pc)min are two limits of the compensated position for an arbitrary nominal position among all nominal positions decomposed on a nominal trajectory. IT we consider a set of (Pc}mu and (PC)min for whole nominal positions of a nominal trajectory, we can obtain a bandwidth which allows us to assure the respect of a given tolerance Ng;;;;;;. Figure 8 schematizes this bandwidth .
Ng;;;,a;. For this conclusion to be reasonable, the following condition must be satisfactory: • The minimum error Emin of the profile (1) must be less than Cmin , and the maximum error Emu of the profile (2) must be less than C mu . (.0. 101 < 0) Unless this condition, the intersection (bandwidth) of the two sets produced by (Tc}mu and (Tclmin cannot exist. We have treated a practical example of simulation. Figure 9 shows two cases of milling operations without and with compensation. Firstly, there is a plan to manufacture a workpiece. The purpose of this plan is to cut a surface indicated by a toleranced dimension 17~g:~~. In case (1), a milling process without compensation is presented. After this milling process, the milled surface stays outside the criteria of the tolerance. So, we cannot obtain the milled surface respecting the tolerance. On the contrary, in case (2), a milling process with compensation is presented. Before the milling process, we have obtained a compensated trajectory by the mirror method . After the milling process with compensated radial depth of cut, we have obtained a milled surface which stays inside the criteria of the tolerance. So, we have succeeded in obtaining the milled surface respecting the tolerance.
In figure 8, as the nominal trajectory T N is decomposed in N nominal tool positions, (Pc):"u is a compensated tool position which can compensate the tool deflection so that a maximum error Emu is equal to Cmu for a itl! nominal position, and (Pc ):"in is also a compensated tool position which can compensate the tool deflection so that a minimum error Emin is equal to Cmin for a ill! nominal position. (Tc)mu is a limit consisting of a set of (Pc):"u (Vi = 1,2,3,···,N), (TC)min is a limit consisting oh set of (Pc}:"in (Vi = 1, 2, 3", . ,N).
We have carried out another simulation for the compensation of the milling operation of a cylindrical workpiece (Depince et al., 1997). We have 367
fore, we choose the second method to carry out the compensation.
,~
/7~:J 1I1m
(Gi,'(" loIu/Ult:t)
..
Sol ~
2O~
-0
C.... (II M,I/ur'.,.."""'"
C... (tJ MoiWtz _ _
WITHOUT COMPENSATION
WITH COMPENSATION
• .,.,
X r"""J
~D
'~'
(a) !>fit/Id f Ur/act ",,"/I!OUI COmptrlSill'Q~
,~
.. /6.668111m
:
" J.JJl mm
IC_ _ """
-
17.1"~'------:
_
-
11Io.1S_-l
_
_
174,,,..,
-
17./J_
::
-~,~'-'----,
'T'
Cmuia a/ tolufPlCt
D
1715 .... _
.
16.1S_ --':
_
tUn-.
-
16.1S1 _ _
:
i'-i_---,
( h) !>f,U,d /ur/act with compensatIon
'T'
Crituia o!lClUrollU
Fig. 10. Predicted milled surfaces with and without compensation
D
Failure
.-
Success k_
7. REFERENCES Armarego, E.J .A. and N.P. Deshpande (1994). Force prediction models and CAD / CAM software for helical tooth milling processes: 3. end milling and cutting analyses. International Journal oJ Production Reseo.rch. D~pince, Ph., J.Y. Hasc~t and T.I. Seo (1997). Surface prediction in end milling. 2rd International ICSC Symposia on IIA97 in Nfmes, FRANCE. Fujii, Y. , H. Iwabe and M. Suzuki (1979). Effect of dynamic behaviour of end mill in machining on work accuracy. Bulletin of the Japan Society of Precision Engineering 13(1), 20-26. HascOi!t , J .Y. , T.l Seo and Ph. Depince (1997). Compensation des deformations d 'outils pour la generation de trajectoires d'usinage. 16th Canadian Congress of Applied Mechanics CANCAM97 in Quebec, CANADA. Kline, W.A. , R.E. Devor and I. A. Shareef (1982). The prediction of surface accuracy in end milling. Tmnsactions of the ASME 104, 272278. Kops, L. and D.T. Vo (1990). Determination of the equivalent diameter of an end mill based on its compliance. Annals of CIRP 39/ 1, 93-
Fig. 9. Example of compensation to respect the given tolerance chosen the average value of the maximum and minimum errors for the reference of compensation. Two simulation results are shown in figure 10. After compensation, we have globally reduced the errors as compared to the figure 10, but some errors cannot be reduced. We could also take a form tolerance as criteria, instead of a dimension tolerance.
6. CONCLUSION We have presented the tool path compensation with respect to a given tolerance. Firstly, we have summarized the modeling and methodology to approach reasonable considerations for the path compensation. Then , we have taken into account the determination of reference of compensation. We have described the choice of the reference of compensation for the two different methods to predict the milled surface. The possibility to respect a given tolerance was mathematically verified in the two different cases of the surface prediction . In the impossible case of respecting the tolerance, we have taken into account references of the compensation which allow to guarantee this tolerance. 'Ve have presented some examples for the two cases of compensation. We have obtained reasonable results for the two cases. But, the second method allows to more precisely compensate the errors and to predict the milled surface. There-
96. Sub, S.H. , J.H. Cho and J.Y. Hascoet (1995). Incorporation of tool deflection in tool path computation. International Journal of Manufacturing System 15(3), 190-199. Tlusty, J. , S. Smith and C. Zamudio (1991). Evaluation of cutting performance of machining centers. Annals of CIRP 40/ 1, 405-410.
368
PATH COMPENSATION WITH RESPECT TO MANUFACTURING TOLERANCES Tae-Il Seo
Philippe Depmce
Jean-Yves Hascoet
Institut de Recherche en Cybernttique de Nantes UMR CNRS 6597 - Ecole Centrale de Nantes 1, rue de la Noe, BP 92101, 44321 Nantes Cedex 3, FRANCE Tel. , (33],2.,0.37.16.71 F.. , (99) 2.,0.7,.7,.06 e-mail: [email protected]
Abstract: In this paper, we present our research dealing with the choice of the reference of compensation with respect to a given tolerance and the compensation tool trajectory in the milling process. The general purpose of our research is to compensate the tool deflection that occurs during the milling process. We consider it under two aspects : surface prediction generated by the tool deflection and surface prediction generated by the contact points between the cutting tool flute and the workpiece. These two aspects imply two different approaches to carry out the compensation method , called miTTOT method, which can generate a new tool trajectory. In order to compare these two approaches, we present some practical examples. Keywords: CAD/ CAM, Machine tool, Path Compensation, End Milling, Manufacturing Tolerance
L INTRODUCTION
pensated tool trajectory. This path generator reQuires three steps, cutting force model, tool deflection model and path compensation .
The end milling operation is very useful in the fields of industry. Despite the multiplication of NC machines-tools and the increase of CAD/CAM software performances, the dispersions due to the tool deflection make an inaccuracy of milling operation. Therefore, we cannot produce what we hope, in spite of the generation of a nominal tool trajectory by the CAD/CAM system. In order to increase the accuracy of milling operation, we have to take into account an approach to determine a new tool trajectory, which allows to compensate the errors of the tool deflection. Our ultimate purpose is to develop an approach to compensate the tool deflection by the generation of a new tool trajectory without reducing the productivity.
--
Cuttio Process
PIli!, Gtota1or:
_ Cutting Fol'I:C Model • Tool Deflection Model • PlIIth Compcnsllion
Fig. L Milling process with compensation We have used a cutting force model (Kline et aL , 1982), a tool deflection model (Armarego and Deshpande, 1994; Suh et al., 1995) and a path compensation method , called mirror method (Hascoet et al. , 1997). For the path compensation, the mirror method requires a reference to search a compensated tool trajectory. This refer-
To accomplish our purpose, firstly, we consider a general milling process shown in figure 1. This process schematizes a cutting process with an offline compensation, which can determine a com-
363
ence of compensation can be considered according to two different aspects of the milled surface generation. We have firstly considered the surface generation by the average tool deflection (Suh et al., 1995; Hasco~t et al., 1997). This approach allows us to simplify the process of the milled surface prediction. We have also considered the surface generation by the contact points between the tool flute and the workpiece (Fujii et al., 1979; Tlusty et al., 1991). This approach allows us to predict more exactly the milled surface. For these two different approaches, we present our considerations of the tool path compensation with respect to a given manufacturing tolerance.
2. CUTTING FORCE MODEL
forces along the tool flute. With a view to improving inaccuracy of the computed tool inertia, we have applied an equivalent tool diameter DE of a cantilever beam, which is determined in order to make its inertia the same as that of the real tool diameter D (Kops and Vo, 1990). We have calculated the tool deflection as a function of the tool angular position because the cutting force evolves according to the tool orientation in the workpiece. We have also calculated the avera.ge tool deflection for several tool angular positions under the same cutting conditions, radial depth, axial depth and feed rate per tooth. In fact, these two types of tool deflection are required to predict the milled surface.
4. COMPENSATION METHOD
Among the different cutting force models, we have chosen the model of Kline and DeVor, which allows to determine oot only cutting forces but also force centers (Kline et al., 1982). The force model is based on the determination of two specific coefficients KT and KR which can make the proportional relation between cutting forces and an unit chip section area. From experimental results, we characterize a tool-matter couple by these coefficients KT and KR, modeled by a function of the three cutting parameters, radial depth of cut, axial depth of cut and feed rate per tooth. From this model, not only the cutting forces along any direction (distributed along the tool flute) can be computed but also a cutting force center is applied to simplify the tool deflection model. The cutting force center is a point where a concentrated cutting force applied for the cantilever beam model. So, the bending moment produced by a concentrated cutting force and his center is equivalent to that produced by distributed cutting forces. We have characterized a tool-matter couple. We have chosen an end milling tool, 4 flutes, diameter 6 mm, 30o-helix angle, 30 mm-used length and a steel workpiece (middle carboo steel). The spindle speed is fixed by 1250 RPM. A set of experiments has been realized in order to determine the models of KT and K R. All simulation and experimental results presented in this paper are carried out with this specification of cutting tool and workpiece.
3. TOOL DEFLECTION MODEL To avoid the deflection model becoming sophisticated, the calculation of the deflection is then based on a cantilever beam model (Armarego and Deshpande, 1994; Suh et al., 1995). A concentrated cutting force is loaded at a force center instead of considering the distributed cutting . 364
The path compensation consists in correcting a theoretical path by integrating errors due to the deflection. Workpitu (Profi~)WI
.DT
------'(Profile)",
(Pro ItJo
(a) without com~ruation
.D(b) with compelUotwn
Fig. 2. Path compensation Figure 2 shows two cases of the milling opera.tions : with and without compensation. In the case of the figure 2-(a), (profile)w is the nominal profile (wished profile) which has to be obtained, and TN is the nominal trajectory computed to obtain (profile}w. In fact, TN is provided by a CAM system without taking into account the effects of the tool deflection. We obtain the deflected profile (profile)D which is deviated as compared to (profile)w because of the tool deflection. In the case of the figure 2-(b), our aim is to determine a new trajectory Tc, that associated to the defects of the tool defiection will allow to generate the profile Pc equal to the wished profile (profile}w. To obtain the compensated trajectory Tc, we use a method, called mirror method. The mirror method is an iterative process that allows to compute a new path which compensates the tool deflection ( Hasco~t et al., 1997). The mirror method is an iterative algorithm to generate a new tool trajectory which allows to
compensate tool detl.ection errors. In each step of iteration , we need a reference to generate a temporary compensated trajectory which compensates previous tool detl.ection errors. With a view to determine this reference of compensation , we have considered the process of milled surface prediction .
consequence, we have shown that the two milled surfaces are different (Depin~ et aI., 1997). We present the consideration of a reference of compensation to compare these two methods of surface generations. When a certain tolerance is given, we must respect this tolerance to manufacture a workpiece. So, we also consider the choice of the reference of compensation regarding to the given tolerance.
Tooi"rt1fiI~-i
5.1 Compemation when milled surfa~ is generated by tool profile
In this section, we present a compensation when milled surface is generated by tool profile. The profile of the deformed tool is a curve and its detl.ection is variable .
(aJ surftlCtl gtlluaud by,~ a ...uag~ loci tkflutiOll
.
" Tool "roJIk~
,,",f-, , ' " ,," ,,
3
Fig. 3. Surface generation
5. REFERENCE FOR THE COMPENSATION In order to apply the mirror method, the question of how to predict the milled surface takes precedence over all others. So, we have taken into account the two following aspectS: (1) prediction of surface generated by the profile of detl.ected tool, and (2) prediction of surface generated by the trace of the contact point between the tool flute and the workpiece. In the first case, we have made an assumption that the milled surface corresponds to the deflected tool profile. We have taken into account a deflected tool profile to generate a milled surface. In the view of calculation time, this assumption gives us some advantages to accomplish the tool trajectory compensation. But , if we want to predict a precise milled surface, we must consider the variation of the tool deflection with respect to the tool angular position. In the second case, we have made a different assumption that the milled surface is generated by contact points between a cutting tool and a workpiece. In this case, we must take into account the variation of the cutting force as a function of the tool angular position. Figure 3 shows the process of the surface generation by the profile of detl.ected tool and by the trace of the contact points. We have two different ways to predict the milled surface, as a
Fig. 4. Levels of compensation The error due to the tool deflection differs along the tool axis, when cutting tool mills a workpiece by a nominal trajectory. Figure 4-(a) shows a case of milling process without compensation. In this case, we consider maximum error Emu and minimum error Emin. The errors Emu and Emin are distances measured from the wished profile. If Emu (or E'min) produces an overcut error (or undercut error), the value of Emu (or E,.,;n ) is negative (or positive). For the calculation of the compensation, we have to choose a reference along the cutting tool. Three examples of compensation are presented on figure 4 ; (b) compensation with respect to the top of the part, which entails an undercut error compared to the theoretical profile, (c) compensation with respect to the bottom of the part , which entails an overcut error, (d) some intermediate situation between (b) and (c) (Sub et al., 1995 ). In the fields of industry, the dimension to obtain is toleranced : one has to produce a surface in a 365
valid domain . In our case, the errors due to the deflection have to belong to this area. We propose to determine the level of compensation (see figure 4 (b) , (c) and (d)) according to the toleranced dimension .
domain {(Prel )mu ; (Prr ! )min} assure the respect of the dimension
N::.:·.
We have tested this choice of compensation reference on an example of cylindrical workpiece (Dl!pincl! et al. , 1997). In this case, we have succeeded in reducing the maximum error by approximately 80%.
I'ffJ/Ilrll)
5.2 Compensation when milled surface is generated by the only one moving contact point E_{:c..... ) (a) T"lt~fPlCt and trrors
E,.., (:c .... )
As mentioned in the previous section , this case is different, but can be considered in the same way. In this case, we choose three points for the reference of compensation.
(b) Domain "f~eftrtnct
Fig. 5. Reference of compensation taking into account a tolerance The first step consists in verifying the possibility to respect a given tolerance N;;:,:x. When we manufacture a workpiece by the nominal tool trajectory, if a tolerance N; ...n is given , we can define a following discrimina;t.a.lol to distinguish between a possibility and an impossibility for respecting the tolerance.
(b) CtwrlpeTUlUUJI1 fo r
W"'"I1,"W>! uro'
M_ _
Ct.tol
== (cmu - Emu ) . (Cm;n - Em;n)
(1)
-
M_
We can verify possibility to respect N;m._x by following definitions. ....n
.6. 101 ~ 0, impossible to respect N;;:;:x. The compensation is necessary. • If .6. 101 < 0, possible to respect N;;;:;:x. The compensation is not necessary.
• [f
(t:) COIPIpenslJ1llJll/o' IMmAXIJIIIDII
u ror
rd} ComJH.flS(Jlion lo~ IM (lVerog~ \·,II.. ~ Mm·u n IM mlUlIIIlDII and nu"""",,,
Fig. 6. References of compensation
we require the compensation , we try to determine a reference of compensation p~e! (see figure 5-(a)) . After the milling process with the compensated trajectory for the reference Pr e !, if .6./(11 < 0, the compensation is successful. In order to consider a domain of the reference of compensation , we take into account two limits of the reference. In figure 5- (b) , two different profiles of the deformed tool are shown. Profile (1) is determined so that the maximum error arrives at the point N + Cmu . In this case , (Pnl )mu is the reference of compensation . So, in order to determine (Prt! )mu, we continue to carry out the mirror method until the maximum error will be equal to N + Cmu. By the same way, Profile (2) is determined so that the minimum error arrives at the point N + Cmin. In this case, (Pnt )min is the reference of compensation. So, in order to determine (Pret )min, we continue to carry out the mirror method until the minimum error is equal to N + Cmin. Therefore, if (P~tt )mu > (P ret )min , the two limits of reference (Prr ! )mu and ( P~d )min make a domain of reference of compensation with respect to N;:::.,:·. All the references of compensation taken inside this
[f
Figure 6 shows uncompensated profile and three compensated profiles. Figure 6-(a) shows the milled profile without compensating. Here, Emu and Emin are the minimum and maximum errors whkh move as function of the cutting conditions. So, we cannot easily predetermine positions of these errors. But , we can compensate with respect to three points as the previous case. Three examples of compensation are presented in figure 6 ; (b) compensation for the minimum error, (c) compensation for the maximum error and (d) compensation for the average value between maximum and minimum errors. Firstly, we can verify a possibility to respect tbe given tolerance N:;;:;:z. In figure 7-(a), the milled profile generated by contact points and a given reference of compensation are shown. This milled profile is obtained after compensation with respect to a given reference. As compared to a tolerance N;::,:~ , if both Emu and E min are in the domain of tolerance (or if .a.tol ? 0), the compensation is not necessary. But, if not , the compensation must be carried out to respect the given tolerance.
N:::,:z
366
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Pndicud milkd profik
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~
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Wis~d
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Fig. 8. Bandwidth allowing to guarantee the tolerance "f--N+c_ _
;~e _ _~'·~~:::,N N+c_ (b) Limiu wirl!
rts~ct
Both (Tc)mu and (TC)min constitute a bandwidth. So, we can make the following conclusion
. •
• If an arbitrary compensated tool trajectory Tc exists in a bandwidth consisting of two
10 rht rokranct
Fig. 7. Limits of compensation with respect to the tolerance
limits compensated trajectories (Tc)mu and (Tc )minl Tc can compensate the errors of the tool deflection to guarantee a given tolerance
We can consider the domain of reference of compensation in the way as in the previous section. In figure 7, two profiles generated by contact points are shown. They are different pattern profiles due to the variation of the tool de8ection as a function of the tool angular position . To obtain profile (1), we continue to carry out the iteration of the mirror method until the maximum error Emp. is tangent to the limit (1). After that we have determined the profile (1) by iterations of the mirror method , we can know a compensated position indicated by (Pc}mu. By the same way, for a limit (2), we can determine a profile (2) and a compensated position indicated by (PC)min ' As a matter of fact , (Pc )mu and (Pc)min are two limits of the compensated position for an arbitrary nominal position among all nominal positions decomposed on a nominal trajectory. IT we consider a set of (Pc}mu and (PC)min for whole nominal positions of a nominal trajectory, we can obtain a bandwidth which allows us to assure the respect of a given tolerance Ng;;;;;;. Figure 8 schematizes this bandwidth .
Ng;;;,a;. For this conclusion to be reasonable, the following condition must be satisfactory: • The minimum error Emin of the profile (1) must be less than Cmin , and the maximum error Emu of the profile (2) must be less than C mu . (.0. 101 < 0) Unless this condition, the intersection (bandwidth) of the two sets produced by (Tc}mu and (Tclmin cannot exist. We have treated a practical example of simulation. Figure 9 shows two cases of milling operations without and with compensation. Firstly, there is a plan to manufacture a workpiece. The purpose of this plan is to cut a surface indicated by a toleranced dimension 17~g:~~. In case (1), a milling process without compensation is presented. After this milling process, the milled surface stays outside the criteria of the tolerance. So, we cannot obtain the milled surface respecting the tolerance. On the contrary, in case (2), a milling process with compensation is presented. Before the milling process, we have obtained a compensated trajectory by the mirror method . After the milling process with compensated radial depth of cut, we have obtained a milled surface which stays inside the criteria of the tolerance. So, we have succeeded in obtaining the milled surface respecting the tolerance.
In figure 8, as the nominal trajectory T N is decomposed in N nominal tool positions, (Pc):"u is a compensated tool position which can compensate the tool deflection so that a maximum error Emu is equal to Cmu for a itl! nominal position, and (Pc ):"in is also a compensated tool position which can compensate the tool deflection so that a minimum error Emin is equal to Cmin for a ill! nominal position. (Tc)mu is a limit consisting of a set of (Pc):"u (Vi = 1,2,3,···,N), (TC)min is a limit consisting oh set of (Pc}:"in (Vi = 1, 2, 3", . ,N).
We have carried out another simulation for the compensation of the milling operation of a cylindrical workpiece (Depince et al., 1997). We have 367
fore, we choose the second method to carry out the compensation.
,~
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WITHOUT COMPENSATION
WITH COMPENSATION
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-
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_
-
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_
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-
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.
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_
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Fig. 10. Predicted milled surfaces with and without compensation
D
Failure
.-
Success k_
7. REFERENCES Armarego, E.J .A. and N.P. Deshpande (1994). Force prediction models and CAD / CAM software for helical tooth milling processes: 3. end milling and cutting analyses. International Journal oJ Production Reseo.rch. D~pince, Ph., J.Y. Hasc~t and T.I. Seo (1997). Surface prediction in end milling. 2rd International ICSC Symposia on IIA97 in Nfmes, FRANCE. Fujii, Y. , H. Iwabe and M. Suzuki (1979). Effect of dynamic behaviour of end mill in machining on work accuracy. Bulletin of the Japan Society of Precision Engineering 13(1), 20-26. HascOi!t , J .Y. , T.l Seo and Ph. Depince (1997). Compensation des deformations d 'outils pour la generation de trajectoires d'usinage. 16th Canadian Congress of Applied Mechanics CANCAM97 in Quebec, CANADA. Kline, W.A. , R.E. Devor and I. A. Shareef (1982). The prediction of surface accuracy in end milling. Tmnsactions of the ASME 104, 272278. Kops, L. and D.T. Vo (1990). Determination of the equivalent diameter of an end mill based on its compliance. Annals of CIRP 39/ 1, 93-
Fig. 9. Example of compensation to respect the given tolerance chosen the average value of the maximum and minimum errors for the reference of compensation. Two simulation results are shown in figure 10. After compensation, we have globally reduced the errors as compared to the figure 10, but some errors cannot be reduced. We could also take a form tolerance as criteria, instead of a dimension tolerance.
6. CONCLUSION We have presented the tool path compensation with respect to a given tolerance. Firstly, we have summarized the modeling and methodology to approach reasonable considerations for the path compensation. Then , we have taken into account the determination of reference of compensation. We have described the choice of the reference of compensation for the two different methods to predict the milled surface. The possibility to respect a given tolerance was mathematically verified in the two different cases of the surface prediction . In the impossible case of respecting the tolerance, we have taken into account references of the compensation which allow to guarantee this tolerance. 'Ve have presented some examples for the two cases of compensation. We have obtained reasonable results for the two cases. But, the second method allows to more precisely compensate the errors and to predict the milled surface. There-
96. Sub, S.H. , J.H. Cho and J.Y. Hascoet (1995). Incorporation of tool deflection in tool path computation. International Journal of Manufacturing System 15(3), 190-199. Tlusty, J. , S. Smith and C. Zamudio (1991). Evaluation of cutting performance of machining centers. Annals of CIRP 40/ 1, 405-410.
368