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Optimization and Dynamic Adaptation of the Cutter Inclination during Five-Axis Milling of Sculptured Surfaces
Optimization and Dynamic Adaptation of the Cutter Inclination during Five-Axis Milling of Sculptured Surfaces Jean-Pierre Kruth (I), Paul Klewais, Department of Mechanical Engineering, Katholieke Universiteit Leuven, Belgium Received on January 13,1994
Summary Five axis milling allows to machine free form surfaces with cylindrical or toric cutters instead of ball nose cutter. This drastically reduces the machining time. Commercial CAM modules however require the NC-programmer to specify the appropriate inclination of the tool with respect to the work piece surface normal. This is difficult especially for complex parts with vaving surface curvature. This paper presents five axis CAM software which vanes the tool inclination during the tool path generation, in order to achieve the best combination of scallop height, workpiece accuracy, surface roughness and machining cost. Machining time to produce some free form workpieces, given a maximum scallop height, could be reduced by 50%.
Keywords Milling, Tool path, Sculptured surfaces
Introduction. The increasing complexity of automotive parts and household utensils makes the potential benefits of five axis milling continuously grow. Some tool and die makers believe that by changing from 3 to 5 axis milling efficiency gains of 5 to 10 times could be achieved [1,5,8,10]. However five axis CAM-software is still expensive and often lacks flexibility when specifying the tool orientation. This paper relates the tool inclination (with respect to the surface normal) and the machining direction with the scallop height, the workpiece accuracy and the surface finish. An algorithm is proposed which dynamically optimises the tool inclination when machining free form surfaces. A global optimisation of the five axis tool path generation will aim for a better workpiece quality at a lower machining cost. This can be achieved by: 1. a reduction of the machining time; 2. a better geometrical accuracy of the workpiece; 3. an improved surface finish (low scallop height and surface roughness); 4. a longer tool life and a higher reliability of the cutting process: 5. a collision free tool path (eliminating the need of expensive test runs). Practical results obtained with different five axis milling strategies are presented at the end of the paper (see tables).
Scallops are produced between the successive tracks machined on the surface. The height of these scallops depends on the type of cutting tool, its orientation, its dimensions and the distance between the tracks. Replacing the traditional ball nose cutters with cylindircal cutters, inclined in the feed direction (alead), reduces the scallop height by sin(q,,d). The total reduction of the machining time may be more than ten times for some parts [8].Therefore in this paper we focus on the optimal use of cylindrical and toric end mills. Results obtained with ball nose, cutters will only be used for comparison purpose.
Tool inclination angle and its technological incidence. Commercial CAD/CAM systems require the NCprogrammer to specify the tool orientation. He usually specifies a constant lead ang/e (inclination of the tool axis in the feed direction with respect to the surface normal) and/or a tilt ang/e (inclination of the tool in the direction perpendicular to the feed).
Fig 1. Machining lechniques Comparing milling with ball nose and cylindrical cutters. To mill a surface the CAD/CAM system moves the cutter along a number of "parallel curves" (usually isoparametric curves or projection curves on the surface).
Annals of the ClRP Vol. 43/1/1994
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Different machining techniques. The first step in the tool path generation is the selection of the appropriate machining technique. Four main techniques can be applied (fig 1): 1. Machining with ball nose cutter; 2. Machining with the side of a cylindrical or toric cutter; 3. Cutting with the bottom of an end mill is a rarely used technique; 4. Ruled surfaces are efficiently machined with the side of the cutter positioned tangentially to the surface; The last method is not applicable for general free form surfaces and will not be treated in this paper.
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Fig. 2 Technologicalincidence of lead angle To investigate the technological implications of the tool inclination a test part made of steel was milled with different lead angles. Figure 2 shows the resulting surface roughness and accuracy. For ball nose cutters the surface roughness increases if the tool axis is positioned perpendicularly to the surface. This is caused by the low cutting speed, which leads to build up edge, and by the unfavourable cutting geometry at the tool tip. For toric cutters, on the other hand, the inclination angle influences the roughness only slightly. When applying a cylindrical tool with no or a small corner radius a large inclination angle increases the surface roughness strongly. However a small lead angle usually
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results in a better surface roughness than the one obtained with the tool axis oriented perpendicularly to the work plane. Lead angle -5 Lead angle -0 Fc Newton Fc Newton .4 0 0 ~ -.-.. -. . , 3001
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The tool inclination and its geometrical incidence: inclination optimisation in function of the local surface curvature.
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The best results were achieved when applying a small negative lead angle. The profile of the cutting forces (fig 3) explains this result. Due to a negative lead angle a larger component of the cutting forces is oriented in the axial direction (Fa). This has a stabilising effect. The course of the cutting forces versus time also becomes smoother. As a result the frequency content of the force signal is reduced. In this way strong vibrations can be avoided. However, the large cutting speed difference along the cutting edge, that occurs when cutting with the bottom of the tool and the difficult chip removal may lead to unfavourable machining conditions. As a consequence, a positive lead angle is mostly preferred.
a) Fig. 4 Tool deflection by cuttingforces The dimensional error on the workpiece increases for increasing lead angle (fig 2). This phenomenon, confirmed by Tonshoff [7], is explained in figure 4. The stiffness of the tool and its clamping is much higher in the direction of the tool axis than perpendicular to the axis. The cutting forces will bend the tool mainly in this last direction especially if large tool cantilever or tool extensions are applied. The resulting error on the machined surface is about: sin(inc1ination) deflection.
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The dimensional errors are not only caused by the bending of the tool but also by the limited positioning accuracy of the five axis milling machine and the reduced stiffness caused by the additional rotary axes. Large changes of the tool orientation often yield large machine slide movements whereby the positioning error enlarges.
From a purely geometrical point of view the inclination of the ball nose cutter can be chosen arbitrarily. This is not the case when machining with cylindrical or toric mills. The curvature of a convex surface to be milled imposes a minimal inclination of the cutter, in order to avoid undercutting the surface.
'hzmaximum scallop height Fig. 5 Contact length Fig. 6 Surface curvature Choosing the appropriate inclination angle is difficult especially if the surface curvature strongly varies along the tool path, if the inclination of the tool is limited by the range of the rotational axes of the milling machine or if collisions are to be avoided. On the one hand a minimum inclination angle is needed to avoid unwanted cutting into the workpiece. On the other hand a large inclination angle increases the scallop height and deteriorates the workpiece accuracy as the tool bends more due to the cutting forces. The optimal lead angle may be defined as the smallest tool inclination in the feed direction which avoids the tool to cut deeper into the theoretic surface than a given internal tolerance. This internal tolerance will be called into/, as usual in APT language. Besides the optimal tool inclination, one has to decide on the optimal distance between the successive milling tracks. The width of the milling track yielding a deviation equal to the maximum allowed scallop height will be called the contact /engtb in this text (fig 5). In this section a mathematical approach will be used which only considers the local surface curvature in the contact point between the cutter and the machined surface. A more robust algorithm to calculate the optimal tool orientation will be proposed in the next section. To estimate the appropriate lead angle of the cutter, the surface in the surrounding of the cutter contact point is approximated by a quadratic equation. The second order derivatives of the surfaces are assumed to be continuous around the contact point. The surface normal is considered positive if pointing out of the material and the curvature is said to be positive if it bends towards the surface normal. The maximum and the minimum curvature of a surface in a point P are called the principal curvatures and are denoted respectively kl and k2 (fig 6). The directions in which these extreme values occur are called the principal directions. The unit vectors in these directions u l and up are called the principal vectors. A special case occurs at umbilic points for which kl = k2 (for example a point on a sphere or a plane). In these points the surface bends to the same extent in all directions. For all other points (kl # k2) there are exactly two principal
directions and these are orthogonal. In the co-ordinate system XYZprincipal [i.e. ul=(l 0 0) u2=(0 1 O)] the shape of the surface near the point P is approximated by: z=% (kl#+k2$). This is called the quadratic approximation of the surface in the point P.
Fig.
Contour of cylindirca. -001 with corner radius located inside afictive sphere
The co-ordinate system XYZfeed is constructed in the contact point between cutter and work surface. The Z-axis is equal to the surface normal and the Y-axis points in the direction of the feed. The tool axis is inclined in the feed di.rection. To simplify the formulae in this section the contact point is assumed to be located exactly on the theoretical surface. Normally, the contact point is translated along the surface normal to an amount equal to the surface tolerance intol or outtol, whichever is most favourable to approximate the surface with a minimum number of tool positions. The effective tool radius is defined as the distance between the contact point and the tool axis, while the effective contour of the tool is defined as the circle around the tool axis passing through the contact point. Determining the intersection of the quadratic approximation of a free form surface and the toric shape of the cutter however still has to be solved numerically. To allow simple reasoning a fictive sphere is constructed which is tangential to the surface and the cutter in the point P and on which the effective contour of the cutter is located. Figure 7a shows that the effective contour of a cylindrical tool with radius R and corner radius cr is located on a sphere with radius equal to &here and whose centre is located on the surface normal. Rsphcre = (&utter cr)/sin(aleod)-+ cr (1) Figure 7b shows that the shape of the cutter is completely located inside this sphere. As a consequence the cutter can only penetrate the workpiece at a certain position if the fictive sphere also penetrates into the workpiece at that position. The intersection curves between the sphere and the quadratic approximation of the surface can easily be calculated. We will assume that the cutter penetrates the workpiece if the effective contour of the cutter (which is located on the fictive sphere) cuts these intersection curves.
If the internal tolerance is zero and thus no penetration of the cutter into the theoretic surface is allowed then neither the fictive sphere is allowed to cut the nominal surface. This implies that the radius of the sphere must be smaller than Vk1. However, usually a small penetration depth, equal to intol, is allowed. Therefore the following approach is used to estimate the optimal ollead (fig 8): 1. Use the quadratic approximation of the machined surface, offset this surface by intol.
2. Use the quadratic approximation of the fictive sphere on which the effective contour of the tool is located. 3. Calculate the intersection between both surfaces. 4. Project the intersection curves on the XYfeed-plane. This generally result in the two dark grey hyperboles depicted on fig 9. 5. Project the effective contour of the cutter on the XYfeedplane. As the cutter is inclined in the direction of Yfeed, this results in an ellipse (fig 9). If this ellipse cuts the projected intersection curves the mill cuts into the intol offset surface at that location. If the ellipse tangentially touches the intersection curve, the cutter touches the intol offset surface and the optimal lead angle is achieved. The equation of the intoi offset surfaces in the XYZprinc co-ordinate system is: 2 2 Xprin Y rin z=ki-tk 2 L - intol restriction k p k 2 (2) 2 2 The quadratic approximation for the fictive sphere is: 2=
Xirin + $rin
(3)
Rsphere
The intersection curve ( 2 - 3) gives the xy-location where the fictive sphere cuts the -into1 surface: (4)
If I/Rsphere > k l > k2 there is no solution and no undercuts occur. If k l > 1/Rsphere > k2 the Solution of equation 4 is a hyperbole. To prevent undercuts this curve may not cut the elliptic projection of the effective contour of the mill given by the equation:
where Refiect-cuRer equals the distance of the contact point and the tool axes. Combining equation 4 and 5 to calculate the minimum lead angle a yields a forth order equation. The symbolic solution of this equation is very complex. Generally it is easier to calculate the roots numerically. However if the feed direction equals the principal direction u1 or u2 a simple solution can be found. When the feed direction is equal to u l then XYZPin is equal to XYZfeed. If a is small then cos(a) may be approximated by the value one and Reffect-cuNer by
Solving equation 8 yields a single solution (I/Rsphere< kl): 1 Rsphere
= k1
-
$&=a
(9)
Resecr -curter
This formula shows that in this case the inclination of the tool is mainly determined by the maximum curvature kl. The influence of the second principal curvature k2 is
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limited since intol is usually small. The minimal inclination of the tool can be calculated using equation (1): = !cutter --CT = Rcutrer - cr 1 Rsphere - cr -cr + (10) 2( k, - b).intol
and u2 is even strengthened if only the symmetric contact region is considered (fig 9 ). maximum symmefric
1-
A second special case occurs if the feed direction equals the second principal direction u2 (fig 9 left). In this case the soulution yields: Rsphere = l/kl (fig 7b).
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Feed direction = u2Optimal lead angle 22.57
Angle Feed - u, = 10 deg Optimal lead angle 23.3
Angle Feed - u, = 45 deg Optimal lead angle 23.54
Feed direction = u, Optimal lead angle 23.6
1 yfeed\
rhnlir Intersection
ere and (4ntol) offset stiface (equation 4) Fig. 8 Intersection sphere and intol surface
Scallop height and contact length (or pick feed). The contact length is approximated by calculating the intersection of the effective contour and the offset surface, which is
Surace kl = 0.04 k2 = 0.01 Into1 = 0.005 Tool offset = 0 Max Scallop Height = 0.01 Cylindrical endmill diameter 20 Sphere cuts beneath intol surface (undercut) Sphere between intol and outol surface, distance cutter - surface < m a . scallop height Fig. 9 Influence of machining direction on scallop height max
Other constraints
scallob If k l > l/RSphere > k2 , which is the case for the optimal lead angle a,the solution of equation 11 is a hyperbole giving the bound of the contact region between cutter and surface. If the lead angle is chosen larger than the optimal value and if 1/Rsphere > kl > k2 equation 11 represents an ellipse. The intersections of this quadratic curve (1 1) and the elliptic projection of the effective contour of the cutter (5) will allow us to determine the contact length. Figure 9 shows the graphical representation of equations 4, 5 and 11 when the optimal lead angle is applied in a specific point on the machined surface. The different subfigures apply to different feed directions. The cutter is always inclined in the direction of the feed. The contour of the cutter touches the hyperboles representing equation 3. The contact length is defined by the intersection of the contour of the cutter and the curve described by equation 11. Figure 9 clearly shows that the best results are obtained if the angle p between the feed and u2 is small. If this angle increases the contact region first becomes asymmetric w.r.t. the feed direction. This might be a disadvantage since only very complex algorithms, to divide the surface into successive machining paths, will make optimal use of the full contact length. Figure 10 shows the contact length when machining a surface with different feed directions. The feed direction strongly influences the contact length especially if kl and k2 differ a lot. Choosing the appropriate machining direction might reduce the number of cutting tracks by ten while still maintaining the same scallop height. The importance of having a small angle p between the feed
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0.01 l k p h e r e = kl 0.01 Tool-offset -0.004 0.02 0.01
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k2 = 0.0 l / m intol = 0.005
0 20 40 60 80 90 Angle between feed direction (Yfeed)and u2 (Degree) Fig. 10 Contact length /feed direction If the angle p rises the contact length abruptly drops. As mentioned earlier, the contact length results from the intersection of two quadratic curves 9 and 10. This gives a maximum of four intersections (fig l l a ) . At the discontinuity the contact length moves from one of these solutions (C) to another (A). If however we would still assume the contact length to go until point C the successive milling tracks would be separated too far. This would cause a deviation between the theoretic and the actually milled surface (fig 11b). A later manual polishing operation will normally only remove the scallop in point C
on the surface [3,4]. The excess of material between point A and 6 is not corrected since it is usually not visually noticed. This phenomenon will not occur if a ball nose cutter is applied. As a result the maximum scallop height should be chosen smaller for cylindrical mills.
only considering the local curvature but the complete workpiece geometry.
Scallop between
\ k c t u a l milled surface Undercut Material removed Contact during manual benching Fig. I 1 Shape of scallops on fiee form surj4ace f
The difference between k l and I/Rsphere is normally small (see eq. 9) and can for large values of the angle p even be neglected. Figure 10 shows that only when the angle p is small, there is a big difference between the contact length when on the one hand the optimal inclination angle is applied (solid line) and on the other hand when 1/Rsphere is taken equal to k l . As a result one could consider applying the rule 1/Rsphere equals k l to avoid difficult calculations of the lead angle. As the maximum scallop height becomes larger with respect to the tool dimension the influence of the feed direction diminishes. The influence of the tool dimensions is remarkable. A smaller cutter diameter might even lead to a larger contact length. This is due to the influence of the tool diameter on the minimum lead angle. For convex parts larger tool diameters usually give smaller scallops. Milling concave parts demand a good combination of tool dimension versus surface geometry to obtain optimal results. The maximum tool load, resulting from the cutting forces, and eventual geometric constraints, restricting the maximum tool inclination, will limit the choice of the cutter. One could consider to always machine with the feed along the principal direction with minimum curvature (u2). The principal direction however often varies irregularly as shown in figure 12. Usually the machining direction is matched to the principal direction u2 as well as possible.
Fig. 12 u2 curves (principal direction)
Software to automatically calculate the optimal inclination of the cutter for finish milling sculptured surfaces. This section describes algorithms which have been implemented in CAM software that continuously fits the tool inclination to the surface curvature. In the previous section a theoretical approximation was made of the optimal inclination of the cutter with respect to the surface. Unfortunately, the surface curvature is often characterised by sudden changes. Applying the formulae of the previous section may result in abrupt changes of the cutter orientation. A more general algorithm was developed not
Fig. 13 Optimisation of the lead angle of a toric cutter
Suppose a toric cutter positioned in a point P1 on a surface parameterised by u and v (Fig. 13). A first estimation of the inclination angle can be calculated by means of the local surface curvature in the point P1. The estimation is used to give the tool a first orientation. To not penetrate the surface in any point the distance between the cutter and the surface must be larger than intol for every point on the tool. The distance between the cutter and the surface is equal to: - cr (See fig.13a). Circle C is the centre of the toric edge of the cutter (radius of C = Rcutter - cr). The circle C can be described implicitly or in a parametric way : C(y) = center + vl.cos(y)+ v2.sin(y) and - n c y c n The distance between the surface and the cutter in the pointy equals: dist_surf(y)= min,,,((lSutf(u, v) - C(Y)I\)- cr
For each y the values u and v must be searched which minimise distsurf. If dist-surf is less than -into1 (the allowed undercut) the cutter penetrates to deep into the surface.Therefore a larger lead angle a is needed to avoid this penetration (See fig. 13b). The additional lead angle A a can be calculated by the formula: dist-surf + intol A a = -arctan( 1 distance- to- rot- axis Discto-rotaxis is the distance from the point on the surface to the axis around which the cutter rotates as the lead angle a changes. A good approximation for the previous formula is:
The value A a must be calculated for each y and the maximum value must be applied to avoid undercut in any position on the surface. If however the maximum A a is negative the lead angle may be reduced to obtain a better fit of the cutter onto the surface. The maximum A a has to be obtained for these values u, v and y where the function F reaches a global minimum. As a result the maximum A a can be determined by calculating the values u, v and y which minimise the function:
The formula can be adapted for cylindrical end mills by introducing a fourth parameter r which varies the radius of the toric cutter from 0 to &utter - cr. The method described here gives the optimal inclination angle of the cutter from a purely geometrical point of view. From a technological point of view however it may be desirable to limit the contact length to a certain extent or to avoid the cutter to cut with the back point P2 into the rough pre-
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milled or blank surface (Fig 13b). This is easily obtained by replacing G with: . k i u r ~ u v) , - C(v)).(SurAu,v)- C(V)) -cr-(l -co$y)) * offser+intol (1 -Cody)) * f RcL,ller-cr)
The use of this modified formula will guarantee the distance between the tool in the point -1 and the surface to be at least (1 - cos(7)) offset.
Contact length for fixed and variable inclination angle. Figure 14 shows the region where the scallop on a free form surface is less then 0.2 mm. The surface is milled with a toric cutter (diameter 20 corner radius 4), first with a variable lead angle and second with a constant lead angle equal to the minimal value that still prevents the cutter from penetrating into the workpiece. This figure clearly shows that a variable lead angle will reduce the number of cutting tracks needed to achieve a certain surface finish. The selection of an appropriate offset in the previous formulae keeps the cutting angle cp more
cutting forces and the resulting bending of the tool and the machine structure. This is import for five axis machines which are characterised by a poor stiffness. However the higher corner radius of the toric cutter reduces the surface roughness and the lower mean chip tickness allows a higher feed pro tooth. As a result the use off toric cutters can reduce as well the machining time as the surface roughness. The table below shows the results obtained when milling the surface of the figure right in different directions (vl, v2, v3, v4) with coated carbide cutter (feed always 0.1 mm/thooth and ae=5mm ): Table 3: Surface finish for different machining directions Scallop Cutter type I Location and feed 1 Ra I height I direction
I I
v2 cylindrical 2 v l (feed dir = up) toric
10.6pm 170 pm 3 pm 3.7 pm
I
0.5pm 0.8 pm cylindrical 2 v3 1.5pm v4 (feed dir= ~ 2 ) 1.5 urn v3 V4
-cutter less than 0.2 Toric cutter: diameter 20, comer radius 4 Minimal constant lead angle Fig. I4 Deviation cutter-surface
ee convex
Table 2: Surface finish for different cutter types I Surface 1 Ra I Scallop I time I RMS height deviatio
toric cvlind.1 I ball nose toric cvlind.1
I
1
I
I concave 10.5pm I
4 pm 4.5 urn I concave I 1 um I75 um convex 0.5 pm 16 pm I convex 10.8 urn I20 urn
I concave I 0.7 um I
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16'30" 20 pm 18'14" I 8 um I 7'15 I >50 urn I 7'10" 25 pm 19'55" I 15 urn
1
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The best workpiece accuracy was achieved with cylindrical end mills with cermets [9]. The positive cutting geometry and the reduced feed pro tooth [9] decrease the
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22 8 40 5
pm pm pm urn
Conclusion. The optimisation of the tool inclination and the machining direction can on the one hand achieve a serious reduction of the machining time while maintaining the same scallop height. On the other hand the cutting angle cp can be restricted to obtain proper cutting conditions.
References.
The resulting surface roughness, scallop height and workpiece accuracy, when machining a convex and a concave surface, are shown in the table below:
I Cutter
(feed dir=U2)
I
We notice an enormous difference of the scallop height when machining the same surface with a different feed direction (up to 35 x). The best result were always achieved with the feed direction parallel to the principal direction up with minimum curvature ( i.e. the most convex or the least concave direction).
0Distance surface
Machining results. Free form workpieces (100*100 mm) were milled out off mould steel HB 330 applying the algorithms 'Oncave described in the previous section (ae = 5mm, ap = 1mm).
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1. Altan T., et. al., "Advanced techniques for die and mould manufacturing", Annals of CIRP, Vol42/2, pp 707-716, 1993. 2. Brunotte D., "Funf-Achsen-Frasenim Modell- und Werkzeugbau", WerksfaftundBetrieb 125 (l), pp 15-17, 1992. 3. Hernandez-CamachoJ., Gehring V., "Feinbearbeitung im Werkzeug- und Formenbau", VDI-Z 132(2), pp 51-55, 1990. 4. Hock S.and Janovsky D., "Freiformflachenim Werkzeug- und Formenbau bearbeiten", Werkstatt und Betrieb 125 (8), pp 597-606, 1992. 5. Konig M., "New technologies in milling", 7th lnf. Conf. of Tool, Die and Mould Indutry, ISTA, Bergamo, Italy, May 1992. 6. Meretz H. and Seeheim-Jugenheim, "Konstruction un Fertigung van Freiformflachen", 124 (7), pp 585-588, 1991. 7. Tonshoff H.K., "Hohlformbearbeitungdurch Drei- und Funfachsenfrasen", VDI-Z 131(9), pp 77-81, 1989. 8. Vickers, G.W., Quan, K.W., "Ball-mills versus end-mill for curved surface machining', Journal of Engineering for industry, Feb. 1989. 9. Wagner, R, "Frasen mit Cermets", Werkstatt und Betrieb, 125 (l), pp 563-565, 1992. 10. N, "Step up to five-axis programming?", Manufacturing Engineering, November 1993, pp. 55-60.
Summary Five axis milling allows to machine free form surfaces with cylindrical or toric cutters instead of ball nose cutter. This drastically reduces the machining time. Commercial CAM modules however require the NC-programmer to specify the appropriate inclination of the tool with respect to the work piece surface normal. This is difficult especially for complex parts with vaving surface curvature. This paper presents five axis CAM software which vanes the tool inclination during the tool path generation, in order to achieve the best combination of scallop height, workpiece accuracy, surface roughness and machining cost. Machining time to produce some free form workpieces, given a maximum scallop height, could be reduced by 50%.
Keywords Milling, Tool path, Sculptured surfaces
Introduction. The increasing complexity of automotive parts and household utensils makes the potential benefits of five axis milling continuously grow. Some tool and die makers believe that by changing from 3 to 5 axis milling efficiency gains of 5 to 10 times could be achieved [1,5,8,10]. However five axis CAM-software is still expensive and often lacks flexibility when specifying the tool orientation. This paper relates the tool inclination (with respect to the surface normal) and the machining direction with the scallop height, the workpiece accuracy and the surface finish. An algorithm is proposed which dynamically optimises the tool inclination when machining free form surfaces. A global optimisation of the five axis tool path generation will aim for a better workpiece quality at a lower machining cost. This can be achieved by: 1. a reduction of the machining time; 2. a better geometrical accuracy of the workpiece; 3. an improved surface finish (low scallop height and surface roughness); 4. a longer tool life and a higher reliability of the cutting process: 5. a collision free tool path (eliminating the need of expensive test runs). Practical results obtained with different five axis milling strategies are presented at the end of the paper (see tables).
Scallops are produced between the successive tracks machined on the surface. The height of these scallops depends on the type of cutting tool, its orientation, its dimensions and the distance between the tracks. Replacing the traditional ball nose cutters with cylindircal cutters, inclined in the feed direction (alead), reduces the scallop height by sin(q,,d). The total reduction of the machining time may be more than ten times for some parts [8].Therefore in this paper we focus on the optimal use of cylindrical and toric end mills. Results obtained with ball nose, cutters will only be used for comparison purpose.
Tool inclination angle and its technological incidence. Commercial CAD/CAM systems require the NCprogrammer to specify the tool orientation. He usually specifies a constant lead ang/e (inclination of the tool axis in the feed direction with respect to the surface normal) and/or a tilt ang/e (inclination of the tool in the direction perpendicular to the feed).
Fig 1. Machining lechniques Comparing milling with ball nose and cylindrical cutters. To mill a surface the CAD/CAM system moves the cutter along a number of "parallel curves" (usually isoparametric curves or projection curves on the surface).
Annals of the ClRP Vol. 43/1/1994
:
:
,
:
I
:
,
:
:
,
:
'
I
3---.---+--
2
Different machining techniques. The first step in the tool path generation is the selection of the appropriate machining technique. Four main techniques can be applied (fig 1): 1. Machining with ball nose cutter; 2. Machining with the side of a cylindrical or toric cutter; 3. Cutting with the bottom of an end mill is a rarely used technique; 4. Ruled surfaces are efficiently machined with the side of the cutter positioned tangentially to the surface; The last method is not applicable for general free form surfaces and will not be treated in this paper.
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0.5. - - -,---: " I t 0 p-L-..L--L->-
YIU"
a.5.-
YIW
~
LU'b."
Fig. 2 Technologicalincidence of lead angle To investigate the technological implications of the tool inclination a test part made of steel was milled with different lead angles. Figure 2 shows the resulting surface roughness and accuracy. For ball nose cutters the surface roughness increases if the tool axis is positioned perpendicularly to the surface. This is caused by the low cutting speed, which leads to build up edge, and by the unfavourable cutting geometry at the tool tip. For toric cutters, on the other hand, the inclination angle influences the roughness only slightly. When applying a cylindrical tool with no or a small corner radius a large inclination angle increases the surface roughness strongly. However a small lead angle usually
443
results in a better surface roughness than the one obtained with the tool axis oriented perpendicularly to the work plane. Lead angle -5 Lead angle -0 Fc Newton Fc Newton .4 0 0 ~ -.-.. -. . , 3001
"
-'0°/ -200
The tool inclination and its geometrical incidence: inclination optimisation in function of the local surface curvature.
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Fa Newton
Fa Newton
1.8 rotation ~
time Surface roughness profile pm
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1 2 3mm 0 I 2 Fig 3. Cuttingforces and surface roughness when applying different lead angles
3mm
The best results were achieved when applying a small negative lead angle. The profile of the cutting forces (fig 3) explains this result. Due to a negative lead angle a larger component of the cutting forces is oriented in the axial direction (Fa). This has a stabilising effect. The course of the cutting forces versus time also becomes smoother. As a result the frequency content of the force signal is reduced. In this way strong vibrations can be avoided. However, the large cutting speed difference along the cutting edge, that occurs when cutting with the bottom of the tool and the difficult chip removal may lead to unfavourable machining conditions. As a consequence, a positive lead angle is mostly preferred.
a) Fig. 4 Tool deflection by cuttingforces The dimensional error on the workpiece increases for increasing lead angle (fig 2). This phenomenon, confirmed by Tonshoff [7], is explained in figure 4. The stiffness of the tool and its clamping is much higher in the direction of the tool axis than perpendicular to the axis. The cutting forces will bend the tool mainly in this last direction especially if large tool cantilever or tool extensions are applied. The resulting error on the machined surface is about: sin(inc1ination) deflection.
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The dimensional errors are not only caused by the bending of the tool but also by the limited positioning accuracy of the five axis milling machine and the reduced stiffness caused by the additional rotary axes. Large changes of the tool orientation often yield large machine slide movements whereby the positioning error enlarges.
From a purely geometrical point of view the inclination of the ball nose cutter can be chosen arbitrarily. This is not the case when machining with cylindrical or toric mills. The curvature of a convex surface to be milled imposes a minimal inclination of the cutter, in order to avoid undercutting the surface.
'hzmaximum scallop height Fig. 5 Contact length Fig. 6 Surface curvature Choosing the appropriate inclination angle is difficult especially if the surface curvature strongly varies along the tool path, if the inclination of the tool is limited by the range of the rotational axes of the milling machine or if collisions are to be avoided. On the one hand a minimum inclination angle is needed to avoid unwanted cutting into the workpiece. On the other hand a large inclination angle increases the scallop height and deteriorates the workpiece accuracy as the tool bends more due to the cutting forces. The optimal lead angle may be defined as the smallest tool inclination in the feed direction which avoids the tool to cut deeper into the theoretic surface than a given internal tolerance. This internal tolerance will be called into/, as usual in APT language. Besides the optimal tool inclination, one has to decide on the optimal distance between the successive milling tracks. The width of the milling track yielding a deviation equal to the maximum allowed scallop height will be called the contact /engtb in this text (fig 5). In this section a mathematical approach will be used which only considers the local surface curvature in the contact point between the cutter and the machined surface. A more robust algorithm to calculate the optimal tool orientation will be proposed in the next section. To estimate the appropriate lead angle of the cutter, the surface in the surrounding of the cutter contact point is approximated by a quadratic equation. The second order derivatives of the surfaces are assumed to be continuous around the contact point. The surface normal is considered positive if pointing out of the material and the curvature is said to be positive if it bends towards the surface normal. The maximum and the minimum curvature of a surface in a point P are called the principal curvatures and are denoted respectively kl and k2 (fig 6). The directions in which these extreme values occur are called the principal directions. The unit vectors in these directions u l and up are called the principal vectors. A special case occurs at umbilic points for which kl = k2 (for example a point on a sphere or a plane). In these points the surface bends to the same extent in all directions. For all other points (kl # k2) there are exactly two principal
directions and these are orthogonal. In the co-ordinate system XYZprincipal [i.e. ul=(l 0 0) u2=(0 1 O)] the shape of the surface near the point P is approximated by: z=% (kl#+k2$). This is called the quadratic approximation of the surface in the point P.
Fig.
Contour of cylindirca. -001 with corner radius located inside afictive sphere
The co-ordinate system XYZfeed is constructed in the contact point between cutter and work surface. The Z-axis is equal to the surface normal and the Y-axis points in the direction of the feed. The tool axis is inclined in the feed di.rection. To simplify the formulae in this section the contact point is assumed to be located exactly on the theoretical surface. Normally, the contact point is translated along the surface normal to an amount equal to the surface tolerance intol or outtol, whichever is most favourable to approximate the surface with a minimum number of tool positions. The effective tool radius is defined as the distance between the contact point and the tool axis, while the effective contour of the tool is defined as the circle around the tool axis passing through the contact point. Determining the intersection of the quadratic approximation of a free form surface and the toric shape of the cutter however still has to be solved numerically. To allow simple reasoning a fictive sphere is constructed which is tangential to the surface and the cutter in the point P and on which the effective contour of the cutter is located. Figure 7a shows that the effective contour of a cylindrical tool with radius R and corner radius cr is located on a sphere with radius equal to &here and whose centre is located on the surface normal. Rsphcre = (&utter cr)/sin(aleod)-+ cr (1) Figure 7b shows that the shape of the cutter is completely located inside this sphere. As a consequence the cutter can only penetrate the workpiece at a certain position if the fictive sphere also penetrates into the workpiece at that position. The intersection curves between the sphere and the quadratic approximation of the surface can easily be calculated. We will assume that the cutter penetrates the workpiece if the effective contour of the cutter (which is located on the fictive sphere) cuts these intersection curves.
If the internal tolerance is zero and thus no penetration of the cutter into the theoretic surface is allowed then neither the fictive sphere is allowed to cut the nominal surface. This implies that the radius of the sphere must be smaller than Vk1. However, usually a small penetration depth, equal to intol, is allowed. Therefore the following approach is used to estimate the optimal ollead (fig 8): 1. Use the quadratic approximation of the machined surface, offset this surface by intol.
2. Use the quadratic approximation of the fictive sphere on which the effective contour of the tool is located. 3. Calculate the intersection between both surfaces. 4. Project the intersection curves on the XYfeed-plane. This generally result in the two dark grey hyperboles depicted on fig 9. 5. Project the effective contour of the cutter on the XYfeedplane. As the cutter is inclined in the direction of Yfeed, this results in an ellipse (fig 9). If this ellipse cuts the projected intersection curves the mill cuts into the intol offset surface at that location. If the ellipse tangentially touches the intersection curve, the cutter touches the intol offset surface and the optimal lead angle is achieved. The equation of the intoi offset surfaces in the XYZprinc co-ordinate system is: 2 2 Xprin Y rin z=ki-tk 2 L - intol restriction k p k 2 (2) 2 2 The quadratic approximation for the fictive sphere is: 2=
Xirin + $rin
(3)
Rsphere
The intersection curve ( 2 - 3) gives the xy-location where the fictive sphere cuts the -into1 surface: (4)
If I/Rsphere > k l > k2 there is no solution and no undercuts occur. If k l > 1/Rsphere > k2 the Solution of equation 4 is a hyperbole. To prevent undercuts this curve may not cut the elliptic projection of the effective contour of the mill given by the equation:
where Refiect-cuRer equals the distance of the contact point and the tool axes. Combining equation 4 and 5 to calculate the minimum lead angle a yields a forth order equation. The symbolic solution of this equation is very complex. Generally it is easier to calculate the roots numerically. However if the feed direction equals the principal direction u1 or u2 a simple solution can be found. When the feed direction is equal to u l then XYZPin is equal to XYZfeed. If a is small then cos(a) may be approximated by the value one and Reffect-cuNer by
Solving equation 8 yields a single solution (I/Rsphere< kl): 1 Rsphere
= k1
-
$&=a
(9)
Resecr -curter
This formula shows that in this case the inclination of the tool is mainly determined by the maximum curvature kl. The influence of the second principal curvature k2 is
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limited since intol is usually small. The minimal inclination of the tool can be calculated using equation (1): = !cutter --CT = Rcutrer - cr 1 Rsphere - cr -cr + (10) 2( k, - b).intol
and u2 is even strengthened if only the symmetric contact region is considered (fig 9 ). maximum symmefric
1-
A second special case occurs if the feed direction equals the second principal direction u2 (fig 9 left). In this case the soulution yields: Rsphere = l/kl (fig 7b).
ZI'
Feed direction = u2Optimal lead angle 22.57
Angle Feed - u, = 10 deg Optimal lead angle 23.3
Angle Feed - u, = 45 deg Optimal lead angle 23.54
Feed direction = u, Optimal lead angle 23.6
1 yfeed\
rhnlir Intersection
ere and (4ntol) offset stiface (equation 4) Fig. 8 Intersection sphere and intol surface
Scallop height and contact length (or pick feed). The contact length is approximated by calculating the intersection of the effective contour and the offset surface, which is
Surace kl = 0.04 k2 = 0.01 Into1 = 0.005 Tool offset = 0 Max Scallop Height = 0.01 Cylindrical endmill diameter 20 Sphere cuts beneath intol surface (undercut) Sphere between intol and outol surface, distance cutter - surface < m a . scallop height Fig. 9 Influence of machining direction on scallop height max
Other constraints
scallob If k l > l/RSphere > k2 , which is the case for the optimal lead angle a,the solution of equation 11 is a hyperbole giving the bound of the contact region between cutter and surface. If the lead angle is chosen larger than the optimal value and if 1/Rsphere > kl > k2 equation 11 represents an ellipse. The intersections of this quadratic curve (1 1) and the elliptic projection of the effective contour of the cutter (5) will allow us to determine the contact length. Figure 9 shows the graphical representation of equations 4, 5 and 11 when the optimal lead angle is applied in a specific point on the machined surface. The different subfigures apply to different feed directions. The cutter is always inclined in the direction of the feed. The contour of the cutter touches the hyperboles representing equation 3. The contact length is defined by the intersection of the contour of the cutter and the curve described by equation 11. Figure 9 clearly shows that the best results are obtained if the angle p between the feed and u2 is small. If this angle increases the contact region first becomes asymmetric w.r.t. the feed direction. This might be a disadvantage since only very complex algorithms, to divide the surface into successive machining paths, will make optimal use of the full contact length. Figure 10 shows the contact length when machining a surface with different feed directions. The feed direction strongly influences the contact length especially if kl and k2 differ a lot. Choosing the appropriate machining direction might reduce the number of cutting tracks by ten while still maintaining the same scallop height. The importance of having a small angle p between the feed
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O.O1
I
0.01 l k p h e r e = kl 0.01 Tool-offset -0.004 0.02 0.01
I
k2 = 0.0 l / m intol = 0.005
0 20 40 60 80 90 Angle between feed direction (Yfeed)and u2 (Degree) Fig. 10 Contact length /feed direction If the angle p rises the contact length abruptly drops. As mentioned earlier, the contact length results from the intersection of two quadratic curves 9 and 10. This gives a maximum of four intersections (fig l l a ) . At the discontinuity the contact length moves from one of these solutions (C) to another (A). If however we would still assume the contact length to go until point C the successive milling tracks would be separated too far. This would cause a deviation between the theoretic and the actually milled surface (fig 11b). A later manual polishing operation will normally only remove the scallop in point C
on the surface [3,4]. The excess of material between point A and 6 is not corrected since it is usually not visually noticed. This phenomenon will not occur if a ball nose cutter is applied. As a result the maximum scallop height should be chosen smaller for cylindrical mills.
only considering the local curvature but the complete workpiece geometry.
Scallop between
\ k c t u a l milled surface Undercut Material removed Contact during manual benching Fig. I 1 Shape of scallops on fiee form surj4ace f
The difference between k l and I/Rsphere is normally small (see eq. 9) and can for large values of the angle p even be neglected. Figure 10 shows that only when the angle p is small, there is a big difference between the contact length when on the one hand the optimal inclination angle is applied (solid line) and on the other hand when 1/Rsphere is taken equal to k l . As a result one could consider applying the rule 1/Rsphere equals k l to avoid difficult calculations of the lead angle. As the maximum scallop height becomes larger with respect to the tool dimension the influence of the feed direction diminishes. The influence of the tool dimensions is remarkable. A smaller cutter diameter might even lead to a larger contact length. This is due to the influence of the tool diameter on the minimum lead angle. For convex parts larger tool diameters usually give smaller scallops. Milling concave parts demand a good combination of tool dimension versus surface geometry to obtain optimal results. The maximum tool load, resulting from the cutting forces, and eventual geometric constraints, restricting the maximum tool inclination, will limit the choice of the cutter. One could consider to always machine with the feed along the principal direction with minimum curvature (u2). The principal direction however often varies irregularly as shown in figure 12. Usually the machining direction is matched to the principal direction u2 as well as possible.
Fig. 12 u2 curves (principal direction)
Software to automatically calculate the optimal inclination of the cutter for finish milling sculptured surfaces. This section describes algorithms which have been implemented in CAM software that continuously fits the tool inclination to the surface curvature. In the previous section a theoretical approximation was made of the optimal inclination of the cutter with respect to the surface. Unfortunately, the surface curvature is often characterised by sudden changes. Applying the formulae of the previous section may result in abrupt changes of the cutter orientation. A more general algorithm was developed not
Fig. 13 Optimisation of the lead angle of a toric cutter
Suppose a toric cutter positioned in a point P1 on a surface parameterised by u and v (Fig. 13). A first estimation of the inclination angle can be calculated by means of the local surface curvature in the point P1. The estimation is used to give the tool a first orientation. To not penetrate the surface in any point the distance between the cutter and the surface must be larger than intol for every point on the tool. The distance between the cutter and the surface is equal to:
For each y the values u and v must be searched which minimise distsurf. If dist-surf is less than -into1 (the allowed undercut) the cutter penetrates to deep into the surface.Therefore a larger lead angle a is needed to avoid this penetration (See fig. 13b). The additional lead angle A a can be calculated by the formula: dist-surf + intol A a = -arctan( 1 distance- to- rot- axis Discto-rotaxis is the distance from the point on the surface to the axis around which the cutter rotates as the lead angle a changes. A good approximation for the previous formula is:
The value A a must be calculated for each y and the maximum value must be applied to avoid undercut in any position on the surface. If however the maximum A a is negative the lead angle may be reduced to obtain a better fit of the cutter onto the surface. The maximum A a has to be obtained for these values u, v and y where the function F reaches a global minimum. As a result the maximum A a can be determined by calculating the values u, v and y which minimise the function:
The formula can be adapted for cylindrical end mills by introducing a fourth parameter r which varies the radius of the toric cutter from 0 to &utter - cr. The method described here gives the optimal inclination angle of the cutter from a purely geometrical point of view. From a technological point of view however it may be desirable to limit the contact length to a certain extent or to avoid the cutter to cut with the back point P2 into the rough pre-
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milled or blank surface (Fig 13b). This is easily obtained by replacing G with: . k i u r ~ u v) , - C(v)).(SurAu,v)- C(V)) -cr-(l -co$y)) * offser+intol (1 -Cody)) * f RcL,ller-cr)
The use of this modified formula will guarantee the distance between the tool in the point -1 and the surface to be at least (1 - cos(7)) offset.
Contact length for fixed and variable inclination angle. Figure 14 shows the region where the scallop on a free form surface is less then 0.2 mm. The surface is milled with a toric cutter (diameter 20 corner radius 4), first with a variable lead angle and second with a constant lead angle equal to the minimal value that still prevents the cutter from penetrating into the workpiece. This figure clearly shows that a variable lead angle will reduce the number of cutting tracks needed to achieve a certain surface finish. The selection of an appropriate offset in the previous formulae keeps the cutting angle cp more
cutting forces and the resulting bending of the tool and the machine structure. This is import for five axis machines which are characterised by a poor stiffness. However the higher corner radius of the toric cutter reduces the surface roughness and the lower mean chip tickness allows a higher feed pro tooth. As a result the use off toric cutters can reduce as well the machining time as the surface roughness. The table below shows the results obtained when milling the surface of the figure right in different directions (vl, v2, v3, v4) with coated carbide cutter (feed always 0.1 mm/thooth and ae=5mm ): Table 3: Surface finish for different machining directions Scallop Cutter type I Location and feed 1 Ra I height I direction
I I
v2 cylindrical 2 v l (feed dir = up) toric
10.6pm 170 pm 3 pm 3.7 pm
I
0.5pm 0.8 pm cylindrical 2 v3 1.5pm v4 (feed dir= ~ 2 ) 1.5 urn v3 V4
-cutter less than 0.2 Toric cutter: diameter 20, comer radius 4 Minimal constant lead angle Fig. I4 Deviation cutter-surface
ee convex
Table 2: Surface finish for different cutter types I Surface 1 Ra I Scallop I time I RMS height deviatio
toric cvlind.1 I ball nose toric cvlind.1
I
1
I
I concave 10.5pm I
4 pm 4.5 urn I concave I 1 um I75 um convex 0.5 pm 16 pm I convex 10.8 urn I20 urn
I concave I 0.7 um I
I
I
I
I
I
I
I
16'30" 20 pm 18'14" I 8 um I 7'15 I >50 urn I 7'10" 25 pm 19'55" I 15 urn
1
I
The best workpiece accuracy was achieved with cylindrical end mills with cermets [9]. The positive cutting geometry and the reduced feed pro tooth [9] decrease the
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22 8 40 5
pm pm pm urn
Conclusion. The optimisation of the tool inclination and the machining direction can on the one hand achieve a serious reduction of the machining time while maintaining the same scallop height. On the other hand the cutting angle cp can be restricted to obtain proper cutting conditions.
References.
The resulting surface roughness, scallop height and workpiece accuracy, when machining a convex and a concave surface, are shown in the table below:
I Cutter
(feed dir=U2)
I
We notice an enormous difference of the scallop height when machining the same surface with a different feed direction (up to 35 x). The best result were always achieved with the feed direction parallel to the principal direction up with minimum curvature ( i.e. the most convex or the least concave direction).
0Distance surface
Machining results. Free form workpieces (100*100 mm) were milled out off mould steel HB 330 applying the algorithms 'Oncave described in the previous section (ae = 5mm, ap = 1mm).
I
I
1. Altan T., et. al., "Advanced techniques for die and mould manufacturing", Annals of CIRP, Vol42/2, pp 707-716, 1993. 2. Brunotte D., "Funf-Achsen-Frasenim Modell- und Werkzeugbau", WerksfaftundBetrieb 125 (l), pp 15-17, 1992. 3. Hernandez-CamachoJ., Gehring V., "Feinbearbeitung im Werkzeug- und Formenbau", VDI-Z 132(2), pp 51-55, 1990. 4. Hock S.and Janovsky D., "Freiformflachenim Werkzeug- und Formenbau bearbeiten", Werkstatt und Betrieb 125 (8), pp 597-606, 1992. 5. Konig M., "New technologies in milling", 7th lnf. Conf. of Tool, Die and Mould Indutry, ISTA, Bergamo, Italy, May 1992. 6. Meretz H. and Seeheim-Jugenheim, "Konstruction un Fertigung van Freiformflachen", 124 (7), pp 585-588, 1991. 7. Tonshoff H.K., "Hohlformbearbeitungdurch Drei- und Funfachsenfrasen", VDI-Z 131(9), pp 77-81, 1989. 8. Vickers, G.W., Quan, K.W., "Ball-mills versus end-mill for curved surface machining', Journal of Engineering for industry, Feb. 1989. 9. Wagner, R, "Frasen mit Cermets", Werkstatt und Betrieb, 125 (l), pp 563-565, 1992. 10. N, "Step up to five-axis programming?", Manufacturing Engineering, November 1993, pp. 55-60.