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Collision-Free Five-Axis Milling of Twisted Ruled Surfaces
Collision-Free Five-Axis Milling of Twisted Ruled Surfaces Fritz Rehsteiner - Submitted by H.J. Renker (1) Received on January 13,1993
SUMMARY Ruled surfaces (generated by straight ruling lines) are a soecial kind of free form surfaces which plays an important role, e g , in aerodynamic surfaces (rotary engines) They are particularly well suited for five axis milling by the flank milling method In the case of twisted ruled surfaces, however, the obvious milling method of having the cutter touch these ruling lines along a generator of its own flank leads to collision problems The nature of these twisted ruled surfaces is analysed and conditions for their collision free milling by a cylindrical cutter are established Efficient milling strategies are found KEYWORDS Cutting path, Milling, Surfaces, Geometry
1.
Statement of the Physical / Geometrical Problem
Ruled surfaces (generated by straight lines) are a special kind of free form surfaces which plays an important role, e.g., in aerodynamic surfaces (rotary engines). They are particularly well suited for five axis milling by the flank milling method. In the case of twisted ruled surfaces, however, the obvious milling method of having the cutter touch these straight lines along a generator of its own flank leads to collision problems. This is explained in Fig. 1. Fig. l a ) shows the overall situation; I1 and 12 are the two generators and g a typical ruling line of the ruled Surface R. A is a surface point on g. It is assumed that the cutter touches R along g so as to obtain "optimum contact" between the cutter and the surface: the cutter axis is parallel to g and passes through a point on the surface normal no in A at a distance r from A, where r is the cutter radius.
Instead, he recommends turning the cutter about the surface normal no in A until it touches the two generators 11 and 12 while still touching R at A; A is midway between endpoints L1 and L2 of g. The design rule is then as follows: At each endpoint L1, L2 erect the surface normal n1 and a second line nz2' parallel to the "other" end surface normal n2 (and vice versa), see Fig. 2. Incidentally, these two lines include the twist angle 26. Now, imagine a third line m l in L l that includes an angle F/2 with n1; m l lies between n1 and n2'. Turn the cutter axis about no until it intersects m l (and, for symmetry reasons, m2). This way the cutter will touch R not only in A but also somewhere along 11 and 12 (provided I1 and 12 are not seriously curved), thus providing optimum contact with R.
t
b) projection on plane
a) general view
E normal to g
Fig. 1 A twisted ruled surface. Its interference with a cutter that touches it along the straight ruling line g A g 11, 12 I l ' , 12'
E
surface point on g ruling line generators projections of 11, 12 on plane normal to g
no r
6
E
u R
surface normal in A cutter radius twist half angle undercut the twisted ruled surface
E is a plane normal to g; the projections of I1 and 12 onto E are 11' and 12'. Fig. 1b) gives an enlarged view of E. It shows in particular the cutter cross-section together with 11' and 12'. Quite obviously this situation leads to "undercuts" at both ends of g due to the finite angle 26 between 11' and 12'. These undercuts persist as long as the cutter axis is kept parallel and at a distance equal to the cutter radius to g. They are least if A is midway between I1 and 12; the more A is shifted towards one end or the other of g the larger is the maximum error. At the ends it is about four times as large as in the optimum position with respect to the "distant" end of g while disappearing relative to the "close" end.
Fig. 2
Sielaff's construction of optimum cutter orientation
C1, C2 Cutter axis position in the planes y = -a, y = +a PI, P2 Cutter contact points with generators 11, 12 6 twist half-angle "1; n2 surface normals in end points of ruling line g
This method still creates slight undercuts between A and L1 and between A and 12, respectively, as sketched in Fig 3 These will be insignificant in most applications Problems may arise, however, if collision problems "in the large' prevent the cutter from being positioned according to these rules It is then desirable to understand in a more fundamental sense the way the twisted ruled surface interacts with the cutter and what strategies exist to obtain error free surfaces while maintaining a maximum degree of freedom for collision free cutter orientation
It has been recognised quite some time ago by W Sielaff /1/ that much can be gained by giving up the idea of touching R along a ruling line, say, g
Annals of the ClRP Vol. 42/1/1993
457
An arbitrary point P(x,y,z) in R may then be defined by
P = LI + q.(L2 h1);
Cutter
(4)
I
(5)
q = (a+y)/2a , 05qI1 where underlined letters designate vectors
The cutter C is visualised as a cylinder with radius r (we do not consider the much more complicated case of :he conical cutter since this would hardly contribute much to the basic understanding) It touches R in A, this means that its axis intersects the surface normal no in A at the distance r from A Otherwise we do not impose any a priori restrictions on its radius r or on the direction of its axis
Error (undercut) Fig. 3 Undercut with Sielaff's construction
2.
Statement of t h e Mathematical Problem
As stated above, the idea pursued here is primarily one of understanding 'what's going on" rather than giving ready-to use cookbook recipes, say, in the form of software programs Therefore, the method used relies largely on classical analysis It pays off to reduce the description of an arbitrary ruled surface to as simple a mathematical form as possible while still keeping all important features These are -
Twisted generators I1 and 12. The twist angle defined as the angle between the projections 11' and 12' of I1 and 12 onto a plane E normal to the ruling line g is 26.
-
Curvature of I1 and 12. It can be said without loss of generality that, as far as the resulting ruled surface R is concerned, any generator I with an arbitrary spacial curvature can be replaced by an equivalent "projection" I' on E. The law of this projection may be a complicated one; for simplicity's sake we do not pursue this subject any further.
-
Fanning out of the ruling lines. Imagine a plane (x, y) perpendicular to no in A, with g" the projections of the ruling lines g along no onto this plane. Again, we postulate that any "fanning law" for the g's may be replaced by an equivalent, though different, law for the g ' 3 .
Our problem may now be stated as follows (a) Investigate the nature of R in a "small" neighbourhood around A, in particular ( a t ) the effect of twist alone, that is, of C ? in equ (1) (a2) the combined effect of twist and curvature of the generators, that is, of c1, c2 and c3 (a3) the combined effect of twist 'and fanning, that is, of c1 and c4 (b) Let the cutter C interact with R such that ( b l ) it touches R in A (b2) there is no intersection between the two bodies (b3) if possible, establish rules for optimum contact while respecting the above conditions To do this, we proceed in principle as follows, see Fig 5
Distortions of a higher degree like, e.g., variations of curvature along the 1's may be left out since we restrict our considerations to a "small" area around the contact point A. A measure of what "small" means will be given later. z; w
Z
A
1 /Ee
Y
7X
I
' I
I
Fig. 4 Coordinate system x, y, z. (x,y) plane is tangent to ruled surface, contact point is A (O,O,O).
For our detailed investigation we re-orient the actual ruled surface such that (see Fig. 4) Point A corresponds to the origin of a rectangular coordinate system x, y, z g (the ruling line containing A) is identical with the y axis. -
The generators I1 and 12 are defined in the two planes E l (x,-a,z) and Ez(x,a,z): 11: {x, -a, c1 .x 12: {x, a, -c1 .x
+~2.~2) +~3.~2)
Fanning: Let XI, X2 be the abscissas of end points L1 and L2 on I1 and 12 of an arbitrary ruling line g, then
458
We introduce a plane N(u,O,w) in A which is perpendicular to the x,y plane (or, in other words, which contains the z-axis) and let it rotate about the z-axis, its angle with the x-axis being a. The curve w = w(u) created by intersecting N with R will allow us to gain most of the insight requested by above (a). Introduce yet a new coordinate system (u', v', w') in which the cutter C is described by
C: un2 + (w'-r)2 = r2 (1a) (1 b)
c1 is the inclination of 11 by the angle 6:
X2 = c 4 . X ~.
Fig. 5 Rotating plane N (u, 0, w): axis of rotation is z-axis, angle between u axis and x axis is a.R is the ruled surface.
(3)
(6)
This means that the surface R has to be arranged such that the tangent plane in A is identical with the (u',v') plane. It may be of value to introduce an angle 0 between the u and the u' axis. To investigate collision problems it is then sufficient to shift a test plane T(u', b , w') along the v' axis by arbitrary distances b. Its intersection curve with the cutter is always given by eq. (6) while the intersection with R is obtained by putting v'= b.
3. The Simplest Case: Straight Generators, No Fanning The mathematical description is obtained by putting the "curvature parameters" c2 = c3 = 0 in equation (1) and the "fanning parameter" c4 = 1 in eq (3) Eq (4) yields together with eqs (5) and (1)
rxi
rx
1 (7)
3.1 Principal directions
u'lr
( w l '-wz')/r
0.20
0.0001
0.30 0.50 1 .oo
0.001
(Wj'-W2')/W1' 0.01 0.02 0.07 0.5
0.01 0.5
Thus, eq. (15) is a valid representation of the cutter cross-section up to a u' value of about half the cutter radius r.
The term for z in eq. (7) which may be rewritten by means of (5): z = c1 .x.y/a
The differences between the approximate value w2' computed by eq. (15) and the exact value w1' computed by eq. ( 6 ) are like this,
(74
suggests a hyperbolic type of surface. Indeed, a rotation of axes by an angle a about the z-axis as obtained by the well-known transformation equations x = u.cos a - v.sin a y = u s i n a + v.cos a z=w
The whole situation has been depicted in Fig. 6. In particular, intersections with R are shown for p = 10" and 30" at distances b = 0.2 and 0.5. All dimensions are scaled to a, the distance of the generators from the contact point A.
p = 0"
7-
yields a quadratic expression for w(u,v): w(u,v) = (c1/2a) [(u2-v2).sin(2a)
+ 2.u.v.cos(2a)]
(9)
The plane N as suggested by (c) in section 2 is obtained by setting v=O in (9); the remaining intersecting curve is a pure parabola! Its curvature depends on the turning angle a with maxima at 2 a = 90" 1270" or amax = 45" I 135".
I
I
(1 0)
It looks worth while to change over to the new coordinate system by a rotation by amax. By definition its axes are principal axes, i.e., axes of extreme curvature. We note in particular the curvature K of the plane w (u.0) at the vertex A(O,O,O): KO = c l l a ;
(11)
the complete equations of R in this new system are (12) We note:
(A) Eqs. (9) and (10) indicate that in the relatively simple situation considered here, the direction of least probability for interference is always at 45"to the ruling line passing through the point of contact between cutter and workpiece ! (B) For stability reasons one always likes to work with as large a cutter diameter as possible. Statement (A) indicates that, for this reason, the best position of the cutter axis is at 45" to the ruling line passing through the point of contact, A. (C) The radius of curvature in A is
p = 1/K = d c l ; (1 3) this is equal to the maximum admissible cutter radius ! (D)We have declared our investigation to be restricted to a "small" region around A. (C) suggests to take p as the unit of scale: "Small" then means "small compared with p". (E) The surface R is perfectly symmetric with respect to u and v. (F) Its curvature in planes u = const. and v = const. is constant over the entire surface! In other words, R can be created b simply moving a parabola w = (c112a).u2 along the curve w = -(c1/2a).v
5
3.2 Other Directions To study interference between R and C if the cutter is not aligned with the u axis, let us again apply the same technique as in the previous section and introduce a coordinate system (u',v', w') which is rotated about the w-axis by an angle P with respect to (u, v, w) This is done by again inserting eqs (8) in eq (12) where, in eq (8), -
a has been replaced by p; x, y, z have been replaced by u,v, w; u, v, w have been replaced by u', v', w'
To examine interferences between cutter C and surface R we intersect both "bodies" with planes normal to the v' axis by attributing to the v' coordinate the fixed value b. Let C' and R' be the respective intersection curves:
R':
WR'
= (1l2p) [ ( ~ ' ~ - b ~ ) . s i n (-22.uT.b.cos(2P)] P)
(1 4)
C' has already been given by eq. (6) in a form, however, which is awkward to manipulate. Since we declared our investigations to be restricted to the neighbourhood of A, that is, to "small" values of u' and v', it is admissible to replace (6) by the quadratic term of a corresponding power series expansion w'(u') in (0,O):
c':w c ' = u'2/(2r)
(1 5)
P=
Fig 6 Twisted ruled surface R (u,v,w) with lines of constant height w (full lines indicate height above origin A, dashed lines below origin) Straight generators 11, 12 in planes E l , E2, Intersection with planes normal to (u,v) plane at distances b from A, b is given in multiples of a, viewing direction includes angle /3 w t h u-axis Also shown is cutter cross-section with limiting radius ro
3.3 Cutter intersection The condition of C' and R' interfering with each other is that w c ' = WR'
(1 6 )
W(U,V=O) = (u2/a).(cj .sina.cosa + a.cos2a. [(c2/2)(1-(u/a).sincc) + (~312) ( 1 +(u/a). s ina)]}
for at least one value of u': -
The curvatures of the generators 11 and 12 in their planes (x,-a,z) and (x,a,z), respectively, are z1"=2 c2 and z2"=2 c3, the radius of curvature is p = l/z" In order to obtain the angle of rotation for the principal axes, a , we use equ (8) and note for the intersection curve between N and R in the Diane v=O
One such equality means that the two curves touch - in other words, the cutter creates exactly the desired contour, Two equalities mean an intersection of the two curves, that is, the cutter damages the workpiece This has to be avoided by all means
Eq. (16) combined with eqs. (14) and (15) yield the following expression for the "intersection abscissa" u':
(22)
A first investigation of eq (22) shows that i t contains a cubic term in u unless c2 = c3 It will lead to a linear dependence of the curvature K from u , this term hence disappearing at u=O In other words, c2 and c3 will contribute to the curvature at the origin only a term [(c2+c3)/21.cos2a Thus, the orientation of the principal axes remains unaffected i f we set c2 = c3
The curvature
(23) Now, the condition for real solutions to exist is that the expression under the dsign be 2 0. It is a condition for the minimum value ro of r; all values r > ro will lead to two real intersections or a damage to the workpiece ! Therefore, for collision free milling we have to request r < '0: r s ro = po.cos(2f3)
(1 8)
has extrema at angles a for which tan(2a) = cl/(a.c2)
-
It is interesting to compute the coordinates of the contact point by eq. (17) and, e.g., (15) for the case r = '0. To designate all coordinates as free variables we replace b again by v':
(24)
Incidentally this is twice the ratio of
-
the maximum curvature due to twist, eq. (1 1): KO = c l / a and the curvature of the generator, ~2 = 2 . ~ 2 .
A series expansion of eq. (24) for small c2: a = 45" - a.c2/(2.cl) = 45"
-
~2/(4.~0).(180"/~~)
(25)
shows the angular influence of c2 to be linear for small c2 We note: (G) Eq. (18) indicates that real (positive) cutter radii ro for damage free milling exist only in the sector [ZPi 5 90" or /pi 5 45". At p = 45" it shrinks to zero. This is a corollary to the findings (A) and (B) above. (H) On the other hand, in a sector of, say, ipl 5 30" we may position a cutter with half its maximum permissible radius po entirely at will without inducing damage. This gives us a surprisingly large amount of freedom of coping with other constraints. (I) In Eq. (18) the distance parameter b is missing. This implies, at least within the approximations used, that a cutter with the limiting radius ro touches the workpiece all along its length ! It looks like this way, we may again have achieved a line contact which we were ready to abandon at the outset. In practice the procedure will be such that a cutter radius r i po is given; eq. (18) is then used to compute the corresponding "limiting angle" PO that leads to a line contact. (K) Furthermore, we learn from eqs. (19) and (20) that this line of contact is a plane parabola, its plane containing the w'-axis and including the angle (90"-20) with the cutter axis. (L) The same equations suggest that this contact line becomes the shorter, the smaller p. that is, the better the cutter is aligned with one of the principal axes. We conclude that, indeed, it may be advantageous to work with medium values of 0 in the neighbourhood of 30°, thus achieving considerably better contact conditions than with a "well a1igned" cutter . (M) Incidentally, this way we return back to some extent to the original layout with the cutter aligned with the ruling lines g ! (N) The optimum cutter orientation as defined by eq. (18) also comes quite close to Sielaff's construction. The latter implies a slightly larger angle 0s (with &O on the principal axis of maximum negative curvature) and guarantees touching contact of the cutter with the two generators 1 1 , 12 but leads to a slight undercut between these and the point of contact, A. The limiting angle PO according to eq. (18) guarantees collision free milling. Since, however, (I) is strictly true only for the quadratic approximation (1 5) of the circular cutter cross-section, the cutter will slightly "lift off" the workpiece surface at increasing distances v' from A. These statements should give enough insight into the possibilities and limitations of collision free flank milling of twisted ruled surfaces with straight generators to allow a valid choice of adequate milling techniques
Fig. 7 shows an example of such a surface with the principal axes as computed by eq. (24).
a) View along ruling line g \
\
'i
\ I I
I /
b) Height chart
4.
Finite Generator Curvature
4.1 Principal axes
Our analysis proceeds along the same lines as in section 2, but now we attribute finite values to c2 and c3 in eq. (1) but still c4 = 1 in eq. (3). Expression (7) becomes
rxi
460
Tx
1
Fig. 7 Twisted ruled surface with curved generators 11 and 12 in planes E l , E2. Generator parameters: a = 50 mm; inclination c1 = 0.5; curvature parameter c2 = -0.005. Angle between (x,y) and (u,v) axes a = 31.7". a) Shows, besides the generators, intersection curves y = const. at the y values indicated. b) Is a height chart. Parameters indicate values of constant z or w; respectively, in mm. Full lines correspond to positive values of z, dashed lines to negative ones.
4.2 Inserting the Cutter
The complete expression for w(u,v) in a principal coordinate system (u,v,w) where c( is given by eq (24) is w(u,v) =
(u2ia) cosu ( c l sinu + a c2 cosu) (v2ia) sina ( c l cosu a c2 sinu)
(26)
Absence of any non quadratic terms in u or v indicates the surface to be perfectly symmetrical with respct to the u and v axes Indeed, this expression is very similar to the one in eq (12) except for the different coefficients for u2 and v2, setting c2=0 and u = 45' restores this simpler fcrm Computing the maximum admissible cutter radius ro at any given angle R is straightforward in principle by proceeding exactly the same way as in section 3 3, although the actual computations are awkward and tedious Let A1
=
(cosa/a).(cl .sinu + a.c2.cosa)
A2 = (sincc/a).(cl.cosu - a.c2.sinu) where u as a function of c l / ( a c2) is given by eq (24) We obtain for the limiting cutter radius (where, again, the cutter circular cross-section has been approximated by eq. (1 5))
These expressions are considerably more difficult to interpret than the ones we obtained in case of the straight generators, that is, of c2 = 0. If we define, for ease of insight, cl/(a.c2) = f;
Fig (8) shows a plot of these dependences The curve parameter is l i f , that is, the ratio of curvature of the generators to the "curvature due to Wist alone" The curve 1if = 0 corresponds to straight generators, it represents eq (18) rOir0,O = cos(2D) Intersection points of the curves l / f = const with the abscissa where ro/ro,o = 0 (that means, where the limiting cutter radius shrinks to zero) yield the angles a between the principal axes and the ruling lines as given by eq (24) In this interpretation, we may consider the ruling line g (which is still our base element) as a 'limiting cutter" with its radius equal to zero Further observations:
(0) A positive curvature c2 corresponding to a 'valley' narrows down the range of admissible cutter radii ro and deflection angles R The effect of a negative curvature c2 indicating a 'ridge' or 'mountain' is to the contrary, large negative values of I f f indicating very small twist lead to almost unrestricted freedom of cutter orientation These observations correspond very well with what one would expect based on the nature of the surfaces (P) A generator curvature c3 # 0 does not introduce any significantly new features compared with c3 = 0 as discussed in section 3, but, of course, the numerical values of the various parameters change This is not surprising since we deal with parabolic surfaces all the way This is shown by eq (26) for the surface (this being a combined result of the parabolic generators, eq ( l ) , and the paraboloid created by twist, eq (12)) and eq (15) for the cutter Any intersections and combinations of such paraboloids with other paraboloids or with planes result again in pa raboloids for which the observations made above, particularly (G) to (N), hold true (with numerical values adapted)
5.
and observe that, by eq. (24), a = fl(f); also, we recognise by eq. (1 1) r0,o = a/cl as the "radius of curvature in the saddle point due to twist alone", then eq. (28) together with eqs. (27) reduce to a form (29)
mlo
Fanned Ruling Lines
Statement (P) suggests that the primary effect of fanned ruling lines will again be one of introducing new parameters but leaving the overall situation which is dominated by parabolic surfaces unchanged As most of our discussion has much more to do with the surfaces considered being paraboloids rather than their being ruled surfaces we refrain from a further investigation of the effects of fanned ruling lines
6.
Conclusion
In the paper, collision-free milling strategies of cylindrical cutters applied to twisted ruled surfaces including a constant curvature of the "untwisted" base surface are investigated This is done using the classical method of checking the mutual interference between cutter and workpiece by intersecting them with planes normal to the clrtter axis For the sake of mathematical simplicity, the cutter is approximated by a constant cross-section paraboloid; also, higher order curvatures of the surface and fanned ruling lines have not been considered
The main findings are:
(1) Any such twisted ruled surface is a paraboloid The point of contact is always a saddle point Therefore there exists a pair of principal axes in which curvature is extreme The surface is symmetrical with respect to both principal axes (2) The maximum admissible cutter radius depends on the angle between its axis and the principal directions It attains a maximum if they are aligned, this radius therefore is equal to the surface radius of curvature in the principal direction of positive curvature (3) In case of vanishing "base curvature" the principal axes include an angle of 45"/135" with the ruling lines The angle gets larger if the base curvature is negative (ridge) and smaller if it is positive (valley) (4) Consider the cutter axis being rotated with respect to the principal axis of negative curvature If the angle of deflection equals the maximum collision-free value then the cutter touches the surface not only in a point but along a line which, in the realm of our "parabolic approximations", is infinitely long This means that in reality, too, this configuration implies a contact quality which is close to the one obtained in untwisted surfaces with the cutter aligned with the ruling line 0
30"
P
60"
90"
Fig 8 Ratio of limiting radius ro at angle p between cutter axis and principal axis of maximum negative curvature to limiting radius ro 0 at p = 0 Parameter l i f is ratio of generator curvature c2 to "cuhature due to twist", c l l a
We imagine the most important "next step" to be a numerical implementation of the above findings such that they can be incorporated easily in five axis milling software.
References
(1) Sielaff W., 1981, Funfachsiges NC-Umfangsfrasen venvundener Regelflachen: Springer Verlag, Berlin (2) Berichte aus dem lnstitut fur Steuerungstechnik der Werkzeugmaschinen und Fertigungseinrichtungen der Universitat Stuttgart: ISW 33, ISBN 3-540-10640-5
461
SUMMARY Ruled surfaces (generated by straight ruling lines) are a soecial kind of free form surfaces which plays an important role, e g , in aerodynamic surfaces (rotary engines) They are particularly well suited for five axis milling by the flank milling method In the case of twisted ruled surfaces, however, the obvious milling method of having the cutter touch these ruling lines along a generator of its own flank leads to collision problems The nature of these twisted ruled surfaces is analysed and conditions for their collision free milling by a cylindrical cutter are established Efficient milling strategies are found KEYWORDS Cutting path, Milling, Surfaces, Geometry
1.
Statement of the Physical / Geometrical Problem
Ruled surfaces (generated by straight lines) are a special kind of free form surfaces which plays an important role, e.g., in aerodynamic surfaces (rotary engines). They are particularly well suited for five axis milling by the flank milling method. In the case of twisted ruled surfaces, however, the obvious milling method of having the cutter touch these straight lines along a generator of its own flank leads to collision problems. This is explained in Fig. 1. Fig. l a ) shows the overall situation; I1 and 12 are the two generators and g a typical ruling line of the ruled Surface R. A is a surface point on g. It is assumed that the cutter touches R along g so as to obtain "optimum contact" between the cutter and the surface: the cutter axis is parallel to g and passes through a point on the surface normal no in A at a distance r from A, where r is the cutter radius.
Instead, he recommends turning the cutter about the surface normal no in A until it touches the two generators 11 and 12 while still touching R at A; A is midway between endpoints L1 and L2 of g. The design rule is then as follows: At each endpoint L1, L2 erect the surface normal n1 and a second line nz2' parallel to the "other" end surface normal n2 (and vice versa), see Fig. 2. Incidentally, these two lines include the twist angle 26. Now, imagine a third line m l in L l that includes an angle F/2 with n1; m l lies between n1 and n2'. Turn the cutter axis about no until it intersects m l (and, for symmetry reasons, m2). This way the cutter will touch R not only in A but also somewhere along 11 and 12 (provided I1 and 12 are not seriously curved), thus providing optimum contact with R.
t
b) projection on plane
a) general view
E normal to g
Fig. 1 A twisted ruled surface. Its interference with a cutter that touches it along the straight ruling line g A g 11, 12 I l ' , 12'
E
surface point on g ruling line generators projections of 11, 12 on plane normal to g
no r
6
E
u R
surface normal in A cutter radius twist half angle undercut the twisted ruled surface
E is a plane normal to g; the projections of I1 and 12 onto E are 11' and 12'. Fig. 1b) gives an enlarged view of E. It shows in particular the cutter cross-section together with 11' and 12'. Quite obviously this situation leads to "undercuts" at both ends of g due to the finite angle 26 between 11' and 12'. These undercuts persist as long as the cutter axis is kept parallel and at a distance equal to the cutter radius to g. They are least if A is midway between I1 and 12; the more A is shifted towards one end or the other of g the larger is the maximum error. At the ends it is about four times as large as in the optimum position with respect to the "distant" end of g while disappearing relative to the "close" end.
Fig. 2
Sielaff's construction of optimum cutter orientation
C1, C2 Cutter axis position in the planes y = -a, y = +a PI, P2 Cutter contact points with generators 11, 12 6 twist half-angle "1; n2 surface normals in end points of ruling line g
This method still creates slight undercuts between A and L1 and between A and 12, respectively, as sketched in Fig 3 These will be insignificant in most applications Problems may arise, however, if collision problems "in the large' prevent the cutter from being positioned according to these rules It is then desirable to understand in a more fundamental sense the way the twisted ruled surface interacts with the cutter and what strategies exist to obtain error free surfaces while maintaining a maximum degree of freedom for collision free cutter orientation
It has been recognised quite some time ago by W Sielaff /1/ that much can be gained by giving up the idea of touching R along a ruling line, say, g
Annals of the ClRP Vol. 42/1/1993
457
An arbitrary point P(x,y,z) in R may then be defined by
P = LI + q.(L2 h1);
Cutter
(4)
I
(5)
q = (a+y)/2a , 05qI1 where underlined letters designate vectors
The cutter C is visualised as a cylinder with radius r (we do not consider the much more complicated case of :he conical cutter since this would hardly contribute much to the basic understanding) It touches R in A, this means that its axis intersects the surface normal no in A at the distance r from A Otherwise we do not impose any a priori restrictions on its radius r or on the direction of its axis
Error (undercut) Fig. 3 Undercut with Sielaff's construction
2.
Statement of t h e Mathematical Problem
As stated above, the idea pursued here is primarily one of understanding 'what's going on" rather than giving ready-to use cookbook recipes, say, in the form of software programs Therefore, the method used relies largely on classical analysis It pays off to reduce the description of an arbitrary ruled surface to as simple a mathematical form as possible while still keeping all important features These are -
Twisted generators I1 and 12. The twist angle defined as the angle between the projections 11' and 12' of I1 and 12 onto a plane E normal to the ruling line g is 26.
-
Curvature of I1 and 12. It can be said without loss of generality that, as far as the resulting ruled surface R is concerned, any generator I with an arbitrary spacial curvature can be replaced by an equivalent "projection" I' on E. The law of this projection may be a complicated one; for simplicity's sake we do not pursue this subject any further.
-
Fanning out of the ruling lines. Imagine a plane (x, y) perpendicular to no in A, with g" the projections of the ruling lines g along no onto this plane. Again, we postulate that any "fanning law" for the g's may be replaced by an equivalent, though different, law for the g ' 3 .
Our problem may now be stated as follows (a) Investigate the nature of R in a "small" neighbourhood around A, in particular ( a t ) the effect of twist alone, that is, of C ? in equ (1) (a2) the combined effect of twist and curvature of the generators, that is, of c1, c2 and c3 (a3) the combined effect of twist 'and fanning, that is, of c1 and c4 (b) Let the cutter C interact with R such that ( b l ) it touches R in A (b2) there is no intersection between the two bodies (b3) if possible, establish rules for optimum contact while respecting the above conditions To do this, we proceed in principle as follows, see Fig 5
Distortions of a higher degree like, e.g., variations of curvature along the 1's may be left out since we restrict our considerations to a "small" area around the contact point A. A measure of what "small" means will be given later. z; w
Z
A
1 /Ee
Y
7X
I
' I
I
Fig. 4 Coordinate system x, y, z. (x,y) plane is tangent to ruled surface, contact point is A (O,O,O).
For our detailed investigation we re-orient the actual ruled surface such that (see Fig. 4) Point A corresponds to the origin of a rectangular coordinate system x, y, z g (the ruling line containing A) is identical with the y axis. -
The generators I1 and 12 are defined in the two planes E l (x,-a,z) and Ez(x,a,z): 11: {x, -a, c1 .x 12: {x, a, -c1 .x
+~2.~2) +~3.~2)
Fanning: Let XI, X2 be the abscissas of end points L1 and L2 on I1 and 12 of an arbitrary ruling line g, then
458
We introduce a plane N(u,O,w) in A which is perpendicular to the x,y plane (or, in other words, which contains the z-axis) and let it rotate about the z-axis, its angle with the x-axis being a. The curve w = w(u) created by intersecting N with R will allow us to gain most of the insight requested by above (a). Introduce yet a new coordinate system (u', v', w') in which the cutter C is described by
C: un2 + (w'-r)2 = r2 (1a) (1 b)
c1 is the inclination of 11 by the angle 6:
X2 = c 4 . X ~.
Fig. 5 Rotating plane N (u, 0, w): axis of rotation is z-axis, angle between u axis and x axis is a.R is the ruled surface.
(3)
(6)
This means that the surface R has to be arranged such that the tangent plane in A is identical with the (u',v') plane. It may be of value to introduce an angle 0 between the u and the u' axis. To investigate collision problems it is then sufficient to shift a test plane T(u', b , w') along the v' axis by arbitrary distances b. Its intersection curve with the cutter is always given by eq. (6) while the intersection with R is obtained by putting v'= b.
3. The Simplest Case: Straight Generators, No Fanning The mathematical description is obtained by putting the "curvature parameters" c2 = c3 = 0 in equation (1) and the "fanning parameter" c4 = 1 in eq (3) Eq (4) yields together with eqs (5) and (1)
rxi
rx
1 (7)
3.1 Principal directions
u'lr
( w l '-wz')/r
0.20
0.0001
0.30 0.50 1 .oo
0.001
(Wj'-W2')/W1' 0.01 0.02 0.07 0.5
0.01 0.5
Thus, eq. (15) is a valid representation of the cutter cross-section up to a u' value of about half the cutter radius r.
The term for z in eq. (7) which may be rewritten by means of (5): z = c1 .x.y/a
The differences between the approximate value w2' computed by eq. (15) and the exact value w1' computed by eq. ( 6 ) are like this,
(74
suggests a hyperbolic type of surface. Indeed, a rotation of axes by an angle a about the z-axis as obtained by the well-known transformation equations x = u.cos a - v.sin a y = u s i n a + v.cos a z=w
The whole situation has been depicted in Fig. 6. In particular, intersections with R are shown for p = 10" and 30" at distances b = 0.2 and 0.5. All dimensions are scaled to a, the distance of the generators from the contact point A.
p = 0"
7-
yields a quadratic expression for w(u,v): w(u,v) = (c1/2a) [(u2-v2).sin(2a)
+ 2.u.v.cos(2a)]
(9)
The plane N as suggested by (c) in section 2 is obtained by setting v=O in (9); the remaining intersecting curve is a pure parabola! Its curvature depends on the turning angle a with maxima at 2 a = 90" 1270" or amax = 45" I 135".
I
I
(1 0)
It looks worth while to change over to the new coordinate system by a rotation by amax. By definition its axes are principal axes, i.e., axes of extreme curvature. We note in particular the curvature K of the plane w (u.0) at the vertex A(O,O,O): KO = c l l a ;
(11)
the complete equations of R in this new system are (12) We note:
(A) Eqs. (9) and (10) indicate that in the relatively simple situation considered here, the direction of least probability for interference is always at 45"to the ruling line passing through the point of contact between cutter and workpiece ! (B) For stability reasons one always likes to work with as large a cutter diameter as possible. Statement (A) indicates that, for this reason, the best position of the cutter axis is at 45" to the ruling line passing through the point of contact, A. (C) The radius of curvature in A is
p = 1/K = d c l ; (1 3) this is equal to the maximum admissible cutter radius ! (D)We have declared our investigation to be restricted to a "small" region around A. (C) suggests to take p as the unit of scale: "Small" then means "small compared with p". (E) The surface R is perfectly symmetric with respect to u and v. (F) Its curvature in planes u = const. and v = const. is constant over the entire surface! In other words, R can be created b simply moving a parabola w = (c112a).u2 along the curve w = -(c1/2a).v
5
3.2 Other Directions To study interference between R and C if the cutter is not aligned with the u axis, let us again apply the same technique as in the previous section and introduce a coordinate system (u',v', w') which is rotated about the w-axis by an angle P with respect to (u, v, w) This is done by again inserting eqs (8) in eq (12) where, in eq (8), -
a has been replaced by p; x, y, z have been replaced by u,v, w; u, v, w have been replaced by u', v', w'
To examine interferences between cutter C and surface R we intersect both "bodies" with planes normal to the v' axis by attributing to the v' coordinate the fixed value b. Let C' and R' be the respective intersection curves:
R':
WR'
= (1l2p) [ ( ~ ' ~ - b ~ ) . s i n (-22.uT.b.cos(2P)] P)
(1 4)
C' has already been given by eq. (6) in a form, however, which is awkward to manipulate. Since we declared our investigations to be restricted to the neighbourhood of A, that is, to "small" values of u' and v', it is admissible to replace (6) by the quadratic term of a corresponding power series expansion w'(u') in (0,O):
c':w c ' = u'2/(2r)
(1 5)
P=
Fig 6 Twisted ruled surface R (u,v,w) with lines of constant height w (full lines indicate height above origin A, dashed lines below origin) Straight generators 11, 12 in planes E l , E2, Intersection with planes normal to (u,v) plane at distances b from A, b is given in multiples of a, viewing direction includes angle /3 w t h u-axis Also shown is cutter cross-section with limiting radius ro
3.3 Cutter intersection The condition of C' and R' interfering with each other is that w c ' = WR'
(1 6 )
W(U,V=O) = (u2/a).(cj .sina.cosa + a.cos2a. [(c2/2)(1-(u/a).sincc) + (~312) ( 1 +(u/a). s ina)]}
for at least one value of u': -
The curvatures of the generators 11 and 12 in their planes (x,-a,z) and (x,a,z), respectively, are z1"=2 c2 and z2"=2 c3, the radius of curvature is p = l/z" In order to obtain the angle of rotation for the principal axes, a , we use equ (8) and note for the intersection curve between N and R in the Diane v=O
One such equality means that the two curves touch - in other words, the cutter creates exactly the desired contour, Two equalities mean an intersection of the two curves, that is, the cutter damages the workpiece This has to be avoided by all means
Eq. (16) combined with eqs. (14) and (15) yield the following expression for the "intersection abscissa" u':
(22)
A first investigation of eq (22) shows that i t contains a cubic term in u unless c2 = c3 It will lead to a linear dependence of the curvature K from u , this term hence disappearing at u=O In other words, c2 and c3 will contribute to the curvature at the origin only a term [(c2+c3)/21.cos2a Thus, the orientation of the principal axes remains unaffected i f we set c2 = c3
The curvature
(23) Now, the condition for real solutions to exist is that the expression under the dsign be 2 0. It is a condition for the minimum value ro of r; all values r > ro will lead to two real intersections or a damage to the workpiece ! Therefore, for collision free milling we have to request r < '0: r s ro = po.cos(2f3)
(1 8)
has extrema at angles a for which tan(2a) = cl/(a.c2)
-
It is interesting to compute the coordinates of the contact point by eq. (17) and, e.g., (15) for the case r = '0. To designate all coordinates as free variables we replace b again by v':
(24)
Incidentally this is twice the ratio of
-
the maximum curvature due to twist, eq. (1 1): KO = c l / a and the curvature of the generator, ~2 = 2 . ~ 2 .
A series expansion of eq. (24) for small c2: a = 45" - a.c2/(2.cl) = 45"
-
~2/(4.~0).(180"/~~)
(25)
shows the angular influence of c2 to be linear for small c2 We note: (G) Eq. (18) indicates that real (positive) cutter radii ro for damage free milling exist only in the sector [ZPi 5 90" or /pi 5 45". At p = 45" it shrinks to zero. This is a corollary to the findings (A) and (B) above. (H) On the other hand, in a sector of, say, ipl 5 30" we may position a cutter with half its maximum permissible radius po entirely at will without inducing damage. This gives us a surprisingly large amount of freedom of coping with other constraints. (I) In Eq. (18) the distance parameter b is missing. This implies, at least within the approximations used, that a cutter with the limiting radius ro touches the workpiece all along its length ! It looks like this way, we may again have achieved a line contact which we were ready to abandon at the outset. In practice the procedure will be such that a cutter radius r i po is given; eq. (18) is then used to compute the corresponding "limiting angle" PO that leads to a line contact. (K) Furthermore, we learn from eqs. (19) and (20) that this line of contact is a plane parabola, its plane containing the w'-axis and including the angle (90"-20) with the cutter axis. (L) The same equations suggest that this contact line becomes the shorter, the smaller p. that is, the better the cutter is aligned with one of the principal axes. We conclude that, indeed, it may be advantageous to work with medium values of 0 in the neighbourhood of 30°, thus achieving considerably better contact conditions than with a "well a1igned" cutter . (M) Incidentally, this way we return back to some extent to the original layout with the cutter aligned with the ruling lines g ! (N) The optimum cutter orientation as defined by eq. (18) also comes quite close to Sielaff's construction. The latter implies a slightly larger angle 0s (with &O on the principal axis of maximum negative curvature) and guarantees touching contact of the cutter with the two generators 1 1 , 12 but leads to a slight undercut between these and the point of contact, A. The limiting angle PO according to eq. (18) guarantees collision free milling. Since, however, (I) is strictly true only for the quadratic approximation (1 5) of the circular cutter cross-section, the cutter will slightly "lift off" the workpiece surface at increasing distances v' from A. These statements should give enough insight into the possibilities and limitations of collision free flank milling of twisted ruled surfaces with straight generators to allow a valid choice of adequate milling techniques
Fig. 7 shows an example of such a surface with the principal axes as computed by eq. (24).
a) View along ruling line g \
\
'i
\ I I
I /
b) Height chart
4.
Finite Generator Curvature
4.1 Principal axes
Our analysis proceeds along the same lines as in section 2, but now we attribute finite values to c2 and c3 in eq. (1) but still c4 = 1 in eq. (3). Expression (7) becomes
rxi
460
Tx
1
Fig. 7 Twisted ruled surface with curved generators 11 and 12 in planes E l , E2. Generator parameters: a = 50 mm; inclination c1 = 0.5; curvature parameter c2 = -0.005. Angle between (x,y) and (u,v) axes a = 31.7". a) Shows, besides the generators, intersection curves y = const. at the y values indicated. b) Is a height chart. Parameters indicate values of constant z or w; respectively, in mm. Full lines correspond to positive values of z, dashed lines to negative ones.
4.2 Inserting the Cutter
The complete expression for w(u,v) in a principal coordinate system (u,v,w) where c( is given by eq (24) is w(u,v) =
(u2ia) cosu ( c l sinu + a c2 cosu) (v2ia) sina ( c l cosu a c2 sinu)
(26)
Absence of any non quadratic terms in u or v indicates the surface to be perfectly symmetrical with respct to the u and v axes Indeed, this expression is very similar to the one in eq (12) except for the different coefficients for u2 and v2, setting c2=0 and u = 45' restores this simpler fcrm Computing the maximum admissible cutter radius ro at any given angle R is straightforward in principle by proceeding exactly the same way as in section 3 3, although the actual computations are awkward and tedious Let A1
=
(cosa/a).(cl .sinu + a.c2.cosa)
A2 = (sincc/a).(cl.cosu - a.c2.sinu) where u as a function of c l / ( a c2) is given by eq (24) We obtain for the limiting cutter radius (where, again, the cutter circular cross-section has been approximated by eq. (1 5))
These expressions are considerably more difficult to interpret than the ones we obtained in case of the straight generators, that is, of c2 = 0. If we define, for ease of insight, cl/(a.c2) = f;
Fig (8) shows a plot of these dependences The curve parameter is l i f , that is, the ratio of curvature of the generators to the "curvature due to Wist alone" The curve 1if = 0 corresponds to straight generators, it represents eq (18) rOir0,O = cos(2D) Intersection points of the curves l / f = const with the abscissa where ro/ro,o = 0 (that means, where the limiting cutter radius shrinks to zero) yield the angles a between the principal axes and the ruling lines as given by eq (24) In this interpretation, we may consider the ruling line g (which is still our base element) as a 'limiting cutter" with its radius equal to zero Further observations:
(0) A positive curvature c2 corresponding to a 'valley' narrows down the range of admissible cutter radii ro and deflection angles R The effect of a negative curvature c2 indicating a 'ridge' or 'mountain' is to the contrary, large negative values of I f f indicating very small twist lead to almost unrestricted freedom of cutter orientation These observations correspond very well with what one would expect based on the nature of the surfaces (P) A generator curvature c3 # 0 does not introduce any significantly new features compared with c3 = 0 as discussed in section 3, but, of course, the numerical values of the various parameters change This is not surprising since we deal with parabolic surfaces all the way This is shown by eq (26) for the surface (this being a combined result of the parabolic generators, eq ( l ) , and the paraboloid created by twist, eq (12)) and eq (15) for the cutter Any intersections and combinations of such paraboloids with other paraboloids or with planes result again in pa raboloids for which the observations made above, particularly (G) to (N), hold true (with numerical values adapted)
5.
and observe that, by eq. (24), a = fl(f); also, we recognise by eq. (1 1) r0,o = a/cl as the "radius of curvature in the saddle point due to twist alone", then eq. (28) together with eqs. (27) reduce to a form (29)
mlo
Fanned Ruling Lines
Statement (P) suggests that the primary effect of fanned ruling lines will again be one of introducing new parameters but leaving the overall situation which is dominated by parabolic surfaces unchanged As most of our discussion has much more to do with the surfaces considered being paraboloids rather than their being ruled surfaces we refrain from a further investigation of the effects of fanned ruling lines
6.
Conclusion
In the paper, collision-free milling strategies of cylindrical cutters applied to twisted ruled surfaces including a constant curvature of the "untwisted" base surface are investigated This is done using the classical method of checking the mutual interference between cutter and workpiece by intersecting them with planes normal to the clrtter axis For the sake of mathematical simplicity, the cutter is approximated by a constant cross-section paraboloid; also, higher order curvatures of the surface and fanned ruling lines have not been considered
The main findings are:
(1) Any such twisted ruled surface is a paraboloid The point of contact is always a saddle point Therefore there exists a pair of principal axes in which curvature is extreme The surface is symmetrical with respect to both principal axes (2) The maximum admissible cutter radius depends on the angle between its axis and the principal directions It attains a maximum if they are aligned, this radius therefore is equal to the surface radius of curvature in the principal direction of positive curvature (3) In case of vanishing "base curvature" the principal axes include an angle of 45"/135" with the ruling lines The angle gets larger if the base curvature is negative (ridge) and smaller if it is positive (valley) (4) Consider the cutter axis being rotated with respect to the principal axis of negative curvature If the angle of deflection equals the maximum collision-free value then the cutter touches the surface not only in a point but along a line which, in the realm of our "parabolic approximations", is infinitely long This means that in reality, too, this configuration implies a contact quality which is close to the one obtained in untwisted surfaces with the cutter aligned with the ruling line 0
30"
P
60"
90"
Fig 8 Ratio of limiting radius ro at angle p between cutter axis and principal axis of maximum negative curvature to limiting radius ro 0 at p = 0 Parameter l i f is ratio of generator curvature c2 to "cuhature due to twist", c l l a
We imagine the most important "next step" to be a numerical implementation of the above findings such that they can be incorporated easily in five axis milling software.
References
(1) Sielaff W., 1981, Funfachsiges NC-Umfangsfrasen venvundener Regelflachen: Springer Verlag, Berlin (2) Berichte aus dem lnstitut fur Steuerungstechnik der Werkzeugmaschinen und Fertigungseinrichtungen der Universitat Stuttgart: ISW 33, ISBN 3-540-10640-5
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