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On the finite screw axis cylindroid
Mech. Mach. TheoryVol. 24, No. 3. pp. 143-155, 1989 Printed in Great Britain.All rights reserved
0094-114X/89 $3.00 + 0.00 Copyright ~ 1989PergamonPress pie
ON THE FINITE SCREW AXIS C Y L I N D R O I D F. S T I C H E R School of Mechanical Engineering, University of Technology, Sydney, N.S.W. 2007, Australia (Received in revisedform 10 October 1988) Abstract--lt is shown that the pitches of the finite screw axis cylindroid are characterised by a step change, and when this is applied to the limiting case of the infinitesimal screw displacement cylindroid, it results in two zero pitch lines of different nature to each other. The characteristics, order and class of the line congruence consisting of all line and point-row displacements which result in the same cylindroid surface, is derived and discussed in some detail.
1. INTRODUCTION Much has already been written on what is called in this paper, "the finite screw axis eylindroid", for example by Bottema in [1] and Tsai and Roth in [2]. The first original contribution in this paper addresses itself to a full understanding of the differences and similarities between the finite screw axis cylindroid and the well-known cylindroid which characterises infinitesimal screw displacements. In particular, a discussion concerning the different natures of the two zero pitch lines in the infinitesimal case is offered. Since the derivation of the cylindroid surface in Section 2 of this paper differs from those in [1] and [2], it is perhaps wise to point out the relationship between parameters in this paper and Bottema's[l]. The origin of the axes in Section 2 is displaced a d i s t a n c e + ( ~ / 2 ) s i n $ along z to that in [1], and Bottema's a, Z, a and d are respectively 7t/2-~b, n/2 - 0, s, ~ cos ~b/2 in this paper. A comprehensive study of all possible line-point row displacements which produce the same cylindroid surface, is the second original contribution offered. The intrinsic physical nature of this (2, 2) congruence of lines is explored in Sections 5 and 6, and a graphic summary of this system is shown in Fig. 8. In the author's opinion, Phillips and Hunt[3] contains the best available summary of the infinitesimal displacement cylindroid. Comments on the finite screw axis cylindroid are also to be found in Chap. 19 of Ref. [4]. Indeed, this paper was inspired by conversations with Professor Phillips in the course of his writing Vol. 2 of Ref. [4], and was originally offered in note form by the author to Professor Phillips, who consequently encouraged its publication. 2. THE FINITE SCREW AXIS CYLINDROID
shown in the figure, and their relation to Bottema's co-ordinate system[l] has been discussed in the introduction. The point P2 on the line position l: is moved to Pt in line position Ii. P2 and Pt are at distances from the y-axis of (2 and (~ respectively, so that the displacement of all points in the line is defined as = ( ~ - (2)- Let the line L be any finite screw axis (FSA) about which/2 can be rotated, and at the same time along which it can be translated, such that line position 1~, together with point displacement ( can be realised. Let L intersect the yz-plane in (Y, Z ) and be inclined at angle 0 to the yz-plane. Since L must make equal angles with lj and Is, it must be perpendicular to the z-axis; that is, any point Q on the line L has co-ordinates (2 sin 0, Y + ). cos 0, Z), where ). is the distance from the yz-plane along L. Let Q~ be the foot of the perpendicular to L, passing through point Pj, and Q2 the corresponding point for the perpendicular to L passing t h r o u g h / 2 . Co-ordinates of Qt, Q2 are ().~ sin 0, Y + ).t cos 0, Z ) and ().2 sin 0, Y + )-2cos 0, Z) respectively. The vectors Pt and P2 (Qi Pl and Q2P2) are therefore given by: Pl = i(~j cos ~ - ).l sin 0) + j ( - s - Y - )-i cos 0) + k ( - C t sin ~ - Z); P2 = i(~2cos ~b - ).5 sin 0) + j ( s - Y - 22 cos 0) (2)
+ k(£2 sin ~ - Z )
and
]P~I--
Ip2l -- P.
From equation (1):
p~ =p2 __ (~l COSqb -- itI sin 0) 2 + ( - - s -- Y - 21 cos 0)' + ( - ~ t sin ~ - Z)'
or
In Fig. 1, a line of points 12 is to be moved to position 1~, such that any point on the line is translated along it by a distance ~. x-, y-, z-axes are as MMT.24r--x
(1)
2~ -- 221(~l cos qb sin 0 - (s + Y)cos 0) + (s + y)2
+ ~ + Z2 + 2 Z ~ l s i n ~ - - p 2 = O . 143
(3)
144
F. S'nCHER
L~
L2
Subtracting equation (9) from equation (10), putting ( i - (2 = ( and dividing by 2r/,
z
~(1 - cos: ~ sin" 0) + 2Z sin +2scos0sin0cos~b=0.
(11)
This is the equation to the finite screw axis cylindroid, relating the angle 0 to Z. Note that if equation (11) is multiplied by ((i + (z) we obtain equation (9) minus the 4s Y term, i.e. (~1 + ~2)(11) - (9) gives Y = 0.
Fig. 1 Equation (3) expresses the obvious fact that given any P(Qt not necessarily the foot of the perpendicular from Pt to L), there are up to two solutions for ;.~, hence two Qt on L. Since PIQt is however, the shortest distance between Pt and L (PI Q~ perpendicular to L), then equation (3) should give a single (repeated) value for ;.I- Thus equation (3) may be written: [21 - (~t cos ~ sin 0 - (s - Y)Cos 0)] 5 = 0.
(4)
From equations (3) and (4) we therefore obtain: 21 = ~1 cos ~b sin 0 - (s + Y)cos 0;
For any position Z along the z-axis therefore, two generators of the cylindroid are defined which intersect the z-axis at right angles. If now Z as a function of 0 from equation (11) and (21, ;.2) from equations (5) and (7) are substituted into the expressions (1) and (2) for Pl and P2 respectively, and if we put: (z = ~ ,
Pt = "~cos 4~ cos 0 + s sin 0 × ( i c e s 0 - j s i n 0 + k ctan ° s : ) ; (12) and P2= ~ c o s ~ c o s 0 + s s i n 0
(5) x ( - i c e s 0 + j sin 0 + k tan c°s 0b) ; (13)
p2 = -(~1 cos ~ sin 0 - (s + Y)cos 0) 5 + (S + y)2 + ~ + Z 5 + 2Z~1 sin ~.
(6)
Applying the same argument to equation (2), we similarly obtain: 25 = (s - Y)cos O + ~5 cos ~ sin O);
(7)
and
p2xpt=2cos0cos~bsin~b(-isin0-jcos0) P2" Pl cos2 q~ cos 2 0 - sin 5 ~b
(14) "
Now P2" Pt
(8)
Equating p5 from equation (6) to p5 from equation
(8): [(~t -
from which
IP, × p,l
p] =p2 = -(~2 cos ~ sin 0) + (s - Y)cos 0) 5 + (s - y)5 + ~I + Z 2 _ 2Z~5 sin ~.
~2= - ~ s a y , then:
is tan ¢,, where ~ is the angle through which 12 is rotated about the FSA L into 1I. Also, equation (14) shows that if we write: - 2 cos 0 cos ~ sin ~b tan ¢, --- cos5 O cos 2 ~ - sin 2 ~"
~2)cos ~ sin O - 2.s cos 0]
(15)
× [(~1 + ~2)cos ~ sin 0 - 2Y cos 0 ] -- (~1 - ~z)(~t + ~ ) + 4sY + 2Z sin ¢(~1 + ~2). (9) Equation (9) is true for any other points P[ at £t + and P~ at ~5 + r~, such that £t + r / - (~ + ~) is still the displacement ~ = ~t - ~5. Writing equation (9) with ~t replaced by ~ + r/, ~ replaced by ~5 + ~/, [(£~ - ~5)eos ~ sin 0 - 2s cos 0] x [(~1 + ~ 5 + 2 . r/)eos ~b sin0 - 2 Y c o s O ] --(~ - ~5)(~1 + ~ + 2~) + 4sY + 2Z sin ~b(~l + ~2 + 2. ~/).
(10)
Then this will indicate the sense of ~. Since we shall use the convention that - n ~< ¢J ~< + n, then the signs of the numerator and denominator of equation (15) are the same as the signs of sin ~k and cos ~b respectively. Equation (15) could also have been obtained easily by other means, for example, by finding the angle between the projections of/2 and il onto a plane perpendicular to L. The sliding or translation along L which takes place is given by/a = ;.5 - ;.i which from equation (5) and (7) is: /~ = 2s cos 0 -- ~ cos ~ sin 0. (16)
Finite screw axis cylindroid
145 i"
x
t
•
L1
enerotor x
k2
z
.>I xz
-~y
Fig. 2 Y
Finally, the cylindroid equation (II) may also bc expressed in the form:
~(I +
sin 2 ~b) +
2Z sin 0~ Fig. 3
= -cos ~
cos2 ~ + s z" sin(20 +/~)
(17)
where tan/~
=
cos ~, 2s
Equation (17) shows clearly that if 0 is a solution for a particular Z, then 1r/2 - ( 0 + ~) is also a solution, corresponding to two generators for any Z lying within the range. 2sin~b
--
(l +sin2~b)-cos~b
cos2~b+ s 2
1
~
¢2
+oos0j cos, o 1. (18) In equation (16), note that p has been defined as (A~ -~.~), so that if~.~ is less than ~-z, # is positive and in the same sense as positive ~#given by equation (15). that is, if 0 ~ ~# ~ + n and p is +v¢, then the sense of both ~# and p are shown in Fig. 2. The pitch of L is defined as for the infinitesimal motion cylindroid by h = ~/~.
3. SPECIAL SCREW AXES AND THE INFINITESIMAL MOTION CASE There exist two screw axes of special interest; these are the cylindroid generators of zero pitch (tt ==0) and for want of a better name, the n generator where ffi ± n or half a turn. The pitch of the n generator can then be regarded as positive or negative and results in a step jump in the pitch on an (h, 0) plot, if we adopt the convention that I~ I ~ n. (a) The zero pitch generator (1~ ==0 line) occurs when tan 0 = 2s/~ cos ~b from equation (16). Substi-
tution of this 0 into equation (11) yields Z = - ~ / 2 sin 4. The second generator at this value of Z i s a t 0 = 0 w i t h # f 2 s and ~ = - 2 0 b . These almost self evident positions and values arc shown in Fig. 3. (b) When ~ ffi ±n, then from equation (15), 0 = n/2 and from equation (11), Z = - ; sin 4/2. As we pass through this value of Z, we change the sense of rotation of ~b but not the sense of # and we have a step jump in h from - ~ cos ~/n to + ~ cos ~/n. Note that this "~ generator" is orthogonal to the 0 ffi0 generator mentioned in (a),and the zero pitch generator in (a) isorthogonal to the second generator through Z = -~ sin ~b/2 at tan 0 = -~ cos ~b/2a(see Fig. 3). An inspection of equation (18) shows that the mid-point of the range for Z (the middle of the "spine" of the cylindroid) is at:
Zc=
(I +sinZ~b)=~I
4 sin ~
~
~sin~bl
2 sin ~b
2
"
That is, the zero pitch generator and the n generator are equally and oppositely spaced from the centre of the cylindroid. If the equation (17) of the cylindroid is referred to axes parallel to our x-, y-, z-axes but with origin at Z c, it becomes:
2z tan ~ +
cos2 ~b + s z sin 200 =
O,
(19)
where
z=z-z~, Oo=IO+~l. (c) When the line positions/2 and i~ become infinitesimally separated, we may substitute 6s, 64, 6¢ for s, 4, ¢ and note that to the first order sin ~b--*8~
146
F. STICHER
and cos q~ --, I. Using a:16¢, &IMp as constants, the cylindroid equation (17) becomes (as as, aq~, 6: --* 0)
lF6:l
~aF6¢T
(16) become: - 2 cos 0 - &p tan ~ = cos20( 1 _ (&b):/2) - (&b):
r&l:sin(2o+,a);
+L j
(20)
where
ira:
tan/~
=
(21)
and
L~" ~aq~]
/~ = 2& • cos 0 - 6: • sin 0. (22) Now equation (21) becomes tan~, = - 2 a ~ / c o s 0 provided 0 does not approach the value n/2. Then tan ~ --, ~, and
"
/~
(2& cos O - 6: • sin 0)cos 0
The zero pitch generator of (a) then is at:
I ra:l,j z=-~L3g
tan0--
=~[~lsin20 1V&-ll~j L ~J -2L-~'f c°s2°
La~j 2 ~ ~..
which may be expressed in the form,
The n generator of (b) is then at: Z
=
-
a;. 6,/, ~ , 2
h=A
a~ /t
to
h=
+Bcos(20+=)
i.e.
that is, we have the classic sinusoidally varying h for the infinitesimal motion case. As 0 approaches re~2, ~k very quickly becomes large and approaches + n, i.e. it can no longer be assumed that tan ~b ~ ~b, and moreover we must retain the 2nd order quantities in equation (21) to see how ~b varies. If we put 0 = (n/2 - ~) where ¢ is small, equation (21)
at Z = 0 as 3~, a~ ~ 0 and 0 = n/2. The pitch of the generators as we pass through Z = 0 then changes from: h=---
+ I),
6(
+--. /t
But as a~ --* 0, the pitch changes from - o, to + o, and the n generator can then be regarded as a second zero pitch line. The existence of two h = o lines corresponds to what is well known about the infinitesimal motion cylindroid[3]. However, we must be very careful here! Let us examine the distribution of pitch along the cylindroid in this case, as a function of 0. Equations (15) and x
/ /
tan ~k
and
-
2&.b
"-'E 2 -
•
(23)
(6~)2'
/~ -~ -~(.
E can be made as small as we wish, and we see that if ~ < &b, we obtain ~, ~ + n; if E ~ 6~b; ~k ~ + n/2
~- Ls-2,¢-3o"
l,r generoT, or I h=O,z=-I / /
becomes:
1-2.39721 _:/
~ . 1 - 2 . o ,
I-0.251+~
~ 1-101
,.__o0.%/_
)_ ,_o.,o, (*0-2) \
1÷1.o) \
11.1472)
Fig. 4
~y
I47
Finite screw axis cylindroid (0)
a=77.Ts" h=O,z=-I
1.e)
i///
• .
Z, /--'-
/
/-/<-'6'
/
///I
<_o.% iii1
IFo.yI/
~o°oS~ u''p'
III
////
--y
(-0.25)
_/
1o)
i
1÷0.41
Fig. 4a and if E > 6~, tan ~ rapidly becomes small in magnitude with increasing ~. Since the constants of the infinitesimal displacement case are really 6~/6~ and 6s/tdp, it appears that we can argue that 6~b can also be made as small as we wish, and therefore, the region where h no longer varies sinusoidaUy also becomes infinitesimally small. This is true, but perhaps the best way of still reminding ourselves of the unique character of the second h = 0 generator is to call it a singular generator, It is singular in the sense that [~'1 rapidly diminishes from ~r to an infinitesimally small value at an infinitesimally small distance from this generator along the "spine" of the cylindroid! The second of the following examples illustrates this point. 4. EXAMPLES OF PITCH DISTRIBUTION IN THE CYLINDROID In the following two examples, the conventional polar plot of pitch has been modified to include the
sense of rotation for ~, which is perhaps of greater importance in the finite displacement case than in the well known infinitesimal displacement cylindroid. Normally, each generator is counted twice along the cylindroid, that is, the line designated by 0 is counted as different to the line designated by (lr + 0). The polar plot of pitch h as a fucntion of 0 is then drawn resulting in the familiar four-lobed figure as Phillips shows in [3]. Figures 4 and 5 show only two lobes, since the system adopted in this paper is that a single cylindroid generator has only one direction defined by the sense of tb, and the length of the lines drawn in the direction of ~bis proportional to Ih[. Ifh is positive it is marked + and if negative, m a r k e d - . The Z value is also shown in brackets. To illustrate this system, let us calculate the relevant quantities for the case Z = 0.6 in Fig. 4 ( ~ = 3 0 °, ~ = 1 , s = 2 ) . From equation (17), /1=12.2163 ° and the two solutions to 0 are
148
F. STICFIER
~-o.o4,s=o.ol,~=z-
Positive pitch Loop (- o.2) //~(- o.3)
h=¢0.013
(O,
(-0.4)
3.3)
Negative pitch Loop Fig. 5
0[ = -27.9723 ° and 02 = 90 - (0, + fl) = 105.7560 °. From equation (15), tan ~1 = -0.7649/0 + 0.3350 and tan ~'2 = +0.2352/-0.1947 or ~'l = -66.3482°, ~2 = 129.6227 °. Equation (16) yields /~ =3.9389, /~2= -1.9197. Referring to the sense of ~, and ~ in Fig. 2, we see that for 01 a line should be drawn from the z-axis (origin of system in Fig. 4) at - 2 7 . 9 7 ° since ~1 is negative and therefore in the opposite sense to that shown in Fig. 2. The pitch /~l 3.9389 180 -3.4015 h I i s - - - ; = --66.3482 ~ = and therefore the line is drawn 3.4015 units in length and labelled as negative with Z(0.6) as shown in Fig. 4. For the line at 02, however, ~: is positive, hence it is drawn not at 105.76 ° but in the opposite sense at 105.76 ° - 180 ° = - 7 4 . 2 4 ° (see Fig. 2). Note that in Fig. 4a, the region near the n generator has been magnified to show the sudden jump from h -- -0.2757 to h --- +0.2757, resulting in a highly truncated small lobe and a slightly
truncated large lobe. Other lines of interest are clearly marked. The case depicted in Fig. 5 is for ~ ..~ 0.04, s = 0.01 and ~b = 2 °, and this exhibits the infinitesimal displacement characteristics. From the diagram, it can be seen that both lobes are almost "complete", the jump in h at the n generator being from -0.013 to + 0.013. In order to illustrate the discussion in Section 3 on the behaviour of ¢ and the pitch h near the n line for the infinitesimal case, the following table of selected values is interesting. The reader should keep in mind that the total length (range of Z ) of the cylindroid is 0.64 units. As well as h ~ , the approximation h,~,,,o~ is given, showing the breakdown of this approximat/on near the ~t generator, i.e. near
0 =~/2. From Section 3: 1 6~'
hsiw,aoid.s= ~ ~
. 1 6s sm 20 -- ~ ~ (1 + cos 20).
Where ~
=
0.o4, 6~,
=
o.ol, ~ )
(2)~ =
]8o'
Finite screw axis cylindroid
Referring to axes x I, Yl, z,, we see that z, = z - zo and 0, = 0 - 0o. Thus, if we substitute Z = ZI + zo and 0 = 0, + 0o into equation (24), we obtain:
so in this case: h,~,~,d,, = 0.2865 sin 20 -- 0.1433(1 + cos 20) Z
0°
hu,~
hz..
]%1err. in h
+0.1565 +0.1767 +0.1653 +0.1173 +0.0501 +0.0368 +0.0231 +0.0130 -0.0133 -0.0288 -0.0551 -0.0987
+0.1564 +0.1768 +0.1650 +0.1167 +0.0477 +0.0334 +0.0173 +0.0006 -0.0014 -0.0238 -0.0525 -0.1002
0.6 0.06 0.18 0.51 4.79 9.24 25.11 95.38 89.47 17.36 4.72 1.50
g,~
-0.4 4 7 . 9 3 9 6 -5.9679 -0.3 5 7 . 1 1 5 2 -7.3600 -0.2 6 6 . 1 6 6 7 -9.8786 -0.1 76.1559 -16.6063 -0.03 84.9864 -43.5615 -0.02 86.5455 -60.1877 -0.01 88.2451 -97.5026 -0.001 89.9397 -176.5475 0.000 90.1403 +171.9754 +0.01 92.3328 +81.2546 +0.02 95.0575 +43.2197 +0.03 99.2852 +24.4243
149
As Z ranges from -0.03 to +0.03, 0s ranges from about 85 ° to 99 °, but g,, passes from - 4 4 ° through + 1800 to + 2 4 ° - - a range of roughly 292°! 5. THE SET OF LINES 12 AND 11 WHICH PRODUCE THE SAME CYLINDROID
We now ask, "What other lines Is and It with different values of ($, ~, s) would produce the same cylindroid?" We do not require the $ and/~ distribution along the cylindroid to be the same, only the cubic surface itself. We shall take as reference the set of axes and cylindroid defined by the original pair of lines/2 and 11, as shown in Figs 1 and 6. At distance z0 along the z-axis, assume a common perpendicular between two other skew lines Is0 and 1,0 tilted at angle 00 to the zy-plane. The quantities Or, ~,, sl are shown in Fig. 6 together with a new set of axes xt, Yo, z~. Now the equation to the cylindroid in the x, y, zsystem and in terms of ~, ~, s can be written [from equations (11) or (17)] in the form:
~[.I + sin2@l
sin~ j+2(zo+Z,)+cot¢
x(~coscpcos2(O+Oo)+ssin2(O+Oo))=O or
sin~ + cot 4~ +
~cosqJcos20o+ssin20o)¢os201
- ~ cos ~ sin 200 + s cos 200)sin 20, = 0.
x,, Yt, z,,
But the equation to the cylindroid using the 4h, ~,, s, system is of the same form as equation (24), i.e.
¢i1_
-~I 1+sinS-fi~'¢l .| + 2z, + c o t $ 1 ( ~ c o s $, cos 20, + s , sin 2 0 , ) = O. (26) Comparing coefficients of cos 201 and sin 201 in (26) and (25) and equating the constant terms:
cotC~(~coscpcos2Oo+ssin2Oo)=cotO,~cos¢~l, (27)
(
cot ~ \ - ~ cos ~p sin 200 + s cos 200)
= cot ~,. 5t (28)
and
2 (24)
2Z,
(25)
~[I ++2Zsin sin'~'] + cot ~ -~ cos q~ cos 20 + s sin 20 --- 0.
J+
sine, +2,O=#k- ~
J" (29)
By substituting ~l from equations (27) into (29), we may then solve for sin2 ~L:
a-b
sins ~l = a + b' Z,IZ 1
(30)
where a = 4zo sin~ + ~(1 + sins ~) and
L1
b -- cos ¢p(2s • sin2 O0+ cos cp cos2 00)
Yl
= cos ¢p~/~s cos s ~ + 4s s . sin(20o + 8) and tan ~ff -- ~cos 2s ~) as in equation (17).
/ Fig. 6
Ct and s, are then found from equations (27) and (28). It does not matter if 0r is taken as + v e o r - r e , since the true sense of the new line and 110 will be defined by a positive or negative s,. That is, the values (Oi, ~t, s,) define the same skew-pair and sense of~ as
pair l~
150
F. SmCHEa b
.A [
A-cos @~/(.;2cosZ@+4s2
-A O
-A
~o
÷A
Fig. 7
(-~bl, -~1, sl). For convenience, we shall take ¢~1 as being positive (0 ~<~bl ~< n/2) and allow ~t and sl to adjust themselves accordingly (remember that the sense of ~, and F is irrelevant). From equation (30), since a-b
0 .N<~ - - ~ ~ + 1 and since
become infinitesimal, i.e. the infinitesimal displacement case is approached. If 00 = - # / 2 (the lower boundary) or 00 = (n/2 - 3/2) (the upper boundary), then we approach cases where ~1 = re/2, sj ~ + oc and ~(1 + sin-' ~) + 4Z0 sin ¢ 2 sin @ As z0---. + oo, the angle ~bt always approaches n/2.
- c o s q~x/~2cos 2 ~b + 4s 2 ~ z , , then a l > b and i f z < z c , then a~
The interpretation of Fig. 8 is as follows. Choose any particular value of z0, Z0 say, as shown in the figure. Then the allowable ranges for 00 are given by those parts of the line z0 --- Z0 which lie in the shaded region. As the values of 00 approach the boundaries of the sine curve, it is found that the values of @1, ~1 and Sl
For Z0 beyond the z bounds of the cylindroid, it is seen in Fig. 8 that no value of 0o produces line l~0,110 which are infinitesimally displaced. When Z0 is within these bounds, there are two such pairs of lines/2o and 110 (two intersections of z0 = Z0 with the sine curve); the two corresponding s, lines in this case are orthogonal to the two generators of the cylindroid residing at Z 0. If0z, 0z are the angles 0 defining the generators, then the two 00s will have values of (01 + n/2) and (02 _+ rc/2), (Whether it is + n / 2 or -rr/2 is best resolved when a particular numerical example is considered.) Indeed, this shows that all infinitesimally displaced line pairs (double lines), generate the original cylindroid surface! This could also have been deduced from Section 3, where the n line (a line of the cylindroid) is virtually coincident with an infinitesimally displaced pair. Figure 8 also shows that there is a singular point 0 0 = - # / 2 or n / 2 - # / 2 , Zo=Zc (the centre of the cylindroid), which lies on both the infinitesimal displacement sine curve boundary and on the @~= n/2 boundary. Indeed, at z0 = z¢, there is no allowable range of 00 other than 00 = - # / 2 or 00 = n/2 - 3/2! (Note that 00 = - n / 2 - #/2 results in the same case as 0o = n / 2 - #/2 in what follows.) Let us see what happens at this singularly interesting point. If we revert to the original equations (27),
Finite screw axis cylindroid
151
Zo
\
0o 2
2 Ze
~°
i
~, - {Z¢~
A
4 )in
Fig. 8
(28) and (29) and put 00 = - / / / 2 or ~/2 - / / / 2 , z0 = zc, we obtain: 0 = ~£cot ~ cos ~l,
(31)
Z
+ cot ~b :-2 - - ~ N/~ COs2~) -I-
I1o. Physically, however, all becomes clear when we look at a converse problem. What becomes of the cylindroid equation (17) when ~ = 07 Equation (17) is then: 2Z sin ~ = - s cos ~b sin 20;
4.$ 2 =
S1cot ¢~1,
(32)
(The signs on the left-hand side correspond to 00 = - / / / 2 , ~ / 2 - / / / 2 respectively). 0 = ~l[ ,1 + sins ~17
j.
(33)
Clearly, equations (31) and (33) are satisfied for any 0 < ~l < ~/2 if ~ = 0. s~ is then found from equation (32), and ranges from 0 to + ~ . Thus, at zo = zc, there exist two perpendiculars (at 0 0 = - / / / 2 and ~/2 -B/2) each common to an infinity of possible/20 and 1~0 line pairs, all with ~ = 0. ~bl can range from 0 to ~/2, so it is indeed true that zo = zc can lie on both the infinitesimal displacement boundary ( ~ 0 , s~--, 0), and the (~. = ~/2, sl ~ oo) boundary. This is quite remarkable, since for any other particular zo, 0o specified, there exists only one pair of skew lines 12o,
or
2Z = - s cot ~b sin 2 0. (34) Thus, the shape of the cylindroid is defined only by s cot ~ and s and ~ can vary provided this product remains constant. From equation (32) we see that si cot q~l is indeed a constant, equal to + s cot ~ when ~=0. Figure 9 is a sketch illustrating two arrays for the case ~ - 30, s = 2, ~ = 1. The first array shows the lines 12o, llo defined by equation (32) at Zo = z¢ = - 0.625. The second shows some lines l:o, l~0 for 0o = 70o, i.e. all the lines lzo, l~0 lie in parallel planes, and in this case 0.2011 ~ zo ~< oo. The construction is not exact in the sketch, but shows the behaviour o f s l , ~l, ~bl for 0o = 70°. The table of exact values for 00 = 70 ° is shown below. Note the small displacement between the lines near zo = 0.2011 (infinitesimal displacements).
152
F. STICh~R
~1=40 ;1=30
~=20 70* ~I=11
~y
Z=Zc,80=-~-,~1=0
: z . 8o: T -
,g~=O Fig. 9
O0= 70°
0.2012 0.5 1.0 2.0 3.0 5.0 10.0 15.0 20.0 100.0 1000
- 0.0274 - 1.3338 -2.1806 -3.2721 -4.0814 -5.3443 -7.6367 -9.3849 - 10.8552 -24.3713 -77.1388
Zo > 0.2011
0.0299 1.5274 2.7987 4.9833 7.0592 11.1280 21.1857 31.2063 41.2169 201.2432 2001.2500
0.5001 23.0430 34.8144 46.2188 52.4651 59.5978 67.6762 71.5242 73.8875 82.6683 87.6722
x = sz sin 0o + 2 cos 4)t cos 00;
(35)
y = s~ cos 0o - 2 cos @t sin 0o;
(36)
z = z0 + 2 sin @,;
(37)
where :. = SQ. Let Q be a generic p o i n t in space, i.e. x, y, z are treated as given c o n s t a n t s in what follows. s~ and @~ can, o f course, be negative, representing a conjugate line o f a n lzo, l~o pair.
x,y,z)
Z
Clearly, the system o f ~ 2 lines 120 a n d lz0 defines a line congruence F o f some sort. In the next section, it is s h o w n t h a t b o t h order and class o f this congruence is two.
S 8o
6. THE ORDER AND CLASS OF THE CONGRUENCE F
(a) The order of F C o n s i d e r any point Q on a line o f the congruence as s h o w n in Fig. 10. By inspection:
Fig. 10
Finite screw axis cylindroid or
From equatons (35) and (36): y cos 00 + x sin 00 = st
x cos 00)tan ~bt + z.
(40)
(42)
f ( a ) is a cubic function with some very interesting characteristics. We shall show that/n general, there is only one value of = which produces two real values of 0o. Since a may be written in the form: . = x/4s 2 + ~2 cos 2 q~ sin (200 +/~)
(39)
Also, by substituting st from equation (38) into equation (27), (x sin 00 + y cos 00)tan tan ~b~ = (s cos 2 00 - ~/2 cos ~b sin 2 00)"
f ( : t ) = O.
(38)
and from equations (36) and (37), using st in equation (38): tan ~b~ zo = (y - st cos 00) ~ + z = (y sin 0o -
153
where COS
tan ~ = - - , 2s it is obvious that for real 0o,
Since a-b sin2 ~bt = a + b
- ~ / 4 s 2 + f f : c o s ' ( b ~<~( ~< + ~ / 4 s 2 + ; 2 c o s a ~ b .
(equation 30), then
(43)
Now f ( = ) --, - ~ 3/4" cot 2 ~ for large values of l" I, i.e. f ( = ) is negative for very large positive = and f ( = ) is positive for very large negative values of =. f ( = ) therefore takes the form shown in Fig. I 1. Now if = is equal to + ~ / 4 s 2 + ~2 cos-' ~ = ¢1, f ( ¢ ) becomes:
a-b tan2~bt= 2b
= 4z0 sin q~ + (1 + sin 2 ~b) - b 2b
f ( a t) = (x 2 + y2)= t + 4sxy - (x 2 - y2)~ cos ~b
or
tan z qbt =
= x2(:¢ 1 - ~ cos ~b) + 4 s x y +y2(=t + ~ cos ~b);
4[z + (y sin 00 - x cos 0o)tan ~b~] sin ~b + ~(1 + sin 2 ~) - b 2b after using the z0 from equation (39). If we now substitute tan #, from equation (40) and b from equation (30) into this equation, after some algebraic manipulation, we obtain: (x 2 + y2)(2s sin 200 + ~ cos q~ cos 200) + 4sxy _
(x 2 _ y2)~ cos ~ = [4z sin ~b - cos
x (2s sin 200 + cos ~b cos 2 00) + ((1 + sin 2 ~b)]. cos2~b ] • I s cos 200 - ~ cos ~ sin 200] 2. ILsin ~b]" Let
(41)
but ~/(,,'-
~ cos ~ ) ( = ' + ~
cos
= j ( ~ , ) 2 _ 22 cos 2 ¢ = ~
$) = Z~,
so that f(:z =) may be written as:
f ( = ' ) = [x~/= l - ~ cos ~b + y ~ / = ' + ~ cos ~ ]2 which is always positive for non zero x, y. This means that if = at its upper limit =t for real 0o, then the highest solution to f ( = ) = 0 is outside this limit (see Fig. 1 I). Similarly, it can be shown that if = is at its lower limit ( _ , , t ) for real 0o, then: f ( - ~ ') = - [x ~/~t + ~ cos~b - y ~/~, t _ ~ cos ~ ]2
2s sin 200 + ~ cos ff cos 200 = :~ and
f(a) s cos 20o - ~ cos ~ sin 200 = ~.
Then et2 + 472 = 4s2 + 42 cos 2 or
72= (s2+-~cos2$--~)
¢1
so that equation (41) becomes: (x 2 +y2)= + 4sxy - (x 2 _y2)~ cos ~b -
e
[4z sin ~ + ~(1 + sin 2 ~) - = cos ~]
¢2cos2~ ~-].cos 4
4_J sin2~ = 0
Fig.
11
154
F. S'r]CHER
which is always negative for non-zero x. y. Hence, from Fig. 11, it is seen that the lowest solution to f ( ~ ) = 0 is again outside the range of ~ for real 00. Thus, we have proved that in general, there is only one solution for zc resulting in two values of 00; hence two lines of the congruence pass through a generic point (x, y, z). The exception to this is when x and y are both zero. This means the point chosen is on the z-axis. A glance at equation (42) shows that the solutions to are: =-+-x/4s2+2cos"(a
or
sin ( 2 0 0 + / 3 ) = +1
(44) and = [4z sin (a + (1 + sin-" ~b)]/cos (a.t" Since x, y are zero, any lines of F passing through the point must have s,---,0. It is obvious that the = [4z sin (a + ((1 + sin" (a)]/cos (a solution represents the two 0o which lie on the infinitesimal displacement sine curve in Fig. 8 for constant z, and this is easily verified from the equation to the curve. For the two 00 to be real, z must, of course, lie within the range shown in Fig. 8, i.e. within the z limits of the cylindroid. If s i n ( 2 0 0 + / / ) = + l , then 0 0 = ~ / 4 - / / / 2 and -rt/4 - / ~ / 2 respectively. It seems at first strange that these two solutions are not dependent upon z, and therefore z is not necessarily equal to z0 as equaton (39) shows for x, y = 0. If z is not equal to z0, but lines of F from z0 with s,--* 0 pass through z, then these lines must coincide with the z axis. That is, (a, = n/2 ( o r - n / 2 ) and tan (a, in equation (39) becomes infinite in magnitude. In other words, the solutions of sin(200 +/~) = + 1 represent the lines of F with s~ = 0 , and (a~ = +n/2. F r o m the remarks in Section 5 on the interpretation of Fig. 8, lines where (a, = + n / 2 occur either on the "vertical" boundaries in Fig. 8 where s~ = + oo or at z0 = + oo. By a limiting process, it can be easily shown that if 00 = n/4 - / / / 2 or - n / 4 - / / / 2 , then for all z o, sl--,O, and when z0= +oc, (a, ~ +r~/2. Thus, any point on the z-axis is intersected by four line pairs of F. Two of these line pairs are the infinitesimal displacement paris (s I = 0 ) at Zo = z. Two other pairs with s~ --- 0 come from + oo along the z-axis. Note that these are not infinitesimal displacement pairs even though sx = 0, since (a~--, _.+;r/2 and in fact ~l -" + oc. As an example of the more general case, let us choose a point x = 1, y = 0 , z = - 0 . 5 and use the parameters ~ = 1, s = 2, ~b--300 as those defining our initial eylindroid. Equation (42) yields:
(, 4) X
2,fit 4
3
'
\
The solutions to which are ~--- -4.2723, +0.3315 and +4.2294. As predicted, when all three solutions are real, the lowest and highest are always outside the allowable range of z~ which is ~
-3.7852 -0.0660 -1.0709 -0.4814
+81.5688. 1.0069. -15.9085. -0.4581.
The interpretation of these line positions is shown in Fig. 12, and follows the sign convention for the parameters in Fig. 10. (b) The class o f I"
In order to find how many lines of F lie in a generic plane, let the plane be defined by: (45)
p x + qy + rz = d.
Then using the parametric form for a line of F as shown in equations (35-37), where the line now lies in the given plane: p(st sin 00 + 2 cos (at cos 00) + q(sl cos 00 -
(46)
2 cos (a~ sin Oo)+r(zo + ;. sin (al) = d.
x
t
,-
-
Y
t Note that this particular solution also is the only solution if (x, y, z) lies on the cylindroid, and represents the infinitesimal displacement (double) line coincident with the cylindroid generator passing through (x, Yt z) and another line.
",,,,~Oo ~ 3 . 7 9 •
Fig. 12
Finite screw axis cylindroid Now equation (46) is true for all points on the line, i.e. for all 2, hence we obtain: p cos q~t cos 00 - q cos ~bt sin 00 + r sin Sj --- 0; (47) psi sin 00 + qsl cos 0o + rzo = d.
(48)
F r o m the above two equations, we obtain respectively: tan St = (q sin 0o - p cos Oo)/r
(49)
z o = [d - sl(p sin 00 + q cos Oo)]/r,
(50)
155
This defines two 0o and hence two lines of F which lie in the generic plane p x + qy + rz = d, i.e. the class of F is two. As an example, the plane defined by the two lines intersecting (1, 0, - 0 . 5 ) in part (a) of this section is found to have the equation: ( - 0 . 0 8 0 9 1 7 x + (0.610898)y + (-2.161834)z = 1. (53) Substituting these p, q, r, d-values into equation (52): sin(200 + fl) = 0.08100;
or, substituting st from equation (28) and tan $i from equation (49) into equation (50), we obtain after some trigonometric manipulation: 20 = -r -
s cos 200 - ~ cos q~ sin 200
tan fl = x/~,
and 00= -3.7852° or +81.5688o as before. The corresponding values of q~, si and -'0 are then found from equations (49), (28) and (50) respectively• 7. CONCLUSION
x~2q2sin2Oo+pqcos20o).
(51)
As in part (a) of this section, tan 2 dp~ = a - b/2b where a, b are given in equation (30)• Substituting tan ~'l from equation (49) and z0 from equation (51) into this equation, after much reduction we obtain: v/~ 2 cos" q~ + 4s2 sin(200 + fl). [p2 + q2 q_ r 2] • cos ~b + ( p 2 _ q2)~. cos 2 q~ _ 4pqs cos ~b - (r~(1 + sin 2 ~b) - 4dr sin ~b = 0
REFERENCES
or
sin(200 + fl) = 4dr sin ~b + 4pqs cos ~b + ~r:(l + sin e q~) + (q2 __ p2)cOS2 t~ COS ¢~ " (p2 + q2 + r2). x/4s2 + ~2 COS2 t~" (52)
I~BER DAS ENDLICHE
By using only simple algebraic techniques, it has been shown that the intrinsic nature of the finite screw axis cylindroid and an associated line congruence F can be explored. This has led to a pattern of pitches and other parameters, such that their behaviour as a function of distance along the spine of the cylindroid can be visualised. The relationship between the finite screw and infinitesimal screw displacement cases has consequently been clarified.
!. O. Bottema, J. Engng Ind., Trans. ASME B 95, 451 (May, 1973). 2. Lung-Wen Tsai and B. Roth, J. Engng Ind., Trans. ASME B 95, 603 (May, 1973). 3. J. R. Phillips and K. H. Hunt, Aust. J. Appl. Sci. 15(4), 267 (December, 1964). 4. J. R. Phillips, Freedom In Machinery, Vol 2, Chap. 19. Cambridge University Press.
SCHRAUBACHSEN--ZYLINDROID
Zusammenfassung--Es wird gezeigt, dass die Steigungen des endlichen Schraubachsen-Zylindroids durch
eine stufenmaessige Veraenderung charakterisiert sind, und, wenn dieses auf den Grenzfall eines Zylindroids mit infinitesimaler Schraubverschiebung angewandt wird, resultieren daraus zwei Linien mit Null-Steigung und yon verschiedener Art zu einander. Die Charakteristiken, der Grad und die mehrgliedrige Zahlengrocsse der Linienkongruenz, die aus allen Linien-und Punkt-Reihenverschiebungen besteht, die dieselbe Zylindroidflaeche bilden, werden abgeleitet und ausfuehrlich behandelt.
0094-114X/89 $3.00 + 0.00 Copyright ~ 1989PergamonPress pie
ON THE FINITE SCREW AXIS C Y L I N D R O I D F. S T I C H E R School of Mechanical Engineering, University of Technology, Sydney, N.S.W. 2007, Australia (Received in revisedform 10 October 1988) Abstract--lt is shown that the pitches of the finite screw axis cylindroid are characterised by a step change, and when this is applied to the limiting case of the infinitesimal screw displacement cylindroid, it results in two zero pitch lines of different nature to each other. The characteristics, order and class of the line congruence consisting of all line and point-row displacements which result in the same cylindroid surface, is derived and discussed in some detail.
1. INTRODUCTION Much has already been written on what is called in this paper, "the finite screw axis eylindroid", for example by Bottema in [1] and Tsai and Roth in [2]. The first original contribution in this paper addresses itself to a full understanding of the differences and similarities between the finite screw axis cylindroid and the well-known cylindroid which characterises infinitesimal screw displacements. In particular, a discussion concerning the different natures of the two zero pitch lines in the infinitesimal case is offered. Since the derivation of the cylindroid surface in Section 2 of this paper differs from those in [1] and [2], it is perhaps wise to point out the relationship between parameters in this paper and Bottema's[l]. The origin of the axes in Section 2 is displaced a d i s t a n c e + ( ~ / 2 ) s i n $ along z to that in [1], and Bottema's a, Z, a and d are respectively 7t/2-~b, n/2 - 0, s, ~ cos ~b/2 in this paper. A comprehensive study of all possible line-point row displacements which produce the same cylindroid surface, is the second original contribution offered. The intrinsic physical nature of this (2, 2) congruence of lines is explored in Sections 5 and 6, and a graphic summary of this system is shown in Fig. 8. In the author's opinion, Phillips and Hunt[3] contains the best available summary of the infinitesimal displacement cylindroid. Comments on the finite screw axis cylindroid are also to be found in Chap. 19 of Ref. [4]. Indeed, this paper was inspired by conversations with Professor Phillips in the course of his writing Vol. 2 of Ref. [4], and was originally offered in note form by the author to Professor Phillips, who consequently encouraged its publication. 2. THE FINITE SCREW AXIS CYLINDROID
shown in the figure, and their relation to Bottema's co-ordinate system[l] has been discussed in the introduction. The point P2 on the line position l: is moved to Pt in line position Ii. P2 and Pt are at distances from the y-axis of (2 and (~ respectively, so that the displacement of all points in the line is defined as = ( ~ - (2)- Let the line L be any finite screw axis (FSA) about which/2 can be rotated, and at the same time along which it can be translated, such that line position 1~, together with point displacement ( can be realised. Let L intersect the yz-plane in (Y, Z ) and be inclined at angle 0 to the yz-plane. Since L must make equal angles with lj and Is, it must be perpendicular to the z-axis; that is, any point Q on the line L has co-ordinates (2 sin 0, Y + ). cos 0, Z), where ). is the distance from the yz-plane along L. Let Q~ be the foot of the perpendicular to L, passing through point Pj, and Q2 the corresponding point for the perpendicular to L passing t h r o u g h / 2 . Co-ordinates of Qt, Q2 are ().~ sin 0, Y + ).t cos 0, Z ) and ().2 sin 0, Y + )-2cos 0, Z) respectively. The vectors Pt and P2 (Qi Pl and Q2P2) are therefore given by: Pl = i(~j cos ~ - ).l sin 0) + j ( - s - Y - )-i cos 0) + k ( - C t sin ~ - Z); P2 = i(~2cos ~b - ).5 sin 0) + j ( s - Y - 22 cos 0) (2)
+ k(£2 sin ~ - Z )
and
]P~I--
Ip2l -- P.
From equation (1):
p~ =p2 __ (~l COSqb -- itI sin 0) 2 + ( - - s -- Y - 21 cos 0)' + ( - ~ t sin ~ - Z)'
or
In Fig. 1, a line of points 12 is to be moved to position 1~, such that any point on the line is translated along it by a distance ~. x-, y-, z-axes are as MMT.24r--x
(1)
2~ -- 221(~l cos qb sin 0 - (s + Y)cos 0) + (s + y)2
+ ~ + Z2 + 2 Z ~ l s i n ~ - - p 2 = O . 143
(3)
144
F. S'nCHER
L~
L2
Subtracting equation (9) from equation (10), putting ( i - (2 = ( and dividing by 2r/,
z
~(1 - cos: ~ sin" 0) + 2Z sin +2scos0sin0cos~b=0.
(11)
This is the equation to the finite screw axis cylindroid, relating the angle 0 to Z. Note that if equation (11) is multiplied by ((i + (z) we obtain equation (9) minus the 4s Y term, i.e. (~1 + ~2)(11) - (9) gives Y = 0.
Fig. 1 Equation (3) expresses the obvious fact that given any P(Qt not necessarily the foot of the perpendicular from Pt to L), there are up to two solutions for ;.~, hence two Qt on L. Since PIQt is however, the shortest distance between Pt and L (PI Q~ perpendicular to L), then equation (3) should give a single (repeated) value for ;.I- Thus equation (3) may be written: [21 - (~t cos ~ sin 0 - (s - Y)Cos 0)] 5 = 0.
(4)
From equations (3) and (4) we therefore obtain: 21 = ~1 cos ~b sin 0 - (s + Y)cos 0;
For any position Z along the z-axis therefore, two generators of the cylindroid are defined which intersect the z-axis at right angles. If now Z as a function of 0 from equation (11) and (21, ;.2) from equations (5) and (7) are substituted into the expressions (1) and (2) for Pl and P2 respectively, and if we put: (z = ~ ,
Pt = "~cos 4~ cos 0 + s sin 0 × ( i c e s 0 - j s i n 0 + k ctan ° s : ) ; (12) and P2= ~ c o s ~ c o s 0 + s s i n 0
(5) x ( - i c e s 0 + j sin 0 + k tan c°s 0b) ; (13)
p2 = -(~1 cos ~ sin 0 - (s + Y)cos 0) 5 + (S + y)2 + ~ + Z 5 + 2Z~1 sin ~.
(6)
Applying the same argument to equation (2), we similarly obtain: 25 = (s - Y)cos O + ~5 cos ~ sin O);
(7)
and
p2xpt=2cos0cos~bsin~b(-isin0-jcos0) P2" Pl cos2 q~ cos 2 0 - sin 5 ~b
(14) "
Now P2" Pt
(8)
Equating p5 from equation (6) to p5 from equation
(8): [(~t -
from which
IP, × p,l
p] =p2 = -(~2 cos ~ sin 0) + (s - Y)cos 0) 5 + (s - y)5 + ~I + Z 2 _ 2Z~5 sin ~.
~2= - ~ s a y , then:
is tan ¢,, where ~ is the angle through which 12 is rotated about the FSA L into 1I. Also, equation (14) shows that if we write: - 2 cos 0 cos ~ sin ~b tan ¢, --- cos5 O cos 2 ~ - sin 2 ~"
~2)cos ~ sin O - 2.s cos 0]
(15)
× [(~1 + ~2)cos ~ sin 0 - 2Y cos 0 ] -- (~1 - ~z)(~t + ~ ) + 4sY + 2Z sin ¢(~1 + ~2). (9) Equation (9) is true for any other points P[ at £t + and P~ at ~5 + r~, such that £t + r / - (~ + ~) is still the displacement ~ = ~t - ~5. Writing equation (9) with ~t replaced by ~ + r/, ~ replaced by ~5 + ~/, [(£~ - ~5)eos ~ sin 0 - 2s cos 0] x [(~1 + ~ 5 + 2 . r/)eos ~b sin0 - 2 Y c o s O ] --(~ - ~5)(~1 + ~ + 2~) + 4sY + 2Z sin ~b(~l + ~2 + 2. ~/).
(10)
Then this will indicate the sense of ~. Since we shall use the convention that - n ~< ¢J ~< + n, then the signs of the numerator and denominator of equation (15) are the same as the signs of sin ~k and cos ~b respectively. Equation (15) could also have been obtained easily by other means, for example, by finding the angle between the projections of/2 and il onto a plane perpendicular to L. The sliding or translation along L which takes place is given by/a = ;.5 - ;.i which from equation (5) and (7) is: /~ = 2s cos 0 -- ~ cos ~ sin 0. (16)
Finite screw axis cylindroid
145 i"
x
t
•
L1
enerotor x
k2
z
.>I xz
-~y
Fig. 2 Y
Finally, the cylindroid equation (II) may also bc expressed in the form:
~(I +
sin 2 ~b) +
2Z sin 0~ Fig. 3
= -cos ~
cos2 ~ + s z" sin(20 +/~)
(17)
where tan/~
=
cos ~, 2s
Equation (17) shows clearly that if 0 is a solution for a particular Z, then 1r/2 - ( 0 + ~) is also a solution, corresponding to two generators for any Z lying within the range. 2sin~b
--
(l +sin2~b)-cos~b
cos2~b+ s 2
1
~
¢2
+oos0j cos, o 1. (18) In equation (16), note that p has been defined as (A~ -~.~), so that if~.~ is less than ~-z, # is positive and in the same sense as positive ~#given by equation (15). that is, if 0 ~ ~# ~ + n and p is +v¢, then the sense of both ~# and p are shown in Fig. 2. The pitch of L is defined as for the infinitesimal motion cylindroid by h = ~/~.
3. SPECIAL SCREW AXES AND THE INFINITESIMAL MOTION CASE There exist two screw axes of special interest; these are the cylindroid generators of zero pitch (tt ==0) and for want of a better name, the n generator where ffi ± n or half a turn. The pitch of the n generator can then be regarded as positive or negative and results in a step jump in the pitch on an (h, 0) plot, if we adopt the convention that I~ I ~ n. (a) The zero pitch generator (1~ ==0 line) occurs when tan 0 = 2s/~ cos ~b from equation (16). Substi-
tution of this 0 into equation (11) yields Z = - ~ / 2 sin 4. The second generator at this value of Z i s a t 0 = 0 w i t h # f 2 s and ~ = - 2 0 b . These almost self evident positions and values arc shown in Fig. 3. (b) When ~ ffi ±n, then from equation (15), 0 = n/2 and from equation (11), Z = - ; sin 4/2. As we pass through this value of Z, we change the sense of rotation of ~b but not the sense of # and we have a step jump in h from - ~ cos ~/n to + ~ cos ~/n. Note that this "~ generator" is orthogonal to the 0 ffi0 generator mentioned in (a),and the zero pitch generator in (a) isorthogonal to the second generator through Z = -~ sin ~b/2 at tan 0 = -~ cos ~b/2a(see Fig. 3). An inspection of equation (18) shows that the mid-point of the range for Z (the middle of the "spine" of the cylindroid) is at:
Zc=
(I +sinZ~b)=~I
4 sin ~
~
~sin~bl
2 sin ~b
2
"
That is, the zero pitch generator and the n generator are equally and oppositely spaced from the centre of the cylindroid. If the equation (17) of the cylindroid is referred to axes parallel to our x-, y-, z-axes but with origin at Z c, it becomes:
2z tan ~ +
cos2 ~b + s z sin 200 =
O,
(19)
where
z=z-z~, Oo=IO+~l. (c) When the line positions/2 and i~ become infinitesimally separated, we may substitute 6s, 64, 6¢ for s, 4, ¢ and note that to the first order sin ~b--*8~
146
F. STICHER
and cos q~ --, I. Using a:16¢, &IMp as constants, the cylindroid equation (17) becomes (as as, aq~, 6: --* 0)
lF6:l
~aF6¢T
(16) become: - 2 cos 0 - &p tan ~ = cos20( 1 _ (&b):/2) - (&b):
r&l:sin(2o+,a);
+L j
(20)
where
ira:
tan/~
=
(21)
and
L~" ~aq~]
/~ = 2& • cos 0 - 6: • sin 0. (22) Now equation (21) becomes tan~, = - 2 a ~ / c o s 0 provided 0 does not approach the value n/2. Then tan ~ --, ~, and
"
/~
(2& cos O - 6: • sin 0)cos 0
The zero pitch generator of (a) then is at:
I ra:l,j z=-~L3g
tan0--
=~[~lsin20 1V&-ll~j L ~J -2L-~'f c°s2°
La~j 2 ~ ~..
which may be expressed in the form,
The n generator of (b) is then at: Z
=
-
a;. 6,/, ~ , 2
h=A
a~ /t
to
h=
+Bcos(20+=)
i.e.
that is, we have the classic sinusoidally varying h for the infinitesimal motion case. As 0 approaches re~2, ~k very quickly becomes large and approaches + n, i.e. it can no longer be assumed that tan ~b ~ ~b, and moreover we must retain the 2nd order quantities in equation (21) to see how ~b varies. If we put 0 = (n/2 - ~) where ¢ is small, equation (21)
at Z = 0 as 3~, a~ ~ 0 and 0 = n/2. The pitch of the generators as we pass through Z = 0 then changes from: h=---
+ I),
6(
+--. /t
But as a~ --* 0, the pitch changes from - o, to + o, and the n generator can then be regarded as a second zero pitch line. The existence of two h = o lines corresponds to what is well known about the infinitesimal motion cylindroid[3]. However, we must be very careful here! Let us examine the distribution of pitch along the cylindroid in this case, as a function of 0. Equations (15) and x
/ /
tan ~k
and
-
2&.b
"-'E 2 -
•
(23)
(6~)2'
/~ -~ -~(.
E can be made as small as we wish, and we see that if ~ < &b, we obtain ~, ~ + n; if E ~ 6~b; ~k ~ + n/2
~- Ls-2,¢-3o"
l,r generoT, or I h=O,z=-I / /
becomes:
1-2.39721 _:/
~ . 1 - 2 . o ,
I-0.251+~
~ 1-101
,.__o0.%/_
)_ ,_o.,o, (*0-2) \
1÷1.o) \
11.1472)
Fig. 4
~y
I47
Finite screw axis cylindroid (0)
a=77.Ts" h=O,z=-I
1.e)
i///
• .
Z, /--'-
/
/-/<-'6'
/
///I
<_o.% iii1
IFo.yI/
~o°oS~ u''p'
III
////
--y
(-0.25)
_/
1o)
i
1÷0.41
Fig. 4a and if E > 6~, tan ~ rapidly becomes small in magnitude with increasing ~. Since the constants of the infinitesimal displacement case are really 6~/6~ and 6s/tdp, it appears that we can argue that 6~b can also be made as small as we wish, and therefore, the region where h no longer varies sinusoidaUy also becomes infinitesimally small. This is true, but perhaps the best way of still reminding ourselves of the unique character of the second h = 0 generator is to call it a singular generator, It is singular in the sense that [~'1 rapidly diminishes from ~r to an infinitesimally small value at an infinitesimally small distance from this generator along the "spine" of the cylindroid! The second of the following examples illustrates this point. 4. EXAMPLES OF PITCH DISTRIBUTION IN THE CYLINDROID In the following two examples, the conventional polar plot of pitch has been modified to include the
sense of rotation for ~, which is perhaps of greater importance in the finite displacement case than in the well known infinitesimal displacement cylindroid. Normally, each generator is counted twice along the cylindroid, that is, the line designated by 0 is counted as different to the line designated by (lr + 0). The polar plot of pitch h as a fucntion of 0 is then drawn resulting in the familiar four-lobed figure as Phillips shows in [3]. Figures 4 and 5 show only two lobes, since the system adopted in this paper is that a single cylindroid generator has only one direction defined by the sense of tb, and the length of the lines drawn in the direction of ~bis proportional to Ih[. Ifh is positive it is marked + and if negative, m a r k e d - . The Z value is also shown in brackets. To illustrate this system, let us calculate the relevant quantities for the case Z = 0.6 in Fig. 4 ( ~ = 3 0 °, ~ = 1 , s = 2 ) . From equation (17), /1=12.2163 ° and the two solutions to 0 are
148
F. STICFIER
~-o.o4,s=o.ol,~=z-
Positive pitch Loop (- o.2) //~(- o.3)
h=¢0.013
(O,
(-0.4)
3.3)
Negative pitch Loop Fig. 5
0[ = -27.9723 ° and 02 = 90 - (0, + fl) = 105.7560 °. From equation (15), tan ~1 = -0.7649/0 + 0.3350 and tan ~'2 = +0.2352/-0.1947 or ~'l = -66.3482°, ~2 = 129.6227 °. Equation (16) yields /~ =3.9389, /~2= -1.9197. Referring to the sense of ~, and ~ in Fig. 2, we see that for 01 a line should be drawn from the z-axis (origin of system in Fig. 4) at - 2 7 . 9 7 ° since ~1 is negative and therefore in the opposite sense to that shown in Fig. 2. The pitch /~l 3.9389 180 -3.4015 h I i s - - - ; = --66.3482 ~ = and therefore the line is drawn 3.4015 units in length and labelled as negative with Z(0.6) as shown in Fig. 4. For the line at 02, however, ~: is positive, hence it is drawn not at 105.76 ° but in the opposite sense at 105.76 ° - 180 ° = - 7 4 . 2 4 ° (see Fig. 2). Note that in Fig. 4a, the region near the n generator has been magnified to show the sudden jump from h -- -0.2757 to h --- +0.2757, resulting in a highly truncated small lobe and a slightly
truncated large lobe. Other lines of interest are clearly marked. The case depicted in Fig. 5 is for ~ ..~ 0.04, s = 0.01 and ~b = 2 °, and this exhibits the infinitesimal displacement characteristics. From the diagram, it can be seen that both lobes are almost "complete", the jump in h at the n generator being from -0.013 to + 0.013. In order to illustrate the discussion in Section 3 on the behaviour of ¢ and the pitch h near the n line for the infinitesimal case, the following table of selected values is interesting. The reader should keep in mind that the total length (range of Z ) of the cylindroid is 0.64 units. As well as h ~ , the approximation h,~,,,o~ is given, showing the breakdown of this approximat/on near the ~t generator, i.e. near
0 =~/2. From Section 3: 1 6~'
hsiw,aoid.s= ~ ~
. 1 6s sm 20 -- ~ ~ (1 + cos 20).
Where ~
=
0.o4, 6~,
=
o.ol, ~ )
(2)~ =
]8o'
Finite screw axis cylindroid
Referring to axes x I, Yl, z,, we see that z, = z - zo and 0, = 0 - 0o. Thus, if we substitute Z = ZI + zo and 0 = 0, + 0o into equation (24), we obtain:
so in this case: h,~,~,d,, = 0.2865 sin 20 -- 0.1433(1 + cos 20) Z
0°
hu,~
hz..
]%1err. in h
+0.1565 +0.1767 +0.1653 +0.1173 +0.0501 +0.0368 +0.0231 +0.0130 -0.0133 -0.0288 -0.0551 -0.0987
+0.1564 +0.1768 +0.1650 +0.1167 +0.0477 +0.0334 +0.0173 +0.0006 -0.0014 -0.0238 -0.0525 -0.1002
0.6 0.06 0.18 0.51 4.79 9.24 25.11 95.38 89.47 17.36 4.72 1.50
g,~
-0.4 4 7 . 9 3 9 6 -5.9679 -0.3 5 7 . 1 1 5 2 -7.3600 -0.2 6 6 . 1 6 6 7 -9.8786 -0.1 76.1559 -16.6063 -0.03 84.9864 -43.5615 -0.02 86.5455 -60.1877 -0.01 88.2451 -97.5026 -0.001 89.9397 -176.5475 0.000 90.1403 +171.9754 +0.01 92.3328 +81.2546 +0.02 95.0575 +43.2197 +0.03 99.2852 +24.4243
149
As Z ranges from -0.03 to +0.03, 0s ranges from about 85 ° to 99 °, but g,, passes from - 4 4 ° through + 1800 to + 2 4 ° - - a range of roughly 292°! 5. THE SET OF LINES 12 AND 11 WHICH PRODUCE THE SAME CYLINDROID
We now ask, "What other lines Is and It with different values of ($, ~, s) would produce the same cylindroid?" We do not require the $ and/~ distribution along the cylindroid to be the same, only the cubic surface itself. We shall take as reference the set of axes and cylindroid defined by the original pair of lines/2 and 11, as shown in Figs 1 and 6. At distance z0 along the z-axis, assume a common perpendicular between two other skew lines Is0 and 1,0 tilted at angle 00 to the zy-plane. The quantities Or, ~,, sl are shown in Fig. 6 together with a new set of axes xt, Yo, z~. Now the equation to the cylindroid in the x, y, zsystem and in terms of ~, ~, s can be written [from equations (11) or (17)] in the form:
~[.I + sin2@l
sin~ j+2(zo+Z,)+cot¢
x(~coscpcos2(O+Oo)+ssin2(O+Oo))=O or
sin~ + cot 4~ +
~cosqJcos20o+ssin20o)¢os201
- ~ cos ~ sin 200 + s cos 200)sin 20, = 0.
x,, Yt, z,,
But the equation to the cylindroid using the 4h, ~,, s, system is of the same form as equation (24), i.e.
¢i1_
-~I 1+sinS-fi~'¢l .| + 2z, + c o t $ 1 ( ~ c o s $, cos 20, + s , sin 2 0 , ) = O. (26) Comparing coefficients of cos 201 and sin 201 in (26) and (25) and equating the constant terms:
cotC~(~coscpcos2Oo+ssin2Oo)=cotO,~cos¢~l, (27)
(
cot ~ \ - ~ cos ~p sin 200 + s cos 200)
= cot ~,. 5t (28)
and
2 (24)
2Z,
(25)
~[I ++2Zsin sin'~'] + cot ~ -~ cos q~ cos 20 + s sin 20 --- 0.
J+
sine, +2,O=#k- ~
J" (29)
By substituting ~l from equations (27) into (29), we may then solve for sin2 ~L:
a-b
sins ~l = a + b' Z,IZ 1
(30)
where a = 4zo sin~ + ~(1 + sins ~) and
L1
b -- cos ¢p(2s • sin2 O0+ cos cp cos2 00)
Yl
= cos ¢p~/~s cos s ~ + 4s s . sin(20o + 8) and tan ~ff -- ~cos 2s ~) as in equation (17).
/ Fig. 6
Ct and s, are then found from equations (27) and (28). It does not matter if 0r is taken as + v e o r - r e , since the true sense of the new line and 110 will be defined by a positive or negative s,. That is, the values (Oi, ~t, s,) define the same skew-pair and sense of~ as
pair l~
150
F. SmCHEa b
.A [
A-cos @~/(.;2cosZ@+4s2
-A O
-A
~o
÷A
Fig. 7
(-~bl, -~1, sl). For convenience, we shall take ¢~1 as being positive (0 ~<~bl ~< n/2) and allow ~t and sl to adjust themselves accordingly (remember that the sense of ~, and F is irrelevant). From equation (30), since a-b
0 .N<~ - - ~ ~ + 1 and since
become infinitesimal, i.e. the infinitesimal displacement case is approached. If 00 = - # / 2 (the lower boundary) or 00 = (n/2 - 3/2) (the upper boundary), then we approach cases where ~1 = re/2, sj ~ + oc and ~(1 + sin-' ~) + 4Z0 sin ¢ 2 sin @ As z0---. + oo, the angle ~bt always approaches n/2.
- c o s q~x/~2cos 2 ~b + 4s 2 ~ z , , then a l > b and i f z < z c , then a~
The interpretation of Fig. 8 is as follows. Choose any particular value of z0, Z0 say, as shown in the figure. Then the allowable ranges for 00 are given by those parts of the line z0 --- Z0 which lie in the shaded region. As the values of 00 approach the boundaries of the sine curve, it is found that the values of @1, ~1 and Sl
For Z0 beyond the z bounds of the cylindroid, it is seen in Fig. 8 that no value of 0o produces line l~0,110 which are infinitesimally displaced. When Z0 is within these bounds, there are two such pairs of lines/2o and 110 (two intersections of z0 = Z0 with the sine curve); the two corresponding s, lines in this case are orthogonal to the two generators of the cylindroid residing at Z 0. If0z, 0z are the angles 0 defining the generators, then the two 00s will have values of (01 + n/2) and (02 _+ rc/2), (Whether it is + n / 2 or -rr/2 is best resolved when a particular numerical example is considered.) Indeed, this shows that all infinitesimally displaced line pairs (double lines), generate the original cylindroid surface! This could also have been deduced from Section 3, where the n line (a line of the cylindroid) is virtually coincident with an infinitesimally displaced pair. Figure 8 also shows that there is a singular point 0 0 = - # / 2 or n / 2 - # / 2 , Zo=Zc (the centre of the cylindroid), which lies on both the infinitesimal displacement sine curve boundary and on the @~= n/2 boundary. Indeed, at z0 = z¢, there is no allowable range of 00 other than 00 = - # / 2 or 00 = n/2 - 3/2! (Note that 00 = - n / 2 - #/2 results in the same case as 0o = n / 2 - #/2 in what follows.) Let us see what happens at this singularly interesting point. If we revert to the original equations (27),
Finite screw axis cylindroid
151
Zo
\
0o 2
2 Ze
~°
i
~, - {Z¢~
A
4 )in
Fig. 8
(28) and (29) and put 00 = - / / / 2 or ~/2 - / / / 2 , z0 = zc, we obtain: 0 = ~£cot ~ cos ~l,
(31)
Z
+ cot ~b :-2 - - ~ N/~ COs2~) -I-
I1o. Physically, however, all becomes clear when we look at a converse problem. What becomes of the cylindroid equation (17) when ~ = 07 Equation (17) is then: 2Z sin ~ = - s cos ~b sin 20;
4.$ 2 =
S1cot ¢~1,
(32)
(The signs on the left-hand side correspond to 00 = - / / / 2 , ~ / 2 - / / / 2 respectively). 0 = ~l[ ,1 + sins ~17
j.
(33)
Clearly, equations (31) and (33) are satisfied for any 0 < ~l < ~/2 if ~ = 0. s~ is then found from equation (32), and ranges from 0 to + ~ . Thus, at zo = zc, there exist two perpendiculars (at 0 0 = - / / / 2 and ~/2 -B/2) each common to an infinity of possible/20 and 1~0 line pairs, all with ~ = 0. ~bl can range from 0 to ~/2, so it is indeed true that zo = zc can lie on both the infinitesimal displacement boundary ( ~ 0 , s~--, 0), and the (~. = ~/2, sl ~ oo) boundary. This is quite remarkable, since for any other particular zo, 0o specified, there exists only one pair of skew lines 12o,
or
2Z = - s cot ~b sin 2 0. (34) Thus, the shape of the cylindroid is defined only by s cot ~ and s and ~ can vary provided this product remains constant. From equation (32) we see that si cot q~l is indeed a constant, equal to + s cot ~ when ~=0. Figure 9 is a sketch illustrating two arrays for the case ~ - 30, s = 2, ~ = 1. The first array shows the lines 12o, llo defined by equation (32) at Zo = z¢ = - 0.625. The second shows some lines l:o, l~0 for 0o = 70o, i.e. all the lines lzo, l~0 lie in parallel planes, and in this case 0.2011 ~ zo ~< oo. The construction is not exact in the sketch, but shows the behaviour o f s l , ~l, ~bl for 0o = 70°. The table of exact values for 00 = 70 ° is shown below. Note the small displacement between the lines near zo = 0.2011 (infinitesimal displacements).
152
F. STICh~R
~1=40 ;1=30
~=20 70* ~I=11
~y
Z=Zc,80=-~-,~1=0
: z . 8o: T -
,g~=O Fig. 9
O0= 70°
0.2012 0.5 1.0 2.0 3.0 5.0 10.0 15.0 20.0 100.0 1000
- 0.0274 - 1.3338 -2.1806 -3.2721 -4.0814 -5.3443 -7.6367 -9.3849 - 10.8552 -24.3713 -77.1388
Zo > 0.2011
0.0299 1.5274 2.7987 4.9833 7.0592 11.1280 21.1857 31.2063 41.2169 201.2432 2001.2500
0.5001 23.0430 34.8144 46.2188 52.4651 59.5978 67.6762 71.5242 73.8875 82.6683 87.6722
x = sz sin 0o + 2 cos 4)t cos 00;
(35)
y = s~ cos 0o - 2 cos @t sin 0o;
(36)
z = z0 + 2 sin @,;
(37)
where :. = SQ. Let Q be a generic p o i n t in space, i.e. x, y, z are treated as given c o n s t a n t s in what follows. s~ and @~ can, o f course, be negative, representing a conjugate line o f a n lzo, l~o pair.
x,y,z)
Z
Clearly, the system o f ~ 2 lines 120 a n d lz0 defines a line congruence F o f some sort. In the next section, it is s h o w n t h a t b o t h order and class o f this congruence is two.
S 8o
6. THE ORDER AND CLASS OF THE CONGRUENCE F
(a) The order of F C o n s i d e r any point Q on a line o f the congruence as s h o w n in Fig. 10. By inspection:
Fig. 10
Finite screw axis cylindroid or
From equatons (35) and (36): y cos 00 + x sin 00 = st
x cos 00)tan ~bt + z.
(40)
(42)
f ( a ) is a cubic function with some very interesting characteristics. We shall show that/n general, there is only one value of = which produces two real values of 0o. Since a may be written in the form: . = x/4s 2 + ~2 cos 2 q~ sin (200 +/~)
(39)
Also, by substituting st from equation (38) into equation (27), (x sin 00 + y cos 00)tan tan ~b~ = (s cos 2 00 - ~/2 cos ~b sin 2 00)"
f ( : t ) = O.
(38)
and from equations (36) and (37), using st in equation (38): tan ~b~ zo = (y - st cos 00) ~ + z = (y sin 0o -
153
where COS
tan ~ = - - , 2s it is obvious that for real 0o,
Since a-b sin2 ~bt = a + b
- ~ / 4 s 2 + f f : c o s ' ( b ~<~( ~< + ~ / 4 s 2 + ; 2 c o s a ~ b .
(equation 30), then
(43)
Now f ( = ) --, - ~ 3/4" cot 2 ~ for large values of l" I, i.e. f ( = ) is negative for very large positive = and f ( = ) is positive for very large negative values of =. f ( = ) therefore takes the form shown in Fig. I 1. Now if = is equal to + ~ / 4 s 2 + ~2 cos-' ~ = ¢1, f ( ¢ ) becomes:
a-b tan2~bt= 2b
= 4z0 sin q~ + (1 + sin 2 ~b) - b 2b
f ( a t) = (x 2 + y2)= t + 4sxy - (x 2 - y2)~ cos ~b
or
tan z qbt =
= x2(:¢ 1 - ~ cos ~b) + 4 s x y +y2(=t + ~ cos ~b);
4[z + (y sin 00 - x cos 0o)tan ~b~] sin ~b + ~(1 + sin 2 ~) - b 2b after using the z0 from equation (39). If we now substitute tan #, from equation (40) and b from equation (30) into this equation, after some algebraic manipulation, we obtain: (x 2 + y2)(2s sin 200 + ~ cos q~ cos 200) + 4sxy _
(x 2 _ y2)~ cos ~ = [4z sin ~b - cos
x (2s sin 200 + cos ~b cos 2 00) + ((1 + sin 2 ~b)]. cos2~b ] • I s cos 200 - ~ cos ~ sin 200] 2. ILsin ~b]" Let
(41)
but ~/(,,'-
~ cos ~ ) ( = ' + ~
cos
= j ( ~ , ) 2 _ 22 cos 2 ¢ = ~
$) = Z~,
so that f(:z =) may be written as:
f ( = ' ) = [x~/= l - ~ cos ~b + y ~ / = ' + ~ cos ~ ]2 which is always positive for non zero x, y. This means that if = at its upper limit =t for real 0o, then the highest solution to f ( = ) = 0 is outside this limit (see Fig. 1 I). Similarly, it can be shown that if = is at its lower limit ( _ , , t ) for real 0o, then: f ( - ~ ') = - [x ~/~t + ~ cos~b - y ~/~, t _ ~ cos ~ ]2
2s sin 200 + ~ cos ff cos 200 = :~ and
f(a) s cos 20o - ~ cos ~ sin 200 = ~.
Then et2 + 472 = 4s2 + 42 cos 2 or
72= (s2+-~cos2$--~)
¢1
so that equation (41) becomes: (x 2 +y2)= + 4sxy - (x 2 _y2)~ cos ~b -
e
[4z sin ~ + ~(1 + sin 2 ~) - = cos ~]
¢2cos2~ ~-].cos 4
4_J sin2~ = 0
Fig.
11
154
F. S'r]CHER
which is always negative for non-zero x. y. Hence, from Fig. 11, it is seen that the lowest solution to f ( ~ ) = 0 is again outside the range of ~ for real 00. Thus, we have proved that in general, there is only one solution for zc resulting in two values of 00; hence two lines of the congruence pass through a generic point (x, y, z). The exception to this is when x and y are both zero. This means the point chosen is on the z-axis. A glance at equation (42) shows that the solutions to are: =-+-x/4s2+2cos"(a
or
sin ( 2 0 0 + / 3 ) = +1
(44) and = [4z sin (a + (1 + sin-" ~b)]/cos (a.t" Since x, y are zero, any lines of F passing through the point must have s,---,0. It is obvious that the = [4z sin (a + ((1 + sin" (a)]/cos (a solution represents the two 0o which lie on the infinitesimal displacement sine curve in Fig. 8 for constant z, and this is easily verified from the equation to the curve. For the two 00 to be real, z must, of course, lie within the range shown in Fig. 8, i.e. within the z limits of the cylindroid. If s i n ( 2 0 0 + / / ) = + l , then 0 0 = ~ / 4 - / / / 2 and -rt/4 - / ~ / 2 respectively. It seems at first strange that these two solutions are not dependent upon z, and therefore z is not necessarily equal to z0 as equaton (39) shows for x, y = 0. If z is not equal to z0, but lines of F from z0 with s,--* 0 pass through z, then these lines must coincide with the z axis. That is, (a, = n/2 ( o r - n / 2 ) and tan (a, in equation (39) becomes infinite in magnitude. In other words, the solutions of sin(200 +/~) = + 1 represent the lines of F with s~ = 0 , and (a~ = +n/2. F r o m the remarks in Section 5 on the interpretation of Fig. 8, lines where (a, = + n / 2 occur either on the "vertical" boundaries in Fig. 8 where s~ = + oo or at z0 = + oo. By a limiting process, it can be easily shown that if 00 = n/4 - / / / 2 or - n / 4 - / / / 2 , then for all z o, sl--,O, and when z0= +oc, (a, ~ +r~/2. Thus, any point on the z-axis is intersected by four line pairs of F. Two of these line pairs are the infinitesimal displacement paris (s I = 0 ) at Zo = z. Two other pairs with s~ --- 0 come from + oo along the z-axis. Note that these are not infinitesimal displacement pairs even though sx = 0, since (a~--, _.+;r/2 and in fact ~l -" + oc. As an example of the more general case, let us choose a point x = 1, y = 0 , z = - 0 . 5 and use the parameters ~ = 1, s = 2, ~b--300 as those defining our initial eylindroid. Equation (42) yields:
(, 4) X
2,fit 4
3
'
\
The solutions to which are ~--- -4.2723, +0.3315 and +4.2294. As predicted, when all three solutions are real, the lowest and highest are always outside the allowable range of z~ which is ~
-3.7852 -0.0660 -1.0709 -0.4814
+81.5688. 1.0069. -15.9085. -0.4581.
The interpretation of these line positions is shown in Fig. 12, and follows the sign convention for the parameters in Fig. 10. (b) The class o f I"
In order to find how many lines of F lie in a generic plane, let the plane be defined by: (45)
p x + qy + rz = d.
Then using the parametric form for a line of F as shown in equations (35-37), where the line now lies in the given plane: p(st sin 00 + 2 cos (at cos 00) + q(sl cos 00 -
(46)
2 cos (a~ sin Oo)+r(zo + ;. sin (al) = d.
x
t
,-
-
Y
t Note that this particular solution also is the only solution if (x, y, z) lies on the cylindroid, and represents the infinitesimal displacement (double) line coincident with the cylindroid generator passing through (x, Yt z) and another line.
",,,,~Oo ~ 3 . 7 9 •
Fig. 12
Finite screw axis cylindroid Now equation (46) is true for all points on the line, i.e. for all 2, hence we obtain: p cos q~t cos 00 - q cos ~bt sin 00 + r sin Sj --- 0; (47) psi sin 00 + qsl cos 0o + rzo = d.
(48)
F r o m the above two equations, we obtain respectively: tan St = (q sin 0o - p cos Oo)/r
(49)
z o = [d - sl(p sin 00 + q cos Oo)]/r,
(50)
155
This defines two 0o and hence two lines of F which lie in the generic plane p x + qy + rz = d, i.e. the class of F is two. As an example, the plane defined by the two lines intersecting (1, 0, - 0 . 5 ) in part (a) of this section is found to have the equation: ( - 0 . 0 8 0 9 1 7 x + (0.610898)y + (-2.161834)z = 1. (53) Substituting these p, q, r, d-values into equation (52): sin(200 + fl) = 0.08100;
or, substituting st from equation (28) and tan $i from equation (49) into equation (50), we obtain after some trigonometric manipulation: 20 = -r -
s cos 200 - ~ cos q~ sin 200
tan fl = x/~,
and 00= -3.7852° or +81.5688o as before. The corresponding values of q~, si and -'0 are then found from equations (49), (28) and (50) respectively• 7. CONCLUSION
x~2q2sin2Oo+pqcos20o).
(51)
As in part (a) of this section, tan 2 dp~ = a - b/2b where a, b are given in equation (30)• Substituting tan ~'l from equation (49) and z0 from equation (51) into this equation, after much reduction we obtain: v/~ 2 cos" q~ + 4s2 sin(200 + fl). [p2 + q2 q_ r 2] • cos ~b + ( p 2 _ q2)~. cos 2 q~ _ 4pqs cos ~b - (r~(1 + sin 2 ~b) - 4dr sin ~b = 0
REFERENCES
or
sin(200 + fl) = 4dr sin ~b + 4pqs cos ~b + ~r:(l + sin e q~) + (q2 __ p2)cOS2 t~ COS ¢~ " (p2 + q2 + r2). x/4s2 + ~2 COS2 t~" (52)
I~BER DAS ENDLICHE
By using only simple algebraic techniques, it has been shown that the intrinsic nature of the finite screw axis cylindroid and an associated line congruence F can be explored. This has led to a pattern of pitches and other parameters, such that their behaviour as a function of distance along the spine of the cylindroid can be visualised. The relationship between the finite screw and infinitesimal screw displacement cases has consequently been clarified.
!. O. Bottema, J. Engng Ind., Trans. ASME B 95, 451 (May, 1973). 2. Lung-Wen Tsai and B. Roth, J. Engng Ind., Trans. ASME B 95, 603 (May, 1973). 3. J. R. Phillips and K. H. Hunt, Aust. J. Appl. Sci. 15(4), 267 (December, 1964). 4. J. R. Phillips, Freedom In Machinery, Vol 2, Chap. 19. Cambridge University Press.
SCHRAUBACHSEN--ZYLINDROID
Zusammenfassung--Es wird gezeigt, dass die Steigungen des endlichen Schraubachsen-Zylindroids durch
eine stufenmaessige Veraenderung charakterisiert sind, und, wenn dieses auf den Grenzfall eines Zylindroids mit infinitesimaler Schraubverschiebung angewandt wird, resultieren daraus zwei Linien mit Null-Steigung und yon verschiedener Art zu einander. Die Charakteristiken, der Grad und die mehrgliedrige Zahlengrocsse der Linienkongruenz, die aus allen Linien-und Punkt-Reihenverschiebungen besteht, die dieselbe Zylindroidflaeche bilden, werden abgeleitet und ausfuehrlich behandelt.