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The Bennett linkage and its associated quadric surfaces
Mech. Mach. Theory Vol. 23, No. 2, pp. 147-156, 1988 Printed in Great Britain. All rights reserved
0094-114X/88 $3.00+0.00 Copyright © 1988PergamonPress pie
THE B E N N E T T L I N K A G E A N D ITS A S S O C I A T E D QUADRIC SURFACES J. E D D I E B A K E R Department of Applied Mechanics, The University of New South Wales, P.O. Box l, Kensington, NSW 2033, Australia (Received in revised form 30 September 1986)
Abstract--The 4-revolute Bennett linkage has been intensively studied by several researchers over many years, and yet there remain depths to be plumbed. In this study, we begin by considering separately the manner in which the Bennett linkage is related to the hyperboioid defined by its joint axes and the one defined by its links. We go on to algebraically inter-relate these and other aspects of the loop, including an associated sphere and an equivalent screw of the overall instantaneous motion.
INTRODUCTION It would be unwise to speculate about what might constitute the last word on Bennett's "skew isogram mechanism"[l, 2]. Since its discovery more than 80 years ago, it has puzzled, fascinated and challenged kinematicians of both geometrical and algebraic persuasions, and there continue to be fresh contributions made to the abundant literature concerning this remarkable 4-bar (Fig. 1). Its closure equations are accorded thorough consideration in[3], where also, along with[4], there is discussed the peculiar propensity of the linkage to function as a building-block for synthesising 5- and 6-bar loops. More recently, but founded on work of 40 years past, it has been shown[5-7] that the Bennett chain has a role to play in developing mobile mechanical networks. Most of the other research on the Bennett linkage is referred to in the papers cited here, except for the spasmodic suggestions concerning practical applications of the skew 4-bar. One of the lesser known of these latter works is[8], in which the relationship between input and output angles of the Bennett loop is shown to coincide with that for a meshing pair of elliptic gears of suitably selected eccentricity. The subject of this paper represents a return to fundamentals, whereby we consider the detailed definitions of the hyperboloids and sphere which are associated with any particular Bennett linkage at a specified configuration. It has been known for at least 70 years that the four hinge axes of the linkage can be taken as generators of the same regulus on a hyperboloid (of one sheet) at a given configuration of the loop, and this knowledge has been used to advantage by various authors. It was only recently[9], however, that algebraic relationships between the hyperholoid and the linkage were established, through employing the device of the tetrahedron which is delineated naturally by the 4-bar. In that paper, and in[10], there is also highlighted the somewhat surprising positioning of the linkage on the
hyperboloid of its axes, a fact apparently first noted by a student of Professor Phillips. For later convenience, we shall henceforth refer to this quadric surface as the J-hyperboloid. It is demonstrated in[7] that the links of a Bennett loop define another hypcrboloid of one sheet, a surface of revolution, by virtue of their identification with two members of each of the reguli forming the conicoid. We shall call this surface the L-hyperboloid. As well, Yu[ll] has investigated the significance of the sphere determined by the four vertices of the linkage. Our purpose in this work is to directly relate three independent parameters and a suitable single joint variable of the linkage with the relevant quantifies of the sphere and the two forms of hyperboloid associated with it. We also intend, by means of screw system algebra, to obtain a simple expression f o r the pitch of the equivalent screw along the line of (physical/geometrical) symmetry of the mechanism. Much of the notation employed herein is in common usage and need not be re-defined. Any new or unusual notation will be explained as it is introduced. Most results will be presented without detailed derivation, for the sake of brevity, but all major steps will be indicated. With regard to the skew isogram itself, numbering the revohites 1 to 4, we denote the two different link-lengths by l~2 and 123 and the corresponding angles of skew by ~2 and c(23, only three of these quantities being independent, because
l~
= l~.
These parameters and the joint variables are illustrated in Fig. 2. The notation is made consistent and unambiguous by using the right-hand screw rule and restricting all angles of skew to lie in the range (0, ~). The reader is referred to[3] for an extended discussion on these matters. In particular, the commonly held belief that angles of skew should be acute and alternately right- and left-handed is false. In this context, the reader's attention is drawn to Fig. 17 147
148
J. EDDIE BAKER
Fig. 1. A display model of the Bennett mechanism, designed to permit full-cycle motion.
of[6], which depicts a network composed of four Bennett loops; the two loops which appear to be in crossed configurations have all acute angles of skew, defined in accordance with the conventions of[3], and as stated above, and the other two have alternately acute and obtuse angles. Figures 3 and 4 herein display two views of a model which may be employed as an aid in visualising the relative dispositions of links and joint axes in a Bennett chain. THE J-HYPERBOLOID The central conicoid known as the hyperboloid of one sheet is usually represented by an equation of the form X 2 y2 Z2 A~+~= 1 +~-5, (1) the Z-axis being the axis of the hyperboloid, and the X- and Y-axes containing the major and minor axes, respectively, of the conicoid's principal elliptical section. A, B and C are, therefore, the parameters which define the quadric surface. There are two sets of
generators which rule the sheet, designated generally as 2-generators and /~-generators[12]. It is known that any three of the four joint axes of a Bennett linkage will define one such set and, for definiteness, we choose the #-generators for our analysis. Clearly, one other quantity must be specified so as to make up the four required to determine a particular configuration of a certain Bennett loop. There are various ways of doing so, and we select here the "eccentric angle" y, (Fig. 5), one for each of the hinges. Of course, only one of them is independent. The angle y is measured in the throat of the hyperboloid, anticlockwise from the positive X-direction. The direction ratios of a #-generator of eccentric angle 7 are[12] (AsT, - BQ,, C),
and the cartesian co-ordinates of its point of intersection with the principal elliptical section are (Ac~', BsT, 0),
whence the position vector from the origin to this point is given by p = A ic7 + BjsT.
We set up the Bennett linkage so that its joint axes coincide with four appropriate /~-generators, and proceed to determine the loop's defining quantities in terms of those of the hyperboloid. Adopting the technique of line vector analysis presented in[13], we can write the unit rotation vector of a joint axis as A is7 ~ Bjc7 + Ck rb = {A2s27 + B 2 c 2 7 + C2}J. 2,
its moment about the origin as v=p×~b 1 : ~ {BCis7 - C A jeT -- A B k } ,
Fig. 2. A skeleton of most of the Bennett loop, showing the essential parameters and variables.
where
149
The Bennett linkage
A
I
t
Fig. 3
I
Fig. 4
Figs 3 and 4. Two aspects of the same configuration of a Bennett linkage, hinge axes extended.
F = {A2s27 + B2c2~, + C2} I/2,
and the normal vector from the origin to the joint axis as
where F12 = {B2C2(c)'2 - c)q )2 + C2A 2(~ 2 - s~ l )2 + A 2B2s2(y2 - yj )} 1/2,
p=t.b x v 1 = ~ {A (B 2 + C2)ic~ + B ( C 2 + A2)js7 - C ( A 2 _ B 2)ks~c7 }.
If, now, joint axes 1 and 2 are characterised respectively by the sets of vectors, dh,v~,p~ and tb2, v2, P2, and corresponding quantities F~ and F2, then the length of the common perpendicular between the joint axes, and therefore the link-length, is given by ll2 = t'bl"v2_+tbl x tb--L'vl tb2 ' The normal vector from the origin to the unit linklength vector ~12 is
Pl2
(tbl" p2)tbl + (tb2' l~l)tb2 l~a, x ~b212
We also note that
El2 0 < S~12= - F~ F2 1 cal2 = F ~
Ill2
{A 2s71s72 + B2c~l c~2 + C 2} {BCi(c72 - c71 ) + CA j(sy2 - s~l )
+ ABks(y2 - ?t)}. Analogous expressions may be written for the quantities relevant to link 23. The eccentric angles may be related implicitly by virtue of the Bennett's parametric relationships, but the resulting equations are fairly cumbersome. We can also determine joint angle 02, say, through the equations sO2dh = 1t12'tt23 c02 = ~12"~23,
S~12fll2 = •1 X £~2' Consequently, we find that
]12 ~
2 A B C [ 1 - c(72 - )'l)]
F,2
but it is not necessary to pursue this enquiry. It is clear that, in principle, a given hyperboloid and eccentric angle determine a certain Bennett loop at a particular stage of its cycle of movement.
150
J. EDDIE BAKER 1
1
In order to avoid ambiguity and inconsistency, we must define axis directions with care. In Fig, 6, a dot is used to indicate a sense approximately out of the plane and a cross for the opposite sense. The consequent joint angle directions are also shown. From[12], we easily write down the cartesian coordinates of the Bennett's four vertices, as follows.
A f 1+'1/1 /
,1+~
1 +,1/1
As: Fig. 5. The two reguli of the Bennett chain's J-hyperboloid, the joint axes of the loop chosen to belong to the set of /a-generators.
THE L-HYPERBOLOID
During the course of investigating the mobility of a linkage network[7], it was found that a "continuum" of Bennett chains could be mounted, coincident with the reguli, on a hyperboioid of revolution. Although the study was confined to the instantaneous mobility of the network at a specific configuration, the result concerning Bennett loops was not at all restrictive. In any case, we shall re-establish here the relationship between the Bennett linkage and its L-hyperboloid in a fairly simple and direct manner, employing more suitable co-ordinates for the present context. We set up the conicoid in the form X2
y2
a ,1
'1-.
1-2~
1-'1/1"~
,1-~
,1-/1
1 - ,1/1"~
+/1'--a~--~,--c-~--~)
(
,1 +/1 - - a 1--,1/1 - c l - - ~ A4: a ! + :./1' 1 + ,1/1' Joining At to A3 and A2 to A4 completes the tetrahedron which served as a point of departure for[9] and provides us with three pairs of opposite sides. The straight lines joining the midpoints of each pair are obviously concurrent on the loop's axis of symmetry, and there is some interest in the x-co-ordinate of the point B of intersection, namely,
a{(,1 +/1)2 + (1 +,1/1) ~} xs =
2(,1 +/1)(1 + ,1/l)
(In fact, the line of symmetry passes through the midpoints of AIA3 and A2A4.) We can proceed in a fashion somewhat similar to that of the previous section to calculate the other quantities of direct significance to us, and we list them here:
2.2
a--~+~-~ = 1 + - -
C2'
whereby the surface is defined by the two parameters, a and c. The Bennett linkage, being one element of a continuum, may be placed (even with fully specified parameters and configuration) at an infinite number of locations around the appropriate hyperboloid. We choose it to lie, for convenience, as depicted in Fig. 6, the x-axis coinciding with its axis of (physical/geometrical) symmetry. To bring the number of determining quantities to the requisite four, we select apposite values of ,1 and /1. If we denote the eccentric angle determined by/1-generator AI A2 by ~,, and that determined by ).-generator A2A3 by ya, then[12] ,1=tan
(:
-
,
/1=tan
(4
-
Z
k 4
.
If we now assign corresponding values ,1' a n d / 1 ' to the other two sides of the Bennett linkage, it is not difficult to deduce from symmetry that
Fig. 6. Location of the Bennett linkage on its Lhypcrboloid and identification of the links with both sets of generators, the inset detailing the conventional measurement of joint angles.
The Bennett linkage
t,2=
(I - 22)(1 4- #2)
/'~
2,
151
we shall designate here as a mid-position, the expressions in[3] may be seen to be equal in value for
t~
(1 -- #2)(1 + 22 )
cot
2
ffi
-
= ce2
t~
Putting
2
G2 = [c2(1 + 22)(1 + #2) + a2(1
- - 2#)211/2
× [C2(1 + 22)(1 4- #2) 4- a2(2 _ #)211/2,
(~23 being chosen to denote the smaller of the angles of skew), whence, from the results of the previous section,
1
42#a 2 = (1 + 25)(1 + #2)c2.
S0~12~ --~2(1 -- 22)(1 4- #2)C %/f~ 4- C2,
Consequently, at the mid-position, we find that 1
H = x a.
cOq2----"-- G2 {22c2(1 + #2) + a2(1 _ 2#)(2 - #)},
s~2~ = ~ 1(
1
-#5)(1 + 2 2 ) e x / ~ + c 2,
1
c~23 = G2 {2#c2(1 + 22) -- a2(1 -- ;[#)(2 -- #)}, col =
a2[(1 - 211)2 - (2 + #)5] + c2(1 + 22)(1 + #2) (1 + 22)(1 + #2)(a2 + c 2)
a2[(1 + 2#) 2 -- (2 - - # ) 2 ] _ c2(1 + 22)(1 + #2) C02= (1 + 22)(1 4- #2)(a2 + C 2) It is immediate from these results that /12S0t23 ~- /23S0~12, thereby, along with the symmetry conditions, establishing the spatial quadrilateral A~A2A3A4 as a Bennett loop. We also note the special case of equal link-lengths when 2# = 1, for which ~2 and ~23 must be supplementary (not equal), and h and ~,~ sum to 2n.
At a general configuration of the linkage, the lengths of AIA3 and A2A 4 are easily calculated, and may be respectively expressed as follows. ltj = ~
l~ = ~
2 2
{a2(2 -- 1,)2 + c2(1 _ 2#)2}1/2,
{a'(l - 2#) 2+ c2(~ - #),pj2.
The knowledge that the hinge axes lie on a regulus which defines the J-hyperboloid allows us to repeat Yu's[9] operation for determining the quadric surface, but with the advantage here of parameters based on the linkage itself, rather than the tetrahedron. In doing so, of course, we shall relate the two hyperboloids directly. We begin by setting up the Plficker co-ordinates for joint axes 1-3, say. "I~hese coordinates are an alternative form of the line vectors (~, v) discussed earlier, and so it is only necessary, in the present context, to use the position co-ordinates of A]-A3 in writing down the requisite vectors. We obtain the following sets of Pliicker co-ordinates for the three joint axes. Joint 1:
T H E SPHERE AND SOME INTER-RELATIONSHIPS
It is demonstrated in[1 1] that the spherical surface which contains the four vertices of the Bennett loop has its centre on the line of symmetry of the linkage, and that this point is also the centre of the Jhyperboloid. Applying the former result, it is a straightforward matter to calculate the x-value of the sphere's centre, and it is given by H = (1 + 22)(1 + #2)(a2 + c 2)
2a(1 +2#)(,t + # ) The sphere's radius is found to be R ----N//-H-~- c 2. Now, special reference is made in[9] to that configuration of the linkage in which the two variable edges (AtA3 and A2A4) of its tetrahedron are equal in length. Expressions for these lengths are given in[3], where the lines are referred to as diagonals of the loop. For this configuration of the linkage, which
-Jr(2 - # ) ~'1(1 +2#) 0
, where
c(1 + 2#)
Jl=
1 - 2#
2+#
(a 2+c2);
c(2 - #) - a ( 1 --2#) Joint 2: ~2(1 - 2 # ) -Y2(2 + # ) 0 -c(,~ + # ) - c ( 1 - 2#)
a(2 - ~ )
,
where
2 - # (a2 + c2); J2-- 1 4-2/*
J. EDDIE BAKER
152
Joint 3: dr3(2 - / t )
Y3(l + 2/t) 0
- c ( 1 + A/t)
,
where
1 - 2/t (a 2 4- ¢2).
J3= 2 + / t
positive, and, for the other, negative. It is readily seen that there are only two essentially different cases, the other two being physical duplicates under variant axis directions. The two basic cases are governed by allowing either sign for tan26, while permitting cos 26 to be positive only. Consequently, the equation of the quadric surface may be written as X2 y2 l + c o s 2 6 Z 2 l-cos26 H 2 + 2 H 2 cos 26 = l + 2 H 2 cos 26 '
c(2 - / t ) - a ( l - A/t)
so that We subsequently find that all but four of the determinantal coefficients in Yu's equation (13) are zero, the only non-zero ones being [ptu],
[qus],
[pqt]
and
[qst].
Curiously, when checking the result by substitution of the co-ordinates of A~-A4 into the consequent equation, we find that the signs of the coefficients for y2 and yz must be reversed. Yu had a similar experience[9]. This amendment must be accepted, whatever the explanation, and so we find the equation of the J-hyperboloid to be a(2 - / t ) ( 1 - 2/t)x 2 + a(2 --/t)(1 - 2/t)y 2 (2 -- g)(1 -- A/t) (a 2 + c2)( 1 + 22)( 1 +/t2) x
(a +/t)(l +2/t) - c ( l + 22)(1 +/t2)yz = O. The relationship may be transformed to canonical type and hence be made directly comparable with equation (1) by means of the substitutions x =X+H
y=Ycos6-Zsin6 z = Y s i n 6 + Z cos&
B=H
C=H
k / 2 cos 26 l-cos26'
We observe that A, C > B, and so can confirm Yu's conclusion that the X-axis (x-axis) contains the major axis of the J-hyperboloid's elliptical central section. We also note that, for the special case mentioned above of equal link-lengths, defined by 2/t = 1, 7[
6=+~, B=C=O and the two pairs of opposite hinge axes intersect on the line of symmetry, the J-hyperboloid degenerating into these four lines. To complete the comparison, it remains only to consider expressions for the eccentric angle in the J-hyperboloid. As given earlier, the direction ratios of the/t-generator defined by eccentric angle 7i may be expressed as (ASYl, - Bcyl, C). From the relevant set of Pliicker co-ordinates presented above, we see that the joint 1 axis has the following direction ratios, referred to the xyz frame. (c[l + A/a], c[2 - / t ] , - a [ l - X/t])
It is immediate that a 2 4- C2 (1 + 22)(1 +/t2) 2H=-a 0- +/a)(1 + 2/t)' thereby confirming Yu's[ll] geometrical result that the centre of the sphere coincides with the centre of the J-hyperboloid. We also see that tan 26 = c (1 4- 2 2 ) ( 1 4- #2) a (2 - - / t ) ( l
-- 2/t)
2cH (1 + 2/t)(2 + / t ) = a 2 + c 2 (1 -- 2/t)(2 --/t)"
After some manipulation, the equation of the central conicoid takes the form X 24
~f. 2 cos 26 l+cos26'
y2 cos 26 + 1 Z ~cos 26 - 1 4 = H 2, 2 cos 26 2 cos 26
referred to the X Y Z frame, then, they become (c[l + 2/t], c[2 - St] cos 6 - a[1 - A/t] sin 6, -
c[2 - / a ] sin 6 - a[l - A/a]cos 6).
Comparing the two sets of ratios, we find that B
c[2 - / t ] c o s 6 - a [ l - 2/t] sin 6 cTi = c[2 - / t ] sin 6 + a[l - 2/1] cos 6'
By substitution for B and C and rearrangement of terms, c[2 - / a ] cos 6 - a[l - 2/t] sin 6 1 cyz = c[2 - / t ] sin 5 + a l l - A/t] cos 5 Itan 61" It is found that joint axis 3 yields the same value for c73, but that sY3 = -s7~. Similarly, c[1 - 2/t] cos 6 - a[2 - / t ] sin 6 1 cy2 = c[l - ),/t] sin 6 + a[2 - / t ] cos 6 [ tan 5 1= cy4
whence it is clear that A=H.
Whether tan 26 is positive or negative, there will be two solutions for 6, for one of which cos 26 will be
and sy4 = - s y 2. For the special case of equal linklengths, it is immediate from physical considerations that y~ and Y2would be equal to 0 and n. Their values
The Bennett linkage
153
are indeterminate from the foregoing expressions, because of B and C becoming zero. Let us now consider the general possibility that eyl "~- ey2 = 0 "
From the foregoing expressions, then, we have (a[1 - 2p] sin 6 - c[2 - #] cos 6) x (c[1 - 2#] sin 6 + a[2 - #] cos 6)
I
= (c[l - 2#] cos 6 - a[2 - #] sin 6)
)o
x (c[). - #] sin 6 + a[1 - 2#] cos 6),
\,,
whence, after simplification and rearrangement of terms, tan 26 = ac([1 - ).#]2 + [). _ #]2) (a 2 - c2)[1 - )~#][2 - #]' After substitution for tan 26 and further manipulation, we obtain [1 + ~2][1 + #2]c2 = 4).#a 2, which identifies the configuration as that which we have termed a mid-position.
I
Fig. 7. Axes 1 and 3 of the Bennett chain in relation to the linkage's line of (physical/geometrical) symmetry.
surface, and so of eccentric angle ~'i. We may check this expectation in the following way. From our results for the J-hyperboloid, we can write the ISA vector, identical in this context with the line vector, for a #-generator as S=(~,v)
THE SIMPLIFIED SCREW SYSTEM Waldron[14] has shown that a pair of screws of equal pitch h, symmetrically disposed (in the complete algebraic sense) about a line and subjected to equal angular speeds will give rise to a resultant "simplified screw" along the line of symmetry, with pitch h,, say, equal to
1
= ~(asy, --Bey, C, BCsy, -CAcy, -AB). The line vector for the x-axis, the line of symmetry, is simply (i, 0) = (tb0, v0),
say. Hence,
h - b tan//, where 2b is the length of the common perpendicular between the given pair of screws and 2//the angle between their (vector) directions. Now, the relative dispositions of, say, axes 1 and 3 of the Bennett loop and its line of symmetry are depicted in Fig. 7, where account has been taken of the necessary consistency in definition pointed out earlier. Whilst the sense of axis 3 with respect to axis 1 is opposite from that which it should take for correct line-symmetry, it is the case, as indicated in['/], that °~1 + ~3 = 0, whereby the essence of line-symmetric screw motion is effected. Because the pitch is zero for the Bennett's joints, we have that h, = - b tan fl
for this pair of screws. The Bennett linkage being mobile, the resultant screw from axes 2 and 4 must have the same pitch as that from axes 1 and 3; that is, the two simplified screws must belong to the same /-system. It follows that the regulus itself is characterised by this same screw pitch which has to, therefore, be independent of the position of any generator on the quadric
s/~ = I ~ o x
051 = ~ {c 2 + B2c2~} ~/2
and 1
c# = ~0" o~ =-:,~s~. /, As well,
tSo'V [ BC BClsYl .'. btfl = ~ , Fcl~ whence
BC Ih, I ffi m A'
which depends only on the definition of the hyperboloid itself. (For the case of equal link-lengths we see that h, = 0, arising from the fact of b being zero.) The sign, or hand, of the screw-pitch is a matter of choice, dependent on whether the angle between a generator and the x-axis is taken as acute or obtuse. By substitution for the parameters A, B and C, we
154
J. EDDIE BAKER
may write - c(1 + 2 # ) "
2H
a 2 + c 2 1(2 - #)(1 -- 4#)
[h,I = Itan 261 = - - - - ~ - , ~ + #)(1 ~ 2 # )
c(2 - # ) ' 1
Now, for the Bennett linkage, the simplified screw thus found can be identified with one of the principal screws of the general 3-system defined by any three of the loop's joint screws. All three principal screws may be determined in the following way. Let us denote the principal screws by the vectors $~, = 0, &-0,
$~ = (j, hal),
$~=~
- - a ( l -- 2g) J,(2-#)
'
Jl(1 + 2#) 0 where K~ = {c2(1 + 22)(1 + #2) + a2(1 _ 2#)2}1/2;
$~ = (k, h z k )
-c(2 + #)
referred to the X Y Z frame. In the x y z frame, then, this set of linearly independent screws is given by, respectively,
-c(1 l
- 2#)
a(,l - #) 9
0
0
0 $ X ~--"
~x
c(2 + u) - c(l - 2#)
$_1
a(2 - #)
0
-J:(1 - 2 u )
c6
-J2(2 + #)
s6
0
$y= where K2 = {c2(1 + 22)(1 + #2) + a2(2 _ #)2}m.
0 hrc6 - Hs6
It is immediate that
hrs6 + He6
K2
K,
2c(1 + 2#) ($' - $3) = $x = 2c(2 + #~--~($4 - $2), where
0 -s6
hx=
1--2# 2--# a2+c 2 2+# 1+2# c
BC
A '
c6 $z =
as above, in magnitude, for h,. We also see, after some algebraic manipulation, that
0 - hzs6 - Hc3
LIKI
2
hzc5 - Hs5
. L2K2._
.
($1 + $ 3 ) * - - - ~ - ( S 2 ± $ 4 ) - - $ ~ ,
in which Directly from the Plficker co-ordinates set down earlier, we next write expressions for the ISA vectors of the Bennett chain's four joints:
L, = - { ( 1 - 3.2)(1 - #2)}-,
x {C--6c(2-u)+S6
c(I + 2#) L2 = - {(1 - 22)0 -/22)}-1
c(2 - # ) 1
$1=~
-a(l
- 2#)
-J1(2-#) J~(1 + 4 # ) 0
' and consequently hr=
H
CA
t6
B
The Bennett linkage Similarly, L3KI
L(K2
-'-Y- ($1 + $3) + --T-" ($' + $4) = $z, in which L3 = {(1 - 2 2 ) ( 1
×{~O-u)
-- ]g2)} -1
-c'~ (l - 2u)} rt
L 4 = {(1 -- 42)(1 --/~2)}-1 x { ~ (1 -- 2#) -- e-~ (~. -- ~)}, and so AB hz = Ht6 = - - . C
We readily deduce that the screw system reciprocal to the general 3-system here touched upon is defined by the three principal screws ( i , - h x i ) , ( j , - h r j ) , (k, - h z k ) , referred to the X Y Z frame. By the foregoing, it is clear that the line series contained in this reciprocal system on its pitch quadric is the regulus complementary to the one which we have been employing in our analysis, that is, the set of 2-generators on the same conicoid. This latter triad of principal screws and the associated regulus on its pitch quadric are in accord with the results stated in Appendix 6C of[10]. In particular, Fig. 6C.07 of that reference depicts some special (4-) generators on the pitch quadric of the (reciprocal) 3-system and its principal screws lying along the axes of the hyperboloid. CLOSING REMARKS Having discussed above the various geometrical properties of the Bennett linkage relevant to the present study and established several interrelationships arising therefrom, we are in a position to deduce many more algebraic connections, among both the quantities already dealt with and others which can be derived. Although some of these relationships might be of interest, there is little justification here for devising a bank of them. It should be possible for an interested worker to use the foregoing as a base for determining any additional relationships or for further development of the topics already explored. In this context, it may be worthwhile for us to summarise the principal thrust of our investigation, and we do so hereunder. Most of the elements entrained in the analysis, such as the linkage itself, the three different quadric surfaces and the screw systems, have been individually
155
examined or partially inter-related in other places. We have here drawn all of the entities together and related for the first time the J - and L-hyperboloids. A second novel feature of our work has been the establishment among the entities of algebraic relationships which involve the linkage parameters and variables themselves, rather than auxiliary quantities. There are, on the other hand, some geometrical properties of the system to which we have paid scant attention here because of their explication elsewhere; for example, although the Bennett linkage defines a certain J-hyperboloid and a certain L-hyperboloid (questions of uniqueness being not at all considered), those surfaces contain, respectively, three other symmetrically disposed Bennett loops[9] and an infinitude of Bennett chains[7]. We have, in passing, been able to confirm or correct prior findings in the topic area and have accorded some prominence to special configurations, for their intrinsic interest. Acknowledgement--I am most grateful to Dr Yu HonCheung of Hong Kong for his valuable correspondence during the preparation of this work, including prior access to his notes for[11]. REFERENCES
1. G. T. Bennett, A new mechanism. Engineering 76, 777 (1903). 2. G. T. Bennett, The skew isogram mechanism. Proc. Lond. Math. Soc. 2s, 13, 151 (1914). 3. J. Eddie Baker, The Bennett, Goldbcrg and Myard linkages--in perspective. Mech. Math. Theory 14, 239 (1979). 4. Yu Hon-Cheung and J. Eddie Baker, On the generation of new linkages from Bennett loops. Mech. Maeh. Theory 16, 473 (1981). 5. J. Eddie Baker, An analysis of Goldberg's anconoidal linkage. Mech. Mach. Theory 18, 371 (1983). 6. J. Eddie Baker and Yu Hon-Cheung, On spatial analogues of Kempe's linkages and some generalisations. Mech. Mach. Theory lg, 457 (1983). 7. J. Eddie Baker and Hu Min, On spatial networks of overconstrained linkages. Mech. Mach. Theory. 21, 427 (1986). 8. S. Ogino, On the Bennett mechanism. Bull. J.S.M.E. 11, 509 (1968). 9. Yu Hon-Cheung, The Bennett linkage, its associated tetrahedron and the hyperboloid of its axes. Mech. Math. Theory 16, 105 (1981). I0. J. R. Phillips, Freedom in Machinery. Cambridge University Press (1984, 1988). 1 I. Yu Hon-Cheung, Geometry of the Bennett linkage via its circumscribed sphere. Proc. 7th Worm Congr. on the Theory of Mach. and Mech., Sevilla, Spain, 1%22 Sep. 1987, Vol. I, p. 227. Pergamon Press, Oxford (1987). 12. R. J. T. Bell, An Elementary Treatise on Coordinate Geometry of Three Dimensions, 3rd edn. Macmillan, London (1950). 13. E. A. Milne, Vectorial Mechanics. Methuen, London 0948). 14. K. J. Waldron, Symmetric overconstrained linkages. J. Engng Ind. A.S.M.E. Trans. B 91, 158 (1969).
L E M l ~ C A N I S M E D E B E N N E T T E T SES S U R F A C E S QUARTIQUES ASSOCIt~ES R6mm6---Le m6c,anisme remarquable de Bennett (Fig. l) fut d6couvert[l, 2] il y a plus de quatre-vingt ans et on continue 6 faire des nouvelles contributions sur ce m6canisme. En[3] on traite d'une mani6re
156
J. EDDIE BAKER
compl6te des 6quations de cl6ture et en[3-7] on 6tudie sa propri6t6 particuli6re de fournir la base pour des m6canismes de plusieurs boucles et pour des treillis cin6matiques. Dans cet article on consid6re quelques aspects fondamentaux de la g6om6trie du m6canisme et quelques rapports entre les diverses surfaces quartiques qu'il fait naltre. On examine le J-hyperboloide (Fig. 5) qui d6signe la surface d6finie par les axes des articulations du m6canisme. I1 est d6crit par trois paramdtres et une g6n6ratrice est d6termin6e par son angle gauche. On 6tudie ensuite le L-hyperboloide (Fig. 6) qui est d6fini par les membres du m~anisme. C'est une surface de r6volution d6crite par deux param6tres. I1 y a aussi une sph6re, d6termin6e par les sommets du m6canisme, dont on analyse les propri6t6s en comparaison avec le travail de Yu[l I]. On continue en rattachant les trois surfaces quartiques et en consid6rant deux cas sp6ciaux int6ressants. Enfin on cite la technique de Waldron[14] pour obtenir le pas de la vis dirig6e le long de la ligne de sym6trie de la chaine. Cette vis est la r6sultante du mouvement combin6 des deux paires de rotoides oppos6s, donc une caract6ristique de la boucle enti6re d'une configuration particulidre.
0094-114X/88 $3.00+0.00 Copyright © 1988PergamonPress pie
THE B E N N E T T L I N K A G E A N D ITS A S S O C I A T E D QUADRIC SURFACES J. E D D I E B A K E R Department of Applied Mechanics, The University of New South Wales, P.O. Box l, Kensington, NSW 2033, Australia (Received in revised form 30 September 1986)
Abstract--The 4-revolute Bennett linkage has been intensively studied by several researchers over many years, and yet there remain depths to be plumbed. In this study, we begin by considering separately the manner in which the Bennett linkage is related to the hyperboioid defined by its joint axes and the one defined by its links. We go on to algebraically inter-relate these and other aspects of the loop, including an associated sphere and an equivalent screw of the overall instantaneous motion.
INTRODUCTION It would be unwise to speculate about what might constitute the last word on Bennett's "skew isogram mechanism"[l, 2]. Since its discovery more than 80 years ago, it has puzzled, fascinated and challenged kinematicians of both geometrical and algebraic persuasions, and there continue to be fresh contributions made to the abundant literature concerning this remarkable 4-bar (Fig. 1). Its closure equations are accorded thorough consideration in[3], where also, along with[4], there is discussed the peculiar propensity of the linkage to function as a building-block for synthesising 5- and 6-bar loops. More recently, but founded on work of 40 years past, it has been shown[5-7] that the Bennett chain has a role to play in developing mobile mechanical networks. Most of the other research on the Bennett linkage is referred to in the papers cited here, except for the spasmodic suggestions concerning practical applications of the skew 4-bar. One of the lesser known of these latter works is[8], in which the relationship between input and output angles of the Bennett loop is shown to coincide with that for a meshing pair of elliptic gears of suitably selected eccentricity. The subject of this paper represents a return to fundamentals, whereby we consider the detailed definitions of the hyperboloids and sphere which are associated with any particular Bennett linkage at a specified configuration. It has been known for at least 70 years that the four hinge axes of the linkage can be taken as generators of the same regulus on a hyperboloid (of one sheet) at a given configuration of the loop, and this knowledge has been used to advantage by various authors. It was only recently[9], however, that algebraic relationships between the hyperholoid and the linkage were established, through employing the device of the tetrahedron which is delineated naturally by the 4-bar. In that paper, and in[10], there is also highlighted the somewhat surprising positioning of the linkage on the
hyperboloid of its axes, a fact apparently first noted by a student of Professor Phillips. For later convenience, we shall henceforth refer to this quadric surface as the J-hyperboloid. It is demonstrated in[7] that the links of a Bennett loop define another hypcrboloid of one sheet, a surface of revolution, by virtue of their identification with two members of each of the reguli forming the conicoid. We shall call this surface the L-hyperboloid. As well, Yu[ll] has investigated the significance of the sphere determined by the four vertices of the linkage. Our purpose in this work is to directly relate three independent parameters and a suitable single joint variable of the linkage with the relevant quantifies of the sphere and the two forms of hyperboloid associated with it. We also intend, by means of screw system algebra, to obtain a simple expression f o r the pitch of the equivalent screw along the line of (physical/geometrical) symmetry of the mechanism. Much of the notation employed herein is in common usage and need not be re-defined. Any new or unusual notation will be explained as it is introduced. Most results will be presented without detailed derivation, for the sake of brevity, but all major steps will be indicated. With regard to the skew isogram itself, numbering the revohites 1 to 4, we denote the two different link-lengths by l~2 and 123 and the corresponding angles of skew by ~2 and c(23, only three of these quantities being independent, because
l~
= l~.
These parameters and the joint variables are illustrated in Fig. 2. The notation is made consistent and unambiguous by using the right-hand screw rule and restricting all angles of skew to lie in the range (0, ~). The reader is referred to[3] for an extended discussion on these matters. In particular, the commonly held belief that angles of skew should be acute and alternately right- and left-handed is false. In this context, the reader's attention is drawn to Fig. 17 147
148
J. EDDIE BAKER
Fig. 1. A display model of the Bennett mechanism, designed to permit full-cycle motion.
of[6], which depicts a network composed of four Bennett loops; the two loops which appear to be in crossed configurations have all acute angles of skew, defined in accordance with the conventions of[3], and as stated above, and the other two have alternately acute and obtuse angles. Figures 3 and 4 herein display two views of a model which may be employed as an aid in visualising the relative dispositions of links and joint axes in a Bennett chain. THE J-HYPERBOLOID The central conicoid known as the hyperboloid of one sheet is usually represented by an equation of the form X 2 y2 Z2 A~+~= 1 +~-5, (1) the Z-axis being the axis of the hyperboloid, and the X- and Y-axes containing the major and minor axes, respectively, of the conicoid's principal elliptical section. A, B and C are, therefore, the parameters which define the quadric surface. There are two sets of
generators which rule the sheet, designated generally as 2-generators and /~-generators[12]. It is known that any three of the four joint axes of a Bennett linkage will define one such set and, for definiteness, we choose the #-generators for our analysis. Clearly, one other quantity must be specified so as to make up the four required to determine a particular configuration of a certain Bennett loop. There are various ways of doing so, and we select here the "eccentric angle" y, (Fig. 5), one for each of the hinges. Of course, only one of them is independent. The angle y is measured in the throat of the hyperboloid, anticlockwise from the positive X-direction. The direction ratios of a #-generator of eccentric angle 7 are[12] (AsT, - BQ,, C),
and the cartesian co-ordinates of its point of intersection with the principal elliptical section are (Ac~', BsT, 0),
whence the position vector from the origin to this point is given by p = A ic7 + BjsT.
We set up the Bennett linkage so that its joint axes coincide with four appropriate /~-generators, and proceed to determine the loop's defining quantities in terms of those of the hyperboloid. Adopting the technique of line vector analysis presented in[13], we can write the unit rotation vector of a joint axis as A is7 ~ Bjc7 + Ck rb = {A2s27 + B 2 c 2 7 + C2}J. 2,
its moment about the origin as v=p×~b 1 : ~ {BCis7 - C A jeT -- A B k } ,
Fig. 2. A skeleton of most of the Bennett loop, showing the essential parameters and variables.
where
149
The Bennett linkage
A
I
t
Fig. 3
I
Fig. 4
Figs 3 and 4. Two aspects of the same configuration of a Bennett linkage, hinge axes extended.
F = {A2s27 + B2c2~, + C2} I/2,
and the normal vector from the origin to the joint axis as
where F12 = {B2C2(c)'2 - c)q )2 + C2A 2(~ 2 - s~ l )2 + A 2B2s2(y2 - yj )} 1/2,
p=t.b x v 1 = ~ {A (B 2 + C2)ic~ + B ( C 2 + A2)js7 - C ( A 2 _ B 2)ks~c7 }.
If, now, joint axes 1 and 2 are characterised respectively by the sets of vectors, dh,v~,p~ and tb2, v2, P2, and corresponding quantities F~ and F2, then the length of the common perpendicular between the joint axes, and therefore the link-length, is given by ll2 = t'bl"v2_+tbl x tb--L'vl tb2 ' The normal vector from the origin to the unit linklength vector ~12 is
Pl2
(tbl" p2)tbl + (tb2' l~l)tb2 l~a, x ~b212
We also note that
El2 0 < S~12= - F~ F2 1 cal2 = F ~
Ill2
{A 2s71s72 + B2c~l c~2 + C 2} {BCi(c72 - c71 ) + CA j(sy2 - s~l )
+ ABks(y2 - ?t)}. Analogous expressions may be written for the quantities relevant to link 23. The eccentric angles may be related implicitly by virtue of the Bennett's parametric relationships, but the resulting equations are fairly cumbersome. We can also determine joint angle 02, say, through the equations sO2dh = 1t12'tt23 c02 = ~12"~23,
S~12fll2 = •1 X £~2' Consequently, we find that
]12 ~
2 A B C [ 1 - c(72 - )'l)]
F,2
but it is not necessary to pursue this enquiry. It is clear that, in principle, a given hyperboloid and eccentric angle determine a certain Bennett loop at a particular stage of its cycle of movement.
150
J. EDDIE BAKER 1
1
In order to avoid ambiguity and inconsistency, we must define axis directions with care. In Fig, 6, a dot is used to indicate a sense approximately out of the plane and a cross for the opposite sense. The consequent joint angle directions are also shown. From[12], we easily write down the cartesian coordinates of the Bennett's four vertices, as follows.
A f 1+'1/1 /
,1+~
1 +,1/1
As: Fig. 5. The two reguli of the Bennett chain's J-hyperboloid, the joint axes of the loop chosen to belong to the set of /a-generators.
THE L-HYPERBOLOID
During the course of investigating the mobility of a linkage network[7], it was found that a "continuum" of Bennett chains could be mounted, coincident with the reguli, on a hyperboioid of revolution. Although the study was confined to the instantaneous mobility of the network at a specific configuration, the result concerning Bennett loops was not at all restrictive. In any case, we shall re-establish here the relationship between the Bennett linkage and its L-hyperboloid in a fairly simple and direct manner, employing more suitable co-ordinates for the present context. We set up the conicoid in the form X2
y2
a ,1
'1-.
1-2~
1-'1/1"~
,1-~
,1-/1
1 - ,1/1"~
+/1'--a~--~,--c-~--~)
(
,1 +/1 - - a 1--,1/1 - c l - - ~ A4: a ! + :./1' 1 + ,1/1' Joining At to A3 and A2 to A4 completes the tetrahedron which served as a point of departure for[9] and provides us with three pairs of opposite sides. The straight lines joining the midpoints of each pair are obviously concurrent on the loop's axis of symmetry, and there is some interest in the x-co-ordinate of the point B of intersection, namely,
a{(,1 +/1)2 + (1 +,1/1) ~} xs =
2(,1 +/1)(1 + ,1/l)
(In fact, the line of symmetry passes through the midpoints of AIA3 and A2A4.) We can proceed in a fashion somewhat similar to that of the previous section to calculate the other quantities of direct significance to us, and we list them here:
2.2
a--~+~-~ = 1 + - -
C2'
whereby the surface is defined by the two parameters, a and c. The Bennett linkage, being one element of a continuum, may be placed (even with fully specified parameters and configuration) at an infinite number of locations around the appropriate hyperboloid. We choose it to lie, for convenience, as depicted in Fig. 6, the x-axis coinciding with its axis of (physical/geometrical) symmetry. To bring the number of determining quantities to the requisite four, we select apposite values of ,1 and /1. If we denote the eccentric angle determined by/1-generator AI A2 by ~,, and that determined by ).-generator A2A3 by ya, then[12] ,1=tan
(:
-
,
/1=tan
(4
-
Z
k 4
.
If we now assign corresponding values ,1' a n d / 1 ' to the other two sides of the Bennett linkage, it is not difficult to deduce from symmetry that
Fig. 6. Location of the Bennett linkage on its Lhypcrboloid and identification of the links with both sets of generators, the inset detailing the conventional measurement of joint angles.
The Bennett linkage
t,2=
(I - 22)(1 4- #2)
/'~
2,
151
we shall designate here as a mid-position, the expressions in[3] may be seen to be equal in value for
t~
(1 -- #2)(1 + 22 )
cot
2
ffi
-
= ce2
t~
Putting
2
G2 = [c2(1 + 22)(1 + #2) + a2(1
- - 2#)211/2
× [C2(1 + 22)(1 4- #2) 4- a2(2 _ #)211/2,
(~23 being chosen to denote the smaller of the angles of skew), whence, from the results of the previous section,
1
42#a 2 = (1 + 25)(1 + #2)c2.
S0~12~ --~2(1 -- 22)(1 4- #2)C %/f~ 4- C2,
Consequently, at the mid-position, we find that 1
H = x a.
cOq2----"-- G2 {22c2(1 + #2) + a2(1 _ 2#)(2 - #)},
s~2~ = ~ 1(
1
-#5)(1 + 2 2 ) e x / ~ + c 2,
1
c~23 = G2 {2#c2(1 + 22) -- a2(1 -- ;[#)(2 -- #)}, col =
a2[(1 - 211)2 - (2 + #)5] + c2(1 + 22)(1 + #2) (1 + 22)(1 + #2)(a2 + c 2)
a2[(1 + 2#) 2 -- (2 - - # ) 2 ] _ c2(1 + 22)(1 + #2) C02= (1 + 22)(1 4- #2)(a2 + C 2) It is immediate from these results that /12S0t23 ~- /23S0~12, thereby, along with the symmetry conditions, establishing the spatial quadrilateral A~A2A3A4 as a Bennett loop. We also note the special case of equal link-lengths when 2# = 1, for which ~2 and ~23 must be supplementary (not equal), and h and ~,~ sum to 2n.
At a general configuration of the linkage, the lengths of AIA3 and A2A 4 are easily calculated, and may be respectively expressed as follows. ltj = ~
l~ = ~
2 2
{a2(2 -- 1,)2 + c2(1 _ 2#)2}1/2,
{a'(l - 2#) 2+ c2(~ - #),pj2.
The knowledge that the hinge axes lie on a regulus which defines the J-hyperboloid allows us to repeat Yu's[9] operation for determining the quadric surface, but with the advantage here of parameters based on the linkage itself, rather than the tetrahedron. In doing so, of course, we shall relate the two hyperboloids directly. We begin by setting up the Plficker co-ordinates for joint axes 1-3, say. "I~hese coordinates are an alternative form of the line vectors (~, v) discussed earlier, and so it is only necessary, in the present context, to use the position co-ordinates of A]-A3 in writing down the requisite vectors. We obtain the following sets of Pliicker co-ordinates for the three joint axes. Joint 1:
T H E SPHERE AND SOME INTER-RELATIONSHIPS
It is demonstrated in[1 1] that the spherical surface which contains the four vertices of the Bennett loop has its centre on the line of symmetry of the linkage, and that this point is also the centre of the Jhyperboloid. Applying the former result, it is a straightforward matter to calculate the x-value of the sphere's centre, and it is given by H = (1 + 22)(1 + #2)(a2 + c 2)
2a(1 +2#)(,t + # ) The sphere's radius is found to be R ----N//-H-~- c 2. Now, special reference is made in[9] to that configuration of the linkage in which the two variable edges (AtA3 and A2A4) of its tetrahedron are equal in length. Expressions for these lengths are given in[3], where the lines are referred to as diagonals of the loop. For this configuration of the linkage, which
-Jr(2 - # ) ~'1(1 +2#) 0
, where
c(1 + 2#)
Jl=
1 - 2#
2+#
(a 2+c2);
c(2 - #) - a ( 1 --2#) Joint 2: ~2(1 - 2 # ) -Y2(2 + # ) 0 -c(,~ + # ) - c ( 1 - 2#)
a(2 - ~ )
,
where
2 - # (a2 + c2); J2-- 1 4-2/*
J. EDDIE BAKER
152
Joint 3: dr3(2 - / t )
Y3(l + 2/t) 0
- c ( 1 + A/t)
,
where
1 - 2/t (a 2 4- ¢2).
J3= 2 + / t
positive, and, for the other, negative. It is readily seen that there are only two essentially different cases, the other two being physical duplicates under variant axis directions. The two basic cases are governed by allowing either sign for tan26, while permitting cos 26 to be positive only. Consequently, the equation of the quadric surface may be written as X2 y2 l + c o s 2 6 Z 2 l-cos26 H 2 + 2 H 2 cos 26 = l + 2 H 2 cos 26 '
c(2 - / t ) - a ( l - A/t)
so that We subsequently find that all but four of the determinantal coefficients in Yu's equation (13) are zero, the only non-zero ones being [ptu],
[qus],
[pqt]
and
[qst].
Curiously, when checking the result by substitution of the co-ordinates of A~-A4 into the consequent equation, we find that the signs of the coefficients for y2 and yz must be reversed. Yu had a similar experience[9]. This amendment must be accepted, whatever the explanation, and so we find the equation of the J-hyperboloid to be a(2 - / t ) ( 1 - 2/t)x 2 + a(2 --/t)(1 - 2/t)y 2 (2 -- g)(1 -- A/t) (a 2 + c2)( 1 + 22)( 1 +/t2) x
(a +/t)(l +2/t) - c ( l + 22)(1 +/t2)yz = O. The relationship may be transformed to canonical type and hence be made directly comparable with equation (1) by means of the substitutions x =X+H
y=Ycos6-Zsin6 z = Y s i n 6 + Z cos&
B=H
C=H
k / 2 cos 26 l-cos26'
We observe that A, C > B, and so can confirm Yu's conclusion that the X-axis (x-axis) contains the major axis of the J-hyperboloid's elliptical central section. We also note that, for the special case mentioned above of equal link-lengths, defined by 2/t = 1, 7[
6=+~, B=C=O and the two pairs of opposite hinge axes intersect on the line of symmetry, the J-hyperboloid degenerating into these four lines. To complete the comparison, it remains only to consider expressions for the eccentric angle in the J-hyperboloid. As given earlier, the direction ratios of the/t-generator defined by eccentric angle 7i may be expressed as (ASYl, - Bcyl, C). From the relevant set of Pliicker co-ordinates presented above, we see that the joint 1 axis has the following direction ratios, referred to the xyz frame. (c[l + A/a], c[2 - / t ] , - a [ l - X/t])
It is immediate that a 2 4- C2 (1 + 22)(1 +/t2) 2H=-a 0- +/a)(1 + 2/t)' thereby confirming Yu's[ll] geometrical result that the centre of the sphere coincides with the centre of the J-hyperboloid. We also see that tan 26 = c (1 4- 2 2 ) ( 1 4- #2) a (2 - - / t ) ( l
-- 2/t)
2cH (1 + 2/t)(2 + / t ) = a 2 + c 2 (1 -- 2/t)(2 --/t)"
After some manipulation, the equation of the central conicoid takes the form X 24
~f. 2 cos 26 l+cos26'
y2 cos 26 + 1 Z ~cos 26 - 1 4 = H 2, 2 cos 26 2 cos 26
referred to the X Y Z frame, then, they become (c[l + 2/t], c[2 - St] cos 6 - a[1 - A/t] sin 6, -
c[2 - / a ] sin 6 - a[l - A/a]cos 6).
Comparing the two sets of ratios, we find that B
c[2 - / t ] c o s 6 - a [ l - 2/t] sin 6 cTi = c[2 - / t ] sin 6 + a[l - 2/1] cos 6'
By substitution for B and C and rearrangement of terms, c[2 - / a ] cos 6 - a[l - 2/t] sin 6 1 cyz = c[2 - / t ] sin 5 + a l l - A/t] cos 5 Itan 61" It is found that joint axis 3 yields the same value for c73, but that sY3 = -s7~. Similarly, c[1 - 2/t] cos 6 - a[2 - / t ] sin 6 1 cy2 = c[l - ),/t] sin 6 + a[2 - / t ] cos 6 [ tan 5 1= cy4
whence it is clear that A=H.
Whether tan 26 is positive or negative, there will be two solutions for 6, for one of which cos 26 will be
and sy4 = - s y 2. For the special case of equal linklengths, it is immediate from physical considerations that y~ and Y2would be equal to 0 and n. Their values
The Bennett linkage
153
are indeterminate from the foregoing expressions, because of B and C becoming zero. Let us now consider the general possibility that eyl "~- ey2 = 0 "
From the foregoing expressions, then, we have (a[1 - 2p] sin 6 - c[2 - #] cos 6) x (c[1 - 2#] sin 6 + a[2 - #] cos 6)
I
= (c[l - 2#] cos 6 - a[2 - #] sin 6)
)o
x (c[). - #] sin 6 + a[1 - 2#] cos 6),
\,,
whence, after simplification and rearrangement of terms, tan 26 = ac([1 - ).#]2 + [). _ #]2) (a 2 - c2)[1 - )~#][2 - #]' After substitution for tan 26 and further manipulation, we obtain [1 + ~2][1 + #2]c2 = 4).#a 2, which identifies the configuration as that which we have termed a mid-position.
I
Fig. 7. Axes 1 and 3 of the Bennett chain in relation to the linkage's line of (physical/geometrical) symmetry.
surface, and so of eccentric angle ~'i. We may check this expectation in the following way. From our results for the J-hyperboloid, we can write the ISA vector, identical in this context with the line vector, for a #-generator as S=(~,v)
THE SIMPLIFIED SCREW SYSTEM Waldron[14] has shown that a pair of screws of equal pitch h, symmetrically disposed (in the complete algebraic sense) about a line and subjected to equal angular speeds will give rise to a resultant "simplified screw" along the line of symmetry, with pitch h,, say, equal to
1
= ~(asy, --Bey, C, BCsy, -CAcy, -AB). The line vector for the x-axis, the line of symmetry, is simply (i, 0) = (tb0, v0),
say. Hence,
h - b tan//, where 2b is the length of the common perpendicular between the given pair of screws and 2//the angle between their (vector) directions. Now, the relative dispositions of, say, axes 1 and 3 of the Bennett loop and its line of symmetry are depicted in Fig. 7, where account has been taken of the necessary consistency in definition pointed out earlier. Whilst the sense of axis 3 with respect to axis 1 is opposite from that which it should take for correct line-symmetry, it is the case, as indicated in['/], that °~1 + ~3 = 0, whereby the essence of line-symmetric screw motion is effected. Because the pitch is zero for the Bennett's joints, we have that h, = - b tan fl
for this pair of screws. The Bennett linkage being mobile, the resultant screw from axes 2 and 4 must have the same pitch as that from axes 1 and 3; that is, the two simplified screws must belong to the same /-system. It follows that the regulus itself is characterised by this same screw pitch which has to, therefore, be independent of the position of any generator on the quadric
s/~ = I ~ o x
051 = ~ {c 2 + B2c2~} ~/2
and 1
c# = ~0" o~ =-:,~s~. /, As well,
tSo'V [ BC BClsYl .'. btfl = ~ , Fcl~ whence
BC Ih, I ffi m A'
which depends only on the definition of the hyperboloid itself. (For the case of equal link-lengths we see that h, = 0, arising from the fact of b being zero.) The sign, or hand, of the screw-pitch is a matter of choice, dependent on whether the angle between a generator and the x-axis is taken as acute or obtuse. By substitution for the parameters A, B and C, we
154
J. EDDIE BAKER
may write - c(1 + 2 # ) "
2H
a 2 + c 2 1(2 - #)(1 -- 4#)
[h,I = Itan 261 = - - - - ~ - , ~ + #)(1 ~ 2 # )
c(2 - # ) ' 1
Now, for the Bennett linkage, the simplified screw thus found can be identified with one of the principal screws of the general 3-system defined by any three of the loop's joint screws. All three principal screws may be determined in the following way. Let us denote the principal screws by the vectors $~, = 0, &-0,
$~ = (j, hal),
$~=~
- - a ( l -- 2g) J,(2-#)
'
Jl(1 + 2#) 0 where K~ = {c2(1 + 22)(1 + #2) + a2(1 _ 2#)2}1/2;
$~ = (k, h z k )
-c(2 + #)
referred to the X Y Z frame. In the x y z frame, then, this set of linearly independent screws is given by, respectively,
-c(1 l
- 2#)
a(,l - #) 9
0
0
0 $ X ~--"
~x
c(2 + u) - c(l - 2#)
$_1
a(2 - #)
0
-J:(1 - 2 u )
c6
-J2(2 + #)
s6
0
$y= where K2 = {c2(1 + 22)(1 + #2) + a2(2 _ #)2}m.
0 hrc6 - Hs6
It is immediate that
hrs6 + He6
K2
K,
2c(1 + 2#) ($' - $3) = $x = 2c(2 + #~--~($4 - $2), where
0 -s6
hx=
1--2# 2--# a2+c 2 2+# 1+2# c
BC
A '
c6 $z =
as above, in magnitude, for h,. We also see, after some algebraic manipulation, that
0 - hzs6 - Hc3
LIKI
2
hzc5 - Hs5
. L2K2._
.
($1 + $ 3 ) * - - - ~ - ( S 2 ± $ 4 ) - - $ ~ ,
in which Directly from the Plficker co-ordinates set down earlier, we next write expressions for the ISA vectors of the Bennett chain's four joints:
L, = - { ( 1 - 3.2)(1 - #2)}-,
x {C--6c(2-u)+S6
c(I + 2#) L2 = - {(1 - 22)0 -/22)}-1
c(2 - # ) 1
$1=~
-a(l
- 2#)
-J1(2-#) J~(1 + 4 # ) 0
' and consequently hr=
H
CA
t6
B
The Bennett linkage Similarly, L3KI
L(K2
-'-Y- ($1 + $3) + --T-" ($' + $4) = $z, in which L3 = {(1 - 2 2 ) ( 1
×{~O-u)
-- ]g2)} -1
-c'~ (l - 2u)} rt
L 4 = {(1 -- 42)(1 --/~2)}-1 x { ~ (1 -- 2#) -- e-~ (~. -- ~)}, and so AB hz = Ht6 = - - . C
We readily deduce that the screw system reciprocal to the general 3-system here touched upon is defined by the three principal screws ( i , - h x i ) , ( j , - h r j ) , (k, - h z k ) , referred to the X Y Z frame. By the foregoing, it is clear that the line series contained in this reciprocal system on its pitch quadric is the regulus complementary to the one which we have been employing in our analysis, that is, the set of 2-generators on the same conicoid. This latter triad of principal screws and the associated regulus on its pitch quadric are in accord with the results stated in Appendix 6C of[10]. In particular, Fig. 6C.07 of that reference depicts some special (4-) generators on the pitch quadric of the (reciprocal) 3-system and its principal screws lying along the axes of the hyperboloid. CLOSING REMARKS Having discussed above the various geometrical properties of the Bennett linkage relevant to the present study and established several interrelationships arising therefrom, we are in a position to deduce many more algebraic connections, among both the quantities already dealt with and others which can be derived. Although some of these relationships might be of interest, there is little justification here for devising a bank of them. It should be possible for an interested worker to use the foregoing as a base for determining any additional relationships or for further development of the topics already explored. In this context, it may be worthwhile for us to summarise the principal thrust of our investigation, and we do so hereunder. Most of the elements entrained in the analysis, such as the linkage itself, the three different quadric surfaces and the screw systems, have been individually
155
examined or partially inter-related in other places. We have here drawn all of the entities together and related for the first time the J - and L-hyperboloids. A second novel feature of our work has been the establishment among the entities of algebraic relationships which involve the linkage parameters and variables themselves, rather than auxiliary quantities. There are, on the other hand, some geometrical properties of the system to which we have paid scant attention here because of their explication elsewhere; for example, although the Bennett linkage defines a certain J-hyperboloid and a certain L-hyperboloid (questions of uniqueness being not at all considered), those surfaces contain, respectively, three other symmetrically disposed Bennett loops[9] and an infinitude of Bennett chains[7]. We have, in passing, been able to confirm or correct prior findings in the topic area and have accorded some prominence to special configurations, for their intrinsic interest. Acknowledgement--I am most grateful to Dr Yu HonCheung of Hong Kong for his valuable correspondence during the preparation of this work, including prior access to his notes for[11]. REFERENCES
1. G. T. Bennett, A new mechanism. Engineering 76, 777 (1903). 2. G. T. Bennett, The skew isogram mechanism. Proc. Lond. Math. Soc. 2s, 13, 151 (1914). 3. J. Eddie Baker, The Bennett, Goldbcrg and Myard linkages--in perspective. Mech. Math. Theory 14, 239 (1979). 4. Yu Hon-Cheung and J. Eddie Baker, On the generation of new linkages from Bennett loops. Mech. Maeh. Theory 16, 473 (1981). 5. J. Eddie Baker, An analysis of Goldberg's anconoidal linkage. Mech. Mach. Theory 18, 371 (1983). 6. J. Eddie Baker and Yu Hon-Cheung, On spatial analogues of Kempe's linkages and some generalisations. Mech. Mach. Theory lg, 457 (1983). 7. J. Eddie Baker and Hu Min, On spatial networks of overconstrained linkages. Mech. Mach. Theory. 21, 427 (1986). 8. S. Ogino, On the Bennett mechanism. Bull. J.S.M.E. 11, 509 (1968). 9. Yu Hon-Cheung, The Bennett linkage, its associated tetrahedron and the hyperboloid of its axes. Mech. Math. Theory 16, 105 (1981). I0. J. R. Phillips, Freedom in Machinery. Cambridge University Press (1984, 1988). 1 I. Yu Hon-Cheung, Geometry of the Bennett linkage via its circumscribed sphere. Proc. 7th Worm Congr. on the Theory of Mach. and Mech., Sevilla, Spain, 1%22 Sep. 1987, Vol. I, p. 227. Pergamon Press, Oxford (1987). 12. R. J. T. Bell, An Elementary Treatise on Coordinate Geometry of Three Dimensions, 3rd edn. Macmillan, London (1950). 13. E. A. Milne, Vectorial Mechanics. Methuen, London 0948). 14. K. J. Waldron, Symmetric overconstrained linkages. J. Engng Ind. A.S.M.E. Trans. B 91, 158 (1969).
L E M l ~ C A N I S M E D E B E N N E T T E T SES S U R F A C E S QUARTIQUES ASSOCIt~ES R6mm6---Le m6c,anisme remarquable de Bennett (Fig. l) fut d6couvert[l, 2] il y a plus de quatre-vingt ans et on continue 6 faire des nouvelles contributions sur ce m6canisme. En[3] on traite d'une mani6re
156
J. EDDIE BAKER
compl6te des 6quations de cl6ture et en[3-7] on 6tudie sa propri6t6 particuli6re de fournir la base pour des m6canismes de plusieurs boucles et pour des treillis cin6matiques. Dans cet article on consid6re quelques aspects fondamentaux de la g6om6trie du m6canisme et quelques rapports entre les diverses surfaces quartiques qu'il fait naltre. On examine le J-hyperboloide (Fig. 5) qui d6signe la surface d6finie par les axes des articulations du m6canisme. I1 est d6crit par trois paramdtres et une g6n6ratrice est d6termin6e par son angle gauche. On 6tudie ensuite le L-hyperboloide (Fig. 6) qui est d6fini par les membres du m~anisme. C'est une surface de r6volution d6crite par deux param6tres. I1 y a aussi une sph6re, d6termin6e par les sommets du m6canisme, dont on analyse les propri6t6s en comparaison avec le travail de Yu[l I]. On continue en rattachant les trois surfaces quartiques et en consid6rant deux cas sp6ciaux int6ressants. Enfin on cite la technique de Waldron[14] pour obtenir le pas de la vis dirig6e le long de la ligne de sym6trie de la chaine. Cette vis est la r6sultante du mouvement combin6 des deux paires de rotoides oppos6s, donc une caract6ristique de la boucle enti6re d'une configuration particulidre.