Mechanism and Machine Theory 35 (2000) 1535±1549
www.elsevier.com/locate/mechmt
Applications of Watt II function generator cognates
P.A. Simionescu a,*, M.R. Smith b
a
Department of Mechanism and Robot Theory, Politehnica, University of Bucharest, Spl. Independentei 313, 77206
Bucharest, Romania
b
Department of Mechanical, Materials and Manufacturing Engineering, University of Newcastle, Newcastle upon Tyne,
NE1 7RU, UK
Received 25 September 1997; received in revised form 10 March 2000
Abstract
7R and 3RT3R Watt II function-generating mechanisms are examined and the existence of a double
in®nity of cognates are highlighted. For both the cases reference dimensional con®guration mechanisms
are de®ned which have a minimum number of parameters, these being considered best suited for
performing a function generation synthesis. Overconstrained mechanisms are also shown to result from
the merging of the input and output elements of two or more such function cognates. The practical
cases of the rack-and-pinion and central-lever steering linkages are examined as planar mechanisms, and
a maximum number of only three and four independent design parameters, respectively, are proven to
exist. 7 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Planar geometrical transformations; Reference con®gurations; Function cognates; Overconstrained
mechanisms
1. Introduction
The occurrence of cognates in planar mechanism kinematics and their association in
overconstrained mechanisms have been studied by many authors, more recently by Dijksman
[1±3]. However, function generation cognates have been signi®cantly less thoroughly
investigated, most of the published literature focusing on coupler-curve and coupler cognate
* Corresponding author. Department of Mechanical Engineering, Auburn University, 201 Ross Hall, Auburn, AL
36849, USA. Tel.: +1-334-844-5894; fax: +1-334-844-5900.
E-mail address: pasimi@eng.auburn.edu (P.A. Simionescu).
0094-114X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 9 4 - 1 1 4 X ( 0 0 ) 0 0 0 1 1 - 2
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mechanisms. In many cases, Robert's theorem of three-fold generation of four-bar coupler
curves has been the starting point for studying more complex mechanism cognates and the
resulting overconstrained mechanisms. The six-bar Watt and Stephenson mechanisms with
turning joints only (7R) were analysed by Dijksman for curve, coupler and function generation
cognates. Extending the conclusions reached for the Watt I mechanism, by applying a proper
inversion, the Watt II linkage was found to have a double in®nity of function cognates [2]. In
spite of the obvious consequences, no overconstrained mechanism obtained by associating two
or more function cognates of the Watt II type appears to have been reported.
The Watt II mechanism, which is the concern of the present paper, has numerous practical
applications due to its topological symmetry, the most common of them being the central-lever
and rack-and-pinion type steering mechanisms, which will be further considered as examples.
2. 7R Watt II mechanism
The Watt II con®guration consists of two four-bar linkages connected in series, the common
member being a ternary link. It is known that for any mechanism having rotational input and
output member(s) the transmission function is independent of the scale of the mechanism. This
kinematic property is valid no matter whether the mechanism is planar or spatial, or has inner
joints other than revolute joints. If a complex linkage can be interpreted as a number of
mechanisms arranged in series and connected via turning elements, it is possible to scale only
some of the loops in the mechanism, without altering its input-output transmission function.
Thus the mechanism in Fig. 1(b) will have the same transmission function as the initial
mechanism in Fig. 1(a), from which it has been obtained by scaling the second four-bar by a
factor k.
The same transmission function will be ful®lled (apart from a phase angle Da† by the
mechanism in Fig. 1(c), which has been obtained from the initial one by rotating the second
four-bar component mechanism and reshaping the middle ternary link, so that the ®xed angle
a is modi®ed by an amount Da:
These two geometrical transformations applied successively to either of the two four-bar
component mechanisms are the reason for the already proven double-in®nity of function
cognates [2]. A particular case among these in®nities will be that in which the ternary link has
the ¯oating joints merged into a double joint. If the length of this middle link is further
considered equal to unity, a reference mechanism with normalised dimensions will result
(Fig. 2). The parameters describing the geometry of this reference-normalised mechanism are
reduced to a minimum i.e., the lengths l1 ±l6 together with the ®xed angle of the ground link b:
A normalisation of the 7R Watt II mechanisms has been used by McLarnan [4] and by
Dhingra and Mani [5] in the synthesis of function generators, with unit ground lengths l1 ˆ 1
and l6 ˆ 1 and the ®xed angle b ˆ 1808: A more productive choice is to have a unit length
ternary link l0 ˆ 1† instead of the ground links, since the degenerate case of a four-bar loop
with a zero ground link length is still a movable structure (with unit transmission ratio) as
compared to a zero crank length mechanism which becomes an immobile triangular frame. To
the seven design variables describing the input±output transmission function jo ji † of the
mechanism, there can be added the values of the initial angles ji and jo of the input and
P.A. Simionescu, M.R. Smith / Mechanism and Machine Theory 35 (2000) 1535±1549
Fig. 1. Input±output invariant transformations of a Watt II 7R mechanism.
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output links de®ning the ®rst precision point and a corresponding nine-precision-point
synthesis problem can be formulated [4,5]. This contradicts a recent paper of Dhingra et al. [6]
in which the normalisation of the mechanism considered in their earlier paper [5] is abandoned
and the parameters describing the mechanism geometry are chosen in terms of Cartesian coordinates of the component joints. An 11-precision-point synthesis problem was formulated in
these Cartesian co-ordinates as design variables, which the authors admitted they were unable
to solve.
Regarding the reference mechanism as that in Fig. 2, after the synthesis has been performed
for a desired function, the whole resulting normalised mechanism can be scaled according to
the gauge requirements. Finally the required position of the ground joints may be chosen by
rotating and if necessary independent scaling of the component four-bar loops relative to the
central ground joint, following the pattern in Fig. 1(b) and (c).
Based on the above procedures, a function cognate of the mechanism in Fig. 2 was designed,
which was required to have the same distance between the ground joints A and E.
Subsequently, the input and output members of the initial and resultant cognates were merged
and the overconstrained mechanism in Fig. 3 obtained.
The method was as follows. The initial mechanism with the numerical data shown in Fig. 2
was scaled by the global factor 50 (to give practical dimensions in millimetres) then the left
four-bar loop was scaled by the factor 1.5 and the right four-bar loop by the factor 1.25. The
two loops were then rotated one against the other at 65.698 by which the distance between the
outer ground joints was reduced to the initial one and the input and output members of the
initial and scaled mechanisms could be merged into single elements (Fig. 3(a)).
AutoCAD2 package was used to determine the intersection point O' of two circles centred
on A and E (with radii equal to 1.5 EO and 1.25 AO, respectively) and then to align
Fig. 2. Reference dimensional con®guration of a Watt II 7R mechanism.
P.A. Simionescu, M.R. Smith / Mechanism and Machine Theory 35 (2000) 1535±1549
Fig. 3. Working model of Watt II 7R function cognates associated in an overconstrained mechanism.
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independently the corresponding scaled four-bar contours relative to the lines O 'E and O'A.
From the in®nite number of possibilities, a mechanism having as many non-fractional
dimensions as possible was chosen. The set of dimensions obtained by this procedure is given
in Table 1.
The mobility of the mechanism is due only to the correlated dimensions of the elements
since, according to Grubler's formula, this should have zero degree of freedom:
f ˆ 3 n ÿ 1 † ÿ 2 j ˆ 3 9 ÿ 1 † ÿ 2 12 ˆ 0
1†
where n is the total number of elements and j is the number of simple joints.
Using the kinematic and kinetostatic planar linkage simulation software OSMEC [7], the
mechanism in Fig. 3 with the data in Table 1 was modelled, but with the element OC
considered disconnected to give the mechanism a topological single degree of freedom. The
analysis showed that the distance between the released joints O and C varies insigni®cantly by
only 0.7248 10ÿ4 mm (0.32 10ÿ4%) throughout the whole working range of the input
element ED ' (from about 1028 to 3898). This small variation is the result of the truncated input
data and of the round-o€ errors in the software. In a working mechanism (such as the
photographs shown in Fig. 3), the accuracy of manufacturing would be lower, the mechanism
being able to move only as result of the joint clearances and link elasticities.
3. The Watt II central-lever steering mechanism
A practical application of the 7R Watt II mechanism is the central-lever steering linkage
used in motor vehicles and towed trailers. In this case, the transmission function of the
mechanism is de®ned, due to the dimensional symmetry, by only four of the seven parameters
of the reference cognate shown in Fig. 4. A preferred set of parameters can be j0 , l, g0 , and lc
(in which as many angles as possible have been included, because the loop closure is
guaranteed for any initial position of the mechanism), while the length lt of the tie-rod and
angle b can be easily calculated in terms of the same. The kingpin track length l0 is usually
prescribed and can be taken as the normalised member l0 ˆ 1†, and the steering performance
of the mechanism determined for a generic vehicle with a given wheel base/wheel track ratio.
Both Duditza and Alexandru [8] and Arday®o and Qiao [9] considered the ®xed angle a of the
central-lever (which is non-zero in most practical designs) as an extra design parameter. The
present considerations show that no steering improvements can be achieved by varying this
Table 1
Numerical data of the mechanism in Fig. 3
xO = 0.0
yO = 0.0
AB = 65
CD = 75
C2D ' = 112.5
B'C1 = 137.5
xA = 70
yA = 0.0
BC = 110
DE = 40
C2O' = 75
AB ' = 81.25
xE = ÿ50
yE = ÿ20
CO = 50
D 'E = 60
C1O' = 62.5
xO ' = ÿ4.12838
yO ' = 46.48906
angle hBAB 'i = 32.0968
angle hC1O 'C2i = 65.6908
angle hDED 'i = 33.5968
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Fig. 4. A simpli®ed central-lever steering mechanism.
angle beyond those obtained by tuning the angle j0 : Although it has the same in¯uence upon
the transmission function of the mechanism as the initial angle of the steering knuckle arm, it
cannot be considered as a ®fth design variable, even for real spatial mechanisms because the
kingpin inclination and castor angles have very little e€ect upon the steering performance of an
initial design planar mechanism [8].
From a simpli®ed design of a central-lever steering mechanism with triple central joint, an
adjacent central joint con®guration that maintains the same kingpin track l0 can be obtained
by scaling all the link lengths by a factor k. For a ®xed angle a of the central lever, the scale
factor which assures regaining the initial kingpin track l0 will be (see Fig. 5):
k ˆ l0 =l 00
2†
where l00 results by projecting one of the vector loops of the mechanisms, e.g. the E'D'C2O left
loop, onto the horizontal axis as:
Fig. 5. Central lever steering linkages cognates.
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a
a
p a
‡
l 00 ˆ 2 l cos p ÿ j0 ‡
‡ lt cos g0 ‡
ÿ lc cos
2
2
2 2
3†
4. 3RT3R Watt II mechanism
A second type of the Watt II six-bar mechanism, with many practical applications, is that in
which the middle ground joint slides (Fig. 6(a)).
This sliding joint provides the same displacement, velocity and higher derivatives to all the
points rigidly attached to the translational ternary link. This implies that, by choosing di€erent
positions of the two pivot joints on this translational link together with a proper repositioning
of the other rotational ground joint, di€erent con®gurations can be obtained which will assure
the same transmission function between the two pivoting grounded members. Again a double
in®nity of function cognates will result from an initial con®guration as in Fig. 6(a). One will
correspond to di€erent points along the horizontal axis (di€erent Dx displacements) as shown
in Fig. 6(b), and the other to di€erent vertical displacements Dy as shown in Fig. 6(c).
As for the 7R Watt II mechanism, a reference normalised con®guration can be de®ned,
having a ternary link of zero length and an input member of unit length l0 ˆ 1 (Fig. 7).
Alternatively, the output member l3 can be chosen to have a unit length, both cases being
preferable to a normalised ground link l4 ˆ 1, which excludes the particular con®gurations
(which sometimes may be of interest) having the joints O and A merged into a triple ground
joint. The total number of parameters describing the input±output transmission function of the
mechanism is 6, i.e., the member lengths l1 ±l4 together with the distance l5 and the angle a
positioning the slider track.
From the reference con®guration mechanism in Fig. 7, with the given numerical data, the
following succession of geometrical transformations was applied so as to obtain a cognate
having the same distance between the ground joint axes.
A global scaling of the mechanism by the factor 150 was initially applied in order to obtain
appropriate dimensions in millimetres. The inner loop OD'C2C1B'A shown in Fig. 8 was then
obtained by scaling the resulting pentagonal loop by a factor 0.6. The right crank and its
coupler were then moved so as to obtain a superimposed ground joint A. The central sliding
element is now triangular, having the track parallel with that of the joint C. AutoCAD2; was
again used to facilitate the graphical constructions (the software calculating all the linear and
angular transformations in double precision).
As in the previous case, the input and output members may be merged into single elements,
and the overconstrained mechanism shown in Fig. 8 obtained, the dimensions of which are
given in Table 2. Working models of this and the mechanism shown in Fig. 3 have been
manufactured in the Department of Mechanical, Materials and Manufacturing Engineering at
the University of Newcastle upon Tyne to demonstrate the e€ectiveness of this method [10].
Once again Grubler's formula has the same form as Eq. (3) giving a formal zero mobility to
the mechanism.
A computer simulation of the mechanism was also performed, in this case the joints C1 and
C2 being permitted to slide independently on parallel tracks. The distance between the centre of
P.A. Simionescu, M.R. Smith / Mechanism and Machine Theory 35 (2000) 1535±1549
Fig. 6. Input±output invariant transformations of a Watt II 3RT3R mechanism.
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Fig. 7. Reference dimensional con®guration of a Watt II 3RT3R mechanism.
these joints varied by 0.4411 mm (1.1%) for the almost whole rotation range of the input
element OD (ÿ478, 2278), again due to truncated input data and round-o€ errors.
5. The Watt II rack-and-pinion steering mechanism
A common practical embodiment of the Watt II mechanism with a central translational
ground joint is the rack-and-pinion steering linkage. There are two relatively di€erent
con®gurations, one having the pivot joints of the ternary link very close to each other
corresponding to the central outrigger rack-and-pinion steering linkages, and the other, more
commonly used, having these same pivot joints distantly disposed on the actuating
translational element.
The central outrigger type can be well approximated by the simpli®ed con®guration shown
in Fig. 9, in which the central pivot joints are merged into a single triple joint. Due to the
dimensional symmetry, the number of parameters describing the geometry reduces from 6 to 3,
viz. the steering knuckle arm length l, and the initial angles j0 and g0 : The kingpin track l0 will
become the reference length as in the case of the central-lever steering mechanism.
Table 2
Numerical data of the mechanism in Fig. 8
xO = 0.0
yO = 0.0
AB = 150
BC = 105
CD = 90
xA = 79.86732
yA = ÿ55.20297
DO = 75
AB ' = 90
B'C1 = 63
yC = 34.41459
C2D ' = 54
D 'O = 45
xC1ÿxC2 = 31.94693
yC2ÿyC1 = 22.36948
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Fig. 8. Working model of Watt II 3RT3R function cognates associated in an overconstrained mechanism.
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Fig. 9. A simpli®ed central outrigger rack-and-pinion steering mechanism.
A real rack-and-pinion steering linkage with distant central pivot joints will have a reference
cognate with merged central joints (see Fig. 10), and consequently the parameters describing its
geometry will be three in number. These can be conveniently considered as the initial angles j0
and g0 of the steering knuckle arm and of the tie-rod respectively, together with the ratio l 0 =lt0
(note that l 0 ˆ k l and lt0 ˆ k lt , where k is the scaling factor by which the inner loop has
been obtained). Either of the lengths l 0 or lt0 or alternatively that of the tie-rod can be chosen a
priori (for the case of an independent wheel front axle, the tie-rod length must be correlated
with the suspension geometry such that the cross-coupling e€ect between steering and wheel
bump and rebound is reduced to a minimum). This contradicts the ®ndings of Zarak and
Townsend [11], who considered four design parameters in the synthesis of a rack-and-pinion
planar steering mechanism. The fourth geometrical parameter chosen is redundant and any
®ne-tuning of its value would not improve in any way the steering characteristics of the
mechanism.
If the design is performed initially for a simpli®ed con®guration with merged central joints
such as that in Fig. 9 (with prescribed kingpin track l0 and rack length lr ), the real mechanism
link lengths can be obtained by multiplying them by a scale factor k of less than unity:
k ˆ l t0 =lt ˆ l 0 =l ˆ l0 ‡ lr †=l0
Fig. 10. Rack-and-pinion steering linkage cognates.
4†
P.A. Simionescu, M.R. Smith / Mechanism and Machine Theory 35 (2000) 1535±1549
1547
This equation results from consideration of the similar triangles EC2D' 0 ECD and CC1C2 0
CAE where lr denotes the rack length.
6. Conclusions and further studies
An intuitive method for determining Watt II function cognates and the corresponding
overconstrained mechanisms has been presented.
A proof of the invariance of the transmission function following the geometrical
transformations illustrated in Figs. 1 and 6, is that the transmission function of any mechanism
having rotational input and output members can be expressed as a dimensionless equation in
terms of link lengths ratios.
It is interesting to note that the pantograph linkage shown in Fig. 11, known as Burmester's
pantograph [2], can be considered as resulting from a particular overconstrained mechanism of
the Watt II type, with the ground joints of the input and output members merged into a single
triple joint. When the pantograph traces arcs of circles centred in O, it suggests a 7R Watt II
origin while, when its traces straight lines, a 3RT3R Watt II origin might be invoked. Fig. 12
shows a proposed generalisation of this pantograph derived from a 3RT3R Watt II mechanism
with the same proportions as mechanism in Fig. 8 (Table 2). A number of supplementary
elements have been introduced such that the ¯oating L-shaped link derived from the former
central translating link has rotation inhibited. The proof that this is a pantograph linkage relies
on the fact that the line C1C2 remains parallel with the ground joint line AO, ensuring
Fig. 11. Burmester's pantograph.
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P.A. Simionescu, M.R. Smith / Mechanism and Machine Theory 35 (2000) 1535±1549
Fig. 12. A 3RT3R Watt II pantograph derived from the overconstrained mechanism in Fig. 8.
similarity of the triangles CC1C2 0CAO, OD'C2 0ODC and AB 'C1 0ABC. Consequently, any
point solidly ®xed with the L-shaped member of the pantograph will trace the same curve as
that traced by the joint C, but with a reduction in scale k = C2D'/CD = B'C1/CB.
Acknowledgements
The ®rst author acknowledges the support of Romanian Ministry of Education through the
grant no. 5843/1996 for his research visit at the University of Newcastle upon Tyne.
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