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Watts Parallel Motion
Jean-Louis Blanchard
Email: contact@techanimatic.com
Web site: http://www.techanimatic.com
c 2006 Jean-Louis BLANCHARDThird edition
mailto:contact@techanimatic.comhttp://www.techanimatic.com
BROCHURE VISUALIZATION USE OF ANIMATIONS COPYRIGHTS TABLE OF CONTENTS II
Brochure visualisationThis brochure includes animations the visualization of which
requires to use the full version of Adobe Reader version 6.0 or
higher. Lower versions of this software previously named Adobe
Acrobat Reader only enable to visualize the text body and static
figures. Adobe Reader can be freely downloaded in many languages
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The Flash Player version has to be version 6.0 or higher.
Use of animations. To activate an animation, click in its frame; to deactivate it,
press the escape key.
. The animations are driven though a floating toolbar which
groups video cassette recorder-like buttons. This toolbar in-
cludes a help panel which can be displayed by clicking on the
question mark.
. In addition, all animations can be zoomed-in, zoomed-out
and restored to their original size by pressing the right-mouse
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named Print what in the Adobe Reader Print dialog box.
CopyrightsAdobe Reader and Adobe Acrobat Reader are trademarks of
Adobe Systems Incorporated. Flash is a trademark of Macromedia
Incorporated.
Table of contents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Watts three bar linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4 Generalization of the Watts three bar linkage . . . . . . . . . . . . . . 9
5 Scheiners pantograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Watts parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Table of figures1 Sketch of the Newcomens steam engine . . . . . . . . . . . . . . . . . . . . 3
2 Watts three bar linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 System of polar coordinates for the Watts linkage . . . . . . . . . 7
4 Sketch of the stroke for the Watts linkage . . . . . . . . . . . . . . . . . 8
5 Generalization of the Watts three bar linkage . . . . . . . . . . . . 10
6 Scheiners pantograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7 Watts parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Table of animations1 The Watts curve for bars of equal length . . . . . . . . . . . . . . . . . . . 5
2 The Bernoulli lemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 The Watts curve for bars of unequal length . . . . . . . . . . . . . . . . 9
4 The Watts parallel motion for bars of equal length . . . . . . . . 12
5 The Watts parallel motion for bars of unequal length . . . . . 13
6 Skeleton of the parallel motion in the Watts engine . . . . . . . 15
http://www.adobe.com
1 INTRODUCTION 3
1Introduction
In the National Science Museum of London, in one of the
numerous glass cases enclosing the mockups of steam engines, the
following James Watts quotation is displayed: Though I am not
anxious about fame, yet I am more proud of the parallel motion than
of any other mechanical invention I have made. The invention
this quotation refers to is basically made of three articulated bars.
Considering James Watts considerable contribution to the
development of steam engines, which are obviously much more
complex than the three bar linkage, this sentence may astonish.
But, after having seen this invention in motion, one understands
the pride of his designer, since the involved linkage solves a difficult
problem with simplicity and efficiency , in short with elegance.
This brochure illustrates what the Watts parallel motion is
with the help of the resources of electronic publishing which enable
to combine in a single and self-contained document explanatory
text and mathematical equations as well as static and animated
figures.
2History
To deal with the reasons which led James Watt to investigate
the parallel motion, an appropriate start point is the Thomas
Newcomens atmospheric steam engine which seems to have been
put into service in 1712 [2] for pumping the water out of coal
mines. The working mode of this engine can be briefly described
as follows (fig. 1).
When the piston moves up, steam is admitted from the boiler to
the cylinder. When it reaches the top of its stroke, water is injected
in the cylinder to condensate the steam. This condensation forms
a vacuum dragging the piston downward. When this later reaches
the bottom of its stroke, a new cycle is ready to begin.
fire
boiler
chain
steam inlet
water inlet
crossbeam
chain
weight
mine pump rod
Figure 1 : Sketch of the Newcomens steam engine
An important point is that the upward motion of the piston is
not caused by the steam but by the weight of the mine pump rod
attached on the other side of the crossbeam which is so alternatively
dragged by the piston and the mine pump. This is why the
connections between the crossbeam and the piston axle on one
hand and the mine pump rod on the other hand can be made by
chains. The point here is that a chain works only in tension. In
other words, a chain can pull but cannot push.
When Watt devised the double-acting engine, with the steam
working on each face of the piston, he had to resolve two mechanical
problems. The first one was related to the transformation of the
alternated motion of the crossbeam into a rotative one, a motion
Watts Parallel Motion
4 HISTORY 2
which was not possible with the Newcomen steam engine. Although
the crank was already known, Watt was not in a position to use it
because of patents taken in 1779 and 1780. To circumvent them, he
devised and patented in 1781 the fly and planet wheels, not shown
here.
The second problem was related to the connection between the
piston and the beam. This is where the three bar linkage and the
parallel motion patented in 1784 appear.
Watts Parallel Motion
3 WATTS THREE BAR LINKAGE 5
This animation shows the
curve generated by the
Watts three bar linkage,
made of two cranks
linked by a bar. Here the
two cranks have the same
length and the tracing
point is the midpoint of
the bar. The curve looks
like the figure 8, but
for its part outlined by
the yellow rectangle, this
curve is approximately
a straight line segment.
This property is the basis
of the parallel motion.
3 Watts three bar linkage
Animation 1 : The Watts curve for bars of equal length
Watts Parallel Motion
6 WATTS THREE BAR LINKAGE 3
This animation shows
again the curve generated
by the Watts linkage
with two cranks of equal
length, but when the bar
and distance between
pivots are given the
same specific length,
such that the part of the
curve which looks like the
figure 8 is the Bernoulli
lemniscate. This part
of the curve is drawn
when the cranks are in
opposite half-planes. The
cranks meet the Grashof
condition meaning that
they can make complete
revolutions.
Animation 2 : The Bernoulli lemniscate
Watts Parallel Motion
3 WATTS THREE BAR LINKAGE 7
The three bar linkage is depicted in fig. 2 in neutral position
where the two bars AB and CD are horizontal. These two bars
with equal length are connected by a rod BC the length of which
is shorter than the distance AD between the two fixed pivots A
and D. Pivots are denoted by closed circles (B, C), specific points
by open circles (M) and fixed pivots (A, D) are surrounded by a
square.AB
D C
M
Figure 2 : Watts three bar linkage
When the shape of this linkage is changed, the midpoint M
of rod BC describes a curve which looks like the figure 8, as
illustrated by animation 1. However, for a moderate change around
the neutral position the animation shows that M approximately
follows a vertical straight-line segment . So, AB being the engine
crossbeam and M the connection point of the piston axle, the
linkage enables to drive M along a straight-line and to transform
its alternated rectilinear motion into an alternated circular motion
around A.
A(a, 0)
B
C
D(a, 0) O
M
Figure 3 : System of polar coordinates for the Watts linkage
In polar coordinates (fig. 3), with the notations AB = CD = b,
BC = 2c and AD = 2a, the origin being located at the midpoint
of AD , the locus M(, ) is [8][1]:
2 = b2 (a sin
c2 a2 cos2
)2(1)
This curve of sixth degree is named a lemniscatoid or a lem-
niscoid [7]. Its name and its shape suggest some relationship with
the well known Bernoulli lemniscate. For a = c and b = a
2 the
previous equation reduces to 2 = 2a2 cos 2 which is indeed the
Bernoulli lemniscate, as shown by animation 2.
The lemniscate is only one of the many variations which can be
obtained by eq. (1) according to the values of parameters a, b and
c. [7] gives an atlas of Watts curves. However, from the parallel
motion standpoint, the relevant part of the curve is located at the
vicinity of the symmetry point of the eight-shaped figure.
By appropriately choosing the values of a, b and c the eight-
shaped figure is stretched to produce a good approximation of
Watts Parallel Motion
8 WATTS THREE BAR LINKAGE 3
a straight-line path. For that purpose, Watt used the following
parameters [3][4]:AJ
36=
AB
37=
BC
24(2)
with a stroke equal to 24k, k denoting the common value of the
previous ratios. This arrangement is sketched in fig. 4 with the two
extreme positions of the linkage drawn with dashed lines.
AB
CD
J
Figure 4 : Sketch of the stroke for the Watts linkage
Watts Parallel Motion
4 GENERALIZATION OF THE WATTS THREE BAR LINKAGE 9
When the two cranks
have not the same length,
the tracing point located
on the bar joining the
two cranks is located
at a position which is
similar to the ratio of the
crank lengths. As for the
case of crank of equal
lengths, the part outlined
by the yellow rectangle is
approximately a straight
line segment.
4 Generalization of the Watts three bar linkage
Animation 3 : The Watts curve for bars of unequal length
Watts Parallel Motion
10 GENERALIZATION OF THE WATTS THREE BAR LINKAGE 4
In the previous section, the two bars AB and CD were assumed
to have the same length. Actually, the linkage can be generalized
for bars AB and CD with unequal length. To deal with this case,
it is worth recalling two basic results in kinematics [5]. First, the
motion of a movable plane is completely determined by the motion
of two points in the plane or a straight line joining two points in
the plane. Then, every plane motion is equivalent to a rotation
about a certain point which the centre of rotation for that motion.
d
d
D C
C
I
A
B
BM
Figure 5 : Generalization of the Watts three bar linkage
The position of the instantaneous centre of rotation enables to
see why BC moves according to a translation near the neutral
position (fig. 5). The trajectories of points B and C rigidly
connected by the rod BC are indeed known since they respectively
move on the circles centered on A and D. So, the instantaneous
centre of rotation I is located at the intersection of straight lines
AB and CD which are orthogonal to those trajectories. In neutral
position these segments are parallel and the centre of rotation is
rejected at infinity. In other words, the motion in the vicinity of
this position is a translation perpendicular to the common direction
of AB and CD.
When AB and CD are unequal, a problem is to place M on
BC. For that purpose, following the approach given in [5] a small
change of the linkage in fig. 5 is considered, leading to the shape
ABC D. Let d and d denote the rotations around A and D,the centre of rotation I being located at the intersections of BAand DC . Since the purpose is to place M in such a way that itmoves along a vertical line, M is placed at the intersection of BC
with the horizontal line passing through I which gives:
C IB = d + d (3)The lengths of arcs BB and CC are AB d and CD d. As-suming that the change is small, the arcs BB and CC can beapproximated by their chords. Moreover, the motion of BC being
assumed to be a translation BB = CC which enables to writeAB d = CD d or:
d
d=
CD
AB(4)
Let denote the angle of BC with the horizontal. This leads toBC I = d. The sum of angles within triangle IBC gives byreplacing C IB by its value provided by (3):
= IBC + BC I + C IB= IBC + d + d + d = IBC + + d
So, IBC = d and the theorem of sines in triangles BIM and C IM leads to:
MB
sin d=
IM
sin IBC =IM
sin( d)Watts Parallel Motion
5 SCHEINERS PANTOGRAPH 11
and:MC
sin d=
IM
sin C IB =IM
sin( d)(5)
Then, the following approximations:
sin d d d sin d d d
transform (5) in:
MB
d=
IM
sin( )=
IM
sin and
MC
d=
IM
sin
which gives MB/d = MC /d or MB/MC = d/d. Finally,after (4), this leads to locate M according to the following rela-
tionship:MB
MC =
CD
ABIn particular, when AB = CD, M is placed at the midpoint of
BC.
5Scheiners pantograph
The three-bar linkage is the basis of the parallel motion. But
Watt designed a more elaborate mechanism by combining it with
the Scheiners pantograph which is probably one of the first known
examples of linkages since it was published in 1631.
Scheiners pantograph is based on a parallelogram ABCD with
the extended sides BC and CD (fig. 6). A straight line is drawn
which intersects the parallelogram at points O, P , Q and R. These
points are fixed on the straight-line segments they intersect.
A
BC
D
O
P
Q
R
Figure 6 : Scheiners pantograph
The straight-lines AB and CD being parallel, the triangles
OBP and OCR are similar [6] and this property holds true for any
shape of the parallelogram. So, points O, P and R remain aligned.
Then, the simitude of triangles OBP and OCR allows to write
that their sides are proportional:
OB/OC = OP/OR (6)
Since the straight-line segments OB and OC have a constant
length, the ratio of the variable length segments OP and OR can
be expressed as:
OP/OR = k (7)
where k denotes the constant ratio OB/OC. This transformation
is an homothecy (a dilation), O being the homothetic centre and
k the similitude ratio. Therefore, by fixing the pivot O, when R
describes a given figure, P describes an homothetic figure and vice-
versa. So, the pantograph is an instrument which enables to reduce
or enlarge a given figure. Generally the pantograph is made in such
a way that passes through A. The previous approach applies for
point Q by drawing a line parallel to AB and CD passing through
it.
Watts Parallel Motion
12 WATTS PARALLELOGRAM 6
This animation shows
the motion produced by
the Watts parallelogram
when the base cranks
have the same length.
This animation outlines
the stroke increase
due to the Scheiners
pantograph.
6 Watts parallelogram
Animation 4 : The Watts parallel motion for bars of equal length
Watts Parallel Motion
6 WATTS PARALLELOGRAM 13
This animation shows
the motion produced by
the Watts parallelogram
when the base cranks
have not the same length.
Animation 5 : The Watts parallel motion for bars of unequal length
Watts Parallel Motion
14 WATTS PARALLELOGRAM 6
The combination of the three-bar linkage with the pantograph
is made as follows. The bar AB is extended up to point E and two
additional bars EN and NF are inserted in such a way that the
figure BENF is a parallelogram and the points A, M and N are
aligned (fig. 7). The straight-line AMN is the one denoted by
in the previous section.
B
E
N
M
C
FA
D
Figure 7 : Watts parallelogram
Since M describes a straight-line, so does N , which enables
to increase the available stroke. The configuration used by Watt
consisted in making the points F and C coincident. This implies
that the similitude ratio is BF/BM = 2 which gives to point N a
stroke twofold larger than the one of point M . Actually, the piston
axle was connected to N while a pump axle was connected to M , as
shown by the animation 6 which gives a sketch of the parallelogram
as used in a double-acting steam engine.
Watts Parallel Motion
6 WATTS PARALLELOGRAM 15
This animation is a
sketch of the use of the
Watts parallelogram in a
double-acting steam en-
gine. The parallelogram
is located at the upper
left corner. The piston
is attached to the point
with the longest stroke,
the air pump to the point
with the shortest one.
The transformation of
the rectilinear motion of
the piston to a rotative
motion through the beam
is made by a crank linked
to the flywheel engine.
Animation 6 : Skeleton of the parallel motion in the Watts engine
Watts Parallel Motion
16 CONCLUSION 7
7
Conclusion
Considering the resources available in the eighteenth century,
one can only be but full of admiration for the mechanism devised
by James Watt and consider as perfectly legitimate his pride with
respect to his invention. These mechanisms were the pure fruit
of a powerful imagination since at that time, as pointed out by
Koenigs [3], the mathematical tools dealing the transformation of
figures were not available.
Watts parallel motion is part of a problem class named the
straight-line motion. Watts mechanism is an approximate solution
of it and for a long time it was believed as impossible to transform
exactly a circular motion into a rectilinear one, even to the famous
mathematician Pafnouty Tchebycheff who extensively studied the
Watts parallel motion.
Year 1864 was a turning point with the solving by Charles Nico-
las Peaucellier of the straight-line motion problem by the invention
of a linkage made of seven rods based on a mathematical transfor-
mation named inversion. The theoretical interest of this invention
was considerable even if it had very few practical applications1.
From that point, systematic studies of linkages were made by
famous geometers such as Sylvester, Roberts, Cayley and Kempe
in connection with the theory of figure transformations.
8Bibliography
[1] H. Brocard and T. Lemoyne. Courbes geometriques (courbes
speciales) planes et gauches, volume II. Albert Blanchard, 1967.
[2] Collective book. Lexpansion du machinisme, volume III.
Presses universitaires de France, 1968.
[3] G. Koenigs. Lecons de cinematique. A. Hermann, 1897.
[4] R. Prudhomme and G. Lemasson. Cinematique. Theorie,
applications. Dunod, 1955.
[5] N. Rosenauer and A. Willis. Kinematics of mechanisms. Dover,
1967.
[6] E. Rouche and Ch. de Comberousse. Traite de Geometrie.
Gauthier-Villars, 1957.
[7] E. Shikin. Handbook and atlas of curves. CRC Press, 1995.
[8] F. G. Teixeira. Traite de courbes speciales remarquables planes
et gauches, volume I. Chelsea Publishing Company, 1908,
Reprinted in 1995 by Editions Jacques Gabay.
1 Animations of the Peaucellier linkage can be found on the web site http://www.techanimatic.com.
Watts Parallel Motion
http://www.techanimatic.com
Table of contentsIntroductionHistoryWatt's three bar linkageGeneralization of the Watt's three bar linkageScheiner's pantographWatt's parallelogramConclusionBibliography
Table of figuresSketch of the Newcomen's steam engineWatt's three bar linkageSystem of polar coordinates for the Watt's linkageSketch of the stroke for the Watt's linkageGeneralization of the Watt's three bar linkageScheiner's pantographWatt's parallelogram
Table of animationsThe Watt's curve for bars of equal lengthThe Bernoulli lemniscateThe Watt's curve for bars of unequal lengthThe Watt's parallel motion for bars of equal lengthThe Watt's parallel motion for bars of unequal lengthSkeleton of the parallel motion in the Watt's engine
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