Watt's Parallel Motion - Interactive Technical Animationsjeanlouis.blanchard.pagesperso-orange.fr/english_pdf/watt_parallel... · 4 The Watt’s parallel motion for bars of equal

Watts Parallel Motion

Jean-Louis Blanchard

Email: [email protected]

Web site: http://www.techanimatic.com

c 2006 Jean-Louis BLANCHARDThird edition

mailto:[email protected]://www.techanimatic.com

BROCHURE VISUALIZATION USE OF ANIMATIONS COPYRIGHTS TABLE OF CONTENTS II

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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

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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


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.


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



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



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



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


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


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.


12 WATTS PARALLELOGRAM 6


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




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



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.



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


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.


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