Peaucellier’s Inversor
Jean-Louis Blanchard
Email:
[email protected]
Web site: http://www.techanimatic.com
c 2006 Jean-Louis BLANCHARD
First edition
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− USE OF ANIMATIONS − COPYRIGHTS − TABLE OF CONTENTS
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II
Table of contents
1 Visualization of the motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Description of the motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Table of figures
1
2
3
4
5
Peaucellier’s rhombus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Distortion of the Peaucellier’s rhombus . . . . . . . . . . . . . . . . . . . .
Angles of the Peaucellier’s rhombus . . . . . . . . . . . . . . . . . . . . . . . .
Spear-head and kite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rhomboid-based Peaucellier’s inversor . . . . . . . . . . . . . . . . . . . . .
5
5
6
7
7
Table of animations
1 Peaucellier’s rhombus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Peaucellier’s rhomboid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
§1
3
VISUALIZATION OF THE MOTION
1 Visualization of the motion
This animation shows
the Peaucellier’s inversor
which exactly transforms
an alternated circular
motion into an alternated
rectilinear motion and
vice-versa. The involved
transformation is a
geometrical inversion.
The use of a rhombusshaped cell at the left
of the linkage is the
widespread form of the
Peaucellier’s inversor.
Animation 1 : Peaucellier’s rhombus
Peaucellier’s inversor
4
§1
VISUALIZATION OF THE MOTION
This animation shows a
variant of the Peaucellier
inversor where the
rhombus is replaced by
a rhomboid.
Animation 2 : Peaucellier’s rhomboid
Peaucellier’s inversor
§2
5
DESCRIPTION OF THE MOTION
2
Description of the motion
In fig. 1, the Peaucellier’s inversor is shown with its most
conventional shape. Four bars make the rhombus KM LN with
side a. The opposite vertices K et L with respect to the diagonal
KL are linked by the two bars AK and AL of the same length
b to point A. The point M is compelled to move on a circle of
centre O and radius r by the bar OM . When the inversor is made
in such a manner that the circle of centre O passes through A, its
distortion enables to move the point N along a line perpendicular
to the straight line AO.
K
remain aligned when the figure is distorted. The diagonals of a
rhombus being orthogonal the Pythagoras’theorem applied to the
right triangle AIK gives AK 2 = AI 2 + IK 2 or:
AI 2 = AK 2 − IK 2
K
A
N
α2
O
N
Figure 2 : Distortion of the Peaucellier’s rhombus
Similarly, in the right triangle M IK, this theorem gives M K 2 =
M I 2 + IK 2 or:
L
M I 2 = M K 2 − IK 2
A
I
L
M
r
M
a
I
b
(1)
H
Figure 1 : Peaucellier’s rhombus
(2)
Then, substracting (2) from (1) leads to AI 2 −M I 2 = AK 2 −M K 2
or (AI − M I)(AI + M I) = b2 − a2 while the alignement of
points A, M , I and N enables to write AI − M I = AM and
AI + M I = AI + IN = AN , which finally gives:
The projection of N on OA is designated by H and the length
AH by h. As outlined by fig. 2, the points A, M , I and N
Peaucellier’s inversor
AM.AN = b2 − a2
(3)
6
§2
DESCRIPTION OF THE MOTION
So, the product AM.AN is constant and this relationship characterizes an inversion of pole A. The relevant property of the inversion is that a circle passing through the pole is transformed into a
straight line. So, if the circle of centre O described by M passes
through A, the curve described by N is a straight line.
A deeper investigation of the inversor requires to set up the relationships between the bar lengths and the mechanism behaviour.
For that purpose, the values of a, r, and h > 2r are assumed to be
given. In horizontal position, the inversor is such that AM = 2r
while N coincides with H, which implies after (3) 2r.AH = b2 − a2
or b2 = a2 + 2r.AH. This enables to compute the lengths of bars
AK et AL as:
q
(4)
b = a2 + 2rh
If the driving parameter is taken as the angle α1 of OM with
AOH (fig. 3), the main point is to notice that α1 = 2α2 . Since
OA = OM , the triangle AOM is indeed isoceles and the sum of
its angles is such that:
d =π
2α2 + AOM
d being straight it follows that AOB
d = AOM
d +
The angle AOB
d + α = π and AOM
d = π − α . Substituting this
Md
OB = AOM
1
1
value in (7) leads to:
2α2 − α1 = 0
By taking O as the origin of the Ox axis, the point A is located at
(−r, 0) and the point H at (h − r, 0). By designating the angle of
the straight line AM IN with AOH (fig. 1) as α2 , the right triangle
AHN gives tan α2 = N H/AH and the ordinate of N follows:
N H = h tan α2
(7)
(8)
M
(5)
The abscissa of N being identical to the one of H, the coordinates of
N are (h − r, h tan α2 ). Moreover, the relationship (5) also enables
to compute the stroke of N , the limit value α20 of α2 being obtained
when the rhombus is flat. In this configuration, AN = b +a. Then,
the right rectangle AHN gives AN cos α2 = AH = h and the
relationship:
h
cos α20 =
(6)
a+b
provides the value of angle α20 , the actual value of which is reduced
by the pivots.
α2
A
α1
O
B
Figure 3 : Angles of the Peaucellier’s rhombus
This equation also gives the lenghts of the curves described by
M et N . For that purpose, the length of the arc BM is designated
Peaucellier’s inversor
§2
7
DESCRIPTION OF THE MOTION
by `(BM ). After (8) this length is equal to rα1 or `(BM ) = 2rα2 ,
i.e.:
`(BM )
(9)
α2 =
2r
Moreover (5) gives:
HN
(10)
α2 = arctan
h
From these last two relationships, it follows that the lengths of arc
BM and straight line segment HN are:
`(BM )
HN
= arctan
2r
h
or HN = h tan
`(BM )
2r
(11)
KM LN with the equal sides KM et KN on one hand and the
equal sides LM et LN on the other hand, while KM 6= LM .
Known designations of the convex and concave rhomboids (fig. 4)
are respectively spear-head (left figure) and kite (right figure).
By designating by a0 the common length of sides LM and LN
and by b0 the lengh of AL, an approach strictly similar to the one
led for the rhombus gives the new equation:
AM.AN = b2 − a2 = b02 − a02
A Peaucellier’s inversor based on a rhomboid is shown in fig. 5.
K
In these equations, the curvilinear abscissae BM with B as origin
as well as the algebraic distance HN with H as origin require a
consistent orientation. For instance, for α1 > 0 et BM > 0, the
straight line HN has to be oriented in such a manner that HN > 0.
N
K
N
L
M
a
b
A
K
(12)
L
M
O
M
I
r
N
a
b
Figure 4 : Spear-head and kite
L
Replacement of the rhombus by a rhomboid. The approach leading
to the basic equation (3) is based on the alignment of points A,
M , I et N and on the orthogonality of the rhombus diagonals,
but these properties still hold true when the rhombus is replaced
by a rhomboid. This figure is defined as a parallelogram such as
Peaucellier’s inversor
Figure 5 : Rhomboid-based Peaucellier’s inversor