Kinematic Diagrams, Kinematic Inversions
Equivalent linkages
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Kinematic diagrams are dawn for ease of understanding the kinematic behaviour of a linkage. Therefore for drawing such
diagrams sufficient information is provided for the kinematic behaviour of linkages. The important information are the
length of the link between two pairs, the pairs with which the links are connected, and the points of anchoring or the points
of fixation. The material of the link, other geometrical details like shape, size, thickness etc are not given as these do not
decide the kinematic behaviour. Two members are considered as two different links if there is a possibility of relative motion
between them.
For ease of representation the following diagrams are used. A linkage represented using these elements of the diagrams is
called the Kinematic diagram.
Prismatic pair between links 1 and 2
Revolute pair between links 1 and 2, pair is at the edge
A bent link
A quaternary link connecting link 1 to links 2,3,4,5
Links 1 and 2 connected by a revolute pair, present
within the boundary of link 1
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Courtesy Ghosh and Mallik
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Example of the Kinematic Diagram of a 4R 4 bar Mechanism
Explanation:
The locus of B with respect to point O4
is a circle. This is exactly generated by
placing a circular guide of radius O4R3
centered at O4 and guiding a curved
slider along the path. So the
equivalence of a revolute pair is a
prismatic pair working on a curved
path. Let us apply this principle on a
3R 1P 4 bar mechanism shown below.
Kinematic Diagram of a 4R 4 bar Mechanism; 3 is the jth revolute pair,
where, j =1,2,3,4. The numbers 1,2,3,4 are the link numbers.
Applying the explanation
The slider is equivalent to an infinite
length of the link 4 and vice versa, as
the locus of point 4 is straight.
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The 3R 1P
chain, (Slider-Crank Mechanism, P4 is the prismatic pair between links 1 and 4)
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An Example
The diagram of a drafting board is shown in the
figure (a), where A and B are the two
perpendicular scales. The circular link connects
four links as shown. The purpose of this linkage
is to take the perpendicular set of the scale to
any location of the drawing sheet to draw.
The figure (b) shows the kinematic diagram of
the mechanism. From intuition we may say that
the degree of freedom of the linkage should be
2, as then only we shall be able to take the set of
perpendicular scales to any point on the sheet.
figure (b)
figure (a)
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An example of a disguise
The figure (a) shows the diagram of a mechanism, in which, the link 2 is a circular disc hinged non-centrally with a revolute
pair to the ground and, as it rotates it drives the link 3 and through that the link 4.
The kinematic diagram is not apparently visible, but is identifiable, as drawn using the dotted line. The kinematic diagram is
given below in figure (b).
figure (b)
figure (a)
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The concept of Inversion
To construct any mechanism, one of the links must be fixed, else, it is not possible to operate it. If the fixed link is
changed, i.e. if another link in the linkage is fixed and the link originally fixed is freed, then a different kind of motion
may be created. This gives rise to different mechanisms, which are the inversions of the original mechanism.
Linkage with no link fixed (not
workable)
Mechanism with link 1 fixed
Mechanism with link 2 fixed
Mechanism with link 3 fixed
Mechanism with link 4 fixed
4 Revolute 4 bar or 4R4bar mechanism, and its inversions
Please remember that in this process the pair between two links is not altered and, therefore, the relative motion between two
links remains unchanged.
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A Conceptual Diagram of Inversion of 4R 4bar mechanism.
1
R
2
R
3
Link-1 fixed.
R
R
4
1
R
2
R
1,2,3,4 are the links.
‘R’ on the connecting lines show,
that they are connected by revolute
pairs.
line below shows the link
fixed.
Link-2 fixed.
3
R
R
4
1
R
2
R
Link-3 fixed.
3
R
R
4
1
R
2
R
Link-4 fixed.
3
R
R
4
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Inversion continued
Some nomenclatures and observations
A
O2
B
O4
The fixed link, link-1 in this case is called the “Frame”
The link 2 and link 4 may be either the input or the output link. If link 2 is the input link
then link 4 is the output link and vice versa.
The input link may be technically called the “Crank” and the output link may be called the
“Follower”
The link 3 coupling the input and the output links is technically called the “Coupler”
The loci of the points A and B about the points O2 and O4 are circular, which may be
complete or incomplete, if it completes the corresponding link is called the crank else if the
circle is incomplete, it is called the rocker.
All the inversions of the 4R 4 bar mechanism, shown in the slide before, will result in a
combination of cranks and rockers, like crank-crank, crank-rocker or rocker-rocker. We
shall learn about them in detail in subsequent slides.
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Inversion continued
Inversion of 3R 1P chain
Courtesy: Ghosh and Mallik
The slide is self explanatory.
You may note that, as the fixed
link is changed, the purpose of
the mechanism changes.
Link 1 is connected to link 2
with revolute pair.
Link 2 is connected to link 3
with revolute pair.
Link 3 is connected to link 4
with revolute pair.
Link 4 is connected to link 1
with prismatic pair.
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Inversion of 3R 1P chain continued
if e = 0
The offset slider crank mechanism,
offset is ‘e’ between the hinge point of
crank (link 1) and line of translation of
the slider link 4.
Slider-Crank Mechanism without offset (e = 0).
This mechanism is used in IC engines. The time
taken by the slider due to its to and fro motions
are same, when e = 0
The offset Slider-Crank Mechanism has different times of
travel of the slider for completing its length of stroke. For
the case shown the slider will travel faster when it moves
from left to right than when it moves from right to left.
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Inversion of 3R 1P chain continued
This is the kinematic
diagram of the Slider and
slotted lever type quick
return motion mechanism
used in shaping machines.
You may note that the
slotted lever reaches its
extreme angular location
when the Crank becomes
normal to the Slotted lever.
The tool of a Shaping machine.
The left motion should be slower,
as this is the cutting stroke, than
the travel to the right as the tool
moves idle to its original position.
Crank
Input
Slotted lever
Therefore, for the
anticlockwise motion of the
crank, the motion of the
slotted lever from right to
left will be slower than the
motion from left to right.
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Output
Shaping machine
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Inversion of 3R 1P chain continued
Purpose
This mechanism find use in making a foot
pump for filling air in tyres. The piston and the
cylinder head have valves to admit air when the
piston moves away from the cylinder head and
deliver the compressed air trapped between the
piston and the cylinder to the tyre through a
pipe connected between the cylinder head and
the tyre.
The oscillating cylinder mechanism
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Inversion of 3R 1P chain continued
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The Hand pump mechanism
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Inversion of 2R 2P chain
The Scotch Yoke mechanism has two revolute pairs and two
prismatic pairs to connect two links.
Link 1 is the fixed, Link 2 is the crank, to provide the input
motion, Link 3 is the slider, which slides vertically in the
frame, which slides with respect to link 1.
The Scotch Yoke Mechanism
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Its motion characteristics are similar to the Slider crank
mechanism, however, due to use of more sliders, it may not
be as efficient as the slider crank mechanism in terms of
power saving.
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Inversion of 2R 2P chain continued
P4
In this, the link 2 is fixed. The mechanism we get is called the Oldham’s coupling.
Its purpose is to transmit rotary power from one shaft to the other, where the axes
of the shaft have a parallel mismatch. The middle member (Link 4) has two shoes,
perpendicular to one another to connect the corresponding slots in links 3 and 1.
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Inversion of 2R 2P chain continued
In this mechanism the
input is given by giving
different angular
orientation of the link
AC, on which D is any
point on its extended
part.
In this case the link 4 is fixed.
Please note that the link 4 (in
the Scotch Yoke mechanism) is
in the form of a frame, which
has two prismatic pairs, one
with the link 1 and the other
with the link 3. In this case, the
link 4 has been thought of as
the hollow prism, which gives
rise to this mechanism.
You will be able to prove
that the equation of the
locus of the point D
about the origin of the xy axis is that an ellipse.
Hence, is the name of this
mechanism.
The Elliptical Trammel
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You will also be able to
prove that the equation
of the locus of the middle
point B of the segment
AC is that of a circle.
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Equivalent linkage
So far we have seen linkages with lower
pairs and their representations. Higher
pairs are those in which the contact is at
a point or along a line, ideally.
Lower Pair
Thinking actually, the area of contact is
much smaller than that in the case of a
lower pair. The figure shows higher and
lower pairs in a cam follower
mechanism.
The higher pair causes both rolling and
sliding at the point of contact, it is called
a roll-slide contact also. For a planar
linkage, a higher pair restricts only one
degree of freedom, to ensure the contact
between two bodies and admits two
degrees of freedom
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Higher Pair
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Equivalent linkage continued
A and B are the centres of curvature of the
surfaces of corresponding surfaces at P
A general concept of cam
Equivalent linkage
It may be seen that for the higher pair, the motion is instantaneously transmitted along the common normal drawn
to the surfaces at the point of contact P. So the points A and B are located, A and B are jointed by links to points O2
and O4 respectively, points A and B are connected by a link. Thus we get the equivalent linkage as shown. This is a
4R 4bar linkage.
The equivalent linkage is, in general, valid for the instant, and changes from instant to instant, as the locations of
the points A and B change. However, if the locations of A and B do not change under some special situations, the
equivalent linkage is same for all instants.
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Equivalent linkage continued
The cam with a flat face follower is shown at an
instant.
Following the earlier way to draw the
kinematically equivalent lower pair mechanism,
1) we should first place a revolute pair at the
point C, the centre of curvature of the cam
surface and then
2) we should place another revolute pair at the
centre of curvature of the other surface, which
is straight and its centre of curvature is located
very far, i.e. at a distance tending to infinity.
3) Whenever such a situation comes, we should
put a slider at C with the revolute pair located
on the slider. The equivalent mechanism is
shown.
The cam follower mechanism and its lower pair
equivalent will have the same kinematic
characteristics at the instant shown.
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Cam with flat face follower
Its equivalent mechanism
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