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Introduction to Theory of Machines
Research · June 2015
DOI: 10.13140/RG.2.1.3347.4727
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Brown Hills College of Engineering & Technology
Kinematics of Machines
Introduction: mechanism and machines, kinematic links, kinematic pairs, kinematic chains, plane
and space mechanism, kinematic inversion, equivalent linkages, four link planar mechanisms,
mobility and range of movement, straight line mechanisms, steering mechanisms, pantograph,
problems.
 Introduction: The subject Theory of Machines may be defined as that branch of
Engineering-science, which deals with the study of relative motion between the various parts of
a machine, and forces which act on them. The knowledge of this subject is very essential for
an engineer in designing the various parts of a machine.
 Classification of Theory of Machines as following four branches:
1. Kinematics: It deals with the relative motion between the various parts of the machines.
2. Dynamics: It deals with the forces and their effects, while acting upon the machine parts in
motion.
3. Kinetics: It deals with the inertia forces which arise from the combined effect of the mass
and motion of the machine parts.
4. Statics: It deals with the forces and their effects while the machine parts are at rest. The
mass of the parts is assumed to be negligible.

Mechanisms is a combination of rigid or restraining parts or bodies from which the machine is
assembled, this is done by making one of the parts as fixed, and the relative motion of other
parts is determined with respect to the fixed part. Example, Slider-crank mechanism used in
internal combustion engine or reciprocating air compressor, where the rotary movement of the
crank is converted through the connecting rod into the reciprocating motion of the slider, or
vice-versa. Fig shows; Link-1 is fixed, Link-2 is Crank, Link-3 is Connecting rod and Link-4 is
piston which slides in a cylinder. However, the term linkage has been widely used as a
synonym for the word mechanism.
 Machine is a combination of the mechanisms which receives energy and transforms it into
some useful work from which we reduce the human efforts. A machine consists of a number of
parts or bodies.
 Kinematic Link
Each part of a machine, which moves relative to some other part, is known as a kinematic link.
A link may consist of several parts, which are rigidly fastened together, so that they do not
move relative with another part. For example, in a reciprocating steam engine, piston, piston
rod and crosshead constitute one link; connecting rod with big and small end bearings
constitute a second link; crank, crank shaft and flywheel a third link and the cylinder, engine
frame and main bearings a fourth link.
 Types of Links
1. Rigid link: A rigid link is one which does not undergo any deformation while transmitting
motion.
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Brown Hills College of Engineering & Technology
Kinematics of Machines
2. Flexible link: A flexible link is one which is partly deformed in a manner not to affect the
transmission of motion. For example, belts, ropes, chains and wires are flexible links and
transmit tensile forces only.
3. Fluid link: A fluid link is one which is formed by having a fluid in a container and the motion
is transmitted through the fluid by pressure or compression only, as in the case of hydraulic
presses, jacks and brakes.
 Kinematic Pair
The two links of a machine, when in contact with each other, are said to form a pair. If the
relative motion between them is completely or successfully constrained (i.e. in a definite
direction), the pair is known as kinematic pair.
 Classification of Kinematic Pairs
A. According to Nature of Relative Motion
1. Turning Pair / Revolute Pair: When the two elements of a pair are
connected in such a way that one can only turn or revolve about a
fixed axis of another link, the pair is known as turning pair. Turning
pair has a single degree of freedom.
2. Sliding Pair / Prismatic Pair: When the two elements of a pair are
connected in such a way that one can only slide relative to the other,
the pair is known as a sliding pair. Sliding pair has a single degree of
freedom.
3. Screw Pair: When the two elements of a pair are connected in such
a way that one element can turn about the other by screw threads,
the pair is known as screw pair. The lead screw of a lathe with nut,
and bolt with a nut are examples of a screw pair. Screw pair has a
single degree of freedom.
4. Cylindrical Pair: When the two elements of a pair are connected in
such a way that one element in rotation or translation, parallel to the
axis of rotation to the other element, the pair is known as cylindrical
pair. Cylindrical Pair has a two degree of freedom.
5. Rolling pair: When the two elements of a pair are connected in such a way that one roll
over another fixed link, the pair is known as rolling pair. Ball and roller bearings are
examples of rolling pair.
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Kinematics of Machines
6. Spherical pair: When the two elements of a pair are connected in such a way that one
element (with spherical shape) turns or pivots about the other fixed element, the pair
formed is called a spherical pair. The ball and socket joint, attachment of a car mirror, pen
stand etc., are the examples of a spherical pair.
7. Planar pair: it has a three degree of freedom. Two coordinates x and y describe the
relative translation in the xy-plane and the third  describe the relative rotation about the zaxis.
B. According to the type of contact between the links:
1. Lower pair: When the two elements of a pair have a surface or area contact when relative
motion takes place and the surface of one element slides over the surface of the other, the
pair formed is known as lower pair. It will be seen that sliding pairs, turning pairs, cylindrical
pairs, spherical pairs, planar pairs and screw pairs form lower pairs.
2. Higher pair: When the two elements of a pair have a line or point contact when relative
motion takes place and the motion between the two elements is partly turning and partly
sliding, then the pair is known as higher pair. A pair of friction discs, toothed gearing, belt
and rope drives, ball and roller bearings and cam and follower are the examples of higher
pairs.
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Kinematics of Machines
 Kinematic Chain
A kinematic chain is an assembly of links in which the relative motions of the links are possible
and the motion of each relative to the other is definite [Figs (a), (b), and (c)].
In case, the motion of a link results in indefinite
motions of other links, it is a non-kinematic chain
[Fig.d)].
However, some authors prefer to call all chains having
relative motions of the links as kinematic chains. A
redundant chain does not allow any motion of a link
relative to the other [Fig. (e)].
A kinematic chain is a series of links connected by kinematic pairs. The chain is said to be
closed if every link is connected to at least two other links shown in fig 1, otherwise it is
termed an open chain shown in fig 2.
fig 1
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fig 2
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Kinematics of Machines
A link which is connected to only one other link is known as a singular link. If it is connected
to two other links, it is called a binary link. Similarly, if a link is connected to three other
links, it is referred to as a ternary link, and so on.
 Plane Mechanisms
If all points of a mechanism move in parallel planes, then it is defined as a plane mechanism. A
simple plane mechanism is shown in Fig. where all points move in parallel planes.
 Space Mechanisms
A space mechanism is one in which all points of the mechanism do not move in parallel
planes. A very common example of a space mechanism, known as Hooke's joint, is shown in
Fig.
 Kinematic Inversion
This process of fixing different links of the same kinematic chain to produce distinct
mechanisms is called kinematic inversion. In this process, the relative motions of the links of
the mechanisms produced remain unchanged.
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Kinematics of Machines
A slider-crank chain mechanism has the following kinematic inversions:
1. First Inversion
This inversion is obtained when link 1 is fixed and links 2, 3 and 4 are made the crank,
connecting rod and the slider shown in Fig (a)
Applications: 1. Reciprocating engine, 2. Reciprocating compressor as shown in Fig. (b), if
it is a reciprocating engine, 4 (piston) is the driver and if it is a compressor, 2 (crank) is the
driver.
2. Second Inversion:
This inversion is obtained when link 2 is fixed and links 3, 4 and 1 are made the crank,
slider and connecting rod shown in Fig (a)
fig (a)
fig (b)
Applications: 1. Whitworth quick-return mechanism shown in Fig (b), 2. Rotary engine

Whitworth Quick-Return Mechanism:
It is a mechanism used in workshops to cut metals. The forward stroke takes a little longer
and cuts the metal whereas the return stroke is idle and takes a shorter period.
Working: Slider 4 rotates in a circle about A and slides on link 1 fig. (b). C is a point on link
1 extended backwards where link 5 is pivoted. The other end of link 5 is pivoted
to the tool, the forward stroke of which cuts the metal.
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Kinematics of Machines
The axis of motion of slider 6 (tool) passes through O and is perpendicular to OA,
the fixed link.
The crank 3 rotates in the counter-clockwise direction. Initially, let the slider 4 be
at B' so that C be at C'. Cutting tool 6 will be in the extreme left position. With the
movement of the crank, the slider traverses the path B'BB" whereas point C
moves through C'CC". Cutting tool 6 will have the forward stroke.
Finally, slider B assumes the position B" and cutting tool 6 is in the extreme right
position. The time taken for the forward stroke of slider 6 is proportional to the
obtuse angle B" AB' at A. Similarly, slider 4 completes the rest of the circle
through path B"B'" B' and C pass through C"C"'C'.
There is backward stroke of tool 6. The time taken in this is proportional to the
acute angle B"AB' at A.
3. Third Inversion
This inversion is obtained when link 3 is fixed and links 2, 4 and 1 are made the crank,
oscillates and connecting rod shown in Fig (a)
Applications: 1. Oscillating Cylinder Engine, 2. Crank and Slotted-Lever Mechanism
 Oscillating Cylinder Engine:
As shown in fig. (b), link 4 is made in the form of a cylinder and a piston is fixed to the end of
link 1. The piston reciprocates inside the cylinder pivoted to the fixed link 3. The
arrangement is known as oscillating cylinder engine, in which as the piston reciprocates in
the oscillating cylinder, the crank rotates.
4. Fourth Inversion
This inversion is obtained when link 4 is fixed and links 3, 2 and 1 are made the oscillates
about the fixed pivot B on link 4, oscillates about B and end 0 and link 1reciprocate along
the axis of the fixed link 4shown in Fig (a)
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Brown Hills College of Engineering & Technology
Kinematics of Machines
or
fig (a)
fig (b)
Application: Hand Pump Fig. (b) Shows a hand-pump. Link 4 is made in the form of a cylinder
and a plunger fixed to the link 1 reciprocates in it.
 Equivalent Mechanisms
It is possible to replace turning pairs of plane mechanisms by other type of pairs having one or
two degrees of freedom, such as sliding pairs or cam pairs. This can be done according to
some set rules so that the new mechanisms also have the same degrees of freedom and are
kinematically similar. This process is called equivalent mechanism.
Ref. Book – Theory of Machines by S.S.Raatan
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