Introduction to theory of machines

Chapter 01- Inversions of Mechanism
Yogesh Kumar Sharma
Assistant Professor
GNA University, Phagwara
11.08.2018 1

• The purpose of this course is to explore the topics
of kinematics and dynamics of machinery in
respect to the synthesis of mechanisms in order
to accomplish desired motions or tasks, and also
the analysis of mechanisms in order to
determine their rigid-body dynamic behavior.
• We will begin with careful definitions of the
terms used in these topics.
C.1-Identify Machine Elements, Mechanism and understand
their kinematics & Dynamics
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• 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.
Theory of Machines
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Theory of
Machines
Kinematics Dynamics
Statics
Kinetics
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Kinematics: The study of motion without consider forces
cause such motion.
Dynamics: The study of forces acting on various
machines along with their relative motion between each
other.
Statics: Branch of dynamics deal with the forces causing
on the various machine elements at rest. Here mass of
machine element is assume to be zero
Kinetics :The study of forces on systems in motion which
occur due to combined effect of mass and motion of
various machine elements.
These two concepts are really not physically separable.
One principal aim of kinematics is to create (design) the
desired motions of the subject mechanical parts and then
mathematically compute the positions, velocities, and
accelerations which those motions will create on the
parts.11.08.2018 5

Resistant Body: capable of transmitting required
type of motion, by the application of force on it,
with negligible deformation
LINK/ELEMENT: a resistant body or assemblage of
bodies fastened together rigidly of a machine, has
relative motion with some other part.
Link
Rigid Link
Flexible Link
Fluid Link
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KINEMATIC PAIRS
KINEMATIC
PAIRS
Types of
Contact
Lower Pair
Higher Pair
Types of
Constraint
Closed Pair
Unclosed Pair
Type of
relative motion
Sliding Pair
Rolling Pair
Turning Pair
Screw Pair
Spherical Pair11.08.2018 8

Degree of Freedom of a Kinematic Pair
No of independent relative motion, both linear
and rotational which a pair can have.
Unconstrained rigid body moving in space describe following motion:
• Linear or translation
• Rotation
Rigid body possesses 6 D.O.F. The linkage of a link with
another impose constraints.
Degree of freedom = 6 – Number of restraints
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A rigid body in a plane A rigid body in space
A planar
revolute pair
(R-pair)
A planar
prismatic pair
(P-pair)
A spherical
pair (S-pair)
A planar
pair (E-pair) A cylindrical
pair (C-pair)
A revolute
pair (R-pair)
A prismatic
pair (P-pair)
A screw pair
(H-pair)
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Constrained Motion
• Completely Constrained
Motion: some definite
direction only. Rectangular
rod in rectangular hole
• Incompletely
Constrained Motion:
more than one
direction. A circular
shaft moving either
rotate or slide11.08.2018 18

Constrained Motion
• Successfully Constrained
Motion: motion not
completed by itself but
by some other mean.
Piston inside cylinder, due
to rotation of crank
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Kinematic Chain
L= 2P – 4
Where, L= No. of
links; P= No. of Pairs
and L= 2/3(J+2)
If above two
equation satisfies
only then form
kinematic chain.
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21
KINEMATIC CHAIN CLOSED
Ground
Slider-crank
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22
KINEMATIC CHAIN OPEN
Ground
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Degree of freedom of a Mechanism
F = 3(n-1) -2p Kutzbach equation
Where F = Degree of freedom
N= no. of links
If F=1 most of the cases eqn become
2p-3n+4 =0 called Gruebler equation.
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24
DEGREES OF FREEDOM OF A PLANAR MECHANISM
1 2
1
2
Gruebler's equation for planar mechanisms
# DOF = 3(n -1) - 2f - f
n number of links
f number of lower pairs (1DOF)
f number of higher pairs with (2DOF)
A planar mechanism containing n links (including the ground link) has 3(n-1) degrees of freedom
before they are connected by pairs.
A lower pair has the effect of providing two constraints between the connected links. Therefore, f1
lower pairs will remove 2f1 degrees of freedom from the system.
A higher pair provides one constraint. So, f2 higher pairs will remove f2 degrees of freedom from the
system.
11.08.2018

25
4 bar linkage.
DEGREES OF FREEDOM: 4 BAR LINKAGE
1 2
1
2
# DOF = 3(n -1) - 2f - f
n = 4
f = 4
f = 0
# DOF = 3(4 -1) - 2 4 - 0 = 1
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26
DEGREES OF FREEDOM: 5 BAR LINKAGE
5 bar linkage.
1 2
1
2
# DOF = 3(n -1) - 2f - f
n = 5
f = 5
f = 0
# DOF = 3(5 -1) - 2 5 - 0 = 2
11.08.2018

27
DEGREES OF FREEDOM: CRANK AND SLIDER
crank mechanism.
1 2
1
2
# DOF = 3(n -1) - 2f - f
n = 4
f = 4
f = 0
# DOF = 3(4 -1) - 2 4 - 0 = 1
11.08.2018

Mechanisms
A mechanism is a device which transforms motion
to some desirable pattern and typically develops
very low forces and transmits little power.
When one of the link of a kinematic chain if
fixed called mechanism.
A simple mechanism has four links whereas
compound mechanism has more than four links
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Machines
A device by mean of which available
energy can be converted into desired
form of useful work.
A machine typically contains
mechanisms which are designed to
provide significant forces and transmit
significant power.
11.08.2018 32

A useful working definition of a
mechanism is A system of elements
arranged to transmit motion in a
predetermined fashion.
On the other hand, a machine is A
system of elements arranged to
transmit motion and energy in a
predetermined fashion.11.08.2018 33

Difference Between mechanisms & machines
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Some of the examples of mechanisms and
machines are:
Moves packages from an assembly
bench to a conveyor
Mechanisms:
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The most important kinematic chains are those
which consist of four lower pairs, each pair being a
sliding pair or a turning pair. The following three
types of kinematic chains with four lower pairs are
important from the subject point of view :
1. Four bar chain or quadric cyclic chain,
2. Single slider crank chain,
3. Double slider crank chain.
These kinematic chains are discussed, in detail, in the
following articles.
Inversion of Mechanism
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Kinematic Chain
Four bar chain ( all
tuning pair)
Coupled
Locomotive Wheels
Beam Engine
Pantograph
Watt’s Mechanism
Ackermann
Steering
Slider Crank Chain (
Three turning and
one sliding)
Pendulum Pump
Oscillating Cylinder
Engine
Crank & Slotted
Lever
Whitworth Quick
Return Mechanism
Rotary I.C. Engine
Double slider crank
chain( two turning
two sliding)
Scotch Yoke
Mechanism
Elliptical Trammel
Oldham’s Coupling
11.08.2018 37

Absolute motion is measured with respect to a
stationary frame. Relative motion is measured
for one point or link with respect to another link.
when one of links is fixed in a kinematic chain, it
is called a mechanism.
Inversions of Mechanism
11.08.2018 39

CONTDD..
We can obtain as many mechanisms as the
number of links in a kinematic chain by fixing, in
turn, different links in a kinematic chain. This
method of obtaining different mechanisms by
fixing different links in a kinematic chain, is
known as inversion of the mechanism
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Inversion of Kinematic Chain
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<
a) b) c) d)
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1.Four Bar Chain or Quadric Cycle Chain
We have already discussed that the kinematic chain is a
combination of four or more kinematic pairs, such that the
relative motion between the links or elements is completely
constrained.
The simplest and the basic kinematic chain is a four- bar chain
quadric cycle chain, as shown in Fig. 1.1
Fig. 1.1 Four bar chain
It consists of four links, each of them forms a turning pair at A, B,
C and D. The four links may be of different lengths.
11.08.2018 44

In a four bar chain, one of the links, in particular the
shortest link, will make a complete revolution relative to
the other three links, if it satisfies the Grashof ’s law. Such
a link is known as crank or driver.
In Fig. 1.1, AD (link 4 ) is a crank. The link BC (link 2) which
makes a partial rotation or oscillates is known as lever or
rocker or follower and the link CD (link 3) which connects
the crank and lever is called connecting rod or coupler.
The fixed link AB (link 1) is known as frame of the
mechanism.
When the crank (link 4) is the driver, the mechanism is
transforming rotary motion into oscillating motion.
11.08.2018 45

A very important consideration in designing a mechanism is to
ensure that the input crank makes a complete revolution relative to
the other links. The mechanism in which no link makes a complete
revolution will not be useful in such applications. For the four-bar
linkage there is a very simple test of whether this is the case.
Grashof’s law for four-bar mechanism.
If one of the links can perform a full rotation relative to another
link, the linkage is called a Grashof mechanism. Grashof’s law
states that for a planar four-bar mechanism (linkage), the sum of
the shortest and longest link lengths can not be greather than the
sum of the other two link lengths if there is to be continuous
relative motion between two members.
11.08.2018 46

B. Single Slider Crank Chain
A single slider crank chain is a modification of the basic four bar chain.
It consist of one sliding pair and three turning pairs. It is, usually, found
in reciprocating steam engine mechanism. This type of mechanism
converts rotary motion into reciprocating motion and vice versa.
Fig. 1.6. Single slider crank chain
11.08.2018 47

In a single slider crank chain, as shown in Fig. 1.38,
the links 1 and 2, links 2 and 3, and links 3 and 4
form three turning pairs while the links 4 and 1 form
a sliding pair.
The link 1 corresponds to the frame of the engine,
which is fixed. The link 2 corresponds to the crank ;
link 3 corresponds to the connecting rod and link 4
corresponds to cross-head. As the crank rotates, the
cross-head reciprocates in the guides and thus the
piston reciprocates in the cylinder.
11.08.2018 48

AI-Inversions of Four Bar Chain
Though there are many inversions of the four bar
chain, yet the following are important from the
subject point of view :
1. Beam engine (crank and lever(rocker)
mechanism).
2. Coupling rod of a locomotive (Double crank
mechanism).
3. Watt’s indicator mechanism (Double
lever(rocker) mechanism).
11.08.2018 54

1. Beam engine (crank and lever (rocker) mechanism
https://youtu.be/1e8ZWsOzun4
A part of the mechanism of a beam engine (also known as crank and lever
mechanism) which consists of four links, is shown in Figs. 1.34 and 1.35 .
Fig. 1.2 Beam engine Fig.1.3 Beam engine mechanism
In this mechanism, when the crank rotates about the fixed centre A, the lever
oscillates about a fixed centre D. The end E of the lever CDE is connected to a
piston rod which reciprocates due to the rotation of the crank. In other
words, the purpose of this mechanism is to convert rotary motion into
reciprocating motion.
https://youtu.be/bmh0a2Ol0u8
11.08.2018 55

2. Coupling rod of a locomotive (Double crank mechanism).
The mechanism of a coupling rod of a locomotive (also known as double crank
mechanism) which consists of four links, is shown in Fig. 1.36.
https://youtu.be/gYmT1M4NyyM
Fig. 1.4. Coupling rod of a locomotive
In this mechanism, the links AD and BC (having equal length) act as cranks and are
connected to the respective wheels. The link CD acts as a coupling rod and the link AB
is fixed in order to maintain a constant centre to centre distance between them.
This mechanism is used for transmitting rotary motion from one wheel to the other
wheel.
11.08.2018 57

3. Watt’s indicator (Double rocker) mechanism.
A Watt’s indicator mechanism (also known as Watt's straight line mechanism or double lever mechanism) which
consists of four links, is shown in Fig. 1.37.
Fig.1.5. Watt’s indicator mechanism
The four links are : fixed link at A, link AC, link CE and link BFD. It may be noted that BF and FD form
one link because these two parts have no relative motion between them. The links CE and BFD act as
levers. The displacement of the link BFD is directly proportional to the pressure of gas or steam which
acts on the indicator plunger. On any small displacement of the mechanism, the tracing point E at the
end of the link CE traces out approximately a straight line.
The initial position of the mechanism is shown in Fig. 1.37 by full lines whereas the dotted lines
show the position of the mechanism when the gas or steam pressure acts on the indicator plunger.
11.08.2018 59

Inversion of Slider Crank Mechanism
11.08.2018 60

3. Oscillating cylinder engine. The arrangement of oscillating cylinder engine
mechanism, as shown in Fig. 1.40, is used to convert reciprocating motion
into rotary motion. In this mechanism, the link 3 forming the turning pair is
fixed. The link 3 corresponds to the connecting rod of a reciprocating steam
engine mechanism. When the crank (link 2) rotates, the piston attached to
piston rod (link 1) reciprocates and the cylinder (link 4) oscillates about a pin
pivoted to the fixed link at A.
Fig.1.8 Oscilating cylinder engine
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11.08.2018 63

4. Rotary internal combustion engine or Gnome engine.
Sometimes back, rotary internal combustion engines
were used in aviation. But presently gas turbines are
used in its place. It consists of seven cylinders in one
plane and all revolves about fixed centre D, as shown in
Fig. 1.8, while the crank (link 2) is fixed. In this
mechanism, when the connecting rod (link 4) rotates,
the piston (link 3) reciprocates inside the cylinders
forming link 1.
11.08.2018 64

Fig. 1.9
11.08.2018 65

4. Crank and slotted lever quick return motion mechanism. This
mechanism is mostly used in shaping machines, slotting
machines and in rotary internal combustion engines.
Fig.1.10
11.08.2018 66

In this mechanism, the link AC (i.e. link 3) forming the turning
pair is fixed, as shown in Fig. 1.10. The link 3 corresponds to the
connecting rod of a reciprocating steam engine. The driving
crank CB revolves with uniform angular speed about the fixed
centre C. A sliding block attached to the crank pin at B slides
along the slotted bar AP and thus causes AP to oscillate about
the pivoted point A. A short link PR transmits the motion from AP
to the ram which carries the tool and reciprocates along the line
of stroke R1R2. The line of stroke of the ram (i.e. R1R2) is
perpendicular to AC produced.
11.08.2018 67

5. Whitworth quick return motion mechanism.
Fig. 1.11
Whitworth quick return motion mechanism
11.08.2018 68

This mechanism is mostly used in shaping and slotting machines. In this
mechanism, the link CD (link 2) forming the turning pair is fixed, as shown in
Fig. The link 2 corresponds to a crank in a reciprocating steam engine. The
driving crank CA (link 3) rotates at a uniform angular speed. The slider (link 4)
attached to the crank pin at A slides along the slotted bar PA (link 1) which
oscillates at a pivoted point D. The connecting rod PR carries the ram at R to
which a cutting tool is fixed. The motion of the tool is constrained along the
line RD produced, i.e. along a line passing through D and perpendicular to CD.
11.08.2018 69

When the driving crank CA moves from the position CA1 to CA2 (or
the link DP from the position DP1 to DP2) through an angle α in the
clockwise direction, the tool moves from the left hand end of its
stroke to the right hand end through a distance 2 PD. Now when the
driving crank moves from the position CA2 to CA1 (or the link DP from
DP2 to DP1 ) through an angle β in the clockwise direction, the tool
moves back from right hand end of its stroke to the left hand end.
A little consideration will show that the time taken during the left to
right movement of the ram (i.e. during forward or cutting stroke) will
be equal to the time taken by the driving crank to move from CA1 to
CA2. Similarly, the time taken during the right to left movement of the
ram (or during the idle or return stroke) will be equal to the time
taken by the driving crank to move from CA2 to CA1.
11.08.2018 70

Since the crank link CA rotates at uniform angular velocity
therefore time taken during the cutting stroke (or forward
stroke) is more than the time taken during the return
stroke. In other words, the mean speed of the ram during
cutting stroke is less than the mean speed during the return
stroke. The ratio between the time taken during the cutting
and return strokes is given by
Time of cutting stroke/ Time of return stroke
= α/ β =α/( 360 °- α) or = (360 °− β)/ β
Note. In order to find the length of effective stroke R1 R2,
mark P1 R1 = P2 R2 = PR. The length of effective stroke is
also equal to 2 PD.
11.08.2018 71

Double-Slider crank Chain
11.08.2018 72

C-Double Slider Crank Chain
A kinematic chain which consists of two turning pairs
and two sliding pairs is known as double slider crank
chain, as shown in Fig. 1.11 We see that the link 2 and
link 1 form one turning pair and link 2 and link 3 form
the second turning pair. The link 3 and link 4 form one
sliding pair and link 1 and link 4 form the second sliding
pair.
11.08.2018 73

C.Inversions of Double Slider Crank Chain
The following three inversions of a double slider crank chain
Are important from the subject point of view:
1. Elliptical trammels. It is an instrument used for
drawing ellipses. This inversion is obtained by fixing
the slotted plate (link 4), as shown in Fig. 1.11. The
fixed plate or link 4 has two straight grooves cut in
it, at right angles to each other. The link 1 and link
3, are known as sliders and form sliding pairs with
link 4. The link AB (link 2) is a bar which forms
turning pairs with links 1 a
11.08.2018 74

Fig.1.12 Eliptical trammels
11.08.2018 76

When the links 1 and 3 slide along their respective grooves, any
point on the link 2 such as P traces out an ellipse on the surface
of link 4, as shown in Fig. 1.12 (a). A little consideration will show
that AP and BP are the semi-major axis and semi-minor axis of
the ellipse respectively.
11.08.2018 77

2. Scotch yoke mechanism.
This mechanism is used for converting rotary
motion into a reciprocating motion. The
inversion is obtained by fixing either the link 1 or
link 3. In Fig. 1.45, link 1 is fixed. In this
mechanism, when the link 2 (which corresponds
to crank) rotates about B as centre, the link 4
(which corresponds to a frame) reciprocates.
The fixed link 1 guides the frame.
11.08.2018 78

Fig 1.13 Scotch yoke mechanism.
11.08.2018 80

3. Oldham’s coupling.
An Oldham's coupling is used for connecting two parallel shafts
whose axes are at a small distance apart. The shafts are coupled in
such a way that if one shaft rotates, the other shaft also rotates at
the same speed. This inversion is obtained by fixing the link 2, as
shown in Fig. 1.14
The link 1 and link 3 form turning pairs with link 2. These flanges
have diametrical slots cut in their inner faces, as shown in Fig. 1.46
(a). The intermediate piece (link 4) which is a circular disc, have two
tongues (i.e. diametrical projections) T1 and T2 on each face at right
angles to each other, as shown in Fig. 1.46 (b). The tongues on the
link 4 closely fit into the slots in the two flanges (link 1 and link 3).
The link 4 can slide or reciprocate in the slots in the flanges.
11.08.2018 81

Fig. 1.14 Oldham’s coupling.
11.08.2018 83

Introduction to theory of machines

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