4 bar mechanism and its inversions

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Kinematics of Machinery - Unit - I
KINEMATICS OF MECHINERY
UNIT-1
Simple Mechanisms
Dr. M Krishna
Professor and Dean
Research and Development

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Unit-1 Simple Mechanisms
 Definition of link, Pair, Kinematic chain, mechanism,
machine, inversion, structure
 Type of motion: Constrained, unconstrained and
successfully constrained motion
 Degree of freedom- Grubler’s criterion for mobility of
mechanism
 Grashof’s criterion, Inversions of 4 bar chain , single
slider crank chain and double slider crank chain

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Theory of Machines
ToM deals with the study of relative
motion between the parts of the
machine and the forces which act on
those parts
The study of the relative motion
between the parts without
considering force
The study of forces which act on the parts
It deals with the forces which act on the various
parts when the mass of the parts is not considered
The inertia forces due to the combined effect of
mass and the motion of the machine parts
considered

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Links or Elements
A link is a part of machine which has relative motion with respect to
another machine parts
Classification of Links
 Rigid link : it does not undergo any deformation while
transmitting motion.
 Flexible Link: it is partly deformed while transmitting
motion (eg. Belt, rope, etc.)
 Fluid Link : it is the one in which the motion is
transmitted through the fluid by pressure or
compression. (eg. Hydraulic, pneumatic)

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Rigid Links
A link is a part of machine which has relative motion with respect to
another machine parts. It may consist one or more resistant bodies
Characteristics of the links
 It should have relative motion .
 It must be a resistance body
 Binary link
 Ternary link
 Quaternary link
Note: Spring is not link

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KINEMATIC PAIRS
 Kinematic pair is a combination of two kinematic
links such that their relative motion is completely
constrained
Classification based on nature of contact between elements
(1) Lower pair : The joint by which two members are connected
has surface contact (area contact). Its six types are

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TYPES OF LOWER PAIRS

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(ii) Higher pair: The contact between the pairing elements takes
place at a point or along a line (gear, cam, etc.)
(ii) Wraping pair: is a partial constraint that
comprises belts, chains, and such other devices.
Wrapped pair is very similar to the higher pair (which is
having point or line contact), but having multiple point contact

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KINEMATIC CHAIN
Group of links either joined together or arranged in a
manner that permits them to move relative to one
another.
Relation between Links (L), Pairs(P) and Joints(J)
L=2P-4
J=(3/2) L – 2
L => No of Links
P => No of Pairs
J => No of Joints
L.H.S > R.H.S => Locked chain
L.H.S = R.H.S => Constrained Kinematic Chain
L.H.S < R.H.S => Unconstrained Kinematic
Chain

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KINEMATIC CHAIN
Typ3 Chain From Fig (LHS) Computed (RHS)
L J P L=2P-4 J=(3/2)P-2 Type
3 3 3 2 1
LHS>RHS
Locked chain or
Structure
4 4 4 4 4
LHS=RHS
Constrained
Kinematic Chain
4 4 4 4 4
LHS=RHS
Constrained
Kinematic Chain
5 5 5 6 7
LHS<RHS
Unconstrained
Kinematic Chain

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KINEMATIC MECHANISM
 When one of the links of the kinematic chain is fixed , the
chain is know as Mechanism .
 It is used to transmitting motion from one link to another .
 There are two types of Mechanism :
1) Simple Mechanism (having only four links)
2) Compound Mechanism(having more than four links)
Four Bar Mechanism Seven Bar Mechanism

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MACHINE
A machine is a mechanism or collection of mechanisms,
which transmit force from the source of power to the
resistance to be overcome.
Though all machines are mechanisms, all
mechanisms are not machines
Mechanisms: a piece of machinery or a mechanical part
or group of parts having a particular function

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MACHINE /Mechanism
Machine Mechanism
1 Transforming mechanical energy Only transform motion without
considering forces / energy
2 An assemblage of parts that transmit
forces, motion and energy in a
predetermined manner.
Responsible for an action,
reaction or other natural
phenomenon.
3 A machine can use one or more
mechanism to perform the desired
function,
A mechanism is a single system to
transfer or transform motion
4 e.g. sewing machine has several
mechanisms
Pulley a`nd belt drive
Similarity
- They are both combinations of rigid bodies
- The relative motion among the rigid bodies are definite

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Types of motion: Movement
Motion type Example
1
Planer motion: A body has plane motion, if all its
points move in planes which are parallel to some reference
plane
1) Translation
2) Rotation
3) Combination of
translation and rotation.
1(a) Translation: A body has translation if it moves so that all
lines in the body move to parallel positions.
1) Rectilinear translation
2) Curved linear Motion
 Rectilinear translation is a motion wherein all points
of the body move in straight line paths
 Curved linear Motion: is a motion wherein all points
of the body moves in curve linear path
 Translation and rotational: It is the combination of
both translation and rotation
Car travels strait line
Stone travels in space
Tyre rolling down the hill
1(b) Rotational motion All points in a body remain at fixed
distances from a line which is perpendicular to the plane
of rotation.
Gear rotation
2 Special Motion: a body has special motion, if its
points move in space
Motion of jet flight

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Constrained motion: In a kinematic pair, if one element has got only one
definite motion relative to the other
Types of motion: Constraints
Motion type Example
1 Completely constrained motion: If the constrained
motion is achieved by the pairing elements themselves
2 Successfully constrained motion: If constrained
motion is not achieved by the pairing elements
themselves, but by some other means
3 Incompletely constrained motion: When relative motion
between pairing elements takes place in more than one
direction

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Types of motion: Mech. constraints
Motion type Example
1 Closed Pair: Elements of pairs held together mechanically due
to their geometry constitutes a closed pair.
2 Unclosed or force closed pair: Elements of pairs held
together by the action of external forces constitute unclosed or
force closed pair

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DEGREES OF FREEDOM (DOF):
It is the number of independent coordinates required to
describe the position of a body.
Freedom / mobility of a mechanism

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GRUBLER’S CRITERION
Number of degrees of freedom of a mechanism is given by
F = 3(n-1)-2J1 – J2. Where,
 F = Degrees of freedom
 n = Number of links in the mechanism.
 J1 = Number of lower pairs, which is obtained by counting the
number of joints. If more than two links are joined together at any
point, then, one additional lower pair is to be considered for every
additional link.
 J2 = Number of higher pairs

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Examples – Problems
Problem N J1 J2 F=3(n-1)-2J1-
J2
Remark
4 4 0 = 3(4-1)-2(4) =
1
One input to any
one link will result
in definite motion
of all the links.
5 5 0 = 3(5-1)-2(5)
= 2
two inputs to any
two links are
required to yield
definite motions in
all the links.

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J2
Remark
6 7 0 = 3(6-1)-2(7) =
1
One input to any
in definite motion
of all the links.
6 7 0 = 3(6-1)-2(7)
= 1
One input to any
in definite motion
of all the links
At the intersection of 2, 3 and 4, two
lower pairs are to be considered

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J2
Remark
11 15 0 = 3(11-1)-2 (15)
= 0
No motion (rigid
structure
Two lower pairs at the intersection of (3,
4, 6), (2, 4, 5), (5, 7, 8) and (8, 10, 11)
4 5 0 = 3(4-1)-2 (5)
= -1
It neither structure
nor mechanism
Two lower pairs at the intersection of (2,
3 and 4)
1 1
1
2
3
4

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J2
Remark
3 2 1 = 3(3-1)-2(2)-1
= 1
One input to any
in definite motion
of all the links
3 2 1 = 3(3-1)-2(2)-1
= 1
One input to any
in definite motion
of all the links
1
1
2
3

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J2
Remark
7 8 1 = 3(7-1)-2(8)-1
= 1
One input to any
in definite motion
of all the links
8 10 0 = 3(8-1)-2(10)-0
= 1
One input to any
in definite motion
of all the links

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J2
Remark
6 7 0 = 3(6-1)-2(7)-0
= 1
One input to any
in definite motion
of all the links

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Grashoff Law
 In a four mechanism, the sum of
the shortest and longest link
length should not be greater than
the sum of the remaining two
link lengths if there is to be
continuous relative motion
beween the to links
s+l < p+q
(e.x) (1+2) < (3+4)

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Type of Kinematic Chains
The kinematic chains are those which consists of four lower pairs,
each pair being a sliding pair or turning pair .
The following three types of kinematic chain are
01 Four bar chain or quadric cyclic
Chain
02 Single slider crank Chain
03 Double slider crank chain

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4-bar Mechanism Nominaclature
1. Gound link (Fixed rigidily to the ground)
2. Crank (rotates about the fixed link)
3. Coupler (ossilates about the two joints)
4. Rocker (Rocks about the fixed link)
4
2 4

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INVERSIONS OF MECHANISM
Different mechanisms can be obtained by fixing different links
of the same kinematic chain. These are called as inversions of
the mechanism.
 The number of inversion is equal to the number of link in the
kinematic chain
1) Inversion of four bar chain
2) Inversion of single slider crank chain
3) Inversion of double slider chain
.

1.Four Bar Chain - Inversions
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1
2
3
4
Crank
Rocker
FFour bar kinematic chain and equivalent Mechanism
The four bar chain has four turning pairs.
 This is simplest types of
kinematic chain in which four
rigid links are connected by four
pin joints.
 The four bar chain has four
turning pairs.
 Link 1 & 2 forms first turning
pair, link 2 & 3 forms second
turning pair, link 3 & 4 forms
third turning pair, link 4 & 1
forms fourth turning pair.
 A link which can make complete
revolution is known as crank.
 The fixed link is known as frame.

1.Four Bar Chain - Inversions
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1
2
3
4
Crank
Rocker
FFour bar kinematic chain and equivalent Mechanism
The four bar chain has four turning pairs.
 The opposite link of frame is
known as connecting rod.
 The oscillating fourth link is
known as lever or rocker.

Inversion-1(Crank and lever Mechanism)
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Crank-rocker : Shortest link is the crank. Frame is adjacent
1
2
3
4
Crank
Rocker
Four bar kinematic chain and equivalent Mechanism
Link 1 is fixed (adjacent to shorter
link), link 2 act as a crank and
link 4 act as a lever and link 3 as
a follower

Inversion-1 (Crank and lever Mechanism)
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1
2
3
4
Crank
Rocker
Link 1 is fixed, link 2 act as a
crank and link 4 act as a lever
and link 3 as a follower
Eg. Beam engine

Inversion-1 (Crank and lever Mechanism)
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Eg. Beam engine
 The function of this mechanism is
to convert rotary motion of crank
into oscillatory motion of lever and
finally into reciprocating motion of
piston.
 Here, crank AB rotates about fixed
centre A, lever CDE rotates about
centre D and at end of lever E,
additional linkages i.e. piston rod
and piston is connected, which
reciprocates the cylinder.
Beam Engine

Inversion-2 (Double Crank Mechanism)
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Double crank mechanism
Double Crank or drag link: Shortest link is the frame.
1
2
3
4
Crank
Rocker
Fig. 24 Four bar kinematic chain and equivalent Mechanism
 If the links 1 and 3 are
made of unequal length in
four bar chain, we get
drag link mechanism
 Link 2 is fixed and link 4
represents coupling rod
which connects two rods

Inversion-2 (Double Crank Mechanism)
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Double Crank or drag link: Shortest link is the frame.
 If the links 1 and 3 are made of equal length in four bar
chain, we get double crank quadric chain which is used to
transmit the motion from one wheel to another of a
locomotive.
 Link 2 is fixed to maintain fixed centre distance between the
wheels and link 4 represents coupling rod which connects
two rods and is parallel to fixed link.
 The mechanism is used to transmit the rotary motion of a
wheel which acts as driver to another wheel mounted.
3
4
1
2 2
Double crank mechanism

Inverse-3 (Double lever mechanism)
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1
3
2
Fig. 28 : Double-lever Mechanism
1
2
3
4
Crank
Rocker
Rocker-Crank : The shortest link is the follower and opposite link is the frame
 Link 4 is fixed, link 1 and link 3 act as
levers.

Inverse-3 (Double lever mechanism)
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Rocker-Crank : The shortest link is the follower and opposite link is the frame
1. Link 4 is fixed, link 1 is ABC and
link 2 is CDP, both act as levers.
2. Link BE and link ED form link 4
since no relative motion between
them.
3. The point E is connected to
piston of indicator cylinder i.e.
displacement is proportional to
pressure of the gas.

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Grashoff Law

2.Single Slider Crank
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Single Slider Cracnk
Crank
Slider
2
3
4
=
=
1
2
3
4
Three revolute pairs and one prismatic pair (sliding pair)
Fixing 1, 2, 3 and 4 in turn, we get four different mechanisms.
1
2 3
4

2.Single Slider Crank _ Inversions-1
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Inversion of Slider Crank Chain
Crank
Slider
1
2
3
4
=
=
Fixed link
1
2
3
4
Eg. IC engine and compression
The cylinder and the frame is fixed (link-1). Link 2 is the crank, link 3 is the
connecting rod and link 4 piston.
Reciprocating engine mechanism

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Fixed link
Second Inversion of Slider Crank Chain
Crank
Slider
1
2
3
4
=
1
2
3
4
Crank
With word quick return mechanism
Fix the link-2 fixed
The link 2 is fixed,1 and 3 act as cranks
Quick-return Mechanism is obtained.

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Third Inversion of Slider Crank Chain
Crank
Slider
1
2
3
4
=
1
2
3
4
Link – 3 fixed
The link 3 is fixed,1 acts as a connecting rod and 2 crank shaft
Ocillating-cylinder engine mechanism obtained

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Link -4 fixed

3.Double Slider Crank- INVERSIONS
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There are four type of inversion of double slider
crank chain mechanism as:
[a] Elliptical trammel
[b] Scotch yoke mechanism
[c] Oldham’s coupling
[d] Donkey Pump

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1-Ellipitical Trammel
 Two slots are cut at right angles in a plane which is fixed (link-4)
 Two blocks slide in these two slots (link-1 and link-3)
 Two sliding blocks are connected by one link (link-2).
 Point on this link at anywhere generate ellipse, when any blocks slides in
its slot

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1
sin
cos 2
2
2
2






















p
y
q
x
1
sin
cos 2
2
2
2






















p
y
q
x
Elliptical trammel
AC = p and BC = q, then,
x = q.cosθ and y = p.sinθ.
Rearranging,
1-Ellipitical Trammel

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 This mechanism converts rotary motion to reciprocating motion (simple
harmonic motion).
 It is used in vibration exciter to produce vibration of desired frequency.
 This Chain has two revolute pairs and two prismatic pairs.
 The first inversion is with a link with revolute pair and prismatic pair is
fixed.
 Displacement = x = OA cos q
2-Scotch Yoke Mechanism

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 When the distance between two shaft axes is small, motion is transmitted
by oldham’s coupling.
 At the end of both the shafts there is flanges.
 This flange forms links 2 and 4.
 Both this links forms a turning pair with link 3 (Fixed).
 There is a diametrical slot on both the flange.
 An intermediate piece have a diametrical outside tongues at right angle
on both sides.
Oldham’s Coupling

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 When the driving shaft is rotates, the intermediate piece rotates as well
as sides in groove.
 This intermediate piece also rotates the driven shaft by sliding as well as
rotating motion.
Old ham’s Mechanism

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4 bar mechanism and its inversions

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