Unit I Fundamentals of Mechanism
Kinematic link, Types of links, Kinematic pair, Types of
constrained motions, Types of Kinematic pairs, Kinematic
chain, Types of joints, Mechanism, Machine, Degree of
freedom, Mobility of -Bar Chain and its Inversions, Slider
crank Chain and its Inversions, Double slider crank Chain and
its Conversions, Mechanisms with Higher pairs, Equivalent
Linkages and its Cases - Sliding Pairs in Place of Turning
Pairs, Spring in Place of Turning Pairs, Cam Pair in Place of
Turning Pairs
Machines are devices used to accomplish work. A mechanism is the heart of a
machine. It is the mechanical portion of a machine that has the function of
transferring motion and forces from a power source to an output.
Mechanisms are assemblage of rigid members (links) connected together by
joints (also referred to as Mechanical linkage or linkage).
KINEMATICS OF MACHINERY
• Deals with relative motion between the
various parts of machine, and forces acting
on them
The design of a mechanical system needs a
proper understanding of
• The geometrical aspects of motion
• The forces involved in motion
THEORY OF MACHINE AND MECHANISMS
• Analysis –
• Study of motions & forces of different parts of
mechanism
• Synthesis -
• Design of various parts of machine regarding its
shape, size, materials & arrangement of parts
CLASSIFICATION
Theory of machine and mechanisms
Kinematics (Study from
geometric point of view) Dynamics (Involves
(Relative motion bet various force & their effects)
parts)
Statics (Study of forces Kinetics (Study of
when the body is inertia forces when
stationary) body is in motion)
Fundamentals of Kinematics & Mechanisms
• Machine –
• consists the various elements arranged together
so as to perform the prescribed task.
Rigid and Resistant body
• A body is said to be rigid if under the action of
forces, it does not suffer any distortion or the
distance between any two points on it remains
constant.
• Resistant body is one which transmits force with
negligible deformation in the direction of force
transmission
Kinematic chain
Machine
Fundamentals of Kinematics & Mechanisms
• Link-
• A resistant body , which is the part of the
machine and has a motion relative to other
connected parts.
• Types –
1. Rigid link
2. Flexible link
3. Fluid link
Fundamentals of Kinematics & Mechanisms
1. Rigid link
• Do not undergo any deformation
eg; piston, connecting rod, crank
Connecting rod
Fundamentals of Kinematics & Mechanisms
2. Flexible link
• Partly deformed
eg; chains, belts, springs
Fundamentals of Kinematics & Mechanisms
3. Fluid link
• Motion transmission by fluid pressure (fully
deformed)
eg; liquid used in hydraulic press, jacks & brakes
of an automobile
Fundamentals of Kinematics & Mechanisms
• Types of Rigid links-
1. Binary link
2. Ternary link
3. Quarternary link
MACHINE
• A machine is a mechanism or a combination of
mechanisms which, transmits force from the
source of power to the resistance (load) to be
overcome, and thus performs useful work.
STRUCTURE
• An assemblage of resistant bodies, having no
relative motion between them and meant for
carrying loads having straining action, is called a
structure.
• Examples : Roof trusses, bridges, buildings
Differentiate between a machine
& Structure
MACHINE STRUCTURE
1) A machine is a mechanism or a 1) An assemblage of resistant
combination of mechanisms bodies, having no relative motion
which, transmits force from the between them and meant for
source of power to the resistance carrying loads having straining
(load) to be overcome, and thus action, is called a structure.
performs useful work.
2) Relative motion exists between its 2) No relative motion exists between
parts. its members
3)Machine serves to modify and 3) Structure serves to modify and
transmit forces & motion. transmit forces only.
4) Examples : Shaper, lathe. 4) Examples : Roof trusses, bridges,
buildings.
TYPES OF CONSTRAINED
MOTIONS
• Completely constrained motion
• Incompletely constrained motion
• Successfully constrained motion
• https://www.youtube.com/watch?v=RMBsBRW
X7QE
COMPLETELY CONSTRAINED
MOTION
• When the motion between a pair is limited to a
definite direction irrespective of the direction of
force applied, then the motion is said to be a
completely constrained motion.
INCOMPLETELY CONSTRAINED
MOTION
• When the motion between a pair can take place
in more than one direction, then the motion is
called an incompletely constrained motion.
SUCCESSFULLY CONSTRAINED
MOTION
• When the relative motion between the links is
not completely constrained by itself but by some
other means, then the motion is said to be
successfully constrained motion. e.g. Piston
reciprocates inside an engine cylinder.
KINEMATIC PAIR
The two links or elements of a machine, when in
contact with each other, are said to form a pair.
If the relative motion between them (pair) is
completely or successfully constrained (i.e. in a
definite direction), the pair is known as
kinematic pair.
CLASSIFICATION OF KINEMATIC PAIRS
The kinematic pairs are classified based on the following considerations
https://www.youtube.com/watch?v=cU1PLmkjwIg
• (a) According to nature of relative motion between the contacting
surfaces
(i) Sliding pair or prismatic pair
(ii) Turning or revolute pair
(iii) Rolling pair
(iv) Screw or helical pair
(v) Spherical or globular pair
• (b) According to nature of contact between the contacting surfaces
(i) Lower pair
(ii) Higher pair
• (c) According to nature of constraint between the contacting surfaces
(i) Closed pair
(ii) Unclosed pair
SLIDING PAIR
• If two links have a sliding motion relative to each
other, they form a sliding pair.
• Eg:
▫ A rectangular rod in a rectangular hole in a prism
▫ The piston and cylinder,
▫ ram and its guides in shaper,
▫ tail stock on the lathe bed
• sliding pair has a
completely constrained motion.
TURNING OR REVOLUTE PAIR
• When one link has a turning or revolving motion relative
to the other, constitute a turning or revolving pair
ROLLING PAIR
• When the links of a pair have a rolling motion
relative to each other, they form a rolling pair,
• e.g. A rolling wheel on a flat surface,
Belt drive,
Wheels of rail,
ball and roller bearings
SCREW OR HELICAL 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.
SPHERICAL PAIR
• When one link in the form of a sphere turns
inside a fixed link, it is a spherical pair
• The ball and socket joint is a spherical pair
According the class of pair
CLASSIFICATION OF KINEMATIC
PAIRS
• (b) According to nature of contact between the
contacting surfaces
(i) Lower pair
(ii) Higher pair
LOWER PAIR
• When the two elements of a pair have a surface
contact when relative motion takes place and
the surface of one element slides over the
surface of the other, the pair formed is a lower
pair.
• It will be seen that sliding pairs, turning pairs
and screw pairs form lower pairs.
LOWER PAIRS
• NUT AND BOLT
HIGHER PAIR
• When the two elements of a pair have a line or
point contact when relative motion takes place,
it is known as higher pair.
• Eg:
▫ A pair of friction discs,
▫ toothed gearing,
▫ belt and rope drives,
▫ ball and bearings and
▫ cam and follower
• BALL BEARING
• GEAR
CLASSIFICATION OF KINEMATIC
PAIRS
(c) According to nature of constraint between
the contacting surfaces
(i) Self-Closed pair
(ii) Force-closed pair
SELF-CLOSED PAIR
• When the elements of a pair are held together
mechanically, it is known as a closed pair.
• All the lower pairs and some of the higher pairs are
closed pairs.
FORCE-CLOSED PAIR
• When two links of a pair are in contact either
due to force of gravity or some spring action,
they constitute an unclosed pair.
• In this, the links are not held together
mechanically, e.g. cam and follower pair
• CAM AND FOLLOWER
KINEMATIC CHAIN
• A kinematic chain is an assembly of links which are
interconnected through pairs, permitting relative motion
between the links
• https://www.youtube.com/watch?v=KBFFwgCCP0U
KINEMATIC CHAIN
• Conditions to form a kinematic chain
n 2 p 4..............(1)
• 3
j n 2.................(2)
2
H 3
j n 2.................(3)
2 2
Where;
n- number of links p-no. of lower pairs
j-number of joints H-no. of higher pairs
KINEMATIC CHAIN
• A W Klien’s criteria of constraint
•
L.H .S . R.H .S
• ….chain is locked/structure
L.H .S. R.H .S.
• ……chain is constrained
• L.H .S. R.H .S.
• ……..chain is unconstrained
•
TYPES OF JOINTS IN KINEMATIC CHAIN
• Binary Joint
• Ternary joint
• Quaternary joint
BINARY JOINT
• When two links are
joined at the same
connection, the joint is
known binary joint.
• For example, a chain as
shown in Fig. has four
links and four binary
joins at A,B,C,D
• When three links are joined at the TERNARY JOINT
same connection, the joint is known as
ternary joint.
• It is equivalent to two binary joints as
one of the three links joined carry the
pin for the other two links.
• For example, a chain, as shown in Fig,
has six links. It has three binary joints
at A, B and D and two ternary joints at
C and E.
No. of binary joints 2(No. of ternary joints) No. of binary joints
• When four links are joined at the QUATERNARY JOINT
same connection, the joint is called a
quaternary joint.
• It is equivalent to three binary joints.
In general, when l number of links
are joined at the same connection,
the joint is equivalent to (l — 1)
binary joints.
• Fig. has two binary joint at F,C two
ternary joints at A, D, and, and two
quaternary joints at B and E.
No. of joints 3(No. of quartenary joints) 2(No. of ternary joints)
No. of binary joints
• Check whether the following configurations are
kinematic chain or not.
1)
2)
3)
KINEMATIC CHAIN
1. Four bar chain (4R)
https://www.youtube.com/watch?v=KBFFwgCCP
0U
KINEMATIC CHAIN
2. Single slider crank chain (3R-1P)
https://www.youtube.com/watch?v=lLHMoRem
mMg
KINEMATIC CHAIN
Double slider crank chain (2R-2P)
https://www.youtube.com/watch?v=-
waxgJT3Kh0
INVERSION
• When one of links is fixed in a kinematic chain, it
is called a mechanism.
• So 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.
INVERSIONS OF FOUR BAR MECHANISM
• Crank and Lever Mechanism (First
Inversion) (Beam engine mechanism)
• Coupling rod of a locomotive (Double
crank mechanism)
• Watt’s indicator mechanism (Double
lever mechanism).
CRANK & LEVER MECHANISM
BEAM ENGINE (FIRST INVERSION)
• Beam Engine Mechanism
is the most popular
example of crank and
lever mechanism. The
purpose of this
mechanism is to convert
rotary motion into
reciprocating motion.
ENGINE ANIMATIONBEAM
• https://www.youtube.com/watch?v=lcJN2qY1g
RE
Double crank Mechanism
https://www.youtube.
com/watch?v=oTcC_x
XfdrA
• This mechanism is meant
for transmitting rotary
motion from one wheel to
the other wheel.
• Coupled wheels of
locomotive is the example of
double crank mechanism.
Watt’s indicator mechanism (Double lever
mechanism).
https://www.youtube.com/watch?v=xsh5H
654llg
Grashof’s Law
• For a planar four-bar linkage, sum of the shortest and
longest link-lengths must be less than or equal to the
sum of the remaining two link-lengths, if there is to be a
continuous relative rotation between two members.
s+l≤ p+q
Types of Mechanisms for 3 different conditions of
Grashof’s Law
1] Class – I (s + l < p + q) :-
a) Grashoffian Four Bar Linkage /
Double crank
Link s - fixed
Link l & q – crank
Link p – coupler
a) Crank & rocker /
Rotary oscillating converter mechanism
Link l - fixed
Link s – crank
Link p – rocker
Link q - coupler
Types of Mechanisms for 3 different conditions of
Grashof’s Law
c) Rocker rocker mechanism/ Double lever
Link p - fixed
Link l & q – rocker
Link s – coupler
Types of Mechanisms for 3 different conditions of
Grashof’s Law
2] Class – II (s + l > p + q) :-
a) Non-Grashoffian Four Bar Linkage
Double rocker / Rocker-Rocker / Double Lever
Types of Mechanisms for 3 different conditions of
Grashof’s Law
3] Special Cases of Four Bar Linkage (s + l = p + q) :-
a) Parallel crank four Bar Linkage
Rotary-Rotary converter
b) Deltoid four Bar Linkage
INVERSIONS OF SINGLE SLIDER
MECHANISMS
• It consists of one sliding pair and three turning
pair. It is usually found in reciprocating steam
engine mechanism. This mechanism converts
rotary motion into reciprocating motion
Reciprocating Engine Mechanism/Bull Engine
(First Inversion)
• It converts rotary motion
into reciprocating and
vice-versa.
• This mechanism has four
links and forming three
turning pairs and one
sliding pair.
Link 1 - Cylinder and
Frame
(Fixed)
Link 2 – Crank
Link 3 - Connecting rod
Link 4 - Piston or Slider
A) Rotary engine (Second inversion)
link 1- Cylinder link
link2- Fixed Link
link 3 - Connecting Rod
link 4 - Piston ,
B)Withworth quick return mechanism
(Second inversion)
A)Oscillating Cylinder Engine Mechanism (Third
Inversion)
• Link 1 - Piston & Piston Rod
• Link 2 - Crank
• Link 3 - Fixed Link
• Link 4 – Cylinder
• In this mechanism, the link 3
forming the turning pair is fixed.
The link 3 corresponds to the
connecting rod of a
reciprocating engine
mechanism.
• When the crank (link 2) rotates,
the piston attached to the
piston rod (link 1) reciprocates
and cylinder (link 4) oscillates
about a pin pivoted to the fixed
link (link 3) at A.
B)Crank and Slotted Lever Mechanism (Third Inversion)
• This mechanism is mostly used in
shaping machines and slotting
machines. This is a Quick return
motion mechanism.
• The slider (1) reciprocates in
oscillating slotted lever (4) and
crank (2) rotates while link 3 is a
stationary link. Another link 5,
connects the end of link 4 to the
ram (6).
• Stroke = (length of slotted bar ×
length of crank ) / length of fixed
link
ANIMATION OF QUICK RETURN MOTION
MECHANISM
Hand pump (fourth inversion
Double slider crank chain
• A kinematic chain which
consists of two turning pairs
and two sliding pairs is known
as double slider crank chain,
• INVERSIONS:-
• 1)Elliptical Trammel
• 2)scotch-yoke mechanism
• 3)Oldham coupling
ELLIPTICAL TRAMMEL
• Here, the slotted frame is fixed.
Any point, such as S on the link
BC will trace out an ellipse as the
blocks B and C slide along their
respective slots.
• It is an instrument for drawing
ellipses. Clearly, the CS and BS
are respectively the semi-major
and semi-minor axis.
• link 1 - Slider
• link 2 - Connecting Link
• link 3 – Slider
• link 4 - Slotted frame (fixed link)
ANIMATION OF ELLIPTICAL TRAMMEL
SCOTCH YOKE MECHANISM
link 1 – Slider (fixed link) ,
link 2 - Connecting link ,
link 3 - Slider link ,
link4 – Slotted Frame
• Here, one of the two slide blocks i.e.
either link 1 or link 3, is kept fixed. In
such an arrangement, the whole frame
i.e. link 4 will reciprocate as seen in
Scotch-Yoke mechanism.
• Here, rotary motion of link 2 is
converted to reciprocating motion to
link 4.
OLDHAM COUPLING (THIRD INVERSION)
• Here the Connecting link in the
basic configuration is fixed.
• Maximum sliding speed of each
tongue is
Vs = w. r
W = angular velocity of each shaft
r = distance between the axes of
shafts
Animation of Oldham coupling
DEGREE OF FREEDOM
• Degrees of freedom of a pair is
defined as the number of
independent co-ordinates required to
define the position & orientation of a
body.
• Point in space have 3 DOF
DEGREE OF FREEDOM
• Degrees of freedom of a pair is
defined as the number of
independent relative motions, both
translational and rotational, a pair
can have.
• For spatial mechanisms
Degrees of Freedom = 6 - (Number
of Constraints)
• For planar mechanisms :-
Degrees of Freedom = 3 - (Number
of Constraints)
DEGREE OF FREEDOM
• For spatial mechanisms
The complete motion cannot be
represented by single plane (3D
motion paths)
• For planar mechanisms :-
The complete motion can be
represented by single plane
Mobility
• The number of independent input parameters that are to
be controlled so that a mechanism can take up a
particular position
• The number of inputs required to produce the
constrained motion of a mechanism is called DOF. So if
only one input is required to produce the constrained
motion of a mechanism, then DOF is 1.
• Classification based on no. of restraints:-
• Zero order mechanism - No restraint
• First order mechanism - No restraint
• Second order mechanism - No restraint
• Third order mechanism - No restraint
• Fourth order mechanism - No restraint
• Fifth order mechanism - No restraint
DOF OF PLANAR MECHANISMS
• It can be stated that an
unconstrained rigid link in
the plane has three degrees
of freedom.
KUTZBACH CRITERIA
• Since in a mechanism, one of the links is to be fixed,
therefore the number of movable links will be
(n-1)
• Thus the number of degrees of freedom of a mechanism
is given by
f = 6 (n-1)
f = 6 (n-1) – p1 – p2 – p3 – p4 – p5 – p6
p1 – No. of pairs having 1 DOF
p2 – No. of pairs having 2 DOF
p3 – No. of pairs having 3 DOF
p4 – No. of pairs having 4 DOF
p5 – No. of pairs having 5 DOF
p6 – No. of pairs having 6 DOF
GRUBLER’S CRITERIA
• The Grubler’s criterion applies to mechanisms with only
single degree of freedom joints where the overall
movability of the mechanism is unity.
• Substituting n = 1 and h = 0 in Kutzbach equation, we
have
• f = 3 ( n – 1 ) – 2p1
f = 3 ( n – 1 ) – 2p1 – 1p 2
PROBLEM M-09
• Q1(c) Fig. shows schematic
of a mechanism. Redraw
the free-hand sketch on the
answer book. Find out the
total number of kinematic
links and number of
kinematic pairs. Hence find
out the degrees of freedom
for the mechanism
[4 MARKS]
• (n=8;p1=10;dof=1)
PROBLEM M-09
• Q 2C)Fig. shows schematic
of a mechanism. Redraw
the free-hand sketch on the
answer book. Find out the
total number of kinematic
links and number of
kinematic pairs. Hence
find out the degrees of
freedom for the
mechanism. [4]
• (n=8;p1=10;dof=1)
PROBLEM M-07
• (b) Figure shows
schematic of a
mechanism. Redraw the
free hand sketch on the
answer-book. Find out
the total number of
kinematic links and
number of kinematic
pairs. Hence find out
degrees of freedom for
the mechanism. [
• (n=9;p1=11;dof=2)
PROBLEM M-07
• Q2C) Figure shows
schematic of a
mechanism. Redraw the
free hand sketch on the
answer-book. Find out
the total number of
kinematic links and
number of pairs. Hence
find out degrees of
freedom for the
mechanism. [4]
• (n=9;p1=11;dof=2)
PROBLEM D-05
• Justify the linkages
shown in Fig. is a
mechanism with single
degree of freedom.
• (n=5;p1=5;p2=1;dof=1)
PROBLEM DO-04
• Define degrees of freedom of a mechanism and
find degrees of freedom for the following cases
(Fig. 1 and Fig. 2). [8 MARKS]
PROBLEM
• Figure shows schematic
of a mechanism. Redraw
the freehand sketch on
the answer-book. Find
out the total number of
kinematic links and
number of pairs. Hence
find out degrees of
freedom for the
mechanism. [4]
EQUIVALENT LINKAGE OF MECHANISMS
• Many times the physical shape of the connection
between the links is such that the actual nature
and function of the connection are not
immediately noticed. This is mainly on account
that the centre of a revolute pair is not directly
apparent..
EQUIVALENT LINKAGE OF MECHANISMS
• Very often, a mechanism with higher pairs can be
replaced by an equivalent mechanism with lower
pairs.
• This equivalence is valid for studying only the
instantaneous characteristics.
• The equivalent lower pair mechanism facilitates
analysis as a certain amount of sliding takes place
between connected links in a higher-pair mechanism.
Equivalent Linkages of Mechanism
1] Turning Pair in Place of Sliding Pair
Equivalent Linkages of Mechanism
1] Turning Pair in Place of Sliding Pair
Equivalent Linkages of Mechanism
2] Turning Pair in Place of Higher Pair
Equivalent Linkages of Mechanism
2] Turning Pair in Place of Higher Pair
Equivalent Linkages of Mechanism
3] Turning Pair in Place of Spring
Equivalent Linkages of Mechanism
3] Turning Pair in Place of Spring
