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KINEMATICS OF MACHINERY
MEE 215
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Can crusherSimple press
Examplefor Mechanism
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Rearwindow
wiper Lift platform
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Moves packages from an assembly
bench to a conveyor
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TERTIARY LINK
BINARY LINK
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Microwave carrier to assist
people on wheelchair
Lift platform
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CAN SLIDE
AND ROTATE
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Front loader
Device to
close the top
flap of boxes
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Rowing type exercise machine Conceptual design for an
exercise machine
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Machines
Machine: an assemblage of parts
(components) that transmit forces, motion
and energy in a predetermined manner.
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Machines
A machine is a combination of rigid orresistant bodies, formed and connected so
that they move with definite relative motions
and transmit forces from the source of powerto the resistance to be overcome.
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Machines
A machine has two functions:
1. transmitting definite relative motions, and2. transmitting forces.
These functions require strength and rigidity
to transmit the forces.
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Mechanisms
A mechanism defined as a
combination of rigid or resistant
bodies, formed and connected so
that they move with definite
relative motions with respect to
one another.
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Machines Vs Mechanisms
The similaritybetween machinesandmechanismsisthat:
They are both combinations of rigid bodies
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Structures
Astructureis an assembly of rigid or resistant bodieswith nonmovable joint for load bearing purpose.
Structure does not have internal mobility (No relative
motion)
Machines and Mechanisms have internal mobility
Usually static deals with structures
Kinematics and Kinetics deal with the Mechanics and
Mechanisms
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Some Important Definitions
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Links or Elements
A link is defined as a rigid body having two or
more pairing elements which connect it to otherbodies for the purpose of transmitting force or
motion OR
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Links (completely rigid)
Crank Pin
Crank Shaft Flywheel
Identify which
link is fixed?
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Links (Not rigid)
Springs
Belts
Ropes
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Links
Links are connected to transmit motion From Driver (input link)
To Driven or follower (output link)
Links Unitary ( singular),
Binary,
Ternary or
Quaternary.
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Links
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Kinematic Pairs
Two elements or links are connected togetherin such away that their relative motion is
completely constrained form a kinematic
pair. or
A joint of two links having relative
motion between them.
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Kinematic Pairs Types
Types
Higher Pairs (point, Line or Curve contact)Gear teeth, cam and follower, friction disks
Lower pairs (Surface Contact)
1. Revolute joint (Pin joint) such as hinges
2. Prismatic pair such as sliders
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Six Types of Lower Pairs
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Higher pair:
The contact between the pairing elements takes place at a point or along a line.
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Kinematic Chain
When several movable links are connected to eachother by joints, they are calledkinematic chain.
A kinematic chainwith at least one link has beengrounded constitutes a mechanism chain.
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Closed Chain
Ifevery link in the chain is connected to two or morelinksthen the chain form one or more closed loops
If the link form a closed loops, it is called closedmechanism chain
IfNOT, the chain is said to be open mechanism chain
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Planar and Spatial Mechanisms
In planar mechanisms, all of the relative
motions of the rigid bodies are in one plane or
in parallel planes
If there is any relative motion that is not in the
same plane or in parallel planes, the
mechanism is calledspatial mechanism.
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Degrees of freedom (DOF)
The definition of theDegrees of Freedomof a
mechanism is the number of independentrelative motions among the rigid bodies.
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Mobility
The number of degrees of freedom of a mechanism
is also called themobilityof the device.
DOFswithnumberjo
DOFswithnumberjo
JJnhjnM
2intJ
1intJfreedom)ofdegrees(twopairshigherofnumberh
freedom)ofdegree(onepairslowerorjointsbinaryofnumberj
frame)the(includinglinksofnumbern
mechanismin thefreedomofdegreestotalM
Where
.12)1(32)1(3
2
1
21
TheKutzbach criterion
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Degrees of freedom/mobility of amechanism
It is the number of inputs (number of independent coordinates) required to describe
the configuration or position of all the links ofthe mechanism, with respect to the fixed linkat any given instant.
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Examples DOF
F = 3(n-1)-2J-h
Here, n = 4, J = 4 &
h = 0.
F = 3(4-1)-2(4) = 1 I.e., one input to any one
link will result in definite
motion of all the links.
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F = 3(n-1)-2J-h
Here, n = 5, l = 5 and h = 0. F = 3(5-1)-2(5) = 2
I.e., two inputs to any two links arerequired to yield definite motions inall the links.
F = 3(n-1)-2J-h
Here, n = 6, l = 7 and h = 0.
F = 3(6-1)-2(7) = 1
I.e., one input to any one link will result indefinite motion of all the links.
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DOF ?
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F = 3(n-1)-2J-h Here, n = 6, l = 7 (at the intersection of
2, 3 and 4, two lower pairs are to beconsidered) and h = 0.
F = 3(6-1)-2(7) = 1
F = 3(n-1)-2J-h
Here, n = 11, l = 15 (two lowerpairs at the intersection of 3, 4,
6; 2, 4, 5; 5, 7, 8; 8, 10, 11) andh = 0.
F = 3(11-1)-2(15) = 0
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Calculate the degrees of freedom of the mechanisms
shown in Figure.
Here , n = 4, l = 4 ( at A, B,C,D) and h = 0.
F=3(4-1) -2 x 4-0 = 1
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LOCKED CHAIN OR STRUCTURE
Links connected in such a way that no relativemotion is possible.
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Unconstrained kinematic chain
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MECHANISM
Slider crank and four bar mechanisms
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Find DOF ?
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(a)
F = 3(n-1)-2l-h
Here, n = 4, l = 5 and h = 0.
F = 3(4-1)-2(5) = -1
I.e., it is a structure
(b)
F = 3(n-1)-2l-h
Here, n = 3, l = 2 and h = 1.
F = 3(3-1)-2(2)-1 = 1
(c)
F = 3(n-1)-2l-h
Here, n = 3, l = 2 and h = 1.
F = 3(3-1)-2(2)-1 = 1
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F = 3(n-1)-2l-h
Here, n = 9, l =11 and h = 0.F = 3(9-1)-2(11) = 2
F = 3(n-1)-2l-hHere, n = 8
l = 9+2(on account of double joint),h = 0.
F = 3(8-1)-2(11) = -1
F = 3(n-1)-2l-h
Here, n = 10,l = 9+2(2)=13 and
h = 0.
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F = 3(n-1)-2l-hHere, n = 9,
l = 11 and
h = 0.
F = 3(9-1)-2(11) = +2
F = 3(n-1)-2l-hHere, n = 7,
l = 8 and
h = 1
F = 3(7-1)-2(8)-1 = +1
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F = 3(n-1)-2l-hHere, n = 9,
l =11 (one DOF)
h = 1F = 3(9-1)-2(11)-1 = +1
Fig. Undercarriage mechanism of an aircraft
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Kinematic Equivalence
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Kinematic Equivalence
Higher pair joints are often replaced by kinematically equivalent lowerpair joints.
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3
2
11
2
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n=2
l=1
h=0
F=1
n=3
l=2
h=0
F=2
n=4
l=4
h=0F=1
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n=4l=3
h=0
F=3
n=6
l=7h=0
F=1
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Find DOF ?
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F = 3(n-1)-2l-hF=3(10-1)-2(12)=+3
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Find DOF ?
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F = 3(n-1)-2l-h= 3(10-1)-2(12)=+3
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Grublers criterion for plane Mechanisms
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Grubler s criterion for plane Mechanisms
The Grublers criterion applies to mechanisms with onlyone degree of freedom joints, where overall mobility of
the mechanism is unity. Substituting F=1 and h=0 inKutzbach equation
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0423
02)1(31
jL
or
jL
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Reference: Theory of machines S.S.RATTAN PP2155KHALID/SMBS/VITU/WINTER 2015165/01/2016
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Grashoffs Type2: s+l > p +q
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no complete rotatable joints
Grashoffs law neutral: s+l= p+q
linkages can be flattened
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Transmission Angle
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Min. transmission
angle
Max. transmission angle
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Transmission angle
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Crankrocker
doublerocker
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Mechanical advantage
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INVERSIONS OF MECHANISM
A mechanism is one in which one of the links of a kinematic
chain is fixed. Different mechanisms can be obtained by fixingdifferent links of the same kinematic chain. These are called asinversions of the mechanism.
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FOUR BAR CHAIN
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(link 1) frame (link 2) crank
(link 3) coupler (link 4) rocker
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INVERSIONS OF FOUR BAR CHAIN
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INVERSIONS OF FOUR BAR CHAIN
1. Crank-rocker mechanism
2. Drag link mechanism
3. Double rocker mechanism
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CRANK ROCKER MECHANISM
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CRANKROCKER MECHANISM
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DRAG LINK MECHANISM
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DRAG LINK MECHANISM
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DOUBLE CRANK MECHANISM
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SLIDER CRANK CHAIN
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INVERSIONS OF SLIDER CRANK CHAIN
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(a) crank fixed (b) connecting rod fixed (c) slider fixed
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INVERSIONS OF SLIDER CRANK CHAIN
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(a) crank fixed (b) connecting rod fixed (c) slider fixed
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First Inversion fix link no.1
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2crank4 slider
Link 1 fixed
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IC Engine
Compressor
Second Inversion fix link no.2 (crank)
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Fixing link No2
Rotary Engine mechanism
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Whitworth quick return motion mechanism 2 Inversion
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Rotary engine 2 inversion of slider crank mechanism. (crank fixed)
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Third Inversion
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Fixing link No3Oscillating cylinder engine
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Crank and slotted quick return motion mechanism
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Fourth Inversion
Fixing link No4
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g
Hand Pump
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DOUBLE SLIDER CRANK CHAIN
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It is a kinematic chain consisting of twoturning pairs and two sliding pairs.
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SCOTCH YOKE MECHANISM
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Turning pairs ? Sliding pairs ?
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Elli ti l t l
Inversions of double slider crank mechanism
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Elliptical trammel
1sincos 2222
p
y
q
x
AC = p and BC = q, then,x = q.cos and y = p.sin.
Rearranging,
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Oldhams Coupling
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FOUR BAR MECHANISM
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Straight line motion mechanisms
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Condition for perfect steering Locus of pt.C will be a straightline, to AE if,
is constant.Proof:
ACAB
..,
.
constACifABconstAE
constbutAD
AD
ACABAE
AE
AB
AC
AD
ABDAEC
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Peaucellier mechanism
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Harts mechanism
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Scott Russells mechanism
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Roberts mechanism
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Intermittent motion mechanisms
Geneva wheel mechanism
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Application of Ratchet Pawl mechanism
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Other mechanisms
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Toggle mechanism
Considering the equilibriumcondition of slider 6,
For small angles of , F ismuch smaller than P.
tan2
2tan
PF
P
F
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Pantograph
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