KoM ppt unit1 pvpsit 2015 Basic Mechanisms

February 5, 2015 1Kinematics of Machinery - Unit - I
KINEMATICS OFKINEMATICS OF
MECHINERYMECHINERY

Example for MechanismExample for Mechanism

MACHINEMACHINE
A machine is a mechanism or collection ofA machine is a mechanism or collection of
mechanisms, which transmit force from themechanisms, which transmit force from the
source of power to the resistance to besource of power to the resistance to be
overcome.overcome.

Though all machines are mechanisms, allThough all machines are mechanisms, all
mechanisms are not machinesmechanisms are not machines

DYNAMIC/INERTIA LOADDYNAMIC/INERTIA LOAD
Inertia load require accelerationInertia load require acceleration

LINK OR ELEMENTLINK OR ELEMENT
Any body (normally rigid) which has motionAny body (normally rigid) which has motion
relative to anotherrelative to another
 Binary linkBinary link
 Ternary linkTernary link
 Quaternary linkQuaternary link

KINEMATIC PAIRSKINEMATIC PAIRS
 A mechanism has been defined as a combination soA mechanism has been defined as a combination so
connected that each moves with respect to each other.connected that each moves with respect to each other.
A clue to the behavior lies in in the nature ofA clue to the behavior lies in in the nature of
connections, known as kinetic pairs.connections, known as kinetic pairs.
The degree of freedom of a kinetic pair is given byThe degree of freedom of a kinetic pair is given by
the number independent coordinates required tothe number independent coordinates required to
completely specify the relative movement.completely specify the relative movement.

TYPES OF KINEMATIC PAIRSTYPES OF KINEMATIC PAIRS
Based on nature of contactBased on nature of contact
between elementsbetween elements
 (i)(i) Lower pairLower pair : The joint: The joint
by which two members areby which two members are
connected has surfaceconnected has surface
contact. A pair is said to be acontact. A pair is said to be a
lower pair when thelower pair when the
connection between twoconnection between two
elements are through theelements are through the
area of contact. Its 6 typesarea of contact. Its 6 types
areare
Revolute(Or)TurningPairRevolute(Or)TurningPair
Prismatic(Or)SlidingPairPrismatic(Or)SlidingPair
Screw(Or)HelicalPairScrew(Or)HelicalPair
CylindricalPairCylindricalPair
Spherical(Or)GlobularPairSpherical(Or)GlobularPair
Flat(or)PlanarPairFlat(or)PlanarPair

(ii) Higher pair:(ii) Higher pair: The contact between theThe contact between the
pairing elements takes place at a point or alongpairing elements takes place at a point or along
a line.a line.

Based on relative motion between pairing elementsBased on relative motion between pairing elements
(a) Siding pair [DOF = 1](a) Siding pair [DOF = 1]
(b) Turning pair (revolute pair)(b) Turning pair (revolute pair)
[DOF = 1][DOF = 1]

Based on relative motion between pairingBased on relative motion between pairing
elementselements
(c) Cylindrical pair(c) Cylindrical pair [DOF = 2][DOF = 2]
(d) Rolling pair(d) Rolling pair
[DOF = 1][DOF = 1]

Based on relative motion between pairingBased on relative motion between pairing
elementselements
(e) Spherical pair [DOF = 3](e) Spherical pair [DOF = 3]
(f) Helical pair or screw pair [DOF = 1](f) Helical pair or screw pair [DOF = 1]

Based on the nature of mechanical constraintBased on the nature of mechanical constraint
(a) Closed pair(a) Closed pair
(b) Unclosed or force closed pair(b) Unclosed or force closed pair

CONSTRAINED MOTIONCONSTRAINED MOTION
one element has got only one definite motionone element has got only one definite motion
relative to the otherrelative to the other

(a) Completely constrained motion(a) Completely constrained motion

(b) Successfully constrained motion(b) Successfully constrained motion

(c) Incompletely constrained motion(c) Incompletely constrained motion

KINEMATIC CHAINKINEMATIC CHAIN
Group of links either joined together orGroup of links either joined together or
arranged in a manner that permits them toarranged in a manner that permits them to
move relative to one another.move relative to one another.

Kinematic ChainKinematic Chain
Relation between Links, Pairs and JointsRelation between Links, Pairs and Joints
L=2P-4L=2P-4
J=(3/2) L – 2J=(3/2) L – 2
L => No of LinksL => No of Links
P => No of PairsP => No of Pairs
J => No of JointsJ => No of Joints
L.H.S > R.H.S => Locked chainL.H.S > R.H.S => Locked chain
L.H.S = R.H.S => Constrained Kinematic ChainL.H.S = R.H.S => Constrained Kinematic Chain
L.H.S < R.H.S => Unconstrained KinematicL.H.S < R.H.S => Unconstrained Kinematic
ChainChain

LOCKED CHAIN (Or) STRUCTURELOCKED CHAIN (Or) STRUCTURE
Links connected in such a way that no relativeLinks connected in such a way that no relative
motion is possible.motion is possible.
L=3, J=3, P=3L=3, J=3, P=3 L.H.S>R.H.SL.H.S>R.H.S

Kinematic Chain MechanismKinematic Chain Mechanism
Slider crank and four bar mechanismsSlider crank and four bar mechanisms
L=4, J=4, P=4L=4, J=4, P=4
L.H.S=R.H.SL.H.S=R.H.S

Working of slider crank mechanismWorking of slider crank mechanism

Unconstrained kinematic chainUnconstrained kinematic chain
L=5,P=5,J=5L=5,P=5,J=5
L.H.S < R.H.SL.H.S < R.H.S

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

Degrees of freedom/mobility of aDegrees of freedom/mobility of a
mechanismmechanism
It is the number of inputs (number ofIt is the number of inputs (number of
independent coordinates) required to describeindependent coordinates) required to describe
the configuration or position of all the links ofthe configuration or position of all the links of
the mechanism, with respect to the fixed linkthe mechanism, with respect to the fixed link
at any given instant.at any given instant.

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

Examples - DOFExamples - DOF
 F = 3(n-1)-2l-hF = 3(n-1)-2l-h
 Here, n = 4, l = 4 & h = 0.Here, n = 4, l = 4 & h = 0.
 F = 3(4-1)-2(4) = 1F = 3(4-1)-2(4) = 1
 I.e., one input to any one linkI.e., one input to any one link
will result in definite motionwill result in definite motion
of all the links.of all the links.

 F = 3(n-1)-2l-hF = 3(n-1)-2l-h
 Here, n = 5, l = 5 and h = 0.Here, n = 5, l = 5 and h = 0.
 F = 3(5-1)-2(5) = 2F = 3(5-1)-2(5) = 2
 I.e., two inputs to any two links areI.e., two inputs to any two links are
required to yield definite motions inrequired to yield definite motions in
all the links.all the links.

 F = 3(n-1)-2l-hF = 3(n-1)-2l-h
 Here, n = 6, l = 7 and h = 0.Here, n = 6, l = 7 and h = 0.
 F = 3(6-1)-2(7) = 1F = 3(6-1)-2(7) = 1
 I.e., one input to any one link will result inI.e., one input to any one link will result in
definite motion of all the links.definite motion of all the links.

 F = 3(n-1)-2l-hF = 3(n-1)-2l-h
 Here, n = 6, l = 7 (at the intersection ofHere, n = 6, l = 7 (at the intersection of
2, 3 and 4, two lower pairs are to be2, 3 and 4, two lower pairs are to be
considered) and h = 0.considered) and h = 0.
 F = 3(6-1)-2(7) = 1F = 3(6-1)-2(7) = 1

 F = 3(n-1)-2l-hF = 3(n-1)-2l-h
 Here, n = 11, l = 15 (two lowerHere, n = 11, l = 15 (two lower
pairs at the intersection ofpairs at the intersection of 3, 4,3, 4,
66;; 2, 4, 52, 4, 5;; 5, 7, 85, 7, 8;; 8, 10, 118, 10, 11) and) and
h = 0.h = 0.
 F = 3(11-1)-2(15) = 0F = 3(11-1)-2(15) = 0

(a)(a)
F = 3(n-1)-2l-hF = 3(n-1)-2l-h
Here, n = 4, l = 5 and h = 0.Here, n = 4, l = 5 and h = 0.
F = 3(4-1)-2(5) = -1F = 3(4-1)-2(5) = -1
I.e., it is a structureI.e., it is a structure
(b)(b)
F = 3(n-1)-2l-hF = 3(n-1)-2l-h
F = 3(3-1)-2(2)-1 = 1F = 3(3-1)-2(2)-1 = 1
(c)(c)
F = 3(n-1)-2l-hF = 3(n-1)-2l-h
F = 3(3-1)-2(2)-1 = 1F = 3(3-1)-2(2)-1 = 1

Grashoff LawGrashoff Law
 The sum of the shortestThe sum of the shortest
and longest link lengthand longest link length
should not exceed theshould not exceed the
sum of the other twosum of the other two
link lengths.link lengths.
s+l < p+qs+l < p+q
(e.x) (1+2) < (3+4)(e.x) (1+2) < (3+4)

INVERSIONS OF MECHANISMINVERSIONS OF MECHANISM
A mechanism is one in which one of the links of a kinematicA mechanism is one in which one of the links of a kinematic
chain is fixed. Different mechanisms can be obtained by fixingchain is fixed. Different mechanisms can be obtained by fixing
different links of the same kinematic chain. These are called asdifferent links of the same kinematic chain. These are called as
inversions of the mechanism.inversions of the mechanism.

INVERSIONS OF MECHANISMINVERSIONS OF MECHANISM
 1.Four Bar Chain1.Four Bar Chain
 2.Single Slider Crank2.Single Slider Crank
 3.Double Slider Crank3.Double Slider Crank

1.Four Bar Chain - Inversions1.Four Bar Chain - Inversions

Beam Engine (crank &Lever)Beam Engine (crank &Lever)

Double Crank MsmDouble Crank Msm

Double LeverDouble Lever

2.Single Slider Crank _2.Single Slider Crank _
InversionsInversions

Single Slider Crank _ InversionsSingle Slider Crank _ Inversions
1.1.PENDULUM PUMPPENDULUM PUMP

2.OSCILLATING CYLINDER2.OSCILLATING CYLINDER

3.ROTARY IC ENGINE3.ROTARY IC ENGINE
February 5, 2015 Kinematics of Machinery - Unit - I 52

4.Crank and Slotted Lever Mechanism4.Crank and Slotted Lever Mechanism
(Quick-Return Motion Mechanism)(Quick-Return Motion Mechanism)

Basic Quick-ReturnBasic Quick-Return

Quick-Return IN SHAPER M/CQuick-Return IN SHAPER M/C

WHIT WORTH QUICK-WHIT WORTH QUICK-
RETURNRETURN

3.Double Slider Crank3.Double Slider Crank
INVERSIONSINVERSIONS

ELLIPTICAL TRAMMELELLIPTICAL TRAMMEL

OLD HAMS COUPLINGOLD HAMS COUPLING

1. FOUR BAR CHAIN1. FOUR BAR CHAIN
 (link 1) frame(link 1) frame
 (link 2) crank(link 2) crank
 (link 3) coupler(link 3) coupler
 (link 4) rocker(link 4) rocker

INVERSIONS OF FOUR BAR CHAININVERSIONS OF FOUR BAR CHAIN
Fix link 1& 3. Crank-rockerFix link 1& 3. Crank-rocker
or Crank-Leveror Crank-Lever
mechanismmechanism
Fix link 2. Drag linkFix link 2. Drag link
or Double Crankor Double Crank
mechanismmechanism
Fix link 4. Double rockerFix link 4. Double rocker
mechanismmechanism
PantographPantograph

APPLICATIONAPPLICATION
link-1 fixed-link-1 fixed-
CRANK-ROCKER MECHANISMCRANK-ROCKER MECHANISM
OSCILLATORY MOTIONOSCILLATORY MOTION

CRANK-ROCKERCRANK-ROCKER
MECHANISMMECHANISM

Link 2 Fixed- DRAG LINKLink 2 Fixed- DRAG LINK
MECHANISMMECHANISM

Locomotive Wheel - DOUBLELocomotive Wheel - DOUBLE
CRANK MECHANISMCRANK MECHANISM

2.SLIDER CRANK CHAIN2.SLIDER CRANK CHAIN
Link1=GroundLink1=Ground
Link2=CrankLink2=Crank
Link3=ConnectingRodLink3=ConnectingRod
Link4=SliderLink4=Slider

lnversions of slider crank chainlnversions of slider crank chain
(a)(a) crank fixed (b) connecting rod fixed (c) slider fixedcrank fixed (b) connecting rod fixed (c) slider fixed
Link 2 fixedLink 2 fixed Link 3 fixedLink 3 fixed Link 4 fixedLink 4 fixed

ApplicationApplication
Inversion II – Link 2 Crank fixedInversion II – Link 2 Crank fixed
Whitworth quick return motion mechanismWhitworth quick return motion mechanism
2
1
2
2
ˆ
ˆ
θ
θ
=
′′′
′′′
=
BoB
BoB
urnstrokeTimeforret
wardstrokeTimeforfor

Quick return motion mechanismsQuick return motion mechanisms
Drag link mechanismDrag link mechanism
12
21
ˆ
ˆ
BAB
BAB
urnstrokeTimeforret
=

Rotary engine– II inversion of slider crankRotary engine– II inversion of slider crank
mechanism. (crank fixed)mechanism. (crank fixed)

Inversion III -Link 3 Connecting rod fixedInversion III -Link 3 Connecting rod fixed
Crank and slotted lever quick return mechanismCrank and slotted lever quick return mechanism
2
1
2
2
ˆ
ˆ
θ
θ
=
′′′
′′′
=
BoB
BoB
urnstrokeTimeforret

Crank and slotted lever quick returnCrank and slotted lever quick return
motion mechanismmotion mechanism

Crank and slotted lever quick return motionCrank and slotted lever quick return motion
mechanismmechanism

Application of Crank and slotted lever quickApplication of Crank and slotted lever quick
return motion mechanismreturn motion mechanism

Oscillating cylinder engine–III inversion ofOscillating cylinder engine–III inversion of
slider crank mechanism (connecting rod fixed)slider crank mechanism (connecting rod fixed)

Inversion IV – Link 4 Slider fixedInversion IV – Link 4 Slider fixed
Pendulum pump or bull enginePendulum pump or bull engine

3. DOUBLE SLIDER CRANK3. DOUBLE SLIDER CRANK
CHAINCHAIN
It is a kinematic chain consisting of twoIt is a kinematic chain consisting of two
turning pairs and two sliding pairs.turning pairs and two sliding pairs.
Link 1 FrameLink 1 Frame
Link 2 Slider -ILink 2 Slider -I
Link 3 CouplerLink 3 Coupler
Link 4 Slider - IILink 4 Slider - II

Inversion I – Frame FixedInversion I – Frame Fixed
Double slider crank mechanismDouble slider crank mechanism
1sincos 22
22
=+=





+





θθ
p
y
q
x
1sincos 22
22
=+=





+





θθ
p
y
q
x
Elliptical trammel
AC = p and BC = q, then,
x = q.cosθ and y = p.sinθ.
Rearranging,

Inversion II – Slider - I FixedInversion II – Slider - I Fixed
SCOTCH –YOKE MECHANISMSCOTCH –YOKE MECHANISM
Turning pairs –1&2, 2&3; Sliding pairs – 3&4,Turning pairs –1&2, 2&3; Sliding pairs – 3&4,
4&14&1

Inversion III – Coupler FixedInversion III – Coupler Fixed
OLDHAM COUPLINGOLDHAM COUPLING

Other MechanismsOther Mechanisms
1.Straight line motion mechanisms1.Straight line motion mechanisms
Condition for perfect steeringCondition for perfect steering
Locus of pt.CLocus of pt.C will be awill be a
straight line, ┴ to AE if,straight line, ┴ to AE if,
is constant.is constant.
Proof:Proof:
ACAB×
..,
.
constACifABconstAE
constbutAD
AD
ACAB
AE
AE
AB
AC
AD
ABDAEC
=×=∴
=
×
=∴
=∴
∆≡∆

1.a) Peaucellier mechanism1.a) Peaucellier mechanism

1.b) Robert’s mechanism1.b) Robert’s mechanism

1.c) Pantograph1.c) Pantograph

2.Indexing Mechanism2.Indexing Mechanism
Geneva wheel mechanismGeneva wheel mechanism

3.Ratchets and Escapements3.Ratchets and Escapements
Ratchet and pawl mechanismRatchet and pawl mechanism

Application of Ratchet Pawl mechanismApplication of Ratchet Pawl mechanism

4.4. Toggle mechanismToggle mechanism
Considering theConsidering the
equilibrium condition ofequilibrium condition of
slider 6,slider 6,
For small angles ofFor small angles of αα, F is, F is
much smaller than P.much smaller than P.
α
α
tan2
2tan
PF
P
F
=∴
=

5.Hooke’s joint5.Hooke’s joint

Hooke’s jointHooke’s joint

6.Steering gear mechanism6.Steering gear mechanism
Condition for perfect steeringCondition for perfect steering

Ackermann steering gear mechanismAckermann steering gear mechanism

MechanicalMechanical
AdvantageAdvantage
 Mechanical AdvantageMechanical Advantage
of the Mechanism atof the Mechanism at
angle a2 = 0angle a2 = 000
or 180or 18000
 Extreme position of theExtreme position of the
linkage is known aslinkage is known as
toggle positions.toggle positions.

TransmissionTransmission
AngleAngle
θθ = a1=Crank Angle= a1=Crank Angle
γγ = a2 =Angle between= a2 =Angle between
crank and Couplercrank and Coupler
μμ = a3 =Transmission angle= a3 =Transmission angle
Cosine LawCosine Law
aa22
+ d+ d22
-2ad cos-2ad cos θθ ==
bb22
+ c+ c22
-2 bc cos-2 bc cos μμ
Where a=AD, b=CD,Where a=AD, b=CD,
c=BC, d=ABc=BC, d=AB
DetermineDetermine μμ..

Design ofDesign of
MechanismMechanism
 1.Slider – Crank Mechanism1.Slider – Crank Mechanism
Link Lengths, Stroke Length,Link Lengths, Stroke Length,
Crank Angle specified.Crank Angle specified.
 2.Offset Quick Return2.Offset Quick Return
MechanismMechanism
Link Lengths, Stroke Length,Link Lengths, Stroke Length,
Crank Angle, Time RatioCrank Angle, Time Ratio
specified.specified.
 3.Four Bar Mechanism –3.Four Bar Mechanism –
Crank Rocker MechanismCrank Rocker Mechanism
Link Lengths and RockerLink Lengths and Rocker
angle Specified.angle Specified.

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KoM ppt unit1 pvpsit 2015 Basic Mechanisms