kom unit 1 ppt

April 11, 2023 1Kinematics of Machinery - Unit - I

KINEMATICS OF KINEMATICS OF MECHINERYMECHINERY


KINEMATICS OF MECHINERYKINEMATICS OF MECHINERY

Unit – I-Basics of MechanismUnit – I-Basics of MechanismUnit – II – KinematicsUnit – II – KinematicsUnit – III – CamsUnit – III – CamsUnit – IV – GearsUnit – IV – GearsUnit – V - FrictionUnit – V - Friction


Unit – I-Basics of MechanismUnit – I-Basics of Mechanism

1.Terminologies1.TerminologiesMachineMachineMechanismMechanismKinematic PairKinematic PairLinksLinksKinematic ChainKinematic Chain

2.DOF2.DOFKutzhback Kutzhback EquationEquationGrubler EquationGrubler Equation

3.Grashoff Law3.Grashoff Law4.Mechanism4.Mechanism

Four BarFour BarSingle-Slider CrankSingle-Slider CrankDouble-SliderDouble-Slider

5.Inversion of 5.Inversion of MechanismMechanism

6.Mechanical 6.Mechanical AdvantageAdvantage

7.Transmission Angle7.Transmission Angle8.Design of Mechanism8.Design of Mechanism


MECHANISMMECHANISM

Mechanism – Part of a machine, which Mechanism – Part of a machine, which transmit motion and power from input point to transmit motion and power from input point to output pointoutput point


Example for MechanismExample for Mechanism


PLANAR MECHANISMSPLANAR MECHANISMS

When all the links of a mechanism have plane When all the links of a mechanism have plane motion, it is called as a planar mechanism. All motion, it is called as a planar mechanism. All the links in a planar mechanism move in the links in a planar mechanism move in planes parallel to the reference plane.planes parallel to the reference plane.


MACHINEMACHINE

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


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


KINEMATICSKINEMATICS


RELEVANCE OF KINEMATIC RELEVANCE OF KINEMATIC STUDYSTUDY

Motion requirementsMotion requirements Design requirementsDesign requirements


MOTIONMOTION STUDYSTUDY

Study of position, displacement, velocity and Study of position, displacement, velocity and acceleration of different elements of acceleration of different elements of mechanismmechanism

Given input Desired output Given input Desired output


Motion requirementMotion requirement


DESIGN REQUIREMENTSDESIGN REQUIREMENTS

Design: determination of shape and sizeDesign: determination of shape and size

1.1. Requires knowledge of materialRequires knowledge of material

2.2. Requires knowledge of stressRequires knowledge of stress

Requires knowledge of load actingRequires knowledge of load acting

(i) static load(i) static load

(ii) dynamic/inertia load(ii) dynamic/inertia load


DYNAMIC/INERTIA LOADDYNAMIC/INERTIA LOAD

Inertia load require accelerationInertia load require acceleration


LINK OR ELEMENTLINK OR ELEMENT

Any body (normally rigid) which has motion Any body (normally rigid) which has motion relative to anotherrelative to another

Binary linkBinary link Ternary linkTernary link Quaternary linkQuaternary link


Examples of rigid linksExamples of rigid links


PAIRING ELEMENTSPAIRING ELEMENTS

Pairing elements: Pairing elements: the geometrical forms by which two the geometrical forms by which two members of a mechanism are joined together, so that the members of a mechanism are joined together, so that the

relative motion between these two is consistentrelative motion between these two is consistent. . Such a pair of Such a pair of links is called links is called Kinematic PairKinematic Pair..


KINEMATIC PAIRSKINEMATIC PAIRS

A mechanism has been defined as a combination so A 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 of A 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 by The degree of freedom of a kinetic pair is given by the number independent coordinates required to the 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 contact Based on nature of contact between elementsbetween elements

(i) (i) Lower pairLower pair : The joint : The joint by which two members are by which two members are connected has surface connected has surface contact. A pair is said to be a contact. A pair is said to be a lower pair when the lower pair when the connection between two connection between two elements are through the elements are through the area of contact. Its 6 types area of contact. Its 6 types are are

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 the The contact between the

pairing elements takes place at a point or along pairing 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 pairing Based 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 pairing Based 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 motion one element has got only one definite motion relative to the other relative 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 or Group of links either joined together or arranged in a manner that permits them to arranged 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 JointsL=2P-4L=2P-4J=(3/2) L – 2J=(3/2) L – 2L => No of LinksL => No of LinksP => No of PairsP => No of PairsJ => No of JointsJ => No of JointsL.H.S > R.H.S => Locked chainL.H.S > R.H.S => Locked chainL.H.S = R.H.S => Constrained Kinematic ChainL.H.S = R.H.S => Constrained Kinematic ChainL.H.S < R.H.S => Unconstrained Kinematic L.H.S < R.H.S => Unconstrained Kinematic

ChainChain


LOCKED CHAIN (Or) LOCKED CHAIN (Or) STRUCTURESTRUCTURE

Links connected in such a way that no relative Links 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 Working of slider crank mechanismmechanism


Unconstrained kinematic Unconstrained kinematic chainchain

L=5,P=5,J=5 L=5,P=5,J=5

L.H.S < R.H.S L.H.S < R.H.S


DEGREES OF FREEDOM DEGREES OF FREEDOM (DOF):(DOF):

It is the number of independent coordinates required to It 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 Degrees of freedom/mobility of a mechanisma mechanism

It is the number of inputs (number of It is the number of inputs (number of independent coordinates) required to describe independent coordinates) required to describe the configuration or position of all the links of the configuration or position of all the links of the mechanism, with respect to the fixed link the 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 by Number 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 the l = Number of lower pairs, which is obtained by counting the

number of joints. If more than two links are joined together at any 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 point, 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 link I.e., one input to any one link

will result in definite motion will result in definite motion of all the links.of all the links.


Examples - DOFExamples - DOF 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 are I.e., two inputs to any two links are

required to yield definite motions in required 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 in I.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 of Here, n = 6, l = 7 (at the intersection of

2, 3 and 4, two lower pairs are to be 2, 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 lower Here, n = 11, l = 15 (two lower

pairs at the intersection of pairs 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) = 0 F = 3(11-1)-2(15) = 0



(a)(a)F = 3(n-1)-2l-hF = 3(n-1)-2l-hHere, 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) = -1I.e., it is a structureI.e., it is a structure

(b)(b)F = 3(n-1)-2l-hF = 3(n-1)-2l-hHere, n = 3, l = 2 and h = 1.Here, n = 3, l = 2 and h = 1.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-hHere, n = 3, l = 2 and h = 1.Here, n = 3, l = 2 and h = 1.F = 3(3-1)-2(2)-1 = 1F = 3(3-1)-2(2)-1 = 1


Determining DOF and PairsDetermining DOF and Pairs

NNbb=No of Binary =No of Binary LinksLinks

NNtt=No of Ternary =No of Ternary LinksLinks

NNoo=No of Other =No of Other LinksLinks

N=Total No of N=Total No of LinksLinks

L=No of LoopsL=No of Loops

P=No of PairsP=No of Pairs M=Mobility or M=Mobility or

DOFDOF

P=N+L-1P=N+L-1

M=N-(2L+1)M=N-(2L+1)



P=N+L-1P=N+L-1

M=N-(2L+1)M=N-(2L+1)

NNbb = 4,N = 4,Ntt=2, N=2, N00=0=0

N=6, L=2N=6, L=2

Sol:Sol:

P=6+2-1=7P=6+2-1=7

M=6-(2x2 +1)=1M=6-(2x2 +1)=1



P=N+L-1P=N+L-1

M=N-(2L+1)M=N-(2L+1)

NNbb = 5,N = 5,Ntt=1, N=1, N00=0=0

N=6, L=2N=6, L=2

Sol:Sol:

P=6+2-1=7P=6+2-1=7

M=6-(2x2 +1)=1M=6-(2x2 +1)=1



P=N+L-1P=N+L-1

M=N-(2L+1)M=N-(2L+1)

NNbb = 9,N = 9,Ntt=0, N=0, N00 =2=2

N=11, L=5N=11, L=5

Sol:Sol:

P=11+5-1=15P=11+5-1=15

M=11-(2x5 +1)=0M=11-(2x5 +1)=0


Grashoff LawGrashoff Law

The sum of the The sum of the shortest and shortest and longest link length longest link length should not exceed should not exceed the sum of the the sum of the other two link other two link lengths.lengths.

s+l < p+qs+l < p+q

(e.x) (1+2) < (3+4)(e.x) (1+2) < (3+4)


INVERSIONS OF INVERSIONS OF MECHANISMMECHANISM

A mechanism is one in which one of the links of a kinematic A mechanism is one in which one of the links of a kinematic chain is fixed. Different mechanisms can be obtained by fixing chain is fixed. Different mechanisms can be obtained by fixing different links of the same kinematic chain. These are called as different 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 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 INVERSIONS OF FOUR BAR CHAINCHAIN

Fix link 1& 3. Crank-rocker Fix link 1& 3. Crank-rocker

or Crank-Leveror Crank-Lever

mechanism mechanism

Fix link 2. Drag link Fix link 2. Drag link

or Double Crank or Double Crank

mechanism mechanism

Fix link 4. Double rocker Fix link 4. Double rocker

mechanismmechanism

Pantograph Pantograph


APPLICATIONAPPLICATIONlink-1 fixed-link-1 fixed-

CRANK-ROCKER CRANK-ROCKER MECHANISMMECHANISM OSCILLATORY OSCILLATORY

MOTIONMOTION

April 11, 2023 Kinematics of Machinery - Unit - I 67

CRANK-ROCKER CRANK-ROCKER MECHANISMMECHANISM


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


Locomotive Wheel - DOUBLE Locomotive Wheel - DOUBLE CRANK MECHANISMCRANK MECHANISM


2.SLIDER CRANK CHAIN2.SLIDER CRANK CHAINLink1=GroundLink1=GroundLink2=CrankLink2=Crank

Link3=ConnectingRod Link3=ConnectingRod Link4=SliderLink4=Slider


lnversions of slider crank lnversions of slider crank chainchain

(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 fixed Link 3 fixed Link 4 fixed Link 4 fixed


ApplicationApplicationInversion II – Link 2 Crank Inversion II – Link 2 Crank

fixed fixed Whitworth quick return motion Whitworth quick return motion

mechanismmechanism

2

1

2

2

ˆ

ˆ

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


ApplicationApplicationInversion III -Link 3 Connecting rod Inversion III -Link 3 Connecting rod

fixed fixed Crank and slotted lever quick return Crank and slotted lever quick return

mechanismmechanism

2

1

2

2

ˆ

ˆ

BoB

BoB

urnstrokeTimeforret



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


Application of Crank and slotted lever Application of Crank and slotted lever quick return motion mechanismquick return motion mechanism


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


ApplicationApplicationInversion IV – Link 4 Slider fixed Inversion IV – Link 4 Slider fixed

Pendulum pump or bull enginePendulum pump or bull engine


3. DOUBLE SLIDER CRANK 3. DOUBLE SLIDER CRANK CHAINCHAIN

It is a kinematic chain consisting of two It is a kinematic chain consisting of two turning pairs and two sliding pairs. turning pairs and two sliding pairs.

Link 1 Frame Link 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 FixedDouble slider crank mechanismDouble slider crank mechanism

1sincos 22

22

p

y

q

x

1sincos 22

22

p

y

q

x

Elliptical trammelAC = 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&1 4&1


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


Other MechanismsOther Mechanisms1.Straight line motion 1.Straight line motion

mechanismsmechanismsCondition for perfect steeringCondition for perfect steering

Locus of pt.C Locus of pt.C will be a will be a straight line, ┴ to AE if, straight line, ┴ to AE if,

is constant.is constant.

Proof:Proof:

ACAB

..,

.

constACifABconstAE

constbutADAD

ACABAE

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 Application of Ratchet Pawl mechanismmechanism


4. 4. Toggle mechanismToggle mechanism

Considering the Considering the equilibrium condition of equilibrium condition of slider 6, slider 6,

For small angles of For small angles of αα, F is , F is much smaller than P.much smaller than P.

tan2

2tan

PFP

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 Ackermann steering gear mechanismmechanism


Mechanical Mechanical AdvantageAdvantage Mechanical Mechanical

Advantage of the Advantage of the Mechanism at Mechanism at angle a2 = 0angle a2 = 000 or or 18018000

Extreme position Extreme position of the linkage is of the linkage is known as toggle known as toggle positions.positions.


Transmission Transmission AngleAngleθθ = a1=Crank Angle = a1=Crank Angle

γγ = a2 =Angle between = a2 =Angle between crank and Coupler crank and Coupler

μμ = a3 =Transmission = a3 =Transmission angleangle

Cosine LawCosine Law

aa22 + d + d2 2 -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

Determine Determine μμ..


Design of Design of MechanismMechanism 1.Slider – Crank 1.Slider – Crank

MechanismMechanismLink Lengths, Stroke Link Lengths, Stroke Length, Crank Angle Length, Crank Angle specified. specified.

2.Offset Quick Return 2.Offset Quick Return MechanismMechanismLink Lengths, Stroke Link Lengths, Stroke Length, Crank Angle, Length, Crank Angle, Time Ratio specified.Time Ratio specified.

3.Four Bar Mechanism – 3.Four Bar Mechanism – Crank Rocker MechanismCrank Rocker MechanismLink Lengths and Rocker Link Lengths and Rocker angle Specified.angle Specified.


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