Kinematics of machines-VTU Notes

IV Sem 'B' Section, Mechanical Department KOM Notes-2016

Santosh A Goudar, B.E, M.Tech,(Ph.D).,Asst. Professor,Mechanical Department, AIET, Moodbidri 1

Unit-I:Introduction

Syllabus:Definitions Link or element, kinematic pairs, Degrees offreedom, Grubler's criterion (without derivation), Kinematicchain, Mechanism, Structure, Mobility of Mechanism,Inversion, Machine.Kinematic Chains and Inversions: Inversions of Four bar chain;Single slider crank chain and Double slider crank chain.

Santosh A Goudar, B.E, M.Tech, (Ph.D).,Asst. Professor,Mechanical Department,AIET, Moodbidri

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Kinematic Link or Element

A link is a resistant body that constitutes thepart of machine connecting other parts whichhave motion relative to it.

OREach part of a machine, which moves relativeto some other part, is known as a kinematiclink (or simply link) or element.

Example: slider-crank chian

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Classification of links

1. Singular Link : A link which is connected toonly one other link is called a singular link

2. Binary Link : A link which is connected to twoother links is called a binary link

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3. Ternary Link : A link which is connected tothree other links is called a ternary link

4. Quarternary Link : A link which is connectedto four other links is called quarternary link

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Types of Links

1. Rigid link: A rigid link is one which does notundergo any deformation while transmittingmotion. Ex- connecting rod, crank etc.

2. Flexible link: A flexible link is one which is partlydeformed in a manner not to affect thetransmission of motion. Ex- belts, ropes, chainsand wires

3. Fluid link : A fluid link is one which the motion istransmitted through the fluid by pressure

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Kinematic Pair

The two links or elements of a machine, whenin contact with each other, are said to form apair. If the relative motion between them iscompletely or successfully constrained (i.e. in adefinite direction), the pair is known askinematic pair.

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Types of Constrained Motions1. Completely constrained motion: When the

motion between a pair is limited to a definitedirection irrespective of the direction of forceapplied, then the motion is said to be acompletely constrained motion.

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2. Incompletely constrained motion: When themotion between a pair can take place in morethan one direction, then the motion is calledan incompletely constrained motion.

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3. Successfully constrained motion: When themotion between the elements, forming a pair,is such that the constrained motion is notcompleted by itself, but by some other means,then the motion is said to be successfullyconstrained motion.

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Classification of Kinematic Pairs

Kinematic pairs may be classified according to,

1. Type of Relative Motion Between The Elements2. Type of Contact Between The Elements3. Type of Closure

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(a) Sliding pair: When the two elements of a pair areconnected in such a way that one can only sliderelative to the other, the pair is known as a slidingpair.

EX- The piston and cylinder, tail stock on the lathe bedetc.

1. Type of Relative Motion Between The Elements

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(b) Turning pair: When the two elements of a pair areconnected in such a way that one can only turn orrevolve about a fixed axis of another link, the pair isknown as turning pair.

EX- A shaft with collars at both ends fitted into acircular hole lathe spindle supported in head stock,cycle wheels turning over their axles etc.

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(c) Rolling pair. When the two elements of apair are connected in such a way that one rollsover another fixed link, the pair is known asrolling pair.

EX- Ball and roller bearings are examples ofrolling pair.

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(d) Screw pair. When the two elements of a pairare connected in such a way that one elementcan turn about the other by screw threads, thepair is known as screw pair

EX- The lead screw of a lathe with nut, and boltwith a nut

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(e) Spherical pair. When the two elements of apair are connected in such a way that oneelement (with spherical shape) turns orswivels about the other fixed element, thepair formed is called a spherical pair.

EX- The ball and socket joint, attachment of acar mirror, pen stand etc.,

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2. Type of contact between theelements

(a) Lower pair: When the two elements of a pairhave a surface contact when relative motiontakes place , the pair is known as lower pair.

Ex-sliding pairs, turning pairs and screw pairs(b) Higher pair: When the two elements of a pair

have a line or point contact when relativemotion takes place, the pair is known as higherpair.

Ex- toothed gearing, belt and rope drives, ball androller bearings and cam and follower.

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Closed pair

Unclosed pair

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3. Type of closure

(a) Closed pair: When the two elements of a pairare connected together mechanically in such away that only required kind of relative motionoccurs, it is then known as closed pair

Ex- lower pairs are self closed pair(b) Unclosed pair:When the two elements of a pair

are not connected mechanically but are kept incontact by the action of external forces, the pairis said to be a unclosed pair

Ex- The cam and follower and foot-step bearing

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Kinematic chain- A kinematic chain is an assembly of links in which therelative motion of the links is possible and the motionof each link relative to others is definite

- The last link of kinematic chain is attached to the firstlink

Ex-

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The following relationship holds foe a kinematic chainhaving lower pairs only,

N= 2P- 4J= 3N/2 -2

Where N = No. Of linksP = No. Of pairsj= No. of joints

If LHS>RHS, chain is lockedLHS=RHS, chain is constrainedLHS<RHS, chain is unconstrained

Ex-

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Machine v/s MechanismMachine Mechanism

1. A machine is meant fortransmitting energy or todo useful work

A mechanism is meant for transmit ortransform motion

2. A machine is acombination of two ormore mechanismsarranged in such a way soas to obtain the requiredmotion

A mechanism is a combination of of rigid orrestraining bodies which are so shaped andconnected, they move each other withdefinite relative motion

3. A machine can use oneor more than onemechanism to performthe desired function

It is not the case with mechanisms. Amechanism is a single system to transfer ortransform motion

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Machine v/s StructureMachine Structure

1. The parts of amachine move relativeto one another

The members of a structure do notmove relative to one another

2. A machinetransforms theavailable energy intosome useful work

In a structure no energy is transformedinto useful work

3. The links of amachine may transmitboth power andmotion

The members of a structure transmitforces only

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Mobility, Degree of freedom(DOF),Grubler criterion

DOF: The number of DOF is described as thenumber of ways of link of mechanism canmove

ORThe no. Of dof of a kinematic pair is defined asthe no. Of independent relative motion bothtranslation and rotation that a kinematic paircan have.

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• Mobility: of mechanism is defined as thenumber of degree of freedom it possesses or itis defined as minimum number of independentparameters required to specify the location ofevery link within a mechanism.

• Grubler criterion :F= 3(n-1)-2j1-j2

where , F = Degrees of freedomn = Number of linksJ1 = Number of lower pairsj2 = Number of higher pairs

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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 canbe obtained by fixing different links of the samekinematic chain. These are called as inversions of themechanism.

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Types of Kinematic Chains

1. Four bar chain or quadric cyclic chain2. Single slider crank chain, and3. Double slider crank chain.

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Four Bar Chain or Quadric Cycle Chain

Link 1- The fixed link is known as frameLink 2 –Crank (the link which can make complete rotation)Link 3- connecting rod or coupler (which connects the crank andlever)Link 4- lever or rocker or follower(The link which oscillates)

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According to Grashof’s law for a four barmechanism, the sum of the shortest andlongest link lengths should not be greater thanthe sum of the remaining two link lengths ifthere is to be continuous relative motionbetween the two links

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Inversions of Four Bar Chain

1. Beam engine (crank and lever mechanism)2. Coupling rod of a locomotive (Double crank

mechanism) or parallel crank mechanism3. Watt’s indicator mechanism (Double lever

mechanism)

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Beam engine (crank and lever mechanism)

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- In this mechanism, when the crank rotatesabout the fixed centre A, the lever oscillatesabout a fixed centre D.

- The end E of the lever CDE is connected to apiston rod which reciprocates due to therotation of the crank

- The purpose of this mechanism is to convertrotary motion into reciprocating motion

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Coupling rod of a locomotive (Doublecrank mechanism)

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- In this mechanism, the links AD and BC(having equal length) act as cranks and areconnected to the respective wheels.

- The link CD acts as a coupling rod and the linkAB is fixed in order to maintain a constantcentre to centre distance between them

- This mechanism is meant for transmittingrotary motion from one wheel to the otherwheel.

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Watt’s indicator mechanism (Doublelever mechanism)

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• In this mechanism, the links AB & DE acts aslevers, the ends A&E of these levers are fixed.

• The links AB & DE are parallel in meanposition of mechanism and coupling rod BDperpendicular to the levers AB & DE.

• The point C traces the shape 8, which will beapproximately straight. Hence this isapproximately straight line mechanism

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Single slider crank chain

• It consist of one sliding pair and three turningpairs. It is usually, found in reciprocating steamengine mechanism. This type of mechanismconverts rotary motion into reciprocating motionand vice versa.

Link 1 – frameLink 2 – crankLink 3 –connecting rodLink 4 – slider

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Inversions of Single slider crank chain

1. First inversiona) Reciprocating engine

2. Second inversiona) Rotary engine (Gnome engine)b) Whitworth quick return mechanism ( will study in next

unit )3. Third inversion

a) Oscillating cylinder engineb) Crank and slotted lever quick return motion mechanism (

will study in next unit )4. Fourth inversion

a) Hand pump

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First inversion

Reciprocating engine:-

This inversion is obtained whenlink 1 is fixed and links 2 and 4are made crank and sliderrespectively

In Reciprocatingengine, Link 4 is apiston acts as driver

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Second inversiona. Rotary engine (Gnomeengine)

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• It consists of seven cylinders in one plane andall revolves about fixed centre D, while thecrank (link 2) is fixed.

• In this mechanism, when the connecting rod(link 3) rotates, the piston (link 4) reciprocatesinside the cylinders forming link 1

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Third inversion

Oscillating cylinder engine

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• In this mechanism, the link 3 forming theturning pair is fixed, The link 3 corresponds tothe connecting rod of a reciprocating steamengine mechanism

• When the crank (link 2) rotates, the pistonattached to piston rod (link 1) reciprocatesand the cylinder (link 4) oscillates about a pinpivoted to the fixed link at A.

• The arrangement of oscillating cylinder enginemechanism is used to convert reciprocatingmotion into rotary motion

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Fourth inversionHand pump Link 1- piston rod

Link 2- crankLink 3- connecting rodLink 4- cylinder(Fixed )

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• In this mechanism, the inversion is obtainedby fixing the cylinder or link 4 (i.e. sliding pair),as shown in fig.

• In this case, when the crank (link 2) rotates,the connecting rod (link 3) oscillates about apin pivoted to the fixed link 4 at A and thepiston attached to the piston rod (link 1)reciprocates.

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3. Double slider crank chain.

1. Elliptical trammels2. Scotch yoke mechanism.3. Oldham’s coupling.

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Elliptical trammels

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• It is an instrument used for drawing ellipses.• This inversion is obtained by fixing the slotted

plate (link 4), The fixed plate or link 4 has twostraight grooves cut in it, at right angles to eachother.

• The link 1 and link 3, are known as sliders andform sliding pairs with link 4. The link AB (link 2)is a bar which forms turning pair with links 1 and3

• When the links 1 and 3 slide along theirrespective grooves, any point on the link 2 suchas P traces out an ellipse on the surface of link 4.

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• Let us take OX and OY as horizontal andvertical axes and let the link BA is inclined atan angle θ with the horizontal, as shown inFig(b).

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This is the equation of an ellipse. Hence thepath traced by point P is an ellipse whosesemi-major axis is AP and semi-minor axis isBP.

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2. Scotch yoke mechanism.

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• This mechanism is used for converting rotarymotion into a reciprocating motion.

• In Fig, link1 is fixed. In this mechanism, whenthe link 2 (which corresponds to crank) rotatesabout B as centre, the link 4 (whichcorresponds to a frame) reciprocates. Thefixed link 1 guides the frame.

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3. Oldham’s coupling

Link 1 – FlangeLink 2 – Supporting frameLink 3 – FlangeLink 4 – Intermediate piece

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• An oldham's coupling is used forconnecting two parallel shafts whose axes areat a small distance apart. The shafts arecoupled in such a way that if one shaft rotates,the other shaft also rotates at the samespeed.

• The shafts to be connected have two flanges(link 1 and link 3) rigidly fastened at their endsby forging .

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• The intermediate piece (link 4) which is acircular disc, have two tongues (i.e.diametrical projections) T1 and T2 on eachface at right angles to each other, as shown inFig.

• The tongues on the link 4 closely fit into theslots in the two flanges (link 1 and link 3). Thelink 4 can slide or reciprocate in the slots inthe flanges

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• When the driving shaft A is rotated, the flangeC (link 1) causes the intermediate piece (link4) to rotate at the same angle through whichthe flange has rotated, and it further rotatesthe flange D (link 3) at the same angle andthus the shaft B rotates.

• Hence links 1, 3 and 4 have the same angularvelocity at every instant.

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Maximum sliding speed of each tongue (in m/s)

Let,ω = Angular velocity of each shaft in rad/s,r = Distance between the axes of the shafts inmetres.

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Kinematics of machines-VTU Notes

Engineering

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