1 Introduction to theory of machines

8/11/2019 1 Introduction to theory of machines

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Theory of Machines & Mechanisms

MCT 2212


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Introduction

Rigid BodyMechanics

Statics

Body at rest

Body withconstantvelocity

DynamicsBody with

acceleratedmotion

under the

action offorces andmoments


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Theory of Machines andMechanisms

Mechanisms/

Linkages

Parts/ Link

Jointsdeals with the determinationof the forces and motions of

links in machines

Susystems ofmachines to facilitate

analysis

Introduction


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4

Links! rigid memer ha"ing nodes# i$e$ attachment %oints inary link! 2 nodes

Ternary link! ' nodes

(uaternary link! ) nodes

Links & joints

Joint! connection etween two links *at their nodes+ which allows

motionClassified y ty%e of contact# numer of ,-.# ty%e of %hysical

closure# or numer of links oined$

kinematic pair! Joints are also known as kinematic %air


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5

Joint Classification

Type of contact!

line/%oint i$e$ higher %air#

area/surface i$e$ lower %air

Number of DOF!

full oint01,-.#

half oint02,-.

Form closed *closed y geometry+ orForce closed *needs an eternal force to

kee% it closed+

Joint order = number of links-1


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Full Joint! %ermits one relati"e motion etween adacent links$ ll of these

kinematic %airs are referred to as one degree of freedom*,-.+ %airs$

Turning pairs allow relati"e turning motion etween two

links$# e$g$ earings# %i"ots# or %in oints$

olling pairs allow relati"e rolling motion etween two links#

e$g$ %air of friction wheels .or a rolling %air# it is assumed that

there is no sli%%age etween the links$

!liding pairsallow relati"e sliding motion etween two links#

e$g Piston3Cylinder$

"alf Joint! allows two relati"e motions simultaneously

etween the adacent links and referred to as two degree of

freedom %airs$

!liding pairs

Turning pairs

"alf Joint

#inematic $airs


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Types of joints


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"ig%er $airs

& Loer $airs

Loer pairs' kinematic

%air or oint with

surface/area contact$


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"ig%er $airs & Loer $airs

"ig%er pairs' kinematic %air or oint with %oint contact or line contact$


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Mechanism

(obility or DOF of a mec%anism! The moility of a mechanism is defined as

the numer of inde%endent %arameters re4uired to s%ecify the %osition of all

links of the mechanism$ 5t also s%ecify the numer of in%ut/actuators needed to

o%erate the mechanism$

kinematic c%ain ' kinematic chain is an assemly of links formed y

%lacing kinematic %airs at each of the nodes without s%ecifying the ground

link$

6inematic chains may e either open type or close type$

(ec%anism! 5t is an assemlage of links and oints with at least one link grounded and

interconnected in a way to %ro"ide controlled out%ut motions in res%onse to su%%lied

in%ut motions$


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Link classification!

7round! fied wrt$ reference frame

Crank! %i"oted to ground# makes com%lete re"olutions

8ocker! %i"oted to ground# has oscillatory motion

Cou%ler! link has com%le motion# not attached to ground

(ac%ine! mechanism designed to do work$


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A simple machine may also be considered as a singlemechanism.

.igure 1$'*+ shows a free ody diagram of

the system used to analy9e the manual force

re4uired to generate sufficient gri%%ing force$

.igure 1$'*a+ The tongs

can e considered either

as a machine or as amechanism$

igure! "nline 4#$ylinder %ngineigure! "$ %ngine Demonstration

(ac%ine & (ec%anism


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igure! A &aper' $ard &unching Machineigure! (uic) return mechanism

(ac%ine & (ec%anism

igure! Slider cran) mechanism

igure! Scotch *o)e mechanism


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Mechanisms are widely used in a%%lications where %recise relati"e

mo"ement and transmission of force are re4uired$ Motions may e

continuous or intermittent# linear and/or angular$

(ec%anisms

:orm;gear 7ear3gear 7ear3rack

)*amples of continuous motion output

)*amples of intermittent motion output

Cam


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>"ery mechanism has one stationary ase link$ ll other links may mo"e

relati"e to the fied ase link$ .rom the same kinematic chain# an in"ersion os

a mechanism is otained y making the originally fied link into a mo"ing

link and selecting an originally mo"ing link to e the fied link $

(ec%anism In+ersion

.igure 1$'? Slider crank mechanism and its three in"ersions

*a+ slider crank mechanism *link 1 fied+#

*+ in"ersion @1

*link 2 fied+#

*c+ in"ersion @2

*link ' fied+#

*d+ in"ersion @'

*link ) fied+$


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Planar motion is restricted to a %lane$ .or a %lanar mechanism# the motions of

all of its links must take %lace either in the same %lane or in %lanes that are

%arallel to one another$ The slider crank mechanism and four3ar mechanism

are eam%les of %lanar mechanisms$

$lanar (ec%anism

Figure 1.5 Slider cran) mechanismFigure 1.7 Slider cran) mechanism with o+set

Figure 1.8 our#bar mechanism

The 7ruelerAs e4uation for the moility#

m, of a %lanar mechanism is gi"en as

n0 numer of links in the mechanism

J10umer of one degree of freedom %airs

J20umer of two degree of freedom %airs

212+1*' JJnm =


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If, m< 0 i.e. ve, Preloaded Structure, may require force to

assemle ! Indeterminate "rolem .If, m# 0 , Structure.

If, m$0 i.e. %ve, &ec'anism.

(obility

e ve re oa e tructure may requ re orce to


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,-Figure 1.36 %amples of mobility.

e. ve , re oa e tructure, may requ re orce to

tructure.

. %ve, &ec'anism.

(obility


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,/

Figure 1.36 (Continued)


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n=5; J1=6; J2=0;m=0

n=5; J1=6; J2=0;m=0 but, m=1.

.ull Joint#

Pure rolling#

no sliding

In case of pure rolling,

n=3; J1=3; J2=0;m=0

In case of rolling sli!ing,

n=3; J1=2; J2=1;

m=1

Balf Joint#

rolling &

sliding

,

--

-

./

0

- - -

. /

0

,

.

,

- -

(obility $arado*es

The 7rueler criterion %ays no attention to

link si9es or sha%es# it can gi"e misleading

results in the face of uni4ue geometric

configurations$


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Idle Degrees of Freedom

n 5dle degree of freedom is one that a%%ears *and is+ %resent ut its "alue has

no effect on the in%ut < out%ut relationshi%s of interest

To identify 5dle degrees of freedom# first identify the in%ut and out%ut links

Then we must determine if a single link or cominations of links can mo"e

without affecting the in%ut/out%ut link %ositions

Like a connecting link rotating *aout its ais+ in a steering mechanism

without changing the relationshi% etween the steering wheel and the front

tires in a "ehicle


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,0'1',4 M% 2320 &age 33

ote! &in#in#slot $am $ontact arehalf 6ointsBere#

The Structure has fi"e ,-. with two 5dle ,-.As$

They are the roller and the cam rocker $

mActual7 M8heoretical # m"dle

75#3 7 2

Idle Degrees of Freedom

C

21'2+112*'

2+1*'

2

1'

12

21

2

1

=

=

=

=

=

=

JJnm

J

J

n


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.igure 1$2? *a+ a %rosthetic hand .igure 1$2?*+ .ingers wra%

around an oect as shown in

5n a s%atial mechanism# links mo"e in three dimensions$ .or

eam%le# in a %rosthetic hand# the thum mo"es in a %lane that is

not %arallel to the %lanes of motion of the other four fingers$

where the suscri%t refers to the numer of freedoms of the oint$

spatial mec%anism

The 6ut9ach moility e4uation for s%atial linkages!

C)'21 2')C+1*D JJJJJn" =


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)*ample of !patial Linkage

) LinksE 2 s%herical Joints# 1cylindrical oint and 1 re"olute oint$

,-. of a S%herical Joint is '

,-. of a Cylindrical Joint is 2

,-. of a 8e"olute Joint is 1

'

2'1)1C+1)*D

2')C+1*DC)'21

=

=

= JJJJJn"


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Four12ar (ec%anism13ras%of4s Criterion

.our3ar mechanisms may e studied y distinguishing the link lengths as

follows!

s! the length of the shortest link

l! the length of the longest link

", q! the lengths of the other two links

To assemle the kinematic chain it is necessary that#

The ty%e of a four3ar mechanism may e determined using 7rashofFs

Criterion#

*i+ *ii+ *iii+

8hen9 only case :i; o+ers all three types of a four#barmechanisms.

l#ps ++

#pls ++ #pls +=+


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Class_I Class_II Class_III

*i+ 5f s is the in%ut link# then

the mechanism is a crank

rocker$

*ii+ 5f s is the ase link# then

the mechanism is a drag link$

*iii+ 5f otherwise# then the

mechanism is a rocker3rocker$

8ocker;8ocker Change Point

Four12ar (ec%anism13ras%of4s Criterion

#pls ++ #pls +=+


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ure 1.43 8ypes of four#bar mechanisms :a; cran) roc)er9 :b; drag lin)9 :c; roc)er#roc)er.

For !5L6$57

Crank1rockerif either link adacent to shortest is grounded

Double crank if shortest link is groundedDouble rocker if link o%%osite to shortest is grounded


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For !5L8$57

ll in"ersions will e doule rockers

Go link can fully rotate

.igure! .our ar doule rockersFor !5L9$57 :!pecial case 3ras%of;

ll in"ersions will e doule cranks or crank rockers

Linkage can form %arallelogram or anti%arallelogram

-ften used to kee% cou%ler %arallel *drafting machine+

3-

Parallelogram form nti %arallelogram form ,eltoid form


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.igure 1$)H .our3ar mechanisms!crank rocker

Let the lengths of the three mo"ing links are r20 2$I cmE r'0)$I

cmE r)0$I cm, adusting the length of the ase link we can get the

following in"ersion of four ar mechanism$


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1 Introduction to theory of machines

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