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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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