Department of Mechanical Engineering-master Copy

8/11/2019 Department of Mechanical Engineering-master Copy

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Department of Mechanical Engineering

Staff-In charge:C.Maria Antoine Pushparaj

Year & Seester: I I I r d - V t h


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RAJIV GANDHI COLLEGE O ENGINEERING

AND !ECHNOLOG"

Depar t me nt of M ec ha nical E ngi neer i ng

#taff-In charge$ Maria Antoine %&'hpara( #&) Code$ ME*+

#&) Name$ D,namic' of Machiner, "earCla''$ III -Mechanical

NO!E # Of LE ##ON

O. JE C! IVE $

!o un"erstan" the etho" of static force ana#$sis an" "$naic force ana#$sis of

Mechaniss

!o stu"$ the un"esira%#e effects of un%a#ance in rotors an" engines.

!o un"erstan" the concept of i%rator$ s$stes an" their ana#$sis

!o un"erstan" the princip#es of goernors an" g$roscopes.

RE E RE NCE # $

!hoas 'ean( )!heor$ of Machines)( C'S Pu%#ishers an" *istri%utors( 19+,.

hosh A. an" Ma##ic A./.( )!heor$ of Mechaniss an" Machines)( Affi#iate" 0ast

- est Press Pt. t".( 3e4 *e#hi( 19++.

Shig#e$ 5.0. an" 6icer 5.5.( )!heor$ of Machines an" Mechaniss)( Mcra4-7i##(Inc.(

1998.

ao 5.S. an" *uipati ..( )Mechanis an" Machine !heor$ )( i#e$-0astern

iite"( 3e4 *e#hi( 1992.

5ohn 7annah an" Stephens .C.( )Mechanics of Machines)( ia #o4-Price" Stu"ent

0"ition( 1999.

Sa"hu Singh ;!heor$ of MachinesPearson 0"ucation( 2


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INDE/

0NI! 1 I - orce

Anal ,'i '

2 +3 Int rod& ct ion

2 43 Ne5t on6 ' La5

2 73 !,p e' of force Ana l,'i '

2 83 %rin cip le of #&p e r %o'it ion

2 *3 ree . od, Diagram

2 93 D6 Alem) e rt ' % rin cip le

2 :3 D,namic Anal,'i' of o& r ) ar M ech an i'm

2 ;3 !& rn ing M omen t Diagram

2


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2 +3 Int rod& ct ion$

0 N I ! 1 I - o rc e A n a l, ' i'

If the acce#eration of oing #ins in a echanis is running 4ith consi"era%#e

aount of #inear an">or angu#ar acce#erations( inertia forces are generate" an" these inertiaforces a#so ust %e oercoe %$ the "riing otor as an a""ition to the forces e?erte" %$ the

e?terna# #oa" or 4or the echanis "oes.

2 43 Ne5t on6 ' La5 $

i r't La5

0er$%o"$ 4i## persist in its state of rest or of unifor otion @constant e#ocit$ in a

straight #ine un#ess it is cope##e" to change that state %$ forces ipresse" on it. !his eans

that in the a%sence of a non-Bero net force( the center of ass of a %o"$ either is at rest or

oes at a constant e#ocit$.

# econ d La5

A %o"$ of ass m su%ject to a force un"ergoes an acce#eration a that has the sae"irection as the force an" a agnitu"e that is "irect#$ proportiona# to the force an" inerse#$

proportiona# to the ass( i.e.( ma. A#ternatie#$( the tota# force app#ie" on a %o"$ is eDua#

to the tie "eriatie of #inear oentu of the %o"$.

!h ird La5

!he utua# forces of action an" reaction %et4een t4o %o"ies are eDua#( opposite an"

co##inear. !his eans that 4heneer a first %o"$ e?erts a force on a secon" %o"$( the secon"

%o"$ e?erts a force E on the first %o"$. an" E are eDua# in agnitu"e an" opposite in

"irection. !his #a4 is soeties referre" to as the action-reaction law( 4ith ca##e" the)action) an" E the )reaction)

2 73 !,p e' of force Ana l,'i '$

0Dui#i%riu of e%ers 4ith t4o forces

0Dui#i%riu of e%ers 4ith three forces

0Dui#i%riu of e%ers 4ith t4o forces an" torDue

0Dui#i%riu of e%ers 4ith t4o coup#es.

0Dui#i%riu of e%ers 4ith four forces.


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2 83 %rin cip le of #&p e r %o'it ion $

Soeties the nu%er of e?terna# forces an" inertia# forces acting on a echanis

are too uch for graphica# so#ution. In this case 4e app#$ the etho" of superposition. 6sing

superposition the entire s$ste is %roen up into @n pro%#es( 4here n is the nu%er of

forces( %$ consi"ering the e?terna# an" inertia# forces of each #in in"ii"ua##$. esponse of a#inear s$ste to seera# forces acting siu#taneous#$ is eDua# to the su of responses of the

s$ste to the forces in"ii"ua##$. !his approach is usefu# %ecause it can %e perfore" %$

graphica##$.

2 *3 ree . od, Diagram$

A free %o"$ "iagra is a pictoria# representation often use" %$ ph$sicists an"engineers to ana#$Be the forces acting on a %o"$ of interest. A free %o"$ "iagra sho4s a##

forces of a## t$pes acting on this %o"$. *ra4ing such a "iagra can ai" in so#ing for the

unno4n forces or the eDuations of otion of the %o"$. Creating a free %o"$ "iagra

can ae it easier to un"erstan" the forces( an" torDues or oents( in re#ation to one

another an" suggest the proper concepts to app#$ in or"er to fin" the so#ution to a pro%#e. !he"iagras are a#so use" as a conceptua# "eice to he#p i"entif$ the interna# forcesFfor e?ap#e(

shear forces an" %en"ing oents in %easF4hich are "ee#ope" 4ithin structures.

2 93 D6 Alem) e rt ' % rin cip le $

*GA#e%ertGs princip#e( a#so no4n as the Lagrange1dAlem)ert principle( is a

stateent of the fun"aenta# c#assica# #a4s of otion. It is nae" after its "iscoerer( theHrench ph$sicist an" atheatician 5ean #e on" "GA#e%ert. !he princip#e states that the su

of the "ifferences %et4een the forces acting on a s$ste an" the tie "eriaties of the

oenta of the s$ste itse#f a#ong an$ irtua# "isp#aceent consistent 4ith the constraints ofthe s$ste is Bero.

2 :3 D,namic Anal,'i' of o& r ) ar M ech ani'm $

A fo&r-)ar lin>age or sip#$ a 8-)ar or fo&r-)ar is the sip#est oa%#e #inage.

It consists of four rigi" %o"ies @ca##e" %ars or #ins( each attache" to t4o others %$ sing#e jointsor piots to for c#ose" #oop. Hour-%ars are sip#e echaniss coon in echanica#

engineering achine "esign an" fa## un"er the stu"$ of ineatics.

*$naic Ana#$sis of eciprocating engines.

Inertia force an" torDue ana#$sis %$ neg#ecting 4eight of connecting ro". e#ocit$ an" acce#eration of piston.

Angu#ar e#ocit$ an" Angu#ar acce#eration of connecting ro".

Horce an" !orDue Ana#$sis in reciprocating engine neg#ecting the 4eight ofconnecting ro".

0Duia#ent *$naica# S$ste

*eterination of t4o asses of eDuia#ent "$naica# s$ste


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2 ;3 !& rn in g M omen t Diagram$

!he turning oent "iagra is graphica# representation of the turning oent or

cran effort for arious positions of cran.

2


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2 +=3 ! & r n in g m o m e n t d iag r am f o r 8- ' t r o > e I ? C e n gi n e $


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2 ++3 ! & r n in g m o m e n t d iag r am f o r a m & lt i c, lin d e r e n gi n e:


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2 +73 l& c t& a t ion of #p ee d:

!his is "efine" as the ratio of the "ifference %et4een the a?iu an" iniuangu#ar spee"s "uring a c$c#e to the ean spee" of rotation of the cran shaft.

2 +83 M a @ im & m f l& c t& a t ion of e n e r g, $


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2 +*3 Co e ff ici e n t of f l& c t& a t ion of e n e r g,$


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2 +93 Co e ff ici e n t of f l& c t& a t ion of ' p ee d $

2 +:3 E n e r g, ' t o r e d in f l,5h e el:

A f#$4hee# is a rotating ass that is use" as an energ$ reseroir in a achine. It

a%sor%s energ$ in the for of inetic energ$( "uring those perio"s of cran rotation 4henactua# turning oent is greater than the resisting oent an" re#ease energ$( %$ 4a$ of

parting 4ith soe of its /.0( 4hen the actua# turning oent is #ess than the resisting oent.

2 +;3 l,5heel in p&n ch ing p re' '$

!he f#$4hee#s use" for prie oers constitute a c#ass of pro%#es in 4hich the

resisting torDue is assue" to %e constant an" the "riing torDue aries. f#$4hee#s use" in

punching( rieting an" sii#ar achines constitute another c#ass of pro%#es in 4hich theactua#@"riing turning oent proi"e" %$ an e#ectric otor is ore or #ess constant %ut the

resisting torDue@#oa" aries.


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2 4=3 E@amp le % ro) lem '$

1 !he ariation of cranshaft torDue of a four c$#in"er petro# engine a$ %e

appro?iate#$ represente" %$ taing the torDue as Bero for cran ang#es 2. Ca#cu#ate

@i Cran-pin effort

an"

@ii !he effectie turning oent on the cran

shaft.


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INDE/

63 I! L II ' A2A3C I3-


2 43 . alancin g of rot at in g ma''e'

2 73 #t at ic ) alan cin g

2 83 D,namic ) alancin g

2 *3 Vario& ' ca'e' of ) alan cin g of rot at in g ma''e '

2 93 . alancin g of a 'in gle rot at in g ma'' ) , 'in gle ma'' rot atin g in th e 'ame p lan e

2 :3 . alancin g of a 'in gle rot at in g ma'' ) , t 5o ma''e ' rot at ing in th e d ifferen t p lane

2 ;3 . alancin g of a 'eBera l ma' 'e' rot at in g in 'ame plan e

2


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0NI ! 1 I I .AL AN CI NG


'a#ancing is the process of e#iinating or at #east re"ucing the groun" forces an">or

oents. It is achiee" %$ changing the #ocation of the ass centres of #ins. 'a#ancing ofrotating parts is a 4e## no4n pro%#e. A rotating %o"$ 4ith fi?e" rotation a?is can %e fu##$

%a#ance" i.e. a## the inertia forces an" oents. Hor echanis containing #ins rotating a%outa?is 4hich are not fi?e"( force %a#ancing is possi%#e( oent %a#ancing %$ itse#f a$

%e possi%#e( %ut %oth not possi%#e. e genera##$ tr$ to "o force %a#ancing. A fu##$ force

%a#ance is possi%#e( %ut an$ action in force %a#ancing seere the oent %a#ancing.

2 43 . alancin g of rot at in g ma''e'$

!he process of proi"ing the secon" ass in or"er to counteract the effect ofthe centrifuga# force of the first ass is ca##e" %a#ancing of rotating asses.

2 73 #t at ic ) alan cin g $

!he net "$naic force acting on the shaft is eDua# to Bero. !his reDuires that the #ine of

action of three centrifuga# forces ust %e the sae. In other 4or"s( the centre of the asses

of the s$ste ust #ie on the a?is of the rotation. !his is the con"ition for static %a#ancing.

2 83 D,namic ) alancin g $

!he net coup#e "ue to "$naic forces acting on the shaft is eDua# to Bero. !hea#ge%raic su of the oents a%out an$ point in the p#ane ust %e Bero.

2 *3 Vario& ' ca'e' of ) alan cin g of rot at in g ma''e '$

'a#ancing of a sing#e rotating ass %$ sing#e ass rotating in thesae p#ane.

'a#ancing of a sing#e rotating ass %$ t4o asses rotating in the"ifferent p#ane.

'a#ancing of a seera# asses rotating in sing#e p#ane.

'a#ancing of a seera# asses rotating in "ifferent p#anes.


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2 :3 . a la n c in g of a ' in g le r o t a t in g m a ' ' ) , t 5o m a '' e ' r o t a t in g in t h e d iff ere n t p la n e $


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2 ;3 . a la n c in g of a ' e B e r a l m a ' ' e ' r o t a t in g in ' a m e p la n e $


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2


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2 +=3 . alan cin g of Recip rocat in g ma''e '$

Mass %a#ancing encopasses a 4i"e arra$ of easures ep#o$e" to o%tain partia# orcop#ete copensation for the inertia# forces an" oents of inertia eanating fro the

cranshaft asse%#$. A## asses are e?terna##$ %a#ance" 4hen no free inertia# forces or

oents of inertia are transitte" through the %#oc to the outsi"e. 7o4eer( the reaining

interna# forces an" oents su%ject the engine ounts an" %#oc to arious #oa"s as 4e## as

"eforities an" i%rator$ stresses. !he %asic #oa"s ipose" %$ gas-%ase" an" inertia# forces


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2 ++3 % r im a r , a n d ' ec o nd a r, &n) a la n c e d f o rce ' of rec ip r o c a t in g p a r t ' $


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Page ="o4n oeent fro the

connecting ro"Gs change of ang#e has the opposite "irection of the ajor contri%ution to thepistonGs up>"o4n oeent fro the up>"o4n oeent of the cran pin. !he piston therefore

trae#s faster in the top ha#f of the c$#in"er than it "oes in the %otto ha#f( 4hi#e the otion of

the cranshaft 4eights is sinusoi"a#. !he ertica# otion of the piston is therefore not Duite thesae as that of the %a#ancing 4eight( so the$ canGt %e a"e to cance# out cop#ete#$.

Secon"#$( there is a i%ration pro"uce" %$ the change in spee" an" therefore

inetic energ$ of the piston. !he cranshaft 4i## ten" to s#o4 "o4n as the piston spee"s up an"

a%sor%s energ$( an" to spee" up again as the piston gies up energ$ in s#o4ing "o4n at the topan" %otto of the stroe. !his i%ration has t4ice the freDuenc$ of the first i%ration( an"

a%sor%ing it is one function of the f#$4hee#.

!hir"#$( there is a i%ration pro"uce" %$ the fact that the engine is on#$ pro"ucing

po4er "uring the po4er stroe. In a four-stroe engine this i%ration 4i## hae ha#f the

freDuenc$ of the first i%ration( as the c$#in"er fires once eer$ t4o reo#utions. In a t4o-stroe engine( it 4i## hae the sae freDuenc$ as the first i%ration. !his i%ration is a#so

a%sor%e" %$ the f#$4hee#.

2 +73 . alan cin g of in ertial fo rc e' in the m& lt i - c,lind er en gine $

In u#ti-c$#in"er engines the utua# counteractions of the arious coponents in the

Cran shaft asse%#$ are one of the essentia# factors "eterining the se#ection of the

Cran shafts configuration an" 4ith it the "esign of the engine itse#f. !he inertia# forces are'a#ance" if the coon centre of grait$ for a## oing cranshaft-asse%#$ coponents #ies

at the cranshaftGs i"point( i.e. if the cranshaft is s$etrica# @as ie4e" fro the

front. !he cranshaftGs s$etr$ #ee# can %e "efine" using geoetrica# representations of1st - an" 2n"- or"er forces @star "iagras. !he 2n" or"er star "iagra for the four-c$#in"er in-


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#ine engine is as$etrica#( eaning that this or"er is characteriBe" %$ su%stantia# free inertia#

Horces. !hese


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forces can %e %a#ance" using t4o countershafts rotating in opposite "irections at "ou%#e therate of the cranshaft @anchester s$ste.

2 +83 %art ial ) alan cin g of Loco mo t iBe'$

2 +*3 Va r ia t ion of ! r a c t iBe f o r c e:

!he resu#tant un%a#ance" force "ue to the c$#in"ers( a#ong the #ine of stroe( is no4nas tractie force.


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2 +93 # 5a,in g Co&p le$

!he coup#e has s4a$ing effect a%out a ertica# a?is( an" ten"s to s4a$ the

engine a#ternate#$ in c#oc 4ise an" antic#oc4ise "irections. 7ence the coup#e is no4n as

s4a$ing coup#e.


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2 +;3 . alan cin g of In line en gin e'$

An in-#ine engine is one 4herein a## the c$#in"ers are arrange" in a sing#e #ine( one

%ehin" the other as scheatica##$ in"icate" in Hig. Man$ of the passenger cars foun" on In"ian

roa"s such as Maruti +gas tur%ines #e" to the near e?tinction of these engines. 7o4eer it is

sti## interesting to stu"$ their state of %a#ance in ie4 of soe e#egant resu#ts 4e sha## "iscuss

short#$. Kur etho" of ana#$sis reains i"entica# to the preious case i.e.( 4e procee"4ith the assuption that a## c$#in"ers are i"entica# an" the c$#in"ers are space" at unifor

intera# aroun" the circuference.


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1 A shaft carries four rotating asses A( '( C an" * 4hich are cop#ete#$ %a#ance".

!he asses '( C an" * are 8< g( +< g an" < g respectie#$. !he asses C an" * ae

ang#es of

9


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INDE/

0N I ! 1II I RE E VI . RA! I ON#


2 43 . a'ic element ' of Bi) rat ion ','t em

2 73 Ca&'e' of Bi) rat ion

2 83 Effect ' o f Bi)rat ion

2 *3 M eth od ' of red& ct ion of Bi) rat ion

2 93 !,p e' of Bi) rat or, mot ion

2 :3 !erm ' & 'ed Bi) rat o r, mot ion

2 ;3 Degree' of f reed om

2


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2 473 Logarit h mic d ecre men t

2 483 !ran 'Be r' e Vi) rat ion

2 4*3 hirlin g of 'h aft

2 493 !or'ion al Vi) rat ion

2 4:3 !or'ion a l Bi) rat ion of a 'in gle rot or ', 't e m

2 4;3 !or'ion al Bi) rat ion of a t 5o rot or ', 't em

2 4


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0 N I ! 1 II I R E E V I . R A ! I ON#

hen a s$ste is su%jecte" to an initia# "istur%ance an" then #eft free to i%rate on

its o4n( the resu#ting i%rations are referre" to as free i%rations .ree Bi)ration occurs 4hen a

echanica# s$ste is set off 4ith an initia# input an" then a##o4e" to i%rate free#$. 0?ap#esof this t$pe of i%ration are pu##ing a chi#" %ac on a s4ing an" then #etting go or hitting a

tuning for an" #etting it ring. !he echanica# s$ste 4i## then i%rate at one or ore of its

)natura# freDuencies) an" "ap "o4n to Bero.

2 43 . a'ic element ' of Bi) rat ion ','t em$

Mass or Inertia

Springiness or estoring e#eent

*issipatie e#eent @often ca##e" "aper

0?terna# e?citation

2 73 Ca&'e' of Bi) rat ion $

0n ) a la n ce $ !his is %asica##$ in reference to the rotating %o"ies. !he uneen"istri%ution of ass in a rotating %o"$ contri%utes to the un%a#ance. A goo" e?ap#e of

un%a#ance re#ate" i%ration 4ou#" %e the ;i%rating a#ert in our o%i#e phones. 7ere asa## aount of un%a#ance" 4eight is rotate" %$ a otor causing the i%ration 4hich aesthe o%i#e phone to i%rate. You 4ou#" hae e?perience" the sae sort of i%ration occurring

in $our front #oa"e" 4ashing achines that ten" to i%rate "uring the ;spinningo"e.

M i' a lig n m e nt $ !his is an other ajor cause of i%ration particu#ar#$ in achines thatare "rien %$ otors or an$ other prie oers.

. e n t #h a f t $ A rotating shaft that is %ent a#so pro"uces the the i%rating effect since it#osses it rotation capa%i#it$ a%out its center.

Gea r ' in th e m a c h ine: !he gears in the achine a#4a$s ten" to pro"uce

i%ration( ain#$ "ue to their eshing. !hough this a$ %e contro##e" to soe e?tent( an$pro%#e in the gear%o? ten"s to get enhance" 4ith ease.

. e a r in g ' $ ast %ut not the #east( here is a ajor contri%utor for i%ration. In ajorit$ of

the cases eer$ initia# pro%#e starts in the %earings an" propagates to the rest of the e%ersof the achine. A %earing "eoi" of #u%rication ten"s to 4ear out fast an" fai#s Duic#$( %ut

%efore this is notice" it "aages the reaining coponents in the achine an" an initia# #oo4ou#" see as if soething ha" gone 4rong 4ith the other coponents #ea"ing to the %earing

fai#ure.


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2 83 Effect ' of Bi) rat ion $

2a3.ad Effect'$!he presence of i%ration in an$ echanica# s$ste pro"uces un4ante" noise(

high stresses( poor re#ia%i#it$( 4ear an" preature fai#ure of parts. i%rations are a greatsource ofhuan "iscofort in the for of ph$sica# an" enta# strains.

2)3Good Effect'$A i%ration "oes usefu# 4or in usica# instruents( i%rating screens( shaers( re#ie

pain in ph$siotherap$.

2 *3 M eth od ' of red& ct ion of Bi) rat ion $

-un%a#ance is its ain cause( so %a#ancing of parts is necessar$.

-using shoc a%sor%ers. -using "$naic i%ration a%sor%ers.

-proi"ing the screens @if noise is to %e re"uce"

2 93 ! , p e ' of Bi ) r a t o r, m o t io n:

Hree i%ration

Horce" i%ration

2 :3 !erm ' & 'ed Bi) rat o r, mot ion$

2a3!ime period 2or3period of Bi)ration$

It is the tie taen %$ a i%rating %o"$ to repeat the otion itse#f.tie perio" isusua##$ e?presse" in secon"s.

2)3 C,cle:It is the otion cop#ete" in one tie perio".

2c3 %eriodic motion$A otion 4hich repeats itse#f after eDua# intera# of tie.

2d3Amplit&de 2/3!he a?iu "isp#aceent of a i%rating %o"$ fro the ean position.it is usua##$

e?presse" in i##ieter.

2e3 re&enc, 2f3!he nu%er of c$c#es cop#ete" in one secon" is ca##e" freDuenc$


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2 ;3 D e g r ee ' of f ree d o m:

!he iniu nu%er of in"epen"ent coor"inates reDuire" to specif$ the otion of

a s$ste at an$ instant is no4n as *.K.H of the s$ste.

2


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A t4o "egree of free"o s$ste ith reference to autoo%i#e app#ications( this is

referre" as ;Duarter caro"e#. !he %otto ass refers to ass of a?#e( 4hee# etc coponents

4hich are %e#o4 the suspension spring an" the top ass refers to the ass of the portion of the

car an" passenger. Since 4e nee" to specif$ %oth the top an" %otto ass positions to

cop#ete#$ specif$ the s$ste( this %ecoes a t4o "egree of free"o s$ste.

2 ++3 !,p e' of Vi) rat or, mot ion $

2 +43 !,p e' of Vi) rat ion $

@aongitu"ina# i%ration @%!ranserse i%ration @ c!orsiona# i%ration.


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2 +73 Lon git&d in al Vi)rat ion$

hen the partic#es of the shaft or "isc oes para##e# to the a?is of the shaft( then

the i%rations no4n as #ongitu"ina# i%rations.

2 +83 ree & nd amp ed lon git &d in al Bi) rat ion '

hen a %o"$ is a##o4e" to i%rate on its o4n( after giing it an initia#

"isp#aceent( then the ensuring i%rations are no4n as free or natura# i%rations. hen thei%rations tae p#ace para##e# to the a?is of constraint an" no "aping is proi"e"( then

it is ca##e" free un"ape" #ongitu"ina# i%rations.

2 +*3 Na t & r al f r e & e n c , of f r e e &nd a mp e d lo n gi t &d in al Bi ) r a t io n:

2 +* ? a3 E & ili) r i& m m e t h od or N e 5t on 6 ' m e t h od $


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2+*?)3Energ, MethodIn free i%rations( no energ$ is transferre" into the s$ste or fro the s$ste. !herefore(

the tota# energ$ @su of /0 an" P0is constant an" is sae a## the ties.


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2c3Ra,leigh6' methodIn this etho"( the a?iu inetic energ$ at ean position is a"e eDua# to

the a?iu potentia# energ$ at the e?tree position.

2 +93 E& iBalen t 'tiffn e'' of 'p rin g?

@1 Springs in series

@2 Springs in para##e#@= Co%ine" springs

@, Inc#ine" springs


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2 +:3 Da m p ing $It is the resistance to the otion of a i%rating %o"$. !he i%rations associate" 4ith this

resistance are no4n as "ape" i%rations.

2 +;3 !,p e' of d amp ing$

@1 iscous "aping@2 *r$ friction or cou#o% "aping

@= So#i" "aping or structura# "aping@, S#ip or interfacia# "aping.


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2 +


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2 483 !ran 'Be r' e Vi) rat ion $

hen the partic#es of the shaft or "isc oes appro?iate#$ perpen"icu#ar to the a?is

of the shaft( then the i%rations no4n as transerse i%rations.


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2 4*3 h ir lin g ' p ee d of ' h a f t:

!he spee"( at 4hich the shaft runs so that the a""itiona# "ef#ection of the shaft fro

the a?is of rotation %ecoes infinite( is no4n as critica# or 4hir#ing spee".

3o shaft can eer %e perfect#$ straight or perfect#$ %a#ance". hen an e#eent of ass is a

"istance fro the a?is of rotation( centrifuga# force( 4i## ten" to pu## the ass out4ar".

!he e#astic properties of the shaft 4i## act to restore the ;straightness. If the freDuenc$ of

rotation is eDua# to one of the resonant freDuencies of the shaft( 4hir#ing 4i## occur. In or"er to

sae the achine fro fai#ure( operation at such 4hir#ing spee"s ust %e aoi"e".

!he 4hir#ing freDuenc$ of a s$etric cross section of a gien #ength %et4een t4o points is

gien %$:

PM

here 0 $oungGs o"u#us( I Secon" oent of area( ass of the shaft(

#ength of the shaft %et4een points

A shaft 4ith 4eights a""e" 4i## hae an angu#ar e#ocit$ of 3 @rp eDuia#ent as fo##o4s:


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Page 8=of 92

2 493 !or'ion al Vi) rat ion $

hen the partic#es of the shaft or "isc oe in a circ#e a%out the a?is of the shaft(

then the i%rations no4n as tensiona# i%ration


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2 4:3 ! o r ' io n al Bi ) r a t ion of a ' in g le r o t o r ' , ' t e m $


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2 4;3 ! o r ' io n al Bi ) r a t ion of a t 5o r o t or ' , ' t e m $


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1 A spring ass s$ste has spring stiffness of ; 3> an" a ass of ;M g. It has the

natura# freDuenc$ of i%ration as 12 7B. An e?tra 2 g ass is coup#e" to M an" the natura#

freDuenc$ re"uces %$ 2 7B. Hin" the a#ues of an" ;M.

2 A steppe" shaft of . *eterine the un"ape" an" the

"ape" natura# freDuencies of the s$ste in ertica# "irection( neg#ecting the ass of iso#ators.


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2 73 orced Bi) rat ion of a 'in gle d egree - of- f reed o m ','t e m$

e sa4 that 4hen a s$ste is gien an initia# input of energ$( either in the

for of an initia# "isp#aceent or an initia# e#ocit$( an" then re#ease" it 4 i##( un"er

the right con"itions( i%rate free#$. If there is "aping in the s$ste( then the osci##ations "ie

a4a$. If a s$ste is gien a continuous input of energ$ in the for of a continuous#$ app#ie"force or a continuous#$ app#ie" "isp#aceent( then the conseDuent i%ration is ca##e" force"

i%ration. !he energ$ input can oercoe that "issipate" %$ "aping echaniss an" the

osci##ations are sustaine".

e 4i## consi"er t4o t$pes of force" i%ration. !he first is 4here the groun" to

4hich the s$ste is attache" is itse#f un"ergoing a perio"ic "isp#aceent( such as the i%ration

of a %ui#"ing in an earthDuae. !he secon" is 4here a perio"ic force is app#ie" to the ass( oro%ject perforing the otionQ an e?ap#e ight %e the forces e?erte" on the %o"$ of a car %$

the forces pro"uce" in the engine. !he sip#est for of perio"ic force or "isp#aceent is

sinusoi"a#( so 4e 4i## %egin %$ consi"ering force" i%ration "ue to sinusoi"a# otion of the

groun". In a## rea# s$stes( energ$ 4i## %e "issipate"( i.e. the s$ste 4i## %e "ape"( %ut often

the "aping is er$ sa##. So #et us first ana#$Be s$stes in 4hich there is no "aping.

(83 #t ead, #t at e Re'p on 'e d & e t o Harmonic O'c il lat ion $

Consi"er a spring-ass-"aper s$ste as sho4n in figure ,.1. !he eDuation

of otion of this s$ste su%jecte" to a haronic force can %e gien %$

@,.1

4here( m ( k an" c are the ass( spring stiffness an" "aping coefficient of the s$ste( F isthe ap#itu"e of the force( w is the e?citation freDuenc$ or "riing freDuenc$.

Higure ,.1 7aronica##$ e?cite" s$ste


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Higure ,.2: Horce po#$gon

!he stea"$ state response of the s$ste can %e "eterine" %$ so#ing eDuation@,.1 in an$

"ifferent 4a$s. 7ere a sip#er graphica# etho" is use" 4hich 4i## gie ph$sica#un"erstan"ing to this "$naic pro%#e. Hro so#ution of "ifferentia# eDuations it is no4n that

the stea"$ state so#ution @particu#ar integra# 4i## %e of the for

@,.2

As each ter of eDuation @,.1 represents a forcing ter iB.( first( secon" an" thir" ters(

represent the inertia force( spring force( an" the "aping forces. !he ter in the right han"si"e of eDuation @,.1 is the app#ie" force. Kne a$ "ra4 a c#ose po#$gon as sho4n in figure

,.2 consi"ering the eDui#i%riu of the s$ste un"er the action of these forces.

Consi"ering a reference #ine these forces can %e presente" as fo##o4s.

Spring force @!his force 4i## ae an ang#e 4ith

the reference #ine( represente" %$ #ine KA.

*aping force @!his force 4i## %e perpen"icu#ar to the

spring force( represente" %$ #ine A'.

Inertia force @this force is perpen"icu#ar to the

"aping force an" is in opposite "irection 4ith the spring force an" is represente" %$#ine 'C .

App#ie" force 4hich can %e "ra4n at an ang#e 4ith respect to thereference #ine an" is represente" %$ #ine KC.

Hro eDuation @1( the resu#tant of the spring force( "aping force an" the inertia force 4i## %ethe app#ie" force( 4hich is c#ear#$ sho4n in figure ,.2.

It a$ %e note" that ti## no4( 4e "onGt no4 a%out the agnitu"e ofX an" 4hich can %e easi#$ copute"

froHigure 2. *ra4ing a #ine C* para##e# to A'( fro the triang#e KC* of Higure 2(


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Hro the preious o"u#e of free-i%ration it a$

%e reca##e" that

3atura# freDuenc$

Critica# "aping

*aping factor or "aping ratio

7ence(

or

As the ratio is the static "ef#ection of the spring( is no4nas the agnification factor or ap#itu"e ratio of the s$ste


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2 *3 o r ce d Bi ) r a t ion 5i t h d a mp in g$

In this section 4e 4i## see the %ehaiour of the spring ass "aper o"e# 4hen 4e

a"" a haronic force in the for %e#o4. A force of this t$pe cou#"( for e?ap#e( %e generate"

%$ a rotating i%a#ance.

If 4e again su the forces on the ass 4e get the fo##o4ing or"inar$ "ifferentia#

eDuation: !he s t ea " $ s t a t e so#ution of this pro%#e can %e 4ritten as:

!he resu#t states that the ass 4i## osci##ate at the sae freDuenc$( f( of the app#ie"

force( %ut 4ith a phase shift R.

!he ap#itu"e of the i%ration ;S is "efine" %$ the fo##o4ing

foru#a.

here ;ris "efine" as the ratio of the haronic force freDuenc$ oer the un"ape"

natura# freDuenc$ of the assLspringL"aper

o"e#.

!he phase shift ( R( is "efine" %$ the fo##o4ing

foru#a.

!he p#ot of these functions( ca##e" )the freDuenc$ response of the s$ste)( presents one of the

ost iportant features in force" i%ration. In a #ight#$ "ape" s$ste 4hen the
http://en.wikipedia.org/wiki/Steady_statehttp://en.wikipedia.org/wiki/Steady_state


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high. !his phenoenon is ca##e" r e ' o n a n c e @su%seDuent#$ the natura# freDuenc$ of a s$ste

is often referre" to as the resonant freDuenc$. In rotor %earing s$stes an$ rotationa# spee"

that e?cites a resonant freDuenc$ is referre" to as a c r it ic a # s p e e ".

If resonance occurs in a echanica# s$ste it can %e er$ harfu# L #ea"ing

to eentua# fai#ure of the s$ste. ConseDuent#$( one of the ajor reasons for i%ration ana#$sis

is to pre"ict 4hen this t$pe of resonance a$ occur an" then to "eterine 4hat steps to

tae to preent it fro occurring. As the ap#itu"e p#ot sho4s( a""ing "aping can

significant#$ re"uce the agnitu"e of the i%ration. A#so( the agnitu"e can %e re"uce" if the

natura# freDuenc$ can %e shifte" a4a$ fro the forcing freDuenc$ %$ changing the stiffness or

ass of the s$ste. If the s$ste cannot %e change"( perhaps the forcing freDuenc$ can

%e shifte" @for e?ap#e( changing the spee" of the achine generating the force.

!he fo##o4ing are soe other points in regar"s to the force" i%ration sho4n in the freDuenc$

response p#ots.

At a gien freDuenc$ ratio( the ap#itu"e of the i%ration( X( is "irect#$ proportiona#

to the ap#itu"e of the forceF< @e.g. if $ou "ou%#e the force( the i%ration "ou%#es

ith #itt#e or no "aping( the i%ration is in phase 4ith the forcing freDuenc$ 4hen

the freDuenc$ ratio r 1 an" 1+< "egrees out of phase 4hen the freDuenc$ ratio r T 1

hen r 1 the ap#itu"e is just the "ef#ection of the spring un"er the staticforceF


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2 93 Rot at in g &n) alan ce forc ed Bi) rat ion $

Kne a$ fin" an$ rotating s$stes in in"ustria# app#ications. !he un%a#ance"

force in such a s$ste can %e represente" %$ a ass m 4ith eccentricit$ e ( 4hich is rotating

4ith angu#ar e#ocit$ as sho4n in Higure ,.1.

Higure ,.1 : i%rating s$ste 4ith rotating un%a#ance

Figure 4.2. Freebody diagram of the system

etx%e the "isp#aceent of the nonrotating ass @M-m fro the static

eDui#i%riu position( then the "isp#aceent of the rotating ass m is

Hro the free%o"$ "iagra of the s$ste sho4n in figure ,.2( the eDuation of otion is


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(4.1)

or (4.2)

!his eDuation is sae as eDuation @1 4hereF is rep#ace" %$ . So fro the force

po#$gon as sho4n in figure ,.=

(4.3)

or (4.4)

or (4.5)

Figure 4.3: Force polygon

or (4.6)


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and (4.)

So the cop#ete so#ution %ecoes

(4.!)

2 :3 Vi) rat ion I'olat ion an d !ran' mi' 'i) ilit, $

hen a achine is operating( it is su%jecte" to seera# tie ar$ing forces %ecause of 4h

ten"s to e?hi%it i%rations. In the process( soe of these forces are transitte" to the foun"ati4hich cou#" un"erine the #ife of the foun"ation an" a#so affect the operation of an$ other achine osae foun"ation. 7ence it is of interest to iniiBe this force transission. Sii#ar#$ 4hen a s$ste

su%jecte" to groun" otion( part of the groun" otion is transitte" to the s$ste as 4e just "iscu

e.g.( an autoo%i#e going on an uneen roa"Q an instruent ounte" on the i%rating surface of an aietc. In these cases( 4e 4ish to iniiBe the otion transitte" fro the groun" to the s$ste.

consi"erations are use" in the "esign of achine foun"ations an" in or"er to un"erstan" soe of the %

issues ino#e"( 4e 4i## stu"$ this pro%#e %ase" on the sing#e ".o.f o"e# "iscusse" so far.

4e get the e?pression for force transitte" to the %ase as fo##o4s:

2 ;3 Vi) rat ion I'olat or'$

Consi"er a i%rating achineQ %o#te" to a rigi" f#oor @Higure 2a.!he force

transitte" to the f#oor is eDua# to the force generate" in the achine. !he transitte"force can %e "ecrease" %$ a""ing a suspension an" "aping e#eents @often ca##e"

i%ration iso#ators Higure 2% ( or %$ a""ing 4hat is ca##e" an inertia %#oc( a #arge ass

@usua##$ a %#oc of cast concrete( "irect#$ attache" to the achine @Higure 2c.Another

option is to a"" an a""itiona# #ee# of ass @soeties ca##e" a seisic ass( again a %#oc

of cast concrete an" suspension @Higure 2".


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2+=3 E @amp l e %ro) le m'$

1 A ass of 1< g is suspen"e" fro one en" of a he#ica# spring( the other en" %eing fi?e".

!he stiffness of the spring is 1< 3>. !he iscous "aping causes the ap#it u"e to "ecrease

to one tenth of the initia# a#ue in four cop#ete osci##ations. If a perio"ic force of 18< cos 8< t

3 is app#ie" at the ass in the ertica# "irection( fin" the ap#itu"e of the force" i%rations.

hat is its a#ue at resonanceW

2 A achine supporte" s$etrica##$ on four springs has a ass of +< g. !he ass of the

reciprocating parts is 2.2 g 4hich oe through a ertica# stroe of 1


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2 4 * 3 E f f e c t o f G , r o ' c o p i c c o & p l e o n a N a B a l # h i p d & r i n g

Rol lin g

2 4 9 3 E f f e c t o f G , r o ' c o p i c c o & p l e o n a 8 - 5 h e e l d r i B e

24:3 E @ a mp le % r o ) le m '


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0NI! - V ME CHANI #M OR CON! ROL

G oBe r nor

2 + 3 I nt r o d& c t io n $

A centrif&gal goBernor is a specific t$pe of g o e r n or that contro#s the s p e e "of an e n g ine%$ regu#ating the aount of f u e # @or 4o r in g f #u i" a"itte"( so as to aintain a

near constant spee" 4hateer the #o a " or fue# supp#$ con"itions. It uses the princip#e of

p r o p o r t io n a#c o n t r o #.

It is ost o%ious#$ seen on s t e a e n g in e s 4here it regu#ates the a"ission of steainto the c $ #in" e r@s. It is a#so foun" on in t e r n a # c o %u s t io n e n g in e s an" arious#$ fue##e"

t u r % in e s(an" in soe o"ern s t r i ing c #o c s.

2 43 %rin cip le of or>in g $

Po4er is supp#ie" to the goernor fro the engineGs output shaft %$ @inthis instance a %e#t or chain @not sho4n connecte" to the #o4er %e#t 4hee#. !hegoernor is connecte" to a t h r ot t #e a#e that regu#ates the f#o4 of 4 o r in g f #u i" @stea

supp#$ing the p r ie o e r @prie oer not sho4n. As the spee" of the prie oer

increases( the centra# spin"#e of the goernor rotates at a faster rate an" the inetic energ$ ofthe %a##s increases. !his a##o4s the t4o a ss e s on #eer ars to oe out4ar"s an" up4ar"s

against grait$. If the otion goes far enough( this otion causes the #eer ars to pu## "o4n

on a t h r u s t % ea r in g( 4hich oes a %ea #inage( 4hich re"uces the a p e r t u r e of a thrott#e

a#e. !he rate of 4oring-f#ui" entering
http://en.wikipedia.org/wiki/Governor_(device)http://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/Enginehttp://en.wikipedia.org/wiki/Fuelhttp://en.wikipedia.org/wiki/Working_fluidhttp://en.wikipedia.org/wiki/Loadhttp://en.wikipedia.org/wiki/Proportional_controlhttp://en.wikipedia.org/wiki/Proportional_controlhttp://en.wikipedia.org/wiki/Steam_enginehttp://en.wikipedia.org/wiki/Cylinder_(engine)http://en.wikipedia.org/wiki/Internal_combustion_enginehttp://en.wikipedia.org/wiki/Turbinehttp://en.wikipedia.org/wiki/Turbinehttp://en.wikipedia.org/wiki/Striking_clockhttp://en.wikipedia.org/wiki/Throttlehttp://en.wikipedia.org/wiki/Working_fluidhttp://en.wikipedia.org/wiki/Prime_mover_(locomotive)http://en.wikipedia.org/wiki/Prime_mover_(locomotive)http://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Thrust_bearinghttp://en.wikipedia.org/wiki/Thrust_bearinghttp://en.wikipedia.org/wiki/Aperturehttp://en.wikipedia.org/wiki/Governor_(device)http://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/Enginehttp://en.wikipedia.org/wiki/Fuelhttp://en.wikipedia.org/wiki/Working_fluidhttp://en.wikipedia.org/wiki/Loadhttp://en.wikipedia.org/wiki/Proportional_controlhttp://en.wikipedia.org/wiki/Proportional_controlhttp://en.wikipedia.org/wiki/Steam_enginehttp://en.wikipedia.org/wiki/Cylinder_(engine)http://en.wikipedia.org/wiki/Internal_combustion_enginehttp://en.wikipedia.org/wiki/Turbinehttp://en.wikipedia.org/wiki/Striking_clockhttp://en.wikipedia.org/wiki/Throttlehttp://en.wikipedia.org/wiki/Working_fluidhttp://en.wikipedia.org/wiki/Prime_mover_(locomotive)http://en.wikipedia.org/wiki/Prime_mover_(locomotive)http://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Thrust_bearinghttp://en.wikipedia.org/wiki/Aperture


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the c$#in"er is thus re"uce" an" the spee" of the prie oer is contro##e"( preentingoer spee"ing.

Mechanica# stops a$ %e use" to #iit the range of thrott#e otion( as seen nearthe asses in the iage at right.

!he "irection of the #eer ar ho#"ing the ass 4i## %e a#ong the ec to r s uof the r eac t i e c e n t r if ug a # f o r c e ector an" the graitationa# force.

2 73 C la '' if ica t ion of goB e r n o r ' $

oernors are c#assifie" %ase" upon t4o "ifferent

princip#es. !hese are:

1. Centrifuga# goernors

2. Inertia goernors

Centrifuga# goernors are further c#assifie" as L

2 83 Heig h t of goB e r n or

It is the ertica# "istance %et4een the centre of the goernor ha##s an" the pointof intersection %et4een the upper ars on the a?is of spin"#e is no4n as goernor height. It is

genera##$ "enote" %$ h.
http://en.wikipedia.org/wiki/Vector_sumhttp://en.wikipedia.org/wiki/Reactive_centrifugal_forcehttp://en.wikipedia.org/wiki/Vector_sumhttp://en.wikipedia.org/wiki/Reactive_centrifugal_force


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2 *3 # leeBe lift

!he ertica# "istance the s#eee trae#s "ue to change in the eDui#i%riu

Spee" is ca##e" the s#eee #ift. !he ertica# "o4n4ar" trae# a$ %e tere"

as 3egatie #ift

2 93 I'och ron i' m

!his is an e?tree case of sensitieness. hen the eDui#i%riu spee" is constant for a##

ra"ii of rotation of the %a##s 4ithin the 4oring range( the goernor is sai" to %e inisochronis. !his eans that the "ifference %et4een the a?iu an" iniu eDui#i%riu

spee"s is Bero an" the sensitieness sha## %e infinite.

2 :3 #t a)ilit ,

Sta%i#it$ is the a%i#it$ to aintain a "esire" engine spee" 4ithout H#uctuating. Insta%i#it$resu#ts in hunting or osci##ating "ue to oer correction. 0?cessie sta%i#it$ resu#ts in a "ea"-%eat

goernor or one that "oes not correct sufficient#$ for #oa" changes

2 ;3 H&nt in g

!he phenoenon of continuous f#uctuation of the engine spee" a%oe an" %e#o4 the

ean spee" is tere" as hunting. !his occurs in oer-

sensitie or isochronous goernors. Suppose an isochronous goernor is fitte" to an enginerunning at a stea"$ #oa". ith a s#ight increase of #oa"( the spee" 4i## fa## an" the s#eee 4i##

ie"iate#$ fa## to its #o4est position. !his sha## open the contro# a#e 4i"e an" e?cess

supp#$ of energ$ 4i## %e gien( 4ith the resu#t that the spee" 4i## rapi"#$ increase an" thes#eee 4i## rise to its higher position. As a resu#t of this oeent of the s#eee( the contro#

a#e 4i## %e cut offQ the supp#$ to the engine an" the spee" 4i## again fa##( the c$c#e %eing

repeate" in"efinite#$. Such a goernor 4ou#" a"it either ore or #ess aount of fue# an" soeffect 4ou#" %e that the engine 4ou#" hunt.

2


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2 +=3 Charact eri't ic ' an d & alit ie' of cent rif& ga l goBern o r $

Hor satisfactor$ perforance an" 4oring a centrifuga# goernor shou#" possess

!he fo##o4ing Dua#ities.a. Kn the su""en reoa# of #oa" its s#eee shou#" reach at the top ost position at Knce.

%. Its response to the change of spee" shou#" %e fast.c. Its s#eee shou#" f#oat at soe intere"iate position un"er nora# operating Con"itions.

". At the #o4est position of s#eee the engine shou#" "ee#op a?iu po4er.e. It shou#" hae sufficient po4er( so that it a$ %e a%#e to e?ert the reDuire" force

At the s#eee to operate the contro# & echanis

2 ++3 a t t goB e r n o r $


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2 +43 % o r t e r goB e r n o r $


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2 +93 i l' on Ha r tn e ll goB e r n o r $


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2 +:3 %ic>e rin g goBe rn o r$

2 +;3 Di ff ere n c e ) e t 5 ee n a f l,5he e l a n d a goB e r n o r $


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G,r o' c ope a nd it' applic a ti on'

2 +


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2 4=3 De'cript ion an d d iagram$

*iagra of a g$ro 4hee#. eaction arro4s a%out the output a?is @%#ue correspon"

to forces app#ie" a%out the input a?is @green( an" ice ersa. ithin echanica# s$stes

or "eices( a conentiona# g$roscope is a echanis coprising a rotor journa# #e" to spina%out one a?is( the journa#s of the rotor %eing ounte" in an inner gi%a# or ring( the inner

gi%a# is journa# #e" for osci##ation in an outer gi%a# 4hich is journa# #e" in another gi%a#.

So %asica##$ there are three gi%a#s. !he o&ter gim)al or ring 4hich is the g$roscope frae isounte" so as to piot a%out an a?is in its o4n p#ane "eterine" %$ the support. !his outer

gi%a# possesses one "egree of rotationa# free"o an" its a?is possesses none. !he ne?t innergim)al is ounte" in the g$roscope frae @outer gi%a# so as to piot a%out an a?is in its o4n

p#ane that is a#4a$s perpen"icu#ar to the piota# a?is of the g$roscope frae @outer gi%a#.!his inner gi%a# has t4o "egrees of rotationa# free"o. Sii#ar#$( ne?t innermo't gim)alis attache" to the inner gi%a# 4hich has three "egree of rotationa# free"o an" its a?is

posses t4o. !he a?#e of the spinning 4hee# "efines the spin a?is. !he rotor is journa#e" to spina%out an a?is 4hich is a#4a$s perpen"icu#ar to the a?is of the innerost gi%a#. So( the

rotor possesses four "egrees of rotationa# free"o an" its a?is possesses three. !he 4hee#

respon"s to a force app#ie" a%out the input a?is %$ a reaction force a%out the output a?is.

!he %ehaior of a g$roscope can %e ost easi#$ appreciate" %$ consi"eration of

the front 4hee# of a %ic$c#e. If the 4hee# is #eane" a4a$ fro the ertica# so that the top of the4hee# oes to the #eft( the for4ar" ri of the 4hee# a#so turns to the #eft. In other 4or"s(

rotation on one a?is of the turning 4hee# pro"uces rotation of the thir" a?is.


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2 4+ 3 Effe c t of the G ,r o'c op i c C o&p l e on a n Aero plane

2 44 3 E E C ! O G " R O # C O %I C C O 0 % L E

!his coup#e is( therefore( to raise the nose an" "ip the tai# of the aero p#ane.

Note'1. hen the aero p#ane taes a right turn un"er sii#ar Con"itions as "iscusse" a%oe( the effectof the reactie Coup#e 4i## %e to "ip the nose an" raise the tai# of the aero p#ane.2. hen the engine or prope##er rotates in antic#oc4ise "irection 4hen ie4e" fro the rear ortai# en" an" the aero p#ane taes a #eft turn( then the effect of reactie g$roscopic coup#e 4i## %eto "ipthe nose an" raise the tai# of the aerop#ane.

=. hen the aero p#ane taes a right turn un"er sii#ar Con"itions as entione" in note 2a%oe( the effect of eactie g$roscopic coup#e 4i## %e to raise the nose an" "ip the of the aerop#ane.

,. hen the engine or prope##er rotates in c#oc4ise "irection 4hen ie4e" fro the front an"

the aero p#ane taes a #eft turn( then the effect of reactie g$roscopic coup#e 4i## %e to raise the

tai# an" "ip the nose of the aero p#ane.

8. hen the aero p#ane taes a right turn un"er sii#ar con"itions as entione" in note,

a%oe( the effect of reactie g$roscopic coup#e 4i## %e to raise the nose an" "ip the tai# of theaero p#ane.

2 473 Effect of g, ro 'cop ic co&ple on 'h ip

!he top an" front ie4s of a naa# ship are sho4n in fig. !he fore en" of the ship is

ca##e" %o4 an" the rear en" is no4n as stern or aft. !he #eft han" an" the right han" si"es ofthe ship( 4hen ie4e" fro the stern are ca##e" port an" star %oar" respectie#$. e sha## no4

"iscuss the effect of g$roscopic coup#e in the naa# ship in the fo##o4ing three cases:

1. Steering


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2. Pitching( an"

=. o##ing


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2 4 83 E f f e c t o f G , r o ' c o p i c C o & p l e o n a N a B a l # h i p d & r i n gp i t c h i n g F # t e e r i n g

Steering is the turning of a complete ship in a curve towards left or right, while it movesforward, considers the ship taking a left turn, and rotor rotates in the clockwise

direction when viewed from the stern, as shown in Fig. below. The effect of gyroscopiccouple on a naval ship during steering taking left or right turn may be obtained in thesimilar way as for an aero plane as discussed in Art.

When the rotor of the ship rotates in the clockwise directionwhen viewed from the stern, it will have its angular momentum vector in the direction oxas shown in Fig. A. As the ship steers to the left, the active gyroscopic couple willchange the angular momentum vector from ox to ox!. The vector xx! nowrepresents the active gyroscopic couple and is perpendicular to ox. Thus the plane ofactive gyroscopic couple is


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perpendicular to xx! and its direction in the axis "# for left hand turn is clockwise asshown in Fig below. The reactive gyroscopic couple of the same magnitude will act inthe opposite direction $i.e in anticlockwise direction%. The effect of this reactivegyroscopic couple is to raise the bow and lower the stern.

N o t e '

. When the ship steers to the right under similar condition as discussed above, the effect ofthe reactive gyroscopic couple, as shown in Fig. &, will be to raise the stern and lowerthe bow.

'. When the rotor rotates in the anticlockwise direction, when viewed from the stern andthe ship is steering to the left, then the effect of reactive gyroscopic couple will be tolower the bow and raise the stern.

(. When the ship is steering to the right under similar conditions as discussed in note 'above, then the effect of reactive gyroscopic couple will be to raise the bow and lowerthe stern.

). When the rotor rotates in the clockwise direction when viewed from the bow or foreend and the ship is steering to the left, then the effect of reactive gyroscopic couple willbe to raise the stern and lower the bow.

*. When the ship is steering to the right under similar conditions as discussed in note )above, then the effect of reactive gyroscopic couple will be to raise the bow and lowerthe stern.

+. The effect of the reactive gyroscopic couple on a boat propelled by a turbine taking leftor

right turn.

2 4 *3 E f f e c t o f G , r o ' c o p i c c o & p l e o n a N a B a l # h i p d & r i n g Rollin g$

e no4 that( for the effect of g$roscopic coup#e to occur( the a?is of

precession shou#" a#4a$s %e perpen"icu#ar to the a?is of spin. If( ho4eer( the a?is of

precession %ecoes para##e# to the a?is of spin( there 4i## %e no effect of the g$roscopic coup#eacting on the %o"$ o f the ship. In case of ro##ing of a ship( the a?is of precession @i.e.

#ongitu"ina# a?is is a#4a$s para##e# to the a?is of spin for a## positions. 7ence( there is no effect

of the g$roscopic coup#e acting on the %o"$ of a ship.


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2 4 9 3 E f f e c t o f G , r o ' c o p i c c o & p l e o n a 8 - 5 h e e l d r i B e $


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2 4:3 E@amp le % ro) lem '$

1 A ship is prope##e" %$ a tur%ine rotor 4hich has a ass of 8 tonnes an" a spee" of 21


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Department of Mechanical Engineering-master Copy

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