flutter evaluation of an airfoil - mcgill...
TRANSCRIPT
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Flutter Evaluation of an Airfoil
NlohaUlIuad Hacli Akbari
DqmrtuH'ut of l\1cchanical Engineering
NIcGill University
l\1011trc.al, Canada
f<'hruary, 1993
TII('~is Sllblllit.tpc! t.o tht' Facnlty of Graduatc Studies and
Ht'st'arch in part.ial flllfilll1('lIt of the rcquircll1ents for the
d('g['('(' of l'vI aster of Enginccri ng
© 1\1. Il. Akbari 199:3
Flutter Evaluation of an Airfoil, Incompressible and Transonic Regimes
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l'tI 111\ IIi1dIH'I,
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'l'II\' pIIJld"llllJf l'III.'t 1'1" 1"',1 IIlIlIIdl/('d 'l'hl' 1'(J1Ii1tiull~ of lJlolioJl of <III cliIfoIl
wJl11 1 \\'11 d<')!,1 ''l'~ III fI (·(·do"l, i/l pi!., h ,111<1 plllllg(', dl(' ul>t dill(,(1. T'h(,ll, tJJ(' 111l~t(,étd'y
dl'I 0") IIdlnil IlwIII i('~ fOI dilfel ('III. /Io\\' J('gi/ll('s ill<' pll'~elll('d. l'Il<' tlilditiollal sol1l
liO/l~ 10 t.Il1' 1111111'/ plold""I. Ildlll<ly, III(' 1'-/.' dlHI li-y methods, arC' [UIIIIUldL('d, ,lIld
1 III' !'"ph, l' t 1 dll:·,fUJ IIldl 10" lIH't1/Od [01 fil/Un élllétly~is i" élisa iIlLroduC'C'cl. Tb(,ll, the
1'11(·( f. uf "lff(·I(·II1. d('~igll pdldJIJ('I('I~ of dll diJfoil U!l the f11JttC'r :'>1)('('(\ is illlaljzeo,
IHII h ill 11/(' illl OlllpJ(·""r11k ,1I1d tJd/l~()llrr Il'gill1('~. F1\rt 1H'IIlJOrf', Ill(' (·If('( t, of the
1'1/111,,'( \'.t11/(·S o[ III(' dl·~i!!.1l pdlil/lwklS Oll t!w (J('fllln/lef of Hutter is ill\'!'sligated.
Fill,dly, "UIII<' gl'II('ldl 1 lIllI 11J"i()ll~ I('!!.dldlllp, II\(' abo\'('-lIwlltiollcd plH'llol11<'lIél al(' d('
lin'cI Th(' go,d ollhi~ \\llik i~ Ih\' f"ct t!J,d, III!' 1111~t((fdy (l(I'odYllal1l1('(l,tI,él ha~ ])('('11
Il:,,·cI. !Jolh ill Ih(· illllllllJlI(· .... i].](' d1lcl tJdll~()llic 1 \'giIl H'S, ancl, Il]('1 ('fo]'(' , th(' obLaill('d
/! ", 1 JI 1:-. d li' 1 d i 1 1 Y pl ('( I:-W •
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Sommaire
Le probl?'lll(, C!(' flollr III( III ('~t ilIl IOdlllt ('Il p(,('lIlll'l 1)('11 Le''' ("'111,11 Î(lll;-. dll III()II\'('
men!. d'ull profll a deux d('gri's de lilJ('lté>, ('II f'"1gd[',<' d ('II 11(',\1011, ;-.uld (lldt'Illlt'''' LC':-,
théori('s aélodynamiqlH's llon stcdiollll,lil<'s p01l1 dirf<"I('lIfs IIl()r!(·S d'.'·( UIII('IIWIlI ;-'0 Il t
alors pr<-s(']Jtées. Lc~ sollltioll~ das~iql((,s clll plobli'lll<' d"~ 11011«'111<'111 (1(';-' 111/'(1)0<1(,:-,
p-k ct, U-g) SOllt fOll1lulé('s ct la IlJ(',thoc\(' de' fl,lll"f()III)(~(' d(' L<lpl,\«' 1'0111 1\III,tI,\'S('
du flottemcnt est alors introduit P. Les df('\,:-, rks dil!'<"1 ('Ilts pd 1,1111(\( 11':-' dl' 1 (l1I( l'pt iOIl
d'lIll profil sur la \'it('s~c J(. flottC'llH'nt sont élllitly:-,{,S rI,IIIS 1(,:-, (<1;-' d"~ 1 ''·l'.illll's illl 'JlII
prrs~iblc rt transoniqut'. Lc~ cfrds des v,t!l'lII's n 11I1,t'( s d(·~ pdl c11111'f ((':-, d,. «(JI)( l'pt.iOIl
sur la pn'sel/ce du flot tf'lllent sont étlldié>s, Fill,d('!ll<'lIt, r1(.~ r r III( 11I~i()l1~ p/IJ/·I,t1(·S ~III
les phénom(~ncs mentionnés ~'i-d(,sslls SOIlt. !l)(·I1{·(,S. Le bill. dc' ('(' tlil\'it" (·~t d'ol,fr'lIil
dcs résultats précis ct. ccci en IItili~alll dC'!'> dOIlIl('e~ ,['o(/()(IYII(J/lIIf/I/( /1011 ,~I(JI/OIIIIIII/(
dans les ras de régimcs iJJCOIllpr('~~ihl(' pt (rilll~()lIi(J1«' .
Il
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A(~kn()w le(] gcrnents
1 wOllld 1 i kt· 1 () ('X III l'~~ Illy ~jll(('1 (' grct! iLIl rie tü my !->lIJ)(,l'vi!->or, Professor S .. J. Pric(',
fOI IIls 1 (JIII,illll(·rI gllidallc(, dlll illg the (,Olll'~(' of this wOl'k. His critical rc\'icw of Illy
WOI k ,I!ld \',dll,tlJlc' ~1Ip,g('~tioIlS <11'(' gl('at ly aPPl'cciated.
'l'hallks <II<' ,t1w dlle 10 Illy (olIl'dgu('s Jafar Sadigh alJd Mcheli Kcshmiri, with
wholll 1 Il ,ICI lI~dlll dise lls~io!ls during 1.h(' cOlllplction of this work.
'l'hi~ Il'SI',II( il \\',tS lIIade possible througil the flnancial support by the Iranian
<;O\'('IIIII)('lIt, who pro\'idl'd a :,(holal:,hip for myself.
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Contel1.ts
Sommaire
AcknowlcdgPIlH'llt s .
Table of COlltent s
List of Fig Il l'C'S
List. of Tablps
1 Introdllct ion
1.1 I1i"t()1 i( ,d H('\ Il'\\'
1.2 E\pldlldt iUII o! tilt' l'lo!,II'11l (11111 1 ('1)
I.:J Tlw"i" (h 1'1 \'il'\\', .
2 Equat iOlls uf M ot.ioll
3 Ullsteady A Pl'odylla/llirs
:~ 1 Ilii lIlll]lII':-,<,il)II' ~-I) F1l)w
IV
Il
III
1 \'
\ 1
\ Il
'\ 1 \'
ï
JI
11
• 1 .) 1 •
1 (i
. 1 l '1 1 d Il '" III ,) 1) l' l,)\'.' I~
-1 1 )ill"')'f'llf A ppl 0;1( 11I'!-> 1 (J f IJ(' P}'ohl('lll 25
) \\'llld 11111111'1'/,"'''- :W
1 .) 1 1''11111''1 \1.11.".111 \1"III"d 11)1 ri dll"lJlll, Flo\\'
1.\
1 \ ,\ 1 dpld'" 'l'Id Il,,flJlllldllt)11 \1('llwd
51
ri 1
~I 1 1 Ilit 011 'Ill l',,,>d,f(· Flow 56
-, 1 .)
TI d Il''\llIlt 1: lu,,' ...................... !J7
- .) , - I·ïlt·t t ur ILldlll" td ( : \ l,dlOII .'J7
~ .) ,1 • Ilit \ Il 1 11'1I·:."tJ ,JI' FI\I\\ ................. dl
~ , ) , - - 'l'Id Il:-'\ 1111, 1· I\!\\ .i8
Fil," t (II t III' )l,,'>It HIll ol (\'11111' or \Iil'>s
" :~ 1 ,')9
60
• \'
• ") I·'!,'( 1 (Ii 1 III' p, "1111I11 ,d l'Id"ll( \ \ 1" Idl
1 1 Ill. (JllljJl'· ...... J1>1', 1 l, 1\\
1 1 .) 1'1 d Il''''III( 1''1,1\\ - Il'
"1 ") Flr,,( 1 ()I FI' ''II Il '11< \ 1\'1111'
.) ) 1 111' '11111' 1 ('" '- Ji ,f (' 1'1,,\\
" " ,) .) ) - TI d Il''''llif l'Ill\\
G Pnl'allll'trÎc Fllltt('1' Allaly ... is
1 1
li:! 1
(' .) .) ).- -
7 COllclllsions :\lld H(l(,OIIIIIIl'lIdat iOI1~ 1 ():~
1111
7.1 1 III( III 1 IJlJ(':-,:-,dtlf' IIUI\' 1 (J 1
.... ,) 1 _ H(·( UIIIII](·ll!l.tf IUI1'o
Bibliogrnphy 1 Il.''\
A Algehl'aic ÂI1(dy~is of Pilll'!,' St IIdy J J:~
B Compllt ('1' Pl'ogl'<tlll Li...,tillgs J:LO
• \'1
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J-1ist of Figllres
.) '1 III' pl'lll.l',ill.l'" :., dlld pltl Il.0, IIdllIlUlli( IIIUllOll'" (lf " 1\\O-t!I'g,I<'I'-ol-
.~I'II('ldll()1I uf (ll( 111.1111111 dlt/llllli tlH' di!loil. .... , , ...
:1.1 '1\ l'Il ,il 1(' .... 1l11~ t'lllIll JUlddll (19:">:3) fu!' tl1(' dl'Ioc!ynamic dl'!'i\atiH'S
("," ('1.., ('1/.,,' dllt! ('1/". of a lIat platl' <It ,\1 == 0 . .1. ,\~ il fll)}ctiùl1 uf
.l.1 rh" d('IUd,\II<1llli( (odli('ll'Iils nI.' ("", ('mil' alld CIIl " fOI a :\1\(':\
hl \(HIIl dllf(IIl dt \/ --- () ~,id'" d rllll( tioll uf Il'dll('cd h('(J1I<'IIIY. l,'
(1 ('(' 1 q~ 1) . .
\11
la
:21
:2:2
:2 !
• I! .\ 1:- pic,d \<1 Ildl 1\111 ul 1 Ill' d" IllpIIIP, Il, 1111' \ 1111\ .1 1111\1\"''' \ d .Ill .Ill \ l.dl.
\ (')"11'" d~ II"III!! \11(· ... "1111" 1/
1!J(, 1/1.1/111,1 "J!11t/1I<",'c\ ill Ill!' (·'\lldp\l1.1111I11 l'loIll~\lIIH Il,,\\' . .t \ \('\
(i 1.\ ()()(i "il r()tI ,11 .\1 .- (ue ('\lIIl!'dll"'"11 !wl \\\ \'111 III' 1"·,(11,,, 011 Il' III",
lhl('I' dilr('II'Ii1 101/'1,,1 "/Jlld 111 ,Ii(' 1"\1 Id!"I\.t1 101"
1.1 1·'luI\ (h.tl 1 I(JI 1 Ill' /1-1 .. Ilwl Ill/cl , • 1 ' ..
,l,l Fltlt!l'I "'P('('(!. \ f. \ ('1<.,11'" IllJII·dlllH'II"'IOlldl Illd"", Il
I1U\\' ('()lIlp.llhUlll)('t\\'·'·1l Ill\' 1'''-.1111--. ,,1,/.1111,',] (ltl1l1llw {"'/.jl 1 ... .111.\
the' L.lJd;lI l' Il)('II\(Id:-.. , . ,
:'::\C'.\ (il.\()(J(j "itlut! dl ,\1 : () Kr) ('UIII!,dll""11 Iwh\""11 Ill!' 1,,,,,111"
('Ollljldli:-'UII 1)('1\\('1'11 1 Ill' 1('''Ir!t.., ()1I1 <II Ilf'cl 1111111 1 Ill' (1-1/, JI-I .. , "II,] 1 1 If'
Laplde(' 111l'1 }lOd:-., , , , (J'"
,jA Fllllt!'1 :-'pl'I'<I. Vr. \('1:-.11:-' Iddill'" (JI ,'2,\ l,11H11\' 1" TldIJ"'IiJi, flu\\' d
;\:\(;:\ (il.\OO(J clilft)j}.t! \1 ~: (J,K.'i, ('IIIIIPdlj'(J11 1Jf'1\\'I'('11 1111' 1(",lill"
.») FI1\tI"1 :-.p('('d. \",. \'('1<"11,,111\' «('IJlII' ur III" ... .., I(JI ,tlIIJIl, l". IIIIIIIIIIJI!
<llld t Ill' 1,ilpl.1< (' Hwl Ilurl ... ,
• VIl!
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II(JI\' ('()llljJdIJ,,(Jlllwl l \'('('1I11J(' 1 (·,,"lt ... olddlll!'r1llulll tll(' {T_ q. p-I.-. clJHI
il
(JIll .111)('.1 1j()jJI t III' ('-l], p-I,. dllrl 1 Il(' Ldl'LI( (' Hwt "ud~, , ,
('(JIIJl'dl i~()11 111'1 \\'('('11 t III' 1('''lIlb ubLlill('rI l'JUill t Il<' li-g. p-I." ;Int! 11j('
r, III FIllt11'1 "'II'('d, \:r. \·('I ... II~ fll''111('JIC,)' IdtlO. w! Trall~ollic fiow, a î\ ... \CA
l'IOlll t11<' (!-!l' JI-I ... il Il ri tlll' Lapld(,(' IJwt!Jor!s, , , . . . . . . . . . j·1
(l.1 "'lllttr'( ~,H'l'rl \'('ISllS 1/.1'", fur diff('J('llt locatiuns of the ('l;l~ti(' aXIs.
( .. , ) -
III( ()lll,H('~~d,I(' flow ............... .
"'"tltl'I ~PI,(·rI, \ JI \'l'J:-lI:; frl'qllt'Ilty ratio, l.<,.'r' Illc(Jm)Jr('~sibk fio\\',
IX
88
•
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b .. ~ :\Iillillllllll rll'qU('1I1 \ Idlill fOI \\1111 Il !llllki 1" 1"1\.1111("1. "',,,,,,,.1.1\ dlf
1'1'1('111 1(,( ,ditJlI .... (lI IIIl' '('1111(' uf Illd"~. l'" ('tl\lIp"li~(l1l Ill'I\\('('II 11\1'
Il'~IIIt,, ulll"lIwd I!UIII 1'111( ..... "llld.\ "lit! Iltl~ "llId.\ III( ()111j11(·~,·,t1"I·
Il li \\'. TI \l' (·1 li ,,1 \( d \ i" d 1 0 1 -, ( II (lI cl., dl ( . d dol 1 li C' "(' Il 1 cl., 11<1 1 III ccc '111 1 1 •.
<l1It! 11lt' (l'III Il' (lI Ill<l:-''' ,11 tltll'('1 ('111 I(J( ,II iUIl'" 111'1 \\ ('('1[1 1 11'1 Il
(,., 1"11111<'1 :-'Pl'(·cl. \f. "('1"11" flt'qlll'lll.' I,diu. cc',. 1IIItlllljlll· ...... t1dl·ll.l\\
'l'Ill' (·lcl~tl(· éI',i .... ,II ll.I!'i-,lluld. ,t!!(,dd of tlll' d('I(III\II<lIIIIC C c·lilIC·. 011111
1 Ill' C c·ldlt· ()f Ill.t:-. ... ,d "dl('1 c'lIt lo( ,t! iOIl:-' 1lI'llilltllllC' dC·JeI".\ Il'' IIII( 1 l'Id, (' IlIl
G.5 Fllllkl :-'pl'('(1. \ f. \('1:-.11:-. 11('qlll'II C .,. 1,t1IU, ",',. 111(llllljl\(· ...... t1dl· 111.\\'
'l'Ill' dd ... til iI"l'" at II\(' Illicl-(llllld, 1II'Illlltl 1111' ,lI'ltJcI.\lldlllh C('I\IIC',
dllcl tll(' CI'IIIH' uf 111":-':-' elt tltlf('I('111 I()Cdliull:-.I)('I\\·'·(·ltlllC'llI
TI](' (·ld:-.t.il iI'.Î:-. ,il, tIlt' !lllt! dlIJld. 1)('11111,1 1111' ,lI'ICIII\,llelllll( CC'IIIIC·.
,11111 the «'lItll' of Ilid' .. :-' ,t! tllll'('I('lIt l()cdIH," ... IIC'ItIIl,lllll' (·I.I ... lic .1\1.... C):~
li.ï FllIttt'1 ~p(·(·cI, \'J. ,'('1:-.11:-' fl<'qll!'ll(Y 1,1110, w,. III(OIIlPI(" ..... il,j(·!IcJ\V,
'l'II(' ('Icl ... tic ilxi:-. dt III\' 1111('('-'111<111('1-( Ilolcl, ll\'llilld 1.\11' d('llIdYlldllli(
Ct'Ilt no, dllt! tilt' ('('Illl(' of Illil~,> ,d <lifl('1 ('Id ICH"t! iClII:-' III'I.\\'C·(·II 1.\11'111 CI:~
G.S Fluttl')' :-,p('('d, \If' \'('I~IIS 11('(t1I('III,Y 1<11 io, w" III( (1IIqll(· ... c,il,I(· Ilcl\\'.
'l'Il<' plastic iI"l:-' al, t,]H' tlll('('-qlle1II('1 (1IOld, Iwltlliri t.Il1' d('I(Jdylldlllll
U'lItH', and t II(' (1'1111(' (JI' III"'>'> ,t! dilf('1 ('111, 1111 .iI illll'" 1II'IIIIIclIIII' (·1.,· t i(
aXIs . C)_I
•
•
(, ,) \1."1/1111111 fll"I'II'I)( \. l,dl" f'll \\111111111111"1 i.., ul,ldilll'd, 1;."", .... lOI ,bf-
1"lf'll1 1,,,,diuII:-' IJI tlll' (f'lill" (Jf 111;1..,.." l',. ('(JIlIPdll"'''11 1)('I\\'c'('11 11]('
Il,,\\, ï III' f·ld·.llf d·,.i.., ,d 1111' lllre'c' C!1ldll,C'I-cIIOld, lH'hilld II\(' dC'ludy-
11<lllllf '('11111'. dllrl 1111' (1'11111' ,,1 IJ1d":-O ,If dilfc.1 c'Jll loc ,II iOJl:-' h_dü.!!1 t Il('
(, II) I~('dl l'dl ~ ()r t 1)(' ,·ig,·JI\dlllf· .... \,'1..,11" llOl! dIJllf·ll..,ioJ],tl "p('l'd, \', Tldll
"'''1111 Ilft\\'. d :\.\C \ (tI.\(J()f, dlr!Ur! dl .\1:- () ,,",'. ('Oll1Pdli..,OIl 1)1'1\\1'('11
1111' fd..,f·'" \\lllr ~I)(H' ill plllllglll,!!, dlld pil(h. 11)01" i1l plllllp,ill,Q,. ,l!ld
Il )01" ill pil (Ir
h.11 Iliidgil~dl~r Pd_lt_1I1 tlll' l'ip,I·II\·,t!III'S \'('I:-'IIS IICtIJ-di1JlI'll"ion,t! spl'l'd. V.
'l'ltllI..,,tlli( /lo\\'. d l'JACA (il.\UO(i dilfoil ,d :\1 -= 0 ~.1. ('UIllj)dll:-'OIl 1Jl'
t \V('('JI t 1,,· ( .1:-'C':-' \\'1111 :!J)O F 111 plllllgin,l', and pit ch, IIJOF in pll1llging,
dlHI 1 j)OF ill pite h.
li I:! I\(',d p~I~()r tlll' ,·igl·Il\.t!lIc· ... \('1:-'11:-0 Ilun-dilllen:-oionai !'>p('('c1, V, Tlélll
:-Oltlll( Ilu\\', a N:\C .. \ (iIA01ll! élilfoil ,d, JI = O)~:i, ('oll1]Jclli:-'OIl IH'L\\'(,(,11
tlj(' (dS,':-, \\'Itll :!DOF in plllllgill,l', dllt! pitell, IDOF III plllngin,Q" and
1 J)OI-' ill plt( Ir
(i 1:1 Illld.!.!,ill.1!" 1~,~I~ul the l'igl'Il\',J!lIcs \('1:-.11:-. IlUll-dinH'llsiulIéll :-.p('PtL V.
Tldll:-'lIl1ic lIo\\' . .t ;\1.\(':\ (il.\OO() airfuil ,II lU = 0,85. CUJllIMII:-'OIl be-
1\\'1'('11 t Ij(' ( <1";(':-' \\ It Ir :!J)O F in plllllginp, ilnd pilclr, 1 J)OF in plung,ing,
<lIHI 1 DO\-' in pl:' h ...... , , , . , .. .
'\1
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97
gR
99
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{Lit FI'ltll'1 "PI'I'II. If' \1'1"II~fll"IIII'I\l1 Idtill,,,:,. TI,III""llll Il'111,.1 \ \t',\
Il 1 \()Ob dil fud dt .\1 -.!J S,i. Titi' "LI"t il .1'\ 1" ,If (l 1 :-)., hlll ,1. .1 lit! t hl'
1'('lItl(' ()i 1l1d"'" ,If dilkl"llt lUI ,tli()ll" ."(11).!, 1111" !llnd !tlll'
(),I .... l F11I1t"1 "p('cd, If. \l'I'''II ... Ir('qll<'II(~ 1,11111,,,', l'ldll''''lllC 1111\\',.1 ::,\l'\
(i L\()!J(; did'ud .If .\1 0 (u"r) l'Ill' ('LI~tl' .1'\1" ,II tlll' !lll.! 111111.1,0111.1
t 1](' ('('1111(' ni Illcl"" dl dilr('II'1I1 l'JI dt lull ... ,"UII~ t Ill' (1)(1/ d lilll'
fi ](i FI1It.t('1 "'P('(·'1. 1 f' \('I~II ... rl('qlll'III 1 l,IIi", ll', '1'1.111""111' !lu\\', 01 '\ \l',\
(j L\()()(;dillll!l.lI \1. () S1 1'I11','ld"tlt 01\1'" dt tlll' tlll('(' (111011\1-1 ,llIlId,
1 Il li
1 (} 1
dllt! tll<' «('11111' ()lllld~S ,If ddr"{(,llt 1()(,t1i()lI'> ,,1(111).'. tl\l' ,IIIIld 11111' III)
\11
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•
List of Tables
.1 1 TIJ('C/r!"I:-,c·ll'.., IIIIICIHIII. ('(l,) -c l"~ 1 . (,'. \\'1)('1(' 1.: == CI.) bll / .
.1 ~ '1,\'JIH ,d 1 (.,,) 111 " rH JI Il lUI ddll ( 1 (jj:n fUI Il)(' dCI ()d~ n,li nie (Ini \'cI t i \'(':-,
( '". .\, (',., . (lf d Il,d JlI,d (' dl .\/ == 0 . .1 ..
:1 ,1 '1\ pH ,d f('" III l " J'IUIII Jorddll ( 1 !l!'i,!) fOI d)(' dl'I od,) IIdlllic dCI i y cl t i \'\'s
(',11 " ,\: (''"., . or d fI,1f pl.i1(' ,d, ,\/ := 0 . .1
1 1 10'1111 kl "IH'C'cI 101111.) fI(lIll Ill(' ./11111,,. 1I/(1I~/11I IlIt Ihod ill thl' tldll~()llic
flo\\'. '\C·f(l.),) Il.111111 d,dol 0) d \'A(',\ h L\OO(i itil[oil hay(' b('('11 lI,,(·d.
1I111~1 Idlill.!.', 1 hl' 1'11'('( t (JI tll<' /0\\'(',,1 ,1Ild t!l<' 11Iglll':-,1 ~p('('d!'> us('d ill tltl'
"\111
lï
18
19
53
•
•
N Olllellclat ure
Symbol
1/ + 1 b
C
I~ + 1 Il
ql.-)
[Cl
Dl'::->cripf ion
l ,11l-dillH'll .... illll,t1 .11 .... 1.1111(· (1111<'1111 .... ul ·,,'11111 Ilt>ld) (d (·1.\ ,lit
"'\1:-- ,dt uf Illl.! (hol d
lilt (III \(' .... lu]H·
(·1.!!,(·1l\,t111l· (JI' ~.\,t(·lll. (/ dlld h .l'.i\'(· .1.1 "I/II Il,!'. .Ille!
1J(·q1J(·II(~. )(· .... 1)('( t i\ (·h'
lil]('dl (u<'fhl 1l·lIt:-. of '1j1]1I1J\ 11I1,iI ill.!!, l,il IOIl,iI 1111111 I()II
~1'll1i-1 I)(JI d (JI' dillu1!
11IJlllilll'dl' Id.!!, t(·llll.., tJf dPI'III:-'lIl1dl.lllg l,di()lIdl 111111111111
(hUI cl of <,ilillii
('IP,I'II\',t1IIl' (JI \\:-.t('III. l' dlld ,f .l',iv(' ddlill'ill,l', dlld
Il l'Qlll'lI1 y. 1 (':-'1)('( t i,,{'ly
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l'lllllP,(' dl"p!.\( ('111<'111.
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il!c-lIllly IJldU i'\ ('2 X '2)
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1('<111('('<1 "('!jlH'1I1~' \ld~('d Oll (hUIt! wc/l:
"Pl ing 1 (Jll~1 ;1111 III \H'l1dilig
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1 \lllll'I,·\ l'Ig('II\,t1Il!'
l'(w/li, 1('111 ~ ill 1 il" 1 d( t ('1 i~t 1\ ('<iII,tlllll!
pd 1 d !lit' 1 l' r p,1 \ l'~ Il d 1 III l' Il fI, Il 1\111, il 11111 1 t 11111
(h Il d Il Il ( Il 1 (' ... ~ dl l' 1 /:!. 1/ l ':
dylldlllil' 1'11 .... '.1111. ,II tllll Il'1
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Chapter 1
Introductio11
,1(1(J(I(/,~III'/'t/J i:-, tll!' ~tlld~ (If tlll' drl'll ur ''l'IPd''II.IIIII( !l'II 1':, 1111 1·1 .. ·.111 111,.1",·,
Ol\(' of III<' illt('lc~t ill,!!, plOldl'llI'" il! dl·llH·I"..,I)I ih' i~ 111\' ..,I.d,ilrlv III " ..,1111111111' III
\\'incl, A Illéljol pl'ohlcll1 is t II!' 11111 ICI (l( ..,1111111111''', ',II( Ir 01.., "il pl""I'~' III :.11"1'1'11''\1111
III lI)g('::, \\'h('/I il :-'I1I,tll di ... lllll'dlll (' i/lcl1l1 ('''' IIIIJI(' IJI' 1(,:,.., \'illlt'lIl, 1J',t III.il iCI Il.., Il l',
dl,l! dl 1,('1 iz('cI I,y 1 II(' illl' " 1(1)' of rll'I ()d,\ IIdlllll, 1·1"..,ti(, dllcl 1/11'11 i" (Il/ (1''', dlle! 1:,
cl proIJ!f'J]] of clY"rllllic il1~ldhdil\, FI"tI('1 1 .III 1(",1\11. 1/1 III<' • ,d,,·,lllJpIIÎ. 1.,,1111(' ,JI
an ailCldft .. 1"11111('1 IJ/('\'('ntio/l (rill 1)(' rllI illljJCJII,IIJ! 1.11 tOI /II III<' di'l I.dl. dl",I~'.II,
,df('( lin,!!, II/(· :-,I.I'1H!,lIlf11 d('~i)!,lI, rI/lcI. 1 Ir".." 1111' \\'(·i.l'.Irt <I/ld PI'I!CJIIIldllll' \)1 Il A. Il'dl
IIIJcI(·I~t.lllc1illp, of Ill(' 101(· uf ('dlll cll'<,i)!,1I 1';1111)/ Il! dll .III 1'0 ri 011 11\1' ,,"~,('1. fil 111\111'1
l ,II J 1 j(' 1 p d ( "., 1 P, 1 JI' 1 ~ i Il t 1 li' 11111 t (' 1 1 Il' dl, 1 Il 1 (' 0 f .1 1\ ,1 i 1 1 1 • rll
1.1 Historical Review
Sinn' tJ](' IJ('p,illllillp, of III!' Ili~,loly \)1 fll,l',!II, d'·IIJC,I",.,lil Jly, dllli III p"l 1 If Irldl (1111.
tl'l', IlélS illfllll'lI«,d IIIC' ('\'ollllif,1l of "ilfltlfl '1111' Wlight IJlfJllwl',' fil',' 111/',111 Wd',
• ('II. \ IJ'/ FI( 1 1.\'1 /f{)UI '(' II{).\
III III!' 1II",'ld,!',I' .In" 1 dll [4 III' Il )('1 il Il JI wd 1\\'0 illll)()ll dll!, 1 ollïepl 'i III Il](' pa per.
1 ) 1111' ()~I 1I1.1111JII" \\1'1(' IIU 1 d Ill' 1 () 1 (,'-,Olld III (' ill"ll( ('d I)y ,l \'lIJl,duI)' ~O1l1 ('C', and
'n 1 II!' (J',I 111,11 iOII" 1 'Jll1d 1)(' pl ('\ "I!kcl 1 )! llldkillg 1 II<' c!"\dtol S ,,1 i rrel' , III anot!)('1'
Il I~ IIJI('I (':-.t ill,l!, 10 kilO\\' IIl,d IIIf' C('llll,l11 ~id(' in \Vol'Id \V,ll' labo h,ul similar
l,iplcllJ(' dll<l 1 hl' Fo""('[ \)-\'11 J lllOIlOpid 111'. 111 Il](' bot It ccI:-.es lllilny !>tluctlll (' failur('s
dl)(' 1 () Il)(' Will!' <1 i \'('1 "('III (' \\('11' ()I)~I'[ \T" [:iJ . • '"l .-.,
,\ft"1 \\\1)1<1 \\'dl l, fi "Ill,dl )!,lUllp of 1('~('dICh('I:-' III Iht' N('t!lPdélllds :-.la](I'r\ a
:-.\:-.1 ('111.11 if' d Il,d.\ 1 i( ,d 0111<1 "\IH'I illl<'lIl,d :-.tlldy IJlI I1H' IlWI hilni:-'ll1 of a binéll'.)" fillltl'r
dll(' ln \\'ill)!, \('lli, ,d 1)('lIdlll,l!, dllt! ddl'I()II d,,!1l'ction. 'l'Il<' J'('sldts of this sludy, by
Ildll1l1lldl11'1 dllt! 1\(llllllg lb]. \\1'11' illlpUlldllt, lJlo:-.t1y b('t'cluse uf tht' Il'cugnition of
Il)(' 1'11'1'( 1 (JI' 111"":-. 1',"01111 in)!, of t 1 If' ,lil('1 011 Oll tilt' PI'I'\'l'l1tioll of flllu('1'.
,\lln 1111' \\'M, Ih(' IUllllwtillOll tu 1 Il ('al\ th(' \\'0I1t! ~p('('(lll'COlds clild tü win air-
Id( (' III i/"s. 1I1gl'd dl,.,igll('I'-> tu g<'l higll<'l' SpI'('lk A :-'('1 ic!> of lIutt('r aceidents, in
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('lI.\I'TFU J ISfUOf)( ('//U.\
Hlil"ill ..... pll/l'd 1'1111\ III 1 Ill' [<I:!l IlItl'llldtil>l1.d ~I IIII!'ldl'l \1.11111.' 1'1"1"1\ 11.111',.1
SIIPl'lllldlllj(' S 1 ~1()Il()pl.llIl' 1 1 <1,,1 il'd , hl'I<lIl"I'lIt .1 \\III.e, 11llltl'l ()I, 1.111'1, 111 11111,
d C!'!' B.'l' Idlt'I, ill dll ,d 1!'lllPI ,II IHl',d\lIl,e. Ihl' \\lIlld\ ',pl'I'd 1.'1111 II. t'\11I'111 litt'.! "
Will.e,-dIlt'lllll /llItl!'I', dllcl Il\(' [lldlll' ,llIcI Il,, [Hlul \\t'Il' 1(1,,1 ['il
13111,. III ~IJIllt' ('''St'~ t Ill' tl1I1 tt'I \\.1'0 IIl1t "" lldllldll1. dlltl 1 Ill' 11111ldt'l (II 1 Iii' [ll.lllI'
IIdd Il](' ! "il II( t' tU! (JII!" 1 1 Ill' d!'"ign, FlIl t'\dlll[.It" III [I},l~), dllllll,!', 1 Il!' :'\.tI"'II.d \11
Bd!!'. 1/11l' ur tllC' Id('('I~ !'\["'lil'III!'cI d '.t'Ii!'" ul \\1I1l~ 1111111'1 ÎllIl.!I'lIh F"III 111111', III
PI('\I'nt Illl' IIl1ftt'l', MJIlIt' pdl'I of II\(' WllIg \\0\.., 1111 utl [sI FIII.dl\', tlll' I,d.t! .111'.1 III
tll!' \\illg \\'d~ Il'dll(('d lu t:~f'tl flOl1l il~ !J1lglll,d \'.11111' (JI ï:";/1 1, 1,,,1 1 lit' [111111 l'II.!.'"
111' with .t /llIttn-flt·(' ail pl,lIll'!
1)111 illg 1 Ill' 1 !nOs, si 111( t III ,dl) ,,1 III I,ipldll!''' \\1'1(' Il'I.ldl 1,'.1 1,\ 11111111'1'1.1111",111'1.111"1'
uf tll!'1l lligh 1)('1 fl/I'nldll( l' ,\l'''(J, "'lwl'lb \\('1(' 11<'1 1111 1 IIl,l', .1 l "11~,,,II'I.t!,]I' Il.JI 1 H'II 1111111'
S[W('rI of "uI!llIl [<)J. A~ a 1 ()1I"1'1[1)('II((', d III'\\, ~I'IÎI':-' (JI' I.lldl .III Idl'III', 11.l1'11C'111'd III
Elllol)!' and tilt' {j S. fUI :-'('V('I'," dll pldll!'~, ,\1, III!' :-'dlll!' t illll', .Ill .11'1'11'1 l.tI.!l' IIiIIIdll'1
uf dt'~igI)('I~ ,iIlcl 1('~('dl(I\(,I~ ill Ill!' \\'IJlld ()r d\i.dioll \\'l'I!' II\ill,I!, lu cil \'1,1,,1' III'W
tl]t'ol'i(·~ to illI,tlyZ(' dlld :-,uln' Ill(' ploIJI!'1I1 III Ill!' Il S, TI)('odolê,I'1I . LII Il'd WI)II,IIII',
Ull tl)(' plUl,I!'1l\ III I<):~ 1, <llIcI \\'ill,ill ,\ fl'\\' IIIIJlill, ... pwdll! 1'" :\/\( ';\ Ikl'I :"-il' 1%
[10], \\'hi! II "ilS \)('!'II !·,II'II ... i\(·ly Il''t'cI ill (·~t.")II"IIIII.I!, lI(('fIIIJlI" "IIIIJlkl dll.d\" l, III 1111'
'\!lH'l'iC<l1l "ill laft Îlldll~t Iy [IIJ. TIj('()rI()J~I'II\ 1111'111 y i.., 1 I,JI",dl If'd d', f IJI' ,1111/'\1".1
!·x"d tllt'Ol) Ofl!'plt'''('IJflllg "II idl';t1I1,II-pl,J1('dilllJil, 0I1It! pl.I)'·' .llIld)!Jl Il,1,· III ',11'1'
1 IwOly willg IIl1tt('1 all.dy:-.i:-. Ill].
Dili ill,!.!, \\'(JIld \Vrli 1 J, IlIdll)' Illdllg(· ... 11<lI'I"'llI'rI 111 IliI' <1111,1<1111' Il1dll ,II Y '1111'
tll'lIt! \\'<1" fU\\',lId IliglJ('1 sp(·('d.., 0111" ,dl-lIJ('f,t1 r1lll'loIIJf" .\/}\'.' , fJI'.IJfII" <llId 1,,111' .
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( '/1 .. \ VI l',/( J I.\'T/(()/)/'( ''[ I().\'
/ d /1 J',!' 1 J(, Ill] 11'/ ~ ( '" 1 !l'ri (' \ 1 ('1 Il d l ,1/ Ill" Il I!'II l,Ii p 1 ri Il b, 1'1 (', F 111 t t (' 1 Il 1 () 1.1 t'Ill ~ () ( ( 111 1 ('ri
dlH' Il) 1"dll"rI":/I"!!,('" C,"I"'ll,!!, lm" (Jf Ill,I"" 1),tlrlIlC('OI I('du(('r! "Iilrl)(,~~ [7], (;(,ll<'lctlly
"/l""I"'1I'" Il('\\, ""1)('( 1.., .1 Il ri ((Jlllpl('\lll<'" \\('1(' ddd(·d 10 tll!' plo])I('II1,
,\ IJwlll!)ll (J11I1llf101 dll.t1y"i" \\01'" ~lI,!!,g(·"I(,d I)y Sllldg (l/1d \Vd~"'('I/lldllll III 191~
[I:!] 'l'IIi" 1111 111(JrI, lI..,ill,l!. Il)(' ,,()-( ,t1I('d V-.'I ill/1ft l' t!W!lIYlIII, ha~ ])('('11 \Vid!'J)' u,>('r! in
Il)!' loll"\\'111,!!, ~('"I"',
\\'11111' 1 IWl)ldl(.d .l'·I()(·I,,..,lw dll,d:~..,('.., \\,('11' l)('ing, f!(·\(,JoIW(1. l('''('dlCht'l~ .tIld dr
"',1'.111"" 1)('\('1 "lupl'cr! (l1)ldillill,l!. «()I/lpJ('llll'IItélly e'\]Jt'lillH'IJtaJ ddt,\' ,\f'IO(,Ja~tjc ('X-
111'1 III !l'lit... ilJ \\'illd lIIJJI)('J." .... llpp())!I'c1 1))' JIlcttlH'lllélticaJ allaly"(''', gcl\'C dcsigll(,IS a
/toc 1 III!!, ()f «()lJlrrlc'lJ( (', \\'hi( h IlI'it.llI'l t h('ury 1101' ('ÀI)('I ÎIlH'lIt alollP rOllld plO\'idp [7J,
.\ I(JI uf \\(111, !)II J1lt'd~lIl illg tlH' (ll'lUdYII<tmic dPI i\'cüi\'('s \\'as done, tlllI'ing the
\\'<11, l,y tll<' Bllli"Ir, (d lu\\' .... ld)~()lli(' "pl'l'cb, and aft('1 the \\'ar, in the li,S" mainl)'
III t JII' tlolll"'uIIÎ( tille! ~11J)('I~Ulli( ~p{,(,c1 rallgr~ [1:lJ,
Hy III<' ('0111,\ llfl i(·~, I,~ illlpl('lIl<'lltatilill of t Il!' tldll~ollir wÎnd tlllllld at Langley
1(· ... (·<11 (II ('111 Il', ('\1'('1 illwllb ill (hi~ :--1)('('<1 rallg{' h('CclI1W él\'ailc\hle with a 1ll1lrh lu\\'e!'
(0 ... 1 dllc!lliglwl ('!lilll'Jl(Y thtlll otlH'1' ,':o.IH'l'illH'lItalll1rthods [l-t],
III l!llï, "111'('1"""1(' "1)('('<1 III 11·\('1 nig,lIt \\'a~ a(hi('vccl by Chad('s Yl>ager in the
\-1 I('''''dl( Il .!ilplcllll', tIHI1Ip,h, iJl Idpid div('s (JI utlwr ailplanes iL hael happellecl
1)('1'(//(', III cl "'IIJ\'('~ I)~ tIlt' :,\,\(',\ SlIhcOllllllittl'<' un Vib/éltioll and FJlltter, the
"('11(111""""'" (JI' tIH'IJlul,)"1II dlJlill,l!. tIlt' t('II-)('(lI p('liod clft('1 tht' \\'ar is indilated [15],
H,\ "PII/II.!( JI Ill,!!, thl' "'1JI)('I"\Jllic "l){·{'t!~, (\('J()('lcl ... tir ~tlldi('s in this Jang!> had 1)('('n
~t dl floc! I,~ (',,11,11 dllel l,y T{'lIlp,," .lIld ,J.dlll in ElIgland [lü], and hy Gall'ick and
1 { III. i Il (1\\' i Il 1 lit' 1 j S, [1 ï J,
\\'Ith th!' ('\(lhltioll .litt! 1JtiJi/dtiull III digit,d compllting m(l(hincs, III the l'ady
•
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('11:\ l'TI:'/? 1, I.\'TH () j) (' ("rIO.\'
,lIld pl'dI'tical ill ~Dh ing Il'al ('lIgilll'('lllIg pluhll'Ill";, Tht' dl·\,·lo\lIllt'1I1 Id "\JIIH' 1\ PI'"
dlw 1 u 1 Ill' (·xi:-.I illg fa:-.I (Dm))111 ('1 s,
1.2 Explanation of the Problem (nuLLer)
ib \\'Ol':-'l' Cé\!->e. f.,ihll(· uf tlll' :-.IIII( 1111(', TIIl'It' ,11(' dilft'J'('1I1 1)1)('" 1'1' fllIll(·I. \\'lli( Il
mil)" OCClII Ol! dilf('I('lIt Iyp(':-' of Slllll 1111(''';, hOIll :-'1I"IH'Il:-.iull 11Iid,l',(·s. lu 01111'1.\111':-'. II)
j>o\\'er 1 rélll<"l11i:-.:-.ion lill('s [1 !J],
TIlf' (élll..,(· (If titis plH'1I01lH'IIOIl i:-. I)d:-.ic,dly (·,11 <I( 1 iOIl of ('IWII''y fi 11111 1 III' !lOI\' 11')"
il f,ll1all p(,1 t 1I11)iltioll uf tIlt' :-.tllH 1 III (" ",hil h, 111 111111, 11l( Il',1''1':-' 1 hl' dlllpltl 1111(, III 1 III'
distUl bdllCl'S 1('clCling to l'lIg(' dlllplitlld(· u:-.( ill,lI iOIl:-',
Th(' IlH'C!JilllislII uf ('n('I,!!,)' (·,tl d( 1 i()1l flUIII (1)(' lIow 1))' 1 II(' :-.t11l1 t III (' 10111 IJI' 1,.1 ('
(J<;C'illal illg ill plllng<' alld pitt h, Fig, 1 l, Sll]J)lO"'(' 1 II,lt t,lj(' fJ('IJ"I'II( i(·" of 1 III' IIllIt 1011
Sinc(' tlH' lift fUI('(' if, p\IJ]JIJlliollct! lu 1111' itllglt' ()f ,dLlIk. d:-'~lIl1lillJ!, tlJ(' ... illlJ,I(··.t.I(/11l1
of ;WI od~ nalllics (i,C', <jlltl:-.i-"I cltie dl·IOd.1"dllli(..,), IlwlI, t111' 1 III \'(' "Ill/will!', t 1 JI' <1 Il J',! 1 ,
of attack al!->u 1 (·]11( .. ..,(·111<, t)J(' lifl fOI «' ct( 1 illg 011 flIC' ,til 1'011. 11, ~,lrlJllld 1)1' Il,,t l,ri t Il,.1
the (·fred of 0, ::, ,lIIe1 :: (0 tlllt! -; éll(' :-.111)\\'" ill Fig, I,:n 1)11 1 1 If' 11ft flJII (' Il.1', IWI'II
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('II/\/I'f'j·;U 1 /.\"//(()/)( '("/ IO.\" (j
11',lllIll'd If 1 1 JI' pllil'(' rIlff(,I!'IJ((' i, ", ,IJ()\\'II ill Fi,!!" l,L, !)(Jo, t1ll'Il. il 1<111 IH' '>('('11 tlJill
Illd\11111I1I1 ('IJI'I!!,y i .... lwilJ,!!, (",lldl 1('" hOlllllJ(' f/(j\\" \\'I)('n:: i~ Z('IO IIJ!' fUlCillg fllnrll(JI1
l', ,1 t. Il ... Illd' 111111111, "llrI \\'111'11 '; 1'" ,tt il ... Illrlxilllllll1 tlJ(' fUI ('ing, fllllel iun i:, Z('IO, It ie.,
('\oIf Il)' t.!1!' ... .1 III!' ,,~ \\'/)('11 ~()IIJ('I,()"y is pll~llillg a ~wiJlg and making maxillllllll U~C
or 1 1 If' ;tpplif'rI fIJI((' 11('11(', (1)(' ('III'I.!.!,y ('\llilr\(od flom Ill(' flo\\' i ... dil'C'rily depcndc-nt
IJII (1)(' 1'/1,,"'(' dlffl'I('II( (' 1",1\\'('1'11 thl' 1 \\'0 lIIolioll:".
AI ... ", flOlli (1)(' ,d,o\'(' di~( w,:-.iull Il (ail ('d~ily 1)(' COlI! IlI.kd thal the \J('lJ(ling élnd
1 (j):-.if)lI,d 11101 )(/11:-. JlIII~1. Ild\'(' t.!W :-'''lIl1' fl'l'qlll'lI( il':" fOI Ihi~ 10 haplwll, otJ]('r\\'i~(', tbe
Il'qllil\'d ph,,',(, dilr"I!'111 (' (',III Ilot IJI' 1)('ld COII~ldlJt fuI' a 101l,:!, !H'liod of tiIlll', This
i ... :-,ollll'Iillll'S 1('f('I(('<I tu as ''fnql/Illey î{)alrM'( nec", \\'hich ir. chal'act('ri~tic of this
Iyl'(' of 1 \\'o-dq-"JI'(,-()f-fll'(',]ollJ ill:-.t.t1,ility, III a more 1:',('1]('1 <d case, tJW[(' can be more
Illdll t.\\'O dq.!,II'('S of fi ('l,dom pl<tyillg lol('r. in tll<' ge'IH'latiolJ of flutter, 'let, ('\'C'1l in
1 Iii ... ('M'l', 1\\'0 (JI 1 1 Will i11(' llsllally dOlllilldlli (j\'('1 the' ollll'IS fOl each possil,k fluttf'1'
"'1)( '( '( 1.
Th(' (11)('1 1 ('rlSOIl W hy /III 1 kl Illay OCClII in a :,y:,kl1l is 1 he fOi malion of a
"lI/gU/il'(' jll/HI dUlllfJlllg", Filst of cdl, \\hal do('s il 1l('l:',ati\'e da111ping mean? If a
:-.)':-.1<'111 h,l~ po ... ili\'(' dalllpilll:'" tl!l'lI :-'Ol1H' of tJJ(' sy~t('m's C'11elgy is IJping dissipated
to thl' ('II\'iIOllllll'lI1.. No,,", a IH'p,<ltin' dillllpinp, IlWélllS that the :,y:,tpm is SÜI11<'how
,d':-'lIll,ill!-'. 1'111'1 gy l'Will 1 lit' ('II\'iIOllIlWIIl. Su t!J,lt, hcl:,ically 1 he main ide(l is tJw ~all1(,
,1:-> 1)('1'01<', l'lit. 1 \l'II' t 1)(' (all"'(' of 1 hi:-. a\':-,ul htiull of Cll!'lgy i:-. diffcl'C'lIt, ar. di:-'lïl:"~f'd
IlI·lll\\' .
('ulI:-.idl'l' .\ t \\'o-dillll'I1:-.ioll,d ail foil, :-.tiflly mOllllt('d 011 a tOl~illllal ~prlllg, and
","I,jl'ckd ln tllI ,IiI Ho\\'. i\'o\\', illl<l!-',illl' an O~( illatül'y di~t1ll'hal1r(' is gi\('n tü the
dilfoil Silln' 1 hl' ail'foil is lIlU\'illg in thl' ail f1o\\', 1 he ail lIu\\' ~ho\llcl adjust ibt>lf
• ('II, \J! TEH 1. l.yruo J) 1'( 'TI (J,V
to tl)(' 111'\\' Ill)llllddl~ ,Il l'\('I~' Ill\lllll'Ilt Bllt, Ill'ldll:-l' of tlll' illl'Iti" uf tilt' .III, tltl~
tll!' \'('Iulit~, of tll!' !Ill\\' dlltl tlll' Ill'ljlll'lll \ III tlll' u:-,cilJ.llillllul thl' "llflltl, :\u\\, tlte'll'
is a ~l','(ili( "IH'I't! lOI \\bi(h tIlt, ph""I' c11f!I'II'IIII' Iwt\\'l'l'lI tIlt' Il''lill,ItIl'I!'' III tlll' "II
flo\\' rtlld tll!' ail l'ai! it:-'I,lf i~:-'1\I h tl: Il tIlt' dilfllil 1 dlll'\t".l1 t l'III'I,I~\' 1'111111 tIlt, 111I\\',
dlll!. ,II (()J'(lillgl~', tlll' dillplit1\l11' uf Il:-'1 tll,lIiulI 1111 Il'.I:-'I'~, III tlli:- "it Il,11 IIII!. \\'IIt'1I titI'
"i Ifui 1 i:-. dt if:-. Il,IIIII,d po:-.itiUII tll(' "il !Ill\\' IS pltJdltllllg 111.1\111111111 lift, "lId \\'11l'1I it
i:-. at Ih Illd\illlll1l1 dillplit lIt1(' t II(' lift, 1'1111 l' 1:-' "t it" Illillillllllll \',dlt", 'l'Ill' ',"1111' t\'Il\'
uf p1Wllllllll'1I01l JIl,I~' hd (>pl'II \\ Ill'II dll "i 1 Iuil i" II/l' Il'1 t dki Il.L', " IU'I!dliig C)',I dl.tlillil.
A ... cali II(' cOllclllt/I·t! rlulll t III' ,.J)()\'I· .11"111:-':-')1111, t hl' 1'1 ('q1l1'111 \' ni Il''1 "l.tllllIl IIf
tlll' c1ilfuil pl'l)'!'> thl' :-',1I1l<' 1111<' .1:-' tlll' .III "1)('1'.1, ill Iw\\' 1'",,1. thl' III1\V p,dtl'III 1 Ildllgl':-'
tlll' air :-,p(,l'd alld t III' frl'qll('111 Y of U'o( ill,iI ltJlJ. '\1 t Il,dl)', t Iti:-. i:-. \\'hl'II' t.J)I' "" 1 dll"d
U,., dune! fI( qU( 1/cy" pell c1llll't(' .. ,
, ..... 11 1 .. =-= (1 ,
!lamie., is 1I:-,('d tu calr1llafl' t,lw (1I'IudYII,IIIlH rUII ('!">,
1.3 Thesis Overview
III tlJi:-. :-.tlldy t 111' d('1 (J('la~til ('qll,üiulI'" of IIlut il) Il 1)[ il t \VI) dlllll'll',Jrl/l.d , I,!'"d "II foil,
a:, t)lP ailfuil 1)('1 tai!l:' tu fllltt('J',
Filst, the' ('qllal iUIl!) of !IlOticJ!J ul il twu dl'gj(·(·-of-fl(·(·drJ/1I air/I,JI ,11(,11)1 IIJ1d,d f·tI ill
•
•
•
('11/\ IITJ';/{ J, I.\''J'/{{)/)( ("/ IO,\'
Illdptl') ~, III (h"pl(') 'L 1 Iii' 11110.,1(',111, ,lI'lUdYlldlliic<, i~ r1i~clIo.,..,c'rI, alld ~(lIIJ(' IIJ('\Iric'~
fOI t!w l,dl IIlatioll {Jf il Iii c1Iffr'I('liI !IU\\' Il'gillJ('~ illC' illll()(llIt('cl. Tlwn. ill dlclpl('1 II.
rldfr'lI'lll. ('XIH'IIIi)('II1.t1 ;llld '111,il,v1 i( ,d dPIJlUdrlW:-' 10 t1lC' nuUc'l dllilly~is of clll airfoil
dJ(' illllodll«'d 'nll'> (1Idpl('1 ,d..,o JlJ('~r'llh IIi(' tlcldilioll,t! ~()llIti()lI~ 10 tlw nlllter
IJlolll('III. IldllJ('ly. 1 II<' Ji-/.: ,llId {/-.'IIIlt'\ IlOd..,.
('1101111('1 :1 i:-. c!('\'okd lu <III ill\(",tig,tll\Ill UII t,lw dr('cL of difrc'Ic'III (ksig,n PilIillll
('1('1:-' III' <III tillfoii (III Il)(' /lnll('1 ~I)('('d. Ilolh in III(' inc()lllplc~~illk dllfl 1l,lIl,ollie
1 (',l')IIIII'S. III 1 Iii.., (Ilil)/kl, d (OIII)ldIJ'-OIl (If tll<' tldditi()II,tllIletllOd~. 11](' p-l' fllld {T_ q
1111'1 hllrl.." \\'11 Il III(' Ldpld( (' IIl<'t hod i, ,t!:-'II IlIiI<lI'.
A l"II,lllll'llic fl 111 II'r all,dy:-.i~ of dll ailfoil, huth III the ill(Ulllpl('~~illll' élnd Ililll
:-'lIlIi( flu\\' Il'.l',IIIiI':-', i~ 11I1'sl'lIl1'd ill c1ldpkl () III this (haptf'l, élll allaly~is of ho\\' t.he
/1/1///1'1 \'dlll<':-' Ill' d(·~igll )ldlcllll<'t.I'I'" of clIl dilfoil afrect Ill<' orClll'/'fllC'C of f1lllter 18
.t.',1 V(' Il.
Fill,dl.", III 1 hilplt'I i, tll(' 1 OIl1ll1:-.ion.., on tlli:-, ~t lIdy ;11(' mélde. anrl al~o ~ome
Il'1 Onlllll'llIl.ttioll:-' )('gdldillg tll!' IJI('\('lllion of fJlIt!C'1 aIl' :-'lIggc~tt'd.
III d]l]ll'lldix ,\ .Ill ,tlgl,llIdlc allill~:-.i~ of lIultel 11\('( h(lIli~lll. Hllder ~un)(' ~illlplifyillg
d~:-'lllll)ll)(lll:-'. i:-, 1)) ('!'-o('lIl1'd, :\PIH'IHlix B (oll1..tiIIS cumplete comp1lter progldl11 li~tillgs
fur 1 Ill' (T_ II .tlld p-l 11H'11IOd .... Illll h ill 1 1)(' illcollll)J('~~ibl(' élnd tréln~()llic l<'giIllCS .
• ('ll.\P'j'j~'n 1 IS fJU)f)l'( 'j'JO.\' ~ 1
ELASTIC AXIS
CENTRE OF MASS
Fig1l1(, 1,1: A ~(h('matl'( f t l' o il \\'()-( 11Ij('II"jf 1 1 r '1 ) Id dl/ (JI, Wlf Il twu d"/I/f"" f r 1 t, ) 1) 1('('( 11111 III
pJlIngc <Lnd }Jit( h,
•
•
•
('/1:\/1" U( J 1.\'/ /((Jf)/,'("j JO.\'
::, ()
,,-,/
"
1 /
1 /
" " /
o mot iUIl (l i ft fOl cc)
-...... 1 .... '"
'- , , ,
10
t
-
Figlll (' l '2: ï'11I' l,JlIll[,ill~, ::, and pit (h, 0, hal mOllie mot ions of a t\\'O-dC!jICC-of-
•
Chapter 2
EquatioIlS of MotioIl
('oll"id('! d t\\'()-diIlW1\~iull.d. I\\'u d(·P!(·(· uf-f!!'!'dolll .III fuil Il,·\illlv 1IIIlllllkd 1111 ., .
1\\'0 ~plillg<, in 1 Ill' t()!:-,iullal dll«l IH·IIIIIII.!.!, dil('llioll'>, dS ... 110\\'11 III l'·i,l'._ 1 1 1)I'lllllillg
Il)(' v('lli( allIIotio1\ by h, IHJ ... iti\(' dU\\ll\\dld. ,llId Il)(' 1()I ... i()lI.d 1IIIlliuII Il\' Il, jlll',lll\'('
1\O~(' IIp. UII(' (ail Wl'it(· U)(' (·qll.1I iUlh of Illol iOIl ur 1111' 1 ip,id "il fil" III ,.", Il dill'II jOli
1/1.11 + 1/1 (,r" li) fi -- -- /\'" li - l"~ •
/li i~ tl\(' Illcl~:' 1)('1 1\1\it ... pa1\ of 1 JI(' clliloiJ,
.r" b i:-. tl((' di:-tdll«(' flo!IlI])(' ('('Idl(' uf III" ...... ,dt. 1)1 1]1I'(,ld"ti( .1\1;'.
L i~ 1 hl' tut al lift l'OH P,
• b is the "('mi-ciiol d,
1 1
•
•
('I1,\I}'J l',/(! j-'(j(',\'J /(),\S 01' .'[()F/O,\
\1" l', fil(' fol," d"I{)r!Ylldllli( 1I1{)IIII'IJI ,tlJ()IJf 1111' (,!.l"fi( r1\I~.
h h'
l:!
(2.) )
\\ 111'1(', li)" dlld ...... ' .. .11(' 1111' IIII1IJ1lpkd 1I<tllll,d fi ('qlll'Il( i(·!> in 1wlldin,!!, ,Illd pitch, lC-
"1)('( li\'I'I,'" ,l',i\'('!> 1111' f()lIlJ\\'ill,L\ fUI III (Jf thl' ('qll,lIiulI<' of m()lion:
(2 G)
(:!,7)
lu\\'ill.l', lUI Ill:
., ) t L ~t-I n+w'"<.,=----
.. ! 111 b ' (L 8)
l".. c' Il) 2 - --- - Il -t r. <., + ----- - u..'" () /II !J2 III h2
(:!.9)
(II .l',,\ l,II 11111 oft1l<' dllfnil al1111l1 1111' 1·111Stlc d,i~, ('hdl1g('~ th!' fur111 uf the l'quatiom of
Illldioll t(l 1!t1'!lll1u\\'illg'
.. l f, L f, t- r n + u..'," = ---, • ! 1/1 b ' (2.10)
• ('Il. \1> 1 J-.ïf:.! Hjl' \ IIU \..,' (JI-' \101 l() \ \.1
} , /' ,~ 1 1 •• ,~' ,. (\
lUI dIffl'll'llt !lU\\' Il',!!,iIlH'''. Iî](''-.(' \\!l1 1)(' .11'-.( Il''-.l'd III t Ill' full,,\\ III~', (11'if"l 1
•
•
•
Cllapter 3
U llsteady Aerodynamics
/11 .III d('I,)('I.t~ll1 dlldl,\~I,"> 111(' 111(1,,1 (lilJ(,t/ Pluhl('1JI lIlay J)(' cOII~l<kl't'd ,l~ IIH'
(.Ilf Ill.d il/II ur III(' 11I1~,l('dd," (I('Wdylldlllif s, '1'1)(' ()Iigill or the tlll~teéldy dred:, \letS
di,,( Il''''1'</ d:11 ill,e, 1 Ill' 1111 wdl/f t 1011 10 1 Ill' plohlt'Ill: !JoW('\'('I, il \l'dl J)(' di~l 1I<;~('d JJI idly
11(,It, gdill 11(/111 dlltlllJ('1 l'()illl 01 \ Il'\\,,
1,'lu\\' dl()lIllt! "Il dill()d (dll II(' "illl1d,t!l'd Jly (,()1I~idt'lillg il \OI(,('X di.,llib1IliOIl
.lIllllg 1 Ill' f Illnd lilll'lIl 1 Ill' dlll()il. 'l'hl' \I,IOf itit'S illdll«('c! by tll<' \'()J'!('X di'itlib1Itiun
~11'l1dd "dli~h 111(' /\1111.1 (Ulldilllll1 dl 11l<' Ilélilillg ('dg(', ilS \\'(,11 as, the cUlldition
Ill' /('1 () \'(,llH il,' illl lJ 1 II<' ~lllld( l' (If 1 Ill' "ilfuil. Allolll<'l' condit ilJJl wlrich ~1. -'lIld be
'"di~li(,d h I\(,hill~ (il( lIJ.djllll t 1lt'1I1 l'Ill, \l'ilieli ~ay~ lllid. th(' t(lled cil< 1I1,tliull ûf a
111I\\' li(,ld 11111',1 II(' (()II~t.IIIt. F()I d !lo\\' '">1.111 illg flom tll<' Il'~t, 1 his will he Z('IO. uf
IIIIII~(' Thl'I <'l'III (" .11 1 lit' 1 II'gll1 llill,l', (JI 111<' !lu\\' <1 ",,/ar/III!} l'or/t,C" i~ J!IUdIlCl'd, \\Itb
(II( Id.IIIUII ('qll.1I dllt! IIJ1J1(I"ill' III III!' Ilulllld (il( III,tlio!l O!l tll!' clilfuil. ~('l' Fig, :31,
Tlli~ lIl()\'('" dll\\'lI,,1 I('dlll \l'il h 1 JIl' 110\\' \ l'!urity, dll(!. 1 11\'1c!'UII\ i" ll"llillly iglloll'd [1 <1],
Hill, ill th(' (,,:-0(' 111'.111 (l"f dl.lIillg dilloii. :-oi Il l'l' tll<' lIu\\' i~ c()lIlillllOlI~l~ dlilllging. a
Il
• ('Il. \ /J / /'1{ (',\,".;J /: \/» \/,'/U)/J) S \ 'Ill '.'1
'>l'Iil':-' of 1 Ill' ",111/111/1," \111111('-.,,1,,111'.\ 1111' Il'Id,'1 /'(1/,/11 -./111/, .III' llllllillllllll ... h llI'llll' .'
uf (111''''1' \l'clkl' \ (JI (II ( .... dl'fll'lld:-, uli hOII' JII"I 1111' 11\1\\ 1 ... 1 hdll!I.III!~, tllI.\. III 111111. il
3.1 Incompressible 2-1) Flow
[IOJ. ,\11 illllJllqll {,:-, ... il,J(·, 111\ i ... ( HI. dlld ill,,1 ,If 11I1Idi nll\\' \1 Il Il "'11I,dl .11111'111 Il.11' 1)',1 III"
/., ilnd '\1",011 1 Ill' (J,tI-pl,tI(· "i,lull shlJ\\'11 il! Fi,!!,. l, l, "Ill d i'll'd /111111 Illi', ',1 IIII\', d Il'
t hl' f(JI!()\\,i IIg:
--l, :::: -- 71 fJ !J2 (i~ - l' (1 - li 1/ ( i ) -- ~ 71 fi 1,' li ( .( / ) (/, 1 1/ (1 1 II ( :~ 1/)11).
illld,
\Ill ~ il fi II 1 ( Il ft 1 ( 1 Il) (1 II( 1 1 1/2)11
~ :-;
1 Id 1 :.! il fi 1/ ( .( 1 ) ( ~ 1 1/ ) ( /, 1 1: Il 1 (1)0) ~
\\'11<" (', Il I~ t J Il' 11\))1 -1 1)) 1 H ') l ... iO)1 ,I! ,,) ... ld)II(' !JI 1 III' <l,) .... /)1 ) 1 J 1.11 )1 JI 1 ,dl /JI Il,,. Iliid
•
• ('II,\l'') l,;/( '1 ('S:-j'II. \/J} ,\/'./(()J)LY,\.\I/('S JG
1', 'f'II/OI/III,"'I/\jlllllf/O/i Wlllill IC, (OIIIjJ()c,('d of il )('dl dlld 1I1Id.!jlllitl)' pillt., dlld I~ cl
/111111 Il !II 1)/ 1111' II·d!11 l·rI fl('(!,I!'I!1 y, 1 .. ,
('(I) .-= F(I,} + 1 (,'( '.) •
Will'II',
wb l,' = fT'
'l'III' \'dlldll!lIl (JI fo(/) dlld (,'(/,) \\ith l,' d!(' ~h()\\11 III Fig, :~,2. ,md tlH'ir llllllwlical
:~.2 COlIlpressible Subsonic 2-D Flow
IIH'tbod \'l'I\' .... illliidi 10 TllI'odu! .... (·II'.;, ,Ille! (·"pJ(·:-,<.,cd the a('!odyllamir COl'ffjciC'llt!'. in
III!' fullll\\'I1I,!.', l'III Ill.
( • (' ft f
1 = n '1., + b ( I h
( '''1 , Il ('
-= n ( III" + -b 111"
'l'III' 1111111l'1 ie.d \',dlll':-' 11111111' tll'IIIIIYIl'lI11ic (odlil il'Ilts ill(' P!I·~I·llt(·d in tilt' fOlm of
1t('qlll'lI\ i\·~. 1,,, .... ('t! 011 t 111' .Iii 10" (hOld. kt = ~t' = 2 h', rrom 0, to 1 A. SOIll!.' of the
•
• ('/1.\/1'/ FH.~ 1 .\.~·rF \f)) ,,../iO/».\ \ \ [/('.".
-- -1-- ---- -- -- -k F <:
IO,OU (J,-JOU() -OOI:.!1
!l.OOU 0 .. -)01 ï -O,O~O()
1 .. -)00 O.,)~IO -O.OT.Hl
J. (JOO a .ri:1!J 1 -0.1 ()(J:i
o KOO (J .:).~ Il -0.11 fi!)
a.fiUO O.Tl Tt'x -O.I:{ïK
o iUU O.:ï!Jï9 -O,I.j(lï
a IOU (l,(i·2.iO -0.1 fi:)O
O.:HlU (J.()(j iO -0.1 ï!!:l
O.:WO () i~ïfi -(l.I~~(j
n.IOO o :-;:3:W -O.17:n
0.0 in (J I)(Jfj(J -11.1 :W!)
o Olt) Il. 1) ,"'12 1 -Il () IS~
(I,(J(l() 1 _ ()()O() () IIIIIJO ~ -- - ---
Ltlll(' :LI- Tlw()dtll"('ll\ 111111111111. ('(Il FI 1 (i, 1'.'111'1" /,
•
•
•
('II Il'1 l,/( : (\'111- If)) IU(()/J).\',UI/('S
i 1
1
/.
1 (j ()f)
Il 1
1 Jill
Il 1,-,
Il ,\II
Il .!.-,
Il 1 ()
() -,()
1) ()I)lII) Il ()I)()I) Il ()I)()() :~ W~(;
(1 Il 1'\," () 1 (,f!f! ,{ Il :~IJ -() 211!)
Il I.!()(, () -,(, I~l ~ 'l~,-J1 -() 1 ï!!ï
(111:-;1 Il(,ill~ ~S~()Ij -()():-;{l
Il Il,% Il ïïll.'i '2 ï HlS () U 1 ï!!
Il 1'1:'1) () %Il 1 ~ -,xï.{ ().~.ln
Il 1 fi:!:! 1 1.IS!l .) 1 ;-'::!ï n. I.i~'i
·,111/\\'11 III 1 \l'., .{:l
:~.3 'f'ransonic 2-D Flo\v
IS
III (10111"'\11111 Ilu\\. 1111'11' .11\' dlll\'I\'1I1 Il'giUI1:-- uf ,,,"l''''Ullic dllcl ""l))('I:--UIlÎC f1u\\' 0\('1
1 1 if' "Îlfu" lill\'lu 1111'" ÎII\·,!.!,lIldlil.\. '1"'\' uf III\' lilll'dil/('d ('qlldllUll:-- Cilll)Jot pl'l'clirt
111\ 1"'\ id. dllt! il, (l1.II illll.t1 \I ... u. III\' d ... ..,lIl11pllllll uf :-'11l.dl 1 hl( 1\Il(':-'~ of tht' ail fuil, ..,mall
• ('ll .. \/'/TU:~ I·SSTF.\J)) \Uf()j))S\.\Il('.· .. ; 1 !l
------------ - --- -~ ----- - - -- - ------- --
0.00 O. O{) 00 U .(JO()(J .() !1(l{i9 (). (JOO(}
o 0:-) ·0 0010 -lJ.OI'.! -) ·Il ,,,:-):J;) () U:Fl(}
0.10 ·0 OIIl2 -(l.Oï% n ~() l!l O.OI!l<i
0.1 !j ·0.0 )-)(; -0.11'2.-) .(J. ïli II 00110
o :W -OOI\l1 -{l lin -ll ï :WO () 0:-,( ,(;
n ~.) ·0 Il:W 1 -Il. ){,Ill ·ll, ïO:!:! ·0 IW','2
o.'w -() 0 l,"'.! -(j II li 1 Il hï/2 Il. 1 -,.-, 1
0,:3.) ·0 () 1 ~h -IJ.·2'2l1lj ·11.(,:) li Il '211112
0.10 ·Il (J()h'.! -IUli) ll.h.~ 1:1 Il '.!:,S. ~
o :-)0 n.O)(iS . \l.i!)'.!;'; ·II,.',~IS·1 o ,Hi IX - ----- ---- --------
.\: ('III" 0 uf cl fLlt pLtI(· ,d .\1 ...:. O.).
ilH id(,IICl', alld "'1 !l,dl dlllplillld(' i ... Illcld(O, 'l'Il(' Il'' ... llltill.e. ('ljlldlillll 1'" III 111.· 1'1/1111.
L \/ .. 2
1· ÔI/I/ - - Ô"
(J, 1/ "
W IWI (', \/ .... i~ t 1 If' fI( O( 0 ... 1 1 ('dlll \ 1.\1 Il 111111 JI H'l , ~I '1 .!!,d'" 1 1111" l " III , Il .. f"", <,II"d "1 \'1 ·lfll Il v,
",', l ,,2 ,l, l' 1 l' 1 1 (,'J. = -;--,' . (,J)) = -, 7, (-te .. dllf (" 1'-, 1 If' \'(' (If Il \' plll ('ll1ld • \\. 1('11' Il 1 J f J
(,', J ,1 JI f 1 IJ (/1,/ '
.11<' tlJ(' \·(·Iu( ily 1 OIIlIJlJlJ('lIh III 1 1 If' ,r dlld '1 dill" 1 ioll' .. , l ''''IIf'' li\'('ly
locit) Oll 1 Ill' dll luil ~111 Lit (', '1 ]Wll. ll~illg Iklllllldlr"" 1'11'1.1 1 iUII, III' l'II", (Ill' , Iwlili Il'111.
Ôt -- ) .
• I/.~ .
•
•
('II \ VI J-;!{:I l ,\,)"11'; \ J)~' \ J-./{()/)) S. L\lI( '."" LO
'1"1 ',fi]...'" 1 Ill' ,tllll\(' /llflld('111 dl'lllllcI dll dll ro", il Ilw"h uf lIud<' PUilll.." \Vith f(u ('IJ(llIgh
Ilfllllld,1I /1'''\ l', }!,< '1 If'l ,d (·rI tI/lllllld t III' dll fuil, ,lllrl a fillite dlff('((!1( (' llwthud clllploy('d
1 fi IlIld (JI lOI (';I( I, Ilurl(',
'l'I,I'" 1I/1'IIIIJrI. \\'1 Il IIldl/y III'JI(' (()Illpl(·\ili(·~. Ild" h('('1J <'llIplu)l'cl I)y ~()II)(, p('opl('
III 1111' Id,," .1('( cld(·, dllc! ,l f('II' ({Jlllpllt('1 f ode~ h,t\(' IW('II geIH'nd('d. The nl~t rode
I\'d~ plf·pall·d l,y H,""ldlll dllrl (;uuljidll [LL]. IJd!:>{'d 011 t hi~ lIlt't!Jod. ,lIld i!:> (',dl('<1
LTB,\i\L 1.,11('1 fllI. 1111.., "dll/(, (fJd(· \\d~ iIlIjJlu\('d l,) I1oll\\'illk allcl \,Ul .1('1 \'(JOI('II
[:!.!]. dllflllli" II('\\, 1('I"itJlI 1" (,dl('d 1.'1 IL\;\L-:\'LH. TIJ(' <.,all](' (UIJ('f1ioIlS Wl'J(' also
11/.1.1(· l,y ('\llbl,," dlld l\II,l',(·lilli [~I] ,It Oi\EHA, Thi:-. ),u,;t code, wlth '!Olll(' Illinor
IIHHldl( ,I1ioll~\ hd~ 1)('('1/ dldil,",lf' 10 ;\:\E [L5]. Jll lad.. Ill(' 1I1111l<'rical rc'~lIlb fl'om
1 Iii ... ("rI(' Il'('((' "III dilWc! J,y 1,('(' III 1 !},'; J for cl NA CA (j 1:\006 airfoil ,tt él fl'\\' ~JcI('h
lilllllill'IS, .111.1 11j(",(' 11· ... 1111:-. Itdl(' I)('('n 11~('d in tlti:-. ..,llIdy. This IlJlltiw' a)]o\\,s fuI' the
(.d( 111.11 iUII or Ill<' d('lOd.\·IIi1I11i(· .1('1 i\'dti\('!:> ni,' ('t". ('''li,' and (,'711'1 for ail o~cillatillg
"illilii III 1 1)(, t l ,,1I..,ollie 1 ('gillH' al "'('WI,I! :\Ia( Il Illlllllll'I's.
)'(1/ ('\.111/1'1(" 1 IH' .1('/1)(1: Il,llIIic «wflici('lIts ('1". ('t .• , Cml, élnd CHa obtailH'd fOl a
~,\('" {iIA()f)(i ,Iilroil at .\1 -.:: (J.~:i dl(, :-.llOwn in Fig. :t·l .
• ('JI \j'TEH ,1, ('.\'.'''1/ F \ j)) \]';I\()])) ,\, \ ,\IJ<'S
ils)
(a)
cil (llJatioll illullllrl 1 JI(' .IiI foi!.
•
•
•
('11:\/)'1 !-Ji .1. /·S.)'/I·::\I))' .\I·f(()/))·S \.\1/('S'
1 .1
,
09
.-.. 0.0
..x:: '-" IL 0.7
06
05
04 o
\, \, ..
(005) , •
.-..
..x:: (0 1) "-"
~
(0.15)
(0.2) 0
- ..... --- ... _-- ..... - ----- ....
0.2 0.4 0.6
Roduced Frequency, k
,~
_ _ l __________ L
02 0.4 0.6
Reduced Frequency, k
~- -------
__ 1 __ _
0.8
____ 1- _________ _
0.8
Fi,I',lllt':I:2: TIlt' 11',11 .Il1t1 ilnaglll<tl'y prll h of the TIH'orlol:'('Il'S flll1diull, C(J.), \'C)'bUS
•
•
CIf..\fJIFH :J. l'.\,'·j'j'F.\j)) .\FH()j)).\'\.\I/('S ".1
1 2 -
, -08 -
006·
04 . ,
/ / .
/ . .'
02 - • .• -'- • / ' -
l' _____ .-_ .. ---
°O':'-_"--~\- 0 2 0 3 0 4 0 '5
005 .. - ----
Reducod Froquoncy, k
--_. . -- ~---_ ..... -o -::- • __ -.-=~-_-: ._- . ~ ,
(0.05) "
.c. (0.1) E
'. [Real ]
Imôglnmy
o (0.15)
(02) -
(025)
(03)
(035) 0
"
. ___ .. J . ___ J _ 1
0.1 02 03 04 05 06
Rcduced Frequency, k
4
3
_Il 2
o
o
02
0
(02) cs
E o (04)
(06)
(08) ./'.
01 02 03 04 (lb OG Rcduccd Prequcl1cy, k
' . 1 npnl 1 IllltlU!l1dry
"
.~-' --
..,.-
1
01 02 03 04 05 OG Rcduccd Frcquoncy, k
().r) 1 fi '> " f 111)1 1 i () Il Il fI' d III l' d Il 1 . 'l' " '111 y, 1 ...
•
•
('/I,\}}') U(.'I (i.\'S') 1-:,\1))' \/':/U)J)) :\';\.\11('8 21
'15
1, n'.1I1 )
1 Ilrn"fl',mry ," "
o 15
0:>
E
0(0:»
(04)
(0 C) 0
, 01
. .
.. ' .'
J J
02 03 04 05
RCdlJCOd Frcquollcy, k
'. --
.......... '.
1 1 _ J ~ 1 J __
01 02 03 04 05
Rcduced Frequency, k
o
06
06
20 -
15 -
10
5 .
o .
(5)
o _, ,
(02) •
(06)
(08)
_ ,_. J • ____ 1.. 1. __ I _____ J
, . , , , ,
01 02 03 04
, • , ,
Reduced Frequency, k
\ , ,
..... ,- ....... .. --"'r' ....... '. , , , .
05 06
(1) ____ • ..L._---1-. ___ ..L __ ..J ____ .J ___ _
o 01 02 03 04 05 06
Reduced Frequency, k
n.S!i d~ a flllll 1 iUIl of Il'dllccd flcqtH'IlCY, 1.:. (L('(' 198·1)
•
•
Chapter 4
Different Approaches to tlle
Problem
Tu uhLll1l t111' l'/'tlll'lrl.fl"I/( /' "/J(/Ii 01'.111 clill l,dt. Il)('1l' ,lIt' IId~il ,dly 1\\,(1 dllll'II'III
appruadw:-" UIW Ita~('d 011 (':-'jH'liIlWIILd It'( hlliqlll''->, .tilt! t!J(' 01111'1 llll JIIl))l' Ilw(J/(,tl(.d
llwlhod:-.. ILI~i( ally, butb of tllI'!->(' 1\\'0 apjJllId( If(':' ~htJldd 1)(' ('illl'ltJ)('d i/l tIldl'l t,o
ha\'(' ét good (':,tillwlioll of Il)(' ~jH'('d dt \Vhit Il fll/tlt'I (Hi 1/10.., [:~(jl.
4.1 Experimental Techniques
4.1.1 Ground Vibration Tests
Th('I(' éll(' :'('\('I,t! ('\j){'IIII}('I!1,d It'IIIIIÎ(jllt':' rOI Ill/ttl'I "11"ly~i:-, J''iI',I, d "y/llll/ld
l'lbm/lOlI I{.~/" lll,tY 1)(' illlplellwllf ('(1. Tlli.., /ll,l)' 1)(' "OIJ(' It) llil \1' ,\ 1)('11 f'l ,",III/I.tI If/II fil'
tIf(' ,J(/III]J1/1g ,lIlc1 1/a/llui! Jf( ql//!If " ~ or 11j{' \'11" ,d JUII,d IlItJ,Jt", of Il)(' "" 1 l,dl [:tl j. III
thi~ t(,..,t. tlJ(' ail naft Î.., :,u/lJ('1 ill}('.., :'1I'-,jH'IIC!<'d IIUIJI il '-,11"j)('/I',ltJ/1 ',~",I f'/lI, dlld ·,II,"WII,
•
•
('II,\I/'} J-,'/( ,1 j)}FF/',/(j·;.\'J' ,\J>fJ/{(J/\('II/~'S' TU 'J'lIE PUOIJI.E.\/ 26
Il'']llg IIII'llldllil,t! ('x(ill'I" ,dl,u]wt! 10 t1il' "tlllllll)«' [~8]
'/'111'1 l' "1 l' ditl'('II'It! lj'pI'" !If """J)('li~i()li ") ,,1 ('Ill:-. Il'''1't! ill ]>1 d( tir C', The' <lil plrll](,
III"Y 1)(' ')II<,PI'lldl·d 011 ~plillg", 01, it llliI)' \'1' ..,11PPUII(,rI 011 it:-. landing gC'dl le-,ting
IJII "1'111Il', pl.lIl(JIIII" ,\1,,0, Il](' ;III]1Ie1I1(' fll,,(·lelg(' ellltl \\'Ing:-. llla)' Ill' "'IIJlJ)()ltcd (m
1" l ",(' dll tdll'e! f1(J,tllllg IJd,!'," [:.n] , T]w)«' eli (' ;t!"u "cil lOll:-' I.Y)1('" of "hcl ken. (l\,élJlaIJk,
IJlII, 11j(' llllJ,,1 1 (JlJIIJloll l1:--(·d i~ 1·1(·( 11IIIJldglJ('lic ~It,tl~('I, Thi:-. t~J)(' ClJIl:-.i:-.b uf il ruil
111,d i~ .111.11 IlI'd 1 () Ill!' ..,IIII( 1111(' \\'1111 d li\l·d dl i\'(' IOe!, TI}(' coil i~ :-'1111 OIJ Il r!('d by il
1II.I,t',III'II( lil·ld .tilt! i..,..,('( ill IIIOlil)1I IJy dll .t!I(·III;t1ill,e, (1I1l!'lIt, EI('I1,)nni( u:-.(ill;dols
.tlld dl"pldi('I~ dl!' IJ:-.('r! tu 1 (JliIIIJlllli" t.ypc· (JI' \y..,I(·111 [~ï].
Nu\\', l,y 11..,1' tif il IIIIIIlI)('1 01 .III (·I(·llJII}('I('I:-' ellld/OI \'C'locit) pit kllps, thc modal
11·"J)()Il..,(· 01 III!' :-.111111111(' lu UlI' illput :-.igll,t1 i:-. Ill('a:-'llIl'rI, c1lld is ('itlH'1 éIlJal~z('d
1 III lilll' (JI ,,1 (llI·rI fUI flllllH' <1 Il ,tlY"'I':-', ))Uillg "(J, dillllping ilnd the nal111 al fl{'qlH'IlcÎes
(JI III<' IlIlllJ)(/III'III:-. ur III!' :-.1111(1111(' (,Ill 1)(' nJ('<I:-.llll'd,
4.1.2 WiJld Ttlllllcl Tests
'l'III' :-'dlll!' sOIt of lIlt'd:-'111('IIlI'lIt !t'Chlliqll<'s a:-. in a glOtlnd \'ihléltioll t('st, l'an be
II11pl('lIlt'lIkd ill il wIlld 1.111111(·1 1('sl, 11(·)(', cl llludt·1 of tIlt' clilclaft UI ~onw cumpollC'llts
(If il (\\ill,!!., fll"'(·!.I,!!.t·, IOllllul :-'1111'<1(,(':-', ('\1',) i:-. plcl«'d in cl \Vint! tUIIlH,I, (t11d iS~lJbjl'ded
1 Il .t loIlIg(' (JI \\ illtl :-'11('('11:-.. ,\1 (',l( h ",illtl :-.p(·('d, Ilw damping (illel fn'quelley of a
111111111('1 ur II:-. \'illl,1I11111,d 1l1ud('" etH' l1H'cl~IJI(,(1. III !:>Ulll(, Cd:'t'S. the mode ~bcll)('~ éllC
.11:-'11 dt'll'llllilll'd [:!9],
•
•
('lI.\PTfU 1 J)IJ.'FVIW.\ r \ Jll'HO \('/lF.')' ru 1111· l'UOU! F\t
4.1.3 Flight Fllltt0r T('sts
, -- 1
(Iitic:d !lllt!!'1 \\'ollid ()«(Ill. cille! tll!' (\()(' 01 IIlIttl'l 111\ld\/'/1, /111111 IllI'.Io.,lll/'IIII'III·,
!'>imi)dl kiwi OfO';CIl!'>(lI ..... pl.I('d cl! ~(·\/,ldll()(t1i()ll!'> uf tlll' .I11/ldl!
\\ellllillg l, Fig. 1.1. il (UIII'( 1 d('/ hilJll )(·.l',dldillg \\ 11l'! hl'I ul' II/I! t lit' .tIl/ l.tll 1', ·,t.d')(,
skp fllll t('1' Il l el)' O( ('\11 (JI IIU!. [~Il. :WJ.
4.2 Flutter Margin Method
t.)1l' "jlllflf" l/Hl/f}IIt", fOlllwd li 0111 t1l!' (''\IJI'lilll!'lJ1dl fll'qllf'II() dllli .1,11/1/11111'. d,dol 1,..
• ('/""''f/';U 1 IJ/J'J-ï',/U,,\"/ \/'/JJf() \ ('IIJ-:S JO '1'/11'; fi/U)j3lJ;,\J
"11,!',!',,",I,'d (1)11111'111111"1111011).',111 illld.., .1 lill"dl llI'lld\IOlll ",ilh tlJ('dYIl,lIl1i,> PI(",..,III("
111101:-' h""11 ',111)\\11 ['{~1111,,1 Illi.., 1111'\ IlOt! i.., n,d Pdlli, IIldlly "'('II..,itl\,(' tu ('II()I~ 111
1111' dd"'1lillt~ d,ild, Ililt 'JII 1 1 If' ,JI 111'1 Ildlld, it j.., \"'I~ ~('II"iti\(, to tlll' ('lrUI" III tIlt'
.1.., 1111' dlltllclI i;, d\\dl", tll\'l(' hd;' lIut IH'l'1l lIlI) ~t1C, ('s~flll fCl/1ll fOI fllltt(,1' IJI,llgin
1.2.1 FIllUer M m'gin Mcthod for Tral1sonic Flow
III 1 Iii.., ;,(',Iioll, tlH' (l'lIli,"ldl poillts (OIlU'llIi"p, with tll<' 1I~(, oftllis Illl'tllod 111
TI/(' ,11.11 d( \<>1 i~t i( ('qll,iI iOIl of cl t \\o-d('gt('('-uf-f"'l'dulll "itfut! is, in g('w'lal: gi\,(,11
l,y t h(' folio" illg:
\\I!l'Il' IJI
(4.1)
•
•
•
('/I.\I'TFU 1 J)/F/·/-,/{i".\ r \/'/'NO \( 'lll-',li 1'0 /'IW /'/UI!H J-'.\I
'l'Il<' ,!1W\ l' ('\Pl ,·""iUIl 1'(11 Jo' i~ .t f, 11 III ul 111\' "" "dll,t! /llllt! , I/llIltll/l [.11] l'h,' ',1 "l,tilt \
IllJlllldcl1~' i" gi\l'II l,y F -..:: Il.
If tilt' ('lg('II\cdlll':-' (Jf tlll' ".\ ,,\t'Ill dt d "11(" dit "II('('tI dl" gl\"11 1,\ titI' 'tllllll,I'"II,'
l ('IlI]>ll'\: Iltllll!Jl'h,
tlWlI, tll!' I1l1tt('1 Illdlglll 1.111 Il(' ('\dllldkd l,.\, '(Jlllplltlll,l~ t Ill' \,tlll(", (JI' III
Ill!' rullu\\'ill,!!, (':-'Pll'S:-.ilJlI'"
,l1ld t)J('1l :-'lIbl it lit illg 111<'111 illlo ('qll,1f i()1l (·1 1).
1'11 Il''' Il,1',
In ill(OIllIJ1(':-.sil,I" fI()w, il 11<1:-.111"'11 :-.IIlJWIIIII,tl. d !III<1dl."I( ('xl IcljJol,iI;, III 01 tllI'
flul \,('1 l1l<llgill V('I!',1IS d~ Ildlllil pl I·"~III'I', (,V('I .1 ".1 lige of "1J('('d:-. lwlow t.llt' 111111 ('1 ~'IH'('II.
l'ill1 gin' Il'i cl g,IJ()d (':-.llllldl;OII (Jr Ill\' 11111 Il'1 :-,pl'('c1 [:l~l . .'-Jo\\'. tlw ',011111' 1I11·tlIOt! 1:,
('1l1pluy('d ill Il illI:-'Olli( flu\\', to 1 11('1" ho\\' \\'/'1111 IIldy \\'/)1" III Illi-, 101'.('.
'1'0 do ~(), tll!' ('if',"II\,dll(':-' uf tlll' ",):,11'111 dll'/(JIIIIII dt d IIl1l1dJf'1 /)1 ·.pr·III', l'l' 1."
Ih(' flllu('1 :-,p('('d, tllltl, tlwlI. Il''1I1,!!, III!' IIlf'lltod 1· •.. pldilll'rI • .l'I)\r·, Il)(' IIIJlII'1 111.11)',111
If> rUlllld ,d. (';l( J. :-'P('('II.
Ali illljJOJI;lllt rd( t.UI ill IIi!' jJlI'dl1 lir)ll I,f Il\!' 1111l.tr'l "I)('I'rlIJY Illi:, Iwtlilld 1',1111
:-'J>('('<I Jétllg!' ill \\,lli(1! tl\!' !'XtliljJ(Jl'llioll i" IJI'lf'Jlllwrl. Il 1 .... jllI·rli l ldl,lr· 111,JI. d', 1111'
•
•
:W
'IPPII !,IJlllld uf III!' "PI'I''/ 11111).',(' 1" 1111 11,,,,,('1/ d1111 "p)J1()dl 1](,,, Il](' ](',t! /l111t('! ")JI'('II.
1 Iii' 111 1'dl l 1('d 1111111'1 "I)I'('eI Ilili 1)(' Ilu"('1 10 111(' It',t! UIII' Il 1" ,t/"u ('\P('( kel tlldt.
d', 1 Iii' d\'I'1 Il,'',1' of 1111' "1)('(''/ IIIII,!!,I' 1" ill( ]('d""'!. l" , 1111' IU\\('1 I)(JIIlld 1:-- illl I('d,,('d, 1 Ill'
IJI ,'dl' 1"eI ",1111'1 'IJI'I'" 1\ rli Ill' lllU](' dl 1 III ,d t',
:'\it)\\'. 1 U illl't",tl,l'.dl l' 1 Ij(' ,t1,I)\(' li lt'Il 1 iUIj('d ph('IIOIlH'lId. t Ill' fulluwing ('d~(,-"t Ilel.\ I~
11111' Idl'wel, ,\ :\ ,\( '/\ l, 1.\(JOli dll foil ,t! d )'1,,1 II IIIIlIlIH'1 uf 0 ~,j le., ('()II:,id('ll'd. <llId
II' ill,l'. 1 lit' 11I1',lt'dd\' dl'11J11, ""lIli( eI"!.1 ulddlll('d 1'10111 O~FIL\ Pl()).',ldlll [~,jl. the Ji-l,'
1Ii"11111t! i', Il',I'd III fllitlll11' ('ig('II\,t!IJ(':-' olllj(' ".\:-,1 1'111 ,11 'o('\('ldl !'ol)('('d :--I('p~ 111> to tlll'
( Il tlil d 1111 ( '11', il) Il dl) 111111 l '1 "I)(·f·t!. 1\ 1111 Il 1:-' fu 11 Il d t (J 1)(' \ f = ï, U '2~ ï, rh (' 11 ·fu Il', !-.( '\'( '1 al
1 dl1p,I':; ur "I)t'I'" dl (' (1I\J'o('II, "lId I!JI' 11111 k1 !-.I)('('d i" )Jl'edickd. IJcI:-,ed on ('dl h ::;j)l't't!
1.111,1',(' '/ /1(' 11I('eliC1<'d 1111111-1' :-,pI'('d i:-. ('\',J!1Idll'eI hy 11!-.(' uf ,1 (j1la(lIat ic extl'apolation
IJI IIlItll'l llllll).',ill, \'('1:-'11:-' 1 hl' :-.p('('d :-.qll"II'<I, (J\'('I' tlJ(' (hu:-,,'n :-.p('('d l'an).',l'' FI1Iltl'1 b
plt'dll1l't! lu IH ( 111', \\'11t'1I t Ill' 11111 tt-I' 1I1d1'gin \'éllli'olll'!-.,
'l'lit' (t':-'Ilit:-. (Jf 1111:-' dll,dy:-.is cll(, "Iw\\'ll ill Vi,e" ,1.:3, The :-.all1l' 1't':-'lIlts are al..,o
jll('''('III('eI III T,Ii"(' 1.1, J\.., (.Ill 1)(' :-'('('11 flOIll t.hl' f)g]1\'(" \\hcll the l'xtrapolatioll Îs
eI'lIl1' ill 1 lit' dtlllid ill \JI 1 ('].11 1\ l'I,\' 101\ !-.p('(,fb. the pII'dicll'd /l1ltt('1 :-.p('('d is :-.igIlificciIItly
dilr('lf'1I1 1'111111 11j(' I(',d /lIIU(') '0 (i('cd , 'l'III' pIt'di( kd flll!.t('1' !-.pe('d do(':-. Ilot fo Il 0\\' Cl
',\'o11'lIldlil 1 !l'lit!, d" IOIl,!.', oI~ 1111' hl\\'I'1 IHIIlIIII i~ Luulu,,", éllId tl\1' Ilpper IlùlIllcl i.., Ilot
IligiI ('llllilgh 111/11,10\\('1 Il(JlIlld i:-. ill( \1''\'0(,<1. ,!!,l'Il('I,t/ly :-,pl'dking. tlJ(' p!t'di( L('d /1l1ttl.'!
''l'''I'tI i', 1I11llllJ\('d II tlll' dllllldill 01 1 !JI' ('\lldl)(ll,tliOll i:-. clo,>e tu tlw l'l'al f111ttl'J spl,t'd.
',d\' 1111' IU\\('1 1l(llIlId Ill'illg ,-IUCX (If tlJ(' 1(',11 /llllt,(,) :-'J>('('d. th('n thl' plcdictl'd flllttl'I'
"111'1'.] \l'dl Ill' gllud ('lluIIgh, Qll,nd iLd i\'('ly "pedkillg. J!' a IO\\'l'! hOlllId of GO%, élIld
.Ill 1ll'pl'I J,Ullllll uf SO(A uf t IH' Il'(tl 1I111 kl :--1H'cd is (1IU:-'I'Il fOl tl1l' "1)('('<1 l'allgt', tlWIl
III<' ('\11 d!,ul,tI iUIl ('!'olllllcltc:-. 1 III' Il lit tel' :-,p('l'd \l'il h ail ('1 lUI of ollly :3%, This dol's
•
•
('/I\JI/TI< l, J)I/V".'NF.\/'.\P/I/iO\('//FS /'U /J//'/l/UUHf.'\/ ,II
Ilot ~('eI11 1 () 1)(, clll d( (ept.dd(' dlld .... d(· l ,1I1,~(' fUll hl' (·\.t 1.,\" d.tllt," \ :'.111111 "IIII,1l tllll
\\olllll IH' \\'IH'II .1 du ... (· (·~tillJ,lIi(l1l 01 11\(,1('.11 Illlitcl "'pn'" 1 (111 Ici III' ,,1"dllll'c1 Il''111'',.\
10\\l'l 1 d Il!.!,(· ut "'p(·vd..,
A~ crin he ~('('II fllilll 1 If(' JlI(· ... I,IIll'd Il'',1111:-0. Il'''III,L'. d fjlJ.lIlldlll 1'\lldpllJ.IIIt'11 Id Ihl'
nuit('l J1)<Irgill !l(\..,(·d Oll tlJ(' gi\I'1I 1"1111. (·qll.1l jUil ( 1.1). 1,,11 gl\(' .\ ,",(III.! (' ,1 1I110l1( d
flul tCI :>(H't'rI, only If t Ill' IU\\('I ('()111111 ui 111t· "'(11'('" d""ldll l', .iI 1('.1,.1 "rI III 111('
Il'al /llltin "'p('('d, \\,lIi(1! i ... t(IU Iligh, III IIIl) ... t III !11l' (01'.1'", tlllo.. IIII'llllId (111'11",111110111'
tlll' fllltkl "'p('('d, ('\('11 \\'I!l'II tilt' "P('('t! d(llIlolllI 1" (11I"(' tll !11l' 11'0111111111'1 "111'1'.1
4.3 Allalyt ical A pproachcs
('''I)('lillH'lll t tl \("·,,t1ts hrl\(' lu lH' '-,\)l'IHJ\I('d h,\ "'IIIIJ(' dlldl)'11I 011 dl'l'llIdl 1)1'"
'1'0 clll,t1.\Z(' 1 Ill' fllllt('1 iw,t,t1)dit\ (Jf dll .1111,,11. "1 \('I.Ji dll.d,tll.1I 1111'IIt"cI,. Il.1\'(
1)('(')] ('ll1p)u}'(·d 0\('1 tlll' y('cll~, '\IIIUII,!!, till'III. tlwH' .11(' tl\'() 1\,/11"1 Ild\'(' 111'1'11 Il' l'cl
('x{,('n..,i\t'!y \Vith 1!,1('at ~1I( (\'~..,. TIIl'~(' fllllltl/II/lII/IIll't 1II)Ih 011(' Il'''lI,dl) Id('111 cl tll dO,
tilt' "p-/~ /lit '!tod" alld tllC' "1 i_g 1111 '''o,/''
TIi('''(' llwlhot!s itll' il('I,di\(' \\'IH'II 1111,,11',111\ dl'IIJlI) IIdllll(" 1'" Il',1''' III 1 1 If' "'I,d\" l',
III t his ca!>(', il!> di..,( Ih!>(,d in c1ldl'l<'1 :L llJ(' .11'1 ud\ Ildlllll " 1'> l ',' ('111 j,dl\' .t (111111 jtlll Id
t Il (' 1 ('d Il ( (.c1 fi ( 'q Il ('11 ( v, 1;, "" d ,II i!>, i Il t III Il j" dl 'I)f' 1/ tI 1'111 "II d l' d 1 l ,,( 1 1 JI' ' 1 III 1 11/ 1/
i,l'., it i" il f1l1l1'1 iUIl uf w, Ill(' fI ('IJ!J('I1( Y !Jlll11' \)',,1('111 '1111 Il'11111' Ir) IlIld 1 If/' '"llrI IfllI
i.(>., tlw ('jg('ll\'alll('s of ,h(, "y~II'llJ, ,11 ('dl h "1'1 (,tI "tl'P, (;1',1 d (II 'l'l' 1/( \' 1" ,1111 (1/
él!> th(' [1<'(!lI('IH'Y of tht' !>Y:-'('Ill. dllli 1/11'11. "fi/'I 11/1' ",,/1111011 l', (flllild. ;1 j" '1/IIIl'dll'"
• ('/I\I"Uf( 1 /J//'I//U'\/ ,\/'/I/U)\('IIL,'1' 1()/IlI.P!(()HU\/
\" 1 1 I, 1 III' 01' 1 Il. d , d Il lit) Il il 1 1 II' \' .1 II' II' Il 1 1 W "d Il Il' .1 Il 1 JI I!I' 1 il 1'1 dl i () 1 1 1" ri Il Il /',
i'l 111'/1" l' "'IJI .\1'011', d 111'\\ IIJl'lllIld. Il''111!!, d Ildll"f'llllldtiIJIJ of Ill(' 1''111,tlll)]]''
,,1111,,11"" III 1 1 If' /II/I/(/(( rI()/IIIIII/, dl'IIl!.!, \'.Itll rlll d\lPIII'\,11I1,t1IIJIl (Jf Ill/' ;tI'l(Jd~llill)]i(
,j,tlol \','1111 "Ilil' /li//f/Ii"/ Jl/lll/IIIII" Il.1'0 \"'('11 .j,·\,.jl)jJl·11 to '-Ilh(· III(' plul,l('ll1 in il
l, ,III l,tll\" \'.'.1\ dlld .d"Il IIJ IJldkl Il l'd''!l'I I() 1I11lutl1l1 l' '-(Jllll' 1 IJIJlIOII,\\\.., IlItu tll<'
/1.3.1 JI-l,' JH('thod
illt II ,.]II( (,tI litt Il Il](' ('lpldIIUIl" IIf 1111,1 iUIl (''111,t! iUII.., (~,I 0) ;tlld (:! Il). il" r()Il()\\'~:
c _ C 1'/ " - (" 0 (
S: ( 1 :2)
,\ 111111(' ,!.',l'Il(·I,t! f(J11ll uf t Ill' (''11Idl illll" (Jj IllO! iUII i~ as f()I1()\\'~:
(h 11/ b '
( 1.:3)
( 1..1)
\\\1('1". Ch 1" 1 Ill' ,'~"II('I,"I/t',1 dl'Illlhlldllli( fil 1 1 (' cI/lill.!.!, ()n Ilw dilfoil. ancl q. 1:-. the
(.lI. "lit! (>. !l1d~ 1 \l1l1.lill "'()II11' tl'11I1" h<l\ÎII).!. tl\(· )!,1'1\t'I,tlized cUUldilldt(·.., ilnd tlwir
.]('IÎ\dtl\l· ... 111 tlll'ill. dt'(lt'lIdlll).!. 1111 III(' t~pl' Ill' d('IUdylléllllÎC \\'hich i.., Ilt'ing u!'-('o .
•
• ('II \PTfU 1 nll·,/- 1·.'UF.\ r \l'PHU \( 'IIL'S ru 1111,.' flunUI 1-'\1
t 11\'111 d ....
[" f (,] { :: } j_ [ll 1\ 1 { :: } II 12 -/2 Il
tl.
\\ 1/('1 (' [/ 2] i ... t Ill' IIlIit Illdt 1 i\ lIf "1 dt,t .) .I1lt! [\/]. [( '].II1t! [/\] dIt' tIlt' 111.1" .... ",.111'11111",.
1
{~ Il } (1 h)
1'~qll,t1 i(J1l ( 1 ."'i) mil.' t 1 !t'Il 1)(' \\ lIt tt'Il d ... •
[1)] iJ -1 [F] iJ - Il • (1 /)
1
~ } ( 1 S)
III tlli~ ('qll,dIUI). Ill(tlti('(,~ [IJ] dlJd [/j'] df(' 1I1 (Jld('1 1. dlId, ill .'-',(·I1('I.d • .III' flllllllflll'>
tl)(' j'olm ur CUl1lpl('x (JlIjll,!!,at(' IJlIIllIH'I'>.
,\ 1,'\ - = (':1- 1 ri .
~U\\'. :-'llpjJtJ~IIJ.!!, lJj(' d('..,i~11 jJdldIJif'tt'l'i f)f Ill!' dIljf)iJ dIt' ""IJi':JI, llJ . 1/1\'1' 1111',
fI ('(!ill'IJ( y. 1; = 1..<.' bl" BIIf IIJ(' h('IjIIC'IJ( y. ",,'. i ... ill 111,dly ;1 P;II t f/f 1 If/' IJIIIfII/1i l.)1) •
•
•
111'!jIWIII i,'c, (1111' ",11(J1(' pl (J( ,'"c, IC, "IIIJ\\ Il il! Fig. 1.1).
'III/' lil..,1 111'(1')('111 y. h. i~ (1)1111'.111'<1 \l'illl f 1)(' f)('qll1'IlCY w = l'i' oIJf.tinl'd fwm
f lit' ;lc.~llllll'd !t·dll( ('" fi 1'(1' 11'11 ( y. 1 ... (\/I\\'. J!' f Ij('~l' t.\\'u fl<'qll('lI(i('s, W dllll b. ale ('qnal,
1111'11" ", (JI If t'/" illl\\\'/'1 Il.1\ 1)('/'11 /0111111 fUI t III' fil:-.I ('ig,('Il\',illll' dt 1 hat ~J!('cifk c,j)('('t!.
Il 11111, .1 II('\\' .fI,llI'C,C, lUI fil!' 1 ('''II( cd 1'1 ('!jIl('11I Y I~. 1 ]1('11, 1 dk('1! é1!'>:
6b 'l' = {!
""l (' i!'> "bu !l'Pl',tll'" fUI 1 Ill' M', Ollt! l'igl'lI\'.lltW \\'it h c11l ill1étgillill'y pal t of <Ï. III this
~P(·' die :-'pl·l'll. TitI' I(',d 1><Ilt of t111' twu luots dJ(' !'>UIlH' indiccltion of th" !'>y!>t(Jll1'S
".!llIpillg, .!Il" t.lj(' illldgillill,V pcll'tS gin' thl' fll'<jIll'llCil'!'> of the sy:-'Ll'll1. TIl<' dalllping
1,11 iD uj' t Ill' ~y..,t /'111 ('tlll Il(' "dl 1I1dkd j'IOlll t Il<' lolJuwillg [25]:
( 1.9)
1111 cl IOllt of t Ill' fllllll of ,\ =- ,i f- /w .
\t Ihic, poillt Iltl' ('('.t! P"l't of tlll' twu l'oots !'>JlOlIld 1)(' c!Jclkcd. If l'itlH'1 of the
1 \l'II i~ l'mit 1\ 1" t 111'11, it 1llt'.!lIS t h,L! t hl' ~y~It'lll ha~ d lH'gat i\'(' dampillg, indicating
illc,t.d,dity. If lIul. allotlll'I "tl'J> ill :-,pl'l'd is Llkl'lI ,Ind thl' ~am(' pIOC(,~S If'l)('atl'd for
tlll' Ill'\\' :-1'('('''. Tllis i:- cUlltill1l1'd Ilntil l'itlH'I dll III1:-.tahl(' !Out i~ ohtail1l'd ur the
11ld\l11l111ll dl'I idl'd :-pl'l'd i.., Il'clc!Jed .
• Cll.\P J'EH·1. /)IFFFIU';J\ J' \J'I'IW.\('Il/-',"" ru J'IIV 1'/{UHlJ'\l
Incompressible Flow
,!j(' IH'I ,II i z('d r() It ('!'> i Il t Il l' l'q 11 < Il i () l' .... () r IIIU t il) Il. ('<111 < Il Il' Il'' ( 1 .l) .lIte 1 (1 1), t lit' 1 Ilill 1 \ \ i ".1',
.. l il /) b .. é, t- l'" (l + ..... 'It f. .:: - --- (li f. 1 l' Il - li 1/ Il )
1/1
'27ipIT, ( 1 --- - - ( (1.) li i 1 l"~ Il 1 li ( -
III .) Il) Il) . ( 1 1 ())
." . l .. . 2 l _ li fi (l c' ,1 . . 1 LI f, + 1 (t n + 1" LV" n -- --- il 1/ \., _. ( li ( . - - Il) Il III 2
~lIplJ 1 (. 1 ---- (li + :- ) ( '(1.) li f, 1 (! CI 1 h (,
III 2 2 (1 Il)
ilia . ..,,..;", Il = III/lipl/, aIld raI tlllizllI,!!" ,l',i\·('~.
Ô - + '2 ('(1.) (- -- (1) --- -1- (1 ( --(!.) -) (li l ") '2(' (12
li IL '2 hll !J2 l' 0,
(lnd,
. a· ( 1 (/ ) f, (.1'(1 - --) + f, -1 (II + :-)) ('(1.) ,--- -1- il (ru 2 Il -) Il
u)
, 1 l ,(1) (2 2 2 (Il + :)) (:; - (/) ( (1.) -, -- -, Il 1'" 0..'"
- - III
1 If 2) '2(11' )('(1) ') 1/
- 1 l' o (1 J'l)
(1.11)
•
• ( '1"\1 ''j'f.;/( ,1 IJ Il, FI':/{ l';,\'T /\ /1 l' /( () ,H '/ /1'.,1,' 'j'() 'l'liE 1) If () H L J-;.\1
[.\1 ] -[
J + l l'
' . • 1'(, - ~
l' - ~~ ] 't JI
1 2 j_ li.'! ~ , cr l'
[( '] [ 2 (II j- 1) ('(1) l'-2 h"
/ :~- (~( '(q (1 -- (/) -1- 1) i: ,; (1 --fI) (J - .! ((/ + t) ('( 1.) )
] ,
1':qll,tI iUII (1 1 1) (0111 (',j..,ily 1)(' Il'\\ 1 ill ('Il III t 1)(' fUI III I)f ('qllel! iUIl (-1.5), (1I1d 1)(' <'fJl\l'd
11"'111,!!, LIli' ,d,u\ (' 1IH'1I1 iUJJ('r/ Il)('1 hocl,
'1"'all:-'OIlÎc Flow
lifl 1 oc/li( il'Jlt dll('llI l'il (II, JIlUlllI'll1. (lIdlJ< !l'lit dll!' to plllll.!j<', cllld IllUIJl<'llt ('ucfO(ient
dll(' to l'il (h, 1 nl'('l 1 i \'l'Iy [~!il, TIIl'~(' (ol'!!J( il'nb ('<lll lJt' calrlllilkd il t diff('l ('Ill ~Iach
1I1111d)('J ~ II~"'~ 11111'\1" i( al 1lI('\ Ilods.
Slll,~litllti()1l of tll<' dpplll)JJielt(, lUI III of d('10dYllélll1ic!> into th!' ('(!1wtiolls of 1110-
1!l)Il, 1'«lIcdillll:- (~,IO) ,,"d (~,II), gi\('s tl)(' follo\\'illg:
.. 2 c (/ (' t. -j ,l'" n + ,-,-'1. 1., = - ----,- (( 'II, f)2 + n .. rt) , 111 }
(LIt))
, ê' _ j- , 2 .. _j_ ' 2 ,2 __ (/ ('2 , ,- , ,/"... 1" Il ", l"'" Cl - - -'-2 (( '''1. f,/ 2 + ( m •• ()) ,
11/ ) (1.1G)
\\'Ill'I\', CI
•
•
•
Cll.\f1/'FU 1. nJHTUF.\ r \/'}'UO \(ïIF.'" 10 /'Ill,: /'/lUB! 1-'.\1
1/ (' 1/ (' , ( ' ) L' t t ( ,) - {- l" <., - /' •• Il - 1 (1
_ 1/1 , 1/1 b n,
)
c '/ (,a. 1 f.... 2 )) ,1 <., - -- -- - (" <, -1- 1 (\ 1 (l' .• - ,,-' • - -
.• ~ ln b2 'l,
1/ ( 1
1 ) ('", Il
1/1 I~
Th(,~t, (\\'0 l'<jlldtIOIl" ('.Ill Ill' It'\\'litl('1I III 1 hl' rtllhmillg IIldtli\ 1"11111
1·1/ J { ~ } 1 1 Il J { <~ } 1/.
l,II J - [ <" /,.,].
,..,1.
alld,
2 '1 ( , HI 1.,
1'1 ( , III 11/"
(J,
1 1 1 ï)
(IlS)
(1 1«))
~ow, :'>ub~titlltillg q = I/'2p{!1. illid 11/ 7r Il fi 1/ illtll III!' ( .. 'jJlC':-, ... IWI 1(11 1/\], .. "d
~im]>lifyillg, gi\'t,~:
) fi} (' w,-+ - ---- 1 , 1. , l' bl ,.
1/ J ('
r I,/,l 1 ..
Equatio)) (II!)) CtllI ,d'io 1)(' 1('\\'litftoll III tl/(' 1(/1111 (JI' ('ljll,l1,lfJlI (1,./) dlld 1)(' ',oh'I',]
u~ing t lu' lIH'thod dt'~( 1 Il wc! IwloJ(',
lllOIlH'IIt codfitit'Ilh alH/llt III<' ('],1,,1/1 ;l':i,,, till'/('fol('," (1)(',,(, IU"'fl' If'IJI', dit' ~,I\'I'IJ,
: (''''/ 1 ((J ,.) t Il) ni , " /4 1
•
•
('II.\I''} Uf 1 /)1""',,-/(/-,,\'1 ,\/'l'I((),\('III',',)' '/ () 1 IIi" l'IUJHU'.'.\!
(' 111'1
- ('" 1 ((),:-j 1 fi) ('l" ' 1", / 1
( 1 :! J )
1"1111', flf ". 1 1 If' ""'JlJI ,llfll d
1.3.2
f/lrIl'l 10 111.11111.1111 d "ill'iJ/(' hdllllUli1I IIIOli{)1I (SII.\I). If tlli:-. It'lIllil!'rl r1ellll)JJtI.!!, If,
Il l '1'"tI 1\'1', JI 1111'<111" 1/1011 III<' \\'-11'111 il:-'I,1f hit:-. i' Jlu-..ÎtÎ\'(· dalllpillg ellHI, t)H'It'fulf', 1:>
:,I,dJ/t', 011 1 lit' utllt'I """d, ri tilt' It'<jllllt'd d,lI11pÎII,!?, J:-' po-..itivC', Ih(,11 it 11H'c1Il<" that
\)f 1 Il'' ... ,)''''''111 i:-., 1 IIIJ~, \\'11t'1I Z,'Jo dl t.ifil iet! dcllllpi\Jg i:-. \!'ql\ilt'd to IllilÎlItain ,illlple
hdIIlIIlIIÎ'"10tioll,
Ilit 1 odlll ill,!!, tilt' arl/fi('/ol dll III Il/II Ij, ,IJ, Îlilu t hl' :-.y:-.tC'll1, the l'quation:; of mut ion
1"'/ (llllt',
t, ( ) 1. c _ Ch <., -1 ,/ ,. 0 -1 1 1- I!J Wh <., -- -b'
III
/',,2(\ 1-.1',.1. + (li /.I/)/',,2u..'1 2n = !lb02
' 7/1
( 1. 22)
( 1.2:3)
1" thi:-. 1 d"t\ "illtt' SII.\I 1:-' d""llllll't! tilt' rullo\\'ill,l!, fOI III i~ C!JO:-'l'Il fOI the ~ollltiol1:
è _ C I~' t <., - - <.,0 ( ( 1 21)
\\ 111'1(' ~" oIlIt! 0 0 dit' ((II11plt'\ 1111111lwl:-', tilt' ,d':-.olllt!' \alll(,~ of wltich (('PIC:-'t'Ilt the
tI:-.t ill.iI iOIl lIr tIlt' :-.y:-.tt'llI,
•
•
('I1.\/)TEU 1. n/I-'/·FUF.\J' \/'/'!\()\('/IFS ru IIlF/'/iOHlI."1
",hi( Il Il.1\ (' 1 Ill' g('Ilt'1 ,dl/('t! (OUI dllidl ('" dlld 1111'11 dl'II\ ,111\ 1':- 1'1111", II\(' 11111',1 ,l',I'lll'I,tI
l'ulm rDI Ih(' 1 (':-lll 1 11Ig ('qtlcl 1 il/II" 1"
[ Il \
\\'IJ('I(' [/1] and [IJ] dll' ~ x ~ IlldIIÎI('" \\'IIÎlh Illlltdlll lUIIIJ"I'\ Illllltlll'l', • .Illd .Ill'
Itlllel iUllS of Ih(' c!('Cli,e,1I pdl cl 11\('11'1" Id' 1 Ill' dllillii .11111 11\1' .11'11"" IldlllÎl 11111 l'" \1 ',{I.
/
\' - - { ~ 0 /1 Ll f
élnd,
,J"l b2
\ (I!) ( l '.'f j) / '1 ,- l' 'II - (/2 •
aH' ohtailH'd \\'!t i (II aH' ( Ul!1 pl('" II Il Illl)('1 CI,
'\"1 Il 1·' b
\\'11<'1(' ([, b. (', ,llId ri dl<' 1(',t1III11JlI)(,I~. On Ih(' ()IIWI 11t/lltI, It l') 1.1111\\'1' 1 Il,JI,
W" L 1/ 1 l' (/ 1/ L
•
•
(,/1/\/1'/ Uf 1 /JII'II Uf'.,\ï' \IJIJUo.\('III·;'s' '10 'IllE IJ!UJHU':.\1 10
'11""f'/fJ/f I}JI' 1/,,,1 dlld 1111.1,",111.11\' jJdll.., /)f 1 1 JI' f'\jJ'(''''''IJIl'' /UI ,\., /.111 I){' 1'!jlldll'd
Jllv...",-2.ï2.- /1' {: - - .. - w .. Ii -
Il (l
Il ::: { il b b
,-,-,'" L 1/ l' Il
u: . ( i l,'
h
~ - -11/') 1 _. • _.
SIIIII'IIIf' 11'," 1'0111 III' ,\,' fdll ullly II(' cl l'mili\(' 1111111])('1. dlly of Il)(' ('\',tllI,tI/'f! ('ip,ell-
.\1101111'1 ... 1'1 u/ I]J(' ;!lHJ\l' PdldllJ('I('I!'l Illa.\ 1)(' l'\'ctlllcltl'd f,u1I1 the utht'I IOOt.
'l'III'II'/lll!'' /UI l'\'I'I\, Il'dllll'cI f1C<j1l1'1I1 \', ()Ile \\'rli l'lit! 11P \\'ilh t\\'o !'ll)('(,tb dllt! llJ('
/ 1III1'''IHHldill.!-'. \"tllll'!'l fOI 1111' dl ti/il l,II d.lIllpillg, TI\(' It!(',l i~ tu fint! il !'lJ)('('d at \\'hich
iï(lb .. _.- - (b ~ + (T (i -- b (( CI)-
/TI
'2iïfl{' (, 1) -II/'-('U,) b~ j-{'n +b(:.2- a )6 (-1.28)
•
•
nlFl J. }U' \ r \J'}'HU \( 'JI} ..... .,. j'() 1 IJJ' l'J\(JU/ 1 "1
,Illd.
;, l' ( , h· (/ ~
III
1 hl Il) II .,
2 l , .. ) ~ il /' 1 1, ( . 1 b (-7 -j Il·) 1\ + -- - ( lit . )( (l l " ~ 1 l' f\ t "(
~ III ~ ,
dlld tl](' folhmÎlI,!.!, ~lIJ,,,tltlltl()lh.
;11]<1,
~l,..' (.
( - - 1 i .! l,' •
lI..'d.! 1/ \, - Il (1 1 1 (/ ) 1 1
( ~ ./'" Il 1",,2 - 1 ~ l,', --- ~~ , •. / - '2 ('(1)
l '2 1
1 1. !. 1 ) 1 f" 1 ". ('(I) f.,. t
1 1 ( '2 (1) 1 .. , ('(1») Il,, Il,
( 1 2 1 2 1,) (1. l •
~ 1'"III,'c - ,1/ kr t- 1 (Il 1· ~) ( (1.) k, ç" 1 1""· Il 1 .. , ,\" l'.,.
( 1 . \())
Il 11 22 1, Il ) - (- - Il) I,'c + - (-- + (f ) l,' -1-:2 (II 1- ')) ( (1.) 1 1 (Il 1 ( (/) ( '( 1 ) 1,., fi" (J
'2'2 lx ' _ :~'2
( 1 :\ 1 )
[ Il \'
/'
\\'1/('1." .\" -= { ~o no } • <11111 t II{' t \\'() III,tI J i. (", [Il] dllt! [fi] 1 dll IJI' • "',Jlv d. t 1'11111111 ri
Il
• J~
TraJJ~orJi(' Flow
\\ dl IH' fil 1111' 1111 III
(/1,,11 / (]/III)'J,,2(:..: ___ I/~/' ((f/"u~+n .. n), III )
(,1. :~:n
Il ('2
- -12 (( '",,, [,/'2 -/- (,"11" ()) III 1
(,] :n)
\\ /1/'11' Il / 10) 1: 2 _/1 ,
2 1/ l' ,) ( l q (' ,) / (1 / 1 II) Wh -I~) ,- (1" (0 + -,l'" W + 1. C/" (la = 0 ,
_III) II/u
1(,2 ) ( J 2 ) l, ,1 2 ,2 2 (( -, ,) (2-( 1/.,. ~o 1- -1" W +(1 -/-lg)l" W" - --12 ( 111" _1111 111)
no = 0,
( 1 2 u..',/ 2 ( 'l, ) ( 1 l ('1" )
l' /.', - \, (- -) - :)- 1 Ça -/- -- ,1'(\ l' kc - -- --- no = 0, ] lJ" _ if' 1 if'
(1.:3G)
(Il ' 1 (',,/,.) C (l, 2 l, 2 \, 2 ,) ('"", ) - 0 l ,r" Il ". - lf-- .... 0 -1- -1- 1" l' he -. <i 1 l' -1- - -;-- no - , (<\.:37)
l' 11I1,ï l,h 2•
l,', ,', :2 /, ,
\.1 r li / 1 1/) ""/' "}'~ •
• (/ - Il'2/d
•
•
('11.\/1 l'FU 1 n[J-'FFHF.\ r \J>I'HO \( 'Il!'S J'U 1 Il/-: l'HUHI ""1
(' = :! b.
\\'J 1<'1< , III thi ... Cd"'(' X = { C '-.0
[ [. \] 1 1
r } • dlld.
1 l' /.. 2 ('I/! 1 • ,I/" 2
l ' 2 :r -1.1 II t
r ,I/" 2 1-( 'mil. 1 • 1 •
1 ' , Î 1 ... Il ".
4.3.3 Laplace rn'ansfol'luatioll MeUlOd
( .
] 1/,
If
) ('''1-1
nu\\,.., t]lI' d('I()<!)lIdllli( dI'IÎ\,tfÎ\'('''' ('1", ('1". ('",,,, dll<l ('",,, , dl(' dVtld,.ldl'
flllldÎ()1l t () ri ppi o;..i,ll;!1 (' t Jl(' d('1 ()r1Ylldllli,,,
• ('II:\f"ff',/( 1 !Jill 1'.1(/-,,\ T :\l'f'lf() \('IIE'-; T() 'fl/l'; /Jf(OHU;,\I Il
'J Ill' fil/III J'/II v,LI( Il 1'-, '''-l',j 1{J dJlPI()\Jlll,t1{' Il)(' dl'IIJ(I.> Ildllli( <., i:-. Il''Il,tlly III Ilw
III 1111' 1111111 fi! [,H]
(Jf " III,!\' 1)(' f Il.!II!!',J, dlld ,tI:-,o. dqll'ildillg Oll IlU\\' \\'{'II 1 II!' dPPIO:-.illl,d illp, fllll{ 1 iUII
'l'Ill' lil:-'! Il''1'{' Il'1111'' in Ibis <'<!Il<dIUII 1<'IJl(':-'{'llt tll<' ljllélsi-:--l{'c\dy éWJ'oclynilmics,
\\ 11l'1I'd<", 1!t1' (Jlll!'r 1('11I1S :--11U\\' 11lt' llll'>kady dfects, (Ille! é!1'(' l'cf{'n ('d to dS "log
I, J'1I1,~" III rdl 1, 1 Ill' Id,!!, Il'1111:-- ~il1llll.t k 1 lit' 1 ill1<' Idg III tl](' (ldjll:--tllWllt of no\\' <llOllne!
(1IIt Ir 111111' '1 • Ii IlIt~"~) i..1~ h""11 "llIpl,,). d III t Ill" ~t Il "y
•
•
•
('l1.\l'J'FU 1. [)/FFfUf.\"r \l','UO.\('[{F.'''}'U l'III' l'IU)}?/ /"1 l,')
.\<.'1"0('1<15t ie Equat ions in t Ill' Laplace DOIllain
Il,
\\lwl<' in llli:-. 1"I""lioll,., = Iu.', "11.1 ["]. [( ']. ,,"d [/\] .Ill' 1/11' 111""'" "'''"11>1111',. dlld
"lifrJI!'~:-'1I1dlJi«'" (illlllldill,l!, .III II\(, ,,1111(1111,i1 dlld ~,I>lIII'\l1 111<' 011'1".)'"""111 :1'1111,),
iJlg flllll 1 iUIl 'l'II(' \,d11<' (Ir t II!' (1'11,,1 dlll ) d''1'''I1.):-- (/11 1 lit' 1/(1\\ Il'1',1111(' TIlt' t.-Ilil
h li\(' Lapld''!' OJlt" at()] (Jf -:;. \\'1)('1 ("
J
:::.-={~ Il}
1 h.v II II = 1 l,' =-= ,.,
[! 1 f
TIl<'I'('fulC', 1 Il<' Ldpla( (' rU11I1 or (''111,1f iOIl (1 Ill) 1":
(1 It 1 )
., __ h h 2 ,
[ F( ~ 1] = [/1 (J] j - {I [ .. 1 tl" + ( -( ,;) [ .' 12] .,. j [/1 sl l, 1 1 .., 1 l, Il
, 1 [:1.,) /1 1 (,1 Il) '> 1 l, Il
Solution of the A('rlJela~t.Îc Eqll:.t,ÎOI1S jlJ 1.111' L"pl;wl' J)IlJII:lili
thi~ ('qll"liu)) i:-. (JI' 1 Il<' fOI Ill.
2 Il Il l 2 ([.\1] ,., + [(:] ,., -1 [/(] 1- Î' {[Aul ! ( ['ld" ! (II) [/1~I" 1
• ('/I,\I'UH 1 /)11 l'I'IU',,\'I \I)I)f({).\(ï/V,~' 'JO '/IlL' }>I«)H1./·;.\1
1',IIIIIIII.lllull Ilf 1 1 JI' dl'/lflllllll,!1fJI'-, III 111(' dlJl)\(' ('qll,I1I<JlI (1IdIIPt" 11 Il!1f) 1/\1' follo\\III.!!,:
\\'111'1('.
1· (' (: 1; HII ( ." 1 /, d ( " + - hl )( . ., + - "d (." + - h d .
h b h li l' l ' 1 .
HI - ., (" f-"hl)('1 - {, \ ) ( ., -1 b /Id· b l ' l· l '
/1 2 " ( ... /, hl) ( ... 1 --- h j ) ( ., -1 -/J- /, d . /,
1 . / ' l ' /I! - ., ( ... h) ( ... -1 -/; h 2) (., + ï,- b tl .
/1 1 . lI'
1/ 1 -- ., ( ... 1 hl) ( ... -1 -- b2 )(· ... -j---h j ).
h b b
011 Il l, h 2
[ \] - c [ H J -+ /. ( ( ,) [:1 2].
li [HJ -= [C'J + /·(,-)[Ad.
[( '] --: [/\'J +./. [AuJ.
[ n J : = .r [ 1 d .
[F] =-= '/' [. 1:.] .
[(,'J =--.i'[ 1.,].
• (.1.1:3)
• ('/l\PfFH 1. J)/FF/'.lU'\f,\/'/'/iO\('lI"'S/U 1III'I'l\OUI/',\/ lï
Eqll,tt iUIl (1.1:q (,III Il(' 1('\\ lit 11'11 .1"
... { /'} [\]{i} . (1 Il)
"}!l'H'.
{ l'} ... [[ I.d , .:; ,
_[l';,rl [/-'-,] [ J';,] 1 [Fd [ 1';,] 1 ["'tl [ 1';,1 1 [l';d
[ 12 ] [{) ] [{) 1 [01
[. \] = [() ] [ 12 ] [ Il] [Il ]
[() ] [() ] [Il] [01
[() ] [() ] [ (1] [n 1
[II ] [() 1 [ 12 ] [() ]
Eqllatioll (J. H) I~ ri lypl! ,t! ('I,!!,('II\rtlll(' pl (JI,I('III, rllHI "1)h'1 1 I.!..!, Il,(',('1111.111-.., I:~ l''''!''
thc ut!tt'I ('ip,ht Colll<' fl()11I t}1!' 1 rit H,lld} <lppl()Xilllrlllll,!!, 1111111 /(1I1. FIl! III!' ',\ ',11'1111"
ur (}l(' III,dli\ "'}IO..,f' ('I,!!,('II\,t/llf'''' <llf' tq 1)(' fqlliid Ilil Il'rI',I''' (1)11 Icll'ldl,l)' III" Ilill!J,11
• taliu!l,t! df'l11dlld fOI ~()h ill).!, 1}1f' "'''111f' PII}\,!('111 1111 Il'''''(''' /11<111\ IIIIW', 1/,II!J'dl/ ri ... ,'11/1
•
•
lS
1 1 If' ',1111111011 111 1111' 1 ill!" dUlll.IIII Ihlll~ Il.11111 iUII.i1 IIWIII(),b BlIl, il "llUlllrlll(' 1)f11('cI
111,d. III( 111'>11)11 of "11.1(11\'(' (()IIIIIIII,,\\ ill 1111' 1''1l1dlllJII~ of IIIUIIlJlI i., Il,I](h 1'.t"i"1 111
1111', 1I1( tllocl. 111.111 111.11 ill d Ily 1 ilJ)!' c!IJllldill IIll't hod .
• CIl,\JJfI-'1? 1 lJJJFVUV.\ r \J'l'HO \('[IJ,' . ..,· ['() 1 III-' l'[((l/ll ['\/
. 14
.12 • Mede 2
10 -
08
05
.04
.œ
o
.. ~
\-----\"Iooe 1
"œ
'. C6 - _1. __ .l--L-J_--1_-l_----L...---1.. ___ l __ ..J
o 2.0 .,0 6.0 8,0 10 0
Dynamlc pressure. kPI
Fig 111 (' ,1 1: ,\ t)}l i ( (Il \.t 1 1 (If Il J " of t J J(' d.t" '1 Ji" I~ i" t J If' ( 1 i t Il • Il 1 1 J ( J" l " 1) f d Il .t Il 1 loir t,
•
• ( '/ /,\ 1 J 'f U (1 /J 1/ J- /' / (/ .\ 1 \ / J / J J( (J \( 'II /-..S' 'f () '/ III,; 1) 1 U ) 11 /. ,.: .\ 1 .JO
0.4 .. • c
en 03 - . L 0 • • E 02 - • L • <lJ
.. J • -4J
:::l 0.1 G: •
• 00
_ 0 1 ..i __ .L_ J -L --1-, _. • J J -L-J--1 • 1 J • • • 1 -I---..l--'----L.~ o 5 10 15 20 25 Non - dlm. Vel. Squared
•
• (,1I.\Jlf FU 1 IJ/l'Jof·/iJ-'.\ r \J'/'/W \( 'J/} . .'.'l rt> 1111-' l'}l, 'In 1-"1
14 IJ M 085 LOWl'sl Spt~cd· 3 00
~-[1 NACA G li\OOG ,Wlllii
1 J
> Ji 50. L OWI'~,I SIN'()d 01 25
""0 r. 05 Lowusl Spcl'd ~) SO Q) Q) a --1 0 \ l
0.. 12 - 02 ne JI nuiter Spced w, C!) x .. 025 "- Il Il Q) ~:j 1 ] ::l I-J H
u_ 10 1 J
E ""0
IJ
1 A f\ 1 J C
8 0 0 - t\ 1 J
Z ~~ !-\ Î) g ti H (1 00
IJ
6 1 1 1
3 4 5 6 7 0
Highest Non-dirn. SpcccJ, V
at.\/ =-= 0 :-;"). ("HlIIJrlli",lll !)('fl'.«'nl!w lt",lIlh (Jf 11',111)', Ihl,e' rldr, JI 1I1./(II/'( '.f 'JII('/
j Il t Il (' (' ... 1 1 ilj j() L t1 i u Il .
•
•
•
IIOW-U] - f-
I ~es-s k~-~biu-J --- -- ~------ ------_9_- _ , _ ___ -------~-------
~ ./ Q n1~tri)(J. . . eigenvalue analysis l
n eigenvaIU,eSp . B+iaJ
1 fO;eaCh. isi.b!~~.~?J - -no -1 ---- ---- -yes------ -- ----
let k=ab/u at u, p=B+ia - - --- - - do next eigenvalue
---- -- - ----,--- - --- -
lany· BPositiVe?]
~n.~(~tsm;J·lin~r~!en~~-
• ('1/. \ flïTU 1. /)1/'1''/-'/(/''.\ r \ l'l) nu \( ·,11 ... ..,· ru J III-' /'I\OH/Y.\(
:~ (lO ---- ----
:~ ,JO 1 :L7ï:~~
LOO Il,U:mi
l,iO 10 (i!)!) 1 ~ Il :l.)
,1. ï.") Il 1 :~2ï ~,
.1,00 10 I:W~ 1 n. ~7 I(l
T)':iO ;-; K%(, 7. !J:\;)()
(i.OU ~, 1 :):{f) 7 h,~ 1 ~ 1'1 1 1 {ï
6.,")0 ï,,~2:3!) 7 (jOUi 7 .:),-)~f)
7,00 ï, ï:n,j 7 Ij II~ ï ,(,(j:):!
-.)~ 1._:) ï ()K7/) 7 .():{ 1 1 ï ():n 1
ï :)0 7.()!>O:{ 7,(,'21'11 7HDO
7.W:n ï (, 1T)!'i 7H2,ii ï (j217 - ----- -- --------------- -------- -- ------ - --- - -
Tahle ,1.1: Fllltkl :--IH'{'d I01llld IHIlII 111<' /1ILft( J' 1111111/111 IIlf 11/fIt! ill 1 II(' 110111',/1111/ !II)\\'
,\('lodYlldllll! d,da (JI (l0:,\(':\ 1J1'\()()(,dlllod 1,<1\1' lJff'IIII,,('d 11111',1 1<11 III,!'. 1111'1 fI"l
flllll('1 ",pl'('cI
•
•
•
Cllélpter 5
Effcct of Dcsigll. Paralneters on
Fllltter
'l'Il ..,llllh 1 lit' l'II<'I 1 (Jf .1111'1'11'111 d""lgll Jlcll'illIH'tcr.., of aIl diIfoil (JlI !Ill' f1l1tt(,1 !'>p('('d.
11111'1' 1111'11111.1.., Ild\I' 1)('('11 l'IIII'I(J.\('d III 1 hi.., ,,111.1\, Il.111]('1)'. tlH' Iladiti()Il,t! l'-Il' illld (/-g
1111'1111 HI~, ,,1111 /111' 1,,, l'lcI' l'II dll"fulllidl iOll 111<'1 hut!. III t hi.., \\"l)', il C()lIlpal i..,Oll of the
/111 l'l' 1111'1 III)d ... 1 d Il ,d ... u 1 Il' lII<ld, '. ï Ill:, ..,111 dy Ild~ 1)( '( 'II dUlIc ill IlOt II 1 hl' in( Ollll)) ('f,f,i l,Je
,111.1 IldmUIII, !lO\\' Il'.t',IIIH'S 1" Ihl' Il,IIl~Ulli(' 1('gillH', thl' cl('wclYllcllllit ddlél fur Ct
:\.\(':\ (il.\IHlb dllfuIl.1I d \Lllh 11111111)('1 uf .\1-= O.~:i lIa~ 1)('('llll~('d. Tbi~ d(da bas
11("'11 t,d,l'Il flllill III(' ~t IId.\ 1)\ 1.('(' [~r)]
III ,t',I'III'I,t1. tl\l' l'qlldllull~ (JI' 111011011 of ,1 1\\u-dl',!!,l('e-uf-fl!'l'doJJl c\ilfud (',Ill he
1'\pII""'I'" III 11t,· 1111111\\ IlIg fUI 111
"I\l'I", ~ { ~ 1
n }
[,\1]: j- [(']:: -1- [/\J.: = O.
\I.Jtlill·" [\/], [Cl dllt! [/\] .tll' tlll' IJl.I~~ IlI,Llli", Ill(' d,lIl1ping IIlcllli", alld the
:) 1
• ('lI.\/)'/TH") /T/'Ecru/-' /)/-':-;/(,'.\ P\U \.\/F//:U,'-j OS]'[ 1 r/T/i 1 l
'1'1 )(''-.{ > " Il> ~I llll< > ft 11I1 ( i UII II f 1 Ill' JI dl Il Il H' 1 l' l '-. h. Il. Il. /', l' "". ,,'l,. " .. l' { , .1 11 dl.. \\ 1 Il Il> ,.
i'-. IIJ(' "\>I11i-I!J\lId'll 1I1111 .\11111'11"1\111,,1111.1"". (/ Il''11 ,llllll'II"I\lII,d di',ldl1ll' t"l Il'1111'' (II
1III(UIIPlcd IIdllllal fll'qlll'III \' III 1)I'1I.!IlI,!.',. , ... '" ll11lll1l1dl'd IIdllll,d 111''1"1'111 \ III t"I:,II'".
,t1)()\C {'![1l <1 1 1 Il Il , l!tl'Il 11lt' 1I111kl '-.p''l'd (11'1<11111'.1 111)111 tlll' .111.11\"1:- \\dl ,d"l' IIC' lillii
djll)(,ll~j()lI(t1 III t!ti~ 1'<1"(' 1I11' ullly l'dl.tlll!'tl'I~ \\ "il Il 1 <III ,tlll'll t III' 1I(J1l dllll('II'''"II.''
It :-,llOlIlcl 1)(' 1l1l1l'cl 1 IIdt, ill Ill<' (IJIIIfJlII ,d 11111 • .\ I\JIII 1111'" l"I 1 III' {' (I.tlld l' , .. 1111 1 Il
k \\'Ilich li(>~ jlC'l\\('('IJ l\Vu :';('(1'1<'1I1i,d 1,t1'lIldl('rI / .. \, .III IItll'll'ul,tllllll, 11'>111[', ''l'ltill'
111 tlli" '-.(lIdy, lu dl>III(JII~II<II{> (1)('1'11'(" t {JI 1 1 JI''>I > dlllllt! 1'''loIlll,'II'I',UII 1111' 1111111'1
Il = 100
f[ = -0.5
l'" = 0.1
w" = :::'11- =-= IJ,2 . w,.
'Ill!' fil}'!' Jill(' (JI d,da Il]('.111'> 111,d III( .\"11'-.11)' ,,11111' dlllllt! l', 1',II'.d' 1 Illdll 1111' d"11 11\
•
•
•
,j()
1)1' dl 1 1 If' rllldllr'l rlJUlri '1111' Ihllrl IIIJ(' of rLda ".1)''> tll,t! tlJ(' I,trlill" (JI' .e,.\I,tfioll of
IIH' dllfll" ,dHJIII, Il)1' d""I/( ci\l" 1" ,110"('11 to 1)(' 1 orx uf tIlt' ~(·1I1i-(h(Jld. The fUlIltb
IIIH' IJI"d Il'' Illdl Il)(' lUI ,11 Iflll uf Il]1' 'l'Ill H' (Jf IIld"~ i" d""IIIlH'd tu 1)(' 1 / 1 ~('IJlI-r hUI ri
1,..111/1.1 IIH' ,·ld",ll' c1\I", ,\1,,0. 1 Ill' f!llh li,lf' (JI d.tld ~how~ 1}ldt tlH' l1.dlll,d fJ('qIlI'IlCY
jlJ llf"ldj'l1!, '" (111/""/1 lu 1)(' () ~ (Jf t 11,If III IIJI"IOII. TIIf' pll)"j( éd hcl~j", of dJOf):-.illg t}w!'.('
III/d/llit j,,,, \\'111 1)(' (''(pldilWd I.dl'I
T(J "tlldy tll!' "'f(·,t. (JI' ('dlll (JI' tlI<' ,dJ()\'C-II]('IJ1ioIlPc! Pél\dllH't('\~ 011 tll<' fllltt<-1
<,p"('d (Jf .III 'illfod, ,dl t,II<' otill'I pdl,lIl}('t.('I~ d/(' kcpt ('OI1~téLllt (('qllal tu th(' r1d,llllt
",dllf''') .Il1d (1111) t.!H' l'dlrllll('1c1 1lIld('1 rOIJ'-de!(·I.d,iOll IS (hdll,!!pl (J\('r ,L [('c!!'.Ollcl!JlP
5.1 J~ffect of N on-D iluensional Mass
5.1.1 IU(,(Hupressihlc Flow
'l'Ill' 11011 dilll<'lI:-.iulI,d Illd~:-' uJ t Ij(' cli\ foI! bcls 1)('('11 (bclliger! ill t 11<' rallg(' of la 1.0
100. ",hIC Il i:-, cl \(',,~(lII.d,j(' 1 cl "1-',(' :\fut(' tllat Il = III/ïr (lb:! ,llld (1 = 0(1) fOl ,m,
.tlld lII/h2 gi\(':-, .III illdi,· .. tiolluf tll!' dilfoIl's 1lJ,\SS dcnsity Th('I(ofol<', fOl éI t)pical
.III f(ld \\ IIi, h IlId"S d('Il"lt~' Idll,l',('" flOlll ~O to :WO, t.1/(' (h()~l'll Idllgf' fUI Il ~1'('ll1~ ta
1)(, 1I'.l:-'lllIclhl('.
'l'h,' Il'''lllts fOI 1 IIi~ ~I'( t iOIl ,\1(' PI('''('lltcd in Fig. ti.l. ,\s Cilll lw ~('('ll fJOm t.1lC'
li,t!,IIIt" t II!' t'-!I cilie! ji-/" Il]('1 IllHb ~I\(' ,t!IllU'it id('IJ1 ic,t! Ihlt 1(,\ ~P('(·r!s U\'l'I thc f'lltit<~
l.tllgt' (If cid!.\. Tht' "Ill.dl <l1If('lt'llt (':-' ",lllrh du ('\i~t cil<' dll!' tu th(' tul('lclIIU''i (oll'>id
t'I('" III tl)(' !,1(ltt'tllI\'(' (JI t'<llh lllt'Ihnd. 'l'hl' 01111.(')' ~]>(·('d flOll1 th!' Lélpl,l('1' Illethod
.d"o .tglt'('~ \\ilh III(J~(' l'rtllll 1 Ill' Iwo olher ....
•
•
II
An'ol'dill!-', to t Ill' IC"ldts uhttlllH·d (IUIIl t Il(' <lll,"~ "1". l\1\t t ('1 ~1H'I·d 111111'"''1''' ,dlllt,.,t
lilH',II'I~' \\'Ith nOIl dillll'lI"jOIl,!1 111t1"" ur t hl' dll(ui!.
5.1.2 'J)'<lIlSOllic Plow
FUI tIlt' lldll:,onic !lm\' !t'gill\(,. tlit' ",lIlH' 1'11}(1'"'' 1:' (olltl\\('d, 011111 tIlt' 1("oIl1t". ·,11(1\\11
111 Fig . .').2, indic,!ll' thdL tll\'1\\u tl,lI!ttlulI,d lIH'tll\!d:-. .!.!,I\(' tilt· "dlllt·lIl1tt"1 ,,]It't'ds
11Iit t Il<' LlpLI( (' lIlt't !tud U\·(·It·~t iIJltlt(· ... It. Titi ... i" dlll' to t Il(' "ill' ul t II!' "111"'" "t(·p"
Ihoscll ill tl\(' Ll]J\,t((' Illt'tltod ('III)(I ... illg "IJl,t1I('1 :,p"('d ... tl·PS, illl\('.[·,I·;, tIlt' IIIIIIIIII.I~
tillH' of tll!' ]JlOgldlll, cOII"id(·I,tld.\'. 11('1('. dg,lIll. tlJ(' IIlItt"1 "1)('1'<1 ill' H·d·,'''. ,dlllll.,t
lill('illl)' \\'It h t Ill' lIoll-dilJ1('lIsi()lIdl 1Il0l~"'.
5.2 Effect of Radius of Gyration
5.2.1 Incolnpressihle Fluw
'l'hi:, )Jdldllld(·I. /'", IlilS 1)('('11 (1Idll.!.',1·d o\('\, t 1Jf' Idll,t',(' ():1 Lu 1 rI '1'1\1' 111111
diIlH'Il:,iullill fi lit Ivl ~1H'('d \(·I ... lh /'" ie, "II!I\\'IJ ill I-'!.!.', rl:~, ,\:-. 10111 1", """'11 Il''111 tl\('
flgul'l', tlH' two tltlditiollal IIJ('tllUd:-. gi\(' ,,11110:-.1 tll!' "'dllll' fllltl,('1 "1"'I'd·), .d·.11 1111'
flllit(·!, :'jH',·d pll'didl'c\ Ily tlt(· LilP].II(' IIdll:-,I"IlIldtilJlI IllI'tllod i~> \,('1 Y Il'1-.(' 1.11 II\('
ut hl'I t wo,
'l'Ill' pll):-.i(,d illl(JoltdlJ('oftlll' l "d 1 Il''' (JI' ,l',ll"IHIII ofdll .. ill'JlIIJII tf·. 111111"1 •. J1II·d
1 all 1)(' ('xpLlilll'd ilS folio",.." \\-,1)(,11 /' .. 1 ... illlll'd·'(·(1. II. dl t Il,dly III"dll" 1.11011, f/)I .1
:,p('cific jJo:-,ilioll I)f 1.11<' ('!.l..,t.i, ,1'.1..,. tlll' 1111'llid fil' tlll' dllfull ill plI, Il dil'" 111111 Il.!')
IW('11 ÎIlU('i!:-'('(1. '1'1]('11'1'1)\1', ill tlJ(' fil"t g,1<l11( (' Il 1111,!.',1d. "('1 III l.IJid. II. Ilill·.1 ,drl'( 1 IIIf'
flllttn !o]>('('d ill tI](' <'<1111(' Wcly ,1'-, "JI illl 1 (',l"'(' III lJII' 111)11 rlillll'II',IIIII,d 111.1',', dl)l"- /lllt.,
• :'8
'l'III' jJ(JiIJl, i.., 1!r.I1, III Il]1' {'I..,(' (Jf ill{ I('M,ill)!, lIoll-dilllclI"iu!1,t! l1Ii\:-'<;, d( tUéllly 1 Ill' ill('1 tra
1)1' t 1]1' ail for! ill I,otll 11w d('.t',I('r'~ ur fi ('('dol1l in( 1 ('<1,,(',,; t IIl'I (.fOI (', t 1](' ('1]('1 gy (dlJdcity
1,111' iJll'ltia (JI 111(' :-'Y..,('III in pitrll dil('( (ioll, whi('IJ i" plOpultiollal tu 1'() 2. illcrC'é!scs.
01 (0111 'l', ;111 ill{ J ('a..,(' III t Ilf' Il Il! 1('1' :-.IH'{'<I is ('XI)('{ t.('d. bllt. t hi" (ill)('. il, call1lot be
{'}.I)('{ (l'd to Ilf' lill('dl
0.2.2 'Il'é\llsonic Flow
TIlt' 1{':-'lIlt" oft.hi" (d"{' dll' ,!rU\\'1I III Fig. !l,1 !1o(!J of the tléldit,iollftl l1ll'thuds g;\'~
I",th (Ill' l'olltilWS), \Vhil(" t.J1I' Laplan' (lilll"follllil(iull llldhorl, élg"in, u\l'l'e"till1é1f,C'"
(Ill' IIl1t1t-1' "')I('('d. ,t!, 1('c1"( (-"Illpall'd \VIth (Ill' ut 11('1' "ulutions, 'l'hi" is mainly 1)('('ctllSe
(JI' t II(' ",IIIll' 1('a~('11 IIH'lltio!l('d ill 1.1)(' pr('\,iol1f-, section.
It. ""('111:-', fll1lll 1 Ill' li,l!,l1l('. th,,( t.1ll' 11111,1<'1' -'P('('<I iIlCJl'clS('S ",ith 1'0 1I10\'(' or le,,8
III 1111' ~ .. IIl(' f .. ,,"ioll c1S III t I\(' (d~l' of incul1l))!('..,~ihh' l1u\\', b1\t, in this (ë.\~(" it tends
(0 IH' 111<11(' lilll'ar. This :-.ugg(':-,ts thrlt, i" the tl'dll"onic f10w l'('gilllc thl' pitching
r:: 3 o. Effect of the Position of Centre of Mass
Fil"(, it "llollld Iw llut('lI Ih"t ,r,. i:-, th!' di..,ttlIlCl' lH't\\'C'l'n thC' n'lltrC' of ll1eiSS and
(hl' ('I,l~ti(' .t\is ill t(,llIl~ of b. th(' :-'('llli-r1lOrd, pu"itiv(' fOI Cl'lltJ'C' of I1lét~S posit.ions
hl'lIi"d tlH' ('I,I:--til ,,;..i~. 'l'hl' dl'f,l1t1t pu"itiun of the l'lastic axis i~ the quartcl-dlOld,
•
•
•
('Il:\PTEU) fFFE(''[' OF U/-'S[(,'.\' [I.\U \.\/FTlï~.'1· OS FU J'J'FU
f!JI'ld()IP, il Il'''''UII,!1lk l ,1 Il.!!,I , fOI /',,1.111 Ill' () tu l, IIlt'dllillt',. 11\1111 th,' I(11.111l'1 1 hllid
to tIlt' thll'I'-qllillh'I-I!tOI I I. thi" i~ the 1 <l1I.!!,1' III \\hHh tilt' Il'lltll' \JI 111.1"" (11.111 .lill,,"
ll~lI(dl: f,dl ....
5.3.1 Incoluprcssihle Plow
TIIP fi 1\ 1 tel ~p('('c1 \'('1:-'11:-. 1('11111' of 111.1"" pll ... iliull IS ~hll\\11 ill 1,\1', :) rI, 1,,1' l'"
flOm U,075 tu O . .), IIlI',lIl1llg. fllJm d littll' Ilit Iwllllld tll(' '111.1111'1.11111.1 1111111' 111111
(hol d, 'l'Il<' figll1(' do('" llo1 in< 11Id(' 1'1"1111 ~ fUI t II\' 1 l'lit Il' Ill' 111.1"'" dl IIJ(' '111.1111'1 111111 d,
:-.ill(,(' thl' f111!,t<'1 ~p('('c1 i" fOlll1d tll Ill' \('I~ IlIgh (ilt/illlt."), '1'111" 1'1J('llllllll'llllll \\ dl III'
di:-,c1\,>,,('r1llloJ'(' ill tll\' fuJJIl\\'illg Ihdl't<,I', Ikll'. d.l\"ill, tlll' tWIl tiddillllll.d 1III'IIIIlIb
and the Laplét("(' tl'itll:-.fuJ'lllatiull Illl'thod 1',1\1' ,dlllo"t 1111' :,.11111' flllttt'l' :-'))('l',}" 11\'1'1 III\'
l'lltil (' rallgl'.
A!'. tll\' c1i:-.laJJ(I' l)('t\\'('('11 tll\' Il'lltH' Il/III""" dlllllllC' 1,1",,111 d \1" illll('d',I"" flllll('1
spcl'd c!('CI('dS('<;, cl" 1 ail 1)(' :-'1'('11 flOlll t hl' ligllll', 'l'hi" 1.111 1)('1''\1'1.11111'.1 il! 1 III' fllllllWIII,I',
\Vay. Fil'st, it shullld he JJo1,l'd 1 hal t11!' olllv jJdl"llH'fcol \\'lli, Il i:-. 1 Ildlll',illl', 1:, I.IIC'
di:-,tallcc I)('t\\l'I'II tll<' ('Idsl.il· ct,i:-. dllcl Il'11111' of 111f1"~, ;11Ie! 1101 1'" Il. 1111'.111', tll.d.,
while the ('('lItH' uf 11I.t!'.S is movillg d\\'dy fW11l 1.111' 1,ld,,1 i( il '\1:-', III<' III"'" 01 1 IJ(' .Iii flld
is Iwcolllillg 111011' ('()IIC('lIlral('d ,t1)()1If. tlll' (('Itfl(' of 11I,1"~ ill (Jldl'I III 1(',}I11 l' 1",11101""
mOl1lent of ilH'1 f ia "holll. 1 1/1' «('lltt(' I)f 111;1',", '1 Iii ... i" 1 III' (JI"Y \\'dy t1111J1I,l',11 Wllllii 1/11'
Il1Ullll'llt ufilWlti,1 ,t!H)lIl. t.h(' ('Ia:-.I,i, d,i ... Illtly Il'iIIdill 11111',ldltf, wllt/I' t'" i ... 11101111',1111',
ft ml'iUle; t1li1t ;11. (,.tch j)o:-.il iOIl U/ (1)(' 1 ('ltlll' \)f Ill""''' .• 1 dtll'f'/('Ilf. ri Il flJil l', IllItll'l IIIf'
cUll:-.idl'l ,tf iOIl. BlII, il i:-. ~till illf ('1 (,,,1 ill).!, Il) kilO\\' \VII,)', .til hOII).!,11 1'\ ('1 Y flllJ/'1 Ildldllll'l f'l
(indlldill,!j 111<1:-':-' <llId fllOlllI'ld uf illl'Itiil) 1'> IIJ(' "dllll', !Iw/(' l', ',1 dl 01 Illdll!',1' ill 1111'
•
•
('11:\/") VU i LI'VU"}' (JI-' IJ/·;:"/(,'.\' }J \ U.\.\1 l':TFHS os fU'TlFU GO
JI 1 JI/' ("jlldlll'"" (If 1111)11011 dlf' ff'/lc,id('j('d, "lfll.ttium (1 J~) élJld (1 J,n,II fetll IH'
'f f'II 1 1,.If, IIlf 1 f',,',IIIP' '" "" 1 f'd',f", 1 li(' off' didgulldl t"llI1'- III flJ(' 1l1ét".., Illdt 1 i x, ,UH!.
11111, 1 III' (III',', f 'll/pllll!.', III 1 1 If' ',\'"lf'll1l'> lJ)( "'d,('d, ,\ ... cl (()I1"('qlj('II((" "'lll«,the 1 ('<tc,ol1
1,,1 Illf /)JI,,"'I ,d rI'III('1 l', 1 III' 11('fllj('II( \ '/I"k"«('lIc<,. (Ill ill'lt'cI""lIJ (IO:-'::'-«(J"phllglJ('lp:-,
1111'> (' 'd 1,",( ( II/ (' 1 (J /I( (III ('tI',II'1
r,.:J.2 'fr:lnsollic Flow
I;I!', ~I (, ,.111)\\" tll<' (,1" ""11',] 11111 t"1 <-(J,,(,(b \ ,'1 ,>11:-' ,l',l' 1I~1I1g t!)(' {'-q, p-I.', cllld
l ,dl'LI(" 1111 1 11< ,d" 1"1 Il cl Il''111111 !lu\\" 'J III' 1 <lll)!}' of ,l',r r!Ju:,.cn !J"J(' IS 0, t () 1 0, The'
',.1111" t 1 ('1111 ,1.., "1 i", UIIlPI(';,,,Jf,](, Ilu\\, (dll II(' "'('('JI in tIlt' 1'<'],d\'ioUI uf th(' nuiter
"1"'(',] \('10.,110., 1111' «'lrlll' \Jlllld<,~ j>lI'>itilllt: lt d"!'I('cI'-I''', in thl' :-'''"1<' fo"hiUll ,1'> Iwfuw,
\\ill, ' .. l'III' Indy IIl1ljl('dl,J1' dill('ll'II1" 1" th,t!, the Ldpl'il(' tl,lll!,,fullllcl(ioll 1111'1],0<1
",1\('" c,lll!,lrI" ""kll'ilt Il',1'''' (Ulll]ldll',j \\lth tIlt' uth"1 llH'thod:-. for ~lJlilJ\ \ <tl"e:-. of
l'".
111'1(', d,!',dlli. 1(,(,t1lillg titI' "ljll.lIiulI;' of lIlUti(J1l fot tht' ttclJl;,ollic nU\\ Il'gtllW.
1''1".1111'1\'' (1 1 ï) .111,] (1 I~), "II(J\\'" 111,d ,r" dPIH'rll'" ollly in tlll' uff-di'lgollal (('l'I11S
1I1 1111' III""'" Ill.dlIX TIll'll'llIll'. tlte' ~,,1I1l' c1i~(II""'IUJl clpplil'~ h"I<', éI:-. in the ri!:-'C' of
III( /111'111 (",~",]l' 111I\\
:'.1 Elf(:'ct of t.he Position of Elastic Axis
\ l "c\o"UII.t1 ,](' 1 clllp,l' fut Cl \VlIlIl" l'l -LU t 0 () 7,1,
1IIl'.!IIIII,!'" 11\1111 tl)(' l''.li 1111 !!, l,,lp,l'tll IJK lhlltt! ,"!l'cul uf thl' ttcliling, (',Ig" (JI tlH' ai,fuil
•
•
('Il. \lJT/·;U). /.'1''[-'[';(' 1 Of [)/· ... ,,·{(;S l' \If \ \/J'/THS 0.\ 1'/ l' 1 [Flf li \
(Ilok Ih,il 1 Ill' cld',lltl( \,\bll' ul r. i:-. O:~-l. 1!t1'll'Illll',.iI 1111-" \ll"ltlll1l. 11\1' 1"lilII' Id
Illd ...... i:-. 1\)(,111'd clt 1111' 1 1,I11illg l'dgl').
5.4.1 Incolnpressihlp Flow
FOI IIIC(J111PI(·:-. ... t1)I(, !lu\\', IlU !lnl Il'1 (I"tlcI \)(' rUlllld fOI \,dlll'" III Il l, ...... 1 11,\11 .dHIIII
-O.ï!'i. \\'11 II 1 II!' dddltlt \',l1n!' uf 1 Ill' ull\l'I I1dldllll'1I'I'" 'l'Ill' \('.1"')11 l'Ill 1 Itl', \\ tll III'
l''.pl,,il}('d ill III<' fllllo\\'lllg llldpl"1 TIt"II·!()II'.111l' IdllL',I' "III \11''''''IIII,t! ill Il,,,, ,·,1\1111
i ... l'IOl11 n 7-) lu (J.7r) , ,tilt! tlll' 1I1111\'1 "1'('('d-..I"lllld fI Il Il 1 tl\(' 1 III,'" 11\('11111.\" .II" ·.1111"'11
111 Fig .. ) ï, 'l'II<' IltH'" Il)('IIIUd:-. ,~J\I' ,""Ill..,t Icll'lIti, dl \(·"Itlt..,
'l'Il<' 1111111'1 "']H'l'cl ,h\Jp-" Idpidl.\ \\111t IIHI'·d'llI,!.!, Il fUI "111.111 \."111':-' ,d'I/, .11111. tlll'il.
:-.LIl(:-' d('(ll'd ... ill,!.!, wlilt ,l "111,111<'1 :-.11l\l" 1111.111,1111'" 1'" II('I dll",' Id tll(' ,·111'11 (JI tlll'
dl .. Lllllt'lll'(\\('('1I 111<'(,1<1-..1]( d\h "lId tlll't!I'II)lhll.l11111 1('ltlll', 'III IIJ('lhlldlllll'.,,!IIII'
:-.)',,11'111 \Vhih· (11<' 1'1.1,,111 d\i:-. i" ,"Il'd<l (J! 1 Ill' d"\(1(1\ 1I00111i( (l'III l", 1 III' oI"I",h IIdllll!
Il'UIl1l'llt alld llJ(' 1l1()1IJ('1I( (JI' tl\(' d('lll(" IIdllli( fUI ( (' ,dllllii tl\(' ,'I"..,til .1\1'" d( 1 III
opp()site diH', liulls. l'hls PI('\'('ltI" 1 hl' sy:-.I ('Ill 1'I1IIll 1)('( "1111 11,!.!, 1111"1.""(', ('.l'.dv \VIIl'II
litt' (,Ja"lic (l,if, is I)('hilld LIll' d(,l()d~ 1100Illi(' ('('1111(',1 II(' Iwo Ill(llill'lJ1',.1( 1 III Il]1' ' .. 1111('
dil('rtiol1, alld Lhi" d(':-,(,t!Jlliz,':-, 1 II(' "y ... I('lll ,\:-,1111' 1,ld"li( ...... 1·, 1I11l\'/", /111 t IIf'I ""'"lIeI 1 11 (' il l' 1 (J d) Il ,1111 i ( ( 'II 1 Il', 1 lit' III (l Il j(' 1 t1 d 1 III (J f 11 Il' d" 1 CJ cl Y Il d 11111 1 (J 1 ( (' 1 /l' l "II 1"', 1" Il.!'.(' l ,
<1 1It1 , llJ(')('fU!l'. III<' d{",ldl,dizlll.~ (·fI(·(1. 1Jl'(llIIW" '.II(lll,'~I'1 (·(lll'l·qllt'lt1I)'. 1 Iii' 111111,"
"P('I'tI of 1 1)(' dll full d,'( 1 ('d"(':-'.
Il ... 1J(J"ld 1)(' ('>,pldilll'd tllrll, Wh('ll \\(' Ldk .d)(llli II\(' (11,·,1 (JI 1 IJ(' 111 1 /111' IJI IJI'I(',
il. dul''' 1101 IIll'dll 1 Iidt t Ill' 11\('( Ildlli"1l1 (JI' j11/1I1 1 Ild-. IW"II 1111'· ('d III' willl Il ... , 'JIll' 1'1
uf dll·(/(/(II((.llllt.lt llH'clll'" tll,LI, ,,!t,dl'\('l tl\('I('d',')11 (Jllllllt"1 l, 11.1') l' Ild"I"d l,y
d l,i,!!,)!,('1 IIlUllll'lll .
• ('/1:\1"1 U(,j H'/'L'("I (JI /JI" .... )J(,·.\ /1.\/(.\.\/VI /',/(,',' ()S VU '1 ï'/~'U ( ,) I~
G.t1.2 TJ'i1l1S011Îc: Flow
1 1 If' Il.1"'1'"11 III/\\' l'!',IIII(' fUI .111\ \.""1' (Jf 11j(' 11('!jIH'III) I,tfi(). ,Illd ,111)' po..,iiioll (Jf
pll",·.tI"" Il'i\\' 'II".., /lIWlllill/('llIill \\'111 III' di..,1 1I',..,('t! Jll(J/('. III tif(' lullo\\'ill,E!, (haplC'l',
Il ('11'11 III l', /" \\ l,i, I, 1', () (l'~-, ( 1111' d('IIlIl\ """lil d,ild cI\ i,iLdl!c ill 1 hi.., ca..,(' i~ \ ,d,cl jll:,t
1(11.1 Il'd,,((·rI 1'1I'!jIl('llfY ill t III' Idll,l!,(' (JI 0 O:2!) 1Ij) tu O.»). FUI thu:,C' \'(d1\(,~ uf a, tll('
I{(·.."dt:, .11(' ..,I!()\\ Il ill Fig ."") S 111'/(', dg""I. th<, !(''''IIIt:, fJUIIl tll(' ['-g alld p-I..'
III tll!' tlolll"'()lIi< 11111,' I('gilll<', l'd~i( ,dly Il](' :-éllll(' !('cI"'O/l dS fOl t!tf' caSt' of iIlC01l1-
1/ 1" tl"Il"'''II/( Il II\\' , Il rI(I(':-' IIul ..,IIU\\' il..,(·11 (''\plilitlj i/l the ('I(Il,tfiulI~ (JI' IIlotio/l.
Ih,' ""lllfhllolllli( ((l('lIi(Il·III .... ('1111, "",] ('",." H('(,dlillg !'qllclli()ll~ (1 :20) cllld (·1 :21),
·,IIII\\'''' 1 Ii.1I Il,,, ill,l!, 1 h(· I,dlll' ur (/ gl(',lI('! 1 hdll -0 i1. i.f' .. thl' ('1,1:-.1 ie clxi~ Iwhi"d
Il,(· q1l.l1"" ,11()If!. Illel"('!'> 1111' IIIUlllf'llt of tll!' c/('!(J(IYllclllli( lIft clllrl th!' ill'l()rlYllcllllic
•
•
•
('11 \f1 TUf") IJ'Î' L'(' r OF U/,','",/(;\ JI \/{ \ \1 f.' 1 /-'H,'" ().\' 1,,/ 1 [' Il'/{ Id
5.5 Effect of FrcquPllcy Il a t io
l'Ill" Pclldlllt'lcr 1I1d~ Ill' (Oll"ld(')('d .1" tll h,\\ l' t h(' IlIl,,,t ''l'II''lt 1\1'11.11' III t 1", 111I',\'t
uf !lllth'l nqJl'lldill.[!, 011 tlt(, 10(.1111111:-' llf 1111' .II'llld,\ Il.1 11111 Il'IJlII'' 1'1.1'.111 d'l'-, .!IIII
(l'1I1((' uf Ill.!"" ill 1IlIIllllj!l(,.,,,il.j(, 110\\, IIl1tll'l 111.1\' (III III 1111 11111,\ '-.IIIIH' 1.111,",1", III
fl{'(l'lI'IlCj' l ,li l\) III lldll.,Ulli( flu\\" .t11111111,!'.,h thl" 1'.11.111\1'11'1 ,,1111 Il.1'' .III 11111"'11.1111
illfllll'l]('(' UII IIJ(' 111111(" ~p,','d, It "l'l'Ill" Ih,lI 1111'11' i., 1111 '-.11111 01 1111111,1111111 "II 11111 III
o!l~d dll(' lu fll'fllI('lle\ Idli", "'-'" d~ III 1IIIlJllljlll",,,d,j(, fllI\\
5.5.1 IncOlllIJl'('ssihlp Flow
FUI 1.111' d(,Lwll \dlw'" of 1 Ill' lllill'I 1'.Ildllll'll'l", 111111<'1 Idll Ill' ()ld,IIII<'.j ,il ,dl
fll'qll('lley I,t! ilh, ('\( ('pl ,d \1'1) :-'IJI,dl \clllll'~, Ik"l1lh 0111' ',hu\\'11 ill 1,'11', :-, q '11)1' 1 \VII
1 l'ildil iU!I,d lIll'llllJd., "JIU\\' go"d d,!!,II'I'III('111 \\ il h 1 hl' 1,'1/ d.1< l' Il oIl1'.I"IIII,d 1"11 1111'1 III" 1
'l'li(' flul kl' ~I)('l'cl dlop'" hU11l cl Iill.!!," \dh\(', fOI "1l1.dl \',dll"" ,,1 Li.-', (\\'111 1 11 l', Il,,1 ',1111\\'11
ill IIJ(' figllll'). cI("'II'iI"('" alrllu"t, lilll'd.l,\ to cl IIlilllllllllll, .llId, 1111'11, ..,tdlt" III' Il'd'''II!',
lill('dlly cl" Li.-" j" ill( 1(';\...,(,<1, 1,'IIIt.lll'lllllll('. Il ""('111" 111,tI .1..., (v, ill' Il'd',('',, tlil' 1IIIItl'I
"))('('" ,t1"u jll('J ('iI~·,(''', \Vltlt 110 IIP))('I Jillllt,
Hd('ll illg lu t 11(' ('qll.11 jUil" or III()! IUII, (''111,iI j()II., (1 1 :~) dlld (·1 1 :q, ',1111\\", 1 Il.iI 1 1 JI'
fl<'qlll'III,)' léllj(J llél" il (IJIJ1lillllllilll 1l1l1\, 1111111' ',ltlllll:,', IIldt'I'{ 1111.111, JI 1111\\", Il'' If
I!ltly ill 1111' fiJ...,1 di",!!,IJIJ<l1 Il'1111 (JI 1 hl.., III,tlll\ \\'1111(' • .11 1 1 If' I)IIJI'J Pd,,,III' Il'1', .111
kl'jlt C(JIl.,1 dit!. ( lldllgillg 1 111'11 ('<jIH'JJ( ,\ l,Jill! 1 Iid Il,!!,''''' JlI',t 1 III" l,II III' Id 1)1 1 1 JI' ' 1 dlll''',',
IIMI 1 ix, TIII'J (,fuit" cl..., il., \,dll(' gd" 1 IIH'I 1 Il t hel t 1)1 Ilw ()IIII'I ,J""'.{JJI.d t, 11/1 IJI 1 1 JI'
,,1 dfll('"'' Ill,tt, j'o" Il j~ (1..., If 11[(' 1 \\'u 1111),)(,,1)11 III' ',\ ',t('111 Ild\" 1 II) ,t'I fi ('(1 1)/ lit Jt", 1'1'1111 Il
ill 111111, III,d\('", tlll' fi 111 1(" ('d"j", tu III( IIJ A 11111111'1 IlIt J('d'I' III ' .. ), IIld/:t-, 1 1 fi' 1'::11
•
•
dldJ','Jlldll('IIII" ''l'I'dldl(' (JIll .1,[',<1111, .Il1d, <I«Oldlll).',ly,llH' fl('qlll'II(,)' uf Il)(' 1\\'0 lll(Jd('~
1)1'( IJIIIf' 1111111' ,11"11111 l, tllld Il)(' 111111('1 (J11~('1 IW({J/Jl('" IIIUI(' dilli( ,dl.
5,5,2 'J'l'allSOllic Flow
Fi).'" :) I() ·,II(J\\'~ Il)(' 111111('1 :--1H'(·d~ IJI,lc1lll('d lOI ''('\'l'I,t! \.,t1I1('~ of w, ('mployillg III<'
1111!'(' 1111'1110,]" III llr1I1"Olllf !lO\\', (;l'IWI,''I\', III<' :--clllI(' 1)('lld\ iU1I1 <1:-- for ill('()lI1pl(·..,~dJJr.
IIII\\' (.1111)(' :,('('11 Iw/I' fUI llw 111111('1 :-'l'('('r! \'('1,,11:-- !tl'qll('II( y Icllio, 'l'Il(' ('qll,I1I()II~ of
111,,11' 'II ill Il.111',' 'III( flo\\'. ('<jlldlIUII" ( 1.1 ï) dlld (1 1 :-l), illll:--t] ,11 (' 1 Il,11, Ihi~ Pdl,L1lld('I.
ll'" 11111111'11('" llj(' lil:--l dldp'OIl,d 1('1111 (If Ill!' :--(ilrll(':--:-' Ill,ltli", ollly, (II(' ~i\llH' :--itlla
liull cl'" ill ill( ()1I1J1I (' ...... i 1,1(· /lo\\', TII,t! i:-- \\'hy Fig, .l,)O "hO\\':, 1ll01(' 0/ I(·~~ lh" :,ame
IH'hd\'iolll d:-- Fig :l,!) r!(w!'> fUI III( olllpJ<·" ... il,I(· lIU\\',
Il :--1 1011 Id II(' 11II1(·d, 111(JII,t',1I. 111.11 Il)(' d('IOr!Ylldllli( (udli,i('llb tll('\J1('''I'IJt ill ail
Il)(' ('1('1111'111'> (JI' 111(' ..,1 drll(''''''' IIld l11X uf 1 h(' (·qll<11.iulI:' of IIIU! ](lll, <'q1\el! iUII" (,I.I ï) cllHI
(-1.1 K), ri 1111. Il)('1 "lUI (', 1(11 ull)(,1 lo( ,11 iOIl:' of t 1)(' (,Iasl i( ,Ixi ~ cl 11<1 (( 'II t 1 (' of I1Id:-'S, 1 hey
(dll lllrlk('II)(' 1)(,!.d\'iolI1 (JI (1)(' ",\ ..,lelll di!l('I('11I flUll1 \\'h,l1 it, i:-. 1'01 t.h!' d('falllt. "d1l(':-'
III 1 III' Pdldllll'l('I" 'l'hi.., \\'dllll' "'('('11 ill 1l101(' (kl,lll ill III<' follu\\'inl!, (hilpt('I,
•
(HIs .
•
"-ID ~ ::1
65 -
6 -
ü: 45
E -0
1 5 4-Z
r.=-O 5 a -~·O 5
w.~02
20 __ . l
40 1
60
Non-dim. Mass, J1
1
80 100
•
•
:>' o ru ru 0..
Cf}
E u
1
C o Z
45
4
3
r·k rv};lhod
u·g r1f·~hod 1 t,plaGol!~('lhod
ru - 05 o ·05 W, ·02
x. Ci 25
20
-,
40
________ L ________ _
60 NOIl-dirn. Mass, J1
__1- ____ _
80 100
l\l'.I1)(· rI :~: "'hlttel "'pv(·d, \ f. \(·1 .... 11!-> ll(Jll-dÎI1l('Il~ÎOllal Jll<1~S, Il. TlansonÎl f1ow, il
0: . \ ( ': \ (l 1 \ ( 1 0 li d II fu JI • Il \l
t 1((' {'·(I. JI-I.:. tlllt! the Llpl.l(,(· lIwt bod-.; .
•
•
Cll.\}ll'VH 'l F/'TE('J'OF IW,I.,'/(,'.\ /'\U "II /Fl\ .... · ().\ FL1'} /'1'11
otis .
8 -
7 -:;:;-u :E 6 -0.
(/') ...... Q)
:t.:! 5 :J u: È "9 4 -c: o Z
3 -
2 02
p- 50. a=:-O 5
w,-=O 2
- __ 1 _____ _
04 06 1
08 Radius of Gyration, ra
1
1 2 1
1 4 1 6
•
•
12
10 >-0 n>
8 n> Cl..
Cf) L-
n> :t:~ 6 :J iL g u 4
1
C 0 Z
2
0 0
P k t-..~y'!lOd U-g Mulhod
(1
1 flpln(.(~Method
f.1 50
il -05 w, 02 x. -025
1 __ - - - -- - - ,- - ---- - -- - - -----
05 1
Radius of Gyratioll, ra
Fi)',lIl1' rl 1 1.'1111 I\'I "pl·l·d, \j. \1'):-'lI~ lMlius ofg)I,diol1, 1'(\' 'nall'>llllic flo"" a ~ACA
(il:\Ollti "i,luil ,t! \1 0 ~.) ('ul1lpclli,ul\ 1)('1\\1'('11 the 1('<:;11\1<, O!>1cIlIll'd f'OIl1 the U-g,
•
•
C/I \PTEH i "-'J'FE('( Of n/"l;](,'.\ J' \IU \lV/FU,"" ()\' FJ /'/ J FU
8
E i? 5 c o Z
4 -
J
02 1
03
Il !JO r. 05 u -·05
w, 02
1
04
Mass Contre Position, X a
fJ k Ml'Ihod " LJ 9 ~(\,Ih,)d
Inpfuco /v1l'lllnd ,\
1
05
Ou\\" C'OIl1 !J,IIJ"UIl 1wb\('('ll Il](' 1I'",dh 01Jtdlllt'<! fl'lIll 1111' {/ fj, /1/;, dllfl 1 1 If' l,dl'l.\lI'
J]J<'t!Jud",
•
•
2
1 o
J
02
70
J _ L __ __ _ _ _ _ 1 _ _ _ __ - -
04 06 08
Mass Centre Position, Xa
Jo' I! ~ \ 1\ l' ;1 (j F 11\ Ill' l "1 H'l' d. Vi' \ ( '1 ., \ 1" 1 II!' ( (' 1111 e 0 f 11) cl ~ s 101 a 1 i 0 Il l ,r Cl' '1'1<111 SU Il i c f1 u W 1
•
•
( '/1. \ V J F n j FFI 1- ( 'r () J' 1 ]f."; 1 (,' S l' \ 1 i \ '1 F lï ï i ,0.,' () ,\ 1 ï l' J J /. Ii
16 --
14
;;--g 12 -Cl> 0. en
E -0 , c o Z
Ilj('f hl)t!",
JI sa. r. 5 W, 02 x. 025
JI 10. '-'" n".j " lf \1 ~htlH .. l
IJtpflff1Mt"htl<f . \
• ('JL'VI J..H ') J'.UIJ"J 01' IJJ.''-,/(,''y /' \1<,\.\1/','1 /-,IfS' OS /'Li 'J rU{
~ \( '.\ ti 1 \(l(l(i .I1t/llil .tt.\/ Il~) ('(I1Ilpillic..UII lJt't\\'l'l'll tht' I(,:-'Ult~ ubtailwd fJO/l1
(lit' {' Il, p-~, .11\(1 tilt, 1 dpl.llt' Ilwthtlt!" .
•
•
•
( 'II \ fJ / TH, i f FI, L' ( , }' () f/)I . ".\[ ( ,'.\ l' \ H \ \/1 l 'Jo' H S ().\' J.' I.l 1 J '/ 1\
5
~ 4 5 -'0 (l) (l)
a.. Cf) .... (l)
:t.:! ::J iL E i3 , c
4
o 35 -Z
3 -o
Il ~)O
r. 05 a - 05 x. 0 <'5
pk M('lhod " U 9 Ml'th .. d Il
I l dpldle MIIIII.ll ,\
1 ~
Figll] e :" 9' FI1I1 1 ('] "'P' (d, VI' \,(,] "11" fi ( flll"11! YI" 110, (,,), r 111'111 il" l' !id, !'.,,\\, ('''III
•
•
('11. \V/'/'U 'ï f[<'VU'T UV J),.,~ï(,'.\' F \lU \J i" l'FUS o,v VI l' /' n'Ii
6
;:;-_ 5 "'0 a> a> a.
CI) L-a> S 4 LL
E -a
1
c o 3 -Z
2 . -_. _. - .. --... 1. -o 05
]1-1:, (wc! the },ilplan> Ilwt hods .
a =·05 x.. -025
Frequency Ratio, Wr
1
1 5
i'
2
•
•
Chapter 6
Parametric Flutter Analysis
III Illi" 1 Il''IlIt'l, 0111 clll,t1Y~I:-' ()f hu\\' tire 1/ lall/'( \'alll(,~ of d('"igll pétl'éllllders of an
tlil f"il ;dl'<-II 1 Ill' 0111111('111 (' ()f flilt tl'I will 1)(' gin'Il.
l''II~I, l'illt':-,' tllI,t1y..,i.., ",III Il(' 11111 uc!1J( c,d, cille! Iii.., IC'~lIlb \\'illlw 1 att'guri,œd. '1'l)(">n,
l,y Ih(' Ill' 1IIl' 1('~ldh flillll 11lt' IOllljJlIt.lti()lldl IOlltilW:-' ill hand. PillPs' J(':-'lIlts, [Will
his ilpPlll'\illl<lI(' dll,t1.\·..,i..,. \\'dl 1)(' 1'\·;r1Il,t\('d. :\!(,cllI\\'lrik, ~()l1le gC'llt'l'al t'Ollclu~ioJls
1111 1 lit' 1'11'('( t of Il'1,rI i\'(' po:-,il iOlls uf the l'!cI:-,t il' éI\i.., (P.éI.). ill'lOdYl1dl1lic ('litre (a (' ),
oIlld ('t'lItlt' of 1 Il il '0:-' (t .!-',.), <1'0 \\('11 ao.;. tIlt' tOlsiullcr! tu Iwnding nat1ll'al flcqllPllCY latio
(u..',) ()f clll dirfuil, 011 1 lit' flnt t('r .... !)('(·d \\ ill 1)(' <11<1\\'11.
6.1 l=»ines' Study
l'illt':-, :-,llIdi('d tilt' plUl,]t'1l1 of the fllltt('1 uf cl 2-D c1irfoil \Vith t\\'o (\togIP(,~ of
r,,·(·dlllll. ill plllll).!,(' ,md pltl'h, h~ llldkil1,!!, t111('(' cI~~III11ptioll~ [:nJ,
1) tilt' .... y..,tt'Ill IIlldt'lg()('S "illll"'UIiI,d motioll ill Lolh c!('gl('('~ uf fltwlom,
Il) fllllt('1 (Hellrs dllt' tll tIlt' (U,r/("'('('IICt' of tIlt' twu fl(>(l'lt'llt'ips of tl1(' ~)~Il'Ill,
• Cll.\PTUl (j l'\H.\.\IFTH[(' FU J n.ï{ \.\ \/ ),l.,'/S ïh
lit) tlll' dl'lll(hlldll1ic fUll l' I~ hlJ('dlh dt'jll'lld'11l1 1111 tll!' dllgk III dt:"l k (11111' Id II\!'
Ba!'-cd UII 1 ht' Id!'-1 d!1!'-llIllpt loll. t Ill' l'qlldl lUI!" of llloliOIl dl (' Ill' 1 lit' IlllllI
111 h 1- 111./ n 1 1\" Il - (/,\ Il n .
II/ ,. 2 li 1- 1/1 1 li 1- 1\" Cl- ri" II/'/I •
\\'ht'IC', u., :!1 tlll' lift 1 tll "(' !1lt)I)(', l' dt,,1 ,111('(' lil t Ill' d('lUd., IIdlllie 1 ('11111' ,d'I 1111111' 1'1"',111
in the' Plcviuus (haptl'I. AS:-'lIl11IIl,l!, "11I11!10Id.d 1lI011UlI, titI' (1Idld( tt'II"II( ('!jII,I1IC111 1111
this prohkm ('itll 1)(' fOlllld cl!'>:
)
, 1/" I{ .r) 1 u.,',. (1 1 1 )
1\ " Il,
\\'11('1(', u) i!'> tl)(' fl('I(I\(,IlI'Y ur (J ... I IIl.diOIl, dlld uJ, i!'> tlll' 1H'lldill).', II) ItJl:,IIJlldlll,allIl,d
eqllation !'>lwllld 1)(' ZI'I(J UI Ill',!.'"tli\·(',
2 ( .1' 1.) 0., '{ 1 w, 1 -- ( ,-) (1 1 l' ,r)' (J 1 \ "
(Ii 1)
: \ --~ (.1' -- .1 / ,
• [ 2 2 ( _(.1:)2)], B =: ~ (1 + w, ) ((' - l') - 1 w r l' 1 -l'
• ('/1 \1 J '1 ,,; /( (, 1'/ \ /(. \.\ l '" 1 /( 1 (' /, 1.1 . / 'J "-' If .. \. Y. \ t 't '<" J ,C'; Il
dlld.
( , ) 1. 1 1. (1 l ",,;) - 1 u/ (1 -- (--) )
, l'
1 {[ .r 2] 1. 2 .. - -11-(--) w,.r--(I-j-w,)(.r--.r) ( l' f' ) l'
0-- - -.. - ----- -------- --- ------- --}
.f' r 2 '.!.w/ [1 -( )1.].1'[(./'-1')+(.1'-( ) [.)] , /. ,.
1 {[ .1')] 1. 1. i 1--(-)" "},.l'-(l +<'<),)(/'--./') (.,. - /' r J'
O - -- - ------ ---------------} ,r ./' 1.
j '.!. Cl}; [1 - (--) 2 ] .r [ (.r -.1') -1- (.r - ( -) .i')] . /' ,.
\\IWII' 11/1'11,1',111 lidllCl ..,Id(· or tl)(' "ho\'(' t\\·o ilJ('(Plctliti(':-' illC' (1)(' roofs uf cCjlléltion
(h·:n·
FIIJlIl ,t!HJ\C" tll!' {ollm\lIlg poillh 1 <111 IH' ('Ullc lud('c!:
'l'III' d~ Il''IIlÎc P/l'","'ll/I' ,tt f11lf,trol. I/f. 1:-' pl()IHJltiOIl,t! tu tll!' lo/':-,ioll,t! ~tifrll(,'o"
" IIJ i:-- ill\'('!'>l'" jJ!lljJl)JtiuII.d (u II\(' HJlIilJ(' of tl\(' di:--t.lIlcl' lH't\\l'(,1l tl\(' c\('!uc!ynélmir
Il'ltfIC' dl\(1 tll!' C(·!tf!C' O{lIla:-,s (ilS 1"11 1)(' ,-l'c'Il fWIll tl)(· 1/(.1: - .1')2 [,\l'(OI' in the t,,"o
i Il ('q Il, tI i t Î l ':-' ),
U! .
.tilt! tIlt' c Illldit iUII t lidt c/ /1111:-,1 1)(' Ilt)'>it i\'(' gi\'(':-,:
• -/J>O,
•
•
(' Il \ l' T /' U 1 il' \ H \.\ 11 J/{I ( . 1-' LI . 1 l 'J'If \.\' \ 1 ) .'i I,~'
01 ,
l l' •
, .... ; [(.1' t .r' -'! l ) /' 1 1 (./
F())' 1 1)(' (' P, pO"'III()Il"" d 1'1 (lI' Illl' (' <1 •
l
\..4.'; '"
1
( /'
.r( 1
r) , 1 )
,-
r) - ()
I~
((, .1)
Ti- FUI aIl rll'!lW'III) lali()s f11111l'1 i:- po..,..,d,j(· illllC''' (. Ill·..., 1"'1\\1'1'111111' 1' . .1 .!lId 1 !'
(j- If tllf' cl r. 1"" art or 1 hl' l·.g • IIl1t 1('1 1:-' plN.iIJll· ()lIh il
, ( r .r) ,. "'-, r ( 1 1 ~ ) ,
(II 1)
, 1 • , ,. .,.
6.2 The New Study
Sllmptioll:-'. III tll(' tlllrtly:-.i:-. pf(·:-'(·III(·d III 1111:-' "('1 Illill IIJ!':-'(' "'.<.lllIlpllllll" .1'1' "'/lIIJ\'I'd,
---- - - --- -- -- - ,----,- -
ITIIP .dg;lhl"ll J>fuII·dllll·I"!';IVI'IIIfJ d.'.lIl III '(1'1"/1"1), A
~Bj \1..,(' of "" .dgl·IJf:lIC illl.dy.,l., l'fI 0.,"1111 d III '!J'jJ' /l,II" ,\, II, 1-' . h'J\'dl lli,lI f"f J" (J IIIJlI'I J',
fp,>,dt., 111 t III" .,1 IIdy
•
•
;9
11I""jl/ll [10] dlld 1111 tldll',lJllir !lIn\' t.dJl!1,t!('d rlaf,l a\',[il,t1d(' fOI él :\;\(':\ (lIAfJOG
dlll,,!1 l:~:)l ,d, cl \1.,111 lllllldll'i ,,1 Il K,-) i" Il..,('<1 III ,uIdilioll. Il)(' ('ll'leltJ(Jll~ of Il]{)111i1l
dll' \vlitt(,,, il/tlll' IfllllllJl ,,1/ ('I.!!,"II\,t1ll1' pluld"lIl. ,'Ild ~{)h('d 1IIIllH'li(,t1I~ 'l'IIi" J('
IIIUV('''' 1.111' If'L,t III f 101101 Ild\'il/,fl, rl 1'1 (''1''('II( Y (lI,rI(',>( ('Il( (' III 01 dl'I lu hit\'(' n1l1 1('1 '1 ht'
fI,JI"'1 1"11111.1;11\ 1'" fllJfdllll'd Il'''III!!, rl tlclrllflulldl (/-q llll'thud,
G.2.1 JnCOlJJ}H'('ssihle Flow
II/ t hi.., ~t Ild.\' ,dl tl)(' 11111 t('1 "p('I'(b IJI(':-'('lIt('d UI )llu!t('d, al(' llo)1-dinwllsiollal by
d 1'.11 tOI 01 h,<'" Il IIWellh t Il,,t Il)(' Je',ll flut t('1 ~J)('('d ill Céll h 1 d ... e is plOpOI t irJllal
It) 1111' l.(Jl"i(lII<lllI,tllll,d flt'qll('lltY, u-\,. UI. the dYlléllllir pf(':-. ... I\](> aL flltlLn. fli' is
IHOp(Jlt i()llcll lu "'". Il)(' t(}I~I()IItli stilflw..,s. 'l'hi:-. agIe,,:, ",ith Ihe ](':-'111t of Pill(,s'
:-.llIdy
AI'l uldlll.l', lu Plllt",' 1(,:-'111",. IIJ(' d~ lloll1ic 1)l('S"l1W al Hllttel i:-. ill\('I:-.C'ly jJlopor
tioll,t1 lu 11J(' '!fI/air (JI III!' dl~t.IIl(,(' 11('1\\'('('11 Ill(' "lct:-.lic d\i:, ond tlH' ('('l1tl'(, uf ll1a'iS •
. 1' .. , (JI, I.II!' Il,,11('1 "Pt't'e! 1" ill\('I"I'''' PI0!J(Jltiull,d t(J t!J,t! di:-.lilllCl·, Now, lftht' f111ttcr
"P('('e! is pl(lII(·<1 \'('l''''IIS tlJ(' It'( iplllcctl uf tha! di:-.ldI1C<', 1;'1'0:' fOI ddf('!C'lJt locatiolls
uf Il)(' (·Iél:-,I il' a'\l:-', il, ('.tll 1)(' ~t'('ll 1 Ital Ihis is Ilot the' g<'IH'J'éll (ase', 'l'hl' J('''l1ltS for
I!lis (c1~(' dl(' !Jr('~('IItt-d ill Fig (i 1.
,\s 1 <Ill Il\' St 'l'II 1'1 (JIll 1 hl' fi,!!,11I ('. fUI ('la~tic axi!> positiolls alwad of th(' éH'lOd) Ilamic
(l'IIllt' (a < lUi), tIlt' (11I\!' i:- Ilot lillt'dl; titis rontléldict.s Pillcs' rC:-'1I1t. FOI ('lastic
d,i~ loc," iUlIs .Ift uf t hl' <ll'Iodyll<lmlc CI'IIt.r<', ('XCl'pt for ~lllélll ,.,tllI('S of 1/ Co, the
t III\'(' is Llirly lill!'a!. II. 1I11'alJ~ that., fur l'la:-,tic a,Îs po::.itiolls aft. of the' éll'l'Odynamic
('('1111'1" .IS IUllg i1:-, tht' n'III II' of ll1a~" is Ilot tuo l'dl' from the l'lastic axis, ~ay in the
• ClI.\PTEH (i. 1'.\U.U/Fï'IU(, Fl.l /TFH .\.\' \/) ."il.~
\'dlid in t hi ... 1',1-;('.
III iI1COnlpl(",..,illl(,!Iu\\', 11lt' <!l'IIJ(hlldllllC 1 ('lIlll' j.., Ii\l'd ,tt 1 Ill' (!lldill'i 111l1I,] 1/1.
d.C" olll~ 1 II!' pl)~il iOIl~ (JI' 1 Il!' l' cl dllt! I.g ..,1 1011 Id Il(' 1 Ildll)!.I'd ,dl)II.I~ 1 Ill' 1 111l1" hl\('.
The elastic aXIS 11/11 ad of t.he HPl'odyll;UllÎC Cl'llt rc
FUI thi~ ca..,I', "('IIM - ,Iudy A", t \te (' d, 1:-. (),~ b ,dll'dd of 1 II(' cl l'., i l'., Il
IOJlga ... IhC' (.g,lil's ahl'ad uftl\(,(,i\, JIU 1I11ttl'i IJ(IIII~ l'dl dlly l'II'qlll'lll,)' I,ilio Thj~
agJ'('<'s \\'ith t111' 1 Oll( III~ioll Ily Pilws fuI' t hi~ !>it Il,11 iOll,
III Fig. (j,2 fUI ~I'\'('I ili p{J~it i()l1~ of ('.g" in 1 Iti~ li,!!,IIJ(', d., IJ('fuJ(', ,l'" i~. t Ill' 111111
din1C'l1foiollal distan«(' (ill tl'llJl~ 01 ~1'llIi-IIt()ld) ul ('Idl(' 01 1II,I"'~ ,dt or l,lit' (·/<1 ... 111
axis, As t'ail he' S('('11 l'Will IIt(' li,!!,lll(', rOI (';l( It (,!!, jll)"'llillll, f1l1f.f('1 (H (1I1~ ,il ("'('IY
fr('qllt'Ilcy latio l)('j'ond il ('It"ill 1IIIIIilllllllJ \"t1l1C' Tlli~ i~ tIlt' ~;IIII(' 111'11<1\'1\1111 <I~
for diff('!('111 lu( ,tll<JlI~ of !'hl' «'111,1(' uf 111.1.,." Il (<III \)(' "('('11 iJl 1 11(' 11~',111" 1 Il.11. 1 lit'
ft m('é!ll~ that f111tit-r i~ IllOI(' po..,~il)l(' lu (J('t 111 111<111 wlt,tf, l'ilJ('~' .,t,lldy plI·dl l h
A~ the c.g, i~ ll1o\,('d fil 1 till'I' lH'hillll 1 III' il,C" dl «(il dillg t,o PiJl('''' ~>t,lIdy, 1111'11' Iflll~t,
('xi st flllttpr fUI ('V('IY wr • TIH' ~dlJlt· J('~lIlt I~ o!J1.til)('c! ill thj~ ~tlldy, III 1',"111'1.11. 1111'
!luttel sp(·t'd d(·( l'('<IS('S f](J1Il ib illitial V,tllJ(' fUI cl ')111iI1I w, fUld I('d( Iw,> il 11111111/111111,
•
•
•
SI
11/1'11 ..,Idll', 111111'd'dll!.!, !illl'dilv il'" 1!1f' fl('(jlll'll(~ 1.11101" lllrJ<·'I..,(·r1, TIl(' J(·~1I11.., fOi
1111'1 (d'.I· dll' /I!IIIII·d III Fig (1,1
','1)(' ('Ia~t il' aXIs 11ft of the ;\('l'odyn:tlJ1ic cent.re
FOI I!ll'" .... ,., liulIl\\() (d..,(·- .... llIdi(·~ hd\('!J('('1I (UJlSldl'l('d. ·'casr-.... llldy H" ,tIld "(([sr-
• ,.( '(/.~r - ~/II ri 1/ If'
III tlli~ ("~{' 1 II<' I·.d i .... ,JI Ill<' lllid-I llUl d (rl -:: O.). Ill'Ililld Il)(' fI.C As lung a~ Il)(' c.g
i~ lOI ,dl·d ,tIlC'dd uf Ill<' 1'.;1., dl 1 (Jldlllg 10 )111](':-.' ~1 lIdy, )l0 0111 t('r \\'Ill 01'1'111 fo!' any
\',dw' (Jf t ... ·). FOI 1,111' 1 d..,l· ~I IIdy 1II1d('1 «()II~id('1 ill IOn. 11111 tCI' (,Ill 1)(' oIJtailJ('d, aL a
\, 1'1 l "ill l,Ill.!!", ,,1' w) , fOI ,tl!local JOIl:-- (JI' 1)](' (' g. Jwhilld the éLC.; th(' )(~:-.u)ts dl(' ~ho\\'n
III Fig. lUi. A .... 1 <III )w ~('('II flOI!l llJ(' f1glllC', nnl 1<'1' (·'\i~b IIp to a ('el'tain valuC' of W r ,
dl'J)('lIdillp, 011 1.11l' pu.., Il iOIl of (' g. A" 1 II!' ( g. Illu\ ('~ c1o,-;('}' to c.a., this Wr",,", lH'j'ond
\ w'lich !lO fi lit 1.1" (JI (III", II('I ()IlW:-' gl(',tI('I.
Now, ,,~ t lJ(' (.p,. p()~il iOIl i~ IIlu\'(,d lwllind tlH' ('.a .. arcol'dillg to Pilles' sLudy,
11111(' IIIIISI. 1)(' cl 11Id'\illllllll W) ))('.\())ld \\'hi('h no HlItter OCClII'S, Lhi::, wrmar df'pcnds Oll
1.111 po~ili()11 (JI' tllC' ('.,\. illld cg., ,lS \\'(·11 ,lS l',,, gi\'C'll 1)) eqllation (6.:3).
'Ill' 1(,:-'111", ()hlalll<'d fWIIl this ~tlldy é11(' ..,llOwn in Fig. 6.6. As can be f,CCI1
~1111 ,Iltl'I(' I·'\I ... I~ 111Itt(,1 f()1 ,dl \',!llll'''' of w). \\hich ('ontladids PIll('S' 1(''''lIlt fOI this
~illl 1011 'l'Ill' IIl1tl('1 ~IH,('d i\l<ï('a~('s lilll'MI)' with wr ' ,lIld él~ t~le ('.g. 1110\,('''' furUlc!'
fl(lll tllC' l' .1., tlll' 1'111\(' ~I(lp(' ill<ï'(·a,,('s. i.C'., thl' fllltt(,1' :-.p('('d illcrcasC's morC' l'élpidly
d~ t 1 • flt'qlll'Il!',\' Idti() 111l1l·,I:-'('S.
f L
J r. 1 ,. ~
• ('II \Pl'f-.'H 6, Il,-\U \'\W/'Uf(' Fil J'TF/{ \.\ \US1S
• "('(/,~I - ~t ur/y ('H
III Ihis ('.1,,('-:-.11111,\ tlll' (',,1. 1" i11 Ihl' Ihll'(' <ill.llll'i Ihllld, fi
\\\\lldl1H' III l'ill''''' "
"llId:.', ilJ titi" "illlallOlI 111t'11' IIlII~1 ])(' IlU 111111t'l ,If dll\ rlt'qllt'ilt \ Idlhl Hill, Il 1"
f(llIlId lli,d, t\lIlII,II)' lu Pilll·,,'ll'~ldt,lll\lll'l tdllllt'tll,ldllll'd ,d t'\t'I," flt'qlll'l1l\ Idtil',
cilcb c.g. po .... il iOIl, Illd Lt'I ~p('t,d ill( 1 t'dSt'~ //l1t {//'/.I1 \\ il Il "..',. 'l'III' 1 III \t' "llIl'(' 1" "i,l'.h,
('~P('( \,dl:-' fui ( ~, }H' .... II iUIl'" f.1\ rl\llll \ 1\1' l',,\. ,\~ 1 Ill' t,l.'" IIIU\(· .... \ \\1"1'1 1 t, t'.1 , 1 hi~
~Iop(' d(·( I('<I~(''', il'., 111l' Il litt ('1 "1)('('" ill( I('d:-'I'~ \\'11.11 cv" Il'ss Idpi.!ly, :\\-,,,, II i" ',t'I'II
thal, C\~ \1)(' t' ,\. lI1O\'('~ 1'111111(·\ 1H'llilld 11)(' cl (',1111' lilllg('111 fll'«III'IICY l,di" ,d, wlli(11
ll11tt('r (H'CIII:; ('xt('I](I~ flOl1l 10\\ \'ilh\(·~ 1." hlglll'i \,,1111(,:',
Thl' !l':'lIlts fur th(' CclS(' \\ 11l'11' t 1](' c,g. \~ 1)('llllId t 1)(' (' ,\ .11 (' ~h()\\'11 ill FI,I~. fi,,";,
by Pill("';' slwly, ",11('1(' 1.111' \<111\(' of U-'"",,, i~ ,l',1\1'1\ J,y l'qll,dillll (Ii:q '\" (.III II!' ',('('11
from tilt' fig,1l1(', a:-- th(' (.p,.IIlU\·(· ... fil 1 1111'1 (o\V.lId:. Ilu' tlcltlillg('d~(·, i,I',IlIIl'I':'; Illlt.llt'l'
I)('hilld the' ('.a., thl' w, 1)(,C<JlI](,~ SIII(lJJ(·I'. TI){' \'illlI(,:' IIf W, ((JII('~'J)/jlldiIlU. t.() "I(I~ '1111' r
~(,v('r<ll Î.g. po:.i1.ion:. o!llilil]('d flll!l1 thi.., ~\'lIdy, .11(' (()J1IJ1.1I(·d wil.ll Ih(I'-,(' p\l'di/ I/·I!
•
•
•
( '[ H vn;/( rj f):\ If l' .\[ f'[ [( 1(' VI, ("1 'J'l,JI A.\' \ 1.) .'-,'IS'
(L2.2 'J'rallson ic Flow
'l'II 1,01\'(' d Iwl1.(·1 IIIId('I..,1 clI/dill).!, (JI' 11/1' 11111.11'1 111('( hrlllÏ:""l1 in 1 1 dll~()IIi( nu\\' l('glllH"',
tl".., plol,I('III 1.., fil!'>1 'IIIrlly!.(·d i" 1 I,i.., !,>('( liOIJ, <1"rI till'Il él IIllltel élllélly<,ic.; !'>imilcu tü
\\'1,,11, \\'cI!'> d(JI/c' ilJ IIIC' JII(·viol'" "'('( II()II, ",dl Ill' pl ('!'>('IlI('d
V/nU,·)' M"c1'fI"i:-rn /II T"allsollic Flow
Tu ill\(·.,tlg,d,(· tl/(' fllltt('1 IJl(·(I',lld'>"1 ill Ildn",ullir flow, t",ü (élS{·-st.1IdiC's ]Iélve
1)('('11 C (1/1é>ld('I('c! ""t! ""dl\ /'c·(/. [II tl/(' IIIll h ('f\:-,('!'>. tilt' a('lut!YllilJl1IC data of a NACA
(i !:\(J()(j cli, f()" ,d, il ~tl( Ir 111111")('1 (JI (J S!), \Vith t",o dq?,J(·{'c.; of fr('{·dom, in p/1Illgl' and
pil (Ir, Ild\'(' 1)('(,,, 1I0.,('t! '1'//(':-'(' d,t! ri hd\'(' h{'('1l "d,{'1l fWIll L('(· [:25J.
1" "('(J.~(-~/lIdy :1" tilt' (·lrI~tl( rI\i.., i.., dt tll(' h'cHlillg ('dge, cllld 111<' C!'lltrc of rnass
lu( rltc-c! Ils (11I"d 1)('1,,"d It, i C' . Cl = -1 0 and .1'" = 0 2f5. The cig(,!l\'a!tIPs of
tlll' ~y:,1.(·111 cll(' «J1I1PIJtC·t! ,d. (',u h "'J)('('d 11~lng thl' p-k m<'l!Jo(1. The Ic,d palt of
(',t( Ir ('lg('IJ\'db/(' (JI tl\(' "y.,t('1ll j.., illl Îlldi(',dÎOIl uf 1/1<' dclll1pillg in that mode, and
tilt' ÎlIlrlgÎlldl' polIt 1 ('JII (''''('111:-, t /J(' frl·q1l(·lle)'. 'l'Il(' 1 ('éd illld il1ldginilly paIl,::. of the
('1,1~('1I","1I('~ (If 1 1)(' I\\u J110des a.., cl l'III/( t iUIl of ~p('('d ill!' ~bo\\'1l in Fig. () 10 and
l'\I!,. (j.ll, I('~I)('( tn'l'ly. ,\bo ShU\\'1l ill thl' ligUles al(, tlJ(' !'('al and imctgillétl'Y parts
III' tilt' ('ig('I1\"""(,~ (jld,lIl)!'d wlwll ('drh dq:!,I('('-ul-fll't'doll1 is cOll:-id<'l'l'd ill a ~illgle
,I('gl('" uf-f'I'('d()111 l'lOhlclll. :\~ ('<III 1)(' :-,('('11 l'Will the figllJ('~, thl' damping in the
phlllgill,J!, 11111<11' 1)('( ()11l('S ll('g,tti\'(' ct! tlJ(' nutt{'l :-,pl'('rl. ft (clll "Iso \)(' S('('n that, the
fl<'q\lt'Il( il'~ of t 1)(· twu llIud(·.., ,lIt' Ilot {'\('Il (Io~e to ('Mh otlH'1' al, the flatter speed.
III 1',11 t, thc., :-'1'I',II;d(' IlIUIt', cl" t hl' ~p('(·d ilj('J('tl~(':- Tlw fl('qlH'ncy in t.he phlllging
IlHldl' (thl' 1110<11' whilh hC'('UlIII':-' llll~tclhlc) Il'maillS tlw ~dnl(> at <L1l tl1<' sp('eds, and
tIlt' fll'qu('llry of tlll' pitdl lI10de ÎII( 1 (',ISI':- \Vith the "p('cd .
•
•
(,lI.\PTFUti P\H\.\IfJ'U/('FI.l'J'IFH \.\ \f.)S/S ~I
:\U\\', wll!'1! llllh ul!e of Iltl' dt'grl'I'" IIf fll'l'dllill I~ 11I1l"idl'll'd ill " ""ïl.lI,11t' "II'''~'''I~,
i,e', 1.\\U ()IIl'-t1('gll'l'-uf-flt'l't1ulll pllIhll'l!l" li ])OF). Il 10111 hl' ',,'l'II 1II,tI tlll' fll'qlll'III \
ill ('a( It IllU"I' ,tllllll" i" ,1I11lO,,1 (1)(' "011111' .1'" ,It,t! III t Ill' (\\11 dq!,Il'I' of fll'I'''"111 1'1'lld"lll
(:mOF): ~('(' FI,!:!. (l.ll Bllt, cI~ (.III 1)(, "'('1'11 IIUIII Fig li 10, titI' .JdIIIPIII,!', Il'11101111''
jlo"it j\(, al ,dl "'Pl'I'(!-, il! ('d( h of tl\(' 1 \)() F pitll"I'III~
TIII'Il'fol<" h~ Inokillg ,tI tl\(' 1 1 H.;J t1b (Jf II\!' '!J)OF jlitlldl'III If 1IIl,l',It! ~I'I'III th,t! Iltl'
111\((('1' UII'>('( i~ d1\(' III 11\1' plllllgillg dl',!~I!'" uf Ilt'l,dtJllI, ''id\' 'l'Ill' 1.,. k 1I1 01 1""'1111'11' \'
(oal('"n'II('(' Ill<l.\ ,d~o Il'illflJl('(' t I\(' id.,.! H1I1,. tlll"HII'IIIIp, ('cl' Il .l'','~II'I' of /11'1'''"111
:-,('pdrall'Iy ill (\ru IDOF 1'1tJ1,JI'III"', ~llu\\" tllcll.llll'" 1" 1101 11\('(01:-'(', Il''ldll~'''III1'',,III\(,
sy~t('111 III thi~ 1'<1"(' du(':-, 1101. 1II1d('rlak(' Il lit It-I ,II dll\ '>1'('( . .1
Th,' "'''Ill'' J'('~1\lt~ "" fllr Ill!' pl(,\'i()l1~ (c1"'I' c1f(' ()IJI dill .. d IIlI "((/,.,( - ... IIII/Y/I", III will,"
tilt' C'la..,til c1xi:-, i.., dl tll<' qlld/'I('I-(hllid cUld Ih,' 1('111.11' (Jf 1I1c1~'" is 1/'i'I IlllIlt! 1H'llilld
Ît, ic>" (l = -0,·) ,lIld ,/'" = O~I. 'l'If(' 1 l ''''1 dl'> lUI Illi~ 1;1',1' 011,' :-.IIU\\'II III Fig (; I~
éllld Fig, (j I:l, III 1 hi.., ca:-.(' "I,,(). 1111' I,',d "lit! IlIld,t',llldly jlctll.., (JI' 1111' Iwo ('I,L',I'II\',dll(':'
ha\!' J1lUj('OI' 1(,;-,:-, tif(' '>clllH' ((('lit! \\'ith :-'I}('(·d, d~ tli 111('111('\'1011'> (d..,(' ..,llId,\' ,\1',0111('
f-.tlllW rUllclll~iuJl (élll 1)(' cOllclllCI('d fl(J1I1 1111' J('~ldb (Jf Illi~ 1 d:-'I'.
Parametric FJutter AnaJysis
'l'Il(' f-.tlllH' :-'Ult of )hlldllll'tlic !lull('1 dlldly~i~ dOl II' fUI illl(Jllljll('..,:-.il,J(· !ll/W, \VoI:-'
<Ibo done in th(' t Idllo,Olli .. now 1 ('gill\(, (UI cl .\JA('A (j I,\(JI)(, ctidod ,11 cl l'V!.I( 11111111"11'1
of 0.85 ll<;illg t 1](' 11I1"'\('clfly ill'llJd,} Ilctlllll (udlJt i('II1.!"> [L'i].
In l!Je tlclll"OIlÎC now I('~ill](, 1 1 If' cI('IIJdY"dllli( (('1111(' pIJ..,i1.ioll j" 111/\ ,t "1)('( dil
point, {,\('Il at a fix('d ;\,1<11'11 1llIlIdwI', tlll'l<'foJ(', il. i.., 1101 1 ('ILlildy 1(11)WII \\·(·1,/11'1 l,If('
(,Iastir axÎf-. is alJ('(jel of, UI, IH'hilld, 1111' ,H'lodyllrl/lliC (C'II1.I(', 110 IIlrst.t('1 wlwH'll1!'
•
•
(·ld·.II( .1 .. 1', i ... JII( d!forl ·/IIf'I,·rul!'. t Il' Ir I(}III'I jJ{J~tll()lI~ (JI 1 II(' '\{'lodJ IIdllll! 1 (,IIUt'.
(·ld·,II( d·i .... dlld 1111' Il'lrll(' III 1 Iltl " .... (till JI(JI 1)(' (''\dllly 1I1dJ>J)('d iJlI() 1 1 If' rldff'II'/It
(,,l'~',lIlil''' ill 1'1111· ... ' ',IIIrly Hill. III'" ,Idl i"I'·I("tlll.e, I() dl) Ill<' "ilJlH' ill\·(·:-,tigdlioll.
111:,1 I,y 1 Il,, Il.l',III!!, 1111' Il), ,d IIlll, 01 t III' ,·1".,1 i( d '\1:-' tllld 1111' , ('llll(' of IIld":-'.
JII 1.111', , d:-'(', t 1](' :'tllll<' 111/1'(' (i!"('-,llIdi(''-, ;1:' ill iIH (JIIIJII('""illk f1uw, 1r<l\'C b(,(,11
('11I'oId('/I'd ~II)/(' :-'J)('( rfll ,t1ly, titi' (·1,,:,111 d'\i.., 1'" ]JlcI( (·rI ,II, (1 1.1-1 hord, tJj(' mid-l'hord,
dlili Ill!' 1II1""l]lIdlt"1 ,llold. f('''J)('( lin·ly. ,lll,] in (';lIh (d!">!'. t./H' «·IJt.J(' of l1l;1~:-' is
P/.J( "ri ,d dilr,'I"1I1 I(J( ,rliOIl:-' ,rlolI.!!, 1 Ill" lruI,] li,l!'
• ,,( '(I.~( - .,llId" /1 .,
'/'111' 1{",IIII:-. lUI Illi" , d,l' :'!,l/Ily, ill \\ II il Ir the ('.d 1:' IOl'il!,!',] ill n ]!)-ch()rd, aIC' :-,lwwll
III Fil!, (i.1 1. A:-, :-.II()\\'II in 1 Il!' liglll ('. IW Illdtt('1 whrtf tlIC' f('lat ive po~it ion of t Il(' (',il,
.I"d { g, i:-., IIl1lt('1 ()( (III, fUI ('\'{'I) u..',. dnd III ail CcI"('S thl' IIl1tt(" ~p('(,d has the saJl1C'
111'lId \'('l'''I/'' tlll' f, ('ql/I'II!',\' Irdi(),
\\'111'11 t11!' CI!, IS lo!',rl ('d rrl/('dd of t II<' (',il. ;1< cordillg tu Pilll':" st llrly fOI incoIll-
1)f(·:-,~d,J,'lJu\\'. th('I(' 11/11,1 lIUt. ('\/:-,1 /l1IIlI·l. \\I1I'II'a:-.. ill Ihi:-. CtI:-'(' fol' ('\'('IY w, fJ1\ttt'r
i ... oltldilll'd, 'l'III' Sdlll!' 1/I'1Id is ,rI"o ol,:-,I'l'\'(,c! in the (,i1~(, wl}('J(' tir" cg. is clft of the
(' d '1'11<' fi 1/1 II'/' ~I)(·('d Illdillly il/('/('dS(,~ i\:-, 1./](' [1'<'quC'llcy ratio is inl'l'('é\s('c!, a/thoug,h
fOI :-'111111' l'.L'. IHI:-.it iUIl", il ~Ij()\\':-' ,1 d('('/('a:-,(' fol' ~/)]élll [l'l'ql/(,llcy latios .
• "( 'o.,t -,,1 ut/'I If"
III "(,0,,1-,,111,1.'1 /r', \\'I!l'/'l' tlH' (',il. i:-, pla«('d rIt tl)(' mid-chold, thC' 1whaviour is slightly
dif]'t'I('1I1 f,u'll Ill<' !Jt<·\iOIlS l'(l:-'('-Sllldy, 'l'III' 1(':-'lIlls al'(' sho\\'n in Fig,. () 15.
F\JI II/(', ,!!,. 1'(I~lt jUil, fOI \\ ,\1'(1 (If t Il<' (',il" 110 J('a"Ollil],k {lut.!.('1' SpCf'rI can be found,
Tlti:-. i~ II('{ dll~(' 1 h,lI IIlltkl (1('( Ill~ ,tt il Il·dll('(·d fl'l'qlll'IlC)' oubide tll(' allowablC' lange
l'li 1 1Ising t II!' d\diLt/I'l' ''l'IlH],' /lallll(' data t hat is 0,025 ::; l,' :::; 0,.1,
•
•
('//.\1' 1 FH li. IJ.\H \ \",' J'H/(' f.'Il Il FU \.\ \1) .-"ï.';
\\'hl'Il t II\' 1 ,!', 1'" 11l()\('(! 11\·1111111 1 Ill' l' Il .1 l' • ptt"'lt I\l' \.11,1(· ... III l',. Il 1'" Il'IIIIt! l ""t
1 Ill' fll1t tel "(H'('c1 II\! '(·d"I· ... IlIft·.l1 h \\lt h cc', • 011111 tlll'i (. ,.111 Il,.1 !)(' ("'"1.1 ""\ 1'1'111'1
ur 10\\," lill1lt tlll It
:\~ thl' (' t',. 1:-' 1l1()\I·d 1111tlll'i 1"'1111\!t tlJt' l' d . tlll' ... 1"11(' (lltlt"llllttl'l "1'"'.1 \,'1',"'"
tIlt' fll'qlll'I\!) 1.l1io 11111(·<\ ... (· .... Illlt . .1'" 1.111 III' "1','11 (1""1 tIlt' 11\',1111" tlll' I.Ik 11111,,·.
illl'i (·a:-.(" d"1 1"<\",·:- .
• ,,( 'a .... 1 - .... Iudy ('"
The J(':-'I\I1" fUll hi" l ""'l' ... , Il,h d\(' ... hll\\ II III I,\!.', (, 1 (i III 1 Il''' , ".,1' 11\1' ('.\ l', l,,, .dl·"
dt tl1t' tltl'·I·qllillll'I-,hllld. '1111' Ihlttl'I "'pI'I'c1 '",('1111'. I() h.I\I·., '1111".11 tl,'11I1 \\Ith
fn'ql\t'IJ<'y l.tlÎO fOl ,dl l ,l!, l',, ... II)UIl .... th"II"II"1 ''l',·,'d 111111·01·1· ... III .1 '1l1dt!I.t1Jt 1I1011111t'1
\\'ith fl('qlll'1l1 ~ J.l1io Fllllll'l ,,(w,'c1 1"11 1111(1)('1 .tli III.tI"c1 dl 11"'1lh'lll \ 100li,,·, [',l''oIt''1
thdll él «'1 t dill ,.11111' l'III ('dl Ir 1 .e. !lI,"'I! 11111 'l'Il<' \('oI"'llIl :-'1'1'111'" II) II(' 1 lit' col1111' 01'" III
"c</.-;c- ... fudy Ir',
]·']I1(II·\' :-'1'1'('c1 1 ... lrlp,ll!'l II!! tlll' ( g !J() ... itilJlI" fl"IItI'1 /111111 tll!' ".1, TI" ... l'.tll\l' 1"1
both (ël:-'('S \\'111'11' t!ll' l'.,l',. 1:-' 1111 "l('d "Ij('dd (JI. IJI. IlI'blllcl. IltI' 1'"
COllÎhl:-,ioll~ Ol! tlll' \(',,1111:-. (JIll dilll'c1 ill 1 hi ... 1 hdpl(" wtl! Ill' l'II ·.'·1I11·d III t III' lolle.\, 1I1~·.
chapt<or.
•
•
('/I:1/)'I/',/( ri /' \J( \ \/L'IIU(' ff.{!'1 '1 VI( \.Y \U ,'-)1 S'
10
.... Q)
~ ::::J iL G
lU C o cn fii 4
E Ü , c o Z 2
o
a:::-O,7 u
a =-=;..0,5 a~-O.4
(J
(]~-O ° . -(]=-=0,5 •
p:-50 ra- 05 W,= 02
o
o o
~ ... ~ ..... +-.
••••••• ... ---. - ... ------ +--------------- ... ----------------------------. - --- -. ------- ---------- .-------- -----------
1 2
_ L ______ 1 ______ 1- ___ _ -l ___ ..l ____ L.._
4 6 8 10 12 14
Rociprocal of the Mass Centre Position, 1/Xa
Fig1llc (i,]: Flllttcr "P(·(·t! \(']"US !j.r", for differf'lIt locations of the elastic aXIs .
•
•
('11.\1' J'En (j fJ.\H \ \IFI lue FI 1'1 l'FH .\S \U 1,1~'
10
:> 8
"0 Q) Q) a. (j) 6
, ' : : , ' , ' , , , , , , , ' i' - , , , ' i'
1 : , ' : : , . , , , ,
o ------- -- -- _.J. o 05
Ji = 1 00 a-.o·07 r.--oO 1
Frequency Ratio, W r
J
15
... .... . •
u 'J
... •
"
... •
"
 1\
()
'1 Il (1 (1 (1 (1 '
2
l'Jastic axis at 0 15-d10I d, alll'ad of 111(' ;u'ludY"dlllir (t'II1.II', ;11111 LI 1(' (1'111 J(' of J11;1'.~
at diffcJC'lIt loratiolls 1L(~t\\'C'C'11 t!Jf'Ill .
•
•
1 ?
J ;: o ~ 08 a: [)' ~ OC :J 0" Q)
IL 04
c ~
02
o o
Il
005
11=100 a=-0.7 r.=0.1
01 015
Mass Centre Position, Xa 02
FiglllC' () :l: i\lillill1l\lI1 flt'qul'lIcy I<ltio fol' which fllltt(,1' is obtained, WrUIIII ' fol' (IiITer-
t'Ill lorat iOlls of t 11\' tI'lltl (' of IllclSS, .I\~. Compal'iSUll bct\\'l'f'll the IC5ttltS obtained
l'Will l'i Ill'S , :-,tlldy ,\IIC! thi~ 'itudy. Il1compl'C':-,.,iblc flo\\'. The ('Iilstic axis at 0.15-
1 IlIll d, ,1Ill'<\lluf (Ill' ,\('I ùdYll<llnir Cl'Ilt IC" <111<1 the ('('111 1 (' of ma<,<; aL diffc! l'lit locations
•
•
CH:\ PïTH (J. P. \ H \.H E ru /( . FU' ]'J FI? \.\' \ 1 ) ,,,,'IS
14 -
-6 ~ 10 0-
C/)
E 6
"9 c o Z
4 -
11"= 1 oC' a=-Q 7
,\~ ~ ,"" il + ·w; +
f~~ 1 ••
f.} A ~ • fr ,- + •• ~-R + ••
 Il •• • .. ",,~... ~.
,vI"! +. 1>.'" l'Jo -M. t. 1>.
ni 1 •• .1>. ô"". ... ~'... •• • AI>. ({-Q, • _ .... • _. ~... I>. ._ à'..,.. 1>.. •• KY.-  .-n9~..· ....
~'T •• • Â ••
. .ê~ ....... .Â' ••• If Via··· ... ~ .... ' .. -. df!! ... ,k •• ,- ..
2 '.a..... ' ...... o t_ -------- -----
o ___ J
05 Frequency Ratio, Wr
J
1 5
l'la~tic axis cÜ 0,15·( hOId, allf'ad uf the' df'/fJ(!Ylld/llic «'1111<', ;tllt! f,)w « 1111<' I)f IIld',' •
•
•
>-I? -0 ID ID 10 a.
en
1:.
Il
: . .. <:>
• · · ·
1'= 1 00. a=OO r.=-0.1
X.r -dl ~5 X..=,§ 35
x.r-8 25
xa=-O 20 --+-X-r;f !5 Xa,,:j.!O xu=,O 05 . -X~:J_~ . . , . .
6 : 1 Â : g 1::. 9!~! : 'i' " , / " c. 1::. 0' .. fo }I o Cl 0 i . i " z 10'''' "
2 • ...., A-j.f',,/ •• - • .A~ ~.-- ~~.IJ h·~·. iP:, ~Aff. ............ • •.• ..Et
.A.u.Lu.liit.'ii.aJI * ••• .' - .' {)' ~"'\!r' . dl u· n---o---n---O---{3---0·_-Q---Q Er
o o
__ J __
02
_ ____ L _________ ---1 __ -
04 06 08
Frequency Ratio, Wr
. , I;J . , .
91
Figllre (i 5: Flllt!('r ~»l'('d, V" \l'l'HlS frcC!lIl'IlCY ratio, (Vr • Incompressible flo\\'. The
(,1",,1 ie ,,'\is ilf f Ill' Illid ,hord, IJl'hind the élt'I'adynamie ('cntIe, and the CC'11!H' of mass
•
•
ClI. \/1 J'EH li J> \ li UI/'" 1 If/(' FU'li'/·.'H . \.\, \ 1 ) ,~ï.~'
14
_ 12 > '0
~ 10 0..
en .... Q) ~ :J u.. E :a
1
8 -
6 -
C o 4 Z
X.--B 94
x._~g~2
x.--g 28
x.-'~ ?2 Ya==~ ?8 X.'-= J 90
w- 100,
6--0 ° r.",O,1
1
4 5
clastic axis at tl\(' Illid-(holcl, ]wllilld the ,!l'lorlj IJdll,i( «'Id 1(', .!1Ir1 t III' «'Id 1(' of IIl,t"'"
•
•
( '/1. \ V) VU rj fJ,,\ lU ,\ 11~ TIU (' FI. 1 ; j"j'/:'H A N .. \ L ), ,)'IS
10
::> 8
U III III 0.
Cf) 6 .... III ~ ::1 ü:
S 4
'0 , C 0 z
2
o o
1 6.
,/ li>
, ~
~
, l ,
, 1 1
P 1 , 1
1
~ , , , li ,0 , , • , , J~
• ,:+' .' 4' C), ' À -
t 1-'- • •
, , , , ô
, , çi
#'
, , , , ,
1 , , , /' • , , •
• .' , )1 ~' ,
À .' ,/-.- ./
,If
- ,1'-~-.------- - ----,-------,----, '
if JI , 1
1
,-
/
JI' • Ja'
r
.,,/1{ Jl=100, a==-O 5 r.=O,1
X:;i~82l x'==;!l,74 x.=-O 50
--0 -Xa=-O 34 --*--x.=:~ _18
xa::i,_10 x.=-O 02 --e-
1 _ _ _ _ _____ L _ _______ .1. __ _ __ --1-__ _
05 1 1,5 2 25
Frequency Ratio, Wr
Figul'C G,ï: Fluttcl' :--p('('d, l'J, \'(,I::.IIS fJ'l'quC'llc)' J'atio, w,. Inco1l1pI('s~iblC' flow. The
t'liI..,1 ie "\i,, al 1 !te 1 hrc(' qllal ter-l'hord, hehilld the élC'lOdynélmic cenLIC', and t hc ccutre
•
•
(,1/:\jJ'j'FH li P\H.L\lL'J'UJ(' l-'LUï l'EH AS \U.I..;fS
4
a o
L _
02
-1 1 , 1
_ ____ L _
0.4
Frequency Ratio, Wr
Jl-100 a- 05 f. -01
Il fJ fJ
1
06
x .. tq02
x. ,~OG x. 0.'0
()
x. 9 ~8 )( .. ~ 26
x" ~38
)( .. ~ ~O
1
08
t'lastic axis at the thl('('-qllétrkr-(hOld, \)l'hilld tl\l' ,j('rodYlIdll\i{ {('IlI.II', illld tll\' ({'IIII('
•
•
( '/1 \ VI "-Ii fj 1':\ /(. \.\/1';1 /( f(' 1 1./ "j'T/~ïf . \.\'. \/, ) 'SIS
07
1 0 fi ... != 0 ~ 05 Cl: () ai 04 ::J 0-C) ....
LL 03 -
~ ~
02
01 a
~ ,
, ,
J
0.1
JI= 1 00. 8=0.5 r.=0.1
--A. -----a~_
-- - --A- ---- -~--- - -6 - --_. 1::. _____ 1::.- ____ -8- -----8- ___ _
_ _ J_ _______ --'---- J __________ _
0.2 03 0.4 0.5
Mass Centre Location, Xa
1·'i~~1l Il' (i 9: 1\ Llxill1lll11 fll'(!1H'l!cy 1,11 io fOl which flutter is obtaiIlf'd, W rmaz ' for different
lUI .111U1IS of tlH' Cl'Ilt 1(' uf ma<;s, .l'". Compal'ison \JC'twecn t.he results obtailwd from
l'illl's' ~tlldy c1lld t bis ~tudy. IllClllllpl ('!>~ih)e nO\\!, The e)a:-.tic axis at the tlu'ee-
<[lldllcl dlUld, !)('billd tltt> ,\l'loci) nilll11C (,(,Iltl'<', and the ((,Illl'e of maf:>S at diffclcnt
IlH rlt.iOIlS ))('hillll the (').t:-.tic axis,
• CI/,\PTFU(i Il'/? \\/FJ'U/(, Fl.l:'j"n·.ï? \.\ \I)SI."·
-----------
M 085
0 - -NACA 641\006 nltrotl
. -Il -- - () illl 11 Il Il
0- -- 0 0-" u- - .{1 - 0
1\ Il ~O. nl·t Ilhn~1 (?DOr) ID 0 r.· 05 P,teh ~\nor} ::l 0 ·10 ct! 0
fi f3ûndII1V, (1 ()lW) >
(01) 0 w, 02 c: -ID 0 x. 025
Pltell ,1IJ0I ) en ill
0 n - ' .. n 0 ....... _- ......... "'!-----
t 0
ct! (02)
'~"'" >1----- t)
a.. .~, ... --" ...... Il
(ij , '. ;. .. .)
'c-, "01- •• ID .... a: A '. 6---' ..... 1\ ..
' . . (\.. .1\
(03) 6. • 1\
t'\ . ___ 1 J J J 1 t'\
2 3 4 5 6 7 8
Non-dirn. Speed, V
sonic flo\\', a N ACA 6'I.\OOG di, fuil aL M == O.X.rJ. CUIIlj)cl/ i"oll 1)('l,w('('1I III!' (d' f", wil li
~DOF in plllllgillg alld pil( h, 1 DOl" in plllnging, rlild 1 DOl" ilJ pil (h .
•
•
•
35
3 Q) ::l ro > 25 c Q) 0>
LÜ '0 t ro
2
l3ündin~J (2DOF)
Plteh ~I?0F)
l3ündin~-, (1 DOr)
!Jltell ~11?0r)
a. 15 -~ ro c cr, ro E
05
(J
o _ 1 ___ _
2 3
!Ji
M-,O 85 NACA 64A006 8ufoil
Il -50
'u -05 a ·1 0
--13 --B---S-o 0 0 0 0
--0-------0--- 0 -----0-- ---08--1 _______ 1 _______ -1 ________ J __
4 5 6 7 8 Non-dim. Speed, V
'·'igulI' tl.ll: 111l"gÎ~~dI'y J!~~ of the C'Îgl'llValuC's vrISUS llon-dÎmeIl!'>Îona,l spccd, V.
'l'tdll:-'OIlÎC no\\'. li NA(',\ GI:\006 ailfoil at M = 0.85. Comparison bct\\'ceI1 the cases
\\ il Il L))OF in phtll,l!.ing alld pitch, 1 DOl" in plllllging. and 1 DOF in pitch .
•
•
CIl. \ P'fJ:.'U li l' \U:\.\ 1 L' 1 lUI ' F/.lii l''}? . \.\ \ 1. ),'-,'[8
ID ::l
~ C ID 0)
li ë t
0.1
0
~ (01) ïü ID a:
(0 2)
BendlntIJ (?DOF) M-oO 85 - NACA 64AOOG allioll
Pltch ~?Ç>OF) W- 5O Bendln~ (1 DOF) r.=-O 5
Pltch \1 Ç>OF) a ~·o 5 w, ~O 2 x,'-'"025
-- 0 0 D {]- D - {] () - n
0 0 0
.... -.... -.*_ ... _.- .. - - •....... -........
. .& ... A ... r:,. . . " . t, ' .. Ô' 1\. .. {\
______ . 1
2 25
!J
'. 1\.
1 J tI
o
• A. !\
1
3
Non·dim, Spced, V
{J /1
1\
IIII-U
1
35
Figure 6 l~: g('a~J?_(.~~ of tlll' ('igt'IJV,dll{'~ \('1:-'11.., IIIJIl·dilll('II',)(}IUd ',pl'('d, Il, '1'1,,11
SUI1Îc fluw, a ~ACA G,I,\(JOG ;111 fud al, A1 .: o.~!), CUlJlpdl i',ClIl Iwf.I\'{'(·JI t 1)1' f d',(", \', :111
~])OF in plllngillg "lId pite h, J UOF in pllll1gillg, dllcl J /)01" ill pitll!.
•
•
( '11. \ 1 J '1 1,; H (i 1 J \ /{ , \.\ 1 J-. 'j j{/ (' V f.( ; T J U f . \ .\'. \ L) .1.; 1 S
Cl) :J
2
-cil 1 5 > c Cl)
Ul üJ 'f-o t (1j
0-
~ (1j C
g> 05
.s
o 2
l3ondin~j (?DOF)
Plteh X'l?0F)
13ondlrl\J.J (1 DOF)
Pltc.11 ~1 DOF)
(t 1\ 1 ~ ..
M--OBS NACA 64AOOG Blrfal!
Il 50 ru 0 S a -·0 S w.--02 x.--9 25 1> _ /!. -1> 1>- ù ---/!. - ___ Â_:.----I!:-:-{l--A--~-- -
A ~ _______ ; ____ -------- .j.--------
8 {J D --Cl _g ____ o_-_G-o--a---o-D---fl
O n fl {} {] 1.1 {J-- - - 0 ---
_ 1 L____ _ _________ .-1 __
25 3 3.5
Non-dirn. Speed, V
Figult' () 1 ;~: II1Idgilldl'y pé~l~ of the eigc'I\\'dllles V('I'SUS llon-dil11CllSional specd, li.
TldmOllir Huw, ,\ NACA GIA006 "ilfoJl "t M = 0.8.5. CompalÎ!'\onlwt\\'cell the cases
wit 11 '~I)(W in plllllgillg .11111 pitt h, 1 DOlo' in plllllging, amI 1 DOF in pitch .
•
•
Cil \ JI l'FU (i JI \ l? \.\1 i" j'lU ( 'FU J IFU \ \ \ 1 ) .';JS
10 - 1 "1 tl
Xa=·n 22 (} M 085
(f -x..- il ,14 ~~~~j7 1/0
NACA 64A006 01lfOII t
>- 8 -
"0 ID ID 0-
U) 6 ... Q)
~ ::l iL E 4 --0 , c o Z 2
o o
x.";:? 06
x..'- 002 1,-
x..=- ~ ,10
x.. ~ R?6
f-'\
" 0
,0 0
1 " ('\' 0
J
05
. '
l'
.' . -
,1
~'
. ' ,.- . '
•
Frequcncy Ratio, Wr
at diffen'Jlt lUI at iOIl'i ,dollg t Il(' 1 hOI d lill('
l'
.. ... .' • • •
J
1 5
1 (lU
A .. ... ... • • • •
2
•
•
('II \/l'}'U( (j /1\/(:\.\/1';'1 Ifl(·/ï./ '1 'IFU . \.\'. \ 1. YS 1.'-,'
10
>- 8 "0 Q) Q)
a.. (j)
E Ü , c o Z
6
4
2
X :=02 . {) X"=flO.4 x..--O,6
'J
M ,085 NA CA E4A006 wrfoll
l' 50 a -00
r. 01
1 _
0.5
- - __ - ____ 1 __________ ~ ______ _
1 1.5 Frequency Ratio, Wr
101
2
Figllll' G.I.!) FIII!!I'1 :-,!wl·d, ,>. \('l'~\I~ fll'<jlH'llI'Y latio, W,. Tlansonic flow, a NACA
•
•
('1/ :\}J J'VU (j P \ 1\ \.\ Il',TH J(' FU ,'T J FU ,\.\" \/. )."'/.,,'
4 -- -- - ------
:::3 "0 Q) Q) a. en "Q)
g 2 u:: Ë :0
1
c: o Z
x,=-0.7 -B-
x,=-0.5 -0
x. =60.3 x,=-0.1
- .. - -x, = 0.1 . -x,_=...Q.3 x,= 0.5 --e--
M =085 NA CA 64A006 airfoll p--=50 a==O 5 r. 0.1
° _ _ _ __ ___ _ L_ _ __ _ ___ 1 ___ _
° 0.2 0.4
j. ,
.. , ,
_1
0.6
•
Frequency Ratio, Wr
1 ():~
• •
•
0.8 1
FigUlC' 6.16: FluUI'1' ~I)(,l'd, Vf , \'l'r<,llS fl('(I'J('llCY Idtio, W r • T'd'J"CJ'II<' flow, a NÂCÂ
64A006 airfoil at :\1 -c: 0.81. TllP ('l,,:-.tir f\:-.i" al tilt' tlll('" (1",Il!.I·) (Ilold, ,lIId LlIf'
l'l'litre' uf ma<;s ;It drff('J('llt JOI ati(J!1~ ,t10Jlg t II(' (IJol ri IjlH' .
•
•
Chapter 7
Conclusions and
Recommendations
7.1 Cone} usions
Thl'I,' ,11'1' fi\'l' dl·~igll jlaldllH'tl'l!'> of ,\ t'Vo-d('g,Il'('-of-fH"'dolll ailfoil wlrirh r<lll af-
1'1'1 t tilt' fi lit 11'1 ~p",·d uf t Ill' .Iii fui 1. lIallwly. nOIl-dillll'l1!'>iollalllldss, radins of gyration,
P""lt iOIl'" IJ! t Ill' "I<I~ti(' "xi ... dllcl 1 l'IltIC' of Illit!'>::' l'l'IdtiH' tu the al'lOdynamic rentre,
<llld, l',tli() of tlll' 11111 ()uphlll,tllll,t! ftl'!jlll'ncics of the ctitfoil in pit,ll and plllnge.
'\lllIldlllg tu tlll' \l''''lrlt~ olltdlll('(1 fWIll thi~ ~tlldy, flIIU('1 ~pl'!'d inC!C'<t~I'" almost
1111'·.t11~' \\'It h 1I1111'<I:-,illl!, t 1](' IIOIl-dinH'lI"'wlldl IJJa:-~ of aIl di, foil, lJut.h III th!' incom
IIII,:-, ... tldl· .. lld tldll:-'Illill Ilo\\'''
1·'llIt kl "PI'I'tI .t1:-.o inl 11· ..... 1' ... \\'it h ail 1I1l11'a~l' in t 1](' 1 adills uf gyl'ation of ail ail foil,
IIU\\I'\'I'I, tlll' illlll'oI"'l' i ... Ilot lilll'dl in tll<' il1c()1l11)J,'~:-,ibl(' flo\\". ft is found that the
tll'lid of tlli~ illl \1'(\"1' tl'lId~ to 1)(' 111011' lill('dl' in th!' t!,\I1S0Ilic flo\\',
• CH. \ P l'E If l, ('OS ('U','i/O.\'S ,\ S f) lU·.'( '0.\ 1.\ J /o'.\' n, \ /'l() .\',,-!' 1 (lI
'l'hl' l'ffl'cl:-- of Ill!' 11111'(' ollll'I' Pd).I11WI(·)s lI)1 thl' 111111(') "'pl·(·tI IIf .111 dilfll!l d\('
011 tlll' n\ltkr :--pn'd, il i:-- Iwlll'\' lu .1I'01JU! 101 IllI'o.,l· Pdldlllt·ll·)o.,.1I II\(' ... .1111(· 111I1t·,
'1'0 \l'I'if\' tll!' 1(· ... 1111:-- ol,ldllll'd 1'10111 Iltl'" ill\l,o.,lig,tllolI, 11l\'\ .I\(' 1 PI Il pol \('.] \\illl
!l':--C('IIC(', ,lIId Ill) d(') od.\ Jldll1i(' fUI ((' Ilao., d qlld:--i ... I.d i( flllll\.
c1il'C,ctioIlS, alld <1 11\111]('1 i( al :--olllllUIl, \I:--illg 1 Ill' ('-q IIIl'I hocl. i~ (·!llpl!,\'!·d III lilld Il)('
f1l1tt('l hOllllcléll y.
[ound to 1)(' difft')('lIt, the)' al(' 1!l('s('lIl<'d <'1'],.11.11.1·1)',
7.1.1 Incolnprcssiblc flow
In il1l()Il1Pl(·"..,ihll' /lu\\' II\(' foJJu\\'illJ.', (ul]('llI:--iuIIS Il,l\'(' 1)('('11 dldwl\.
]. FuI' the (d~(' \\ Iw/(' Il)(' ('('111 Il' or Illa..,<; i~ ,tJJ(',1I1 of t111' (·lrI ... ti( d ·:1"', II() IIldl \'('1
IJOW tlH'i) ll'ldti\'(' po..,il I()IJ:-- \\'it Il )('~P('( t III tlll' rI('IOdyllrlllli(' (('/dl(' .11(', l'IIJ('<''
élll,d'y~i~ ]J\'I'c1i( 1 ~ 110 fllIt 1.('11. III t111~ ~tllrly tll!' !(Jllo\\ illg l ,d,(',!',OI il· .... 11(' fOltllcl
----------------- -- ------1 fil tllf' alg,·hralr dll.tI)' ,1'0 for tlll., (,L'o", 11/1''''''/lI,'d III "PI" IIdIX" 11.1'0 .. b'J\V1I Ib.II l'III'..,' f 'JII' Iii Iflll
•
•
•
('II/\I)'f'/~f( 7 (:(J,\'('/,(J,l.j/().\',"j .. \'\'J) /(/~('(),\1,\1ESJ),\TI()NS' 105
• If 1 II!' ,·1"..,1 i, rlxi.., i~ ,d~() plé\! ('ri élllt'ild of tll!' ,wrodYllillllic (,(,Iltre, 1 he'll 110
111111('10(( III"', 'lbi.., dgf('('~ \\'Ilh Pill(''''' 1I·"'lIlt for tllis (I~!',
• Il 1 h(· (·1"..,1 I( ,lxi~ i.., pld( ('d 1H'llilld t II!' .H'IUdYllélllIic ((,llllC'l t)J('1l if the
(('11f 1(' o[ Illd"'~ i~ .d..,o 1'1.1< ('d lH'lrillll 1 Iw d('IOclYIl,lIllic celltl(', i.f'" hl'l\\'c'(,1l
1111' il (. dll.! 1111' (' il" /llllln (dll 1)(' oIJlail)('t! al ail fIC'qlf('I)(Y latios IIp to
il (('lldill \'édill', Tlli~ lIpP('1 léUlgp 1Jl'(ollH's gl('atcr, a~ tJ](' C('IlUC' uf IllélSS
J!.(·h (I()~('I t." Il)(' (·Iél<,ti( ct .... is, Hlld the ('Ia:,t.ic axis movt's flll'tlJ('l' Jwhincl
1 lit' éIl'IOr!yll.11I1Îr (('Ilt 1('. Tlli~ i~ (hlr(')('llt fWIll what Pines has !Jn'dirtcd
ill hi.., ..,t.lldy.
2. Now, [01 t IIC' (tlSC' ",IIC'H' UH' ('('1111 c' of llIa~s is placC'd h('hilld 1 he f'lastic axis,
dl'jH'IIr1illg "Il Il)(' 1('l.lIi\!' po~Îli()lIs of tlll' élC'rodY"élmic ('('IÜrf', Pilles' é1Il,t1ysis
plC·di( I~ t h,,·(' di Ir('1 ('lit IW)ld\·itJllls .
• If t 1](' d('lOdYlltlllIic (l'III 1(' is )ucalcd ahead of the f')a~tic aXIS, f1utt!'r is
p()~~ihk olll} fur fI ('qllC'IIt.y 1 dtius J,cIo\\' a Cl'I tain value - dcpClldallt on
tilt' di..,I,,"! ('S of tll!' ('('lItl!' of ma ... .., (wd tlH' (leludynallllc Cl'lItr!' flom the
('I.I~t ic d\i~, a~ w,,11 <lS, Il,,, 1 <I(hll~ of gyratioll. III this :,t1ldy it is f01llld that
:--11('11 cl Illd .... llIl1ll11 fll'qllCII(y I.tf,io (dll 1)(' obtail1"c! ol1ly \\'111'11 the "la:>tic
il',i~ i~ LII 1'11011,1',11 (of thl' llld"r of tlll' ~e'llli-( bOld) floll1 tilt' (t('IUdYlléll11ic
(('lItll'. E\'('II ill tlll~ ('dM', thl' \,dllC' of Ihe Il1t1XiJl1111ll flCqU('llcy ratio at
('(wh (lo!'>il iUIl uf tlll' (l'III re uf ll1él~~, i~ gll'cltc-r thclll th,1I pII'dict('d by Pillcs'
<llldl) :-;i~; !'>l'(' Fig. {Ul.
Whe'll tll<' t'\.l~lic d,is IS Ilot fdr c'l1uugh flom the aC'loclynamic centre,
Hnt 1<'1' l'an Iw ohtaine~d at ail [r('q1lcncy ratios .
•
•
7.1.2
IOh
• 1 f 1 Ill' ,1( '1 ody Il d III i(' 1 l 'll 1 1 l' i:-. IU( ,li l 'd 1 H'I WI'I 'li 1 h,' l 'I.I ~ t i, .1 '\ i:-. "11 cl th" ( l'II t l"
lJf111d:-' .... ,\c('oldinp, I() Pilll'~' dll"IY"I~. IIl1ltl'l on 111~ ,11 ,dl 1'11'1(111'11' \' l.tllll~
TIH' :-..l1lll' Il':-'1I1t i" lIhl dilll'd i1l 1 Ili~ :-.llIely
• If tilt, d('llJd~'llalllll «('11111' i:-. lu(,tI(·cI III'bille! Ill!' «'lltl(' ul III'\~~. l'll1('~'
<L1l,dysi:-. prl'di( b .1 111111 i 11111111 fll·qlll·lIC., l,II iolH'lu\\' \\ hic b Ilollllt tl'l UIIIII~
Th(, \,t1Il1' of 1 hi ... IU\\('I hUlllIe!. ,d~u. dt'jll'lIc1:-. lIll 1 Itl' di~t.III( ( .... lIf t III' 1 ('llt Il'
of 111<1:-':-' .IIHI tl1<' a(,l\Jd~ 11<llIli( 1 ('1111(' flOl1l 1111' (·1.1 ... 1 il' .1\1 .... d'" \\,(·11 dS. tlll'
ladills of l!.\'Icüion. Tlu' ~d1l1l' Il':-'lIlt i ... IJld"il)(,c! 1'111111 IItI:-' ... tll,h. \Vilh 1 lit'
dilr('I('II(' t!t,t!. 111l'llIillillllllll fl(·qll(·II<.' I,diu 1:-. "'111,dll'l Ih"ll 111,tI jlll'cli,I"d
1))' Pillc:-,' c\lIdly:-.i'i; :-'('(' FIl!., (LL
'n.'ansonic Flow
III th" Il cl Il:-'011 W now J'('glllH' t 11<' pU:-.itloll uf 1 III' <I('llIdyll<l11l11 «'1111'1' 1.., 1101. ri
sp:?cific poillt. éllld, tlll'J dUIP, nu ('('1 tdill 1,1:-'(', ,1:-' 1 .. I.(·gol i,wd ill l'ill(':-'' "lldly:-.i:-., (ail
1)(' disting1llslH'd for ally ll'lal i,'(' po:-.it.ioll of 1.11(' ('lél ... ti( dxi:-. ,11111 1 III' «'111 1(' or IIldS:-'.
13111" it is:-.t il! of g](',lt int('I'1·:-.1 to du th\' :-'<1111(' :-'011 of 'lll,dy:-.i.." ilS f(J1 1.111' illlulIllJI (", ... d'/I·
flow.
ft is ba:-.il aIl)' rOlllld thctf., in thi:-. (a:-.p, fllIl ('1 i ... !Jo:-.:-.d'/I' ;11. ,dl fll'qlll'lI('y 1,11.111'" dlld
al! po:-.itiolls of t.11!' da:-.l il' axis <llId tlJ(' 1 ('111.11 IIf III<IS .... Fil 1 1111'1 1l10/f', if. (.III III' ',1'1'11
from the ](':-'1\11,:-. 1 !td!" fOl "'))('lllJc IIJI ,tt iOIl:-' (JI 1 III' l'I" ... t il 01 X 1 ... ,llld 1.111' 1 l'IiI l" IIr 11101','"
fllltt(,J :-'P(·(·r] ill( lI·a:-.(':'o, d:-' t.1}(' fi ('qlJ('1I 1 y l,cl i() i ... 1111 IC·dC,l·d. Jt. 1:-' d!',O 1 IJIII llldl·rI 111.d.,
as tJ](' e!a ... lil <lxi:-. IllOV('''' fil 1 t!1('1 1 1}\\'oIld:-. 1 Il<' Il d i!ill~ l'd).!,I·, \\'11.11 il :,))('1 dit II)I fi t iOIl 01
tl1f' «'ntle' of Illd:-'~ alld dt (J <;1)('( ifil rI ('ljlll'ill y I,dio, 1I1lf.t1·1 :-'JJI'('d illl II·d',I· ... II, i ... ',1'('11
also that, at ct spl'cifk lo( atiull of (Iw c·l,t:-.til ilxi:-. alld a "'JH'( ifir vdll(' /JI fIC'IJII('111'y
•
•
('/1 \II'/ï':/( ï. ('ON('/,(,'.'·j/(),YS .\.\IJ UE(,'O.\IMEt,rn;\,f/o;\,s 107
l,di". Ihlll"1 ~p(·('d 1I1(1(·rI~(·o." il'> 1111' «'IIf/(' of IIldS~ i~ lllU\('d fll111}(,1 fl'umthe e~rtc,li('
li ,10., ill ('If 1/('1 dll ('( tioll.
7.2 Rccommendaiions
III (Jld('1 1.0 ill( 1(''''''(' tlll' fllllt('1 .... ]H·(·d of ail "ilfoil (01 ideally, to totally pn'\,('IIt
1 II(' (II (1111 ('II( (' ,,1' fll/t 1('1'), 1 IJ(' follo\\ illg puillb d\'(O ~lIgg('!>t('d:
1. 1\1,,0.,:-' of t/w "i, 1'011 ~I'()ldd 1)(' ill, 1(·éI~(·d.
~, JL,dill!> of p,yl ;t!i()11 ~1j()l/ld 1)(, illlÏl'éls('d.
:L FII'q1l('lIcy l'atio ~llOlIld Ill' Ld';:('11 dS Iligh ilS possilJk glcat<'l' than unit y,
1. ('('/111 (' 01 IIWSS :-.h(Jldd 1)(' plan'd ,t/wild uf 1 II(' 1·lasti. c1xis,
.1. EI,,:-.1 i( i1\is !>llOuld 1)(' plan'd ahl'ad of t Il<' (l(,lodynamic Cl·nLrt>.
(i. Fol' illly Il'lali\'(' positiull:-' of tlH' ('Iastic aXIS, éll'rod)lldmic c(lntrf', é1nd centre
uf lIlass, l'ldsl il' ilxis :-.llOl/ld 1)(' plan'rI as far as possihlt, frolll the a(llodynamie
c{'n1., C', a/ld, abo, «'III Il' 01 111<1% !>hullld 1)(' placed as dose as possible to the
(·1.1:-.1 ie c1xis .
•
•
Bibliography
[1 J :\'lr Fal Jand, !vi. \\'., I·d., 'J'II( pajJ! r:; of Il'dvlll' (1I1d O/'l'dl, Il'rtf,hI, i\ld:1 ol\\' 11111,
New YOI k, 19!):t
[2J I1ill, C.T.n" "Arhilll(,('s in Ail! r,.rt Stllll tillaI 1)1·~igll," AIlp,lo "1I1I'lil.1Il AI'11l
lIi1lltical ('UII[('1'<>\1('(" 'l'Ill' Hoy.1I AI'!olldll! il.d SOI i\'ty, HI ip,htuII, S"I>!.. I!l;ll.
[:3] Lanc!wslf'r, F.\V , "T()r~io\l(t1 Vibratiol\s of t,]1I' 'l'tlil (JI' élll A(·)lJpJ.llw," JU\'M
276, Part 1,1916.
[<1] Bairstow, L., Fagf', A , "Os('illélti()n~ of t)H' 'l'.IiI Plalll' uf .11l1! BlJdy or .11l J\I'IU
plalJe," H&M 2ï6, Palt 2,1916.
[5] Fok].;N, A.I1.G., 'l'hl' Fly/II!} DuldIlHfl/!, Ikilly 11011" :\l'W Ylllk, I!):~l.
[DJ NACA 'l'l'vI n:3, Un:L
[iJ C.tl'l'ick, I.E., f{('('d III, W.IL, "I1i:-.tol i( ill J),'vr·lojJlIH'llt of A il! l,dt. FIIII,1..'I,"
.Jol/I'//al of:1/1'( rajf, Vol. I~, Nov. I(J~I, pp. :-;1)7-!))~.
[8] Tolv(', L., "lIi..,tCJlY of Fligltt Flnttl'! '1(':-.1 ilJj!,," h(J('(((I",q~ of IIIf I~,'JX Fllfj/tl
FI Il Il C1' Tu,ltng S!}1/1]JO<'1111/l, NASJ\ SP-:~K.j (IJ{'W pl 1IIIlIl).!" 1Il7!)) .
108
•
•
J~JIII,I()(,'I(, \ IIIIY 109
[!J] lIoU, j\ .J . "'1 Ilill SIII'II" III ,\('II)"'Ptll l' St J'lIl'tlll('S," A/A:\ \'011 /\;11111(111 Ll'ct ure,
;1,1/1///(/111/1·,/:: 11(/()//(J/lllt .... Vo!..l, FI,I). l!)(jï. pp. :W-,l:i.
[10] '1IworlfJl'-"I1. 'l'., "CI'III'I,t! TII(")IY of A('rod) Ilillllic Jn~tal)ilil'y ,IIHI th(' ~I('chél
lli"'llIlIf j·IIII.II'I," ;,\,\(;/\ H,'pt. I!)(), 19;1.').
[IIJ Cd/'Iil k. 1.1':., "An '\PPI('( 1"li(J1I of t!J<' ('UlltlilHltiol1 of TIJ('odOlc TlwodOlS(,I1:'
1I/'O('(((llIIq.~ of IIIt '/'Il/odrJ/.'t/1 ('OI!OqUilllll, TlOllrll)('im, NOIway, 1976.
[1 ~I SIII;lp" B . \\''':-:-('1111<11111, L.S., ":\pplll al iUIl 01 TIII'('('-DillJ('llsiollal FllIttl'l' Th('
(Jly lu Ai/( l,dt Stllll flll('<' Ail COJ'ps '['('chnint! Hept. .Jï98, 1912.
[1 ;{J 1"11 1, " .... ,\ H(,( (JI d of 1 Il rUI 111,1 tion 011 O:-ci Il <tt o l'y /\ ('1 odyllamÎc D('l'i\,él t i \'(' ~lea
..,111('111<'111..,." ILl:.: ~I .3:n~. 1 !)(j~.
[l'IJ (:dllilk. 1 E.\ "I\,·J'(),'I",>ti,il.\' Fl(}lItH'I~ c1l1d B('J'ond," AII\A vun IÙl.l'I1liln Lec-
11111" .lourl/II! of ,111'1 mjl. Vo!. I;{, S(·pt. 19ï6. pp. (j·ll-ü57.
[1:1] NM'A SlIl"·UlIlIlliq(·(' (J1l VilIJtll101l élnd FIllttf'l', "A SllI\'<'Y and Evaluation of
10'1111 fi')' Ht'M',IJ (II "nd ElIgil](,('l ing," NACA R~I-')6112, 1956.
[IIi] Tt·llIph .. (;., .J,t1IIl, II.A., "FIlItt!')' cil S1\pl'l'sollir Sp('('ds," R&M 2HO, 19'1.5.
[Iï] (;;Illirk. 1.1-:., HIII,inu\\'. SI. "FllItt(']' etnd O!)cillatillg Air Force ('a/culations for
"Il .\idllil ill cl '1\\'o-Dilllt'lhiull,t! Sll)l('r:-ollic Flow," NACA H('pt. 816, 19,16
[lxJ '1'111111'1'. \1 .J., ('1\l1Igh, H.\V .. ~1"Jtill, Il (', Topp, L .. 1., "Stifrll(,~~ and J)eflection
\ Il,d.\ ~i.., uf ('olllplt'\. SI 11II't lIll'<' .folll'I/a[ of At l'nT/alltieal SC/CI/C(s. Vol. :n,
St'Pl. I!l'i!-i, p. ~():)-~'n!i .
•
•
H 1/3 LI () (,' H. \ PI [)' 1111
[I!)] FUll~, \.< ' ... \/1 1/11mdl/dloll loI/II /'lu ()l'I/ of \( l'Oc 1(/~I"'lfll, .I(111l1 \\'tll" ,1..: SUII ....
N ('\\' Yu 1 k, II):j.'i.
[:W] Do\\'(·11. E II.. . \ .\lodt l'II ('(}I/I'~( 1/1 \, lOt I(/,~t/( Il,1/, ~Ilt! Ft! . I\III\\'C'I ,\, .Ic1,·IIII'
Pllhli~h('ls, ])OI,lr('(!Jt, \!)K!).
[~I] .Jordall, P.F., "i\('IOc!"",llIli, 1-'11111('1' ('(J,·lIi, 1"111:-. fOI SlIh"'Cllli" S'lIli,. dUcI Sil
IWISOIlÏc Flow (LilH'ill '1\\'o-J)illlt'lI~il)II,r1 TIIl'()I'\')," IL\'~I ~!):t~, 1!lr,:l.
[n] Béll1!Jall~, \V F" (;uoljldll, P :\1., "llllpli, il. Fi"il,' l>ill,'Il'II' ,. ('cri, 1I1c11 iOIl~. ,,1 l'II
~tcildy TlélllSOlli, Flow l\hulll \iJl\>I\:..," '\/.L1 .Jol/I'/llIl, \·u\. !r'), 1!17ï', pp 17'~K
1ï:~5.
[2:l] IIul\\\'ink, IL, "on d('1 VOUl('Il, ,/" "lllIjll()\'('d V(,I~i(l1l of L'I'IL\N~ fol' 111l'-.It'cllly
Tran:-.o"ic Flow CompllLd,jolls," :1/:1:1 .JOI/roll!, VIII. \ K, :\ li,!!, 1 !)KII, pp. lOOK
1010.
[~/1] l\lIgl'lilli, .1..1., COlli-lton, 7\'1., "NlIlIl<'l'iccd Solllt.iulis of N()I,~t.(',JlI'y Two
Dinwn:-.iullai Tl,\,lI:-.oni(' Flu\\'s," JO/ll'I/ol of FI,III! .... 1-,'/l9/11lt l'il/y, Vu!. 101, I!rt!',
pp. :341-:3017.
[2.5] L('(', B,II.I\., "A SIndy of Tlilll:-'(Jllic FIIIII('1 of il Twu l>iIJl(',,·,iOJlill Ail l'fi Il If...ill).',
the {l-g illld p-k ;'vT('1 I!uc!·"" N,d i()II,d H(':-'(';II (l, ('011111 il ('''''ilrla, IIJH'1.
(2ü] lIt1idl, G., Stl'illill~!}'r, ;\1., "Ex( Italioll illld '\Iltdy:-.i·, '1'1.,1111111 111' fOI Vlj~J,ht. ',1,,1 t.,.,
T('sts," !\(;AHU H(·pt. 7\0 (jï:l, .Jrlll 1!)7(J.
•
•
HI/JI,/O(,'/(;\ l'II)' 1 1 1
[:n] /I/JUpll>, 1-:.1/., "'1'1)(' SLd.(· of-t1H'-Â,t of FllIU('1 Sllh~I.r1"tiéltÎ()JJ P'OlCdll'('C,
ÂIIIIlIIJ.', 1: S C('II,·"II AvÎatJ<lll Mtlllllf(I<111J('I~," AGAHD ('ollf('J'('llcc' Pro((·(·d
jllV,'> .\(J. ~ïK, S('!>t l ')7IJ.
[~K] 1.('1', HILl,., CI)\)(]"Y, .J.IJ . "/\ ('dllildi,1l1 ÂPPlOélCh 1.0 FIIlt.tc·r ('h',llance for
1'::-:1.I·lIldl St()I"~," ACAB)) Bq>!" No. ïO:J, .Jan. 191'):3
[~!J] .J 01 111:-'011 , '-:.11., 1 h\'d 1lJ.'" C., (·t al., "Ad<lptiv(' 1"1111.1,(') S1lPP)('~sjoll - Alldly~j~ (lIld
'11'..,1." AC,\]{ J) H,·pt. 010 ïlJ.$, .lall. 1 <)~:$
[:1O] Wolfl',~] O.W, 1\lIkl,.\. \V.T., ",\ Hl'\j,·w ul Flight Flllttcr '1(·:-,ting ']'('chlliqul's
jll (:1('.11. Hltldlll." :\CAHn/lgldph .~(j, J lth ~fcdjllg of thl' J\CABI) StructtJlC's
dllt! 1'"1.11"11.11,, Pdll"!. ,\t h'·Il.." CII·(·c(·. S(·pt.. ] !J(jQ.
[:1I] WI·i~""·lIlllllg(·I, .l.T , Zill III 1<') IlIdll. :'-J.I1 , "Pll'di( tioll of FllItter Ons(,t Bas('d 011
Fljp,ht. '!'t·..,till,!!, .11, S1I],t J'jtj( al Sp(·(·ds.'· .]o/l/,/1a[ of /l/1'('/'(/fl, Vol. l, .Jllly-Ang.
1 !J(io!, pp. 190-:W2.
l:t~] PI j, t" S ,1 • 1.('(', lU 1.1\., "l)t·\'t'lojJllll'llt dIld Allcllysis of Flight Fluttcr Predictjon
~1t'lllOds." AI,\.\ ])Y"rlllIjC!'> Spt'rjaljst COll [('J('11 c(', Ap1Î11992, AIAA-92-2101,
pp. 1 ~~-~()n.
[:U] HU,!!,t'l, h. L., ",\jl pl.1I11' ~Iat h ~Iodt·llillg ~kt"ods [or Adj\'(' COlltlul D(·..,jgll,"
AC.\f{]) (,P-~:.!~. '\11).!,. 19ïï.
[:~ 1] Ad.lIll". W.I\I., Tilr .. ".", S.II., "Nonlilll'dI' PlOgr,lIl1Jl1illg ExLenf..ions to R,üional
,\ppltl\j1l1.t1 jOli ~I(·t "otis fOI llllstc'ady At'J'udYlldlllic Forc('s," NASA TP-2ï76,
J%~ .
• HIIJUO(,'U. \ /' Il)'" II~
[:lrl] Pille". S., ",\11 EI(,lIl('IlLlI: 1':"pLlll<ltiulI ur tlll' Fllltlt'l ~k('h(llli'>lll," /'/'(1/ (( dlll!l~
.\'all(llllll.r..,'jJ(('/(//"f" \/((/l/II/ 0/1 /),/111111/11',' tllld 1/1(1(10,111''''1, II1,IIIlIII' III IIlt'
:\('1 ulJétlll j('a 1 S('i(·llll·~. \'1)\. 1 Wi~, pp. ")2 .")~
•
•
Appendix: A
Algebraic Analysis of Pines'
Study
JI. \\,,,~ ~11()\1 Il, ill 1 hdp\,(,1' (i, 1 hal Il 1It.t ('1 i~ pus),ihle (IIIHIPI' t.he> a~sumptiol1s made
.r 2 Jo [ w; (./' - ( -) j') + (.r - .1') 1 2 0, (A.1)
l'
,llId.
ï' 2 w; [ (.r + ./') -- 2 (-) .r] + (i - .z:) > 0 . l'
(A.2)
'1'0 dll,t1.\'i'I· 1 I}(' ~()IIII iUIl~ uf t1)(' sd of ineqll,t1itics, wc ),honld eXélmilW IJoth the
1)('( (.:-. ... ,,l'~. "!lei ... ,dlil il'Ill 1()IIe1itilll1~. A II(,«,.,:-.al)' cOlldition is ('Il1)Jlu)ed to t''\r1l1dc the
dOl1l.lill ill ",l,il Il iW'qll,t1il i('~ 1)("'('1 llUl<I, !luI kt i~ ill1po ... siblp. On Ihc ot.lwl' hand. a
"lllli( Il'111 1 (lllliitioll ~Il()\\'!'l tilt' dUllldill ill \lhich tl](' illt'qllalilic~ always hald, flut.te>l'
(li ( III ~ at SllllH' :-,p('('d.
III ,1!.,(·Il('1 al. tu ~ho\\' (/ i~ t he lit C{ .... MII·Y cOlldttiDIl of p, il ~hollld 1)(' shu"'l1 thdt:
• "-' tl ==> "-' JI •
•
•
:\ PP ES J) 1.\ . \, , \ 1. (,' ['.'1W, \1 (' \.\' \ L' .'i l.~' () F [11.\ FS' SI 'l ï l) III
ur,
[1 - ~, '1
[1 ~, '1
III IIH' folllJwillg, tlll' pu:-.:-.illility or 1111111'1 \)( (IIII"IH (' r()1 ,1111'1'1('111 1(·1.111\" 1"""
\Ve wililly \lO\\' to fintl 111(' Il,·((· .......... I.\' (f)lldilillll l'Ill ("1".1111111" (:\ 1) dllt! (1\ :/) ICI
lroh!. To do :-.u, <1:-.:-.1111](' I!I<J'-,(' LWI) ('«II(dIOIl'" h()ld. 1111'11,
r l ( -) r) 1 (/' l') (J.
(11,2) S.: (:\.:n
(1l,·1 )
,. r 2
~ uJ/ ( l' - 2( ) r) 1 (/ ,. ;- uJ, l / 1 (1'
/ L l') .--- ( ) / w, l
1
1 1.
(.r -.1')./ 2( ) / w/ /'
/' 1. /' L (11,6) lt.: (A,7) - -}- ~(--) FuJ, l /' ( ) /W r 1-
,. 1
(A.~) ,v. (A :q i'J'/'{J
l' ) , Il ,
/ l / . ( ,\ ;-.;)
• \/1/1/.\'/)/.'\.\ :\IJ,',..UU.\/(' .\.\ \U,I,;/,',' ()F /11.\/-;,'," ,',"'J ('J))' 115
Ilwldu/!', 1 1 If" Il''( ''',',dl \ 'U/ld,l'O'l I()/ "CJII,II i()ll~ ( \ 1) <111.1 (:\.:n 1 (J IlIJld (flllttel 10
III dll(III)!'1 \\IJI'1. 1111 111111('11'> IJI)"""II!' lOI .r > () Thi~ i~ in cl.!jI'·('IlH'lIt \l'illr Ihl'
llJ(' \',dll" ul 1 III' 1)1)1<'1 pell elllwll'I" 110\\'('\"'1. 11, j~ ... !JIJ\\·" 11I'1f'. Il)' 1IJ('an., of ,\ "jIllP'"
1 11111111'1 '·\eI/llpl,·. 1 Il,11 1",> ... 1.11"111"111 i ... Ilot III!I' l'Ill (1)(' 1,1"'(' .r < O. Slll)~lilllti(J1l
(JI Il)(' I"II"WIlIl!, 'fileI/l1 Il l" .... -,-'r - 1 . . 1'/" = -:~ 0 <' O. dllli 1'/1' = --05 < O. into
:~x[( (Ji -( () .. i).!x(-:~.O))+(-:{0+05)]=
.{ x [ () -1 j () ï-, -- ~ i] == --:~ x (--~.~!)) = 6.ï:) > 0,
t (/111111011 ( t./) Iw/ds,
ellld.
[( - O}i -:~ Il) - ~ x (-.LO)2( -O.:))] + (-:3 ° + U"ri)
.) - 1') ( 1 -) ,) - .) 0 --: -.J •• ) T _ X.:) -- _.:) = .) > .
( Cf Il (/ Il (/ Il (A . .l) ft 01 ri,."
,\lthUll.!~h lIu, Iu",,'d IUIIII "olIl/i, j,'111 (UlldiljUll (cill 1)(' ohlajll('t! fur tll<' case.r < 0,
.r "O. \\(' (clll ~ljll ('\,P('( 1 III Il,,\(, 1\I\IIel' 1'01 ~()IllP c1JOil'cs of the palélml'!('IS l'and
•
•
•
. \ P Jl ES [) I.\: \. . \/.( ,'[-' BU. \1 (' \ S. \ n .~'/ .'1 () l' JI 1.\ /. 'S' S /'lI» 1 \(i
ofiLol.
.1' '> () ,1:,' /' '. ll. ( . \ .t 1)
Il CciII 1H' ..,hU\\·11 Ih,t!. IlcI\ill,!', t'qll,tlillll (\!l) .."tli..,li(·cI. !lO "clclili, ""111 1 ." .... ,,,1.\
(OJl(lit ion \\'1111)(' i 1I1J>()~'·(\.
Tu ~hu\\' tltat Ilw ~1\f1i, il'lI1 «()lIdllitJlllul IIlltll'l ln U(llIl III tlli.., Id'." l~ tll 1i"\I'
.1' .. ,1' }, < """,,', - -
- .1' (1 --- ~~ ) •
Id 1I~ a:-'Slllll<' Illcll ('qll,tli()n (.\.10) )/(JlcI..,. 1111'11'
(.Ul) .1' /.
}.l' . ( -) ,.. ' Il, l'
(A J ()) S .. : (. 1 JI) - ;. I,i,', }, ( l' .1' 1.
( -) .1') ::' ( r
"
l' ) ..
/. l
( A. ) :!) I\: (-,1 ()) " ? ./ [w, L (1' ' ( -) .1') 1 (.r -.1' ) 1 . () . l'
Oll t)1(' ot)lI'l lielll(!.
(:LIL) ,} W,}, /
(.'1.!J) .": (11. H)
l' .! . w, }, ( ) / 1 (.1' - 1) ---:: () ,
/'
;- W,2 1 1 (J' / ) / (J,
;- w,2 1 1 2
:U-)r;;-O, /'
r 2
( A. 1 :j) & (A. J (j) =--=} vJr 1. [(.1' t .1 ) -' '1 ( .,) .f') 1 -t (.1' ,.1') --- (J . ,.
(.\ 1 ())
(A Il)
(A.I:q
(A 1,1)
(A If;)
(A.rl)
•
•
117
ClI JI" Cll',
() < .r < ./: . (A.I~)
'l'Cl lilHI cl 11l'( (,~~tli \' 1 olldil ion fOI t hl' ~d of ilH'qlléllil )(':-', ('qllit! iuns (A.1) alld (A.2),
h·l, Ils cI:--:--I"I\(' 1 Il;11 t Ir 0:-'(' !'Qllill iOIl:-' Irold, llwll,
(·11) S.: (\I~) __ -'0 - -, .C 2
W, 2(.r -- ( -) .1') + (i' - .1') 2: 0,
011 11)(' ollJ('1 Irdlld.
l'
.r 2 (A.I~) ::-:=} (-) .1' > 0,
l'
( .\ 19) S.: (.'\.:!O) -::-} (./,.', 2.r + (.f - .1') > O.
.i' 2 (.,\.J~) =::} 2(-).1' > 0,
l'
( 1 ~) ,1..: ( \ ~2) :--=-~. W, 2 (.l' + .r) + (.ï· -- .1') > 0 .
(A.19)
(A.20)
(A.:n)
(A.22)
('UII"jd!'III1,!.!, cqll.rliull (.\ IK). O!J(' fillds tlrat !'C!uatiolls (A.~I) ,llld (A.2:3) ale
l'his \ ... in cI,!.!,\(·('!I1I'llt \\ Ith Iht' Il' ... \llt ohlctillcd hy PilH's .
•
•
.\J>PE,\[)IS \. \U,'I-.'HU.\J(' \,\ \1.) SIS OF 1'1.\'1-'.'1' .'1/'1'/» II~
uf it, or.
o ..::., .1' <. .1' •
IL l'an \)(' :-lw,,"1l t hat, t1S in ('d<';(' :J). hd\'illg \''111.1\ itlll ( \.:.! 1) '-,dl i:-.lit'tl. 1I() IIt't t'~'"ll)'
condit.ion j.., impo:-,('d.
<lnd,
Abo,
) r -- l' w,· > .
. r (1 __ '~")
1'.1' --- <: 1. ,.2
,
1'.1' (A.:W) " '. --- 7 .> (),
,.2
1'.1' (ALlI) & (A.L7} : --} .1'(1- ;-;-) --,0,
1'~
(A.'2;») Ir.;. (A.Lfi)
(/1 :10)
(;\ 2 1 ) -=}.r :2 > .1 .i·
r 2 __ ~;. w,2(.r - ('--) .1') ~. (r· .,.) -' Il,
l'
.r 2 ( --) .r) 1 (/' ./ ) J .' Il. ,.
-} (qIlO/101I (:1 1) !told ...
;.. ./,2 __ .1' r :::-- () f' 1.
~ ( ) ,.
(A ~~Ij)
( \ ')") 1 •• /
(/\ :~X)
(/\ .W)
(/\ :~ 1 )
• :11 1 Il 1' .. \ J) 1 \ ,\. ,\/,(,' l';/J J(, \ 1(' :\ X \ L Y,'''iJS' () Jo' J) L\' ES' ,':,1' l ; J) y ] ]1)
(Il n) l,: ( l ,n ) .r 2 ,/'," ':} 1 + ( -) - 2( ~) > 0 , (A.:~2)
l' 1' •
• 1' 2 ./' 2 (:1 ~ 1) .'V (Il :l~) ',., } ( ..... , 2 [,1' + ( -) .r - ~( -) .r] > 0, ,. ,. ( :\ .:3:3)
1 2 .r 2 .l' 2 ;, ....J,
2 [,1' . (") .1' +./' + (-) .r - ~(-) .1'] + (,r - ,l') > 0, (A.:H) ,. l' l'
• 2 2 [ .r J (/\.:H) ,,} wr .r 1.1' - 2(-) .r + (,r - .1') > 0,
l' (A.:3:"»)
/11111'.111" 1 Ir ,11 1111' !'>Idlll jl'Ir! Ililldlli()11 fUI Ill(' fll\tl('1 tu OCClII' i~ <'q1\é1tiuIlS (A 2Fl) and
•
•
•
Appendix B
C()mputer Program ListillgS
:\.-1: TIH' U-g ?\I('UlOcl in hH'()lJll>l'(>ssibl(' Flow (2])oF)
A -2: The U-g ?\'l(>t hod in Trallsolli(' Flow (2DOF)
A-3: TIl(' p-I.: l\,Ipthod in 11l('oJ])}>I'(>ssiIJl(> Flow (2J)()F)
A-4: The p-I.: ~,IptllOd in Trèlll:-,()})j(' F'IO\v (1 nOF ill Pllllllv»
A-G: TIl(' p-I.: ~I('f,h()d in TrêlllsoJlÎ(' FI()w (1 /)OF il! l'JI ('h)
:\.-6: TIl(' p-I.: ~.J(>tllOd in Trall:-'()llÎ(' Flow (2/)()F)
J~O
•
•
A p)H'lIdix A-]: TIH' U-!J ~I('t hod III Ill(,01J1pn'~siblc' FIO\v (2DOF)
c
c
c
c
c
c
rompl cx*16 n(?, 2), b(?, 2), v(2), elg<l( 2), elgb(2), elg(2) cornplcx*16 1(2,2),wk(4,8),lm9,ck chardeter pdr*40
'f'ül*8 ckr,ckl , enl <,(4,4)
rf'ül mlu,kk,k"tcp,kmnx,kc,kkl,kk2,kk3 'f'ill ko(50),rck(50),lCk(50) renl u,ck(50),blCk(50) , eal rr~p(4,'i0), lckp(4,50) lntcgcr 1,lge,optn
datl1 111, lb, n, 1 Job, 1 z/2, 2,2,2,21
open (unlt=7,fllc='uglnc.d<lt' ,~tatus='old')
read (7,*) mlu,ra,ah,wr,xa,wa rend (7,*) optn,kmax,kstep,acpter,nc
,ead (7,*) par,parmln,p<lrmax,npar rend (7,*) nprln,nprnt2
elo$e (unlt=?)
If (optn.ne.l) then open (lHllt=B,flle='teo.dat',~tatus='old') , ead (B, *) ndat l'l'ad (B,*) (ko(l),rck(l),lck(l),i=l,ndat)
rlo"e (ul11t=B) end If
open (unI t=9, f lle= 'ug 1 ne. out' ,err=60)
wrlte (9,64) mlu,ra,ah,wr,xa,wa,kstep If (optn.eq.l) then
wrl te (9,65) goto 33
end 1 f
121
•
•
AIJPESDI.\ R. ('(J,\1Pl :TFH j'HOCU. \.\1 US 11.\"(;S
wrlte (9,66)
33 wnte (9,67) par
e
e
e
e
pJ=4. "atan(l.)
Img=(O.,l.)
If (optn.eq.1) goto 44
eaLL csa~m (ndat,ko,rck,brck,rckp)
raLL csakm (ndat,ko,lck,blck, Ickp)
e The first Loop:
e 44 contInue
e e e
e 23
24
c
e
do 50 li=1,npar+1
pramet =parml n+rea L ( II -1 )* (parmax- pnrllll n )/npar
1 f (par. cq. 'nll u') 1111 u=pramet
if (par.eq.'ra') ra=prarnet
if (par.eq. '1011") wr=pramet
1 f (par. eq.' ah') ah=prarnet
If (par.eq.'xa') xa=prarnet
kk=kmax+kstep
ige=O
i=O
Ire=O
Main Loop: flnding the VfL for the set of paralIIetero;
1=1+1
kk=kk-kstcp
If (ige.eq.l) kk=kk3
if (kk.gt.l0.) then
i =1-1
goto 50
end If
If (kk.Lt.O.) then
\ointe (9,69)
\ointe (*,69)
goto 50
end If
kc=kk*2.
e CaLeuLating aerodynûmics:
e 1 f 'optn' IS 1 then the aerodynarnles WIll be caLculatcd
e uSlng a ratIonal functl0n approXlmntlon.
e
lo':!
•
•
c
c c
c
c
c
c
c
18
11 (optn.cq.l) thcn
call tcod(kk,ck)
gota 18
(·nd 1 f
ckr=c~val(kk,ndat-l,brtk,rckp)
ck1=cwal(kk,ndat-l,blck,lckp)
If (nprnt2.eq.1) wllte (9, *) kk,ckr,ckl
If (nprnt2.eq.1) wntc (*,*) kk,ckr,ckl
ck~dcmplx(ckr,ckl )
tOntlnue
c Calculatlon of matrIces [Al & [8]:
c
c
c
c
a( 1 , 1 )=.25* (m1U+ 1 )*kc**2- i ffig*kc*ck
a(1,2)=.25*(xa*mlu-ah)*kc**2-2*ck-lmg*(.5*kc+(.5-ah)
*kc*ck)
a(2.1)=.25*(~a*mlu-ah)·kc*·2.1mg·(ah+.5)·kc*ck
a(2,2)=.25*mlu*(ra*kc)**2+.25*(.125+ah**2)*kc**2+2*(ah+
.5)*ck+lmg*«ah+.5)*(.5-ah)*kc*ck-.5*(.5-ah)*kc)
b(1, 1 )=dcmpl X(WI **2, O.)
b(1 ,2) = (O. ,0. )
b(2, 1 )=(0. ,0.)
h( 2,2 )=-dcmpl x( rd**2, O. )
c Solvlng the cigenvalue problem:
c
36
43
c
c
call cI91c(a,la,b, Ib,n,1 job,cI9a,cigb,z,IZ,wk, infer, icr)
do 36 l=1,n
v(l)=elga(l)/clgb(l)
continue
J=O
do 43 1=1,n If (dreal(v(I».gt.O.) then
J=j+1
elg(J)=v(l)
end if
contInue
•
•
,\P])EN J) /-'\ H. (,O.H P l ''j'En ]l!?O(,' U. \.\1 1./,1..,"1'1.\'( ,',l..,'
C Calculatl0n of 'u' & 'g' cOl'rcspondlng to the chosen 'k': c 'u' is nondlffienSlonal velo and 'loi' 1!> nondlmcn-;lol1allzed by 'wa'. c sC*,1)=g c sC*,2)=u c sC*,3)=w c sC*,4)=gam c
35 c
c 55
59
61 62
c c
38
do 35 k=1, J sCk+2,1)=dimagCcI9Ck»/drealCclgCk» sC k+2, 2 )=sqrt(lI11 u/drca l (elg( k») sCk+2,3)=sCk+2,2)*kk sCk+2,4)=-s(k+2,1)/2.
continue
if Cnprin.nc.1) goto 55
wrlte (9,68) pramct,kk,(C-;(lo+2,Jy),Jy=1,3),lo=1,J) wnte (*,68) pramct,kk,CC-;(lo+2,JY),Jy-=1,3>,lo-=1,J)
gmln=1.0e10 do 59 k=1, j
gmln=ffilnCgmln,dbs(~Ck+2,1»)
contmue if Cgmin.le.acptcr) goto 55 do 61 k=1,)
gO=sCk+2,1) if CabsCgO).cq.gmln) goto 62
continue if Clgc.cq.1) goto 55
do 38 k=1,J If (sCk+2,1>.gt.O.) then
igc=1 goto 34
end If contmue goto 61
c If a posItive 'g' has bccn found: c 34
c
1 f CI. eq. 1) then write (9,63) write (*,63) goto 53
end 1 f
Ire=ire+1
c Interchange S(old) & SCnew) if neccs5ary:
C
1 ~~ 1
•
•
/\ !J/J/.,,\' /J/ \ H. ('(),\/I J( 'f'I';U JJJ{()( ,'U. tH UST/.\'(,'S
83
c
If CglOlnl.lt glJlln) then
do 83 110=1, J
do 83 IIrm=l,4 <,C 110+2, Hrm)=~C Im,llIln)
contInue
end If
c Check the convergance condItIon:
c If Clre.gt.ne) goto 53
c
c rlnd the new kk:
c
c
57
39
c
If CI.lt.3) then kk3-~max - k<. t ep/2.
goto 37
end If If CglOln2.lt.91OIn1) then
kk3-g0/Cgl-g0)*Ckk2-kk)+kk
goto 37
('nd If
kk3=gO/Cg2-g0)*Ckkl-kk)+kk
kk2=kkl
kkl=kk
g,"ln2=9mln1
gl111nl=9mln
g2=g1
g1=90 do 39 kl=l,J
do 39 k2=1,4
sCk1,k2)=sCkl+2,k2)
cont mue
goto 23
c The end of the maIn loop
c
c
c Prlnt out the re~ult:
c
~3 If CJ .le." goto 54
If Cab~Cs(4, 1».lt.absC~C3, 1») then
wrlte C9,68) prûm('t,kk,C(~(l0+2,JY),Jy=1,3),lo=J,1,-1)
wrlte (+,68) prilmet,kk,(C~(lo+2,JY),Jy=1,3),I.,=j,1,-')
gota 50
l'l1d 1 f
54 wrl ~(' (9,68) pr,lnl('t,kk,CC~(l0+2, JY), Jy=1,3>,lo=1, j>
wrrte (*,68) prilnl('t,kk,(CsClo+2,JY),Jy=1,3),lo=l,J>
•
•
:\PPE1\"D!.' H. ('O.\Il'l rFH l'HO(,'/U,\1 /1,l.,"lï.VUS
50 continue
c
c End of the flrst loop
c c
60 close (unlt=9)
c 63 format (' Kmax IS not bIg enough')
64 format (2x,'mlu=' ,f7.2,8x,'ra=' ,f7.3,Bx,'ah=' ,f7.3,1
65
66
67
68
69
c
c
2
3
format
format
format
format
format
stop
end
3x, 'wr=', f7.3,8x,'xa=', f7.3,8x,'wa=', f7.3,/ 3x,'K~tcp=' ,f7.4)
U/,2x,'The ratIonal functlOn arprOXlIlIiltlOn 1<, helng ,
,'u<;cd' .I,2x,'to calrulate Teoclci ~on funct IOn.',1f)
U/,2x,'The tabulatC'd data 1'> be1l1g u',ed to ullculatc'
,/,2x,'Tcoder"on functlon.',//l
(t7,a9,t16,'K',t?8,'g1',t42,'U1',t55,'w1',166,'g2',
t78,'U2', t93,'w2') (2x,f7.4,2x,f7.4,6(f13.4»
U,2x,'k=O. & no acreptable rc<;ult lS found')
functlon Ign(x)
If (x) 1,2,3
Ign=-1
return
ign=O
return
Ign=1
return
end
1 ~li
•
•
j\PIH'JJ(lix A-2: 'l'Il(' U-y ~Jdlwd III Trall~()ni(' Flow (2DOF)
c
c c
c
c
(Oillpl ex*16 a(?, 2), b(2,2), v(2) ,elga(2), Cl gb(2), el g(2)
cUlllplex"'16 z(7,7),w~(4,8)
l ompl ex· 16 Cl h, C 1 a, Clin, Cma
rOlllplex*16 l'lgV(2,2)
choJrac t cr p..r*40
,pal*8 (lh, ,clhl,clar,clal,Cl1Ihr,tmhl,cmar,cmal
, ca l '. Cl- , 4 )
rr,r! ml11,kk,k,.tep,I<rIk',kr,kk1,kk2,kk3
'l'al ko()O)
l'cal rlh(50),llh(50),rla{50),lln(50)
rNII 1 mh(50),IHÏ1(50),rma(50),1fna(50)
rCill b,lh(50),bllh(50),brla(50),blla(50)
n'al brmh (50), bllM (50), brllla( 50), blma(50)
1 eill rlhp(4.50).1Ihp(4.:;0),rlap(4,50),llap(4,50)
rral 'lIIhp(4,50),1fnhp(4,50),rmap(4,50),1fnap(4,50)
rcal rç 1 (2),1 c1 (2), rc2(2), 1 c2(2), rmo( 2), lmo(2), alTTn(2) ,phm(2)
lntegcr ,>g(2)
1 nteger i, Ige, optn
data la,lb,n,lJob,IZ/2,2,2,2,2/
open (Ul11 t=7,flle='ugtrl1.dat' ,status='old')
read (7,*) mlu,ra,ah,wr,xa,wa
read (7,*) optl1,kmax,k~tep,acpter,nc
re,ld (7,*) par,pnrmln,parmax,npar
1 ead (7, *) nplln,nprnt2
do~.e (unlt=7)
if (optn.ne.1) thcn
open (unlt~B,flle='Dcro.dat',statu~='old')
reild (B,·) ndat
rcnd (B,·) (ko(1),rlh(1),llh(1),1=1,ndat)
reild (8,·) (rla(1),lla(1),l=1,ndat)
•
•
c
33 c
c
c c
c
c
44
c
c
c
c
23
read (8,*) (rmh(I),lmn(I),1=1,ndat) rend (8,*) (rma( 1), lm<l(I), l=l,ndat) close (unlt=8)
end If
open (unlt=9,flle='ugtrn.out',err'60) wrlte (9,64) mlu,ra,nh,wr,xa,wa,k~tep If (optn.eq.1) then
wrl te (9,65) goto 33
end If wrlte (9,66) If (nprln.ne.1) wrlte (9,67) pnr
pl=4.*atan(1.)
If (optn.eq.l) goto 44 calI csnkm (ndat,ko, rlh,bl lh, rlhp) calI rsalem (ndat,ko, llh,bllh, llhp) cali csakm (ndat,ko,rla,brla,rlap) call csakm (ndat,ko, lla,blln, llap) calI c"akm (ndût, ko, rlllh, brlllh, IlIIhp) calI c~akm (ndût,ko, 1II1h,blmh, lmnp) calI ("sakm (ndat,I<o, rma,brm .. , rmap) calI c:>nkm (ndnt,ko, lma,blma, 1Il1ap)
The flrst loop:
contInue do 50 11=1,npar+l
pramet=parmln+rcal(ll-l)*(parmax'parmln)/npar if (par.eq.'mlu') mlu=pramet If (par.eq.'ra') ra=pramct If (par.eq.'wr') wr=pral1Jet if (par.eq.'ah') dh=prdmet if (par.eq 'xa') xn=p'dl1Jet kk=kmax+k<,tep
If (nprin eq.1) wrlte (9,71' pùr,prdmet 19c=0 i=O ire=O
Ma ln loop: fi ndl ng the Vfl for the !.>P.t of P,lI iJIllf't cr <;
1=1+1
kk=kk-kstep
•
•
c
c
l' (Igt Pq 1) ~~-~k3
If (Ir~ yt 0 5) tllen
wnte (9,69)
wr 1 tl (* ,69)
']rJto 50
end l' If (~k It 0 ) then
wnte (9,70)
Wf Ite (".ID)
got 0 50
end 1 f
~c ~k*2.
c Ca 1 r ul dt 1 ng uf'fPdyndJJII co;.
c 1 f 'optn' 1<, 1 thl'fI the aerodynanllcs WIll be cûlculated
r u',lng il riltlonal fumtlon approxImatIon.
C
C
C
c
c
c
c
If (optn eq 1) then
call traef o(kk,Clh,Cla,Cmh,Cma)
goto 18
end 1 f
clhr=csval(kk,ndat-l,brlh,rlhp)
clhl=c~val(kk,ndat-l ,bllh, Ilhp)
If (nprnt2.eq.l) wrlte (9,*) k~,clhr,clhi
If (nprnt2.eq.l) wnte (*,*) à,clhr,clhl
Clh=dcmplx(clhr,clhl)
clar=c<,val(kk,ndat-l,brla,rlap)
clal=csval(kk,ndat-l,blla,llap)
If (nprnt2.('q.1) wnte (9,*) kk,clar,clal
If (nprnt2.eq.l) wrlte (*,*) kk,clar,clal
Cla=dcmplx(clar ,clal)
unhr=-c.,vnl(kk,ndat-l,brmh, rmhp)
crnhl=C.,val(kk,ndat-l,blmh,lrnhp)
If (npf nt;' ("11) wllte (9,*) kk,cmhr,cmhl
If (npfnt? cq 1) Wf Ite (*,*) kk,cmhr,cmhl
Crnh=dc-rllplx(cmhf ,cmll1)
crnar=c~v,Jl (k~, ndat . l, brrnil, rmilp)
cmal =c ... "a 1 (H, ndat -l, bl mil, 1 map)
If (npf nI2.cq." wrlle (9, *) kk,rmar,cmal
If (nprnt2 eq 1) wnte (*,*) kk,cmar,cmal
Cm,l=dcmplx(crr.lr, cmal)
•
•
\l'P/:,.\'J)I.'\ H ('().\1J)I'1 FU l'IU)(;f{ \,\1 IISI/\(,'S
c
c
c
c
c
la cont 1 nue
c CorrectIon of acrodynnmlcs for the loratlon of E.A.:
c
c
c
Cmh=CnVl+C 1 h* (O. 5+ah )*0.5 Cmn=Cma+Cla*(0.5+ah)*0.5
c Calculatl0n of mûtrlCC", [A] & [B]:
c
c
c
c
a(1,1)=.25*mlu*kc**2·Clh/(2.*Pl) a( 1,2)=. 25*m1U*kc" 2* xa'C 1 al (pl) a(2,1)=.25*mlu*~c**2*xo+Cmh/(pl)
a(2, 2)=. 25*ml U*kL" 2*ro"2+2. ·Cllla/(pl )
bCl, 1)=dcn~lx(wl**2,O.)
b( 1,2)=(0. ,0.) b(2,1)=(0. ,0.) b(2,2)=dcmplx(ra**2,O.)
c Solving thc clgcnvalup. problcm:
c
36
47
43
calI clgzc(a,la,b,lb,n, lJob,clga,clgb,7.,l Z,wk,ll1fer,lcr)
do 36 1=1,n v(I)=eiga(I)/clgb(l)
continue
J=O do 43 1=1,n
If (dreal(v(I».gt.O.) then
J=j+l clg(J)=v(l)
do 47 JJ=1,n
end If CO'1t 1 nue
Cl gv( J J , J ) = z e J J , 1 )
If Cnprln.ne.l) goto 49
do 48 k=1,J rclek)=real(Clgvel,J» lcl(k)=lm~geclgV(I,J»
rc2ek)=rcaleclgve2,J» lc2(k)=lmag(clgv(2,J»
dcnam=rc2ek)*·2+1C2eK)**2
5g( k) =~
l,Hl
•
•
48
c c
rrllo(Ie)=r c1(n lmo(r )-1 c1(r) If (denam.nc.O.) then
~g(k)=1
r mo( k )=( rc 1 (le )*rc2(r)+ 1 c 1 (k )*1 c2( k) )/denam lmo(r)~(lc1(k)*rc2(k)'rc1(k)*lC2(k»/denam
end If iIIlfll( le )=',qr t (r mol k) **2+ lino ( k )**2) phm( le )=0. lf(rm(J(k).nc.O.) phrn(k)=atan(irno(k)/rlOo(k»
cont lnue
c Cnlrutatlon of 'u' & 'g' corrcbpondlng ta the chosen 'K': c 'u' 1<' nonrJ101cnblonat vet. and 'w' 15 nondl1OcnSl0nôtlzed by 'wa'. L <'(*,1)=g c '.(*,2)=u c b(*,3)=w c ~,(*,4 )=gam
c
35 c
c
do 35 k=1,j <'Ck+2,1)=dllOagCelg(k»/drcalCclg(k» ~Ck+2,2)=~qrtCmiu/drcôtCelgCk»)
b(k+2,3)=s(k+2,2)*kk sCk+2,4)=·s(k+2,1)/2.
cont 1 nue
If (nprln.ne. 1) goto 55 wflte (9,72) (kk,sCk+2,2),~Ck+2,1),reat(elgCk», IIIlôgC e 1 9 (k» , rc 1( K) , 1 c 1( K) , rcZ C k) , 1 c2 (k) , 5g (k) , k = 1 , J ) wrlte (*,72) (kk,<,(k+2,2),sCk+2,1),rcat(elg(k»,
imag(elg(k», rc1(k), lc1Ck), rc2Ck), lc2Ck), sg(k), k=1, j)
c Check the convergance condItIon: c
c
61 c
If Cgmln.te.acpter) goto 53
do 61 k=1,J gO=b(k+2,1) If CabsCgO).eq.gmln) goto 62
contInue
c Flnd the n1\f1101Um g: c
55
59
gln1n=1.0~10
do 59 k=1,j gmln=mlnCgmln,nbs(sCk+2,1»)
continue
•
•
:\]J /) ES f) /X B. ( '(J.' 11) 1 TFU JI If ()( ,'If \.\ 1 /1,1.) lï .\' (.',~'
62
38
c
If (Îgc.cq.1) goto 34
do 38 1e=1,)
If ('(!c+2,l).gt.O.) then
Îge::1
goto 34
cnd 1 f
contInue
goto 37
c Replace S(new) by S(old) 1 f nece~~ary:
c
c
c
83
If (gmln1.lt.gnl1n) then
do 83 Im:1,J
do 83 i mm:: 1 ,4 s( 1r1l+2, Inln)=s( lm, inTo)
contInue
end i t
c If a positIve 'g' hûs bccn found:
c
34
c
If (l.eq.l) then
write (9,63)
write (* ,63)
goto 53
end If
l rc::i rc+l
c Check the maximum numbcr of Iteratlons:
c 1 f (1 re.gt.nc) goto 53
c
c Flnd the new lele:
c
c
If (i.lt.3) then
Iek3=kmax-lestcp/2.
goto 37
end If
lf (gmin2.lt.gmln1) then
Iek3=gO/(gO-g2)*(kk2-kk)+Yk
goto 37
end If
kk3~gO/(gO-g1)*(~k1-kk)+kk
c Inl tlallzation:
c 37 kk2=kkl
kkl=kk
•
•
9ml n2=glnl n1
glnln1"glnln
g2=g1
91=90 do 39 k 1 = 1, J
do 39 k2=1,4 r,(k1,k2)=r.(k1+2,k2)
goto 23
c
c The end of the main loop
c
c c Pn nt out the rI",ul t:
c
c
53 If (nprln.eq.1) wrltc (9,67) par
If (J .le." !Jota 54 If ("b~«,(4,1».It.ilb ... (q3,1)) then
wr Ite (9,68) pl"J1Iet,H,«~(lo+2,Jy),jy=1,3),lo=J,1,'1)
wr Ite (*,68) prillllct,kk,((5(lo+2,JY),Jy=1,3),lo=J,1,'1)
goto 50
end If 54 wr Ite (9,68) prilrnl"'t,I<K,(h(10+2,JY),Jy=1,3),lo=1,Jl
Wl'Ite (*,68) prarnt't,kk,«b(lo+2,JY),Jy=1,3),lo=1,J)
50 continue
c End of the fI r!. t loop
c
c
c
60 close (unlt=9)
63 format (' Kmax 1 s not bi g cnough')
64 forl11ilt (2x,'miu=' ,f7.2,8x,'ra=',f7.3,8x,'ah=',f7.3,/
65 formilt
66 format
67 fOllllat
68 format
69 forlllilt
70 format
71 fornl;!t
72 form;)t
3x, 'wr=' , f7.3, 8x,' ~a=' , f7 .3,8x, 'wa=' ,f7 .3,/
3x,'K~tep=',f7.4)
(fI ,2x, 'The ratIonal functlon approxImatIon IS being ,
,'u,>ed',1 ,2x,' ta calculate the aerodynarnlcs.',11>
(1l,2x,'The t~bulatcd data '5 bClng used to calculate'
,I,2x,'the aerodynalnlcs.' ,II) (t7,<l9,t16,'K',t2B,'g1',t4?,'U1',t55,'w1',t66,'g2',
t78,'LJ2', t93, 'w2')
(?x,fl.4,2x,f7.4,6(4x,f9.4»
(f,?x,'k,,, grater than the l,m,t; no re~ult.')
(/,2x, 'k=O. & no acceptable result has been found.')
(f,2x,'for ',a9,'=',fB.4,I,t4,'k',t13,'U',t22,'g'
,t28,' r lall', t37, 'Ilan', t47,' rc1', t57,' ,e1', t65,' re2'
, t 7~, , , e2' , t 82, ' h 1 9n' )
(f6.4,8f9.4, ,6)
•
•
\PPESDI.,\ H ('U.Hl)( l'EH l'UU(,'1f \.\lll~/ï,\'(,',~'
c
c
!>top
end
funct 1 on Ign( x)
1f (x) 1,2,3
19n::-1
rcturn 2 ;9n=0
return 3 ;9n:1
return end
1 II
•
•
c
c
c
c
COfllpl ex*16 (\(4,4), b(l.,!.), v(4) ,elga(4), clgb(4)
cOlllplex*16 z(f.,4),WK(6,12),elg(2),ek
f eill kk,""U
rcal ~o(50),rck(50), Ick(50)
l'cal bl ck(50),blck(50)
rcal rdp(4,50),ltkp(4,50)
1Ilteger optn, Iii, ln,n, lJob, lZ
dnta la, lh,n, 1 Jon, lZ/ I.,4,4,2,4/
opf'n (U111 t=7, f Il e=' pk Ine .dat' ,,>tatus=' old')
rend (7,*) mlu,rn,aa,wr,xa,wa,bb
rend (7,·) lJmln,umax,u~tpp
fead (7,*) pal,parrnln,parmax,npar
rend (7,·) optn,ne
cl o<,e (Uni t =7)
If (optn.nc 1) then
open (unlt=8, flle='teo.dat',status='old')
reild (8, "') ndat
redd (8,*) (ko(I),rck(i),ICk(I),I=l,ndat)
clo'-.e (ufl1t=8)
end If
open (U111 t=9,flle='pklnc.out',err=100)
Wf lte (9,64) 1II1lJ,ra,ah,wr,xa,wa,kstep
If (optn.cq 1) then
Wf 1 te (9,65)
goto 33
end 1 f
Wf lte (9,66)
33 wrltc (9,67) par
c pl=4"'atan( 1.)
Il11g={O.,l.)
•
•
\}JPE.\'J)I.'\ 13 ('().\/Pl J'FU IJHO(;U. \.\1 LIS 1'1.'\(;8
c
c c
If (optn.eq.l) goto 44
call csakm (ndat,ko,rck,brck,rckp)
call csakm (ndat,ko,lcK,bick,lCkp)
c Declaring parts of matrices [Al & [Bl:
c
c
c c
a(1,1)=(O.,O.)
a(l,2)=(0. ,0.)
a(2,l)=(0.,O.)
a(2,2)=(0. ,0.)
a(3,1)=(-1.,0.)
a(3,2)=(0.,0.)
a(3,3)=(0. ,O.)
a(3,4)=(0.,O. )
a(l.,1)=(O. ,0.)
a(4,2)=(-1.,0)
a(4,3)=(0. ,0.)
a(4,4)=(0. ,O.)
b( l, 1) = - ( 1 . ,O. )
b( 1 ,3)=- (O. ,0.)
b(l,4)=- (O. ,0.)
b(2,3)=- (0.,0.)
b( 2,4) = - (O. ,0. )
b(3,l)=-(0.,O.)
b(3, 2)=- (O. ,0.)
b(3,3)=- (1.,0.)
b(3,4)=- (O. ,0.)
b(4,1)=- (O. ,0.)
b(4,2)=-(0.,O.)
b(4,3)=- (O. ,0.)
b( 4,4) = - ( 1 • , O. )
c The flrst loop: (parameter change)
c
c
44 cont Inue
do 333 jcfm=l,npar+l
prarnet=parrnl n+rea l ( 1 1 -1 )* (pa rmax, parnll n)/npdr
If (par. eq. 'nllu') mlU'-pr arnet
if (par.eq. 'ra') ra=prarnet
If (par.eq.'wr') wr=pramct
If (par.eq.'ah') ah=prdrnct
If (par.eq.'xa') xa=prarnet
wb=wa*wr
1 :Hi
•
•
,\1'/'/';,\/)/,\ IJ ('(),\1J11 J l'J( l'/(()(,'U.\.H /'/"";'/'1,\'(;.')
c c C"lculat1ng the re~t of miltr 1)1( [Bl:
c
c
c c
c
c
b( 1 ,7)~'xa
b(2, 1)~-xa
b(2,2)=-ra*ra
U'Ulnln-u',tep
nt=O
1=0
c The ~f'cond loop: (~peed chilngc)
c
11. I=i+l
u=u+u~tep
acc//=O,Ol
1 f (u,gt.urnax) goto 20
c
c Inltlol gues~ for kk:
c
c
c c
kk~(WD'wb)*bb/(2.*u)
If nk.gt.l0.) kk=10.
lc=l
1 ront=O
12 If (kk.gt.l0.) thcn
wntc (9,64)
goto 100
end If
1 f (H.l LO.Ol) kk=0.01
icont=l('ont+1
If (lcont.gt.l0) goto 14
c Calculût1l1g acrodynanllcs:
c If 'optn' 15 1 thcn the aerodynarnlcs WIll bc calculated
c uo; i n9 a rdt i anal functl on dpprox lmilt IOn.
c
c
c
c
1 f (optn.cq.1) thcn
coll tC'od(kk,ck)
goto 18
end 1 f
•
•
. \PPESUIS. JJ ('OJ'1I'/' J'FU PU()(,'f{, \.\1 US 1'/.\'(,','-,'
c
c c
c c
ckr=csval(kk,ndat-1,brck,rc~p)
Ckl=csval(kk,ndat-1,blck, ickp)
If (nprnt2.eq.1l wnte (9, *) kk,dr,ckl
If (nprnt2.eq.1) wrlte (*,*) kk,ckr,ckl
ck=dcmplx(ckr,ckl)
18 contInue
c Calculatlng the rcst of matrlx [Al:
c
c c
a(1,3)=wb*wb+u*u*Clh/(2.*pI*bb*bb*mlu)
a( 1,4 )=u*u*C l a/(pl *bb*bb*mlU)
a( 2,3)=- u*u*Cmh/(pl *bb'bb*mlu)
a( 2,4 )=( ra*wa )**2- 2. *u*u*Cma/( p 1 *bb*bb*ml u)
c Solvlng the eigcnvalue problcm:
c
c
c c c
call elgzc(a,la,b,lb,n, Ijob,clga,elgb,1,lz,wk,lnfer, 1er)
do 6 l=1,n
v(l)=elga(l)/eigb(l)
6 continue
j=O
do 2 K=l,n
If (dimag(v(k».ge.O.O) then
J=j+1
elg(j)=v(k)
end if
2 cont 1 nue
ww=kk*u/bb cr.=abs(ww-dlmag(clg(IC»)/Ww
If (err.gt.acerr) then
kk=dlmag(eig(lc»*bb/u
goto 12
end If
If (dreal(elg(lc».gt.O.) nt=nt+1
If (nt. gt. nc) goto 20
If (lc.eq.2) goto 14
ic=2
Icont=O
acerr=0.01
•
•
H - (wô<wu)*bb/( 2. *u)
1 f (~k.yt.l0.) kk=10.
goto 12
c 70 cont 1 nue
wrlte (9,62) wr,~k,u,drcal(elgClc»,dlmag(eI9(lc»
wr Ite C*,62) wr,kk,u,dreaICelg(lc»),dlmllgCeig(lc»
c
333 LOllt 1 nue
c
c
62 format (7x,5(f7,1,,5x»
63 formdt (21,f7.3,5x,f7.4,5x,f7.4)
64 forlIIdt ClII,5x,'K excLds the upper llm!t whlch IS la')
65 format ClI,2x, 'The ratIOnal fUl1ction afJproxlmatlon I!. belng ,
,'u~f'd',/,2x,'to calcutilte Tcoder~on functlon.',11)
66 forilldt U/,h, 'Th" tabulned datd 15 bClng lIsed to calculate'
,/,2x, 'TC'odcr',on fUllctlOn. ' ,Ill
67 forrndt (tT,'K' ,tlB,'U' ,t29,'RC'al' ,t41,'lmag ' ,/)
68 format U/t6,'U',t16,'bC'ta1',t2B,'bC'ta2',1l
100 clo<;c (unlt:9)
'.top
end
•
•
.\PJ)E,\'/)/.'\ Il ('O.\J/'lTEH ,'no(,'If.\.u J.I,'·i Iï.\ (,','-,' 1 III
ApP('lHlix A-4: TIll' p-k ~r('th()d in T'rall:-,ul1ic Flo\\' ( 1 ])OF 111 Plllllgt')
c
c
c
c
c
c
compl cx*16 C lh, p, aaa
charactcr par*40
Ical*8 clhr,clhi
1 cal kk,mlu
rcal ko(50)
rca l rlh(50), i lh(50)
rea l brlh(50),bllh(SO)
real rlhp(4,50), i lhp(4,50)
1 nt cger optn
open (unlt=7,fl lc='pktrn-h.dat',btatu~='old')
read (7,*) miu,ra,ah,wr,xa,wa,bb
rcad (7,*) optn,umln,umax,u<,tcp,acew,aced,nc
rcad (7,*) par,parmln,parmax,npar
rcad (7, *) npn n
close (unI t=7)
If (optn,nc.1) then
open (Uni t=B, fllc='aerr dat' ,btatus='old')
read (8, *) ndat read (8,*) (ko(1),rlh(1),llh(I),i=1,noat)
close (unlt=8)
end If
open (unlt~9, flle='pktrn-h.out' ,crr=100)
wrlte (9,61> Inlu,ra,ah,wr,xa,wa,bb
If (optn.cq.l) then
write (9,65)
goto 11
end If
wnte (9,66)
11 If (nprm.ne.1) \oIntc (9,67) par
•
•
/1/'1'/".\'/)1\ JJ ('().\II I( "1 i',f( IJ/{()(,'/{. DI /'/S"f/.\'C:S
c
c
c
c
pl =4. ·"um( 1.)
If (optn.eq.1) goto 44
c,lll c',ahn (ndnt, ko, r lh, brl h, r lhp)
call (-.nhn (nddt,ko, Ilh,bllh, Ilhp)
c The fll<,t loop: (piJrOJmeter change)
c
c
c
c
c
c
1.4 CUlltlllue do 50 11=1 ,npo.Jr+l pl dlIIet -pi.llIlln' 1 ('al (11-1 )*(pdrmdx-parOiI n)/npar
If (par.':q. '11IItJ') II11U-pl drnpt
If (par.eq. '1"') 1 .. -pl .. met
1 f (poil "q. 'WI') WI -pl ;lInct
11 (par cq.' .. h') illFprdrnct
If (par.eq.'xiI') Xil:prillllct
u=tJnlln-u<;tcp If (nprln.cq.l) wrltc (9,71) par,pramet
1=0
1 flu=O
1I3=0. u2=O.
ul=O.
d:1rnp2=0. dan,)l =0. n:1mpO=O.
(' The -,ccond loop: (specd chilnge)
c
c
12 1=1+1
lIn=u+u"tcp
1 f (lflu cq.l) un=u3
u=un
c InItIal gllC~<; for kk:
c
c
kk-(wn+wb)*bb!C2. *u)
1 cont=O
i conv1=0 olg=O.
111
•
•
\J'/)/~'.\/)f'\ H ('().\II'1 '/ vu /'UO(;/f \ \111.'if 1\'(,'''';
e e The th 1 rd loop: (Ide change)
e
14 lcont=lcont+1
e
e
if (lcont.gt.nc) then
lconv1=1
wnte (9,70) ne
wrlte (*,70) ne
goto 35
end If
e Caleulatlng aerodynamlcs:
e If 'optn' is 1 then aerodynamlcs will be calcul.lted lhlllg
e a ratIOnal functlon apprOXlIlIatlon.
e
e e
c
e 18
If (optn.eq.1) then
calI traero(kk,Clh,Cla,Cmh,Cmil)
goto 18
end 1 f
clhr=esval(kk,ndat-1,brlh,rlhp)
clhl=csval(kk,ndat-1,bllh,1Ihp)
Clh=dcmplx(clhr,cl~l)
ront i nue
aaa=wL'wb+u*u*c lhl (2. *pl *bb*bb*ul1 u)
p=csqr t (- aaa)
if (lIoag(p). l t.0.) p=-p
ww=kk*u/bb
err=abh(ww-lmag(p»/lmag(p)
e wnte (9,*) lC, Hnag(elgva(lc»,ww(lc),err
e wrlte (*,*) lc,kk, irnag(elgva(le»,ww(lc),err
if (err.gt.acew) then
kk= Hnag( p) *bb/u
goto 14
end If
c c End of the th 1 rd loop
e
e dampO=real (p)
35 1 f (npnn.eq.1) then
write (9,69) u,P,ww
wrlte (*,69) U,p,ww
end If
Il'.!
•
•
c
c
c
c
42 If ("b·,(d,.lItPO) , le ;,(('d) gr:>IO 20
1 f (d"llipa gt 0 ) th(n
1 f lu-1
('nd 1 f
c Clwck th" c..ürWf'lgfncc ((}rtdlllon:
c
c
c
28
29
c c
37
c
c
c
c
20
If (l(orw1.(·q.1) tl,,-,n
1011111' (9,6~) u
1011 Ile (*,63) u
goto )0
end If
If (Iflu.rq 1) then
If (1.1 t.3) then
u3 c u'u·.tt'p/2.
goto 37
('nd If
If (Ign(d~mr?) rq.lgn(d"nlp1» then
If (ab<;(d"fflp2) 1 t ,,"·.(ddrnp1» goto 28
goto 29
end 1 f
If (lgn(d'Jfup2).ne 19n(diJnlpO» goto 28
goto 29
u3' Udl11l'O/ (ddmpO· dJlltp?) * (u2' u)+u
goto 37
u3=d"mf->0/(darnpO darnp1 )*(u1·u)+u
goto 37
(,fld If
u2=u1
lJ1=u
d.!I11fl2 c ddrnp1
d:'lIp1 =d.ll11pO
goto 12
[fld of the '.('cond loop
contInue
If (nplll1.eq.1) wnte (9,67) par
10111 te (9,62) pr ilmet ,u, p
loir Ile (*,62) pramet,u,p
•
•
.\/)/JL'.\'f)I.\ H. ('O.\II)/'J J:'U J)HO(,'U \\1 US/ï\U . ...,·
c 50
c c c c
61
62
63
64
65
66
67
69
70
71
72
c 100
c c c
2
3
cont 1 nue
End of the first loop
format (2x,'nllu=',f7.2,8x,'ra=',f7.3,Bx,'ùh=',f7 3,/
3x,'wr=' ,f7.3,8x,'~il=' ,f7 3,Bx,'wù=',f7.3,/
3x, 'bb=', f7.4)
format (4f10.4)
format U/,5x, '''10 iln~Wer for any mude at u~',f7.4)
format (///,5x,'K L'x(,l'd~ the upper llmlt WhlCh 1'> 10')
format U/,2x, 'The ratIonal furICt10n ilpprOXlmdtl0n 1'0 belng ,
,'u;.ed' ,/,~x,'to cùlculnte the ncrouyn,lmlc'o.' ,II)
fonllJt U/,?x, 'The taiJulùtcd diltn 1'> bClng u~.cd to Citlculntct
,/, 2x, , the dl'I odyn"fIll r ... ' ,/1)
fOrlllilt Ult6,a9,t16,'U',t?6,'RC'al',t36,'llIld!l',t/.6,'Rf'ill'
, t56,' Inldg')
fOlmilt (4f10.4)
format (2x,'kk dld not converge after ',12,' lteriltl0n'i.')
format U,2x,'for ',il3,'=',fB.4,I,t6,'U',t16,'lleal',t?6,'lrnag'
, t36, 'Om"9', t46, 'Real', t56,' Imilg', t66, '01l1C'9')
formilt (2x,'U cxceded UflIilX but il f lutter wa;. not foulld')
close (unlt=9)
end
functlon 19n(x)
If (x) 1,2,3
19n=-1
return
19n=O
return
19n=1
return
end
III
•
•
)-15
:\pIH'}JClix :\-G: TIiC' 1'-1.: '\f('lh()d 111 Tl<l1lsemi(' Flow (IDOF III Pitch)
c
e
c
e
c
c
(ompl LX· 16 Cl il, Cilla, p, olilél
lh •• 1 ae ter pul"'O
IC.tl*/J rlar,d,l1,rrnar,rllla1
rCdl kk,nlllJ
1 cill ~o(50)
r('dl rlh(50), 11h(50), rmh(50), 1mh(50)
rpal 1 la(50),11il(~0),rrnae50),1ma(50)
rcal brla(50),b11il(50),blrna(SO),bllna(50)
l'pal rli'p(4,50), 1Iap(4,50), rn'''p(4,50), 1map(4,SO)
1ntcgl'r optn
open (un1t=7,f1Il='pktrn-H_dat',btatus='old')
reéld (7,·) rn1u,ra,ah,wr,xa,wa,bb
rCéld (7, *) optn, UI111 n, umax, Ub tep, acew, aced, ne
rend Cl, *) pur, pdrrn1 n, pal lIIax, nrélr
read (7,*) nprln
rlo'.c (unI t=7)
l' (optn ne.l) then
op ... n (ull1t=S,flle='.wlo.ddt',,,tatub='old')
1l'<Id (B,·) ndat
read (B,*) (ko( 1),1 lh( 1),11 h( 1),1 = 1, ndat)
rL'"d (B,·) (lia ( 1 ) , lia ( 1 ) , 1 = l, nda t )
1 l',ld (8,*) ( 1 mh ( 1 ) , 1 mh ( 1 ) , 1 = l, ndd t )
rl~du (8,*) ( 1 lIla ( 1 ), 1 rn;) ( 1 ) , 1 = 1 , nd" t)
clo.,e (unlt-S)
end If
open (unI t =9, f lle=' pHI n-i1. out' , err= 1 00)
WI lte (9,61) IIllu,ra,ah,wr,XiI,wa,bb
1 f (optn_eq_ 1) thcm
wrlte (9,6~)
•
•
.\P]JE.\J)].'\ H. ('O.\Il'('n.'U l'HO(,'If.\.\1II.~'lïS(,·S
c
c
c
c
goto 11 end 1 f
wrlte (9,66) 11 If (nprln.ne.1) wrltc (9,67) par
pI=4.*atan(1.)
If (optn.eq.1) goto 44
call c!>akm (ndat,~o,rla,brla,rlap) call csakm (ndat ,lco,lla,blla,llap) call csakm (ndat , ko, rrna, brma, rrnLlp) call csakm (ndat,ko,lma,blma,lrnap)
c The Ilr!>t loop: (parametcr change)
c
c c
c
c
c
44 contInue do 50 11=1,npar+1 prarnet=parnll n+rea l ( 1 i -1 )*(par Ill;])' - parmI n) /npar If (par.eq.'rnlu') n11u=pranrct If (par.eq.'ra') ra=pranrct If (par.eq.'wr') wr"pramet if (par.cq.'ah') ah-pr'ilnlet If (par.cq.'xa') ya=pranrct wb=wa*wr
u=umln-ustep If (nprin.eq.1) wrrte (9,71) par,priJmet 1=0 1 flu"O u3=0. u2=0. u1=0. damp2=O. damp1=0. dampO=O.
c The second loop: (speed change) c
c
12 l=i+1 un=u+ustep if (lflu.eq.1) un=u3 u=un
c InItIal guc~s for kk:
1 1( i
•
•
/\ P j'/','\'I JI \ /1 ('().'II'I "1 I- j( Ilu{)r ,'j(, Ul J,j.l.; r!S(,'S
c
c
c
H- (wn 1 wb)"bb/(2.·u)
Icont=O
Iconvl~O
Clg=O.
(. The thlf d loop: (~k change)
c.
c
c
lI. lcunt=lcont+l
1 f (Ieont gt.nd then
Icolwl=l • ; 1011 1 te (9.70)1: ne
loi lit e (", 70 f ne
!Jota 35 end If
c Cdlculntlng ilrrodynnmlcs:
c J f 'optn' 1<; 1 then <lerodyn,lfnlcs wIll be calculntcd uSlng
c n rdtlonal functlOn apprOXimation.
C
c
C
c
c
c
18
c
C
If Coptn.cq.l) then
cnll traero( kk. C l h, Cla, C!Oh 1 Cma)
goto 18
end If
clnr=rsvaICkk,ndat-l,brla,rlap)
rlill=c~val(kk,ndat-l,blla,llap)
Cla=dclJl)lx(clar,clnl)
cmar=c!>va l (kk, ndat -1. bl ma, rmnp)
cmal=csval(kk,ndat-l.blma,lmap)
Cma-den.)l x( clIlar, rmal)
continue
Cma=Cmn l (0.5+ah)*Cla
<1,1.]=2. * u*u*Cm<lj (pI *bb"bb*ra* 1 (1*1111 u) - , •
p=c-..ql t(<Jaa)
If (1111.19 ( p) • 1 t • O.) p= - P
ww=~k*LJ/bb
cr r =abo; (ww' 1 mage p»/ llIlJg( p)
If (err.gt.Drew) then
H.= Imng(~ )"bb/u
J li
•
•
:\ J> P E.\ J) 1.\ H. ('0.\1 P l "rU? P HO(,' li U 1 1 .... J'/.\' ( ,'S
c c
goto 14 end 1f
c Check the convergence cond1t1on: c
c
if (lconv1.eq.1> then wnte (9,63) u wnte (*,63) u goto 50
end 1f
c End of the th1rd loop c
c dampO=real(p)
35 1 f (npnn.eq.1) then
c
c
wr1te (9,69) u,P,ww wr1te (*,69) u,P,ww
end 1f
42 1f (abs(dampO).le.aced) goto 35 1 f (dampO gt. O.) then
c
c
28
29
c
c
37
1 flu=1 end 1f
1f (iflu.eq.1) then 1f (1.1 t.3) then
u3=u·ustep/2. goto 29
end if 1f (lgn(damp2).eq.1gn(damp1» then
11 (ab<,(damp2).lt.ab~(darnp1» goto 28 goto 29
end 1f 1f (lgn(darnp2).ne.1gn(dalllpO» goto 29 goto 29 u3=darnpO/ (d,HopO -dmop2)* (u2 -u)+u goto 37 u3=dampO/ (dUlupO-ddmp1 )* (u1- u)+u goto 37
end 1f
u2=u1 u1=u
Ils
• r.
r.
c
c
20
c
~O
c
c c c
61
62
63
64
65
66
67
69
70
71
72
c 100
c
c
e
•
d;",,!,? - rj,JnfJ 1
u,lrIIpl-d.,tr(lÛ
90to 12
rr,d of the ',l'C onU loop
(Ontlnue
If (npnn,f<q,1) wrlte (9,67) par
wllte (9,62) prdrnet,u,p
wlIle (*,62) prarnet,u,p
(ulIllnUe
rnrl of 1 he f 11 ., t loop
fUII"dt (2X,'1II1lI=',f7 2,8x,'riJ=',f7.3,8x,'ah=',f7.3,1
3x,'wr=', f/.3,8x, 'XiJ=', f7 3,8x, 'wa.::', f7.3,/
3x, 'biJ=', f7.4)
fOllllat (4fl0.l+>
forilldt UI,5x, 'No dn'wer fur any mode at u=' ,f7.4)
fOlnr~,t UII,5x,'K e-<cl'u' the upper 11l1lit whleh IS 10')
fOlfll"t (//,?x,'The latlonal functlon appr'OXlrnatlon 15 belng'
,'u<.,ed',1,'2x,'to ,alculate the ilerodynamles '.1/) fOllll<lt UI,2x, 'Th" t"buL,ted ddta 1<; bLlng used ta calculate'
,1,2x,'thc ,JI rodyniJlI1lc<;.' .II)
fUIIII"t (1It6,.19,t16,'U',t26,'RC'ill',t36,'llIIag',t46,'Real'
, t56,' Im,Jg')
fOlilidt (4fl0.4)
forflldt (?x,'kk dld Ilot converge ilfter ',12,' Iterations.')
fOlllldt U,2x,'for ',il3,'=',f8.4,/,t6,'U',t16,'Real',t26,'Imag'
, t36, '(1l11eg', t46, 'Redl' , t56,' Imag', t66, 'Omeg')
fOll1101t (2x,'U C'xC'cded Urnax but a flutter was not found')
elo',e (uni t-9)
l'nd
fl/mtlon Ign(x)
1 f (x) 1,?,3
1 gn~-l
l'l'HI! n
• \l'PES})l.\ B. ('().\IJ>1 TI· Il l'U(}(,'U \ \1 1/.-'; 1'1.\'(;'-';
2 19n=0 return
3 Ign=1
rcturn end
•
•
•
:\pJ)I'llrlix i\-(): Tlj(' Jl-k \frotl!()(!lll Trall~()lli(' Flow (2DOF)
c
c
c
c
c
(Ofllpl ex*16 <1(1 • .'.). b( 4,4). v(4 >. Cl g<3(4), Cl gb(4)
COIl,plf'x*16 z(l.,1.),wk(6,12),Clg(2)
(011'1-' 1 cx * 16 Cl h , Cl a, Cmh, Cma
(ornpl f'x* 16 c 19v(l.,1.), t'l gvaC?), Cl gvc(2, 2), Cl gvct(2)
Chdl cletpr p,JI*40
rcal"B elhr,( lhl,( lar,clal,cmhr,crnhl,cmar,crnai
1 cal kk ,Ill1 U
1 Cil 1 IIIdgr1f't (?) ,l'h' "c( 2)
1 (',li ka( 50)
1 t'al r Ih(~O), Il h(SO). rla(50),lla(50)
r('ill IlI1h(50), 111111(50) ,lllla(SO), 11l1i1( 50)
ICill uIlh(SO),bllh(SO),ullaC'iO),blla(SO)
1 NIL bl mh('iO), blll~l( 50), bllna(50), blllla(50)
I('al Ilhp(4,SO),llhp(4,50),rlap(4,50),llap(4,50)
rcal rlllhp(4,50),llllhp(4,50),lnlap(4,50),lInap(4,50)
real ww(?), d,l/lIpO( 2), cJ"mpl (2) ,damp2( 2)
llltCgCI aptn
ddtil la,lb,n,l Job,lZ/4,4,4,2,4/
op"n (uI11l=7, f llc='pktl n.ddl ',&tatus='old')
lend (7,*) rnll/,I n,ah,wr,xil,wa,bb
rcad (7,*) optn,lImln,u<,tep,ilccw,accd,nc
Ip"d (7, *) pal, p," 1111 n, pdllnax, npar
1 t'ad (7, *) "1'1 1 n (lo-,c (Ulllt=/)
If (Uptll IlC. 1) then
open (uni t=B, fllc=' aero,dat',status='old')
rL'nd (B, *) ndat
1 (',ld (B,*) O,a(l),rlh(l),llh(l),l=l,ndat)
/('ad (8,*) (llil(l),lla(I),l=l,ndat)
1 t'da (8,*) (rrm(l),lmh(l),I=l,l1dilt)
Fi!
•
•
.\PJ>fSnI.\ H ('O.\/I>{ 'J'VU PU()( ,'U \ \1 I.I.~' 1'1.\,(,'.'1
c
c
c
c
c c
read (B,*) (rma(l),lma(l),I=l,ndat)
close (un1t~B)
end 1 f
open (lJn1t=9,flle·'pktrn out' ,err=100)
wrlte (9,61) mlu,ra,i1h,wr,Xn,w3,bb
If (optn.eq.1) then
wn te (9,65)
goto 11
end 1 f
wntc (9,66)
11 If (nprln.ne.1) wrlte (9,67) par
open (unlt=10,flle='pktrn.cgv',err=100)
wnte (10,7/,) pal
pl=4*atan(1.)
Img=(O.,1.)
If (optn.cq.1) goto 44
call csakm (ndat,ko,rlh,urlh,rlhp)
call c"akm (ndat,ko,l lh,bllh,llhp)
call csakm (ndat,ko,rla,brlil,rlap)
call csakm ( ndù t , k 0, 1 l ù , b 1 l il, 1 l ilp)
call csakm (ndal, ko, r rnh, br mil, rllilp)
call csakm (ndùt, ko, lInh, bllnh, lnl'1p)
call cSùkm (ndnt,ko,rrnù,brlfla,rmap)
call csakm (ndnt,ko, lmn,blmil,llnùp)
c DccLarlng parts of mntrlces [A) & [B):
c
c
a(1,l)=(O.,O.)
a(1,2)=(0.,O.)
a(2,1)=(0.,O. )
a(2,2)=(0. ,O.)
a(3,l)=(-1.,O.)
a(3,2)=(0.,O.)
a(3,3)=(0 ,O.)
a(3,4)=(0.,O.)
a(4,1)=(0.,O. )
a(4,2)=(-1 ,O.)
a(4,3)=(0.,O.)
a(4,4)=(0. ,O.)
b( 1 , 1 ) = - ( 1 . , ° ) b(l ,3)=- (O. ,0.)
b(1,4)=-(0 ,O.)
1 -., 1.
•
•
c
c
b(2,3)::-(0. ,0 )
b(Z,4)::-(0 ,o.) b( 3, 1):: - (O. ,O. )
b(3,2)::-(0.,O.)
h( 3 , 3 ) :: - ( 1 • , o. ) b(3,4)::-(0.,O.)
b(4, 1 )::- (O. ,O.)
b(4, 2):: - (O., O.)
bU" 3) = - (O. , o. ) bU,,4 ):: - (1 . ,O. )
c The fI r.,t loop: (par ùllleter change)
c
c
c
44 cont 1 nue
doSO 11=1,npar+l prolllct-porlllln+fcal(II-1)*(pdrmaX-panoln)/npar
If (par.eq.'mlu') mw-pramet
If (par.eq.'ra') ro=pramct
If (par.cq.'wr') wl-proillet
If (pdr.eq.'ah'> ah-prarnet
If (par eq.'xd') xd-prarnet
wb-wa*wr
c Calculdtlng the re<,t of matrlX [Bl:
c
c
c
c
c
c
c
bCl,2)=-xa
b(2,l)=-xa
b(2,2)=-r a*ra
u=umln-U.,tep
If (nprln.eq.1> wrlte (9,71) par,pramet
1=0 If lu::O
Jca'>L'::O
u3=O. u2~O.
u1=0.
do 15 k::l,2
ddmp2(k)::O.
dompl<k)::O.
15 ddlllpO(k)::O.
133
•
•
c 1hc second loop: C<;pccd ch"nge)
c 12 1 =1+1
c
un=u+ustep If Clf\u.eq.1) un=u3 u=un
c InItIal guess for kk:
c
c c
kk=Cwa+wb)*bbjC2.*u)
le=l ieont=O leonvl=O iconv2=O Clg0)=0. elg(2)=O.
c The thlrd loop: (kk change)
e
c e
14 lcont=lcont+1 1 f C lcont .gt .nc)
wrl te (9,70) write C",70) If (ie.eq.1)
ieonvl=l ic=2 lcount=O goto 14
end if leonv2=1 goto 35
end If
then ne ne then
c Calculating aerodynam1cs: c If 'optn' is 1 then aerodyniJrn1C~ will be udrulotcd U'.1ng
c a rational functlon approx1mat1on.
c
c e c
1f (optn.eq.1) then calI traero(kk,Clh,Cla,Crnh,Cma)
goto 18
end if
elhr=esvalCkk,ndat-1,brlh,rlhp) eth i =csval O:k, ndat-1, bl lh, 1 1 hp)
•
•
.\jljlj~.V/)/X JJ ('O.\I/ I (,"j FU jJf(()(,'U.\M U8'/'1.\'(,'8
C
C
c
c
c
c
c
c
18
Clh=dcrrrplx(clhr ,clhl)
d ar=c·.val (l'Y, ndat -l, br 1 a, rI ap)
clal-:cwnl (YI' ,ndût-l,blla,llap)
Cld=dcmplx(clnr,clal)
CIInr=c·.vill (H ,nddt -l,br IlIh, rmhp)
Ulfl i "twal (kk ,nd.lt -l, bllnh, lIInp)
Cnn-dclflpl1(rlilhr,unhl)
tlnar =c~v,tl (l'Y, ndat -l, blilla, rrnap)
Cllli' 1 ~c·.va 1 (H, ndat -l, bl ma, lmap)
CllkJ=rJCfllpl x(cmdr ,Clllill)
contInue
c CorreLtlon of aerorlynûmlch for the location of E.A.:
c
c
c
Cllfl~Cmh+C l h' (0.5+ ah )*0.5
Crna=CrnaiCla*(O 5+ah)*0.5
c Calculatlng the re~t of matrlx [A]:
c
c
c
a(1,3)=wb*wb t u*u*Clh/(2·*PI*bb*bb*mIU)
n(1,4)=u*u*Cln/(pl*bb*bb*IIIIU)
n( 2,3)= -u*u*Clllh/ (pl *bb*bb*rnl u)
a( 2,4 )=( 1 a*wil)B 2- 2. *u*u*Crnil/ (pl *bb*bb*nl1 u)
c Solvlng the elgenvalue problem:
c
c
calI clgzc(a,lil,b,lb,n,1 Job,clga,clgb,z,lz,wk,lnfcr, ier)
do 36 l=l,n
v(l)=elga(l)/elgb(l)
36 contInue
47
J=-O
do 43 k=l,n
If (IOlng(v(k».ge.O.O) thcn
J=j+1
cigva(j)=v(k)
do 47 JJ=l,n
elgv(jj,j)-z(jj,k)
end 1 f
1 ;j.)
•
•
.\Jl}'[·;'vDI.\ H (,O.\/P1'TL'f{ l'H(}(,'H,\.\lII.''';/ï.\'(,:;
43 conttnue
c c c
c
ww( t c) =kk*u/bb err=abseww( t cl' tmag(clgvù( tc»)/llllag(clgvù( IC»
lf (err.gt.acew) thcn kk=tmag(elgva(lc»*bb/u
goto 35
end If
c End of the thtrd loop
c c
elg(lc)=etgva(lc)
do 48 JJ=3,n 48 cIgvc(jj-2,tC)=CIgV(JJ,tc)
c
c
dampO(lc)=rcal(ctg(tc»
if (tc.eq.1) thcn
Ic=2 icont=O kk=(wa+wb)*bb/(2.*u)
goto 14 end If
c Check the convergence condttton:
c
c
c
c
if (iconv1.cq.l.and.tconv2.cq.1) thcn
write (9,63) u write (* ,63) u
goto 50
end If
35 1 f (nprtn. eq. 1) then
c c
wnte (9,69) u,(etg(k),ww(k),k=1, tc) write (*,69) u,(elg(K),ww(k),k=1,IC)
cnd If
If (Jcase.ne.O) then
k=jcase
goto 42 end if
k=l If (iconvl .eq.l) k=2
•
•
c
c
1.2 If (db·,(o,lfrpO(~».I!?.rJCtd) goto 20
(. If (d.,"Ij:lJ(~).gt O.) dlu=1
c
c
'-
c
c
c
C
C
C
r
c
;>8
29
If (d .. fIIpO(~).gt.O then
1 f lu"1
J t d' e=k
""d 1 f
If (lgn(dafllp2<n) (''1 Ign(d"rnp1(k») thcn
If ("b ... (ddlllp?<~» lt ab~(d.llnpl(k») goto 28
goto 29
(nd 1 f
1 f (lgn(lbmI'2<k)).n('.lg':(d~llrpO(k))) goto 28
goto 29
If (1flu 1'<1.1) then
If (1.1t.3) then
u3~u'll~t(>p/2.
goto 37
end 1 f
u3· d<JnpO (k)1 (ddrnpl (k ) 'Oil/llpO (k) )* (uZ'u)+u
go~o 37
u3=du11lpû( k) / (dd/llp2( k), ddlupO( k» * (ul, u)+u
goto 37
end 1 f
If (Jca-.e.ne.O) goto 37
1 f (ICOIW? eq.1.ol k eq 2) goto 29
k=2
90to 42
37 u?=ul
r
ul=u
00 38 mn=1,lc
dilll1p2(mn)=dalupl (/lin)
38 diJl1~)l (mn) =d.lllipO (rnn)
goto 12
c End of the ,,~rond laop
].')ï
•
•
.\PfJJ:.\'f>l\ IJ ('().\IP( n'Ii JlHOUU \ \111 .... · 1'1.\,(,' . .,'
c
c 20 contInue
If (nprln.eq.l' wrltc (9,67) pnr
loin te (9,62) pl Dmc t , U, (c 1 9 (k) , k ~ 1 ,le)
wrlte (*,62) prJmet,u,(cIg(k),k=1,lC)
do 26 J J = 1 ,le
elgvct(JJ)=elgvr(2,JJ)/elgvc(1,JJ)
mQgnet(JJ)-nb~(elgvlt(JJ»
ph~Le( J J}=O.
If (reJl(clgvet(JJ».ne.O p~~"e( J J) nt,m( 1IIl,lg(l'lgvrt
(JJ»/rcal(clgvct(JJ»)
26 cont lnue
wrlte (*,73) pri1met,u, (m3gnct(JJ ),ph~~e( JJ), J J=1, Ir)
wnte (10,73) pr.'lJ1pt,u, (1J13gnet(J J),ph'."" (JJ),JJ=1, le)
c
50 cont 1 nue
c
c End of the flrst loop
c
c
c
c
C'
c
61 format (2x, 'm1U=', f7.2,8x, 'ra=', f7.3,8x, '<lh=', f7 .3,1
62 form3t
63 format
64 format
65 format
66 format
67 format
69 format
70 format
71 format
72 format
73 forlfldt
74 forlflat
3x, , wr=' , f 7 3, 8x, ' xa=' , f 7 3, 8x, 'wn=' , f 7 .3,1
3x, 'bb~', f7.4)
(6f10.4)
(11,Sx,'No "n'101er for iJny rllod" at u-',f7 4)
(lII,5x,'K oCf'd~ the lIfT,cr lHIIl! whlch 1',10')
(11,2x,'The IJt1on,.[ functlOn i,pprOXlll1atlon 1<, bl'Ing ,
,'u~ed' ,I,211,'to CillclJli,te the ,JPlodyn,Holc" ',II)
(lI,2x,'The tdbul"ted dati] l'. bCIIlg u·c'd ta c<llculate'
,l, 2x,' the dcrodyr,,,rnl cs ',11)
(1It6,39,t16 'U',t26,'Real',t36,'IIfIJg',t46,'Real'
,t56,'lmag')
(7fl0.4)
(2x,'kk dld not cOllverge ~fter ',12,' ltt'rJtlon',,')
(/,2x,'for ',a3,'=',f8.1"I,t6,'U',t16,'Rt'nl',t?(J,'llIIdg'
, t36, 'Oll,eg', t l,6, 'Renl', t56,' III1<>g', 166, 'Olllcg')
(211, 'U CXCL'dl·d Ulllax but a fluttel wa', not fUllnd')
(6fl0.5)
(t6, a9, t 16, , U' , t 26, , rndg 1 ' , t36, ' ph~ l' , t46, , 1110192'
, t56, 'ph<,Z' ,Il
100 close (unI t=9)
clo.,e (unI t=10)
end
1 l~
•
•
\/'/'/',,\ /JIX H ('(J,\-! jJ( "} ru l'U()r ,'U, \.\1 I,I,'-,"}'/.\'(;.'·;
f unt t Ion 1 gn( x)
If (x) '.2.3
1 gn=-' retur n
2 1 gn=O r eturn
3 19n=l
retu! n
end
I.j!)