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an University of Teheran's publication for Continuum Mechanics (in Persian)
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tt+i.r-ro j) La.,1l:y C.? 6.:*,lr$ d/-{ d}+ .+'rb pl*.. -rlrl 4, J-t5.rrr-q ,-J-rlTr.";f 6.:-,1 j1+, c.ll:y J, C:?;l.ry 6 ,S:-h,:y,-r L-l e^L" OiU &t
.J-iU ,Li J.oj +t t r r-fb.JL^0t<*, 6 et:F,g ,j-: ,_/Jr,Jt{Jlc,*,1 -tj, Jl .-r::.,^,o
't'P c.tle:;- olK:*,r rr;l
"[i:.,.r La;|.:y 6.u.--re
1 tlrJ c'""'l gj) .i*J c.*Jr. -l ,r-t' .+F : j> ;il-a:;,.,.S1n*J-f gJ.rlr; 4tet s..aav-o *5rlLr.g- r.l^r-)3J.*r,o J.,-, jl ..:J..i o:l: {:sl ,-& 6La1;o.." ar-9 -7, d
oaztu Utf ,:U GlaS"A 6la,olK:-,r e)J. tu r-9Jreo ls rf sc*,| attr 6Lr.l4b
.$l-s, rr ct'# * s.gtll:,-r{;..r 6tace-,r;'j'ff';];'r3 t't
OUi O! 6ly l,.o1l:y ;t;)tr q-r ,-*\.t 6bc..aL I jtt 6 "r:.,.,o 164,t) 6ta-tg"rr!. ,.r r.i J,6lF 4L-)\^ i+ 6.i-Ft .s-S,y slt lts-*t; 4t13 r, *f.r-,!.f
)yb oaV 6La.1y,*,b .r-l ..L,!"f ,F.S <Stosr* b 3 ,,i:; )9*b t+i ay*i-rr-pkq' €y s l-r-'1i-f .51.411*,1-; .J-LiU (f tJ o:1 6!b tf g 5"r:--a f lJ *ynt.f ,St^t9*jb Glsl-p q clt-^isr-3 Ll c"r::*; Jy'."} &;r.,, 6Lr l.*, ,51it^(,.1:
€-t r-l-b )Jbb.;tic,r-l tr sy? -iJ;l -L; SU osLi:-,| -l-r ./l ;l .c",*i p-lr
lJr +J, 6l) ,-;;-5 s ,F 't:Suai* 6b;y*,b olslr-r/ J* t tii ..r-3!,r.. f-lr
-r-t-b d ,r*V Jti,. 4 rg.-as"r.o 0l,,ii .U-.' osj,al lr; Ooi isf ! -r-.1!,-f ;'>*'l T
,6g tnllr; .l:j1. Ll cJi::.,.,o ,->l-aS,tolK:-,r g;f F.,..o La;l.ry -r:L Larr;b
13,lay-'. te1** q ,li,l.rl, 1t+tt f sp r{ .g;,A r:"-b.il-1, lil-l -r-l-b f.^lJ+JU::t ,fl-i *t.jL;- *y )l gt^t:r-b &.!r.-o,-';L;.-t-"!"f l*,[:, et
.f.,i J^lJt 4Jbl dU ,>l-,":;.- 6ls;j-- a-eJA-*.c J) C,,,::i 0tl.,_LadilJ*
6)9.,e a(*L.t:l ,J,.s
F ;l4lrJ - ,j)U ,9tl'6tL; )) 6fr tFE f - f.gta-t 2'" ..$tS" CU3
6c.*,1 46 ,,*-,s',)js 6n: dJ*i r-, '*yl 6l-as
3 c.:*,1 .f+bl dJn ,,-' 6Vi !:l* tji A-l 'd\s }tJ;l .St"f dUil a*'J,'-
.5y lJi r& d(,- Oyry..rb &*i rJU.rtaS." 6lasj^. olSil-.,, q dlid l-;.li o-,-r:,
t1 ,*J:l: "f-,olr; b Jl$*. 0L-6 ,.>l-,aJtr 6lssj-- olK:-,, y' t) c:;lr rr.ir-. ^-ilJ1
-rirlr; l-asj:''. l.l-J-.u..- -f el-t O3U:"' elily J)U, cto;pb 3 [a;by gLn"iJ-f
..:*ilr
olK:*,, 4 .FF dL; ; .1: ,61+i1.9-, J.(..i ;r l"a1;;L; 3 L-o;l-:y ,11* ,Sly
LAJJ;..- C;,-l ,Slf J €g G-r &fift tf;-'lt dU,Jtrh..- .>l.a*- 6lassz-
lgLa,l:; S.€,-n oa;:s;Jl t - Y Js.i 4t-t"1, 6' , 6r, E ,ulfn-,&- 6Ln;l:y
J+J) *q.$lo-Li a*,t)d er, ^ ,-'c)Le siSfot*,.rLj Os2eu.YK..*lt, &,l+i,r-lt ,oll J.r-y cJ.Lor d,hS;; l, ,.(t:. G+,4 t+iL.,6La;lsJ,..^ldrJ,r rJi..t
.,1:v,g;t-:il-,1by.{ t:., ,t-l *t.&U.f ,JLi yts(|, + sJ::..^^ &-.gta.ll:rr
;-*g a3ltl y Osj,ett 6la.il-5". ,$;Y
v = z,o, * zr|, * 1116, = X vi6i:- rt- I
cy.:Lo.j gfi.gl;.-r:1..r;orl.rot*ii.-.r \ -Y J*,-,1rr.lT;Lo:;gln-J-f 3;lsyrrl
! rlry. .;l-Ul 6rrJb;s e li;i 6. 43U ola' ,->;j"or. J *-f Ui -!;f !'), *. tall.:y
.c.*,1 v ojl-r.rl, G-'-.:yr-f r.l! e|: t#s-t Otyed.ir> rlLr
,Sf r-,!.f go^> tlssl{ ,. t* 6 e*8 .:-,1 .r"L e.:- ;l & c./-l -rr
!t*,i ! *-l*r, 6b-t9-- .<JK" .li:L, 6! 1,,Sldi->\" ,lfU rA r^,.U yl c.:.1i
.J-? 4 ss gsaf .l'Lo*.: r:..iyy-; 4, .t5ro 6 €g Gli ' r.a)B clt* q .r-rL"f
,.,r t +Lal- c:y^?-9 cJ5-,1L:il ,)l* t-rb.lt \ , Y, f y-rti, tt;y1-tyIoyi,;at!
,:L rfi.x q b \ - Y - Y aJst - Oli,f)t- r^,li; d/l jl o.:Liil q JU, 6ly .lp g"+
.r-S-ii.- f-r > g"e ,U; ilt5; :.!.y 1s
(\-Y-Y)
Y : v;Qi (Y - Y - Y)
O?-ti,+.4 1gly a;*921 <Sb L* .S*K" A
(rJl)
(v)
&fiv\ rrE,J&..- 6La.2;*-. (.-,iJt) t - I J6.t
v rlrr, cSl-lciJ;" (y)
' €S - g[nt-l-b .,1.: c,\-^iL..r, bd_t 61L ,.>; jrt-j I jt* Li3 , dU gtorf*,b ,Slf
?rg u?;U L '$.1*i g"-- 6Lac; tJJ .r-*rl orJ.1t{ *rY! _, u>-:,rt1_rJr y'g-? :l:;p ds e, c $loJ; oU;l5i 6elt isf a-z 6 c.,,ai tr l$ cJ\:,.U,r..
1:-x st )l ,-fu,Jr_t ,,r,itJj e* cfJj .J*il, ,-/ yr6*\ri6iJrt . \lS u161r-r.i o.rl.c:.,|
OS^" e36 ua;l'^i c^1" 4, ;l,-id c:l: f+lf clLlj 6.13; t S i -frL.6 l-l,r.rrf'nt!
./Lnc-ii1j c*,1 O<^- g.+ .rl.rllri 6rF a+1' t3;.1'.na ..t.trSr,nU; .l^+ -<_ -,1: r.,bl
.Ji.jJFq J*l-.i l.r 6r& *-l* J &- .5La.,1lry 1l
l-,yL.ljrSt:,_f .<rt<. !LXr6;qb$i uf ttt:*el rl;{ l-f -f (JU
.-t.o.: lr-,_li g.; ll:1|j .;!(V) TtYiili(JJl) u;v;wf1 (rr) Ty;61
6)9.,,0 ctQ;Lt:l ,J,.s
j es; cF si 6s.s C"? d,JJl (-all) cbuyiwi4 = (urv, * urv, * u{) (w,6, * wr6, + wy%)
{ 6to.,1l.t J tSJs.r)\^-7
",-= 'tKt1 LSil t# si ,Sss C:e (V)
Ty;ii = T,jr,\ + Tryr61 + Tryrii
= (Ttrlr * Tt(t * Trgr) 6, + 1l,rvt * Tttvt + Trrvr) 6, +
(T r{, + Irtrrv + T;rrr)et
i es: Lri., si cs: C:a (v)f*i\ = (T,r * Tv.,, + Td Q,6, + v,,i,, + ,r%)
.'y qi (9); (-iJl) c* qt*i; q
.r:-. *,--r* J+ dlJ L ..r-rU .g;l-rf:Lj jl o.:tc;"1 OFt , ,3g,-Jt.t,;l ,r:.:.os t
d,-*.- *,J# ;j*tls -l -,lrJ -F )l: la s(s, Co+- I
w=u*y W;6,;=(ui*v)ii (Y - Y - Y)
?)--lJ.lc 4 L*; jt,, 4 ( rsJ( )
)rv = )w;i; (f - Y - Y)
/"nrJ6rra?)p(+)u.y: y.u:upcos0 (0-y-y)
cri g*,;i ir.. 4 jl t+i .,j-l .:*,1-,1by 3r dH qJlj ,"--;5o -f 0 ol ;t 6,. cl
.JdJU
.&l.S 6HJ6, 1i - \, \, \)qsbj>-r.s\r.!-\-\ A*'"1\
.,l >Je jri, = i >.re j.)e" F\j >sr.f .ri,. * I >sc rt..i^ ft
6r.0, : \o
\o
/ >Je.f .ri, = I >.rc jJi^ F\jl >Je.iri, * i >.re jJi.. F\
5,i =
fo
q.0;-6ij (i,j=\,\,\)6 t-*.* osill*
(\-\-\\
6ii= 6ji=d,, *drr *drr= \ + \ + \ = \$ irif V .srt e-) t*,. U-' -S\F .9$r G,-Fl" g...a\; 0,. \-r c?.t ca1 ,,. .r";^
6,j|j =d,,0, + d;r6, + dn6' 'do *&
j** d>\'" c*,\.y (-.s,r .1> F\.S .gbS) i\ .,, -," Li! c d\>\r^ .rr) t> i .f si, 61 .9\y. \^\
UJr.U. .$\..* 6*" $ls" 5 t*,\6;f1 = 6;
a". \.1 c.,.fp J )-* itJ--\", \. ej 1>/ s-.,l,$rrp e* t*.* -J d;;6; .l> t-\t ,.E$> r
Eijk:
\-\
o
.s: ;uK6; .r\- Jil
u . v = UrE;. YjEj = uy|;i = Uli (V _ Y - Y)
: sls, et ( <st*.r; ) 6/"x ?.e ( + )uxy=-vxu= (uvsin0)6,
f*j;|.:y .,1: .ri i4" 4;l .,:i; .:-,1-11.:y 3r ,y,ur-J1 o < 0 . , ul-r, +f
Jy tf.,*,'l-1 ubg 6 S:-h,.:-,l ,9: 6t ,-,;.. J. r-r* & rlr-,t 0 3 cri;i.t-rS tt-i Y ,Sssu r-tj *-f 0 ,-s\ o-1l"rrl r,0
o";aoh7 .sb
.tfr.* \,0.J8 yp i.,tSnpou. S trjO c-Kr}7 tf .,if \.
t5.3., lS; \ \ \ \ \ d\t c.,.rJnps.l.;k t* y.:U. f\r-1, lU \ \ \ \ \ d\" c.,.rJ^rr. ijk >v 1r\i, ft
t-*.*U jr9 t.ir .i.r* A\t ir;g.ar.,.Uk t* r1rU^ f,t\N - \ - \)
1p tr.llpds. t\- \ - \ A>\'") o>\i,:-.\\. S\$.*t.; q (t = \, \, \) .S\". g.it.;, yrc
.)s', b\.
il 6)9.rp <>l=;L_1 :f ,J-s
€;x€y = e;jt&t (i,j,k:\,\,\) (\-\-\)
*: :+:+ erjr :t {-hitJ 9) Ui. .Jt\t G"UP\" 6 t-* c-3> t.-o'..p Gl-r -i i,:"'A
Jt" g;\;. c>-*.* c-*-Ie
eii*= -e*ii- ekij- -e*j
.j*. tf g fu e;lr t sr\-$ r9\st--l.5'r'p t9\y. o\ ;.
€\\r = €y\r = €\fr : €yyt = o
\-t"
..:,ii r; j.e d/{ b Jg 6s1ry. ?-t' 6li cl 0JSlf^ t-lJquxv : u;6;xvfi = u/i(6ir0r) = e;i*t/i6* (\. - Y - Y)
: ( :g(ry ?.,p ) u& a- 6J-13 7-.,8 ( o )U.yxW=UXV.W=[Or"rWl
tI o, ", w ] = ui6; . (ri6i xwkek) = ui6i . r1r&w&q
= e1${Piq = e;ifltlYP (\\-T-Y)c$ or[i,:-,1e,77..:-)\r -# ,flf sdii ,*F-V ,flf :t eA ^J-t'J) 6
..:*rl
: 6iE o- ag2llS. ,..r8 ( 4 )u x (v x w) - u;i,;x Qi6,ixwkel -- u;6;x (epnay&q)
= e;*eipqu{fp6^: e^ifi*/iwp6'^ (\Y - Y - Y)
e-d 6ek
'3.\rS .g\b$, s....-r+. S\y** \) \ \ - \ - \ A)L- )> tmiqerl,u c-.131* tU yr.o
.tJ". O\. f-.1 l9f\j i);g"aou.
e^iq ejh -- 6^j 6p - 6^p 6;i (\\ - \ - \)
J*1^l .$\..* -Si $$ sbJyJ \r-rr^.(,^ l\ J-- g,J .t *,\ {rS\ J,.\i S3:.* t*,.\. l5
e4p 9o-)re J#r, .lp\P )
e*;q ejt q = tmie Eqjt = e*i tqi* = erm; eitcq
6 >\: o\s o\$** \ \ - \ - \ A:L" lt ;U\ ao. cs-l\*,.
O|-J:*4 agly a:-92-= csla L,r-" cS*19 tr
ej*qe*tq - 1d-i
eptept = ?
.J.,iu.-st l-j os-nt \ Y - Y - Y drt .. dFl u-,u x (v x w) - (6^1 6p - 6*t 6;) uty&*
: (u{^wi-u{;w^) e^ - uiw{^A^-u/iw*A^ (\f _ y _ y)
L-a;l:; -rt* -rr CJF 6i*s n5 r_t' .c:,j: f, Jj J* q .:-,1 OS^.
ux(vxw) = (u.w,) v-(u.v) w'"t-'trtr( "(r" ) /", 9J agsl*b +;- ( g)
uv-uieyjej -uyf,;6i (\0-Y-Y)'$'./ 4*! i ,Sss g"a l-u,lrr c,ry 3 oJ-i o:b L,t ,fi.: -1: 6
uyf;6,1 = u,vf.,6.1 + urvi6r61 + u,rulirili
.$r.,/ 4+:J / 6S-s g+ ,lKd
uyiiliei -- u,,vr6,ril, * urv,,6,6, * r,r,216,0,"
+ urvr6,,6, * urv,,6r6, * urt,r6ri,
+u,rur6r6, *ufrt46',, + urzr6r6' (\p-y-y)UV u,.:ttu6;l-rf.rL-i .&U.-f uv .lf , ,gb a: ,Jg:;b -t? e; g^a,,, 6t_l
.t-d,l orJ -rE lj O:-rel .:[-: ?f 6llu 8 v - u;6iA"i4 = u;vf;8 e1 (\V- y - y)
J-::..^.6 &_lL-l +t ijy- -riii,rrt: ls Js-*+, *f Lar[.r p>u,v, * u,vy * ... * u1; v,p (\^-Y-Y)
.rlr- ,,lt;, cS6+>- ( g)\. u (vw,) - u;6i . Qieir*o) - u/;w**
Y. (u9.w - (u;6y1e).w**=u{;wjdi
f . u x (vw) : @iei "rp) w&r = eqruyf p6r61,
f . (uv) x w = u;6; (ri| "wkeil
: epp/iw$;6n
(\q - Y - Y)
(Y. - Y - Y)
(Y\ - Y - Y)
(YY - Y - Y)
tr 6)9.,p c.UL-;:l ,J'.s
'(c*'l.rlr- tLt +;- ( e )
(o").(ws) = u;6;1vf,i .w&k) triq=uyypu6;6u (YY-Y-Y)
)t*b- /t, <s(a+:- ( t)\. v.T = v;6;. Tpi1i1, = v;Tpiji* = viT#r (Yf - Y - Y)
Y. T.v : TijAiili .v*i* = Tiiilligr = Tti'j6t (Y0 - Y - Y)
A;.tr; TV 3 vT o:1*, J* q Juili,-/ O+"^A 6q-e di'l 6:/ ei).($;-
)yb- )yb +.,- (, )T . S = Tii6i6j . Spqepeq = T;i\ri;6', (Y2-Y-Y)
ad,i.)r6rFdhv = (a.ff)ff *ix(axi) os-t'\v;l-ry ftf -f -f JU,l-*ul o.:-; j 0t i ", dq" 6b;lsy *J.b " .-t-'!,-f &,l'; ff 3 .5.l1":;t;l:y d
(i. ff -, ni6i.ni4: nilP;j : rt;rti: \ 6rF lo.i .-65orL ttttt to:l':'t'*'
;-f ,l.ll1 $ aJrL, L-'1' v;l:rr q ,SU." 6La;lry YJ. Lb
a = (a;6; . "i4)
n&* + n;6;x (af'1xn*6)
,p frls-,1t i Lt cs-9,-i.r,.fU; n:-," tV k Ji , i,SLara-Li 66-llr a*j ta':'-l.,rr
4L-)\- JjV;-,.r>\^--7 sllo,ti lj+" 1.€- U- nJL;t+r.:-)lt t 6;X^r..:-,1 oJ--i.i
drJe ?t' ! 5rf-brl:y.,.:*,1 Utl5"[rti rll^it-p cr:i-rr g*r -:l:.llri Jl;-r d .lt,-r,-/
q.-r.^iq $i ! &- -qq g.7 rl.:.,1l-ri; c-u,-i!.r^, $".rX^ry r:..'A 4y g*l c;;l'; U-
.,.:-^il: f+lf o$ o.:lr clti 6Lr,-, / ot)^s Slfl
" = (a{r{il n,&o * n;i; x (ey4ay*6r)
-- afl*&r * e;*e14np/t**
-- ailfl&* * e^;f4pjnke^
= aflfl** + (6^p* - 6dt) nPf&*
= aflil&* + nPy;e1 - nfr{t&*
= nil4i\ : aj6j = a
aiJ3- d a-,. -elli.l v;1.:y d-ta.:.r" 01J JU" yl t.r--i;b v ! g;L- -t-!at.l;a
.r-re,-H ffyr.1,^o ffx(v ,i) nily.i--l irl"r:"lrs(v ' i) fi
O)-Jr.+ a5ly ozyi <SbLza" *K"
sedcqe1q =, 6^i(,-Jl) 5.r+, clLiir, \f - Y - Y ,JrL^11 orl+:-"1 t:f - f - f{6alU -- 6*i s6** = f 6rF +i) epu ei4 = ?
.,'',--,,€k u.i u;V.it+i[ q b \f - y - y .JrL, (-ill)
e^tqejts - 6*j 6bu-6*6rl
=Yd-r-6^j=Yd-j
f=+lJii aqm "p;B abrt U.7--5,;Ji r, b (dJl) .:^J.1, ,rJlt d.:L. (.;)
..- -.11:
eitq ej*q = 6ij 6H. - 6* 6*
= (YXf) -djj = 1- \" = P
.S_l-,C -at--p "it'j y-t, [.r.rL_l 6lrLA g sg,ll-tr] f _ f _ f
(-iJl) (a) ..(ws) : (v . w)(a . s)
(v) (uv) : (ws) = (u . w,)(v. s)
.J:.S+*r-Li, e q l, kii t3LA,LiJ J, SSa J 4,-J,,i l.llaorj ;r,-l
J-(.-Jl) (uv) ..(ws) = (ri6i . wp)(u**. sr6q) = v4iufi*(v) (uv) : (ws,) = (u;6i . wp1)@**. sriq) = uiw{$r,
,'erb ag2l.K:L"if - |, C1; ,.r$^s ua;t:611-rf.rLi cLa.:;;y_ ) t €V 6L,, rl.:lr ,,ot-a;l L
jt- .:YlL .rr 6; t t, 6t-o11.:-r1 tl* '? s o-rls-,| oJ+ b c-* J 6-S,j;lA, c (aJ.ia
,r.+," xrl;i; ,l.,ti-; U +L+ .i_ (*:-r) g-r_l-b o.:1 *5 €;. &li,.rL,A .r1l.r,r,
FU-* tl;i r-;{; oJJl 4L? 4, .rLl ,l-l 1..:y.: tl[k +L7;s Lr.:*.iyy ) :l-ft*tlti ,.;l; Jr q 4L? .-i- ,-r-,!.f rlr-r: .i- rl;i r-;L.i .i_ ! ,J^? ..i- r,c*,| .:-r.c .(,_
lr
du(v)
d-
Jt:"
ld 6)9.,p c,Ub:l ,J,.s
.fflr b rj gLo,l;'l*;-;.. tl*i Or! ! -l 9-l: F -r^ s,*b 6'.,-
t, |p l1J -r s-*b):-t.e : 1
.'.ldyY JrL" b- cv (J-el 4y s-*b) lby = u;
,-iln;t -r q JrL. L- t uv (g-lr rl:r JJ*,ti) sl s =u/i
,-ild_rl JrL, L- cT (g3.: 1J-t s-*b),K1-:U-: - 7;i
,rlnilytv JrL, t- , Q Qr gril*,b) 4.rL-y = Q;ft
,-rld_r I\ JrL, l,- 6C (jrl+a 1J-x s-*b) 4,rlr = Ciir,^
Y , Y , \ -,,,-.:Li, ..;j{,. tti g:;.pti , tJJt "'' 6Vilr oJ'i 4f gt'.l-f-b tslf
:l;i 6taro-1-i, :l-uN ft *r;*u"Y a;.1-: ,5!lr bra-ti a-t e-f ,g i'J5u'-'t+;l tl
.,-:-.il.: J^lJt 6J*r. a-, 6V) s' dy f, rl-b 0i Sq ,-*E 6'.1:
.t-r-,,-rJ^U; di $ ty 9t"t;1t o'at1 6rt"rfsL; -r: 5'-5r5b +'!
gQ -ur"r ilJ 4l^? &- -r, :(& L;; 6 "lsr
L-'i" 6'[i1 .5[ara-Li - \
.ttf ,-ld^-? 6, .1, a*i.r 9.1 Li| 6 ,jJf L-,1' c n,-r3t50 U- oC*, .5Lara-t'-i - Y
.ttf e rlt1. c-rj-/; € -uV
:[i 6Lnr'o-L! ,Slf. l:8-f l) 4f JL! J.,-t -P drL' 6-.1r tl^? y cOi J 4y.r-f -p1! .l^+ rr;.: r*ir 31 1l ,:'*' {qJ ,ts, JJ € '-r':;U t,1l:
.t-r,,llfrl .ratt:; .5rl-rf-:L;;l orU:-'t 1': iLi q 1';;-b U. ,f\: 'rtJ^'"ui I v; -',ei : s; ua;V.r)-rL, .rLtJ o:l t' 6ta.;3*'b 6c. (t--P -l) g"e d/lJt-.|
i,t uri, ;y,l cJ!, g-l: nl-r,sto-r,*:b slt Tu -v,i * Si = Qi1 s t'r-,1try.6ly
-l gr r5.rr to.;-9*;b .:L^., 0rl: ,ly t JtV ,. tS.r rt+{l tJ (F ;P.r-t-b 4' 6-,-l-b
,i)V ?t' c ,sz;.J;- Jti, 4. otJ:oq .t:f ,! l,>t ,-1315 6l-aua;V.l"r;91} tl-l{
oa+:;6lra .:-,1 ,S-u,')f .i- gb:l .-r-,!"f v;Ti* F t' )J*'V'i- q* l-t-b ssv; ;l,s1.
y*,\:.rbt: ?-t' . ,Ss-*V d^? 6'- ua-t; lr y' (F-J{ cl:l: ;lr! 4;il-"" c.i-)
gLAr-l-b;t bra;U gb.rt .u; t- .i- L-' j: ;s*,,,? 'fr)V ?-t' 4;l :yg &(3gsrgr gti.:l,.l dljlq s-*E d o-r1 6g,-.rb qi ' 'l)V ?t'.;J o-tj els 6lels,tsrr-.rl-l: ebq-e s lr.y.l gb:t 6
'F)V ebq-e,5rl.u'{t-,-r ":"t5
.J.,lo-t-i ,tSi -$1'r Jl':,r'o JJ..jJ
O:- t' *.0 agly a:-92i 16L6 L!^," <S*& l?
,ftts c6?ruliei1*tqv*
e;i&tqvnwn
pG.:l
i=jj=q,k=mi=q,j:m,k:n
,#lt cs6+ru/; (-,lrl drJe ?:*)eil*tiv* 0lr-,: ts.sl:J ?_r-)eil&t{yp (,Slo-'- ?rp)
dt+: "a;V
Jr 0i oJ- L11t? t rSl .rt !Jl.i; u,a;V Jr JA;r;_l*ib eicJ-:--ib Cti* = C1i* sSi : Sjt fl,JB,6!r.$L*,
"/qJ"..J rll*,;-bJhL-&r*-
L-) OrUt t-; l.r -r.1r-b 4, ..r-,[.r j ji 6lar9;b rr d;Lii tt Lnrr-b dl_l ca;, y'
--r u;V rJ drl df )lt-6rS- eV f t +f ,,.-b l, y' -rr (.*_rl - rr;l-crj ji 6lar9;[.l, )J*'V AlAii - - Ai. fl e,.)J4rl-b )l$,-1r r.:^.oj.c_i'rL*j +5 f#ry 4+Si os:r 3eijt = -ejik = ejk; c-r-FJr O+"r4.A..-*"1 clllcr[
.-t^ltf .f d{1,6uatb q$,Jljl. ,l.rU-U ,y8.5{ c-:(,-F
.(,;.:Si : {, urU- s.y?.i-,^S;; A;1 };ls ?t' d.r.or 6Lj.; I -f _ f (JU
.c*,1 ,n-.a A;1 - - 1l,!.,1E:,U )J*rlj.Ftl.tSi .*-rl- Ott - )J-b JAij osli* )J-.V.qJrJ &I b
S;j Aij = - Si.r Aji = - S*, A^, = - Srj A;j
)l J-V dJn pl-ril y.r s $'rlrll, O,U'.U J dlUi .-r,;ld.iiiiq fl$l d,JJl 6
\ s;iA;1 : .d/lJq -u,lf"f bra;l^i urLi Ory ,_.,3K
t- s;1A;i = .
;.: l)YrL-" dt* -lr ol,i .rb.,l ,-5t-* Lsatv 6;l-rfik g.J_j, d/_-r+ il .F_4;l1lu-'14,6tot9"'. .{t<" ua>V.rY.rLr;l ,5lrr)t- 4 .-r-,tf .f 6-1n r-, 6Lii
.c.*,1 oJ*i 4ibl J-s st Lyy i,1l 3.F(tli.:*,ttJlt, +L- tA sal)l-. .5-_) O = Si Ti-SrTii - \
(gl5r^ .:-,1-, i1l.l, ol^+ u, <, aJli-.. ,-,,)ti : eij ni - \(plS"r.:-,! ilt, *L+rLea.,.JrL.. qTii :
^A;i E** * \ ltEii -y
ty 6)t.,Ec,l;|l:l cJa
,SL^-J;.
-J.v; slsr, Cl^nil-fu; : eij* \* o:Vf,S.,- I*.: q f -f - I JA
.-tS;,".n \Wij )J*'V
*iry a*s jp .:)\^? -ii- ,G-il 3k ,ss; LF si ,sss g"+ U l-r:,1 ,-b
Ui = tilkW* * e;t*Wtr * e;ypWy1,
= tiry W* * sirrWv * e;rr Wt * exrWrr * t,tr Wr + eyWyy
d/lJll/r : tryr Ww * er*Wr, = Wr,-W*
/y = ty\r Wr, * erl Wr, : Wrt -Wg/r : tr\v Wr, * erl Wr, - W3 -W1
.J:,sA (k) 4; ;l;t &-r; .slsy o3tu',OJU;. Wij )J**'V fI ty- +i
lAJl+,j).t lAd.jlif - |[^.r-l-b y L-a.;br-, ,f-iu OU )l J.56" l-.o:il :&l .,;t*t ',. bV^ 61L
l, t^,-r--yL t--h CVJU.:[Lr J s)--p C/-J+ t-*:,-l -1r .g"Ll .*l -t ..:1i orl-a:-,|
.:; 6 ,,*,:f L-,1' 'F JIJ:*.Stoqbi 1l c*,1 .J.i f:!' (y-i:*
ttt
-W,i ;r,-*-tt .{ A;1 a-,"}..i.. Slfl .ty€ €l:'*\,1 .:tJ^." oartu ;jl-f
Ui jl,.:*,| drS^- 6c.*.1 oFA;i.Sl-rlr..r--yt",4.$.11.: :ltQSts.*)r.13;-, i :Q];D
drL; \[Aul os-r-\ +f-uq g\.t c-.,o[" c/J+ jl .i- -,r l+ , 6l-,; cLar*r,. ([A)J,
i)r-,-r"-tl- 6 er-*V U--llr-f .ttf u.;.-:,. "r-,!.-f 6;:{^:*-(;' iSf ,".iyr. aS
A*J:6)N , M ,y-i| 4, .rr.f o:lr oLlj ol*' r>;yar 4r--fu rL; !-,y J^iq 0i
.r-rr.J! ,f l^4.:*,1 ir<^- g-r^:tf.f dP N :.bM 6lsl:(M xN r-l,"i,r,
Attt
Art'r
Auw
I t,,A,,
l- ln,,n,,1",L Au' Au'
o:{ : I A;1 (\ Y-Y)
O4-.ti4f agly a:-92; <SbLz-" ,S*9"
6- [r4,arl ,F- \ xN.r--jt".K, .s*,1 &.p c-n.fU c&.r--tt ,-lJ! !lt[ -, N fl.-u:tr.f €t* qt.rU 6"I Aa, I .,iu- M x \ e-i\ 4, -r c.:.-,1 eL'* ,xiL4 f x f C/ .r--iL 6 Jt- .1-r e-ti::f.r, 0t": b La;l-:y, jP J c"b.sLo.r"-yL
u,i{L"., (ta.c.i- .Sblr) \ x \ ,.+,-yt .<r- ! b ,.t" .i- .-t^r.f OtJ b lJr +;Jr r-r-bY x Y e-f ,-t'"jt.<r- ,.Utl,./t5c.*ltr cr"-Jt c"J;l l; SU o.rLi:-,l .J:,o:oo
, g-lr +l;r )J*'U.i- Ft i e-JV;o- c-r-:;b,r,
'6.# .r_iL4 .c*,t ripU.4;jl.,,a J-"lb q5.Sl-rl, ,p tz .p ,_r-_Jt" .{(J:'U.Lr"Arv,,r b1,r jl 5) *( .F rSstjr CtJ * -t'V 6c*,1 L*t'.r"-jL..i-L- 69- ,r-i6.i * jcsljl+f a;i = . q;-F ,-f-ii 4 gtf dF*,- .l.::*^ -kLL"6ss -t'V !6F.r--tl-.i-r,.:...^,n eFF,JIJ, -ftU..r--Jl- fuUt nS o! s->(9
..:*,1 .i- t[L q J-(M
" N) a;{ u--rL ,SLn.r p s LaF
"/t?t? jl *rt_ J+fu N x M .r"_jt
3J .:r.:',-/ 0t .t I Ati l' \ adr .>syoa.r 3 r.:,-,1 ad 6(tf2,-p-JU a' i3r,.,51 ."r::.-r Q-t ft.A,j' = Ajr Q! L--r' i AltfL s d g4-jL ,.(r r-"lb c!-qJl
c-u,.U,r.. o;U:,. "r"-jt-
l-l r.li -r-lU aii = Ari' jlft J.(,,.i' st \ ad : adT 6 ,.*r-jV.*--rl -0rU:-t+orU:"U,y-jib.li"uU a;i = -A,i'\ a{= -atd'S,-;l ,,
e-ji3: .J.i:.,^.o -k'U*.*-1l- d;U:.. ,-p-JU,4 ,-J..ot F s, €ls ,-V .+f 'l
ot-*i_ d,:-, s .b.rl-rw ,glrlr ,SLn.r"-jL .J-iU ot*-i_ Je ! p f I s.,-^ 6s1.,.,,
a{ r.cV aH-% +V fl"r",.-r-,Ji(&Jr,t-)C.? t'\f:t1'.-*-alig.S:-'rf a3l2l y-t'{tt L--y
4ii:Bij+Cij (Y - f - Y)
J:.i!.r F rd.sF r ag + q) - fi + d/(, t_{t?t+ b,_r--yL-. ci+.ei{+ (% +v0 - (e+%) +V
.r"_jl- S_ g"? o17ea'oligl: a,{ 6f "r"_Jt 6.r+,OtJj I - f - | dU
.rJ-*, 0! $ yH ! .*_rt- tJ;Ui .r"_JL 4,I 0.1ti-
, dd+adr &-&r-P,: Y
T Y
lA
It 6)9.,p ol*Lt:l ,J,.s
rq) = | f*u+ &r) *fr-n,-,I+;l dli€4{=q) +V osyqbd'-j?- cbjl -u;rqo ,-jlrl r.1."et't W s%).lKi.-r;q 7: + (ed - nt')
A;1 * A;1r A;i * A1i Aii * Alir::YY
^ A,j - A,j' A;i - A1; Ait - Ai{-,: Y = Y = r
Cji : - giir
.-r-,!"f .*--ll - 0JU:" V i,ssli';, 4) t ;1^lll:'.
Jr ?t' )-rLt'ItAii I t- f d .*-iV 4. ).;4 )v il'e L-'g ad u--/V ?t',*b."^{yL" 6c*,l ori; r.-oy;'PJ & oF€ JL d{'% \ 6 % s & r.*-iV?.,p wr,y-jU 6-Liq oy rl.u .lL,n 6|fi a'd ?.,8 ifu: 'P-iU r:*'-
clr:*:rb
M xN.r"-j|." 6.- a "N.r"-jV s:M "Q ,-*-iV qf s*',..1*'l F.5!b %
.,.:*,t Jsf Or{ ',l-l.ral
6!lr V = a{VJ ,/-iV ?t'.$r,-/ 4+:J
g,i : A* B*j (Y - f - Y)
.rt!t tt;A'.V "r"-jt"
C;1 ya*f d/tJq ..:*,1 eq LFt*i .-i- *.Jtk tli -;s 6.?t',-*,,-p_judJ;-, vlj .f ?-t' (*,r*-jl-p-' vli ultlr ?t' d;lc*,1: ,S-t 4,cS-t- drJli Vl , oM'% * 4)a/ v..st{ti:iF b.""-yL ?t' t.rl5cJt'- .1:
"r"-jt- gyeftrr) cr"-jL &-:-*.,-:*,1 -rltt b.""-yL elt:* t5l{3jr
Al e-tL" .-.<, ,.*j 0.^.a +, . a:{a/ - el Y .:-,1 Js3'e cf{ ; ty.t ,-}tf.J;,f6^ €.itJ cllr; O;U jl ,-r"-yL ?t' '#th.
j ada{e( : dY U c.*,1
&^ ag" : &n 6d^ -- ss/mtn (f - Y - Y )
6p*lr 4?i Or+.J::.,.^ ;"-\c..1 f :lLcln, m 6-11 6
(d")'=(n{')" (A-Y-Y)
.tK;i -ut, %gB = ad flqJ-,{-u=(n{)+ (t-y-y)
.c,.qi :tit t azr;,.t li-, ,-t7 Ll
Bii = : Bj; : Brj'
O?Jj4co agly a:*9;1 2gbL,'o-r' <S+9"
: 98 s cil 6ja;l .SLn.""_r,'V S\ 6+o, 01.i f - f - f JIA(*iJl) (ed + %) r -- a-y'r + gJr
(v) (a-{'%)r = 98ra'{r
(V) effi ='%at - %
o:VV' ,t_l-t\ sC;j : a,i + Ba ,jlgljii.1r ,lKi a{ + q) -- W fl (,--Jl) &;l .:_l
g,i':Cji=A1i* Bj;=A,,r + B,,r \€ -, (a:{+%)r =a{r +'qJr
jlfl,$,:;.,.: ,l#l a{'% : W fl (:)gii : A* Btj - A*' B1,r -- BIr Awr - ciir
.( a{i%) r -- Q}r e.,f , s, ,t-l jl
jtrt r[i.i 1r otKJ offi - V ft (V)g;i = 6;r B*j - Bii -- B* 6rj
. aWJ : !B& - % ,Ki , ,flr,Jrl-r(,-b q
n5'y,_/ "[5..ir.r"-it- r*"lb )1 ,*.-t,4 Jfu ! ey e_iV 4 O 4;"
L-det a{ r>ryao-.I, ad A-;J 0U:4,.;, .ttf,g,_r:L_jtl g\: ;;l-r, ,* I.{.{-rlr (Y x y; a.,{ r'--tV 4.Stf .J;,or.,,. Jl}; I Aij I
detad- llUl =
A,, A,, A,,
A,, A,, A,,
At, Afi Ayr
(v - f - Y)
d,:." elj t .b Vli ii-. jl 5l M,i I Ji *l e-&-r oUi./, cdet o::{:W
.J.lLj rlr.:.")\-" rtaS.:,tijl ,r;.'.6)Aii )eW,-SV-5..:*,1 ,J-i J1(*ii I a,i I
.t:_$,-l -ar-* JJ Os-r-l (rr..fu,/", _ (- \ )'*, I ME I (^-f-Y)
fo
."r^l!.r ad u-_;l- '-fC_f -fiV,@ *.i-t)q.rir: at
n 6)e.,p oULt:f ,Jr^s
|ral:3.xu" oJ*Li rbt#6f 3*n *f J_r* er-l; ! Oli€ L,lL.l:rr.;t-jttt, tOrt- t-) F )r )..a:e y' -t o:9"j rt";l tt ,lL.ry, (,lg-, t{) F } Gs: c,rtlt: .r-9*j
L,r,Jtjr 6lJ.J^r"/ b,-rL.lyrrl$,.6:-e d/-l C"? €,l ?t' c;-f6-f -tt
1l c*,t .ttl* -b,|:t L-'3; V - f - Y drL" 0Li4i,
det a{ = AttA,,A,,
lx,,AYY 7:,', 1l*",,
Al Ar, I
A^ A*I(1 - f - Y)
.:-rg 4" -..ji rl:-r5;$s jl .r,,det ad - A\, (Ardn -ArvLrr) -Ar, (Arv4n -ArrAr') *r4'r (Ar/rr- Arv4r)
Jr y' "/-hF
clJrl;: ..:.-,1t'3 ot-ti ^J-.t'J| -rr Lacll;;r :Jt? -,jtr*
Y - Y )
j .z -,-.,"V d/lltj*: .t3J 'l ,:tslli'- tr lt^o)lrr F *y JU.ft, .i- (,1;t-, t-) F
.g-,11-r5:Lj !tlli"f tt v * f - Y rJrL- ssdet dlt8.-F rL-jt: .:,oi.o p ualt;.rJ.*, Jtd (\ \ - V +lI.,- ,.Lf)./;L:, tSt<li, +'. 6a;V
det a:{ = t;jkAi, Ai, Ah = t;jrAr;Arj Aro (\ \ - f - Y)
,l gl5g cSljl.. ,itji.jr c.-)\'e r*.'; t .lr^.a Lril_9,*,11 ,3sl-w ,51L:;l .t_t?t? r, o-9)\*,
AJtu -. L*.,) a.J.:b oLJ bli,t' \ \ - f - Y aJrt-- c*;j JS-l ;t .u!"f bl.(\ Y _ Y
(\Y-f-Y)
:l: r-rLi 0l;rd ;l*,i I ..1;,U,_r ,*l).r"-jt- det a:{ -- " f | 5g,rb qi '.-,t)tlq S-E ,y-jj Ot+ai, c6i'r-a--o .c*,1 Or* Y x Y .*_-ll - rJ.rU:". ,-r"_.fl- ,,o af
.det@ - D; Dr, ...Dpu,-ri!,_f -6r<- :t J_b t'lb ?t' "$tl'.,,+,.
, Q3 s o:d 6lau-_tV S\ 5.r.+, .itJ f - f - | ,JUdet a{'% = det q36/: det o:ddet ?3
e*, det oJ{ = e;j* Aiq Ai^ Au,
O2*J:,.k4 ,gl, a:-9* CSLaLz-" clJg" fr
\ \ - f - Y aJ:L, s Cij = Ai* Bkj olK{ W = et{?J -Sl rJ-det V : sijk C^ Cjy Ck(
= e;j* A;, Bqt Aim Bn Au Bnr
= eijk A;, Ap Ap, Bqt Bt Bnr
\Y-Y-Yalrt',;l Ll
eij* Aiq Ap Au : tq,nn det ad
'.r-rl,
r.tFl .flfQdet V = det d{gJ = tq,nn Bqt B*r Bn, det dd = det Q) det ad
.det u{'9n} = det gBal c % J dd F;tu,-dt+lF q
+.f-t-r.r6t*t \o - Y - y.9 1- f - y.r).:t -. jl orU:-,l !f -f _ f ,Jl' o
.det ad = detesdr
o1-' ,b
lnt'l=A,, A,, A,,
A,, A,, A,,
A,, A,, AYY
.$r../ 4+ii 0-1;., d:l Li ,-f V-f L ,: t-'.il;:
ln{'l=A,,
."t-l!,_f \o -Y - Y aJ.rt -tf .S-lt*. Js6t-l;l -9c-*11-Y- Y d.:Lrtf Ot--<,_+SIs)-JA JJ ,-rt-pt :_f r-l orl.: ,igL-j A{ -\ Js.9,ett ;61, eSd u-_jV ,-r&
.t:f ,-rA;{e(-\ = 691-\ e;{ = oj (\Y-Y-Y)
:',:XI* A,, l:',', I':l7',:,1 ";')
ad= 4J -' scB -- &-'rtK;i,.r;b 6{% = oJflu4 c.:*,1 J-l-l .rr-.rru &6 y 6
IY 6)9.r,o c:l;lzl:l cF
.t-rg.r{r- r-f|-f"rr-il- dlt-St o.s-re! al rlto' ,-*JU 'r-9r ral3;
al = I ettarr (\Y-f-Y)
;-f 'S.l!:a-l
Jsgett ss| "r"-it-
fJ. r-t-fu.r"-it-
,-t d/6{ -_-det oJ:{(\o-f-Y;
6-r+,otJ\Y-f -YaJrL,;:u'iS',-4fjl d-f -l JU
(.-cJl) (a;{'?3)-\ : %-\ dr(:) (a'(r)s : (ad-\)r
c-i.l.r €lf q^ ad-\ L-,3,a{'%,-*-jUJr?t'rlrJ- ?-t'.J+"q(..jJl) &.(.:-,1 o-\..i orli::*,| \ f - f - Y aJ-lL, ;l)
Q)r a{\ a{% = ?3-\ effi = %^ gJ - Pf
.qj-\ et{-\ = (a{'%f\ .f_ty.Q -l
(V) y - Y - Y Jti-. e{ti 1l o.:ti-,| -l \Y - Y - Y *J.rL,. U-|-h cl:Jd Altfl ! (.f)
f:l',( a:da( -')' =' 1a,{
) 1T adr = oir = oJ
.(ntd^)r = ladr)-t ,;*,- --;;\d' ,-,'-f*" 41.(a-{-')' ,rd/l-11
+J.:-,t e-t' e-il- 4, -r:rr"r bl':;g os-t'{.5! .lha.r"-Jt"i-
,-r-fu.uh:. OrM crti6 .1 S 1--Srf4+J ,-iJ-J*, C/-tjl .sL..-f I -\ -- 9r.f{tlr cJt-- .-./-l-lrlra; .r-.,t .-i'f
9-\ =9r =Q (\2-f -Y)
6Jt-:-i[ -r,L:, ,SLn"r"-yL % : e fl g#"^-"
O:-Jr,{r. ,g(, a:-9* <SbLz-" *&
(":{'%lt :93-\ a,{-\ - %r &r = (a{'e})r (\v - y _ y)
Oli€ "r..:;U -r,L:, adfl ,ti l4_f .-r-rtf"f -r.tg, f^,-f_yU;r,ri ?t' dilJtl.(\t - f +[..- \(z) 6rl.r clti
det ad = -F \ (\A-f-Y)
.r-_jL- 4 t-1 Oli.t' b .rlr-r 4, ,$ ortil \:**i U-l Uf tr +5 g-r+hb6! .rJ^,Otd Y x Y Cr, .r--jL" 4 l-,-r l-, g-1, r1 -r -s-*b .<.,- : j-* U- .SFe-J;,fK-i-,1r 1.1 ,gsj*rV [- gtlrf ,.rYrl-r +fc*,| ..JiJr.:r-y QU;pt LcL6-;l-rfsLj) u : T . v c -21:r- 1.: .,1;ib ?t' r.tL .r!z JU, 4, OlJlt .S-t f+i
.t-t-e.f -*lL il t)sy.at;. '*-JU r$i;r (ua;U dJlJ<rL^;) ui = Tijz; U_ (O{sLt
I u;, I = lT;j11 u;, I t_
I,, I*Itt T*
4t TYY
(\q - f - Y)
?t' Slt,t-).rk^ 1 q--g
w=v.T t_ wi=ui Ty
.ttfg-fVnt2s,g.,oau
Iw';]=fvv]t 4, I t_ l*u*r,wrl=V,,\,rrl
(Y. - f - Y)
/6 ,sb*jb Jl;;;lO - fotK:*,, Ji jl ,1i.....-. d--f 6l-,11r La.;y*jb y La;by c-t; o;t-il yj 6 r.i-1Kjt,
.4F 41a22^-, u u*rl; 0tl-LadJJ.-,r"*j rr.rG, "rJ;,-r-l! € ,-fV .>l-a;*^
.-t-,!.r ,Jful c> U16 r". -1g* c-ru-i! o.t .i .-c,-r; ,>l.a*, 6l-a1'_9^- 6l;a. a j-e,a
fr
lLrl[;,]ll
|lilil]
rd 6)il,or=bb:f ;J,.s
,JJd-...-,.iU 6La1;'." la'y->.a1.: c t'[i gLn.;-lib glrf Jti;t r:):L. yl ar*,-9j -,1:
g*i t+i p s:i-r, ,1--S,fFt, J::...^ ,Sp-i.r=, ,g!l.t 6Ox'rx'rx', jOxrxl4
f+!* dJ! r rlr''+lrl 6,1^sJ^-, crl tJbcr h*S c.{? 6.:*,1 ,Slif 1F-Ji" I.$lor-i 4ibl ..jJl Y - Y J<-, atL s;i : cos (x';, x) o'jet'.
6t-a;1.:r1 o:ls tL.;;l.2r./ Y - Y ji:.1r d-#c.{? i;t J<.":" e-t' \lj2l x; 1x'; p*lf, ,l-l+ , rlrg*l/, .rLa:; rlrb !L;-,,1.,1r ;"i2,r-c-a -l q ,SL., t 6" ,SL., &Q f-*I cl-l{ r3la.rby ,.-...-y 0'; ;bp*L{ ,Jt1," 6b1l:y ,d/lJi1 ..1*,1 Jp. *tej ,4,$.: ;lryj ;.r t-a.;l:y. 4-,J.:-6 )l j;l \ 6 g-J,t,L$) Jij,*, u" ,3\l y-j cr)rL, L-'t:
'G-rg o:l: ,''a+j"ii ;Jl f - f
.toc ttr*;
.loot I tr
I
(JJl)
(v)
| -t JS.t
c,-(r? oJ.:,o.l ..Jt..i.j Jtiil Jr.r7 (=)Ox' tX'yX'y q c'iL. ,sl22"Oxp14 JJu oV:'vt 'iK*s (-,il|)
x1
0̂i 0̂2I
03
xt *z x3
^ter.txl trr 8n 8ro
,aoz
tx2 8tt N2 8n
^te3txt 8sr asa 830
.v0.#
O=nJ:4e agly a;-9;1 ago L2-" cS*lS" f7
6',:a,,Or+4',r6y+ar&
6"=arr6r+arilt+at&
(-ijl \-0-Y)
(: \ - 0 - Y)
(V\-0-Y)
= avej
= avGj
= arj 6j+a&
l,bu"Ji^Jc.{? Ol _fW 6 a,/ "r"-jt"
.-r-JrJ 11
.!:-iJj J) e_jt JK^! q tr Y - 0 - Y
Il] [;ll t:;;t ] [l](\'- 0 - Y)
0'r:ar,6r+arrilt
.J:L-. Oti.f c-t-i!"r- aij j)tJ-
[ .r' I = f, aij11 0i, I t_
g.+ rl:llrj jl os$:...| U orr.i.i ,[-.1-,1r t-
6',=a;j 6j (Y-o-Y)
,xi6 t, ,-jt3r'- e,_*-iV.J::..^.6 *K- 6Ln-1lr I jy gt^.-r--yL )-"W...1i;, 6t+) rlrr .1, gtonil,r Jtj:jt -rrt+i jfr; Lr$L/ 4L-y" 6-,rhb ly-; cc*,l ,Jfulv3.+z'a.i-tl (rJ*jb [-)ttr; clL^"r 6tr..d j.,,.t1u-sy-, lcyt>u 4 q L*f by\
..t-,lf .f .uiI cll-2-9.:
P ^Li;;l t l.6't. 6'i - 5ii.:-ib f+tJt rl-t6t,l, 6ls19^.0ry.:-p-e rttJ"tl 6 6';. 6'j = ondu. aj^em -- a;raim6* = a;qajq = 5ii sY - 0- Y aJrL,
: u",a;V rfii 3.r Jl q t+!-l;:) rsl,-r" -tF-'l Lau"i*Jc.{? 6ss $b L*Wt.(t-'J,g a5lrl ,*,_;L
a;q ajq - di. t- a46/r - I (Y - 0 - Y)
fY 6)9,,8 c:l;lzt:l <Jrcs
9 *-, Vli ultlr ?t' o;V d/l 6 t-r€ t)ar:- jLl q +f-rjq u..xb 9josLi:-l U1."l Y-0- Y oJ.rt-- at ugL4oLi-.Jl:t.i-!.t*!"f a:{o*'-j\F,ylj
.;-.:,1O g*'Ju a{ u--;V rgLaclY-' ;l6; = aj;6' j (0 - 0 - Y)
[6'r ] = [6',j 1la1il t- [6,,6r, El = [6',,
t)Js-fet, ? -O- Y.JrL, sr{3,oJi,:*.la{T
[ 0;' I -- 1 a;ilr[ 6'1, I t-
?t' 4l')\, tf .Sr"f orJ tE 6', si,6La;1.:rr All jp,St^J-;U 6 S:-h,.e:iCr:-,!br.lrl .r.L:,p;:!-sd-A- Y.J:L-. 16r.6i ' 5;i G-slrJ
aij at, = 6j* t- a-dT Ad = oJ (^-0-Y)
.d,{'*-tL oj-, Vlk Jr dj;-, Wlj ,rLb ?-t'otK:-,, )s!'i s f-ll.l;-t, otK;-, r .f u;.SLI-J;. ,5!l: v g;.;t :-ltlr-t 4 0J:5l
O - 0 - Y aJ:L, jl ortj:-,| U 't<ji .4-& -h :t 1r rlr'rl,l,
Ir]
;l .:*,1 .r.,;F ,t-J|}ii.1.r 6
[o,, atr o,. I6'r, 6'rl I o. atr atr I
L"' arr '* I
,:lrft,-p-JV O O{ rj*.:.a:*,1
[;l t::;l ] [:'i ](v - 0 - Y)
O?J:L4e ag(y a;-9;1 cSbL':^".5*9" tA
v = v' j6' j = tt;i; = lrpjie' j
$l-b d.*rLi J s,a;lJ "[i.i 3.: y'.rr) p-.rl-:0;
ut1 = ai/; t{ v' = Ay = vA? (1 - o - Y)
cryJ, 6LauJ y -J
6;1";ltlr: 4 rl, p-lf, gLn.iJy 'J50t* d6tr( Oy6 u--+f rrS-rfb Jr-q .,riJ-, ,-t &ya;1 Jsyeq JUJI .r"-jl. r^"W q-51."r-,!"f ui f-lr*JUJI cr)rL, .st JJ.-r-,tf"f Uls 6c.J,::*i rJlJ g-l: {-f s_*tJ.<.'- 6t"r;Jy ta;i
4.5 A Jt,- :l r-;-ryjL;y 1l rr.:L-e J.,-b-| i..lt-'e &J--,, 6tr-L.-- .fu|--(..01.-i a-,3l:*,1 JUrl ,-p-j|.li.f-yL -l J-.ilf,_f aii G:'4-t^U.J;Jc.{? 'i-ei\6;1*'l;6rU5.rL.; ,ss,J-ljl .r-t'rf o.:Li:-,| ,*sbgl+-il
q - O- Y aJ:L. yS gsf,Slt & gl-$l ylr 4 1*b 6 €'n 45tr 1,1 ,€g o.rti:-,| JU::t ,-*-jV ,Slt
;Fglr-l 6;lK:Lj rft,cA - 0 - Y JrL, d:y J-ct'':"c L* :to.rti:-,| 3 I - 0 - Y U:t y ap ?r-L.
f-tsl € c*"r l, ,*<cao OiU
u1, = ait?ti t- y = v'A = A?v' (\.-0-Y)
.$r../ Ctrlll:gr_l:* ,Str-i-r, yJ.l, f*ly" rt3! 6LareJ J, 6..:-.il:fr^lf uv.:U-.:6ly \" -0- Y eJ.rt -F^-..:-l-161t
u/j : au;u' qamft^ = aria^iu' qv'^ (rr-0-Y)
;l ,L(li 4U; ,4 11 d/-tJq c,.:-,1 g-lr +t-l ,-l-b ;t ,rK-, 4, .*,-l+ -,1: rL-r ..i- 1-l
."t-,tr.f _t)os-ret Tg..lr *-*s-*b.1 ,JIJJUJI O.,,iU( \\ -0- Y d:L,
7ri =aqfl^iT'q,n t- T=AIT'A (\Y-0-Y)
rt 6)e;8 c>Q;Lt:l cJr^s
.$r"/ +rrrJ S :Pcf 'f -fu.rrL:, L-lri L-'y
",3'>l-' y 6
T'ij : a;qaj*Tq t- T' = ATAT (\r-0-Y)
{ay c.ir tf tf - O - Y : \Y - 0 - Y .:)rt-- )t ffi iq 6laua;t^i C"? c...*iy a"'
c.:-l oJ--i orb a*,3: .lt.fL; 6 JEtit O.ili -A- rrrb c..r-9 r ! , .f)! .-t 1L;
.f{:t: Jjq 6 drl€-t'o;t .5.11-:;t dU rr'tt d.l- ,slt 6 '-50t* Jli r-/
^'ii ...* - ai{i^ ... ap, R* -.n (\f-0-Y)
)l gS-; L-,j' t-).:ta:;.. 6,l;+>J.ajl "rd
-2.r t-r;1*,. ok( L-,g U- Tr." ;l ,t*-J91.o ojl.rjl -'. tf;oV F*., o.rL Jti" 4. blpy..:b tb.;l (.:lr-ir ir+"
F* C-l S\t6cr/J+Jc^{? e-JU .4-rK. -b st tr .iJl f - Y ,f5-1 .;:xy sF1l c*,t o:V
Ioiil =
,F.uh;" ,Jfu(i, c.JL- d,rdl .rs )J*'V ,Sl^-Jy JtiJl .-t-,!"f det (A) = \ 3
o:V JUJI ,f-iU, ? f - Y J*' &tlrx#r l'Je;r b1-,--. u,llS'it glf .-t r["f1l .:-,1
Io;il =
a.Srl: ot*il Oli!t'.,*;b 6;rsqr* -r.&:" ,Jfu( F- rdet (A) = - \ .)ir, 5Uj +*,-t )l -b ir*"vj = a;iv'i J u'i = a,ivjoJli L-'j' ( ,# ) ,f(t agLaslss.
L;l:]
[,];]
Oe{J:*4 agly a;-g21 ObLz^" ,S*K" fo
,i*l- agl +sls -: ,/s .t1\e Jl;lt r-iL s;rf. * \ ,-# t^rJft,. Ji;;;t.J.rt-"f JEit LAJJ'., ,-r;rf.'e JUJI c.r.i J il bls-t * lh; 't+, ( cnl^ )
u'i = det Aayl
vj = det Aay'i
x2 rx'2
r -t JKJxl s2r-^ J.5,- 1.o oll.rjla, uJ+:*. g;6.2y- u>r7 eitl
IZ ,Xi
r _t J6;Xrty uJe 2s u$-- 6,1-a.2r^..f 6;l (:)
rt ,ri
6)9.rp oUL_s:l J,csil
,4 t F?.fJ cll{ db-l Jrs u.\:*i Ox'rx'rx'r rt,p-l/, 6la1i-- ft t ' 6 -f JU
,t, p+l/, ebnit-r.J.ib oJJ, ryaz,.1,.' J<-,4U" \ 21'* Jy tf'*L fO' cF*.-rrS,;.r:, b " = 6' + 0, + 6trl:Y
1l .:*,t o;V Jtiil ,p-iU t-.i-l ;r ,J-
-\ /.ry].l\/r-Yl
,*-i!ji-i 1.r q - 0 - Y drL.. ;t -e
:::l Lll Fls-*E rlrg,-lf, gLo";J;, f - d -I JU
Io;il =
IT,il:
L ;
rifr-o
t /V r
r/ffo
r /V r[r][?
A
Y
.gS,l.+ c*.1 oJ-.i, o.rl.: 6.al \ - O - y JL:, y 6 6lrr,j!. LaJJo.,.,_F* r:; l,
,*-iU "$i;.rcr-, ,;1rE ul_r.r b rf - O - y d.:l-, [.'.i_l-2r ,-b
rr;,
f:::l :;]f ;l:] | ,;:l :i:l[-.,, r//v rftt=lL\ ^t,tt .J
g-,.t;<p ag(y a;*9a 66 L,g" cS*& rf
1.1: {/ 4b aglay*'8 'kl 6lar:.aq I u/.,ot ,:tb f - f
u 6-,1L1;l .lbr-, L T g.lr €y -ty3 4, ,rlit, ?_t' *f ,"r;b e;r l-r:rlrr(,q..!-yr_l.lrLj J u,,,at1-j,.g;1j5.:Lo.i t>.1_y,a;r y' q t+!_l.), 6) .r-,!._f 1) c>y.et t
Tqui:v; t- T.u=v (\-t-Y)
s(-t.-r.r bo f 'd o tl"x ^S 5L; J&rt,S.,- Olpq rl,;'r 1,, 1-r, {-t s-*V -f sT oju--rr-b -* Sll. (f p..r-r'g-ry $r"/ JUJ|,rr."+b g_rt- 61.-t )rv 6rrEJG#r^ ! t+r",r-!.f &;; 6l-nll,gl.Lirrr o-r'.i,*.F JTij ;ii>.SLn.i)y 6!b 6
.r-,11.r .r;-3 l) Os_f-a.v; t1y-j"o: -rlr; 6- (c*,1 oli .*.F ni &.rlrf ! 6) +tl .ri.,.r
T;iki=u; t_ T.i:v (Y-P-Y)
,;',_ .r.-.itl n; sr )b 4 V* i;Ut r - ? - t drt-,;l ,.r,i g*,! v; ,l:r, Jl
(Y-t-Y)T,ini-)vv; t_ f .i = jri
rr 6)e.,8 obLt:f J,.s
,,/.i" \ oJ-t,,tr5.' b )v :J-e s l;;V.fT *t qplrn; L-,g ori r.-o1rr; c^6-7'tK;i
o;-yel1Y - ? - Y +J:L." c.:-9'i o:[ir;-,t Ftt'AtJ t'flf;l 6,r,tK:," .u--f g T oig
.t_f'g a::;t J-)
(Tu-),6;)n1 -" t_ (t-1t).ff-. (r-t-Y)
1l c. .l .:tLo oJ-.i o:l: L ".t Ji-,i; J: [-
(T1t- )r) n, * I* ht * Tsnl = o (.-4JlA - p - y)
(v0- t-Y)
(V0- 2-Y)
Ttt fl, * (Tr, -l) n, + Trr ht = o
Trtftt +4r\*(Tn-L)nr: "
fl:rl: ;* * J-| JJ41.y.)U.. jn; ).- aa.5Lac^a7 61L;t$.^s.:)rL. otK:-,: grl
iu- c.:3J te.a .7t-lt a OUiaJ,
I ru-16,i l_ o (?-7-\)
o,a;-;o aJ.l&-" tJ 0l) s_rd
1'-Ii.' + II.fl-III7: o
(.-u-ti"f
(v-2-Y)
s!.rT frL gtn-J-r.
-J*-lr OJ:SI f^ ui-rr +5
Ir:Ti;=trT 1,-.cltn - 2 - Y)
= + lftrD'-tr (r') I (v^-2 - Y)* ,n, r11- ra rlII7 :
III7 - e;jr Tv Tri Tro : det T (Vn-2-Y)
OJ{-L.6. agl2 o;-921 ObL=a" cs*l&
;!el -W *l- g"+.-r:-1.;6" T syv f y t f sJ t ,)s( ,sbtoQ *;yttvt :&tl- ;Fg 4i.ii tr T t>-1j,eq T l-p'b 6ll s cJu..t-r6 ill-t s-*u y' .r"-yL
.X_?_yaJ.rL.
,sLn.il-1,6T u;U-JJ,*,tj 4Le\:Y - ? - Y aJrL, lrtsJLs>, 11,y6Laa*i--1
.i-r.ll-rrf )'6 @ = \ , Y, \") Lufu;1-l;l .i-_.,r dljlq.r:*^ & c.:;1.:,i#-
0:y.r-e -e L;\'!^a 1-?- Y..r)st-,,FLir{d @: \, y,\") Jtelc.{?.S:-,,gO".b yj ifY.rL, L-,y Y - ? - Y,.r!rt . cr-., .rJ- O:iS fl{t; = \
I T,i - ltrt 6,j I n!t) = . (q=\,Y,Y) (q-r-Y)tl
(\"-P-Y)n{d n/at - \ (q:\,Y,Y)
Ll c,J,-::*,t rJ,^-e t' f -l*-l) J JJr.: razuA ;Ipl6Laca; c-U-.iL ;L:, -&r*_ 3llv6 f I
s.r;u. raa:al6 L: [-t c^{+ L;; r3il(r) : irv) ,:.ll_ l:,.iq 4;jL,.,, g tf .ul,-,r Jt flJ::*.4 lJ-r-e e J ;c clt-.is9; Jrr"' J. )f 6 _fl_t c..6? ys r,.:Ji-- c./-l.ls ..:y -r.alf1(,r = lto = 1<r, fl.glr,-r ;tel.gLac^a; lf-ru.arr, &* lr:$,;-,1, otKi., ,K.tKi "r*t, r*b ,**.,-U JLel 6uyr-..lltrq tf.;-,tr-rrL:, 6lasja-. a-eJ.,z"A r^
.rl+ lU Jt-l c.{+ Oli€l_1 u;a7 g
r-t JS.t
ax* rx*rx*, ).al 61,u22^, a,.axrxr4 6)-a.s2^u 'F*
rA 6)t.,o c,bLt:l J'os
i3*. T 1*E ,SLndJ;, {4: ojb-l "f e i l) ;t,v,l13Lac.a-7 lJ+- ^*,-l ,Slf
U Y - Y J,*i .;1tJ'" f ,*l 6l-as9^-; l*! '-t-i dbl &$th tiE 6La.,.5,'." a.r '.:*iOrJJi g-,! b. b-l.,r.,. l-eJ.r,.'-r) Cil cttf JU:;l .P-iV.J-ib o-Lj orl'r Ol-:; ax* rx'*',x*,
q fiJlJ:ea1 \" -2-Y-rq-t-Y.rYsL, tl 6u-&U*n{d tau".;;Jc.{?
rs 6l-,!"f a;i = nl') *'-* oU C-lrQ .tt-|rf f.!" J.rlo-t-i a*,t-.r A 'L yl.c.*,1 oJ-i o.:l: rlLf .rt* f l. * J;-t+
x\ t_ 0, xt t- 0t 4t-E
rt 'lel lr_Ir 4tt = ltrt(t) att nr(\ ) Arr = flr$\
r* +
€y lr_rv att n\(Y) afi = flt(') ar = flr$)
^* *er 1.,_xt aYt = llrff) 4n : flt() arr = llr()
T" ij = a;, aim Tq,n t- T* = ATA?
(\\-2-Y)
,li "r"-yU ^5
tJ T lJr +jJr ,-l-b 6lel
lJr qJ-. dLr-r_l-b ,Jlt JU:jl OiU ! aU, 3 .:*,1 JrL:, cJ$::t .r"-il- Lr:r-| ,,'$U..r
(\Y-t-Y)
.-i-,Ji\,-tltql ,-J.vJ1rU, ttl -ei6c*,1 ,S-b.r"-it- .t'-t' diir d
6bc.aa J JelJrU, | -? -l dU
.l"l
IYl=l?;L. .
If,il
Op.J:*4 agly o;-9a 16Lb L:gr" cs*g" r?
6-j, t o
Y Y-i O
o o Y-),
.r.i r,6lF (y fu.(^So 0i Lr +5
(r -1111\o -v ), + )rY - Y) = o
t_
(r -t117 -1)(\ -),) = "
,Jrt'."ei(r) = f 6lJ.J::.,.^ T &l,1rU-olr,) = Y,lrrl : ?,)rrrl: \ e -l-rd/ljl.$r./ q;s ! 1j .:):t-- 0 - 2 - Y
Yn'*YzY:oY/rr + nr : o
hy= + \ 1,rll:n{ti= f jl .f-l,1Q s*_*,go.,tOUi"r;Uft\:fry = " ftL;js.$r"/ +ap 0 - ? - y aJ.:Lr, )trt : ? elt
-Zr*Y/ty:oYnr-ff\:"
-tzr: o
Jkt: +\ /fi-U- (1 n,,)'f nrY = \p*tlr.:*,lnr= o bnsft\= ynr.xl:ie.sr.-/q,":;0 -?- YalrL-,l1r): \ elL.fl\: -F y/y'-O
Yn, * Yru,,.: o
Y nr-flt:o
6)e.,pobLt:l {,.s
h\=t \/y'|""r,nrr * (-lnr)'- \ t+!-t sr3k1 = o i:-r*r\kr=.i,rr+-o,a
;t .:-,1 o:VA JU:;t ,-r"-Jt eb al;l .-r^l!"f ny : + \ lr-O -l
Y/\6- +\/6\/rfr- +Y/ft-
I oiil=l]
I'l-r.- nf \:*i 6J,'-t U.t<'il ji.t- IJSL5" oattul) JLel 6las;r,- ae-9*->,a;, nj
6.:-,1 '-r; *!; 6lo'f o'.JUJl ,*-tV.f ll- p-, ;t La.:-)\r 5rr ei .u\'lo:l: ,,a:i-;i .il*,i a-r cg",:qto.,a .-r:as rf$i; l, tf;-,1, lr-j*->,a,{ .rl.l 6l-a1.y-
. - ,y \:-,Jl -vls e-jV ;l;ltfy t:,-i)-9,.- ?t' q ",:*l-
.:-,1 "r-L:, A 6 tyr-f+ .:r)\.c 6!b arj -..";l orlq;-,| 6 t-rr-foJ.aLL \ Y - 2 - Y d:L..;l(, .;)[
;fs cc,*l il e-tV 6l.,111 6 6s_*E 6lJ. JIelJrU,6-r+, O[.il; f -? -l JA."t S,;..+ \ ;l,,l6tac.a;3 rllr.-t-b-1,
.t'frr JeV J-i a-v*
[,* ]ffi l][ll:] [i rffi'*]
[,ii]
rr;t=f I ; [],'r' L* \ry r j
O?J:4e aSly a:-9;i CSLaLza" *lS" rA
.f:!.-f JtO:_teet?-?-\ d.:L,6l-.i-l _:t cb
l' 6-1 \ ,rllr 6-^ /rl :.t_tL/t \F ,-^
)
.J--X JAIF 4/ya;-;,A aJ.rL, di;l at
,1r- \?)r\ + lt. )v- \yA = o
t_
11,-x)Q.-Y)Y=o
.$r"/ rn1 0 - ? - Y aJ.:L, c.l11y = A
-ynr *n, 1,,/Tnr= o
nr-Yku*Fnr=o
slt
,'-:.tr*.6l)- ?
,l \ n, + fr-n, - y n', = o
jl o.:tpul ! .f-l11t*: Jtur : r[-\ n, ly J f-lr O).rtg )l Snt = fly c Jjl +J:1,. 3r;l
7 sls n;n; : \
(nr)' + (rr)' + (Fn,,1Y = 1
U1I+r- J;elc.{?-rr&rlrf t.1;i )lcny:+ \ lF1n1 = ny = = + do;t .:*1 .:_,!, (-Utf,-f .rt--j oy_1 1l:L*'.ij_rr- 6) lr,) = A
ff(\)-+G,+6,+f."f-0,*
rt 6)t.,Eobb:l ,J,.s
;t c*,1 s:VrlK..!i ./tstil 4 f.5"t ?till l, ff,r, J rJ*-e &- rlrf y' , ire dLt
ff(Y) - (-0, + 6,) lG - iiy
6 e.:-b. o-; c/ J',, o_fi1,,; : itr r x i,r, &.1, ;li,r, otKi
ff(r) =: (-6, - 6, + frE) = er*Y
tJ Js-req \ \ - ? - \ d:L-. L-'y.J^i dbl JIel 6l-asy- S\A JtiJl ,e-iV '*,.a:*rl
:I \/Y \/Y
-\/fr \/fr-\/Y -\/Y
\/
:l\/I o;il
o;tJl o;U:, lJr +J, 6l.a;_1*,b +t? gtlf d/-l+ c c.'o*j .rl e -tig"+ ,,T-\ uLt,1)U, (f ).r*:!.f 0L*<,- Tt-lT ;t,,,l6Lac.a-7 :
")..tJrU, (\ ).c*,1 .-r-l
,JLnt-l-b ?-t' (Y) J::..^A 01-*i- ,gLac.6-7 6!l: 3.r y' 3 J.::*r JrU:" T rJ.rl Jrtl,. t1
b--S *4 ( ,p ) +t' It tstt;* -,-l-b 4 tr) Jrll.r 0l--<,- ;tel trU, QT ; TQ
q -SleU,-f O2t4 a+!i ( ,* ) *-r "r^ilr \-P) c..:, A -UJ)U, €W -Sl
.r.i[ (,-lr) c..l, fL -r -fu ,*lrl$,
lt.t-t*C Jtl--.- e agyi| ,gLa,.ll4v - |)J*tU 6,t- t 4l*J y' tr x c...*i-1,..P 1 oli orb ,.at.a:il cSlS*b O/"a. 6-
GU:4trU *,Sst t lVijl :3-ta. ^.-U
51 6s_s x.;".1:6S!.,_f T;i...1, $, t)
Tii..* (* , t) 6lalil_f -Sl ,>-,1 6,f )-;lrr,, ,[:U ss ,J-l ;l j a:-,y- Ohi. .tS6 6:,JL ,Jli "s! .t:il_r.f el€-r -f )l Cs-*b .gLarll.t t .-\.iU t s x *y, ely, Q $ , t) 6,slKsL; .t-,-r- .jjt ls 6,sj*'V s 6-tltL c6rJ-e
";t;.5LaOl"r.1.rls oL:; Oli d T4 6 , t) sv; (x , t)
O:-Ji,,k4 ag(y o':-921 c1b L:* ,S+9"
a I u ,rltl L-,9 I, t ,;> e-, ,.:*j 61-*b 01J.4 e- jf 6$S4ty,-,6;,j*;lra: ,fl.: -p :l .5'* {.i:t b 0lit-^- JJ,^.'-. dl-p 5-ur:.r, u&t^jdu osf*, J-s-re! o<^- 6-ru"or.r, OL..ii d I avq;iltl l.x, t>l.a:-- a-, c*i CfJ<^- 6 8,10! dt I &ru,* ltl, g_r,
'jf FlrA, Jlp r,i.-a q.rJ4 r.,r)B
.i- ,llPn: ."t^l!"f .-i#j d,*^ q dq ,JL^FLr{.: -e c:J,..i r,al; 0* t>.1j,,ott
L& o,:;g,r) r;.-,_t-- 6t""1.2-" .<rt<r 1r .- *,1 .jJ, '-o .gllj5rL.*l 6tr*,rr l,.-r,. ;jli. JL ,Slt.;.,l,_t -'E r.o ,>l-o*- q + jf 6-5,I;l;lq1 orb ,ltl; glf !.,
,Sl-e,u;jr s aTij I M* Gl-t Tii,k , avi I ej 6lL vii , Wl M,; 6114 Q,; t-gLt'4, .*-1, ,>l-a:;- 4 q c...- ,S;rj_;lAr 6,,,.lb 4?-f Jtj. c,/-ltr AY u; I tuj Mt,
-e-s Js-*! Axi I Mj ,j:;.,j| J.4, ,,+*:j ,4 O+"^i .tFg -r_l-b +y-," -r, .rlltl.a:*rl
(\-v-Y)5cLall.ry Otit-- ;: (deI)Y _,iltl c.:.-,1 oJ-.i o:l: .hb U'.;_1r, *56rlif:L,.; r:
s3..{'* fV il O;,2,etu. d/_rki 6.rlj5.rL;.1:
fo
eilAxj-xij:5;i
v - *u,*dA^d^
-GvT-tr=-E.iA\ &y ' &.; '
(Y-v-Y)
r.sj-.e dr{ bYQ : gradQ J.rs oL_.t$bli€ #_trq .r-Sel: d;,.rL r[S..i
.c-iio# = Q,; (Y - v - r)
;l c*,1 ,.r;L" Vv /rx 6L-:lf}yi : vi,i (Y - V - y)
;l .:-,1 o;V v. V ;*,_ v n$y-"\yi=vi,i (d-V-y)
;l c.*,1 .r-1!o V x v ,-:'l-v Jfe;1p\ivt : eift vtqj (2 - v - r;
os[i:-l U L5l,st-, 14 '-J1..4;;l:y JJ- csL^tS;L;;f .-rb d,.:;l-r 4?y +,!
f:+r Otii 4*,--l 6ly">\:-,,.c*,1 e)J.._!U l+i e J :-w ua;V6rljfiL;;l.!tr*r} Olt€ 0 -V - Y 3 ? -V - Y .r)sL,;lv;lsy -*,SlLcr.ll div (curlv) = o
il <StS.tp c:l;(:: : l rJ,.s
0; (e;j*vU) = e;ipvrel; : "
j ji 6lar9;L.i q,- -*; bjq c:gu" J-V ut-lr ?t'..*l J;l +L-rt t;,* a;;6,8,i-f dJ").:*,1 diGi" bra;Li 0L^ q p;, o1^+ 6JL-.,r.r:*,1 .*--rl- rJ-,1t-c..
.(-r;!.r au vt Ltl;3.5t^ot-rtf 6\$ v p
,.{*l .t,/f JUel ,st^q.bA -fI -t3!,-r 6-S,L;|A,
"liE &p, t-h,6 Tii...r,,ssVl .J.,-l-b Ol.ttr 6,
)r cff ,S'-V ;t .7-f eF t, ! .:-,1 oJ-i r-.e,-r,ri ,-5:-F 6l.oi;l ,:.- +l ,Sss
it-, #-l ! Ett- t-b. JUt 3 ..:*,1 otf 29,."-,u tt g+- S a:*, CF q 6 -Liq Lii .f-l
gtd ds, = nls rs-re\C!;l Gr oKJ ,,s-xln; Jll o - Y J(,:, 4t-t sFJl-Ssl $ o.tV.5l.,1b 6 & Jt-S;sl .rr: .sl.lqb 4 of nliy: qb.'F,gr,.r;t.: JlrSil ,'-1 Olie,rl\ T;i..1 @-, d p:.. 6 f,- Jl-S-il r r:*i ,:*,lT;1...1,
."t.:.f"rr:ta1l
(\ - A - T)
3 ..:*,lf I rlt-L.i; gsltJ1 dr& gLn.ll.r* e\ r--b al f<" oal; cJt- d/-J+
.c.*,1 .-r-i eSfl d./-)Li S u,;V,g-,11-rf.:t^, Jsye J) y' a-, l*i-lrr
J,Tij...r, nq dS = I uru
.r, on
I {*,ds:
| )tnrds- f l,qdv t- |. tids- |. grad),dvJ5 .rV rr5 rry (y-A-Y)
| ,*udv t- l' v.ffdsrlyrl5
=[ div v dV(Y-^-Y)
e;i1rvq dV
s,rJr Jr oJ*i 4-ibl ,--b Y - n - Y eJ.:L, r,J-..i!.-f of nBsy: +.b ,". is-f.-t*,tf.f La;lsy OqL-
d)-r* p s:t Jlf$l4 q b **r g+-.i- e:: Jl-fuI 6" ,yf a.-2L-
uJI ;'*'rds : I ff"vdS= I curlvdVJv (y-^-y)
O?J:4f csly $-9A cSLaL:-" cS+&
4 q ls G-F u. t-tjr*.o)jt C! &- es: Jl-S.ull A- gSyl +.b cJ.a.:,-r,o tlirl l,
& tf 6'-l-;3 'p C 0'-lJlr..$r./ ltort l) C*'sp 'p Jy Jlf-sl
JU"..4, Al_* q) V - 0 - Y J.*,4t1, C + (fb 21.s, ,l=-(*-" tui s,sqSni Jti n; fl.(c*,| C 6j)t sp '# 4,,-El-rl, ,Sl,-f d C!.i .{i"ys
rf
.)t+1,-{*t -,_J- C ;S ggjs oJ; 6*F,J-rlrJ JUif _f vi s-Sq S & eloi:^.,
eij{rlrsj ds : I.r*ot / i. (v,. v) ds: J:
v. dxs
sc
{(0 -
^ - Y)
0ltlt "L;t
J^iL c"rJ--, -,lrr-i v;l:; dJ ! aJ:L- 0-_l c*,b Jlf{l.th-t lU Orr- Uf.,*
dd81'nP],2
(-ill)
a-t JS.t
rr 6)e.,pa$b:l l..s
J3t*.r-!y-j ott'! ua;t:rl.;-}y .1.: L-v = I x b Jt I - f
v;i; = oi4 "b*t = ei1&1b*6i
r*lr cll-L;
v.v:aYbYsin\g(,;r)txb. a: , (:)axb. [: " (t/)
i.llf-,,gto.;lry .(-u,-; nk- 6trll:y LYfl o1- J uy r uq ,5b2l.ry *Ku, gt-f q c...j f - f
nS.r+., dL-.e = ty' = 6ii ft,j = \, Y , f) ft t;.U J1BJ" g (c..*\ u' sut , u'
..r-:j*r! tl-pt. t) b-lr1
r UvXllr v UrxUr r UrXllv--t---Iu,rllyrur]' - [orrurrurl Iorruyrur]
.rJa* ,.:-,[ or-3 i-y; t) ot-t'e,;6 gV,Si"+ ,St tltl.6lt. t, .l.U", 6t+
u\:16,+6',uy=f6t-qur=6'+6t+6t
ul=
uY=
ul=
-0r-ver)f rra'
* (-4,
* (-e'
.. ,;*LL=
+ YeY-ef)
+yey+rE)
,K-b = b;6; jx = x;6;11 .:*,1 o1l-2P (xr&,rh) 61L:;l ,L;:.51- c.i.iJ,- )$Jf - f
O:-Jffo agl2 o;*9=1 ,gla L:,c-" cS+lS"
0.5c.*,1 ,SJf P ci- 611:; .J;t'. (x -b) , x : o nS-r+, trt-lJ .,:*,1 c:,-b 2(sS.
.-r-,!.r+ b,iiLU_,* : + b r Jf-r
nS.r*^, ot;Ayi11 : ] &4-Al sAOil: ] @i1 + \),5ltj3rL;Jt o:ta;,t !r - rd.lGi SA(ijl rl.lU:" 6Lnr:*i Lfr t Jtjliu/ /.j.r^ A;i 6!a[3a t,4 ,r;b (.--cll)
.tj-i, a,-i7;Aftjl ?-tl -aij=A611 *Api1
't-r-,C 4'-P t) ot-f\Aqil --rA
-;t (:)
A;; = 46;1
B sA.S.rl*-l ,gL^.rr,[- tSll (V)
A,Fu = Ar;iFriil * Apil Bp11
.+-* o.rL:.;l: dtil 45 jt- t, o:l: L*y l) -fJlf,5LatJ: "f.lJ.rf;Lc d - f
6i74i* Q) 6;p1* Q) 6;,6piu e) d,y',j (..ilt)
Ap(o), dru(v) , y(:) , r(-ijt) :*q
-l.rr crt-l .rJ , &-F-l-a;olrl.2 yl1b1,-r-iLb; = eiigrt* JAi = eii&ft ftf -f
ai=8{&i-h*Eia : (c. g) h - (c . h) g ;.:L; ,51tjS:L; 1: L-
'+5Or.r, lt il -ltV "fS.-i
y-;':L . 6ttF ,gbc,*l;r-.t 6st -.t L v - f
rr
etirlriv* (o) erpa{ry (a)
- ut (o) at (Tr, - Tyr) (V)
eii&U Q) eriyapp (-it)t)
. (v) . (.iJl) : *E
.r:*,1 l,11 - J)lib Bik = eijtyj rf-b nj qt, dLl; (.-AJt) A - f.J,-f,$ ,b-;1: lt ui : eilflir !*,.r; .r.i .a-'^,11:y .-r--3[ .*_.ll - dlli:, Bri ft (:).r-l-:..r B*q = ] e*q/; n5{l, clt-:;.(.ub eB u-E uE9.: ,("rrr,li,*url
t-!.rJ.rl sAil ,---"--tBU ,F 6,laii J (;;1; l:B**,l-J,\ Aij = 6,F* + Y B;j ftt - r..tsL; oslL;-,lA;; G*-
rA 6)e.,8 obL::l !,.s
Bw = |n* ; Bij = iAti- *6,y'* : *q
oirG.-- L:,,.^*t T4 SQ f t st-.,- "lt-i..,t*i.;
b:+'VT,fXi )lri. 6 '4ra: cll-.lr l' - I
f':r'-rll tti (T,i + Til ,P"
\-,,I-tI=eijt&;bicp6,la-.-2.?r.,4t+rcrb(f--t,>tt*Jrqt-)f+t"'"tl:bL"...UII-Y..:-ly; .l[yy,
a\ ar aY
b\\bYct ct cY
J#qb \\ -T-Y aJ:L ,c1r6,la1Ar*sb1 SQAti , ai,Sl+Ar;,S.rliK-h!..*
..li:'.lL dd f*.Ai1 isi 1, nf -r,-rji \:-J$ det (ad) = eiit Ary'',y'r*
S U*-r-r J* t :K\\ - Y - Y al:l'-;l lf - r
det a:/ : €;j* A;Aiv4rn
.e*il: €lrA;ig;Lacl3;-,611.;l urtirtz Jq c<^<1 nS-r..': tll'l';
eo*, det ad = tiiy'$i*4u
S:f ,re*^:K-t- .lL; 41.r 1.1 aJ:L ,',-l ,lira !-r .slr"r \ Y - f - Y oJrl'- +J
.r-i .r-,t fi Jy"S l;
? det ad = e*ne;ia4i/rtu
.:j.,i o:1.: t) ottn9Aij e-iV.iU."i, ft tf - f
A,, A,, A,,
A,, A,, A,,
A,, A,, AYY
det o:{:
O,pJr.ho agly a;-9,71 66L!,r" *9"
t'- c'rQ"i" T,T,;j :; :;;:::,:.; :T :;A*, A^, Ar
A,u An, A^Ao, Ao, AF
= t^rp€q,s dgt ad
t),-,tV,ii l* \ :det ad : \ r-#.f nq..;; .r-,[.,4 ;j = 6;i6y rlt4;, I fl AsSt
."fe J-V
s^npsqrn : 6mct @r'lp - 6,'"6pr) - 6^, (6r&* - 6*60) * d^, (6rq6p, - 6r,npq)
..t"-1.91 c,*,-t,-c--r J<-: q b \f - Y - y aJ:L JrLp - Q
t^rfqr, = 6^l^-6rrr"6*
fi gr,sb.r-.lL nS.r+r dL:; lr - r
r?
lbiil:f:ii] slciil=[:,rj]
.J\ii..^_r .ul_9 ,*,-;L tJ.1 lr y'
nS.r.^, ,lt-l; ,r,-; C-l db.r- jV )l o.rlli,,l I t a - rn=l u{s0 - Y - Y ,JrL) <ft Cr" Li r:-,1 ylr,.r-rL 6- C, Al)f/ (djl)
'(r.itl.t".rcrtjl q Y -f -y Jt!..,r+fcfL-n"r-tr.r ( d{gJ ), =%lr adr 1=y
rY 6)9.18 c,U|::l !'.s
I o,il :
[:li]rb,:Lllf]
.r-l3 r-,-;L ef Ji y ^S
&6/T = u'day' -\ = & rp'-.ui[ "r'L:' .r-yL el ft l' - f
deta-{= t \ 6r*^rdl3 f -T-Y tY-t-f Jt:"Ctt;t orU:-l t'., .tKisl:*,1
."u,tr.f
c>l-a;;r-o otJ,r-,, p r: 'l"i: c(,- ,StJ .,--bd -, ft *1,"r ttl"'on tr.r.t-b gf-- ty - r
T;i 6,6oil3' tr g-rr nrr rft-.-^ .p-ls "t-T +S ef .!:t Jt*{y-rb,5l2l: 'JE
,_ts-* L.Axrxr4 q ct-; Ox' ,x' ,x', 6,|-nya. -it .-lt &$ft 6,Gy^- q +oJr:..iL o-r,i .:*,-r, ff= (6, + 6, + at) t '/7 ttt-l 2; 6,19.u Jr \ 1 '" ollll * tr9*'12
[lst lA 0#"4.rl J I,r : Trr = T,t .:;l: f."lr t^JJ."' .r-l C", Jl,L;t q -gor cll-lj
L-,; &$ft 6Lay',,or c^*; d'\l',&'y 6,1-ay',-, ft iy oi).c.[^-J.l, r*
6,Ggti.nJ-r+,,:L5.tK:io-r-i[ o.t ir:*,.t,,ty ,:* Jf q'o ollsl * tf*'5gtsn
4i,e t, ,li.uU glr +r rfL-*.,.1-b -*T ft +5"t-n:,r" +'".: JU:l i,,'-l .r'i.l
.e*,1 .r.-13 ,.1-b I 3 rJ.c S--), ;:i 1: nf c*:; )rl ,>y,-a
l.te;jt, sd;i c;tl*! oOx'rx'1x'r 5Oxrxr4 6l-a1.j^-" rf. P -r,L:, Jtfit 6,- Slt lA-fnS.r*^, ub J*,- ."r.,nr dl-.i;
t'ijk : e ;jr (=) U',i : du {utt)
Y - f - Y aJrL jt -l .r*-L; tt*;l b a'iik : ai;ajnab&q r, (=)s*; y : 6b(S.r{ o:lc*,|
jl -u;.llro &fir\ yax' ,x'1x'r LAI'*L;-.- ,>l-.":;" ol3-t-,, 6,G1ja-.r* rSUlj I t - r
.rl xr 4
X,, foo 1oo YOo
X,, ?oo f00 \Yoo
x'r \ Y.o YO" ^olo
OJ{-t'r",+ a|(y o:-9;', 6bL** <S+&
-t'Li Jt.i;jl "i- njr*^, ,lt.lj: -r.sc,*+ b LAJJe.,. a,..o3az;aJr crl.t Jlir;r .r:-lt ("-eJr)
.-u*:l..r.f ,?:i,oy^'oau.,-511:1t-,-ti 6,G2j-,-yl1x, t x,, * xy - \ /r'T A,*is alrt -(._r)
.+5i,..1,- r-ti.r brx', + b{'t * b{'r : b
rA
l.rt;r //i
:,-)
det a;{ = €ijt
Tl,fi': t \ \
Ar4ry'r*
I'?',?
(v)Y x', * x't * r'r = \(.jJl) [ aii =l
- f - Y aJrt-. jl o:ll-*,| ! f. - f
.+-_-r.i tJot-re\b f _ p _ y oJ:L
I f,i-16,i | = eijr,(T,i-1.6r)(Tri-lld(Tyk-16r; - "
6+,o, clLj; oJrL ,at .b*i ! j
),Y -7,1r + t+ QiiTii_TaTfll),_e;ieT;71iT11,= "
.tr_F er_rvl;Sl _ ? _ y cJ:L tj.e*,lt-., r4.eu..lir--;t ui-L.J 68 ,r;l: 6lr,f I - f
I B;il =
o
-Yr
YA ill:.+5Cr*' \ )J-E (oj,-3 6La;l ;) u!-.16Lrc.a-- .r (o;1-r -r-r[i,) ]-cl ,,-;li,
rt
1r = \V \ = l?ff(t) - (r 6',. + v %) l,l-ra
lr:-Yl :*qi(r) - (-v 6'. + r er) I..lVai(r) : 6,
.l'-.& -b tt l, ,-; ;stl;;..,--it, f f - f
l- o rllB;it:l : ,ll
L+ +J
.:11: .rj.-3 .f-rrL Jr-l ,J-rll-rb 1.: ,r,-tji: j* .(,- n5.r*^, dt5 (.-i:Jl)
,11 JG;;t ,-t--tU nf ++, cl[.l; .-ri-i[ $a ra-,.- ;l*t tt:,j.]tr ft (.:)
t rAtr . - '^r' IIo'il: L\.r!ry-
] ,^l
ni.r*o, ot-:; ox,x,4rr;: trtU, {{ ;}i,! r. r"r.,
l- r 'lr'1-t--l : ,ll
L+ +J
..:*,1 [ Bii I -,+ dt+^a p e-t\ 4S,u,5 ary (o)
[-.- -' Ircut:l : ,ll
L + -+ I
fog* -uur+ ag( y o;- 9;1 <S I'd L,:a" .*K"
-r.-o,: rll-.i; c^-4 .:La; olS,i, e, u,*,13 ,!bd.l, +f l:*,1 c.rtj "l.r-b .1,- D f a - fV(*. D) = D
.r: .,l c..+i.1.. ll:y x = xi6; ai 1., nS
.)*4."r,-rf1 -hrtllxY = xrY *xrt * +'or-t-e,o.,r-ur c-r6r.lrrx= x;6;1rsr,.fp-f
div ({ x) (s)'{5
grad "r (iJt)
* T )r'. nli curl ({ *), (.!) grail (x*)1.1
V' (r tx) (v)
" G)f (n + r) (.:) . (v) - taihb+t) (=)*, I x 1-;sty : *q
.r;5 ir,,r,' lt A ,y-t\ uLel y-:li,. ff - f
t'r o .llKul= | . \\ -fi
IL' -u= 'J?," ub- 6L-r&&{r.11 ..r,ig-J$ &*,,x,'1x*y }-al 6,La-r^ n5-r+,.lb_l
'c'-'l ^:sf J13"o\ l:* JfLt=Y,Lt:A,4-- \Y:d,-t
styttltd g = \, y, r) ,r,-"^,;t6Laca-.1 slrq) @ - \, t y) J*tlrb rr-r.r.;5 Cr..t' y) bji:^.f-yL
I Y -rA/i- \Atr Ilr,il= + | -'rrf q/y r/v
I
I ''tri rh rlv JL :*E
1t9=\ritrl=yrltrl=y
ff(r)- {(lfr 6, + 6, -"r), ff(Y)- + (VT d, - 6, +er), ff.,)- ( 6r+rr) tfr
6l 6)t.,o<:l;b:f ,J.-s
l-q !:e, .# ,gl't\ y;6itr1gL^,S-lL; .'-r:--lLx; '>l-.o:;'-.;: gir.re elif :1 flfv - f
."q-L; JF, fi o$ ,sa;-:-,SL^-,it;l oijl-:i o:p ,F ,taB ,5rljj:t; q b '])rt'-.n-irl g-,! r:-f r:"^-, e-l; ^5.43r cl[,i.:
vx(V'")=i V(v.v)- (Y.V)v (-iJl)
v.uxw=Yxll .w(v)
V x (V x Y) = V (V' v)-Vtt (V)
Y . &V Ql = |VYO + VL. Vd (.r)
v\0il - rYYO + | FD. fvL) + OYtl (.!)
V. (vxu) = (V" u). Y-u.(V " v) (C)
! n5.r+, dt-! .J-!Li,>l-,o:;.olKi, a, nu-13 b 4S!!U SW-6 + v = b x X .,l:y ft f A - f
.c,-.il: f+lr ua;l-',5IUKIL; 1l o:Lu-l
divv=. (*) curlv=Yb (.-lJl)
el-*l L 4J # et-.; "r*-L; ,F ,r*Li 6.,U3rL; *lt fi.rl:L g.z r.i+r f l - I
V I n 6L-o1l:r- SQ t* 6! .:):l-.- c'*,! r:*, .rtJLf -;: or! oa;h gL^.l.lrly,l
;'fc JeVdiv (Sv) = Q div v * v . grad d (*all)
uxcurlv * vxcurlu = -(u. grad) v-(v. grad) u * grad (o.v) (V)
div (uxv) = v. curlu-u. curlv (v)
curl (uxv) - (v. grad) u-(u. grarl) v * u divv-vdiv u(..r)
curl (curl u) = grad (div u) - VYu (e)
.2ltrax; -St,u\,-, tti ,t; Lb, 4r ni k-Jt"i q S )J*.-,- C-' 6,- 612ltV f-*-f " - f
nf r*^, rll.l; r-!U i 4r-'6ssl-g+- "i-b .;: oLi;-P c^:+;"
xin$S = 6ijV (Jji)J,
Jroo.x). itds=rv (:)
O?J:EF agly a:-921 66L*. *lS" Ar
w=curt y,),:1(*),rir,nr {
lw.ffds: I w.grad ldv (v)
6[:-".51 ^rb-r67; qrirr-iJ [6;rry ej ,ildS=\V6ijG,)
(..x-i!.f .:l*":*.JnSJA:. ua;l:6-rli<rL*; t J +-iJ,.t lt t) ott-*.,glo--- ?-ra : 6(e()
Iq"x,6j, i I : ft,"x)(i1"i)
4 .sl.rl, aSC o-t ; i p J:- cFJtFt t nJ.r+, ub ,lr-,1 *--b c(Jr rt - r'c'"-ilr f"^lr Q 8) gt! .:.-,1 fu; o.t-t-\,r.,t^,1lr.ra
"pl-4:
d Q,tu;= "r,C
dr:lJq sxj = xllx oS t+r r.ll:; rxL.u^x o-llsl ,g!rr asxi t4J4 1lsr1 6111rr - f
x,ii=*,", *'-'),ii: ry-$r*l x,ij= !-ffrr;lo.p":l,r b 1a 6i.;;l 6l-r1l:y .l B ;A .Srl*;l 6t-r1j,-L- Slt. Sl+: ..Jlj ff - f
(A.a). (8. b,) - a.(Ar. B). b (rlt)
b x 8 = + (B _Br) . u ,
(v)
(v)
a. A.
I b; - e;i*B*1
b = b. A". a
."uf orfo:,,| \Y - f - y.l \\ -T- y,.r):L jlJri,SLn6iL;..rlil 6ltpslotf pff _f
[Aa, Ab, Acl : det a,/Ia,b, c] (".ail)
AT 6)t.,p obb:f ,J,.s
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.r.:..,i ,Stt*-lA .lr-b s4 b, c 6l.o.,1by
^S p-ierrlU- .r.-:U '>V:;,. olKir l: glrJ.e elirp = tlt (x) sQ : Q @) f tra -r
;:*; r'-l3tj'5 a*)i dJifl .r-tts" Y'0 f"qf,; s YQ Ft-LJ Q; u*;l-' 611j5:L;
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| (f\,i - 1po,) n; d.s : I (f\,ii - q,oi dvJS JV
y-.t ot:.at i.irLJ 6rtr(:L; I +5 Jtu .i
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\dudn
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(Y-\-Y)
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pb;=pi t_ pb=p (Y- \ -r)
p=p(x;,t) l* p=p(x,t) (Y- \ -Y)
Lm = dmAV dV
."tsr*;; _htt_r f."? 4 pb-,1.r -&t M +,,51"L;r jl tli,-l e.,
dy ,F cJr( :y cJ,.s
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JL"-j L-,3, 6 'it
bJe;lA S* c"-L,.. gbl, .(7-6 e f yP 4l'ij .(.jJl Y - Y JS,..i)
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jl .'x[ ;r. a'.;-o yV y.r, d[- JUJI ulilr t5b_l:+ .$r./ ,lLlj ./ Y - Y ,[5.i -1.r
A r..:i GJi.<r- Lt Jrt-, A S* G, J_t Ui,5- I y;-J .LJ 6ssII &.'*j J'^e
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t ;bf A S* .:.-L- is bt, 5 JSs" l.tl oFf ,-i-; dr./.1:;*+ P
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lim Lfi = -4t- -t,(i)AS*-ro A S* dS* '
LM,llm :oAS*-ro A ,S*
(\-Y-Y)
J
(Y-Y-\")
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"l*-f ,Strr.*, e-r.r g;lJ.$t-d C\t-t-,CoJ.l[ Lyf.-f 6 ,hf gi-
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O:-J:'*.A 61, a:-9- 66 4:rl" cS*lS" 6A
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ss ,J-l;l s:r! j,a l.-t^b J,-q JlFr I t) o:Vt,6 = - tr(il (2-Y-r)
a-,";;.,c ,* ,fry J-:,6A,-/u*il ,lji f*? Jlpyl c.^*i 64II c.^J fl .r.o.16 rlLi
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t@) - r,Gt) e, + rrGt) e, + tr@) q
1,-cJl t-f-y)
(v\-Y-r)
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ft.n-S tlS r>l-*,.41A.e,;e +;J y uessVAPB J CPA , BPC * - 6-u"lF oJ.eli c..-L., )l el* C,L.il
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;1,t' l'
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d-r J#P ALii.r,
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f".olf rr>;'ul dS .-ij-3 0-y_f aJ:t,.-lrcliclrf
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td - t,6 i, - 6,(i') 6,.
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.*l ..rra Vl ,t:'-f u+ ,)19E..*- tr et' t' J. U -rr.i- .s!"r t4<J fry st (psi)
Jt<-,t-t3 q'lr r- l, L^,J,l; ,-/+ P ^L;J-11 .:-,1 ;t; qf lt - .rl-- 4
.156" Ol*.(l<si) et' e-l t tt -rlf,f ,1<;t.gl,a-r-t3 9\(MPa)
, _T JK;
+ .rJ el.lot-r .SiL-r rlrp
Jt<-,t,t<" *J'#-JU;$i 1: P ALrt 1s ,il; r;-b glu'dl-r | - f -f cJU
.-t^l!"f s$ CtSj..at t
1o1i | :
tl.t ;r 11.:y (ill).ln'
,ss: 'F rl:y (9)
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f' Y\ -?Y rYl
I tt . ^Yl
L YY ^Y -Y\l
JL"-i 6\B 6 P +tii rr 61l2'it a 63
:+(y6,-yAr+t+)
lJ-' A-SP +tii tr elJ g.la-.;e S{-
o*{ -u!"r J-i ,F q -i-
n
J-fu 4\L-ABC a''.i-a ,31
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<i- t'q,a agly a;-9a 66L':"," <S*K" ?7
o.ti.i o.rl: .rtr>ltl 6ly \ Y - Y - Y aJ:L, jl (,JJl) dtt
Itr@ , tr6 , trd | = - ltq, af , -fyl
blJ rr J', dLn{ij -r slrJ.d/l t(D = tq 6, + 0y 6, - f fer p_-rb C;-rlr:* ;[il yr t_
.J56. Otd l-, t +, - +, + I l-,-f oJ^i urr-rri A-',;e 6:: e:',5) Ct-,c*,1 Y rr * \" x, # xr: Y ^frF € #,JrL* )9-i ;sABC {r,;eal.rL, (e,)i.r.r*r.r,, i = + (y 6, + y 6, + q) ;l c*,1 .r.lt* .e.;e eV ih &_ JUt ,
p;ls \f -Y-YaJ:L,;l
Ir'@ , tr@ , tr6l = [+, +,+l -[ -\Y, -\Y, vv ]
A',;e ;3"*'.tt CyJt JB, "/-l-r,
.td = - \ f er - \ f ey + VV E,Srlrf J*,rr t_
.;-S -b ,- aL)).i_ -1, o3V;., otJ r.eJ,"etr r,t-ri-l , ;l €oJ,oLi-. ,ll; -,1l.ry y ,-it
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t-'-J- dlf JrLli.-Jl-y
grr-li g*,$ Slt :F ri-S -Ftr e,t!i- Js:e g-..r t '*:l:5 ,9ss-l:* JsL;
+Scft^,"."r-,!.;l,rf c*,1 J.rb Jt-.1r aS o:-,24,Lg'-J'1"-,^7;l Tli:/ rJ-l-r r-!
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.:-r,-rotii f-.+J r-rl-r 6,L- J tr db-r-* ,f f -1.qtJtfitl
l' q6ds+ I pb;d'v=.,S VV
(\ -Y-Y)
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v -r JK-t
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(Y-f-r)
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o (f_f_Y)
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I e,,1, | @prr),u * xpb1, I dv - "l,V
Irrunrori) ds. / e,rt,pb1,d.v =
/" ei* I *i,roro r ,j (qrnn + pb*) | d.V - ,
71 # ,Jrt :y ,J.es
.t-# ;* J,-t .JlFlfj l+:,-l;r cr.!l 611-:;t V f:> 03,; , ir"r^.
e;7, oik : o (o-f-r)
plb (-/,a sy--" .!i-- !)i : f rl;i r-;U gtf , oJ:L, "f
l :r^a \b f.it.- I*.: q
,Sly.&-J 0L-b!. orr =ofict--oyr-orr = o t{ srvtoyr *trrrorr = o
,rlsaJL>Jrcfr, O\y = Ott JOrr = or\ l:Si,-l\:*,! i = Y;i = Y
01* = oq ('-f-Y)
,-p syV , f f-* tJLoclL.- "/tii ,, f--+ 4 ,Stf LaOL, JrL; .,,r ,cfl;Q
.g.ii lj o;-r-q 0li./ l, f - f - Y aJ.rL, OFl 3 .:*l 0)U!.
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.,-501* rj )S:,ti6 = o,j,j t- t(i) = o. ff (^-f-r)
n1 s t;@,5b11.:y c A - Y - f aJrL.,*-;Vrfii .l.r
.$lo-t-i
F Jt;;il ,iti 6 -r,,1-b L-'y A - f J<-,aU" Px$#r rjE lgLa;3"*" dr ,- *,i P aLfi;r ,i; cltl
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dsL. yO-p+J:L." d,oqkq*pbp= " grjb Y-Y-Yd:L, slsxi,q=6jrcly6Ll.{t-.f ubK-l-r
t e;11, oft d.v = o
vv
O.uJ:# agly a;-9;1 6bL1D" *8. fo
I o;i I JUll 'fli\
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L-,3: ! -9:r.li
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o'ii=a;&^oq t- o'=AcAf (\ -0-Y)
Ol*,i ,F ,p ,l: p+l/, glr-J-r *t-." glf \ - 0 - Y aJrl-, ,-*-j;.rtJ^r.rFcl o.:1.: 0t*i.j tj JU, Jr L-i 6-r-,lfrf
x1
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r.u, =L i
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\
o
Y
o
-Y
cos'ta,.,
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.-rJ O*+ ls o' ij ,; ,5Lra'iJ -f .$tf 'l
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sin0 o
cos0 o
o\
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6.:*;i fIU,,- bl-1'.,. f 0" ,r-
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.ryLf ,lt*y.j ,y-i\ L-,9 Px{t)
I o;il =
[:
-\Yo
YY
4lLrx' ,x, a2Je )rX' | ;F; ,jl*f g,lf ,l;+ -tF - ! g;t- c1\lSsx' , ,-r* f I
. -r*5 O..n, -r-, lf .f t fu5 px' ,x' ,x' , o tK-, e) ,r. _e tf t, o ,S Ln dJ:. o_9-: g ly ,fil
I x'1.2j^- +5-r.^i'q 6;-.. y":\fl fl .t-S 01{ l.t I oii ] JWt .""-rrL +! l-rft <j-L; sl s a,\ : an = a\y - cos B ol.,KJ,rjl*l J* at!" g,lf r.l;{ 6l_a.2y.
u-:\Q-Sl ,-ry.cosB = f //i;fl.r|.:i = j: f glr y-O-yaJ.:L,d.:yJ-.L:.o
tl 6 a'f;t^," .-r-,!"f a\y = cos Q : sin fl : I I {V .tK;i .-Ut, xr Jx,,r s'tu rtc.:jL"r-- Xt JXt !,S;L- A\l$x'1 _s1-- tt_*€ oJ-aLL LAJJft,.."tr! rt*i 6rl";
5;lr/: \,i= f ,Sll f -O-toJrt--;l ir-r*, tt',.a\\:atr6"r:r.dJrJ,o-t-i ;L;il ar, 61L c..i l:-)\." ;aj c*,1 ljY .,.ii .:.r)\.e) ar, = at\ = -l I {t.:-,t jti ctt-rr &si; It tfu'l.t otK;*,, .<r-;lr5-lx 6la:'.9^-' 4*,--t g;lr .(.*,t.ayr : o Jayr = - \ l r/\, art = t //f-+54++i dp_t q s! 0',, = 6,,
" 0,v
..:*il.:f.olf \ -0-f d-rl--jl .rt)!
vr ,i; cJs-l ;7 ,J,os
fo'ijl=
\ifr \//i--\lF \I'IV
-rA/?- o
rlff o
YA -?t'/Y_?NY \ Y
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tni JVt -sl:LP *til -rr AS c^-L-. a;.6laaJ* ,F f 611 c-*,J- oJ.aLl- \ - y - fg+lf .'-..1oS 4.-f'A - Y - f aJ.rL. L-ry, 6 ti,n) ,i;.;l:y ,t)-4r,;e eirbn; tg ,re yo rlJ;by grlcJ,or,rr JL:j .-jJlq - f
"[ft 66;JtJtr e o3)\*, ..:-.il.r
d/lJti s S,l )^e n; c.{--r, il-r;.r r}i; sltLP st,fV \lrJ" 6lJ.q-Sl.c--i* ,Sll I - f J<-,aU, r, Lr,. .)f .rQ JL";.;l:y jl 6r& ,J.rA ! t, r.li Jli,l
t,6 = *,{*l:p"(\-2-r)
aJst-- 4<i-l 6l:..n; L-,g oJ-^i oll.: clLlj 6Lnca; ..:*,1 n; 4jsJ-e etyi- o J s: 6
GL( ,jtJJ )lJs4\o il.ej c,.:-,1 gLl ,jE i5iolc1,o.!is-r -r*,!-rlyf \ - t - f;l .c,*,1 *( ,.FJ 6^io tt i3r'- l-3,1.n; 1 :y-eP g 6 6,la-r,i.a,f6+^! .la,,Vu^o;1
.gtrri; c ;lpl dlaa-.)e J {i) Jsy 6.:_p-e }it".:Fg oJ-al*i.,. .., 1 - y Jr(.i.$Jl$ tFJ lAat ;e #_l .,1s ,f t
-1, ciL- 4r-,Ji Jl3J dL-6'Wt oUt r;.5Loc^a7 : ")--t
riJ rr:tc.. O:ftJtl, t, lJr +lJ.0JU:, ;j*E -f ,*l,gl.oc.a7 : },.tr)U.0..*, Sll ? - y \}*;s ti6 = oilxj g-lL; jl
"Jl:j )J-U tsl|-,j-s;h,-. aJtu.t):L, J.<jr 1.: .-t:5,r,
l) Js-nr, i-r-t*, \ - t - Y e.JrL, b f=5.t o.:tj:-,| -S;lftJlJ, ,l u-V. 6l-Kb.-t-b,
IrIr,*
1,, I l'] [",ff",?'*]
l
O4-Jt*4 6(, a:-9n 664Do 8*K" vr
(oj; - 6ip) n1 : . (Y-2-r)
(n)tr
I _r J6:,
;Ipl .4-- 4 (V) d;-,krl ca.7 ,-f.- (.-iJt) .5t ,,P a.Lil _rr ,j-:S 2l:.,;
1-sltoil = ojijl o:[.p'.,.| U . o:;*5r$l1r t-
(orr-o)\ *d,r\*oyny= o (JJlf -t-\.)o*flt* @r- o)\*o1lny: o (Vy-p-f)ornt*or.flv * @n-o)nr: o (Vy-t-f)
fJt"o;1 -1j*tb.J[^dJJ, cY-? -Yd.rL, L-,; ogcll* P,f^"d.:L^ar,J."r-,li,-f Oi ti tt-r" o *).al j* )l$4 jni u}.al JV-; ^-i)J.u, LaJ;a.., eJlo-r-j o.JtSs-tls drJsJL";! L; c Jg-*- -tV U-l el_*,..])st -. otK:-,,,lrf ,[+i Al:f
f-J€;t$ c^67,Sb.r-rJ
t\li = \ (f-r-r)
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Yd ,i; dr( :f J-s
I o,i -6,p | = o (o-r-r)
,rr-r;b dr.;.:;a aJl&. 6) t:-',; JeV o'.*,,.r-tll9- e-.i: eJrtt dl$ .t*1 ! 5
o'-rl +rW-III'= o
(:1lr gt,
(? -? -Y)
.*:r1 IIL : llo , lo +-l-e.J::.,.^ oa )'al r.xJ y ste,. ot, , o(r't, o(r ) O!-t'64-l-J 6
l) Js2eq clt*rta.;J y -J
6-r-11f,-f oij f 9* t f 9r , !91 agb|Q y, i1f
ro=oii:trct ,-i;;-'l;
tro = i @,pii-oft): + t Grc)' -tr Ft)l (vv- 2 -y)
UL = ei1qtgrf1 : det c (Vv-t-Y)
dt"t-t, +-, cJ-iU6 ,r;*- glr*l;, 6bls o;U:, JJ*,b 6.oii 'F I,*jE o*
i-; ? - ? - Y nJ:L-. dLon i.,--l ,P- 'kt dt"bu:J a-,Ll, A-h 4; cr,.i,,..a ;; -Li> ;:;:
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olK:-,, c.rrl -lr.ri[ n;(> ,.>g J)r'r 6J.5 ,t-r PX'rx'{'r & ,gttg-tU,gLa;-p.
;l trrr"*, Cfl q g*j o';j ,*z*fu +J.:[., 6 6lotf ,".d'ry = o'\r = " ,it.5Lar;,F,-* JeV fj.lLyy, L
".t
Ott) -O o o
o 0'l\-o o'ff
o o'ry o'rr - o-o
(^-2-Y)
t_
Ior,,r-oll@')-(o'n lo'n)o *o", o'tr-@'rr)' ]= o (1 -2-y)
O?J:,F ,gly a;-9a 66L* r$J&
;pl: fJr r>.;r aJ:t '"6bd*l o<o Jorr1,r:.r-.$L+tl ;tel r.i; 9.:r, +J.:t.. d/l jlJIJ 6.:*,1 c..i (ls-rA;-lr r-1.r nJrLr.l-l .JQrb -tj o:V Ll .-ra*,o .*,:f
'g!.f & o<rt Jo11y J.ar,r.o
(o'r * o'n)'- f t o'tr o'rr-@'rr), | = (o,r,,-o,n)t * y (o,rr)t
tf 4r" JIel,gLac.X; cJi:-iL gr jl leL:, o<1 J o$r, o(g )..t ui; ylb flni$) Jnl" ;l c.:;1.: .:*rJl-l
"tl l,r" n<r_l oJ-ak-. a;!y.-U:.9r-c fAy. Jt-l: tar* yl
tl ,Ki sJ:,.iU o<o Jorrl ! it-.i*.rq,ri L,Ly (rl.-s,56-tl:y) rlal c.{? 6La.ll.rr1
! rj-.*,- ,rLl: ?-t' &(,..i.; L.6 o;p1$) = oslri(" s orfl.('1 : o1r/e;(t) y - ? -y *J.:l--
o,fj")n.$ ) - o <rfii()n;$ )
orflj"'r." ) -- o <t ftr(r h-(r )
_l-,);W.JrLr.-t?.t*.ve ,.-eJl \ o -f -y drt .o;t V
o = [ d(r) - o<rl I n,(\)n.(r')
..:*il: f.=flf li:-,:* n;l) Jn;$' ss r>r.y
(,--4ll \.-t-f)
(9\" -?-Y)
\ . - 2 - Y aJrt-,,jr_rrt q .t,j*r_ C.,*,1
(\\ -t-Y)
4++.,1) -\i:.-.o ,-.,3t5.gt"tra;Vj Si 'Jht3
c*.1 0-rl"- U:J JJ,*it cl;r; trli J.4y
otfljt')n'$) - oiplt)n,() - oiyt,$)n.<rt
.$r,_/ 4+J o<tl * oal b- er:...a raL:, ;lel.Jtorr-J 5,*lrlJ ,yf b* Vl
n;$)nr$) : o (\Y-t-r)f Jnir) 6rl.: ot-:j 0liu" rLL ,j? q 5,-10t1 l, ni() Jn,rr, &.clry J,.L:,. 6
.c*,1 rJ,.$r i ni$) Jni$l d.J'
,i; cJs-( :f ,j..s
crrq o(r) : orrl gP- cJr3,.Q 4;jl,.,tf, ! .,iU I ,-Ji. Jtpl cJiJ JIL J, f I
c\ - ? - Y d.:L. ,l:y ",L;rtE..t3.:*,1
:rftr.a>,-- -s-Po<rl ! Iir" n;K) ").ol
c..e--
c u,,Ll y-lr .ttf yL:- ;t,v,lc^fi ,-i- 0lPq sl3: u"nlt'y ,,9 , a>,i'a .2: ;? folK:-,, 6 \&. tr, <-G,::l\ni$' Jki$) o6l J tf 0*.t t;,l ,.a>;-lsni() Jli,l
r$ !,s-lt-, Jtelj.:; +- Lr"L^t 6Jri*,iti;l fl .lJr$1t tf;-'tt Jtel 6laya-
,A *Ji.> d,rtltr a.a.1.1.:3.:1511 ;l obl c.g7.i-qlt-r.oq dlt€\,;aa;e c'lrir
J=-(*XJ It ul*t 6l-ayr- t-eJ.r's.{P +ti.j -l: tfutt g-tU.5t'o.1-p... 1l 6lacyz*.J:,o.:.t
oath -h -tt JIel r; ,5Lac.a-7 Gss I,t ol-."*- 6la;.jz- ,,fV c1*jj;:
JU, dlJ ,u-,.{,g Pxt*tt;* r.113 tr t, Oi ..lJl \ o - f J<-, atL J 7-},-7 xl
c..l c.{? -1, fl.:-,|(ri.*l5) c.; S S€ )^-e xr* y s j.* aa,,ie ,3i-loslsrlrrrrlt)
aqj = Jt g+"^^ ..r;,.i JL"clx, * cLi;U ;L. W t fl (,S-rL:J) ry -tal-9i uit* j t j-'
6,la.p-'o-nbr ltrrl JUJI,-flJV ogl (Q : \, Y, fl ut-t.JrJ q 1 -t^k- UU;
cJr,:..ib,r, n/d L-i o-t-i .-o1rr; ,SJ-V 6|-2lspxgyx16;!:;l sg.asy,- \ ,rbl 'p,S;ail-f oJ50t* JUJI .J.:Lr.r"Ll C-l L.$r./r.lLij ./ \ . - Y J;"re lf4iFL,baSltl 1tJ*q \Y-2-Y.J.rL*. L-'gP aLt;.;:611-=1 6Lar* yJ,*t,-p
o*,i = ai&^oq t{
.S-l-r'l
o*=AoAr (\y-t-y)
6,s:-J1..tFe,yEorq) sn{d L-'y\ -? -Y *J.rt, ,6tylo.i ro3)\*,
our/d = ok)nik
dtsl .o,flA = ok)aqiJ-i JllJt aui= n{d esl; ,SrU.6(q = \ , Y, fl Alt,€g o.:ti:*,| ,JiJ -rr-b UlJs oalf )l s *-r ami )r ! ajrt - el i* J) y'
.,:-il.r f+lJto1Qqf^i = oglaq4mi
Jrl: J,t :.- L-'; \Y - t - Y eJ.:t'-.;l c-r;U,r, o** Jt4.lft, C-l !--- c;-od,
YACr:-tho agl7 a;-92i <SLaL,:-".*8"
(v)
l. -r JK-,
. ;l4l 6,1-a22^- 6,\ c:.g 6lau.y-.,5 J-l-r+ (V) ,P oL;; s"pxt*xrtrr* .,l.ol crLAJJr.,. (,_.Jl)
;l .:-,1 .:-lk ,-t-t+ n7*i c.r_f € ,J-6_q r:-! c,\..d, t, *5 aqi ami = dr
o*w=6*o(d (\y-p-f)
4,-t:.lj -ryj cJ-a.:,r-o cJ,i{t^j J-.iU oJ.i 'rlr 1to,l ;lel 6l-a1y,- \ eJ t, Oi nS
,*-iU,15-i.1r ..t-,!.f ;tpl -F er, ;iel ri;; yrtr." tl ,-"r" ,yE
ro*4r=f-:'",;,:]L-ro*;;r=f: ;,,:]L. . "n,.1
, L ': ",,,)(\0-r-\.)
(-iJl)
xl X2 xr
x *t
t,r=ilI a,, =dll(ft
fu=Q
xi ar,=nf uo=4 %=t{
x*t ar, =tfl ao=nF, a.. ={
YI ,F cJrt :y t|es
?t' o12 oy2 ont os-te\;Lef6Lar; 5-tor.r, cll:': .5;t-r5:Li t p-l: oJ'rL' -l'd."r-,!"f rlLi-:J-c t)U, )l -b*-e .,'el dt"b,Jij :l -{tL c c":,. 6ta,r; lJ-t.llo't'i
..:-ti Js-ta d/{,Jl-r.,,-/ b.ri:j ,Jtrt-q ' }-"t,Jt^.,J YJ.
L = or',t * otrl * otn = oI * oY * oY (..jjl \7 -? -Y)
IL = o(rlo(r) -l- orrlorl * d(r)o(r)= olon * ouom * oPoT (V \2 - t -Y)
IIL = o(r ) O(9 Og't = O1 O1 oY1 (v\P-t-Y)
yJ.P .1: ui;s sy| gtnniJ;, | -? -f JUpxfi{r 6,1As.92,-,, 1 r:*;
loiil =
MPa
[::
.Yffo o
.Yr
.llo"t-.X asli g '*-tV
L-';
II
l..t$O..* 11 P y ;tpl r;6tac.a; :
")*t6l,or;..:11.:-rrj J* q.iL.t-1, d- ? -Y aJ.:L, i oJ-i o:1.:9l;.rf-b elt cJo
|V-o o YY
o f o -O o
YY o fY-d
aJrL, \-* -b,|:t Jr t-.;Sb-f \\6
(0v - 0)(6. -o)(yy -o) -(Yf)Y(a o -o) = o
;l .:-,1 o:V ,J:' ,$jS )JSV rfK.i 1.r jL! t-
(0" - o)(o-\6)(o -vA) = "
CE-Jqr" agly u-9a1 cSLaL:^" *K"
r-+j +i.g!.f oK) = V0 3o1yy = 0o ,o(t): y0 ;lel r.i;yrtj, c as;rr-l 1l
J.:,b rFl J-ll,St-t, -ijl t ? - ? - f -l -.njl v - ? - y .r).:L, O:,rlr ** rl-pt;r 6,5-F,g
Io:0V * 0o + yf = y0 * A" f V0: \0.
drl*-.,li6l;lo-, ^S 1-#'l:h-rro(r) = yO l"rfl ..rLpl .gLocary o.:*,dlJ
".-ou-c1l: o1, t j-.tc^{? !Ly.-rc^{?6Lar.-pJ6ly.Jrt .a-,y- ? -y
YY n;(r'+ Yf ,rtt) =Yo nY(r) -
q.lr^^
YY nr(r) + \A nr(l) = o
;l -r-"!.r ,r(" = - f ,, tt ) ;1.: aJ.:t - -lr jl -l ,r,t, = o
,,,tt) : = oI
r.3.;.j .s:(nt t)t = * *Sr-* u.oJal.!.. tnp;
.$r.,/ qdf - ? - Y nJ.:L, o11y = 0o ,Sll ,W
v z,(t) + yf nr(t) = o
YY nt(Y)- V nr(t) : o
{d.;l.r k{ti : \ tF ,l ,13;i.J^iLr,(t) : nr,t, = o e.5 sy€i'*,tj eJ &.nr"' = t \
g*tb Y - ? -Y dsl- jlor' = VO,Sl:,,fYq
lso
o
o
- \A n,r) + yy nr() =
- Y0 nr(rl -
Yf nrtt'-fY nr"'=
g-lr al:t ..;l 6c*,1 (\: \ ,-.r_ ,::f JVy L;.-r-tf.fnr"':=f,
o
o
cr-'?s J
Al ,j; ds-l :f aJ-s
,l 6 ty,u" {" -di f ,rtt) : Y /arcl 1(r d:L." -lr jl-l nr"' = . fJ, nJrL. jl tt!-l))
''r(t' : = * J hr(t : = *':*j;l: r=rlr rttt; : \ lr'
./ \. -f #tu Jr-r+ * -tlsl a;i I JU:;f e-iibtSl n{d 1'lt d/l jl
Jr-t-1 ,/-.rl 6La.13"*.- ol51*,, d:.11 tf;*'|rjl OU.ltl d!.i 4(!-l cl.:rr-., pq q
.r;lr ! rj F J.(,.i J[,:;l ,-""-yL,-r",.p:5.;r f*!, J^i[ ffr) = ff(t) x ff(Y)
Ioul
+f-6-o
+f-6
.rJ; ort ;E .!y ,f-JV t, "f-Y!
,514.:-.)to ..y*-l \ Y - 2 - Y aJ:L, jl .,.r-l;Q
I o4* ]-
o
VO
.{lu I ubl"t- ui:i tt:Un V - Y
I JU; ^-ilJ.{, q ,.:-,1 OS^.. P ;: 61.;;l ar.;e 4, .S-l-r t{i) un ,tr"JS6-, J^- aa;,e
"[--l-r.1: 6 os ';lSl L dt^-ilJ..i- q
'lr^"a ory ';l"rrt U ar,;-c
b"il-1, d)& 6La';lSt Ll (r;*j ;1.:y 05 -9
o1,' t-*:r-l 1r) .rrf +,F \ \ - Y J* atLtl c g.al3 r3"L.1 .(r;t'J^i rtatl Sloily 6La:L*j Ol1rq S -lN dLnc"..iifj ,lu.!!.-f
o* soN = t{i) n, ,J.-i,-ls*,! drJt ?t' L-,y o7,r 6 :3-,,r, oJ-aLll, \ \ - Y J*,6 ry.l o-a4ti6) = o;j nj
. =* I+\ " I
o *Y Ial:L
f6o
r6
o
o
\
o
o
fo
o
ON : orjnjn; t- ON : t@ . ff 1r -v-r)
O?J:,+4 gg(2 a;-9;1 6b L*;," *lS" hr
oYs = tr6 t,6 - "t"
oYs = t@ . t6 -o'N
glb c;iLl.q u,'u.r;l #+^,r
(Y-v-Y)
oiy l.ql , ,jg.SLn-J-r (Jt-jJt t i-t:"..) j5l-l- -r ,[rt-r-,rU". 6-* ttc,.:*,!l.r l1P y J<^..6Lac-J9" s"L"t n; JVj sF,-l ,ff Jr.€'l ,gsJls;l cr.Ll All.:F'f bQn{r; = \ t-rir.>.i j \ -V-yaJrt.. L-,yoy1:V
d/J-: €'l65tl: .^l oy JV*SIy.:U, rrrrJi g-,r ,JlJ _tF) ?-t' ,si,ss:-r_y+f ..-il,* o tJ-c.)i .1-t r5f*r./ J-<-i.; t: f(n) = o,jnflj -o (nfli- \) db )JLb
!,-t-,1 -rv lb c*,lnp o r.:*i f(n) .>Ea:- filt,- 'iss C-\.-r-,!.-f yLSA *;E..-:-,!ls frolJt 0n; I ilnp = 6* *3_l 1 q3,
* 6pn) -o (l n;6p)
.{t-.f "i"tSyi
Js-tel aJ.rL, d,*l "r",
.-t-"lf"f 6,rfj = nt s o;j = oj; clg Ll
(oU - o6ry) n1 = " (Y-v-r)
#= oq @*ni
rjui
lt -r JK.t
P y ,;-:S.1lt-r. ,it s JVt 6idts,
Ar ,i; Js-( ;f Jcs
c/lJq..r-,tr"f 0L*<.,- ;tel6torJ 6ly.,j-J-rhL J-rt r.*'- Y - ? - Y aJ'rt'- b 6
.gLa,r;.)ir,t,-ryr s S'l f4l ,,Ll ,JrJ.<r- ,i-L;o ttji ?-t' S p-SC 4+*j
.-r-,!"f J*ti l, 15lt-; ,frl.r- Jt -i ui:i Jl.tL Jr y' ;lpl6la;S-- L-,;.r'l,J.l!, os r.it dJr -flv; ft-r-J'U-.a1 c'"-Lr U.
J;j.etut 6lall.:r -1,>
")'al,JL^,JJ.c.*,1 +L \Y - Y J<-'aU" pxr\r*r* ;lel
jl .:-,1 o:Vt3tlrJ. ,t5--i 1.r t,6 o4tr d S-r'f ?J. 01 ) oY ) ont
{i) =o. ff = ofltilt* * o1p16r* * onttrit* (f -v -Y)
= t6. i qLi ,-b. s
ON = OFlt\ * oa/trr (0-v-\")
f+f;; y -V -y rJ.:L-,rr 0 -V -f 3f -V -Y.ri:L. dt-S A&L- q.Ki..:-il.r
osY = orrnr\ + oytnrY + on/nrY - (ofir' + ofii\Y + oilfly\)\ (P -v -Y)
nr' : \ -t?rY -/tr' ,-p,ft{ti =\ L:l .JSu, bQ.ni c.^{?,3Lar,,;;J *Jbost +5
.J-i $tF nt Jkt;l .*b Lii,lffl6c:y,^, .ji- ? -v - t aJ:L, ,ll, n, bli,g J
osY= (ol-nr') nrt+1ont-oill) nl +on1t-l(o1-oa) nrY +1on -oa) nr\ +onflY
(v-v-Y)
g'J_t **.! .3*.rtj:-i .rr-! , "i JU,#/rr.:U, ,iu- c G.-ic.,*,-lorr3i.:-{ 6tJ
.u-7;l crr, .)-# J- OU;^- s t$ sl-ts .b e:L,flt Jf,t dJr JA e.r r:*i tJ.:L,
a @st)
.f:ir-;\:rr! 6fr .r[Lr
= nt (or - oa) { oronr\ [@yofi nr' * (oroa) nrY11 = "
1-iltn-v-y)
.-.",lylrr o7y11
* otltrl
0n,
07- r..t.o agly a:-9a1 <Sb L,:^" S*K" Ar
a @st)ont
: \ (e - oa) { oronr\ f@7ofi nrt * (on-om) nrYl; = "
(VA-v-\")
lr -r Js;J.al 661y-" 11 c--;P -;.r ,_i5;ls -t ')t.
: JV; C.:,+jt_r
4+Si.ss6ckl: flr: o jl c*"1 c:t!o A-V-yaJst-. )lrylSJ-.<.,_-fu6t*"r,oJ.ol--i; v-v-f drl--;l Oi! W-t ory ,l$._eriJ^lJrzr= t \
6 Sg oa;b\, ,*l a-*;e 6 nr: -{- \ r,r.yry r:*,lrU.jl.r-;y r.::.:j 4, .f-l ..:*lpt La*:,Olirq hr Jr\ ! l* n1" sn, \nf qLi, r*,1:... d.c*,1 -p ,-fJul ,Sr,
J-l- LfJ,ir 6lsa*i-a Oljoq 6_r*: JrlJt -S--, ,*l aa.;e 3.r a.r r2& c.:-9-i;
,l*;l Un-V-f ..:Y.:i-- 6lr:g_rrJ-6rr € c*s.^jt! )y€c;t*, (k)Jlty : -F \ /y'T-rar,r. a-,.:i.xfu" JeV\ ostj s*,I ,Slt,F J kt :. ort
.:l.r-tol-e';brj6h: v-v-Yd:t ,.J*.rl elt.ky = t \ /y'T,lry.r-t-u,;l
dst = |(on-oa), t_ os - x, ](oa_on) (q_v_\")
$: y ,'J.rA.li- }l3;^r.lls nr s\ \nr 3 n1 4;l.sls.r)sL, c^f-r q ft, 1; tst_
n JsE J<*,q lr,J.ts ,F di,ghy : o Jny = " r'f 'es q;r. j r1-rK.
Ad ,i; cJs-( :f dos
.rJ.*, Jl*
ltr\:o, tx\= +\/y'T, fly= {-\/y'T, oS=l}@u-on) I
(,.J1 \"-V-Y)
fl\= t \ /fr , hy= o , ftr= t \ /\ry, dg- I ]@m-o) I
(../\._V_f)
flt= +\/ff, flt: +\/y'T, fly:o, os- |
-V_Y)
bf .Sg 0t* l, o-tV.!LyrLi" o" 6lJ.c!y,.r):L, )l 6:j c Lrt; ui;:6jl .:-,1 .r-,F ,rlu$ )[L a-S,t]6.:*,1 Cls o7 2 on > oilr
uS(max) - ](oru-o) (\\ -v-r)
c c-^4ri crr-l -r: .J*i 4-ibl ,-F tt Lfi t;L:" ;lel,5tari; Slt 6.:l: rlLji Jli,JJ;Scr" Crr"tt b A - V - Y aJ-rt--
,F cSU ty 6lnojj: A - YJ 4-&,Jitri.:-r*, l, (f f -f #ii) ;Lel 6l-a12-.r r.:*ip;: r.l; clll
-f tSs.tls os sox rtli.tr' $i -ru,;L .t$.i/ o7 ) oy ) om ,* t\ ;rel,gtar:;.r-5.i1*;aj rt).:L, L-,y t^ie Oi fi Jt-j 6Lar.iJ y yJ. P .s: ,Sla-,.ie
](n-oil I
(:/ \'
oN = olxrY + o1prY *o'N * ots = ot lftt' + o' ,fl/
ouy'trY
* o\ nrtr'
(Jr \-^-\")(:\-A-f)
.-r-,lr.f $ L-i q,lr^^ 6
nr'*nr'*nr'= \ (',\-A-f).J56" t'ltr, Jflv , n\
G)-rl_ c^{? 6l.au,, j:J ,-, 6lL+JrL, u, 6
O2pJ:'te ag(y a:-92q 6bL'g" *lS" h7
;lsl r-f g-,I .:Y:L, Jr l Ji- jl
llrl
flrY
-_ (on - oil@N - on) + oY s
(o1- ofl(o1- on)
-_ (oy - oa)@N - o) + ot s
(oa-ofi(on-o)
n_t = (ox - o)(ow - oil + ot s'r (oru - o)(om - oa)
(,--rJl y-A-f)
("fY-A-\")
(VY-^-r)
ni 'J,g-c.{+
6lau,3:..,5 )l ,f.li os Jou cJLi....o f1Jr-o om JoII, oI c,.r).rl.- d,rll.1:
eJry 6La,*i7 .Jbt q 'xt
t-.Jr1',lt dy .,r).tL, 5c.*,1 d.,{l 1- J.4i 1'.!-lt, .-t-.i!"f
LF, a*;-a tl aLi; 4,r.11-F q l, t - A - Y nJ.:L, 6 e:_h, iS --,-J os J oxJr)U,
.( \ f - f JS-i {. Qt).ri Crnlt -t3!"f os gs j.*e tF s ox ,}l sF.5!ls 6
tr -r JK-t
,-is Slttt' ua;V g;Lao;-ls
*55;l: *y l"url r, os Jow yJ. dJn u, ,.l; 01* .rJ O+l ui eltJot-on) o Jor-oa) o tJJ: f 6X c,*,1 c.;;Jl f -A-YaJ:l-, e-f
rgf 1.: ,S-- cnlY ) o O#(on-oil@x-on) * osY ) o (Y-^-Y)
Av # drl :Y jos
6-F.:1"1^c J+ jl ,-*, oli,-t ! aJ.rL, uJ-l cc*,i .rltt. rJ;L; c.-)\e 6 jV 61L
..:-ii lS JS-fea'r o:L
Iov- i@o + on)lt + osY = t+ @a-oill' (Y -^-Y)
.:.eu6-l-r{ @a * ov,) eLi;.tt }fy,-r.i!,-r oN, os a2..;e-,1: oy-l: d aJ'rL' 6
\Y-f r[i.i1.rJ f=l[.-f C1 ! oyl: yl..:*,1 ] (oa-oa) o;l-r.,,1 I 6el'-r.51-1lr3o7y
c.:-,1 .:lty. Y - n - Y aJ.rt'.- 61, 16il'.-l ..:-.>\-" 6.lt- ClL.€,y oat':*d,J. )a.;4 JSrt" Criab tJ .lqt, y-t 6os so1,'y-rU.' eJty 6La.!7 6 tygl oJ'aLi;
aJrt - 2r +5o5 sox #:- 't', o;dC1 orr-lr )l el-;.>l-.a;;Jt L. ,* 11;
.S-$'lrty \Y - f rlii;.rCq oyf: :l eV\ess JS6. ota sjJl Y - A - f
isjq.>-,1 ulv e-fu +5,*-',:l: *y, ? Y - A - Y d:L-' o)-r50L*:"1 U
^S 61:\J:-:*r! €Z-"J,> rl"t-el c^{?.gLarryaJ. oy- ot 1 o J oll-om ) o
.Ftb n,,' 2 o
(ow-oal@w-o) * ost s " (0-A-Y)
..:*il: f+lf J.S6^ 4-p oN, os ae'.ip;.: ! oyl.r d;L; .:-)\o 6Jb gt l
Ior-i("r + oa)l\ + osY = I+ @r-odl' e -^-r)
O.ftJb O-A-Y eJ:L,g-lL"-li6.ptUj-l f=nU"f Cy c \f -Y rfS,i-,1:!.ybgrll, V Y - A - Y aJ.rt--. tdf JIJJ JL! rl:r5JLjr L..t-S,g tl-i {l pb.ls JSL1"
.rr !r;'.t-,| 3i;lY - Y "fii;r
C, lyoylr al:t--. b f.,5"f €-t 6lo'f ,'.
Iox-]@+ or)lY + osY = t+ @r-oa)l' (v-^-f)
oyl.r r-, S-SC;!J oyb Cf-t CrB l-,Sss s5 oN, os e-*i"e-lr.-i.*!, f[rU rf lU;
<Slr lf c'(aor-ls a'. iire, Jlo-t-3 o:l: 0t-i,i \f - Y ,$i 1: ^S o-f -,1J o-tJ w;'-a;
JS6" O*.b b Y - A - f aJrt', 6P s;os, oNtrti, O<^.6La.:i7 pL- .-u:..^,t ,:s
O*J:4f 61, o:-9* 6Lb L:+" <*K"
,r., it^-o.rl.;b.;ly rlUi L-,g )p;lra",L 6b.:-t-,- rpb;.: I Laoylr ,J-l tsss
6 t{.l oJ-aLi^.o c (.:*,1 \f - f F,gtf ,-5 -hJ 6U:" a.5) \ \ - y JS;;t c^i,i
fJ q jt+ 6loyb lls_j,.*, .dY! 4^i LA C/lJq cc*,l 6.1t-:;l il ,-,*, d_f r:*)te:.-'t .;l j )-f .l.!oJ-l.r;1.:;,^j jl c o3)\*, ..:.-ilr f+lf c*rj Ol l-,, i-r*: ,-tJ^J ,5- .:-,1b
nJst-.--;) or-.i dlrl 4*! *Sc.*,lCy oyl:1t-r., OL.o P y;Sls- dt u::;JIl.i., +5
.rs6"J.'trb\\-v-rJ-(-i "X
+*;-a;slsos 2 oy c2l^,a;La 6\ls ua;l; r* 4t.ij.{ 43_l ,JlJ
6(l:*,1 oJ; u.a>-).-'n; ! \Y - Y,[K.i1.:5)AS rf r.]->1,*o c,..r.iJ,, t l_, \f - fi q-f GJf c*J.{,f.^, lFrt .:-,1 oJ-i Jt^.ot ul ,stros , oNr;6laa.iJyL9tAc.{+ u"t^i n; &JL"-j 6 a.:-b..*#ryF.r, l.1p ;: f-pc*;,_ I* ,*..rf -r,ol,; ,.Sf ,f C*, 6ss ,f q ,ijl rl_u,l j),\r +tij ,_ruf ; -Ji, Irp .,r g(^..jl .*f_l, os, cryJ)U, t_*ry 4L-y, ? - V - \" _l O - V - y .:!.:L,,t S u_fi1
;!sl 6l.aa>,}-a;r 6l.i-i tt-;l. n; t1l.;^ ss d/l jt 3 cJ.i.:,,..a c^{? 6Larr.pJ )g+&U- 6J; f*? 5.;.ll.i- a1 ryJ.a,o t-r,lL.tr; gi r.J"Ll ,t-l l."r5.',; 4.,*;,s
J-(; ;IABC ,-S C*, L.n; ;l t; ',jX, ^Li e Ctt.€,y.jJt \ r _ y Jft,
.{"{:b,JJl \f-fff : co.r Q6t,* + cos B6r* + cos 96r* (A - A - y)
c,.:"3 ar .J; J^lF oN = o7 cJ;\ JJbA tr o ;)[; aLrt 6 6,1.,1" ff = ir* f I
;.r .r.J -tl,rllJt op - oru )F ,JJ:-C \ fl JoN : on c:1i JJbB L.e es ,Jl-r,iJrr:tl.fl . (./ \f - Y J*) sjn oyls-,11:_g,^.r1.:..-*ul )t.aoS c slYL-.,1l fk
iV;"-,l 6: q jt+,A ri 661-,,.4 ftO6t.$,bllri c rb, a Llirrr.r*j..,JrlJt 6f B VA ,lAB oylr 6t ,",ei rl-r:"1 stQ ,Gss,/ -p ,u.f ;t>tfUr.B) _r.itl
os Jox r>l-.";;- 6\b (Q aLrt y.a,,)q F eLii r./ \y - y JS*i ,-r*, e6.r_ftQ S ,)t- -r, rF 4?i) .rr5 "r,olp ,Sf b V a;l C1 ,y l_r^; rl_u,l ;r clL;^.o .rrL,oy-l"t^^j :l-url..r.: \Aoo ojlrl +'.q ,lSu" .Sf ,F-ii 6l-os .s:AB.:l-gl-,1J l oo o;l-[l
oyls 6-r ,ss-tQ et'-l qL:. :-b..G-5 JrtJt 'Sf &ti 6l.oJ.1sb ata o-u,o.:JL.a;l
rss_slsoi t-.: \_e +tJ-, !i#,,4-.: q aLi: rrr*3 j; siJl \y - f )SaCA \BC.rr5"trlr; cjb;t ./ \ f - f J<.i &U, ca jbc 6La.rrl-r^.;
Al .* cJrl if ,Jos
S tF sQ : O,"y,,el:f;t:;l l) + )J;J.af; jJJ". +O ,-:l: flJrSt
,iJt \0 -Y;f#;t I{D 6loy-lr cr..rs:l-til ;tQ c-l-t7 t1t5;LJrU,,r"L"i 0: fljl .:*,1 .rt!o ;ll f - A - \" aJ.rl.- c,Jt- Cfl gll ,rj;q a;.:l.r l-1
J-.(..ia-,.:*,1 Ofu 6 (ow-oil@x-oa) +osY = (q-oa)@r-on) cosY p,
."u_!tr oyl.r aJrL-" s;l-rrt:-,|
[""-* (on * on)l' + orY = (oroa)@r-oa) cos\ p, + t+ @n-odlt - R,t
(1 -^-r)
(v)
f;^:,6.i_ ,Sltt-t" r:--.5Laay_t.r** (V) ,P 2" g,,.--
lr -r J<.t
,*f .-56.u.i_ 1ultl
Oe-J:,.,te ,gfy a:*9;1 66LaD," c*9"
Jlt.L,R' LU J .F 6Vi 1.: C1 oyl.: 11 ,jJi Sf-r 6l-11.r oy-l-: Orl 6:J,.i.r.. oJ-aLL
J-(.i &ll- KD 6loylr.rt 6ssQ dJt-rrr,C/_lJl,i.-t-,!,_f 1 - n - \" aJ:L.
1,, ..'' \0 - Y JS^i &t!- I<d 6lor.-lt cF q j.,-t aLi; t-6u".5.2-,...eJ1 t0 - f
] (oa-on) arRr , cosQ, = o r-r!rrsldr = +fl6r-f qil.J,-Q._/
,f,Uy.rLi, lL;0 sQ 4solKii,-r'.!U fr : f, . + flt .+U-,-r u:AKCr .yl: Ll; ":*,-
.JSL." 6'z._eJl \ 0 - f )S; tl EG 6l.y l: r,,9i .:l.r:"1 st Q +tU c.ruf ;t ttl b u-JJr,
.rJd ?J,. L4. J; )S.s,".olid! V y - A - \" aJ.rLr,cJt-- y-l .Slr
o
(-;Jl)
(v)ta -r J6;
2jt 6|aoy.ls;ls-pi (9)', p--"7 i.:r 5.- (,-il|) , tsst| :Q 6,t ,Sl-lst
lo
il .i- ,Jr( :7 ,J,*
[or*i ("r +on)lt + osY =(on-on)@a-o) cost fl, + t] (or -oill' - R,,t
(\" -l-Y),Ft = i etcl-.i-l.):.-rus,r.. i--*Rt Lb bCy oyl.: J.iJb Cfy ! t'llr,5- 6
,tf ,Ss.:Q 6 Jf r, ..:*,lCy oyl: LUj 6{t-"f "irt5+
(or - on) a.,.Ry 1L:J5; st eg 6lor-_lr dt' q j:; oLrt
'.tSu" :Sy -,iJl \ 0 - f JS3 ;t EG 6lo1l,>
.r,-Q./b*\A-ffi;l :l-r:,t 6 jt?l Q ^Lrt,,...-?;l P aLd ;s u.a),-L,i J(r- ,SlJ., o-r>B rA6loy-l.r gb.r,r 6p* +tii rr [(,_eJt \ O - Y J(,;) JS6"
C"J !,
''...7 ;l .f g-;.,, ;l-
.ls (os joy Jl-.a:ln,gt.1lr) Oi ! !+r, q ,:*' 4tA 4Lr-J riL-a.1r;:.1lr )l;iEG 3KD
r:lt {jeJj Jti..r.,l.r;lr!./ \0 - f Js:.i ;;s aa;.a.tse7 Jkd 6lo1l: Slttf j)t;.$r,-./ e3lslls ,stss
L-r p*rxt) g1as q + MPa pJP,Lrt;: rlr,)Q | - A - f JU.c,*,1 oJ-.i orl: -$ ,,*-iV
Ioiil =
o
-fio
0
o
-/o
0
[': ]:
.r-:!.r ff = + (Y 0, + 6v + Y E) ,-Jl& Jt--i 6 61.*i-a 6ssls r:.:;lby (-iJl)
..r;5 o.r_
.+5C".- l) az.ie tll-a 63; ot dL e d-f J ox Jl--i ,-,- d-f Q).t-S ,5rsl. .,, \ 0 - Y Js".i, ;tl" .21n ,1lt d;;L ! tr (V) .!J Cb (V)
. -gg-,J.r tr@ ,11L,-,.1-iirfii.;-: A-Y-YrJ.rLrll o:ti:-,| q(.jJl) d-
lIi] [{]l [': ]:t{i)t16
tJ^)
trO4pJ:,.,tf ag(51 a:*gai 66 L!;," .S*&
oN= trq). ff: + (oo 6r - \0. 6v -0. ar/.+ (r 6, + 6, + r er/
t6 ,o1yy-:U-,. ,Sl-e,6 csyi,u" a3lrl y - V - y .Jrl-, .(^(, os ,it a-i)J.
.:F'f rr; d:L, t *-.!t r, cri a*,!..o
osY =YOoo + YYO.o * Yoo o YYoo o
{n) -+ (oo 6r - \0. ey - o. E) t
.r-n, *l*,. e-j\,--tr.,a 11 6;>lal.s ott Oli€ \ - V - \" drLrll orli;-,| ! (y)
[++'*'f': -::
)o] fi]
='N
L o -?"
oN= +._+_+
t) 6t-v ?-t' t {+J yl n'-ry.i o?i .J-,tl cfox = - l-0"- =
.rr;, rf:tfr:
-=A\
.FYtlos : 6\ lv MPa
,-l-b ,Sly.6 # ,rtt t;iliC Ot-r,a;ts \ - ? - f cJE Jls.: jl orLh-,| fr (V).oilr = -v0: oil = Y0, o7 = Ao jl g;!, ;tel r$y:U, c o;L;l :-2yoo;1 up
;l c,*,1 otVpx,*r\r* \ px${r 6,619^.jl JU:;t .r_-rL dr-+
Io;il
tvvv/t1
l.::l
rr ,i; Jr( :Y chs
ff ,gba"J;* 4a.;j 1r
,Q: cor-\ (+) = V"/of r,'-iJl f -f J*qc..j,#-lJqJ* Ot;U ,Slf .S 01J., Jl91 cl:J^r':*n; tf t++ , 'f : 0 = cos-\ (Ir) = f A/\ q"
,;tsl 1., q F^LJ;,>l-*.,Cti: . ;l o:J,.,J JeV \2 -YJK-i,,; \0-Y
.r.1,..*, .rjb l-r os : OY/V , oN = - \ 2/V JrU, : uf
t, -r JK;
\ - A - f Jt:, ,Slt. t-f 6Lao.,,-lr 2l'r1-
gla.r.;aF- t -f
,,,r1, l-r ag(o,>3- d;J Jtd r'-t.iq -fu '.L-ldtn,.i;
jt ,F- Lil J L]3 6 6"1<b
6ll-jc..^,rt t:,J.l; .lt* "f.l
.eU"f a*fx aai,o l) y-* ;Iel CJ,is;: a-io 6
..r-,!"f ,-1j J.fu ,'r.
")'al 66y^^ 4J. s.*'
hl Lr : :llil L il
q(-16.7,52.7)
O"-J{r. 612 a:*9,11 <SbLza, 8-8" ff
OlJ b uJ s-*t f--? 4 sp gl e::-r, _rL: tJ-er a!i; 4 t, ,-&* t4;J6Laoyl.: L-'2: o-L,i o.:l: Otij ,JLl ,i; +-, -ll .F_ c.*,1 OS^. -k ,*l r,U .Jar,r..
.JJU \V - y r$,,.:-l: rti t LXt :y
(v ) o*-o
lv - r JK.t
6,lA2J,,a ,j; 6,111sy 4;Voy_ls
xt 611";;lrlt-i -k ;lel ,.::; c.<+ J r..itt 0.r.il .;; ;tel ,Su.sZ:; -Sl
(.Jt) o, {
(v) om -o
IA ,i; Jr( :y J..s
JS Os7et .J.- ,-P-JV-I'fdL.r/ xrXl a'>'i"'a L. ,Sjl;" 6la''.i'o r'U c-f^iL. o-t'.x '-;tjuil.a:*rl
(,.-4ll \-q-y)
..r-,!.f J: ,F a.'.
")'al 6la:'iau l, c.i U-
(v\-q-t)
L*- t' )l U"tu.-r(, '? &- L-'; 6leJ-a F (t ;t'l 6,1va: ef i
,-8 ,yoyl: ,J(, L-,y clUJI ,P.3 c*,1 oJ-i c.rl.: otJ .-iJl \A - Y jii;r '';-'J'-
,sf.rlr) JS€- oat:a L,Ji; -bUi 45 G-i6.(./ \A - Y JSJ) 5F,€ orl: ,l"Ljjii1.r oyl.: nJrL, .;i -5,; rli,xrx, le,ie;-: !-i.i.3v C;br.Jt t Slt(os sort '>l'a;;u
.t-,t"f l) Js-fq ./ \ A - f
I orr*or, \t* (o*lt=(orr-or, \t* (orr)' (y-q-r)(oN-T)*'ot,--\ , t
b,;e;rr5l-r- ,ft ,F :.:-,1 o-t; €lso* = { (otr * ovr) , oS = ' ortlt f;"olls,gis B sA-ttii . Y - q - Y d:Lr.:-,! c.-ed' )+ P-oy -t'al.;r'ylr lti"rtxv
dtJ .$rd oLi b 0, -l0r & gL"Jty rSl-rtr t1t,iq ,Slfl .:eL"' elt r-JJ,JLn0ql
Jr Jr-q c-tJb,r,P e"ti;.1: 6;L:;t c.{? 4, -r: ^*- t.lU; 4, ,Sbl' 5.:'-L'- )l 'f &
.: j+,, ua;.*y"fSrii 0Q tJ 4; yb- l, V \A - Y ,i(':'.l't c':j;;i;t'a oyl'r
fu$rh 4-.t \:*j.rr sF Jf 0 rr-i[ .:^>; 6 Ox'fi'{'r 6,la2Sz" r'r c'*j
dcJ4 2t 6l+r,i,,.c ;:,:S eW UJt .flrL, G\!l a;,.i|: .ll-l-1, UJI t q - Y J<-' ;.tL
;t ,rli:-,| !.rr,-r dbl \f - O - Y aJ.rL,,t-- Jbl-r-l-b €,f d-rS L''i; xrx,
crr-,| oJ..i .:-,5j ./ \ q - Y Js:i; .;t 6 ';f;* YJ Al elt c'{? 6tar";;JJ-lre
4t.t, g-lf, .u{ gLn"ilr" t0 ,-F-* ,-sD *-/.1r rl: p*lr{ ,:'; 6t'ra;J -f €li€.r-1 ,, .l! -lj
r.,=1,,: I :][ot"
o,,, : ]ro*4 1=
L ' ..l
O..,,J;,,F agly a:-y 16L6 \^" *9" 17
(dJl)
q)-
b.)
lA-r J,<-t
Sll ctf 1y 6l-oS-1, (V) s 6la-".)-a u; 6111 a,aJ4 ,.)y oy_ls (,;) , 4;la-*i-,or;- ,ra (t-iJl)
. 6lat-ie j:;;
t\ -r JK-t
Ol',^* ur- 6)S JEit ..l-r.t- (r-t) , gSli-Jt a ,ys 6ly.a:iL- rll2jr .51.o.,1r-a, (u,il1)
'qx, xr
xi o0 In0 0
4 -$0 or0 ori 0 0 1
(v)
ry ,i; cJs-( :f ,J-s
o'\\ ={ (orr * dvr) + + (o\r -ov',) cos Y 0 + orrsin l0
o'\, ={ (orr * orr) - } (o'r -ovr) cos Y 0 -o* sin \ 0
o'\\: --| (orr - orr) sin \ 0 + o* cos \ 0
O(r)= YO11yrd(r):O1fl=
1r n-*-:-l;: (rl: 04+-:6
(,-Jlf-q-Y)
(vY-q-r)
(Vr-q-r)
orlr rlf;i Gl-,Iq .)F .-/b"jlrt, p*lr, 6Lac.a7 elL'F ;l,,,113141,9"-, fl , .r\jl -srq" ;* * ;lel uir rl"t.L Jr 6 )-r-'L/
..:*,1 ,J^i 4it)l *'*-tU L-'yt 6laa,i-a ;X 6,jjt-,JlJ | - I - f J&
l'. o. .l10 t:
L 1
pr:"y l.s -t,f6taoyl: -l .r*L.i O":{ \ J)elr$ rr:Ur,.r-i!,-f c.ltj ",J
o. ul.;r 6
(Y-q-r)
.ln'
1l c*,1 ..:.1!o 6 - ? - Y aJrL. ,lbr.y, ",.xJ
0t* U C-l Slt cb
o
o
*o
. (")-t r.i.;rr.:tc,) gLn.tu: q :.'sdrL.4 q yt:- aS
f - q - Y aJrL- ;l ol1:u..>ef +.,6C-FL*I tfic-l
o"-o o"
o" o.-ooo
O?J:'.4f agl;1 o:-g21 aglaL2a" *& lA
! *l L1L;t; ',--jU ;r,r.l,gLa.1.p,,fS-:
lo*iil =
;t ,r5r- +-L);s 0:J.l -*- *U. tr.il.1.: ..:*,1 oJ-l o.:lr !Lj^i Ji*, st s-f orrlr;l:3^j
.s-F.l JJb e J ;i., oylr s: s :Fc/ ti-r.,1 4t.i! ...(r- J-f+ L'.b'y 6taoyl.:
j c*,lx,* c.47tr 6ln.<r_ Ui,i56_.f.1 o.:L 3 ,_J.1-tJ,_b.6,.r;l: a+j c6a,,r*,o
,>L.a;;- dl"Ae.i.r.ls (c,*,loJ-i osl.r OL.jiB JA Lld L-,y 6)r5l.r- ,-ftr$y.rU...Sl.f dLltl sjL.," xr*
")*cl c^{? q yO" ,u jls i(.x1 3x,
t-tl.ra;l F ,tl:t 6$dp Jl*. l" - f., JL; 6L^,Jr-, sA*-;.**
oM= {(o,r *oyv *on) =*o,, (\ - \. -r)
-t.itf .S-lt,- JIel ri; u, LrrL"i S ;:;: jl ,lt- ..:-,1 isy ,Jl"i ,F g&t* yu.:r-t^j ,S:b e-ji L-2' s:F,goJ.lti ,F As.S lc.JL>(ouL.,S;.,.,,..f.l-r:e;)
Ii"
il .i; Jrl :y tJ,.s
J* ,: otKi ,-S rs_25J-Su 6lrt 6liu^ tl JrL, yl ..:*.1 -SJl-{,JtJrd;; ..li-rr 5.t -* cf r'nt!tirU:r,fliv
[r,, s,, s',," I l' otr - oM drr orr Ilr,, s,, s,,- I = | o'., ott-oM otr I ,t-\'-Y)
[t,'. s,. t* I L orr ow on - "* )
.rrt* .ilFl ,.l:; t-t,.rl-ll 6 ;.y,y +tb nLrU\ Y - \o - f aJrL,;l d,+"^,!
;l c,.."1
S;;:o,t-*6;;oPr= " (0-\"-Y)
? - ? - Y drL, !L:*) .-ilFl ,F elt d.:L, 4+r:i .,: (c.*,ld;; = Y oJ")
ry Jrl3i c>2j,e d/-+ ( oij 6lJ(2-r.-Y)
jt -l;tL* ,; 6Lat-t, 6
lo, o o I ':g'''oo:l:
l"' I
Io;il: | . o74 " | (Y-\.-f)
I o oM ILJo$ o:1.: epi ? - Y.:^^J .1, 5 G:y,lL^ q J::"'^ ;tef t'oc'a'7 '.ttt 6 e:-.;?t'
0F Jt- t, o-rt..i 4..1-r ,ii;l .l,l .rt!o c cJl-' Cf-l ,Sl; '.(.t>\t 'fr;!- JE' '..:*'l
'<Jtt*,| rt-ij ou = - P. ,Sll. s +-Sg dt'lb(tr'r:a ,i;rr\ ui'-9- o'.i1 'u.6.-u!.-r
.-is q is-rst .# ,5- ; 6if c'^-,s 4 q 6li €l: oU ,*;t lb rl.'-f leyj aJsL, * ^-!lFl
o,i = S,i + 6ij ou = Si + |6;1 os (Y- t" -f)
sr + rIJ-III" = o
O*".JJ.,k4 rgf, a:-9ri 66L:r. cS*K" loo
IIo : -i S;; qr = SrSz * Sr Sar * SzrSr
III' - eijrSV Stj Srr - S/SzSnr
v-so
YY
1.-.allV- \" -Y)
(Vv- \" -f)
o?-r )lr;A drt-- 6 +-&.F.,, ,sll;_&.lro,1 Stnfd ;tpl c..6? 4, . ,ryt:
fl.rbS- +"p;l 'tKi.r.trS;..b lo;i -o(r)6;jln/t': "
t_ t S, * ou 6ij - o(q) 6;1y nfd = o
Isu-@@t-oil6;jlr/t) = " (A-\.-\")
.,^.*-r.l Si JIel '-rU,
,,li I -lr-_," c c*,1 Stj ;iel c^e- Of:"^i nfd J^ru^ OLii 4t
.ty€ ailtl y_t.r-r!o L-,3; o;i ).al JrU."S(q) = o(d - ou , (q-\,Y,r) (q*\"-r)
+slj+"*ilFl j6jf ,,>**,,3J)qb \ -2-\"JU,;:o ul; )J*.li l-t -f dU.-trr-L; O* b -ilFl .:*J Jel rl; y_rir..3
1l c.*,1 .rt!o ui;s 611014t\ - \o -yaJ.rL..(J, cJ-oM:{(Ov*0o+ff):6o
't -l't,g
lou o
lfo1,, .L
rat \"- \o -YaJ.rL. L-,g.g;Lly."cr*,-
::]+[':.:.:]
0+,,) .t*1;t ._ilFl c.^J ;tpl r:; y.:Lo
t-i g+.r--yL q
::l[:."fs JeVoYYl
: -;-, I
- - s [ (v -s,)(*v -s) - (vr)' I = o
lol ,-i; cJr( ;y ,J".s
e-t;j ",*l .St t = -YO JS(rl = o , S1t) = Y0 -t'a:,r" 4+#-, e$:-f! u'-1L6
d:L-. cnts _l \ - 2 - f Jtj- 1.: oJ.-.i g-,!. ;Lely.:U-. jl orU:*,l q JrL.t."f€#tr1-\.-Y
sgt i:ia it,P ll -f,.: .:jL,r,. JIel r; 6bca; ! gtL- cSt-l-lj .,.il{ Jt--i +S P st 614'Je
d, P ur-n*. ol *r* 6"riU6 €e9 0.:6 dai,o l) bip U-t .+-&. -hc:#J c.i,b.*-
'rt*1r,4.1, GrL- 6 t-J r-l sF -l .t"it: Y" -Y )5J ;:ABC
{o) - o*. ff =
jl .:*,1 .rtl* lr,;p ,J-l ,SSs r.::*511.ly .J-it uJJ,^..-o
or,r 6,* * otrr 0r* * ot' 4* (\-\\-r)
i c^a-7 -st "),|+ily'
I
oN: t@ ' i : (Y-\\-Y)
\ry
! *lr:l11
^ a.+6,+i';=
v\ I vY I af
Vr
* [ ",'r r ot,r * ot'r
] = +"n
xi
aaa. otlat+alor-
,E
f" -r JS-t
. ust c--5;/ : JV;.rLno3- t u#s t* a-;e
OI-J:\,F agly a:-g2a 66 L:.r" ,3+lS" I oI
,j,:; *. iSf s r544J!:*:l^ r'a;e 6Ss ilrs Cr" c Y - V - Y +J.rt .. jl, .XlfQ..;*,1 l) Os-req 6J-,!"/ €et O.tb 6:,
atnt=t6.t6 -otp= + [oY.r*oYt,r *"tn,
I + [dr,,r *orrr *o,r,]t
ooct= *V to(r)-otrJt * [or'-orrilt * [otrr-o.Jt
=ff
(Y-l\-r).{t_ .J:^tS -$ os:,e J, jl ,F r c*,1 OS^" -,
\fy
(Y-\\-r)t_
(0-\\-r)
!:^iA d,'.)p 6s:fft_b.lt ooctJopt u^X1 :JVt,SLndJ-f l- ll -f JUont 6lLb {++i -r -r'S.j*{'r^i dll \ - p - f JL:" r, 5..p ;l CV. Slt'rFJ
."t1[j r/)J. f - \\ -f nJ:Lr.5^S,
$ltl y r*_iV ?t' L*,9 csasc-ir^ ao.;p 6ss t-Flby c \ - t - f JU, jl &
,; ][ffi] :[|]F]
.: oi -o
{o) -(vo 0,* + oo 0r* + yo E*)
t_
a-:r,J,j lrfr
oN=t(i). i = {(vo * oo + yo) = oo Mpa
f": ..
L. .
Stt,,*sY,r,+,sY,.,
Y - V - Y a.hL,;l i,++^,r
lof ,F ,Jr( :f ,J.cs
ctnt=t6.t6 -otN= * [fuo)t + (0.)'+ (yo)'l -(0")' = Y\?fY
g)L"l4l,,-.Y- \\ -f aJ.rLr.5^S, ooct='lffi = 1'7f \ MPa d/lr.ti.,.:-.il: f.^tf J €'l
toct=+ :Yo/Y\ MPa
toFO'.-,J.{r4 aSly o:-92i GlaL*q *lS"
$t*eSt.tJ,\,f o;i 6Gai,J3-" 6,1.11:pxgfiy ,gl-alj-,- tu (:s; ,S -l_l-b ,P oLld ci- -;.: I - l,
c...-t*. csst s..:*,1 t(il) ,.:; 1l:y .-r-!tr.r i, &- Jt; ,5!b nSdS(l ) ,y c>'-L-n
1:1(ir) dy s-ron: rrl-.1;..r-,ti.rtrir) .-r* ,br, o,- -,1ff.,, & Jb;6fl., 6 dS(r)".,-
...*,|ff' e,.e.- Jr {i') n;,)t" 6jl-.,- ffy .a1
d)J rr...-i -r:j .SL^Jt; 6l.sb 6 Jgl c,^--L-- tSlt.b \ - f 4l.r-.- 1s oJ..i (:-r! aa,?f, f - f.rs
i,: {fva,+yey+?%)
ffr: $fra,-r4 +rE)
.tr*I, a:\l y) ot_f\fu$,rxr 6,G.p-,- q c..*; P y u:; 'r-i| f I
lro ,'lI
Ioiil= | yq I"r'
ltt . \rILJt)y.ea-:.MPA p'- gbd3- 6,ljspx{{y 6l-n13.,-- 4-i ca--j P .1s u:; r:-b f -f
f y \ I '-t-'tr'rAe'iu
|"''I[o'il= |
t . ] |L\ Y 'J
;l :f;ifi -t t-* a*;e ^S+JO*,-P lri ,tr ,;+ ci- .c*,1 ot ,l; ua;-i.,,ol1 r.li.1: aS
9 !:t, -r,-lr -,tl3 b-,- U-l ,sl,-otr Jh,a, ..r^1t,1(i) - " aa;e is'l 6strt*,- .J-,t, ,:;
ff=*(r0'-0t-lE) , ou=yMPa tfQ
ot^X ua-.L' il o*-gl. Li 6 r:-,llri .:...-J "jbdl" 6)jtP o.tfi 1.r ,g r.r*,b f -f
I oA ,.i; ,Jrl :f ,Jos
Io;jl =
,*.11:r, y-l .llsl (...')
Jt"; c.a11: ,-r:i ,lrr, .r!.5* (:7)
JL; ;t:r, S '*1lrJ .r.: Q; t-es: q.ltj (.r)
vyroAo {o)S (.-7) t@ : y'-\o. (v) td = 06r + \\ 6, - Y., tJJrl : *q
c{tj "i- o. 6,r*l or.i *ltlo;i -- + o"nfli Jt-re! 6,1^Li;1r "r5-
JJ-ti,gl.*1.1, 6 -f
.-r-!.f n; o4a st J ! J. o;l.ll tt ,_.,5 6)p.* ,)t*t c(,- 4S,44r cll-i; .J*:L.," c..-l,
c.j.rJ-. )l ,fu t>l-a:;..o 6,6^.-La GSI .F 6ta.1l:y ollll oLr,. C"- 45.r..n.r clt-l; , - f
.c*,1 !L- e) rq nJ-r*^, clL:; ,rl*,-.lt-i!.f .>l-c;;.^ 6,1r-43.,.-rl-,(e.
,.Gi ,.Gt) * ,.Gt) ,.(it) 4- t,Gi s,Gr)
[ -* r ^ '1 '*:*i
l; ";lL' 1 -P.J
+s;x*r''
J*\ dUJ j;r' oSP y 6lo-*La Gs-t cF 1l:y ('.-cJt)
ff=+(0,+Tay+^er).$! y-j .:.r
O?J:4f ag(2 a;-92a cSo, Lza" c3lJg" lo?
L-'i ,->l*:;. *J r,l:,',Jl* OXrX,,X, 6,Gy>' q + V -f
[::' x" ;-l
L''" .i:
:,',,1
.g-,1 o-t -! aSltl ,; e-i\
Io,il:
+5"t.*^,'
1.: e*,ltly JrL; .r).rL f t ot*t;- )l ,*lt blrb\l) u^.7- Ss';,gbacJ.y (..-il|)
.t;tf ir."L- h roJo)J-.t os:*.\:i&,a,-k-Jt-; aS,g,la-,i-e,SssltP (1, l, f) ^LAlrrl;rbr1 (l7)
.rA r-l .>l-a:;,' c.,:",5t o;3*, ti g.ll-.-,SL-l;j
,by=ftr.t'l:*qp-l Ir
=+ pbt = -YrY
p ,b,
-r
-r
t@ = (r 6' + tq6t + rt6r)
(.,)flF
fi e-tV L--; Fclhia 4 ,Axrx14 t;ltag:-, 6l-ay-,. a., l:*.; A
Ift-x,t)xt**r,' -(r-xl)x, Il-rt-x,')x, -{(x,'-\Y.r,)
|
L ' ' (r -xl1'' J
J. $fr g y ,it-c (F gLn-Ui dl; Jr|-,' .:Y.tL n5.r+, dt-l (.--cJl)
' Yrr = \Y aJ.:[.. +,,g,\a-J.a;tP (1, -\, ?) rjz,ji,lrrSrl:ra(9)
.rj5 ,:,._.,
t(il : { (-116, -r"0, * ver) tvl : *E
jl Jurqe t LL..i :L J* t tlr, cl..:-,| .(,-.1: ,; ,gLadl3* I
.r:*rl o-\-i o:l.l .
Ioiil =
."U:...rt .rlrl-
l.r Yrr * ?\ *
loY ,F cJs-( :f ,J.as
L
Ax, * 84Cx,,
-Cx,
Cx, -Cx,
o
o
Ioul
.J.:.i-,! tlll JrL; .r!.tt', ,. 6.e-.-,Slr.l.f ./Lg .l: 4t J:5 u',,U (._iJl)
.tfig .ii.- cl..:-,1 ;;^,. 7\ e :t ttij .,"L.; 1: ,:;;l:r, ..f ,4.4r c-rt-l; (.-r)
L; Sn- tf*tt.lb.l, 4 L-,; pxrxrx, 6G1ja- ;lpxr'xr'4' uiL- ctf3r 6Lo1j-., | . - f
, .1t.r f-! 6b"iJy .slo.r,i s-r! ,jL,-f Px${r 6,1-ayz,- 41 g*i 6,11-"- gL-l;j 6 PQ
.r.5a.*r,.r-:!t"o (:) t- \Y"o (.:Jl) cl!-e: a,-3! d: fi(MPa..*J) r;.;r'*,b,5111
I oiil = :lL:
loAU*J:*- agly o:-9a1 cSbL=^" cS+K"
[-o \. \'l [. " .l :*q
ro";l= i l : -'] :1"" (1)to'i1t : | " -"
1 Mpa e)')
L\" -y'tr1 L.r tl
"lj JG:;t ,f-t\ L-i:pxrxr4 L5bJJ.." *.px'rxtrx'r -iL- clf3:6Lr;.9,..,. oP oLfr;: il *t,
.sla;i[ lli.,l
l- a r-r'r ,, *filIoiit=+ | '*vt b r-rI
L r-f r+f ' J
Pxfi{r 6,1r-a.p*- a: r.i*i rS r--rL ft.nr-;",^,- r:*,1 -tl-p C Sb , A j y oS
.$5 J.i., lrl. o'ij I ,tKri .r-ilr y-j ,.>.ry,,oa'.MPa pJ
l'\ o 1
to;it=l f ;]lL\ ' 'J
l'\\+rfi o+fi -\ I[o';i l:+l .+rF o o-fluro:*q
L -\ o-rfir \\-YrryJ
-,11 c.*,1 otVksi r*J.px,;,;y'.,61r1j,*, q + ,-1;.f--lL lf -f
f'' - 'r Ir,o;ir: l: y\ I
L '' U J
lol ,.i; cJr( :y J,.s
L.-.6Lr1l:; J.l.rra L-,1pxrxr4 6,1sy.,. a1c.?;px'rx'{'t a;iL- cll2j: 6,lsy^t
Slo.t--i ,-e'--f l) ol-fur\
or^i clt t) ot-*\Px&rh 6,1-n13-,' q c..*; MPa YJ. u5'o*,-J,L ,P oLfi 1': tl'-t'
.r:*rl
\ c.Jl- '.I oij | :f
7
,5 i,,*1"- olL ,^ 6!,.L-l 'p r,-:li" (.--,ell)
-bl 'p,gLrcaa, (..,)
:*E
u,*,- .J.J!.,^ ,l.tg,-tlr 6,La.t2a' y t3 e P ' L;; ,: aS 6,i-x^-*i.e 6u c.F 6t'olby ( jJl)
'rl,5,:..+ E 16r, 6t .stiv.5t-r-,1by YJl,tG'il j 1G'r1 ' 1G't1
i U- t$y-.1,*, ,lrg-lr, 6,6ya' 6l.tlt (-ilt);t o-r.i r:*'.t ,5 ,5t'r1bJ jl 6.,- -f (*)
.sy' A*+ I o';j I dt-r
.t :S 4.tt \ - O - Y aJ:L, f=it- )-p15..1 !-,i fi (V) 1.r oJ.oi ti*'! qp (.-7)
f' \rr Y? \\rIlo'ijl=+l tr \?? rlrr',rQ
L \\T r -tol
:]Y':JL:Io;t:LIil]L
et aY er
6,, LV
tV
7V
6,, rv -?
VY
V
6,,- LV
LV
_rV
O?J:,k4 agly a;*92,, Agl.oL2a" *K" Ito
: 'l c-JL (.ijl)
:YeJt.-
: \ c.JL (.-,)
:Y eJt-
o(r): -lMPa ,o$)=lMPa , o(r)= \,MPao1r) : ot) = \ MPA, O(r) = f MPA
ff(r)--6'*=6r, ff(u=teii, fi,r,: +e,.Vy lf ' --
ffo) - 6' + gr-+ "r, i,r, =Vr-6,-+ 6', ffff) _ -6, -6, + t6,Vr ' -- ,/,
.:tl*- I<si ,-..,-.,-. ,F ,:--lV ,s3Ju^ P ,Lil y u!.cl 6l,asr-- t 1L.rt nS,r,t53o lf - f;l c'*'l
l'Y o .lIo*;jl= | " v "f
L' ' ''J..t-!.lj ot-re\ Px$fir 6Gy^. t &l 6Gy-,-.xi JU:jl ,f-ti fl
f-r \ -rlIoiit= * l": o,i,, ":, IL* -\ fJ
..rsL; a*-,[-.. trloU | .t<ri.$_F 6.,r.,- c*,1 tl]arr Jatr , ar1 ,si ,, n!
l'v Y "lIo;il= I t v t l kri:7*Q
L' r uJ
;l c---'l otV ty L4ipxfig,1 6,lsy-- t "/.1
MPa *Jr:; ,--rL I d - r
ill ,i; ds-t :f cbS
Ioil |
rrS O.*t,
Om, oil, oI u,rtet 6Lrr.-' (-;lt)
,rL-I'p,5bc.a.7 (.-')
or = Tf MPa, oil= . MPa,6ilI = -fV MPa (.-c.Jt): g". E
ff(t) - - "/rqf 6r * .rvrtl ay + 'ITVY "" tvlff(Y) - "rl\Y 0r * "rYVr 0, + 'ri'T 0.
i(r) - .7\\. 61 * "roo\ 0, - '/AYV 4
1l r:-,t o2WMPa y1h${r 6,1-a22^.4i g*; cJ:J d*'-lL 'P oLfr y lP -f
,:;; d*,-J,t PXr*l*\* 6,1^y.'tu c#i t'a; ;Lr.;: 'rilo-t-Li oe;;rb jA i y oS
.t-:-,lt; JS-' q
:'L::,,:]
toiit =f : o. ; l
L-' b ''l
l"' o olIo*ijr:
I " ,"
,rlL "'J
.r;5 J.r., ltom j01 **\ 6A MPa -,rlt.P 1: .r5l-r- et '* ';tsl ,51
or=YMPa, CIil:-f MPa:*q
o..u! a-Jl.1t -f),-f-ii L-,;pXrXrq 6,lay^- 4i g*i l<si t-'"'-s1P ^Lk 1: r; tlta
.g*rl
lv -r
C,i-Jn6. ag(y a;-92a 66L**.5*& ilr
Io;il =
.t*si*+P ), uJel r;- ,gLrca- , J.l r*;"- yrh" (-,iJl)
P y ,Slt-- ,rl .,* -llrL (:).t-y'g JLrl;51.r- ,*t ',*
6l)P,L;: rr '-.;p t stlsi = n;6; Jty (V)
drrl : I (lsi, o(y) = \ kJi, oF)= -f &Jt (..-1l) : *E
ro;t=f I
.rrSCrt+ $,\Q = | oil som = Y MPa ft.uto.r-:;. ilF b j f nl
,*t,-pystn(jJt)D,k;" (-,)
on )*l r; c.g (u)or : |r MPa, oil = y MPa, oilI = y Mpa (Jry : *E
b=o(v)l4r) - 6, (v)
llol
\o
o -l
ff(r)- Y6'Jq,fi(r):6r, fi(",:-6, IyEVO '-- -I ' " /t
(or)** = t Yio ftsi (:)
n- H: (v)
.r:*,1 o"r-i o:lrlMPa
-J s e-il J#r, &$,rh a, t:*;p.,1:,I; tyi lA_f
l:l
I rr ,F cJg-( ;y ,J,os
t -ro)p
r1
LP
ti,
Ax-r
LAJ
T
a
ri
ksi
a,
fi{r 6
q
r
a-^i.e a,
: \Y I
ff*-
n-lj r-;.J*,-,;L L-'t'lrc;i
-J. fi\Px
Ioiil:L
os - f kJi JoN = \ Y ft:li isi ,g,i2 ^S 6,1 4J
oI= \^|lci,
.Jgi,r-)JY - A - Y aJ:L jl orl.e:-,| l,,. b 6i.l)./ \q - f a.k'..-;l o-Lol r:*,J, t-;; l" -t'
6La1ro.-, :l.li.l 1: o.5I^'5 g;31 oJ-i orl: tll--li .r:J dt..-.t {tl Slt. t-l .lr" 6l^'y-l: I I - f'
.r!5 f*t .ui!"f u>\..a;;*t
(v)
2q
ilrO*l:4e agly a:-9;i 66 L*r." cS*&
.c*,1 o-t-! $lj r-.,.r--jL L-,i;MPa plpxtx{1 6G.p>,., ,.r!|,t,F-,Jl* f f -f
f 1 \Y .lroiit=1', q .l
L O O OJ
+s ctt'-"
iri&- Jt--; o.SP y 6la..L.a,sstlsos so1?*yq ,-*t sJt;,5Ladl.r (*Jl)
fi=*(r0l +r0,)
\t - f J#t, o'r:'l y# tf oy_ls c(,- g; L-i L (-ijl).1: or.oir:-,r nap (.-r)
.J+S '4J
s*,1 oLi orlr tJl-j;
oN = \T7"; MPA , OS : 6r\X MPA : *q..qi-Lj tj -ulor-! oslj t; eLre\ +Sr::,- o.:L 6t-ocJL Sl_tb t.f .3boy-1.: lt" - t"
1". . o.l 1". . .l(.ir)[ o;jl= | o. . | (v)[ oijl= | ro. " I
L" ' o'J L' ' -o.J
.+S Cr.+ l, ;Sl.t- et '* .:JL 9 .1.:
(oi** = + o. (:) (os)^ax = o. (JJt) : *E
.r,:*,1 o-r--i o.:lsy-j.f-yL b-y O 2.: r$ blaaxrx14 ,g,lsy-. q g*j f f _f
f ? -Y 'llo;il= f -r ? l*t
IL' ' 'J
1l c.*,1 o1V oS.r-ry[ &r\r\"1* 6,1^y^,4J \:*-; *S"r*.os rlt-.];
lld .i; drl :Y rJe,s
r . .l. q l*to o 'l
r
[o*;; | =
,,--J;t t, e-b .r-l .$lr"r \ ::* J
[,, .l
lo,il= | ' -\?
t?L
fOo ojlsl 1 Al-l.lr jl J-L b.rJ-.,. i,ril nt
.t';Sg-tt.Y-1-YoJrl"
.r.:-,1 oli a:li ,-i J.ye*.P 1r ,:; .l--it dt* f 0 - f
i]ksi
L .ilFl -L-l r; r1:8, .t3;i j r;Slj+" -ilFt s 6sf 6b.:*J q l.r .r-rL .r-l
.+-tJ,j#..t
Sr = VO lsi, Sz = 2 l<si, Szl = -f\ l$i : *q:*,1 or-! .lt* -lj o13-eau. )-cl y-:V YJ -ilFl ,-'5 ;.1' ,5,\\f f -f
II"=SlSn+Srfnr+Szdr
,,b, &l '*t dt^.r;r.:1+f C- f.;r-.,J) dl-ei !,.:*,1 lt g. iil nS'r.'o:.rLl;
..r-,!"f V - \" - Y aJ.lt -" r' *;i
d)J.11 u;-.Jc.-:.i sil ;; Sll,0- \\ -Y;f - \\ -Y ..r'i:t'-1: oJJClrl -t:jfV -f
.$t
a1 c*i MPa
-t u:urr;L'6L^*Jf c &Jr,4 f- "i- jl P ^Li;
yf A-f
-f ol-i o.:lr y-j ,-r--JU L*'; furxyx1 6,ll ya,
l' -1lr -r 'fr IttIoiil= l-r \ -fTl
l_ll/Y -fi Y
ILJ
6x- t.*a 6(y o;-9;i 66L*" c*l&
+5a.+$t-t *bt 'p
g;trca"- t, .lr^ lt oru JaII , o1 )-l r;- y:Li, (jJl)
III' J llo , Io;;" 6t^t-! (:).t;-*,g JL"t Ai ,Sst E ,91a^i.a q Jk JL.. :Flo ,*t ,_,*.rlr;^ (V)
.ilFt _L-l ,;; y-rli" (.r)
,j-,t,Sl^nily s bi JU tr lli ',1-*ss,i+, c-.i., tSst u:;s1l:y (&)
\PQ yav at s*J t$r'; dy6 P.o ojl-ul o.;6 ;l-a1r-,, ,Sl'.bF.r-r,t (C)
.-tJL a--ilr 6lljtpxg{1 ,>l.o:;.. 6Gy>- a1 e*j,S.9l..._ tSL-l-lj
or = ? MPa, oil: I Mpa, oilr - _y Mpa GjJl) : e*q
fr(I)=+G,-G,rfiE)
iso - JG,-ay-/y"rlY'
;lQI)- 6,+6,/T
I" = ? MPa , IIo - -f MPa ,III.: -yf Mpa et)
(or)**=yMPa, i^*: ,",
Sr=f MPa , SU=.MPa, Sm=-f Mpa (.r)
ff,.,
lt?
L-s/F,1._y
, oN: lrooct=
:;l-\Y
rr++fii -\Y + rfr
rVy yy-\tfTrr'T -y
Io'ijl: MPa 14
I ty ,.i; ,Jr( ;f Jas
Ioiilt-'
]
:LxJxt r, (\ -xt')
', :r
-x,') + O: -v,)
bt=bt:
O1 -
$r- \? 4'r
o, br:-f +lp (.-cJl):e*q
^A , Oil=A, OilI=-A(=)
(os)^*: +fro a(V)
Sz: -$a , Su = -\ a(o)
6jk
6Pk
6^t
'.5 i,,*r''.Utr C,rr"b .ll+r
"}5 .r, r:*,lrlrr Jrt; .r!rL ft 6o- 6Ji Cjj (JJl)
P (a, ", \ {a) y }-al,gL^r,-' (9)
P y iSlt-,y-r ",* (V)
P y r-tlr^,1 ;lpl dLaL,ir (s)
"l.,lO
o,i = eiqt& jp Qbn,qp dt! L i Qii clll;:" 61*E blt ' tl o;i ,.i.; ,r;b gtr-J.f t"' - t'
)l .t;-yrf ;.,1; JrL-; .r):L-. (F,SL^-l* "/F -l: n5.t*o.: clt.-lj .g-,1 y';:-'t
g,-tl:\f-V+k-
eist&jw :6i, 6i,
6pi 60,
6*i 6mtt
,Sl^-J.r 4..t c*i ,-lFt ,.!.; g.l: ,5t-t, ,;..- ,r,*,- dIIo I \oii = - Si +f -r*.o.r cll*l;
67tJfu.#$ l',;?' f$7 IJS*2*ela6-b.1fucctl3 I -f
1.r Ln;153 tao.;lJ c i:l.t.l- r[ii1.r ls agsL f(+l&J-,6l'4t9.".(1.'lSr.1.r
A-,-,f-J.r/-,t4 gt-,ql 0t-" .l! glr 6.J- gPi L.t1 g,ai"f$c* -h\ ottJS X A;al;l 6,1l-e3^-r OlP! ! B 6rL p-.* 6- tj,ai. Crt .Slf €'l,-<,- .[,Uj q 4 1 .-i- o.s-re\o:l: ib! .J-,.U
6f*5.-f 6-f s-Jg oJ.r[ a9:6 Lb:
;r 110-;3n 4, r-# qy.-r-,!.-f &hi 6l-";jl R. ,>l.a:;*, otK:-,, Jf rgti
^L;:4 Lt*,l ((a:-,J*r, Lg.- o-13p .-i- c.:-,1 o.f.:,a p? 6- ,gf b .-{3yS ,srt<"
.\:-,1 oJ^i !a{J'i ,l ;q 6c.*,1 13:L
l-, Ot-; ;l 6laLJ .r, c,t13 ir." .(1 r, c.*l B .r!3 ,r"L; c.i,jJ," ol-at:-,L-,; Lyy dr,-l , ,-f\.: ,h .+-S 4L*J ..li;, p-*7 <S+.& J-tn Lyf-f
)-r-,€,1! fj 6+P .tFrt
x:r(X) (\-\-f)
6 €'*,ft.S'l Oi*'J ttp--* tlX "3J^
h:, +rl:*jx c"..*i3*K 6;l-:y Ctr *fjtt 5 Sl*-rJ.a tu !:.-,t A$|-jti, ,hU : t-t,gfu 2iu. ta*:. r.;J, tLirl O,,l
.rF€ dJi oy'e..*{ .r-5* .ry. -r.ol;i ,;S €-r u, L- 3.: Ll-Ftrh ,ri!
X = R-\ (x) (Y-\-f)
.r;lr;lj x c..r*i_;* _1sX os's 6 S,l uat *b J
,/f c,JL-- r, JL 4;l; c.:*,1
'"* Ol&:rr:,-jl ,y'U ,J-t;1,(9'.1.: r*x
,t?.:,1 b r,-r? ,JJi,l1,' .<,- l.fritt € (i,ll.ur -r JU:;t6- J.b +.b f+ 6l9,2t*L- c.aiJf .1: .:lr*:; Lli r r-t'rf f.-? J<-,U- ';l.Ul .;s.:lr*; gy Ll cJSu,
,rtK,r.i; &ti ')r '.. 6;pl cltir"p; ,5- F-, u* :I.-r-,!"f l: -r^ U- g.-e ,-;--lj
O41Ji.,k4 agly a;*94 6Lb Lf," *K" t Io
dJr rA Ut -l t_ J.(-i t- o;l-tjlrr F *_r 6-t-,lf.f cJS,tE* dr_+ J .)e f..:/,_rr t.6,l
ttx ,>1.2': e-e jn-,- +fc*,Ladl5,-; )l Ci) &p, iJU: ,518 tr ,:S*yj nJ.:f', ! ul-r"f lt ,Ff 6;7 .tSu" J^- .t't 6l.ot 4 -r, .iJ*i, .SL^g-ri_S*.
x:K(X,t)''-501*
(Y-\-Y)
.$r"/ dljl J-iLu. * o b - - jl 6 t.>lL^J lk-r, X o;3 -f 6lllx c=ri_9,. *faJrL, L-'t' 'J-i Otd '5;> Cb f=5cf oif tr>-,t A-S,F Jl-tr cJJri_r(*,
"r"r-lt, Oi r"r<'. e*,o .,.:.1lr 6-5|,-;lAr .a!E : lu'gfu $t ra^;-rA f - \ - y
.:1.l lLij tj os-rel dlid
X - r-\ (x, t) (Y - \ - f )
.c.*,1 '* i,, t eLA 1r x c..ri3* ;sX o-1t JS6" oatb 66+.& S.- Jl*ec -r:-i!.r, oce lJ4l-6!l: ,".*7 ,SL^g-r-rt j ..;nAl-e, .$yd ?2@ C.-t' drr_l jl taOtirou u"L^i 6 €'*./Udl ls FrJ*it cJl-- ..4 i.eJ Oi 6c^-.,r g.jV t:-,rf o;sf '>.5 \ et'6qfr, 6 6:j.ai.t : o .,L*Jr,
1,*.+ +f;b ,p_, +J/ 6+.&, Ji,l .JS6.JE;|'>t^" r** -S
Oi jl .r", 6 GSf3 Ln.l[ir*,:; J r_*c/./hl e:-f6qf40ljr1 ,S€JE;|JU-rl t e.tV 4L'"Jr, f-.a 6c*,1 ul ,S,r(+ <S+.&..:*,1 Oi! tir" tFg JeV
'Js6",-f *.1-lrf oal;,Jq,l4 Jr d' UJ.*, ls rqi,Lnrl-5gU, ar-,j 1.r
c*,1 Jt** .f{.lb- jJL'-.:*,1 o-t-.i JeV J-lljl g-1, a-rit 66ldLir q ,,iL-lJgE_-is 6+.& b g-r, cJl- 3 a.a6 Q !b.r*E 6+.& b (g.,"') Jrt c,Jl-- +f
cc.qj.jL;- ,-p;6La1y*,1; dr;9i c.*, \-.t ,t ,*b 0t-j, o-9)\*, .l:rl;u. o;Sl
.ty€ $f -Er: crt i Jl J;:* dqF' -rr y' d/l:t-l.g!lr ta';t-i crg r;l$ pp OlJ+ oclt,st^gli-S*, !-trl cLaoll-.! 6jti;.r
o.J" a.5c*,1 'rL3 &- es)- Oh:. t Lf y d/-llti1 cJi;*i r,rrJ +-,.u,on g(-
til cz99 e rj(,lr* *(+:f ,)'.s
.:11,r I C/)qi,-#t^t* ,"!,;; 4Su"
alJ!.t LS)ln '2L',at;tl - fF ,: b O b 6,t:1,j:,i; Cb .r+ L-'y o$ o.:l: epi eU dq-S4 0tSt
d/-{ \ -f J6.i, st Oxrx14 6la;jz-q+ Xos3 Xc.irJJr"ll.:y6,5:-b..4.K.
X=Q (X)
.t:,\ os-*(\*Y-f)
oJ.i c:l: Ot..ij Jft, -lr 6 I, , Iy , tr .5!, 6l-a;1.:y
-J. tJ X 0l3;u, c.Jt-- #-l tr.rf 0t*;aj drt-. L-'1'
X=Xt\ (Y-Y-f)
| -r $tx cr:b dqA tX Cy c5q.5c .2: ua-+V o1'! &- 4.r'
isf .Jbl;6.X or3 E>n c:tn;;* i:U3l ,r;'*.U- 616 c:ln:;"ltXn 61u.ailje s
d:L .rLar; *J6,J:.JJ^ 6S:\ lgrL .rLa:;1: c*Xjy; OlPq 6 3:|.JSu- d^:'J It u,..;l-:',521-rf.tLJ 13laoJ-eU urL"i sr{,-/ orJ -lt{ J^i[ .J-, 0t*
.1.,...-:J L f b OlPq l-l 't3 y' (X;l:y. q*iJ'. c .1*,-) 6,rL .r[^ar; 6.:-,t Jt*-Olid ! ,ti y' Oi;t :" 6L^,SJ.:,(.9' Lr"t^i.;s c r.7iJ 1.: .J.l-ar,,-c ,zcLa;;l.Ji Oi
xr,\
O.aJ:,.ld ag(y a;-9;, ggla L:c- cA*& trr
..:;1.::.1-_9 ,_/-fu dquii 6 €'*
)( = o-' rx) (t-r-r)
.lsSlr.f!:-,J.r f - \ - f aJ.:L. -lr Y - Y - f eJrt',,S-rl-rl-ta ! """,
x = K IO-t (X, t)] = y(X, t) (Y - T - Y)
Cb l*,f ol-.i o.:1.: Vi e:'-f dqF, o c*j '#_F 6,VJJ, C..* 6f 6
.JSs^ -4'-p tt O g-ti.FIt et,Jqt,-rr X c^r.rJ,, ! i', o-r3y' f - Y - Y d.rLr 6.:;b 4?i Jr-t1
0.8 gaL.!,., \ - Y rfiil.r 16r.j-3Jt (t ALIJ f 6sV dJitJrx.L;'i +t t:)6,j-l;;l c*,1 .:-r!o 6:V qriJ," sls1.Oxrx1q,jE,gLa;9r.." r ,:,,.*j.c,*.1
x = x;Q; (0-Y-f)
6l-as1*-,,irr5 ,J+b 5-rq -p ..JbU.f 0.23 o.]Ai c:L..;;nlsx; 6,1-a+ilJ. Jl-r: 6
-p\ Ll cc.","*l 6sssurl:-,1 o$ lr-l t - f Ji*, st 6 ;4,-r. stl;J J drb ,.>1,,":;,
.:).:t*-- rJrlji g-,J., .1r gl-ril C/-tjl rS-r+ gLa.r-.r-9J.*,. J LrF gta.l ;r.t :y -r,olfo.r1.rE eU 15-ti",S*, !'lr^^ Lil tJ:L.rLa:; 6 aa53".r5b..l11.r, ttrJ dJn'}[.; S u-fiL, .rJ"r;!i o.:lc.,l ,sy lo,s-ri;511 rL glr aV; ,->l*ab- : :F,-l
,5; i5rL',,4 t ::.,rff ?:;#il;:j:
xi = Xi (X,q., t) t_ x = X(X , t) (P-Y-f)
trr
xi= xi(Xe,t) t{ x = x (X, t) (v-Y-Y)
t sXr.li,,St"r-trt,6c*,1 *b -ftU.aJrL-..:-,t, ;r (x U-)"r; )L- 43-t 5r'!. Lii.,.r 4Llj ,-i,- ,j*,- cr-:*,| Cb Jh5" -S"q!-q' \b.,.: :L+j 0t'6 5JL-.,: 'JdU,r'
'f-J.rl-'E "r'+ rV q b,JrlrS'Lj..'*l
rL$l lt €, dq-& ! !ir" ,E;\,ra:a;; t)U, X 6 uf s.:".xlr 4?y J,-tl
.s:l 13jtL6b c:r5J"lJt -r"j;6to.r.i.-r{F CV:4trU g;i.; tlL;'"-o t-h,t ,5,/dblf*.?11013 t el.-bOt-;;l ..L-Jy't)oJu.i, JUjlx
"'-lji+iJ, ?'\ - f '.J:L.
.JS'",-,-J*, J;2edl{ l-,1 d-urJ<*,I db t : ttrri\:, oav.Jt-j 6.-,1r.JA.:,rr
x, = X(X, tJ (^-Y-Y)
OtP{,..-J.i,l 6J:5u" q"-r*bqJ;l dJiS#, ? -\ -f aJ-:L, ct = o;:ru""V2l\
!.ll-rL .EtJ;l +J/ g.lA3 r>L.":;.r, e-?b yl-9 cr:*,l o-t-i n:iy-4 ey d-ur.F*,
t = o d,L-J dlJ dt- "/.1-1,
crr. ,.l^l:...'a Ot--<r- 6.rL .rLar;
x=X(X,o)=X (q - Y - f )
, € f* c*'lXP 6.:t- c4i -f )ltJ,5!tt 6 f eV o.t3 ,SJ.stt qi fl,-S,gbrrJ<, ?-\ -f aJrL'
{=X(f,t) (\.-Y-Y)
.:l-u.f ;r o.13 { ,tns-.$r.,/ eni clL;;l u-ti 0lJ.o1 lr rri Oi ).- L6 la E*6.c*,1 o-\,.i e-c,-rl c,.iJ* .rlr^; GU: iiol*t G*
='f=(#)-{
(\\-Y-f)Pd{Y-
dt
d3 -ts X c-r jrs urf lt<^ *, X -i:, 6-t-o.r..r, ot*i;J 6ri 6rtjfi[j di j.r 6
O=--U{f. aglSl a;-921 66L:"," *lS" lfr5c*,1 oJ-i ,rsrr,. is? ag,j-s ^Li; clsb,f c r;[jL-,| gL r ri_+ .c.*,1 oJ,..i a:^il.:
,f *- ol+* Jlr-g/, -;l(.li clrlr c..r"-l , .(r-;.: .$r./ OtJ b dt j q d*i Jfu.t-S *--* ,*.,o JSj.eu l, f*
v=i -d* = ax(x,t) = ax(x,t) /dt at T (\Y-Y-r)
1l c*,1 o:V 9b:; Ol+*, ,.1L.i.. -rA
a' x(x, t)atl
(\r-Y-Y)
dra-.*J \\"- Y - f d:t -. lr,lidrt- sLe*.Stlj(-F \,fC 0t3y'./S.r-,.:-*C
$'-S; )l ,Jt:* b ,S-rt r;l oSr -ll;li,l .9r!*;l ,h. g.,a 4, .:l-rs ., €Jl
,-4,A Ot-j.(,- _r, b t*ii;.r c..*i_1 .i- ul_p; 61013 l, d ,'W.Jlslrel'p-SeF-r'$i-l-1-r 6rv osr st ol t -',.y,(.ltu3lrt*i c.,.,>) -usJtijrJ, gA ,(t*^r:rt dli,'ra,t J:-:--^ eUA d-urJSt,.1, ,(,-ri- ,(r-t .^^r 1r 6.9lo;3 ;r y'5 o.llti...tJ J,_q c? - \ - y nJrt.trI db,J*), Cf-{._[L.f "llf
t^,g.rftfL'Yr -,c*,1j1# 66,1r.,:r"JiIlJ,lLi;Liia.,\:*i &'o.rli:-i.rr-!-l-t-i!u-,_1r1,
..r-,tf.f I-t ;"- ,-r-fu Cb q jl* , ,-l)tt ..r-lL .;.ll: €_f u !- -1, t
x=I-\$,t) (\Y - Y - f )
ls6r r1b,_4 jl+, \Y - Y - Y nJ:L. ,lJp -b )|..t*inu, l.t1,:,,lfoL.l a,5
eU.JJiS*";: r,:*,lx.rr cl.l5l 013 6lt Gq J eLq'1 l) i:,KJ 'p J { Otr;".
0r-r.i ;"-* c J;Su- O.r-"oj \ o'js-.6b d,+" )Fs 6,_iU, L-i .€d:5r,-ll l;q -* eJ:t .. *<lt gtr ;*,-.?V OrSl3OLr"j,
a=i, =*.=d'!:dtt
0X,i
dX.a1J- (\0-Y-f)
t16
.J-i J.6lJt a:-tJi l) o-l-f1'l-r + clLy;: yl
(\2 - Y - Y)
tJ ostoy g:lniJ-frlii lr t - Y - f aJ-:L, L-3'1'^"a 4'57 I - f -f dU:-,1 o-Li oslr
x\=X'+lX'xr=X'+lX'xr=Xr
..r50"+
.-uq X (t , Y, \).r);ir." 661013 .r" (.:jl)
.ri! "JtJ/ = Y ,i-l o.l3 r,lL,o .-,1S J.t-eJ't 6t'aril;, (9)
jl c*,1 o;V,SyrJ.rLrX (\ , Y, \) o;3,JlJ ('Jt) J-
x\: \+YtY , xr-T+tY r lt=\
&JUr.r.13*o13 ,f,,,\ = \ Ot*,*^-a JXt- Yrr : -Y JArL/4+t'j ct ri:a._ii-t 6
.JSg" t5;- xr : 1 *.e.,,l-a 1r .:*'! E * &- eS:
g:lnil y jii 1.r ,-,k .i J \>-ef 6totll-r... \ Y - Y - T -l \ Y - Y - Y .r).:L, t-,i tVljt -t;'JF
= YXr
: YXI
g+tb/ = Y 4L-JJ: X (t , t, \) o;3 6lY. q.:.1'r
=f at:Y
=: r
1)r=: "
r- l#
a\
aY
A1
yl
vr
vY
yl
vr
vY
= \ txr- \ tx1 J
:o
O.i--u.h{ agly a:*92i OLaL,:>" *K" lr'Jsj-e d/-J-i LJ +;..fdJi-tJ \ - y - y Jti-. ,S)--lr):1.- f _ f _f ,J&!x (\ , " , \) 1:6il-e o;.!,-.,h-.i JtJ-ef S6oit-f,1K;i.-ri-ig*,! X : I-t (x, t)
.-tJC-';.:*,1 u;Lif = y ,.:tS.fi:Si O \t*,$ 6;-,.)Y.:L.o otf u.iS', I 1,, ,rtu JD
x\- *,xu= ft|f , xr=xr
.$r./ 4+rl \ - Y - f JU" ..r[:-X J J-eJp! b.]Ss ;s 6j,t3.t? tl +f
rr: \(x'r-tYxr) n: \@r-ttxr)
vY:W
vr=o
vr=*
vr= o
q a,= \(xr-{-xr)J --r
\ -tr
aY= o
.c*"1 gti/ : 1 r,lJx (\ , o , \/ .l.: il-l .r3.Sl:
v, = -i!- c'\AJ
a,=*
at=-*
aY: o
6flFY.t AJ.tl ,gtari*.as;f - f,-#rg5lL"+-, o.lr 4 t- 'p Jqfu, n tr ,_51-l; rftr.i_-51L-'9' uolf J €f O c.r;J cl! t ,Jl.)J X ,-/t- s Jl.,a;at
-J tv c:,-er-, )3t
dq6i, Ot.;.-c u. e+y.JJit u#-t .c*,1 oJ.i dbl 6.rL L- Fn <-V.9;,,51
s*,,.Jr,;,.1.r,-r,. 6yl.StJ lq-,. 9:'yr .'o l.s, ):c! grlo.ld.lL:;lr = . +L-J;r r-il_9
.lY.rL,
trv c.Sg t JS.E* *(+:f l,as
p=p(Xe,t) l{ p=p(X,t) (.-.cJl \-r-f)J
(v\-r-r)ri = t/i (Xa, t) t{ Y = v (X, t)
;l',j.fu sr.Sr./ dl* r-*;l or3 -f ,sl1.ls,flr i/l "/t-j 4+n-rti evt Cy
,flr A 6f ,fe:;,i o.:l.r t dl4 3x 4Vi r>l-,"*' il eli os-r-u.Y sP 'flr<Sl{ c*.9; L-,i' ,>-,1 g)L..'.rU-el -*'.6 t-FJ \ ,gB 'VoF 6" l-'t
-';;li,g\ - y - f nJrt--. 6c*,t (lsc \f - Y - f .J.rL, P ^Lt;11 .Sl.-t-i cl!
.i:.-ii 0tjL/r,urLl Al l.rJ.l 6'-l-;i,;ra2' ss ,flf OL.l 0t* I ,j1{
p = p (X, t) = p lX^ 8, t), tl = p* (x, t) 1;llf -f-Y)J
v = y (x,t) : th-t 6,t),tl=v* (x,t) (vY-Y-f)
4" -ai Lari:- .,3t+.tt? ;l -alcu., ,-*.1i g r-!"f r5b JJL! a', !'oi 'rU-' 0i .r' 5
.rl.r: ,Slf. rlrlit dLitl Jt--;, .2;i 6s.t qi 6:V -i4',"i st ' 69*-rl'a?i 'JrloJi-i
u.6$r>aq r5t aE i?oi;l 6,rJt-' st c ti.,iu' f -* '- *'lc5-2- Jb-rr ;j:'.J^r,,/ Ol';t , ,fV tbi;: 6c*'l u"-UJ
jl -r;rl..e.J}FI 4ci !.Si"il-r" rtiil-t .Sy.rY.rL, I -f -f JA
x\ : Xret + X, (et - l)xY = Xt * X, (et - e-t1
4:Xr.$0..- l.t6r c-l elslwi
4L=J Jr rr : Xr Jxt = X, , x, - X\ Js-re! '{f
6.1.6 si- qi J';; JD
ptr-.t .c-,1 o-tJ.1l-31 e:'l d-rrJS+, 0lpq qJ-lt ,S-t*5*, 6 tyul *:it = o
tr f.dl .r1l.r ;p rj-ll alrt -. Jr rrxr elt l.:+ 6li € , cSf ,SilflJ.:!rL' '$tV.rf b tf*.r).rL. d:;-ei s*! 6l1.la.it"*
O:-J.l1ro agly a:-9a 6laL*. <S+& trA
Xr = xre-' + x, (e-t - \)Xr = xt * *, (r-t - e')
Xr=xr
giy-r5A.*;lrVH JtJ-ef6tocll"r.r.(, \ -y- f Jtj, ,Sf ,S\f_f _ f JA,."t lt.J r.rl* .SJ;l : ,Sgl-Sl ,;J.l,, +-i rfi.i j: y' rrl-r
,bttettfl r-ef,Sl"*l-r, \Y - Y - f aJ.:t . L-,g Je,-.4r,J s.Sf.ry.:t .11
''--:sin c*r+vt:Xft*Xr"'
vt : Xr 1et + e41
vr: o
jl -t-.r!.e.-'t:-i,S[n.iJ_f \f -y-yal:t -;l3
at = (X, + Xr) ,,
ar = X, ('' - e-')
aY= o
.:Y.tL-.s \ -f - f JB-. 1:o-t ig*,.t, €-fudq.1ij.r):t-- ,C-f\ ,-l_ffe.+i't \:-,! S) sl rfii .1r,-,t:..i
v,:"r, *;, a\=\*hvr = xr 1et + e't; J at = x, (e' - e-t)
Vf=o Ay=o
,ts,#j0t+,f_r6&2r* \r-'.i f..- ,l ua;V 61013 oFcl oJ.rLL \ - y JKj;t 6..;f L."
u = x-X (\ -y-y)
tn oS9 e Jf;,,;;,;;tr, *(+:f les
dlrt -t-.rb ,-l-l .r;a .c*,1 '6if .:l; c:V dq:S*' y et'.Jq:S*' )l s-* Jb rr
6/-ro- *lls.St U- J<-r*1- 6c,*,1 Ji. :+l ,Sr-/ oJ-o.:115 ..r'1.''
3: jl f y elnil-r "fii 1r ! u Otir."i;.ll:y tllyr" .ri; ,l$ f ots"r4a.g.ii fj JrL." o:V
u=u;6;:ulll (Y-f-f)
6:Ljii;:l-,1 \ -Y -f aJ.:L. 6li€ L5'-l;tJdrL 6l-a';=ag 1.!-1:cU,o9)\*'r
u (X,t): x(X, t)-X (,.-Jlf -f -T)sf ,*-;s G-l;3 rfi.i 1r U-
u(x, t):x-X(x't) (YY-f -f)
,s u* x 1l 6,51,t3 alr rhil "r.otr; Jbl 6 Jss"';:-i b Jt<"r.'r Jjt aJrL,
,St:irL- ! .-rrr"f dbl ,J.i Cl-l Oi c.-i x 1: cl;51"o 013 dt, .,,t<"r*l ,g:r oJ'L' -l
fu st ,J-ljl3 cJ-or,r.. tFrl eU dqft a1 b Lac'r"i J1 Lit 6:L ,.rLa;;, nS;-l
,Sl-;. ,t-F-l+ .-,-J- ,-i- .lt-r.eq ! Ot<,r*.u .:lr"l GU: aj bli € cr:-'l clL; ;l
, ,-t', rif F :t rer-'
d (x -x)dt
(f - f - f )
3:y'-;r \ -Y-f JU.-,1)o-t-.i,11,-il .:5-; Slt\;Ot<,"*ijOlJi. l-f -f ,JU
.aJ s*,r Lr{;3; r3rb w;=ag
7-'lt'l.o;,*c \ - Y - f Jtir,.Sy.rY.:t-- )l J-
xt = Xret + X, (et - l)xt : X, * X, ('' - e-t)
xr=Xr
dxdt
da :dt
Cr:-.L.t" agly a:-92; cSbL2a, *9. lfo
ut = \ -Xr - (X, + Xr)(et - l)ilt = \ -Xt - Xr G' -r-t)uy = 4-X, = "
,r:r.,_ c \ - f - f Jti, ,, r** ir)rL, 1l r.:ta;-,t b
X\ - xre-' + x, (et - l)X,r : x,r * x, (e-' - e')
Xr=\
.4'* \:-,! 4:loil-f "$i -,1r cltir*g otJi. aG J*i
= (xt + +)(l -e't)
= + (e'-e'')
LgbJ*6-fl-r ! a:-,-1*, +- f*? 6- ,-$jL"t* t- .F-j.t oAf "-F * c.:^*i c.*l-rr,Lj t-.t L r'li .Ut, 6U*,V [- g:tr: c6ir& .:*,1 OS^, c-.,r\- d*l ,,lla .f--Sg -h
6rL ui..ay _,y.: c*,1 O<^.o di_l .51r tf Pu ... dr g
ul
ul
uY
Pt... -- Pii ... (X, t)
Pt .. - Pii ... (x, t)
(,JJl \-0-f)
aVi r-j..a.r; -rl U
(.:\-0-f)
4 ,Stf ,_rt i.li.:lr*i; GU: LJ )l c*,1 .rrt r elf ;ri7 6s6 ,5;e,.r-r; Jl*}li.t' aiJgu-l b #.L gr_l . c-,.1,9, L--,r*.7;l (.;.*, t_ 4).rl-,13 rrl-- aoSa'e
,>lti'2-3 l+ o-r3 L,i*,o.-6 J .L$ rLU .i_ L-,1, rls PU... -*{ A) S :f sya.,
.,E,srL J;-, altl:j-; 4ys aLii orlr jlj \d I dt,Ft n_, ,iltl .:1- 6f ,;lsl
til
.J::...A d)L ..ft&-L # t, o-l-i r--c1-rJ -rb-i J t>'e)tt 6'-f ei tg-tgf
,rh grt ,*bc.r;,.i dlt ;Jl t - O - Y aJ.rL,.;$ 15'rL 'g.,'i -r'Pi "' uisJ
..-*.1 OLj tu r!.i ,J-f '#':r OL-a orL
da I P,r ... (X, t) l = (Y-0-f)
q + 3 J::*r *JLUf X 6:L ,->l..att c$ orl.r eii )U 6cfb clg
.:J.-.i osl: .-r \ - O - Y eJ.:L-. JJ C-l;t rti":r.1r Pt "' ft Jt 'J::f,r.i J.:ii 0t*j
s lS.l.i:i; t*t1r.ll;i 43J, Ftt-r ooV.>ls3 a'ei -' lS':ls f.otf o4+'3.)y..rtS"t \ 6lo.r"v's 6-SPIA-: olcti -t-L
S tYti... (x, t)l
*n,* [Pu...(x,t)l= Pr ... (x , t)l + r*,tr# (Y-o-Y)$r
f-,, -urf"f it'e;l x t43J. Clt ,flf -r, 6 ,Seis '- "'L tb JJI dt"? t eJ':lt- c./-l -lr
c.:'rJ"..rlr*i; )l ,PV g-l: nJ^+ c.:-,1 c:(t-t: ,h ti r.isf 6J:t-r" 0t:
- y - Y dsL.. L-'y d.;q ..:*,1 e>(s-t: ,/fu| ti ,. isr -l I3!.-f tji 1r cr!3
tr f - O - f aJ.:L, o4€ +.f v = dx I dt 1U-v* : fup I dt) oye+., c.-cy, \ Y
..:..ii 6li€ *-S ;,-t
*, IPu... l= $t
daa,da-
= |.!_- - : = -dt il *&k - dt At
(x, t) pii ... (x , t)l . *[ Pu ... (x , t)l v1, (r - o - Y)
.r-rct!F".l 6'-l*i3 .i?,.1t 2J o-$ bV.,flf Slt <S:L ,5:;:* tylxl' crTll lf
+ v.V,
ua;hdel ,i:l1l.,r c,-.iyrrj -l c-t.i!"f d/-)U ,5.;l"rfiLi Jsi.ou;;r f[S; 6Jb
.r:*,1 6M3.5La;;, a-, c*i ,J-f .rti:-i 6 6'l
(0-0-Y)
4'1L.:.;,4... agly a:-9;1 66L*r. *& lrro.:1.:yj Clqit_f .rY.rt--!*** L-* 4,St : od*r,.Sf 4 t _d_ f dA
- I rg
Jl : Xr€-', xy : X(t, xy - Xr * X,, (e-t - t)
t) Os-*!'f6;rag J, f*-+ oslf 4?Jr dhia Lgl,
T=e-t(xr-\xt*yxr)
r--J.: ol+, dT I dt.g:L .;:.L,..ri;l orU*,| t .45ir.*, \ 6,-l;;,fS-.:;;.r \>cr 6lJi.
..t$+t.*, l-1r>.1..r>
.r' 's!"f er dJ..tts#, ..rlPq qJ;l ,S-ri-& 5c.*l;lr {?i J,-q ir-r*.. t-.i-l.,s a.,jo
.S[n-Jr-c.:;;o.rrr-rt{Y -O- y nJ:L, ;J:.:r,g,yU q - y - f drt 4+J
'x:''g O.ts osf d/+ d-lJ ,5rL r$l -;) r.L.c-,p.,
vt:-Xr€'' , v\:X,rgt ,vy=-Xre-t
."rr-i .r-,+ b ),5 u"Jt, 6li € L,Jr*. \ -Sf rr)rL. C;+^A
Xt : xret, Xt : xte-t, Xy = xr - xt 1g-\t - e-t)
.::-,cf4+;J 0i a6.Sl^-J_1, t cl-e f \l:r.,1.r,g1ljKt q a.f
Vt=-Jt , Vl:Xl , Vy=-Xre-"
.rf l-,1-, f - 0 - f +J:t--;l orl+:*"t !.l_E ;[K...i1r r)dr I dt ali:obFl ,i/lJt..l
dr : - r-' (rr- y rv * y xr) - xr€'' - \ x,,e't - y xr€-r,dt
trr e=59 e d$,Eis *l+:f dps
- \ Xre-rt - y& (\ r-" - e-t) - y Xrc'
,.lJf ,g 6c*,1 6"1<a Sl"f ibt 6 6r-*. ra2;4"1+; J 1J+ t+;J;r rlt5,,.r*I 0hif tSssY - 0 - f aJ.:L, f=4.-. .l,1-rtf! a-6 "$:1.:
lsc--er* Ohra
.156. e-* l-, 45j .r-lyi Jti.. .piL- lt l ;14tji JK^i
r,1ttiL.ul.-rris-,! \ - O - f JU"-lr 6l-1 ,>-cr-,6!l;t gLn"iJ;, f -d -f JA-;) o-tJ
-SS 6f 6lr'. 41-ai rfS.i 1.r 0tiJ*iiJ .SLn*iJ;. 6ss Y - 0 - f nJ.:L." f,it."
.l.S "5e-:|.
l.,1 JU,' Ot^,
;r o-LS olt-.il cSr-.:):t-, jlblir-li-.ar**, l-,,,SrL r[S.i-,1r OKr]+t g:u,on-f ub.ry;iJi#6 \-0-f JU,
u\ = xt -Xt - X\ (r-' - l)tty = Xt -Xt = X, (et - l)lty : Xr-Xr : Xr G-' - l)
\ -O-f Jtj-.-;.:o-t-3a*-l-.-. X = I-t $, t) €J**.rYst -..- jl ,rG ..1 i t
.pig !:*J! lt 'l-G LtLadK".:iiJ
u\: x\ (\ -et)
uY = xr (\ -e-t)
uY : *', ('-" - e-')
.$r"/ 4+J f - O - Y drL, t, Pi ... dlJ u; 6;ltK-t+ ! .,_*lr:ti
dT
-:dt
du' \lt' Eu'!r.=- !r-r-_--j' dt il *&k
O?Jq4 agl2 a:-921 6b L!;," clJg" lff
v\ = - x\et + v, (\ -e')vy = xf-t * vt (\ - e-t)
vy = - xt (\ e-" - e-') + v, (e-" - e'')
4"-.;j ! alr[, i];ry' st -bts-f J.ef ,Sto*iJ-f 6!l: rrY:l-. a-ey*".{ 6
e-f .l"il, ..4 orJJi cd,! .Slf ,.1L;^, -rr-U .f!.:L, a-eJ-'-^. Ot ,"F:-p...$r",/;:f e JeV O;i iltSJ'- r, cJl- ,jr*l.rr .JtiJFi ,-F *y ,Sl-t
vt : -rt vr=xu _ltvr:-xl€'-
.rsLi.-t;16 b \ - 0 - f JU, et:.s
{Jaz.a ,f-5rs6tt;l; ,JK;}; ,g!a,tL-:lf f - fb dl4by, -'r -l gSL dq15*,,9r a.,
'*5rr, -4ls +i c;fi,r*i; ,f-L;;r
ql-rl o:Sk Jk4_-i,t 6+.&)l .;_t+ o;SL (F,r* 6+.&6'*V oJtlr,Slr
-8,;.6- ..ll.9;q 0l-j I c*,1; O: * 6b ,e"Ll y-l; .f-:lS7llSorl-ei \:-,I..tr l"r+t: t-s O;-fa? - Y - f drl.. a2.J.1r cc^-.y
x; = Xi (X) t_ x - X(X) (\-2-f)
_,1.r Jj;l: tlj *U J<-,r*+ ,JJi;fu )l Q : P Lli:r 6 g.-+ 4 ;l qt -^,o oJt -rr gr.,_
{.afrJ,- -,lrr-: L-'y Y - f J<-, aU" P +i c=.,j Q 4;J 6 e:_b. t#,g -h.rJr 0t*1 $ uL-;lAs
dx--ilQra (Y-2-Y)
1l c.*.1 o;VdX .;l-ul 6r,
(il}' -- dx. dx = d)Qd)Q (Y-2-Y)
trd aS9 e (Jgrit, $(+:f !,as
,.:iU Jgr#
xl,xt
| -r J6.t
JS-r.*- Ljq-J* .t"Q sP LV ;r-dx a,.r-i -ul.p ,f,-r+ g-.,r Su.-*,:tQ :P tla; Cx dX rl"y.
.P :p !tr Crn ,i*l 6211' u Olir.;; -2lsy . a:iL-
Q sP t: Crl; r:!3 c \ - 2 - f d.:['-. Xi 6l: L-'y oJi.i g..*; cltir*;J tll-t.. c.t;
tu;: fuurn*n (7 -? -Y)
.:,*r-l {,.ii J.st' d/+ Ub 5
(v-2-f)oi J, at
llry 6,5lnJ<'. J;5," ,5;z *it- Js:.:,r*! ,g"t-.-rS4 st Q sP dLnc:^t-1, I .i*Jq..:*,1 Jsye C/+ Ot5l t+l ,J+ c.iJJ,"
dx : dri6i
@9Y - dx. dx = dxi dxi
(f-2-f).J^i JAIJ' si.;lJrl et' J
(0-2-Y)
tui= #d)e-xiAil(z
Or.,,J4''+ ag(y a:-92a cSbLzoo *9" ff?
xiA = Fu (^-r-Y)
F JJ*,Li.Jitju.- JS.E_JJ oL::l.S c o.:t*,rlh t_ dg,E!;t oL;lf )pbl)c/l-r:.-S!X 4r q.:*,ll e-t'.7p.r>-'l O<^- 6 Sg oa;:a bX_r, -; J<-,r*;,g*A t, J<-,."r*, .s!X;l #:*, F ;|.&U.r <fa* l, J<-,-*r c.Jt--
.+f't.t_r'/-.lV t-jJsfJr jl F.1r V- ? -y drL,cyrL*,6rljS:L:-1.:
dx=F.dX t dx=FdX (1 -r-f)
.ii-- eJrt-- -s: ,g;--lr 4;t;n1:+l &rt $r"/ clt.t g;r cJrt ,. +rcfL,.,..1i;.r 6
f{.r-6- F dpjb"r, sl 6t;1jti; CE Ss,,ri JL^-"| €J \l-.;' 4,r jl .:F,;..r;l: .:;*-9 F-\ u,,JKr-o +#.1: i.*"1
d)(a = Xe,fu; t" dX:tr*l.dx (\"-?-Y)
tsf t-, J<-,,=.- ';l.Ul d/J+ JJ*.,-b.' lrJ<-,r+ s oSf g*i sr
o4;*1r ,fii."..'.; Jyt .1.r ,*iJ 4;+ J 6 b OUi jl .,(.1 l.trl .-rlrrrr.. .,lri .:l-c-,,l"W- r6f eF -r, -\::.-.t Q sP t, il-l .rb3 Crl {.it,- o1(s:l
(dx)' - (tuY)' = dxlxi - d)QilQ
l tj o;j.et.t sli,-l 6* -3'l1l-SdtJ, c....rt-, S.:l&V3 V - ? - y aJ-rL,;l +f
."trl,
(hc)' - (d"Y)' : (xiA d)(e)@i,n ffiil - 6A8 d)ac d)G
-- (xiA xi,B - 6Ad dXA ilG
= (C.qn - 6ad dxq Nn (rr-2-Y)
rrv
CZ3 : X;,e Xi,B t- C : F? . F
dJU:, syE dl jr 6
(\Y - 7 -r)
E,qa 6Ft.SU seso." ,i".5 )*b nL."Ut g,-l jl .eU,_f &.5 JSzEt t*bl:
\ E.qn - CAB -den t- Y E = C - I.rF€vr.,_F(\Y-?-Y)
(d*)' -(fuY)' : \ E,qnffitffin = dX. Y E. dX (\f - ? -\)
.rj.i oslt .t*: qU, ,y-*n'.6,-l*;t,gtor.;;. yJ..:-,| O<^-o (dr)t - (d"Y)' .J-ll.i;
(dr)' - (il]' - 5ii dr; tuj- (X.t; hc)(XAj dx)
= (6ij - XA,i X,q) dr; fuj
= (6ii - c4) dx; dxi
sii = xa,ixej t- . = (F-t)' . F^)
(\0-?-Y)
0JU:r rf-b ,Ji j, 6
(\2-?-Y)
-cr-F cS;J/ JrJ.a-o ,F.S )t*b 4+*i C/_l jl .eU.f irg (tfi4_-is tybls
\eij =@ii-cU) t_ Ye=(l-c).tyrf
(\v-?-Y)O3.Sl +4-.1.1r
(\^-?-Y)(d*) -(il)t-reii dvituj =dx.Ye.dx
O2n.$# ag(y a;-92i <So,,Lzo" cS+8" lrA
Ol.iJJLt 1l 6 ^t,9K;L-o
cx:.*.o o.;1L.. lJr t'J, 61a1.2*,b eij s Een ,yE Jr J, €Jl
.rF€ oJ,aLL
r Ji,arrr.c J<-,r.* .i*rl ddX(Y) J dX$ ) 62t :;l J..-t rr-, ;1.:y 3r -f csll
f*lf \f -2-Y..r \Y -t-f .r).:L,g"-+.e 1- ? -Y d.rt -;l dx(Y) Jd*()..:--il-:
dx(t). dx(t) : F. dx(\). F. dx(r) = dX(\). F7. F. dx(t)
: dX(\). c. 4att) = dX(\'. 0 + y E) . dx(Y)
: dX(r). 4att) + dx(t). f E. dX(Y) (\q-?-Y)
6 S'l oqtl \ I - 2 - Y eJ:L, ($! a:;l-u r7r .F-S) --J,U. -*-"W- E -Sllrdxt') - 4att) = dX Oli,_fl gt"f r/tlr*.r dJ+ ut; )gl u"*,Jl^JJtq3lj dx(t). 4*(t) = dx$)4*$) tot 0 4"-F -r,_, jl JI@r)' -- (ilgt 4+J-r.:-rr5,-/Bsl
,ltir*;; .({ !;i ,-p-5 !/tii Jr crdo .$t J.blJ* "/q
r-,at cl;{ clL3^,r e f JsJ,t dr::
.Sl"f itlJl .-I.a f,.-"J \f - ? -f .rY:L, L-ry' 'g clLi ej:lJ.JiF) s;Jrr.c ,-;;-S 6tr;_p,i;
b Ot--S,- ,5b1-9*,b OJ I .$loJ-i a3l)l u;;;- J<-,r.* 6[a,lL-slrf
-J. \ V - 2 - f
F-*l ctj.."i. C-l e\.rl.r.u-,j 6l&2of agla6l:$-J.0b"^^ bli,-/1:.pS,r-' U-i J<-,r.* JJ" !rl_l-l J,JL"j jl Ji't* r[i.i1rf - f - f dsL,
jt *,1 o-tV 6:L .-i..., ! c4;l+il j',5;l-rfiLi
u;(x) :x;(X) -x; (,.r.ll Y. -? -Y)
!.-l J.IJ.6'-l;;,-i-,ei: st j
ua @) = xA-xe @) (:Y"-t-f)
.J^i JAIJ; \Y - 2 - Y .,J.rL..r, J-91 aJrL-.;l
lrr oSg 9 J,9.lr-i'r, *(+:f J,os
\ Een: xiAxi,B-6.tn: @;rq * 6u)@i,a + 6;il -AAB
.{t".f .-,bl5yi o;:-\ 6
\ tr n = UA,B + UBA * U;r4U;,3 (rr-t-Y)
.$r./ 4+*i \Y - ? - f +J:L,r, g-lr oJrL, jl -l
, e;i = 6rj - Xn,, Xej = 6ri - @a, - ue,)@ej - u.q,)
\ eij = Utj * tli,; - llZ,; U.aj (YY-?-r)
i6x, =Xr, x\:Xt * kxr,\ =Xr * Wt.:L.,..i,yJ<-,r*+ l-?-f dU
Jft'.rJirr Jt^ot J<-,at!" dL t*'.q 6"-6.-'S".i- q c.:-,1 c.1tJ 4 k rsl st
.Jl^-E slt l.: (dx)' - (il]' o_rLb ; {i{ l, :r,(" jABGH C; "it- J<-,.+.r50..-" .-r<" OG sBH , AG
.rl-r3 -l sb,-f .-/q ta ss O i+, 6 )F,g oJ-aLi.. d+.F aJrL, :t"t*;t* ,l-,b (dL, dL, kdL), a (dL, O, O) ltejq?y1 H sG, B,A !la,-;.: AVt
{+3J stc;j1i6 ht?h (dL, ML, dL) : S IdL , (\ + k) dL, (\ + k) dL I
et' P -9 cJijL,r" ,,it Laa'.i"r |La.1.: X$r ac.,;.o1. ejl-f SlaA;ie;.r cr!3
.'y''r.J*r", J<-, at!" abgh JK.i, d.,Ut 1)UYltS )l* yABGH )*'ot+rF 6 t-rrf oIaLL c h - ? - Y aJrL, j 6t:1,j+ii.:):L. ;l dri.+^!
...:*,1 J) ,*-il- JKj 6!l: F J<-,rr"t
[::rllFul =
O44JJ.4e ag(y a:-9a 6bL:"," cSaJlS" lFo
,p, fi!.'t'C = FT. F 0J"
.lYk I
\ +/ctl
.tycto# YE: C-I t+i;t
o olkt \k I
rk " .l
xt ,xt
tcent:
[ "
e(f.(l+$tl.(t+$41
r *"YK
[:
[tPr1:
tn cS2 e !5,1'4; *(+:f J,.s
1-slsAG -E elt s 4-d; s' (tu)' - (il}'
Io(dr.)t-(tuY)'=[o, dL,dLl I "
LO
=Y(Yk+kt)@Lf
E. dX , r.Ftill.
'j I L'lBH tLt 611
=., _dL,dLt
[: ,i ,j
] L :)
= Y (- t k + tctYatloG p; ,st1
jI lil
: dX. Y
o
7rr
YK
Otl' - farl'
t. .
Utl' -filff = LdL, dL, dLl I
kl
L O YK
: Y (Y tc + tctYafl
"f-f ,Slf a:.rt-a J<-,r"ir.-'-<" {-,J:-^ j't**;:li,-f l, eb C.*l "fb
: qi.,-5 n -'t-'"':L ;$lr*;;
&-f.,*J*^;+-bv -fr,-r:-i! &f ,-P Otir*lu ot-rlrf ,Slrr-l-b,Str"il-1,it^j 6.r-re j-:l;' f|4+ti ,: r':f ,E i-7a t^Ot-rFrf ! e*,-tr, 1r t'ac -J Al qf s J6.-r )l bli,-f
.+t+.f ,-;.l;S t-i Jsyeat YY - t - f .,l Y\ - 2 - Y'r)':L'
OF!-.):,.,tf ,g(, a;-9;, 66 Lz^" cS*K" trr
J,-j.. q € G):l : silfl 'pf o.:.:; ,/;.slo-r_l-b a, i3ra..#iq }..l-l-r.rl
cSjL ?-* ry-r ob q ,ti:r
ou; _tu, M.* _tui (_WrA- \_@i,x" :&T"=d (ffi tdu )=fru^ (Y-v-r1
.l^? )l -hu* q *Jl y" - p - f +J.:t-,. ,t rS-S,.*;lrr{, ! r6ls t).lnb t+r_l-rr 6.,qlo:; -;[i c.:*,I o.t i.:-,-t, OtSrJ#J ,lt-rf6r,
#=#*6*e5c*,* g<. , 67t'0t<"r*ru 6Lacrb-:tJcJt- Alt A:6 ?-* €-t ,|:l b Orlfed/l jl .-t-,tf"f 46 [_ 6.rL ,>1.,o:;- o r-:*j Jt<rJaiij cSt"*il;. 6;S;14r \i.<"J5J<-,-"*, ^'-p; el-r'.1: J:t*- ,s*, cltir;; 6tnoL-rf Jli€, otjJr_,
u.+j Jue,a u,L; ^,t_-h;s 5.:lr 0t-j-i Jli./ rl*L;A.rl.: tsl.ij u;r\ u;,a os-re\
.,lr \ -V-f .:YrL.;l c6;L=-rp +y_,r.lb ;6t{,-*rragj . ;l ._r:*,rJrL.
\trn:uAB+uBA
, e;i = U;; * ui,i
E1n = e;1 6;.a \n
(\ -v-f)t
(Y-v-f)
Y_V-Y(f-v-f)
,1"; ,ri Sly.6 s4,f .-,-F tj:,- *f /*, &.5 :*e,{ JJ,*. ,_b. s.ttf .t Gf 1) c>s-9..au e;1
Y'i - haei * # 6ni : k . # = uil # u1,i (o-v- r)I
e*Li (-tI*A lJr ljr.- ;t 3 0;tc" 6,* ei1 s e;1 , E.na ,-;;f ,gLa.,1.pjb clyq
-uil- 't+ Oli € lt ,rbl r:,;f .5Laca7 s ,Ff.JL^t-t, ,. Jpl6L",.rijf ,JlJ :JU,
trr oSy t J,g'E;s *l+:f cJ""s
.l c .l-O,DV' )Pu
:l .r\-a ^> J_l ,-lS
[e*;i] :
I" : €;; = tr (e) : tI * ey * ey1 1,--allv-v-f)
II" - i G,tuii - eqefl = eFil * erFur * enFr (VV - V - f)
III" = €i1&teVQv*: eFtFnt (Vv-v-f)
;.: .E 5.r.- d)/.0t51 6 -r;!.,r" oar:r 6-f tit 611l: e ,5Laa.eJ;*
,r[-: f-j oJ-i C)\-.rl "t-f-^l 1: ! \f - t - f d:L, , &-f J<-'-*'; qJa, 'SF
€,-t(dx)'-(il{)'- Yri il*ili =dX. Ye . dX (A-v-Y)
.,.:'j,l-r f.^lf r:/ ,ry;; (d,rYf y eJ-:L, : ,S-S:-6U eJ:L, ?? \:-a4r tl 6 u"l<a
dx-il( dx+il(-yr..e&dJ( il( ' "'t d){ d)(
*-f .5tajSJ,rs- ,sl-*, s -r-,lq,-r dX:"aa;, & rl1 .i- il(; I d)( = Ni t-t
r.7.11r c (dx + d)0 t d)( = \ tf e; 611,,l
glP{r. e; .-,tiu.,l ! .rr, .r9,*i L-l r.'11 Y }-ai s: 'r* :-*u .JLi -?i
f\ Jtel 6,lay^. q + r.li "r"-yb
r.lrb trl.-Lilf b 6rtS 4l ',jf
[': ',1 ,: ]=L:
,,,,: ] (P-v-ry
ef ,sL^t-t, q .lr^^
drc - il(il( =,NNi-il't'N (q - v - f )
OzrJ;,.,tf agly o:-921 <SbL:^" cS+& trr
oF GIJ;Lpl Jp -t-lj :t J-* ,>l#: Cls:l' d.rL, ,-r-l :-? rb, dJre c*iosl.: ,l, L.J rrN) q 3 c,.-,1 Jt-j
"rr-St- ,;-P ef ,. osr 6-r-,!"r N ,l-r:.1 -r, Cl-lolSJr-r;UN = i, ,f-tp-,t*;l& cAs?rrfr., JV,61Lfl .r*g
e6t):\.e.ir:rn
,-*,.b-'lr-l c*,!srr Jevr JL_,-.Jt^rii-sN = i, U_tl = i.,, ,Slr.: ,Jlyd,r,l t.St^.i;-5, ,>l-o:;' 6l-aya- rl"u,lrr (.("JS_,[*f ) *_f ';;f ,_*E 6-b t'lb
.156" 0[: l-, Jt-rr$S3JS )A.,_;;f ,yE 6-b t' r-W &.^, '** Irp t cr_r:,:6tr
J<-,."*, d*"' Jdx(t) ,gt-n.,,lrr: q 6\P c.$J..,rdX(Y) Jd[g, J";ti, 6tr;1.:y.c-ii t) os_nt .rt-r,_f tr \ q - ? - f aJ.rL, ,.:Jb i/l .rs .f#,gJi;, {l*"f
dx0). dx$) _ 4arr).dx(r)+dX(,). ya.dpg) (\._V_f)
'tK;i..S./t;:;l F-ri_ J. t-re \ dX$), dX$ j7l
dx(r). dx$):6$) 6$) cos 0: dX(r,. y,. dx(y) (\\ _V_f)
-St tti*Sl ..r-,tr.f f - T J-fu 4{-t- dL_ J<-,r-"r .gb;|.:y O*: qJlj g il :, 6l)dK$) sdX"'d-l J-.t o^iE q-llj +f <>lrr;uy q.s\c.:-9,i;L;il 0 -- ft-f
u*tr.:r"?u" orlJlll
coso=cos(+ y)=siny=y
,f-t',F rjL dp-lJti c\:.-,1 -hi* "1.:U
.<"-f j[e dLAJS-i F ,Slt T oX..!:.r^A .q-95 L^J#J*j.r l5l,4 ) d*r', = fl15.$) Jdx() = dX$, €n
T =cos0 = #. y e #=fr,,y. y e. ilr1,y (rr-v-r)
tra <zS9 e J9ir2is *(+:f JA
P dx(t) A
r -r JK;
,"bp , ap L;ht.:,[Ji ga| a.,l; q .-],-+ e:-/ 6+h,.s'BP , AP,rL; -,,t''J* 6 a-i[i t-jl;
."-r-'/ a;iU- Jf-r"., c5+;*
f.^fr cs, ! ,€ ,rn* Tr os-f a.lsT u-Jl J fr<rr = i, ,fr,tr : i' fl'4) "-
':l-ll
[r', €,r ,', I l-'Ir":Yr\""r
l:,, ,;. ::l Lrl :Yer(\Y-v-f)
6l-as-9^-fr c^ee \ \ - V - f aJrt - t, ci$ J<-,-'.* cstl,J ci4' ?btil L a':-';i .,ls
'f+{ qtel) \Y - V - f aJsL' "1,li,Y olt'e;;A
(\f-v-Y)Tii :\eii, 0*i)
6:)t &.5*.is-r T;1 ala+ilv se;i sPV 6-b,t dLadJ -r &+Lf drL';,',l
,Jq$:-, ;: ol.a;hs 6l-ay-- e. 'Pt 4^5U r.SL-l-lj 1': trlr"'; a'ls cfJ{r4
r^ 1 cr*, 6ly,Jlse Jfr +5rr esr .rjL,-/ etJ '5./ 0t* b'u;! J<-'-r..1'
..lit, f, J.rJ.e,gb.1l:rr &i- -f tSlta+tj ylc:r5.-a'y;P ,'dE 6la.1.2zu l..r-t'+',4
,|5-,i -1-, QU f, *-S t:^..,.c*-5 ,*b c
"r*,Jle- ,g)5.-5-,1r .ry J'ol-f )llJ.aL;i
O:-J"1a 6ly az-9;1 66Lg *K" lr7
[€rl=
J-lF\y-v-fa-J.r
l-
| '" {2't
I i t't €Yr
L {z'' }r"L'" c-Ur;./li;jl ;lel '*f A
)' = frtrr. Y e *. fr,r, :
JL--tLt' t-s t>
iilbc^a; J. J.Lb
s:re{.,-#-JV
(\0-v-f)
frtrl -l N,r, -jlr!
..frJ
(\P-v-f)o
at
x;
X;
r -r JK;
=v g4le.tl-* &- t 5-.r, dl"!',Sa ,rd,X') )dN" , dN', ,l,l*ti ul$*. AbJt6ti-."t-S,g
"l*r+..iu_ J<-r.*- dq,so ," fu(,) sdx$l , fu$, *, Urt q
t rJ.e 0U+*A ;lel ,,.'1;-S,5toca.7 urL"i 6r-1J, o.p b rl.r.:.-"3 "e Olir;.f+i;f
CJLJ16:|-* " 4, ,.xlrrQ .riq .<" -$ ,VJ<-,r*+ S gr.:y -r:rl*; .,ltr ec,tj*,b -,1r o.lij Jt+tl dN" ydX') , dN,, *ltj Jift'*J ,tnl ! q-f ,;U"tu
"t,-r+ u;U- J<-,r*:r &*;J s:6-frr cjl+:*,. C:L)l6rlp q Jtelri;r56rac^6ry
.-r*,!.-r !l: f - y j*, 4t!, \ $,sVjb 6 xJu"
rry
tu(i) - [\ + e61ld./i) , (i:\,Y,Y) (\v-v-f)
;lplf.2.' J>lJ J. .qo.;r J*# .J::-,.4 Jlu,l6Lac.a7;: JL; 6L^,F-S 'frl ul -r' 6jl ..:*,1 otV aJLJlglP
[\ + e,,1] dtrIt'[t + elry] dft)[t + e,r,1 dN') -dX') d{\t d/.')
dx$)dx$')dxr'
= t(rl * €ttl * Sty; (\A-v-f)
e 6t{L, d:l LV I V c-; Jtel ,r;-S 66-,'r))-.t- J-tJ .r>\-7 )l -b rr- l.
,.*lr ; f.^r.-/ b$e rLi q t, cli -r eU"r ,# eklS'i J^lJt
e:LVIV=tii=1" (\q-v-f)
,-;;; ,Sl-L sy 6laas.ls a-*p'i tri'!"f d.lk;r lJr *fJ,. r-l-b 6- e oJ"
:, ,y--h rlL.o 3 rjl-,cf O<-,. l, .!lFl S 6S-5,g[nl-l-b '".e '"-F O+"^] : &-Selt 6.:-,1 .liyL^-.- .);t i.i- .-r:5"f Jt1, b Y J',j -,1:,'lJ ,Slf qU''"'rU"
.-r-,!.'r o - Y Jfr &U" + ), -r-13 6!l.r ,rt ,Ff (6.:.e .e) sF c .ljr c;Lao;4b
tM =f;={e ot--i- reWl. e-b,-flJV4 .t-,t & -f tV 6 e.f ,*.5 tyb
d/J-iq u(*l ,F.S )9*b."r-,!.r L,-9:" rJLi rt".Stes-r 5r,f.r" 0t*.
.tyg w;u-t'; c>;2e
rlii :tt-+6ii tt*-*ii 6;i eu (Y'-v-f)
e-rv jii.;:3
AV
-:V
](YtI[ 0.,, etr
l;:: ::
4rrlw
4r
ttt - tM €rt ttr
trr 9lt - tM €yr
€rr €yr €fr - tM-v-f)
Cr:- ,'.t o qsly at-g:? 6L6L:i, cst& trA
0-r $,t-S Slt. ua;V sy 4;6oy-ls
..:*,1 ;"- il-frit 'pf S\r, .rJ_rl c ri; tr.rlai_t;\:iL c eS sy rqi
Qii: " (rr-v-Y)jl rs.rL.- ;lpl .-ilFl ,5lu,_*f :
4(d:e(d-tu ' @:\,Y,Y,) (Yr-v-f)..:*,1 4-f .,Q'*-5:_*U -Vl )lJi-tUl j.rr 6
a:ilr +tl.j OL"o .:t -Sl r.;b s3-3 P ;:Xy X\ lsJe\rSjl-f 6(ao,S,. ,;;.SdU.cl
.ar*el Ul-
(Yr-v-f)trr=/rr=Trt=Tw:Tyr: o
f^-? dlJ:r! dJ 4:-rJ*r- L-, y,.,,7 ssXt Xt u).o 1.-:-i y'.la->,;e ef6:3^ac.:1i ;r.."b ,-,.7 tl GV -e1: YY - V - f aJ.:L. fl :rl: )frJ t-* uif -b-rr aa.r!{OX'tX'tX'. 31a.1j-u dlJ.Ji:.!! X, :l -W.rL
"s!t, f-* ,S''nil-r f tvra^.
c..ljl f - ? )Sa;ll- oX\X$y 6l-a:.y- a-i c: i 0 r_s\l.Xr Jr a:ilr_ cr!3r
J-rt GldeJp "i;
6laY - q - Y aJrt - tf qU..) g;lle,"ie ,-;--S cSlf JUit .tj:t -6tar,,.pJJ;l+ ! ll*rl;r.-r5,r, Jll, l-, \\"- O - y aJ.:lr--,1J or;fi JUit sy!
trt
t'l\ =€rr * tvr a srr -€rr ,o, 191/rt sin\0
Y
t'Yr = trr * evr - €rr -€vr cos y 0 -LJ- sin \ 0\
\0T' tr = - (r,, -€vr) sin \ 0 + yr1 cos
il -u; rLp ^rLi
, ,-;? - f aJ:L..1: .5.:LaJ J. (j--
(..ajl T0-v-Y)
(9Y0-v-Y)
(.rY0-v-f)
(,-ijl)
,-r JK-'
.6lazJ.a ,yf SItJE;:l Jl.r+ (V), a:iL- cs11j 4;61y"a 6,ly 6la-J'c.riit(JJl)
.r*l!.r !i.ri ;r,,t 6le-,J-c 'pf slt -e* Jtel &-f *lb o.*^}
(v)
(crY0-V-f)
.r, .F-5,r1*'YJ*. .-ri!,,r ,fu ,J.-* t-b, ,rt .l.u.c;-S er'< ';lsl dt7
;l ortr;.| t) l5.r o# nt; r.lir, t' :lle J* 'pf '^-, L*) '1.."r l, -to'4a*,1*, /rr YO - V - f aJrL,.1: yrt-ir c/l Orf u-F"q u 'Kji (ssly ,F ycf
.Jor"/ Vi \ uis.sJj JU, ;tf ct
xr x2 x3
xi cos 0 sln 0 0
xi -sln 0 co! 0 0
xj 0 0 1
O2-&f ag(y a:-yi aglaL2-" cS*&
4;l; "r,a.r,_r.-JJ..-iJb 1*-;lt6jl.*t- d^h" 6,s.sly "|5
gttlr 6, t -y- f JUft..>t ^.f ,l-,t J* at!" X', JX', , X, .gLac.a7 :t,)*.Jt^"t f rlrf L:-
, e1 = - f x \o-Y jl l:.^xt-1 ,>tl=s Lac^a-7 cp-l tr oJ-; 6f r-1ll;l 6L^l-r-5aJ,;-;.,.:*,1 1"r*1 rl;{)m lmVil-lJ t"rr = Yx \o-Y J€'rr = f x \o-Y
,Jt-I, oJ-ll o.5cfL^-"r,e,1 * rr, = r',,r f r'y, +5-t-a.:Ot;j. !\r J/rr, €yr
.JSs. ?t4""1 ,pf
ff_-H X'r JX', 6lJqt q br,,,,-rlr jtt L .jJl Y0 - V - Y aJrl,- t.i; t+!_l s, <Jo
.i#jqg = \Yoo-r0 = ?.o 6lJgr",(g,5"f -!j-l.,1 \"-Y ,S;;-t-f6 rfl:,Sl)
+ /,, f f;l'r
YY
d)J.S e--* !.1-1.-f !:.-,!€rr = 0 s t-t-€ ._!i-- y11 drL-o Jr d./-l rl:f dl-,"| U
U..ry .T; = f /y'TrJJi 61113- \:-r! olKierr -ii- )Jtb 1J-ll;l 1'-p aJ.rL,
etig \:-,!l.r t'r, JJI YO - V - f d:L, Jl 0 : \0.'11 or[i:-,|
cYY-
tdo
y = -Y * evr + -Y-€vrYY
v : -Y * et,t * -Y-svrYY
( +)
( +) Ts Vf-TY
+.# (+).#( ^;)€c* c/)Jl-, eb *-f ,F-5.:-*b,St*t, d:l Li
err *tyt : -f + 6 =€'rr *t'lr = f-Y = Y
F-r, *iU J<-,Ji:t, dJurJ+, s: Q3P 6Lac.=J f ssf qt,,^,t o;3 -9.:i:-r*,
tdt <:59 t c\;r=* *(+ :f J,os
g!,-f dt{ J<-,ri*, 6J,1r(- .1.r Q S P tJl.ac:+,f -rr .j;,! 0t5l S p-6,5g;b,rtir*;; J (,J:;-s) J<-,r.* -rr y' 6.1t.;;l ot("r..i; 4. r#t-Jo...(Y - Y J(,"i)
F: ;.r trdu rltir*J ,flror 1t.:y .,l+i Ort' lra, 611 .t,t-,c/ J*t.i b :& f--?
,F;t r+, *6uu J*;J l) i* )FJ,clttr*; :'*
tSs: u*t \ltl .efrf.t-td ls o.s,*y\dui 0t<.. ":*i;
(Y2-v-Y)
cJt-.c,*,1 oJ; ,ra;J.-',g.;l-rfiLi L-'y 6CfL." tyg,.flrjtl P s: "2:,L,
di Js 6dsL, L-'j' P s: {li ot3 q c*i Q st ilJ 'J3 Altl: :ol9
"*: 6l&2t*dli€
'tf *,* t)
(Yv-v-Y)
;r crt.(,r;;,lL-rF u-';.; l rs,|l"5eQ t P )1 ,,# t -i-rlrr:4.li.,1r *f.f:irfc-,r.a--ll- 0JUiJ cJtU:..614c.^*i I YP -V - f aJrL.
du;= (#),*,
@i = ittior,l.$r"/ l.a.:., f^,, r>-'l a4q - - @jt
)l*,
(Yl -v-f)o:Z t{.t Oltft"r,Li.i 5
(r"-v-Y)
dui _, AUL il(i = % ^/.il( - il(t il( - -Y, "i
t+(h.k)++ (# kdui =
: (eij + a4) il\ (Y^-v-Y)
.F* t*bl.ta;j s )-*cl u;L;5t-5rt ."r 'tf :_*V olp\eij ..li;,*5
.eU,, *f ;(*,,g-,-.7 cltir*; 1.,, Oti.*+ 3 rll$ tFJ ef oKJ,:3,,:,t rta .,itjl t;i fl
.€,y 4-P cty'e c/-{ L g\n /tx {.Jt- y-l g! sU.r -J--
,ri : egi ar
Oe-.:u,*4 agly o:*921 Glb Lr:r," cS+& tdr
-r-:,-tal3i s;i- o,JlJ YA-V-f d:L-.d/_lJq
du; -- e*ji @p Ni = e;14 otp d.X1 t- du = o x dX (ft - V - f )
J- f.-? .i_ r..lL-lr jl oJ.,i JeV *7";j ,-.- dt<",":ij J.*jli, 6 ty.f oJ.aLL
4+Si 6.*; dt<rriii ,ij = " f I t F-, or!'f .-r-,tf .f o ;:;r7 j:1 sy* J1>
F .1: rr to - v - r alrr-,.) .rs.r+; - r'-s,*i tr;: 3t::.*'.nS
| -t'
-' , eii
,f J\)ui ols.-r*J o-iJJ. e-, drj-ri c,*.J, elll.: d-fr J**jb4l .:):L, otK:-,.r .iaJrL.;F;l ui-.1(-,.r g.,lc.J-ta.:€,W (-r::...,o l1bxi il ,*.li 613,r,\ e4 eSi-l
*J.:L'" ui*X 6r.,13i *Ju tti = tti @) ,;lti- & Cb r-, cll-fcf ,rls.Jb-,1: -9 r.1[:
,_fr; elonil_r r3_l-r s_l-t oJS:JJ.*-n L* + C/lJl.:1.-^5Crr"l, It jy J.-trrr,0l+r L-g cly .r):L, -r-,tt -'lj f I Q-#f 41.l,dl-r.:ti:-i 6s: {ltr.r; .:.:r5JL"""l
ol2eau ,F.S cS$)L..:).:L, L-'y ,ht;;"i-7 c.:rif g,-.ti o;lJL.* Ot5,r*.u
.Stf 'l.rl* fj(\"r-v-f)
j*, L;3 Ll .(rl;i oat3f ).:;1.: rFJ YY - v - f +J.:L,-,r'i5eJ.rL, YY : A\ .:l-u
.i Otirp 6lJi.4,S! L*,-pdl-l 6rl: rrLi:.j dli€.Jr.-^ lj+, l+i;l aJrL,
"t-,!"f ;t5; gj) JScr-' Jti*il ! ort*, &-t,r., +;-.1.r.<r- 45 # 4,;t o;lJui,
. (Malvern 1969)
aJ.rt*- f rf f $t-. raz,:,-' d.:L. jl;x, x\ t\t^;e s: 6la->,i.o g*f elt.{t_f
'*S $ J* q f aJrL, 'i_ o1 Yf - v - fsrt,yr * tyt,,rt : Y srr,tr (Yr-v-Y)
,r o - v - "::;- Tjff'*} ll:1;; iJ; ::;i
0u, du'
-!- + --r-axj &4
eil,nn t €bn,ij - e*j* - tim,ik = o
tar
,j#clar:-*; A - fd# o-;\dX 'r.dx 6La,;lSl c-e J €,-l a'llr^ Y - f J* q i:-r-'
;l:y .:.6-7 -1, .rJ.;L+ r o? GlJ, oatl th. .€,-l 4p ( csfris'rL ,A t-) A
..:-i3j dli,gP y {l3N &A^ dxrrlN) - il(
L 6:;lr eil Jl ...r:-1, }q .J:,U6"dX = d)( lg.lU- J<-,.+ 4lSldx,rii.t 5.f---iti c-il.r .tS s r '5$ af
A'rnr: (#)' (Y-^-r)
\Y - P - Y -r q - 2 - Y .rY:L'jt.f,lfQ
(\-A-f)
(dxf = dx. dx:F. dX. F. dX : dX. C. dX (Y-l-f;
J.,,l-N ..*-r r, Cil-l ;Lel '? tSlJ il o.tV , (ffi)' J d:y.r f=-- U a+f 1:
)yL-/
(f-^-Y)A'd)= # c.4#=f,r.c.N
aJ.:L, Lip t, Cl-1ff : dx I dx *aa.tt)'Gt u.31jJ o; a-,LL e--h u.
,-U),-lg-,! tj
\ :il(ltt^l dx
(0-^-f)
\:-,J.r \? -? - f *J.:Lrll .rUil L.JdX = tr*t. dx,"r.ll: \'- 2 - Y aJ:t rll t-*i-l;:
'*ti't(fuY)'=dX.dX= F*r. dx. F-r. dx--dx.c-dx (2-,r-r)
O4pJ:,.4f ,gly a;9a1 aglaL2-" cS*K" tdr
\dxdx.r = =.c.-:n.c.n)r'61 dx dx.
\^.7 -e\ C.€r=C,r\:\-Y€rr^ (it)
(v-n-r)
J<-,r.* .Jq-fu r, N 6 ,;#r, s! ,Jl*i- -11:11i -St Jt. A& * t 6t '{r-b.. fr : i, c1ll .ttJ^lJt ,-ll--i- ;it*S ,:.tKi r,-r-.i J^lJt ,f f ,";\
At(t't - i' . C. i' = Cr\.= \ + Y -grr (^-^-f)ff=0,,Slts
(q-^-f)
..--il.r t*'t .>l^as.-6Lac.a; JL .rr ff , N .Slr: qU,,-b.lr,
e-l .€e d)JlsP .s: Ct-r N c.{? y' rr (J-b
"frf) -t-lj u1-5 tr
el_e,6.j_6t - oj.xu-, ;-- c*,1 -r-,13 q# ,5*s -vl1 ,;;S S r-r +iN : Ir ,5t .rtst Jt;l yJ", f-.+ dt<"Ja:ii
fi-. i- \ = y'",,- r = /r * rr',-t (\\ -A-f)
.€'t FE,t ,Sltls o:V
Err: e(rr)* itt6,, (\Y-A-Y)
-:t _hir r[Urtrnr r.7.1 .;s sE; ----) rrr 6 GV.q-5J<-,-*o 9F ell
..ry.tolr;Err = e,,1 = €frr,1 6-u:.r, Ot,ii Jt d.rL.c-t-i!.f e111,1 q r*r_t.l-.
13La;lrrr 6ty..:b $l)l Jtr*S*JOli,-t'1, P ef, Js 0.i qJljtr-*
tda
b .u-i,yr-1.:dx(t) Jd*$) ,>.sy'oru J<-,r.* JX-.lr {ye.r. oSdX$' sdX") 6-,1t :;l
u,j:-,.S;llrd*n' Jd*"'crogq-ltj,rl-r"rdx(t). dx(Y): 6(t)6$'t cos0 dl4t.a ?-t'.t_SgrJ' q lr rj J* Y - n - f aJ:L. .<J," 16':fe*,t'.,
F. dx(\) F. dx(t)
;*(t) 4*$t ffi ffi.9f -A- f .rYrL, jl orli"*.| JdX') dxt!..rt!o J.Vit>.1j,e Ort'p.=J;t t
pi.l !:-'! \ Y - t - f
^ N,.C.N,cos0= '
Arxtt ArNtt(\r-^-f)
(\f-A-f)
jl o-:lt- J<*,-** d-\:".ts:,, -r, l+i Jd tu-Jlj , i, 3 i, 6Lac^a-7 ,, itlL.ot 6 ;lyl 6lJ ,y,.:-*'l O# y; .J.:L,
coso= ct' =
ct'
Af,,rAovt \f c"c*
4att);dXtt'd,n 4,-Jlj @ 6 iq
, dX$) . 4;1tt) = d./') d/t) cos @ 1l r.L:; ,'-rL...:-23t a.rl-9;'.:-,! V-A-f -9 ? -^ -Y.rYrL-;t c-fL.r,
4atr ) dx(r)co-lu:- : n\.c.nY
d/,) d./,, fi-'..}, y'4. - ff,
= 1(i,) )1611 (ir. .. it) (\o - A - r)
.$r"./ l, grt+ 6Ji-$..1.,1, ffy -e ff' 6bca7 ;s J;zl e ,*l ,"-J\ 6
cxt - X\ -X, * Xr 6t:'.;*i;.,:Y:L, L-'t; gK^, JS-,r*+ 4 t -A- f J&'c.*i (-,iJl) l;-fO#J ..:*,1 oJ^i 4jbl xy - X, -X, * Xt Jxt : Xt *X, + X,
d,* ,irt- J<-,r..n, ,JJi-5r.,-rrg,r q;lt (V),fr, - d, + ir) lrl I c.47 ,t ",S.t=:S
.-ril+r^il: ,ly fr, = i, , N' 6bc.a-a ,, itJLrt 6 ;lyl
O?J:# ,gl, a:-9- agb t-+. cAaJK" l6'
(J:5 c/)J.t, t4;t ou-;+;b) '*,;l-r
.J^i 4itJl J<-,r.* )l Ll-
lFul = sICzal =
Y - A - f aJ:L.' jl, "f,lr:Q
(...cJl)
Atrr,t: [\ l.F, t /Ft, .l =
\rx,,r = ff
!rr-A-f +J.:L, )l ,gJ,Atrrrt - ir. C.i' = Cyr = y , fr, = iyc;lr(v)..:*ilr f.^lf (-iJl) .:-o*j.l.r oJ-.i g-,l., q.p;l o.:Li,:-,|
cos 0r, =0' + 4) t:F. c. r, :g
,/VvYvY
..:*,1 ol; 3;y.1rV" ojl-tjl t f O' JLe] tu_;lj ,r.*,_ .:*,19,.. = 0frVo s*,,
C,rblLgLac.{+.1rr.}t-aii,. 6,lasy'fl 16t>-,1 :1,6.L \f -A-Y d.rLr;l
Crr: . iJL-- cp-l -r:) .:*,1 a;Li 4-,-Jlj .i_ dL- J<-,rrnr 0rr tu_-llj c$-f ./tsdl
,P:.r.r;l$ sj+sXt 9X1 6toc.a-7JJ CIJ y "\71drii 4r-Jtj.;s 6r*$; ("r-"lr"f
J, f, J JJ-".e J+Jr sF d- du.p.l, il-l C g J. )F J{J) gL()Jr- L, y' c rrLL
,JJd.-. C:L*Jj6;l*'i- os-ro\ u,*+'F .i-,Slr,.r-l;b. .$r"/,F* p
db l, JA C,.i ,f t ,f f On c (E ;r 0,'"^^) .rlb llri C &l6bc.ary 1: ,l.l Lar.J 61r -e riL6 .,!q utJd.-,. C:L;i6;l-r,. 4 jJb Jr-J? g>e 65115 tF J [ar.J ry I ;l
jl -srF ^*- .Strl-5N, (i = \ , Y , Y) rJ ,gLac.a.7
[l,i][ll]li ,ll,flIr
edr=AfxA-\, (i=\,t,Y) (\2-^-f)
lav oS9 s c$tr* *(+:f ,J'.s
*2.;j;:
dxo -a/i) +d.{t)IArxa-\l: a/i) tt6a (i= \, T, Y) (\v-A-Y)
.ri $lF ;Iel f+- q *it- J<-,-* f ,-:*j 3
4*tt) 4*tt) fir)dJffi;r'Ti;rl = Ao'tArxrtArrqrt 1Jt t A - A - Y )
,*,tlr rF oU-rE Y - A - f drL-" 6 6"19&
: /c,ucn,c,.r = frrrc (? \A-A-f)
-l "ri!"f t-t,.:rF..{ 6c^.,.1 .1i1, ./ \A - A - f eJrL, lJ, J3 c-.^rl
.c*,1 4*,l-. "hti
C gglayru f {atl cU{
eJrL, jlortc-,| [. \ -A-f r.lU"JK-,r..1, 6\lzdV ldlP g-rer::-if -A -f cJA
.-r.1|.j L-f)J.\il c....o./ \A-A- Y rJsLr,J.i uf -l-rJC,..l, JJI \A-A- Y
JIel 6,la;9^. lu-3,e,*.cc(., cJ:o.: 0t fi # Jtjr -lr crL--;4;l: rt sl-f S ^:t** ,1c
, fr,= (i, + i' + +l tlf :l -rrt-.--e \ -A-f Ju-crJit-i cl*f -A-f aJ.rL,;l a:5":;-rr.frr: 0' + i, - tU t,n -lfrv = O' -itt llf7C/l 6l;.-r::...,n Afxrt = Y JAfxO : Y, AG*r) : \ .*y1 ;Iel,5t""rfgSrJst-- L-,g cdV I dlP = Y $r"/ 4+J dJl \A - A - f d.rLr rllpo:r, .,t(, 6tJ
.^r,lsgr\A-A-f
IIIc=detC=
dV=dw
dVdIP
\ -\Y -\\r
r_\
-\
dVldIP=lT?=yr
= \2
tdA
Jg agln27,,-rb ( o:;Jq tFU t - rydi.* l-,1 6;$l g;r +::- tyE g 6J-i 4?J,4:(, oJl 1, y ).ai ':
J.Le -9-: a-, g-lj qj". ,t-.i-l .l-: .rl,*l a,-j"" .l--ll - 0-1u! J cJrLi:.. .:.^*; s: e^.,.aq,rlr.r l-, O.*r.f u-U 7a 6c,-,1 # +F \is-r S €'n G-r\ *-e,r-b.4 r J*,...', o\tV 4(!-l .sr;irL- tr.r-Jqj+ 6;yE,;JJ. J, ?t','-'. bli'; b J<rr.# rt_rfc 0:-1,1 e** #"1 pU.t cl:.-,| (1t69<,-) ir.trf
.tf u';;,J-b _r, ?t'F=R.[J=V.R (\_q_y)
,L:*^ Cr*, e;:- 0Jtjn:r.Sto-l-f-b V -lU -l ,S!"f _f"Li cFn lg*bR Oi.lr 6; U , ttl t 4-r .t:-,,-loJ:tt ,S# * tl*b : f :1 <z-11 )r*b.ii/iq 6
.(f V - t .,k.- t )F Ud J-:-:...^ OL*(,_ oj1 sq;,li,- sglbVtldX dtr5r!-r .rlrr.i 61.1 6t4;-;;,J-.t' ..S{ 0li"q b J(,;-rJ.r clL--,!f
qi .tf .t.9,a: 6;V dqt .,:dxdAF-rt *:rlrJ t rJid F S er.JqtJrqbq-P-f .rJ.rL. dx= F. dX )};JUit \ -q-yd:L.. sstti';l.9l6-r/
lso. u"6-q $ Jt*Julr
dx'=U.dX
dx - R. dx'
1;Jl v-q-f),li."rL:, 3
(vY-q-Y)
lE ,*l66CrjS6-r-,tr.r (I<1 sUrl , 4rr .:*i oj1 1lJi., r-, 6!l: U -r_l-byl.r;l: rtrJ \ rllr, N, ,rt rl J1J?:5r4+4.+;q t+i;f 4rr ! !;r,.r;bcJi.a: It C# <z-12 c-Ll6Lac.a7 a., i3-rr-. -rrL:. ,sKo*,Jj-tJ &_ 6ta;l:r,\(; = \, y, f) 6toc^6.7r1.u,1 yu!;ol-.lcdJl y-q-f JUJI aJ:i.- L-g6tl:*j,:'6. o-t-,,iSc c^6; ltJ ;l 6,'j: ,y\.U6 ft = \ , y, f) ,th, q liiiJtrr:.Jr."t-,!"f .-.r Y - I - Y eJrL. L-'t:.J^i dLl \& f-*+ tll-l-1s,4 ! Jp LJ;J.,jS.,r-J R t-,J".1!-rr 4, l-rfl ,$,g ,-r-fub Jl-r.r c \ - 1- f aJ:t-- tu-j",
s 'Ff &ti c*,1 g5^- :& f-..? Jti:jl c u"1 -c J<-,r*+ -lr .rJ.l J^lJt V t-,i
tat
.i- 45p+.r1., ey,;iJ-S J cF-* cStrl;-b ;t v;611-+"rL tllPq
f.il-t a:-il.r V * " ,3L-a.,l:r-, urL"t Cll 6 gt': r:*'l c"l O.r' T 13;$;l s-*V
y:E-"+Sc;ir.ibc..lO.r-" Tl.3-l ,JlJ GSsgj)lr'.{'V' T' V > o
d"',u; U.g.-5,;F:,1..,c = Fr' F.,-l-b'tL;l| d-l-rr'.li:.*iuc"l ;l'l's5r-rr-14+;J,-t.i!V* o ^?JL"F'V* " 3(detF* " i*''-)-:-'t g5iJ:iaF
.rr, .ri!r/ -F il -S:l s.t ,J-l;t -9 **,1 '?.-f
teyo'tt 6- (F ' V) ' (F ' V)
(F.V).rr.9 =V. Fr. F. V-V- C. V ) " (Y-1-r1
Ul t *-y..1*,1 c..i,o 69, C S
F' .D'= Fr . $')'= F?. F (Y-q-Y)
a,5.:1.:6ti OliL.t'e ot-i.o .,,-Lae.*, t1 .c...*^" foll-lUt' iE+^,! C S'l trLil 6
.c.*,1 c..l F JoJU:, ? c = (F^)r. frt)
.ff ,;F.r, ,p-Ju os-t'u. Jtel 6l'as3>,'.-r'l-r c Ot5l
;tpltr$, Jfrs U. U = c U- U =ffrP-€gq.*C-rJ+ osj"'ou.\U
.o".iJj 6lid ( J:;.,rl c.5 u"L"i C61,( = \ , I f)
I ct'r o olIC*enl= | cr,, . | (o-1-r)
L o o t"'I
toj.evc-rJ-rsC.gLruJy n1 tJC* gLn*Jy lf-t,3q 6-r.L:, ,-fli\l ouxl fl:.ttirt tSs-*E : u,.a;V rtii 3-r y' -rr J-j o$ crQ aJ'rL' e--h;l t'orr'^' )l e-fr-t
..:-.il: f-.^tf otKic-t-n:
C*an = aAe anp Cw t- C* = ACAr (2 - I - r1
l7o6,.-.:;<a ag(y a:*9a agtaL2a" <S+K"
[.u,q, o olt{ryBl-- I fc;,, " | :IU*znl tv-q-r)| /2.., IL -r)J
1l c,*,1 .rt!o (U*)^ Oi r"r<r" s\:*,1 :gi,a nfrrhb
ltrrq, . o I[(u*enl'l: | . \/lq " | (^_q_r)' | \.,
I
Lo o \/14-J
ei\l tt-t.'.t'tl 3 -t.!.^,r c.i, g.r, drudr,SLnr_l-bU-t -lU,J-lrrl a.5 r--J- +y.s-t-rr-l
Uza= aersapnU*e, t- U =ArU*A (q-q-f)
JUnn-' : aeA aps (U*gt)-\ t- IJ-r = lr' N*ft A (t. - q - y)
f:b \ - 1 - f rJ:L.. -rr J-ll qF jt , ..Xlf QR:F.U-I (\\_1_f)
a.:..:J .;.:
Rr. R - fr.u-',)t.(F.u-') = (v^)r. Fr. F. u-r
= IJ-r. C. U-r = IJ-t. U. U.IJ-t= I 1\y_q_f)
.SU..f ,-# "r.l-:, R -tos,r, otJ 6;l o.:l.n; *l j e.,L-L6[-,f !Otr;,r-t, \ -q -f ..Jrt _.-lrfJ:4J_jeL,
.rf -tlU C-r = F. Fr = VY
xy =Xr+Xr,x\ = yX, -yXy.r)rt - L-,1r,f^rJ<-,-r*+4 t -l- f ()U.-trSg.ir J<-,*; Cr-l glr l-l F = R . U
'8 qj"" .,-:*,1 .-t.i 4jblrt = Xr J
czSg e JSr/2;r, *l+ :f !,osl7l
;y€ O*i'J Jtq,Fu = x;ra -ry*tl ,-P-iU ,F ,Y
r "lrFut:l I I "l
L' ' 'J.€,-*n*'1*'L C = FT' Fdi;l 6
l-o -Y "lrcent=l: : ,l
Lo o
J
.-r,i.ft, t) os-fq .r"-yL 0'-l ,;\pl 6,lasy. rfS-.i.1-r
f ^ o oltc*ent=l . :;l
Lo o
J
6*:r.; C o;{J 6L-o;l:y a*,i*, ! """-yL .,1l) jt .:-'l c''.;Lr -r'L:' JUJI e-t\ 6
l- r /r'r -r /vr " I ('r../
Iouut= | 'i- '1^ ;lLIgr--rb n-q-f ;V-q-f .r):t-"rl .UlfQ
tv/Y-. .l l-\. 'llu*.enl =l " y'T- . I sle*.qn)-'l=*l . i o
I
L. . '.J rYrL"'.ml
O:-.u.F 2g(y a;-94 agl.aL2-" *& l?f
(J5.-/)JJ.,_q0t_J+.:jt.r6) \" -1-f 3q -q -YgjLerl .rYrL.;l o.:t,:-,| !3
f=S"f o:r'J
tt,[-\i\rY .lIUenl: | -rl/l r/r'y .
IttL O O ''J
tr \ 'l r
lv'nnl=.,+lt Y _:lL' ' t^J\ \ - q - Y alrL,-il 'fYtt
,-[t r:]fl ; ]*loL 'JL' ' '*lt r/ff -r/fi " I
=lrtfr r/fi "lL ' ' 'l
.s-;.,_* .r;b Rr. f, : I -r F = RU e_h;l osGil f.r Jrt*: g,-l
u'.lsr,(, J** Cj, sef JL.:f I . - f'rr*v; =vi (x' t) L'i t,3:i.;'#',r"T"Zn"T,f!,,_o
Lii= + (\-\"-f)"Mj
l7r
Uii=Dij +W4 (Y - \ " - f )0JU:, .:o*t r, di i, 6
(Y-t.-Y)
+ drf
v -r JK;
P aLii.,, #l:,r,-s c..er-,,.11J=r
tvs-t,lt-rF Su,nil-r 4.ss ueS d qj?. yl ."u-.tir^ cru-l \ *ttPJJib b
..:*,1 -p 0v; I e1 gLo"il-r" Slt s :;lSu"; clsy .fu-95 J eP g ,iV Jt- -rr ,J3 .-\{F F.1, l.sQ sP qL*^.r +tiJ 3:.1r r.=.l." ,St'"il-l'
t, il-l o-r3 .rr, ..:11: y.1: JS; ;.U" lsv; * dv; ,t-er-,P 1.: Ejl3 o;'s jv; '>ef 4, gl"rl'
.Sb.rf tJ O.t-*r, P o.13 t' r:*i dy ,{1 6\l: q
p,i:+ (k.#).a,l- OJU:, .:.^-.J J d9r* gr s-*V ls
*,i =+ (k #) (r-\.-r)
O.-J:4f ,g(, a:-g7 16Lt L:cw <S+K" t7r
d.v;= **, t-'Uj'
dvi : ov; lIL == d
Ucj lXe Axj dt
dv:L.dx (0-\.-f)6 t:-'ei
(#;)V (-t?-\'-Y)
d/_rt^i 6;U5.rL;.,-: I
oJ..i o.:y 1l-i -u:.,.a e .J*+ 6-rL 6Lad\tlf :,grL ;L; .rthi $ G;+A-gi ), 6
F: L. Fd/lJq .':--'l
(v-\.-f)
o-uic-c1rrj f -A- f aJ.rL. ..;"tA el .1r*5A - flss ldX uft:S,:=.u W.-r.;lf"f i rl-r:,1r, ,l_rSlf! 3 N :l-r:,1 t, qjll ,_h,ffi
'b "y j4:Sjr.,_ c,.:*"1
J ni : dxi I dx &- gL.rlrr t-: .L^-i dxi - x;4 dxn ,;[ir.r.ij ,lt-rF.-,_F tl
.!:--:i 6l-y rg Ne : dXA I dX
dxn; : xi4 d)( Na
tt;/\ : x;,a N.e t_ iA = F . N (A-\"-f)
.(p"*-*rt{ b c/-rLj 6.1lj5:Lj '?l) el) e6.! alrt . g,t,5rL Jtu ftiA + fiA - F. N = 1,. r. N - L. ffA
gr;lr r+r ts
L - F. F-' (V ? - \ . - Y)
i+ffz\/A:L.i (q-\"-Y)
176 czSg e JS:,]':2lg *k*:f cJ,os
.pi,y g-,! ff ! d.rL, #l ultlr ?t'.lrl, ,f.ir !
^;:n.ff+i.nA/A=fi.L.i
(\"-\"-f)
,F N c^{-" r, hlL-l 6 jf elt S,g Ot+. b d# b(t ,2t uf A:,5 t, 6aJ:L, 5 fOt-.-f;: ;i=1 {?i q ..:*,1 oJ-i ilJ fi c.a7 tr d-rl+ ,J-t11(1' -r: J c--il.r
6 u'- u{{ c-t-iL.L/.a--)l- 0J!.:rW 6\;.:*,1 O-t-i..:L rlrU \ " - \ o - f
L,/rti= (Dij + W;) nflj = D,inlri
"*lr:Q -r
A/A-ff.D.i U A/A=Diil{ri (\\-\"-Y)
3 fr = 0, cxt c..,€? t: Cl-l 'F tSlJ., Jl:. ,Slf
A/A =vi,jnflj
A/A=[\,o,a l:;Il LllI
Drt
D't
D,,
: D,,,
. L I L : Drrg,:b ff = Al1l.;^ s A / A = Dtt p*b ff : 6t gl; c arLi-o e-*u.
;, .r3,: ;f 6latF t- oi;9 6beF, f$:=p LJ.:*V 6:b )eW c cr!,
.J56" 0! b (.-;-ljj) ,>l-a;;.,5Lac^6;
o-Ulr!., .;l:y yr c:f ,*i: l, Jftr.i; Ll :y'V 6-b* ;eW nSi-l glt
U ;lr-r ;r .,r-l ,rLl.' ?t' 6:t, ,;;. .ff ,g F :t l)P aLrt stful') :fu|' ) g;.lt =l.J!.i Ol.-j a-r r:*i ,S-5,).-;lA.: 6,"..a: otJ b *.ii)q;: +tij srb.1lr! -,11
o:t;:-"|
.(-t-,!"f a.tej .--* ,, CIJ "l:f ,-Fb e4J
O..,.J:4e ag(y a:-94 66 4rl" ,3+& l7?
[ tu,u) 4*,$) i = t dx.$, I d*l', * tui$, I at{', ]
- dvi(') 4*.$) + d*l') 6r.$) - v;j 4*.<t)4*!r) * fu;(" ,rj fu|"
= (vii *\,) dx.$) dx.$' --, Dii 6,$) dx,$) (\y- \" -y)
.rr, ,s!./ 4*.$) 4*.$) -- dx() 68) cos 0 Vl
I dr")4*$)ror 0 l = [ dtt) 1 4*til cos 0 + dx,(\) 1 ta<rt i ror 0 -dc(\t6<rt g sin 0
(\r-\"-Y)
.f+r;lygU;L-b \f -\o-f -l\Y-\o-f .:):t rfl
tD;y'x;o)tu(\)- ({ #.%J} ,o,0-0sin0) *")6('rt(\Y-\"-Y)
6(t)-6$):dx j sin| = o , cosg = \ 39 = " otK;i dx.() - 4*,8) = d.vi fl.{t_"f ubKfj os-re\ \f - \ o - f d.rL. 6 S:-b.
=(t #.#) "' o-bsino) *")4*(t)
dx,i dx; (dx)D.. . J-D:i1:1, =-u dx dx 'r't dx,
(\0-\"-f)
!) -t-,!.f n; &g tr Cl-l *V Jt- tr "? d* s,>(9 s.t ,-i-:.9 gr tFrl o.taLi- 5,0 = 1 o_2je6i-,1r.,-t-.iL dr(.$) J. tft*.$) $U.(l.1Lj a*,_U. \ \ - \ o - f nJrt,
.ri JolJt \f -\o-f d.:L.otfti,.:*,1 sin|: \ ,cos 0= o
\ D6ni(\'nj"' : ir . Y D. i, = -b (\2-\.-f)
frt _l ii,, 6rJ."-e 6Lrc.X-7 -r, il-l Jfl ,t 6i-)r e.r {lSl 0 u_s81.: ,lot5 Ll C-t
t7v e,S9 9 $':'E* $(+:f dqs
Jt--;,-J i a i JlJ:.eq lt ,f * t.r- *5 c.^."1 JJ* cL/+ ,JLndi t, .$!,-f
-6rq q + n jli .gbl3;.2s ,5la.li:.J t-h,,StV ,F ef Js drd (,Jilt5t- .r!-ltl)
g,:1, fir = 6v Jff, = 0r ,SlJ.r-t',.J;5't i-y;
-e,.D.6t=Drr
c;*+ 4-, dlt 6:)1 66Cr, r[<.lr-4 Ll:*l: 6-b* t'W,,rlf.:Jt- 1-:3
.JS6r 0t* b .>l-a*,., 6l-a;.y*
J;,;,l6Lac.a.7 c ;l"^,,lt)tl, jfu,$!.f lJ) +.;r,Ort-r:"r-l-b 4,D r.l;a.rJi tJ-l ttl sll oli € ls ,* 3 c"t5.-i.r# aJ il;-;l -;-l-b .s1,' 61oostb ;l F-Lj,s-uirb- .-r1! a.*,1,, E ,-F-5 t-*L; 6rL ,P J D Oil .Jt.l"b .ry .r,alf J-.0'. c;"ri,"^r
6 tyeYE = c-I = Fr. F-I
.t-r,torJ -rE V - \ o - f aJ:L, ^S o"g$+7.1 1-r
yE - rt. r + Fr. F = (L.Dr. F + rt.rt,.r)= F?.Lr. F + Fr. L. F = F?.(L'+L). F - rt.(r D).F
E = Fr. D. F
t-
(\v-\"-f)
r:.tJ g;t-nrlt-rf fl tui{ + 6;4 = x;4 g7:.l;u; * X; = xi )l6rFa-i Ot+^,!
s (I = F) 6irc = xiA ,Ki , -bi-e "!U , u;r4 11 \ r,-t-ib *-6 ,F u;4 019,.,
JE=e sf -v- Y aJ.:L, L-,g*-6 r-b u;A;a,Slr,: .lL;*. F? = Ir = ta:Jl
{t_"f ub6rj rsy.au.6t-5rL* ,-h6\ \v - \ o - f aJ:L,
i=I.D.I=D (\A-\.-f)
-i o''
+3-l q qi r @x)' - (ill' = dx. Y E . dx oib ;l 6.:Lcf;;* !,'r)!
Or*Jr{f. ag(2 a;-9a <SbLZ^" *9" l7h
.f::lg\:*,! "r-t!.r I@'Y)'I : o ,l;a[ @*)' - (ill'i = t @i, I
t@4'i -- dX. y E . dX = dx. Fr. y D . F . dx = dx. y D. dx
(\q-\"_f)
-tt Jl Lii'3 ft.>,t:& tr 6r,51x Ji,slntil.:..rL, 6f $r"/ otJ 6-\.iq D : o 6x
.g-ij; di./ \\ - \o - f aJ.rtr-.;l orlil-,i siti,sllq - \o - f aJ.:L-.J- tl
it; = v;j n1-n;/t I /\ : (Dii + Wil ,i -Dq*nrkpk;
r_r-'d 4+Si c.:,1-.i.-,ti;-il D ;l4l c^{-.+,5_.rl"g-.-.1 .lsn; /t OFJIcJ-a.),r,o OtJ !D uLrl:lJ* &-DOt rsl y 6D;1 ,l') - D@)n,@ (p = \ , y, f,)
o6J
ir; = D(p)n,@ + W,jnj@ -Dblnr@nr@nfn) : W,jrj(o) (Y.-\"-Y)
aJst -,"rJr;; ;,;,li,-t c^{+ }tJ;l !.ii ^3i;1.:y o3f .nr@)nrb) = \ 6lr;
d--ltJlD J-.t 6la;.y-,,SDI: c4p 2s sro:i iiwf 5-r"o.r,-r, OIJ yo - \o - ft-j sltlW lt lFrl _2r co-9)t*, .:-r,l o:l:W ,". 64*(\ ,;;t lU ,-l-, d/_ljl .$r../
.:;ls .:_1-3
wi=it,iur,ri t- *=+curlv (y\-\"-f)
..-:*.ilr f+lf ..}t *,t*, rlrf Jbr ! ,rr./ o{.U ,Fn ,(ss.6
epqYr : ] eorie1*v41 : + (dy' q* - 6p*6a) vai
: i0r*-ro,)-Wqp (yy-\"-f)
3dv; = L,fii = Wif,ri = eipw1y'x1o.,ij;: .Uri = Wij ots'*l cl,iLD = . fl ,-*,
eiik=-e;j*=ttoj,tsjq
171 <:59 9 J9it!2it *lr* :f J,.s
dv;-e;1pwi dtcp t- dv=wxdx (Yf-\.-Y)
),* )3tu.i- Jf :& f.-? 4 r.ll-.r-1, ,". jt:tp u7 ;t 6;*o Jef A ,* t S
.J56" .)ar;s l-r.rl-l-1, I,t, -l c.{? t 6laais J-efw;lry .-L;.\,-lP ;l ,sf
tsjLl-r*.r) c.dl 6p.;b ei '" *.:.1U:f ;-1 .# crsf a-,)B 6lrr
-r.r D o!; ,-rU, p-Dfol, (p =t, ; jr';X:; fri:T::J!y./r.11* fj Js-re! c D -t Li -l{;r ;iel 6l-a.2j2., rl-url
A/A : ry: pii n,@)n,tr) - n{rt D@)n,@) : D(p)
.w 6lqJlj ,.teJ-, ti D .rL.l 6l-a1jz.u:& clf3: - Y
L^Fr rla,:zb, F 'lal 6)lu J&' I I - fe;tr .l#gF;, tsdx = F . dX J.*jtq, ,*t *9 4i:V p \:*;'.j
V-\o-f d:L, L-,.1Ps@x) =F. dXc.*,\
(\-\\-f)
;l c.*,1 .:_,!o dx. dx 6l,La ?t',grL.p,L. \ - \ \ - f aJrL.;l 6.:;11 4?i Jr_l1
(dx.dx) - \d.x. @i = dx. YL. dx = dx. Y fD + W) . dx: dx. Y D. dx
.c*,1 .Flir, \1-\o-f oJrL."Uef
4rri u*+ o? J 6*L- tf, gtLo.rt-L-i ,S\ 6 ,-b.ltt 6'1. g; j. ;r'l
$71 L-'9 eU dq,(""' .;J oJ-i A,* *.:.-L- ,d/-lJq ..:*,1 -,1;t r-2..X o:l:
c..-1-... .*$e $;;: L I - Y JiLi 4tt- d){n(t jd)(n(\'J*;tF-t "l;
.c*,1 0i"1 "!U
t) sltJ L-, j: dS" Cl\iYl6l*dS"n : ensg dN\)p dFt)g (Y-t\-Y)
OenJAe agl2 a;-9;i 66 L:+" .S*lS" lYo
A - r J,<-'
6L-a.,1lrr- U-dS + "rg rl,-r+ g-.,, dq-s* .2, dX4,$t 1d){1$),5[.n;lr-r1 i;dso c.-L*.
. .uit J<..,"*; d-u".fi ,, fui$) ,fur"'
J+rn-f J<i&U"dS;esV c.eL,..4r.t^>l.*oC/l x = x(X, t) 6yjl-r,1l i:*,1 o:V J r,*,g
dS;= rij,&j") 4*o$) = sijt xi,nd)(n()xhgd)(g(') (Y- t\ -f)
.$r"/ .++;xiA :t ,** ?:- \ 6
x;4dS; = Eijk xiA xj,n xlog dxa$) 42718) = s;jt F;n Fp Flrg dxs$) 4yr$)
gr.rlr \ Y - f - Y aJ.rL. )l : Q-e,-flj)det F = "/ t{,t,g;jir[-
eij* Fu Fin F*c = rABC det F = eaac J
tyl
.:*,J, XA,q ), Ji Oirt ?-t' 3x;4 dS; u. Ll-t nJrL' -,1: r2';l ;11 drJSr1l-l ! ty-lf Q
.:-1-9i,r9t;-
xirq X.q,q dS; = ez,nc I ilnu) dxr$) Xn,u
a1{t ;:X;ra XA,q : 6iqUl
\iqdsi - dS, -- XA,qI dS'n (Y - t \ - Y)
.JS6' J!.-J..a| c.eL- PJl,t e-tV.:--L*,6grttr gjY lt *: SsV 'dS; 6:V.r*'O**' ,JlJ
(det A') = tr (L' A-') det A' (O - \ \ - f )
.f-:si O!:dr! A,Slr: tr'clr-S U-f--V !,Srl"t;l,;-b..5.,, A oi r' 5
@et E) -- i = (det F) t (i. F-',) = I tr Lt_
(2-t\-Y)
)-r-,L/ a.^-ii l)Js-fr,.:*,l t(.o"'f - \ \ - f *J':t'- e5.-;:;l:4?i +.q
dS, : JXa,u dS"n
.ttf ,-l JeV Y - \ \ - f oJ.rL, ,Jt-tj" C/-)Lj drli<)L- 1l ':U:-,1 !
ds = j $^)'. dso = ids'. F*t
I=Ivi,i=Idivv
ds.F=ids'
O*^,'J:LF agly a:-9a <SbL=-" cS+K" tvr
..:-.i;l.r f.^tr ,J:L. #-tjl e6"L*JlAt U.
ds. F + ds. F = i ds" : I (trL) ds"
dS + dS. F. tr*r = f GrL)dSo. F-r = GrL) dS
'FYqds = (tr L)ds -ds. L t- dS; = v4&s;-dsiv1; (y- \ \ - f)
4JblL ,Sl^"iJ;, J L .)ef .lt-.tf;t ,6sV c.-t,., .3..1J I, eb c.r7 *j; aJ S..IA:, -r
$: st ! 6lue ?t' L-,g e:-f d-r".t.lJ o-$ .4J* F oF (fr.*$,s
dlP = 4;1tt). dX(Y)xdX(r) : yABCd)(Ao)dxor)d)(r{tl = [4att) , dX$), dp(r)y
F 'j-'- dJ+I-$-. -9 .:*,1 oJ-i oJ.5 J--F r I - Y J*, -r, 5 G.ty.lLo q
.s*"1 '-t^i .r5fl 1s OsS,eu 1 - Y J.(-.i,.;.r +:iU_ J<-,-+
dv : dx(t ) . dx(t) x d,xr) - t;j* fui$rfulrrdrorr, : I d*rrr, d*rrr, dx(r) |
c*,1 .Jh-> ?t',StV g+-r+# ssdx - F. dX1_sltcx: x (X , t) c-S;> 61L
dV = [F. dx(\), F. dx$), F. dx(r) I = t;i,rxiAxj,nxhgd)(n$rdxon, d.]{"<rt
= det FIdx(t),4{r),dxrtl= I dr (A- \\ -y)
lt*-r-fuF)I * " Jfr.$r../ *ltl;|")'el ';lsl yJ.tr.s;ta L.F'? 6U-l'Ja}-rAtl .:-,1 ir<^r/ < . cui'L1 llJ;l .I >. U_"I < o,*,.-:bc(c,*,|
f"5.f .t.l ! ,1i,..1n," t J.iL. ,-/ '* f+- q iI.- Lf y
tyr
1-V J<.t
,-L-y; g.-a { e:-,' ,s+-fu, ydJ(1,() sdXn$l , dJ(a$' a-i a-t-: ul-.,t'i Ub)trg1tt:" ,-'-.ty,-td-rr, cru- J<-rro6.q,s* ,tfui() ,fur"' , fur"' ib;tr|,;l;t I "; o'tJ'
f?., o;l-uil tdv = d.lP.lK:i"r;b"r = \ rSl !!t/ > " €ur rft 4q c*t jl
.c.*,1 &-tf *tl,3y g-f ,y.-t,3q l.-11 13,i'.'uX ,-fb elt I fl 's!u' bt;"'
(dV) :I dW: I (ftL) dlP - f v;,idlP =v;,;dV (q- \\ -f)
6cr,;i r-,lr 6--r-51A 6f l.3-l ,Slr ,tS s p) .tF 6- "r",
vi,i=divv= "(\" - \\ -Y)
,ju; "? 4 J# tn J*r="; .lL-slrf 6 trg oJ-aLL ca.."j'- jAL c-er-Ot-rf tt .Js6.d^if5t- tr '? &-.r$; s u-l'-'? &..:,lr".i;
.J56. J.,; $l"f etl;l .rp; d/l 5\,?J 6c.*,t
CE- ,'* o agly a;*y.9 <SbL:^" cS*K" lyr
$l*.,',.,1 ol-) oa;* fi ,Sld-f .r):L L-5; ^:*,ry, -b.*, d,- tS)- I - f
xr : + 1x, + Xr) r, * + (X, _Xr) e-t
xy = + (x, + X,,) ,, _ + (X, _Xr) e-r
4=XrC-fu.:rY:t -. Clrlt rli3 ,:j*i"r; .ij.-- J e--5l),)U.,';, *.f-r.^, dt.I (.jJt)
'+5J-X=X(x,t).rJo.-,1-^ 6rL oLa-... !t',4 l, VS J tte)-J Atr"il:, (V)
.lH -l .>e)a,Sto.il.l, .!--;i .lE l, (-iJl) ri , ,,i .lr .:ii: 1-,-t j ,-f -&-oY:L (.-,)
.JjS,ft,t ,J,,-[ji .:t^a:-.
-l]lJ=coshtt-sinhrt=\ (.*ilt):*E
Xt: { (r, * xr) e-, + } (xr _xr) e,
Xt = + (tr * xr) r" - + (x, -x,,) et
Xr=\v' : * (x, + xr) et_ + (x, _xr)r-, (:)v, = * (X, + X,) e,+ + (Xr _Xr) ,-,
vr=o
a\ = + (x, + xr) r' * + (x, _x,,) e-t
at = + (x, + xr) r' _+ (X, _Xr) e-t
aY: o
vt =Xlrvl =Xt,vy= o rat =Xt,41 =Xlray=" (r/)
.,:*,1 oli $lj ,-r.rY:L L_,; 6l&l.5, Jfu 1.: a:-,.1*, !1.*, c<,- tSf f _ ?
Jr=Xr+X,,t+XrlJI=Xt*Xrt+Xrfxy-Xr*Xrt+Xry'
IYA
.r,rJJ- ; fu.r):|.- Gl1,l-t!i. J * " aS.rg.r tll-:; (..-'cll)
..r;S.j{ia, ,'-;6,1-ac.g;s 6ll. t, :H 3 t-;,(9)
o.:yqil.1 t = "rdsi(YrV0, Y,VO, f)4t42.:+f6lo.l!,Sl-tt = \ sdL^J.r:(\)
.r:*'l
. J (\, \, -\) nt.i:1: 156lo1!,Sl-tt = Y s dE"J.r' (Y)
=#,xt:#,xt:#(.;I):c-q+Oer , r=Pa,*far+f6r (:)
.,.:*,1 o.:y il, f :
J = (\ -t')' , X,
(\)v=16,+06,
(t)v = -16,, + 16, + fOr, 8 = -r0, + r6t + r 0t
..-:-,1 y-j or-i .-0,-y.; .rY.rt , l. 'ff
6,\l: a:-'3- 4 f* c('- f - f
xt =Xr + I Xrt', xr = X,, * \ Xrtr, xY - Xr
4la-Jt,b(Y, Y, T) rLiiaS6,lo o6,1.r112t = \r0 +1"-.J1: c*s-t61..a"J9"(iJt)
.J3S J{.f" r:*'l o:lo' Jti-flt : \ J
..r.5,-*.,' :;.j -r-olg ,rt (.-iJl) s-*i .,1: oJ'i ;;Ltti o1! 15 lt 6:* d:t" (..-'l)
.r;5 q-t-, t = | S aI'.'J .1.r ! o.;i trLr .-'b.i .51"+'iJ;" (.-7)
Vr=Y ,Vt: AlVy=o (.iJl):e:q
Xr - t a-J-.a1: T-f1 -Xt = O (.-:)
a\=I,o, =+,ay:. (t/)
.slo.t-i cl.rlr-j .r):L L-'5s 6,lilt,, Jfu r) x = x (X , t) '>St- f - f
xt = X, (l + t), xt :X, (l + 0t,h =Xr(\ + t')
!a*,JJ ,SlSl S,S;I-jY.:i rti, J)y' -J.:H
-l tus-t gt^-J.l' 6)y. u,Lllj1
.+ti
Or-J.qf ag(y a;-921 <S6"Lzo" *& lv?
Vt:Xr= '':,Vy=I I \ +,
Ay= " , Ay
:*qI Xy (\ * t) --fft, vr = | Xrt: #=yX= Ytt
= a-=tX-- Y,rr' (\+t)T ' aY =Yxr \+lY
jl c*,1 o:V ex L?- \Sf 4 6;tfy -iyr: A _ f
r,r = Xr€-t *X,(t-t-l) , xr:Xre'-X,(l-e-'),)=X*t
.+SCr,.*,- \ tSf 6,j31 -iryaj 3 cJU.i-.o l*'r_fu.rY.rl.- i/-l nt.r=r: crl-l;
:*qXt =xr€'-xr(et-l) , X, =x\€-t +xr(e-rt-r-,t),Xr-xp't
.,-*ul o-r.-i o:lrl ,-; ot-re\ ,S;lflJ J*.r, t e)r blt , f _ f
Vr = Y t +Xt, Vt =Xuet, Vy:Xr-t
eb.xl;t3.r,-i(:*Jtt = o rsx = Xlrx - x (X ,t)l+_&Jlf-rl.r):t-,;,-t;t..t 5.:tr'""'l (gL;i) dJJl J.(.i r) ./H s w:u.sLn.iJ_1, aa,l-,- ,grr1
u.-\ *lt+tt v-=_y r, _lxr-lt-tt'r \+, t vl-*l t vy -llG'- : A-L
At=l,Al:Xy,4f=-\
.g-,1 o.t-i orl: yj \ls: Li ttuf,, L"", c<,- tSf V - f
rr:Xret-Xr(r'-l) , xt=Xue-t+Xr(l-r',) , \=Xy
ryy
u\ = (X, - Xr)(e'- \) = (x, - +)(\ - eu)
ut = (X,, - Xr)(r'- \) : (x, - 4)(l - t')
uY= "
:*q
.i:-,1 of-i .ll* y-j ,.rtJqt L-,9 o;-'3X, L--" c<'- , Ojf a'-.-,1: tll't " A - f
T : e-Yt lxY ..ri;, nf xY =;'t * xrY + xrY
gLn*!.1" 6,lit u-'g- Lo-.,>-or- bll1."
vt: xt * Yrr , v\ = xr-xt ,vY = 'r\ + f rY
.$irr.," \ oir- 4>.r) trl+rc dT I dt 6':L .;-:-" $!.r
-e-" (Yx' + ?lcrxr + xxfir) : *qx'
,Sl^*Jr, 6lis ,.zor- o oll-J .lL-.f )l gV oi-tj "i- 1r I - f
vt : A (xrt + xrxr' 1 e-H, vr : A (xlx, + xtt) e-H, v, : o
.St^.ilJ- lr r-ro -,l5(1 b (r,i-tJ) .5rL ,::ri tiltl.$-:-*a .'iti k tA A 1: n5 l-iL,,"
s.r,J*lt = o ^J;A611.(t, t , ") oLfit:./S
at:-\A(k-dA), at--A(k-OA),ay=" :*q
.,.:*,1 o-t.; sl:l fi,.:).rt , L-'; a1l J G-6614;..:" -J
dt<"Jfu JlI,f t'- f
ut:x{\ , ur-xrt , uy:x$
dT _-
dt
6x r1(ra agly a;-9;g 16tb Lf," <*&
.r:-,1 or-i o:l: y-j .rY:L Li 3 6f .gly I t - f
xt = X, cos at + X, sin at
xr = - X, sin at * X, cos ott
h=(t+kt)X,
ryA
ut =.rr (\ - cos ott) + x, sin ott
ut = - x, sin at * xr (\ - cos ott)
uy=xrJctl(l+A)
:*q
6J.tl J-5J 6!b \ - f 4k,..-1: or-i JJ*i .Sr- 6!,-rt5";"'Au." n5-r=.nr rlt.ij II - r.c*,ly; OIfo\
ut = r' -+ (x, * \) e-t -{ (rr -\) e'
u\ = -\ -+ (r, * xr) e-t + { (r, -xr) e,
J\re)r gt"*J.r @ui I dt = a4 lu * vfu; lq,5rL.;i, tjltl;t o:[."i"t u
'JtS q4J b \ - f +Lt.*r .1.: o.t-.! n*,1-., -rl-d.r.:*,1 oli o:l: yj .lbk l-,-" J<-,r*.o .i- 6;tJ) s-e j lf - f
rr = Xr + X, (r' - e-')
x\ = X,, -X, (r' - l)xr=Xre\
tyt c^59 t dsi2zit *l+:f cJes
J-xlxf
.+ i4J ,J.;13+.f-t--or,iti,itii bi t rFJ.i., lt Fu J<*'r.*' .lL-rtf ,Sln*l-r"
.ttf S*V J<-,r.*' 6jJl JPi b JiS ,-r-fu tr,gq.r"'":'irl--
:*q€r , X, = xt -\ (l --t-')
= xt * & (\ -e-')-Y=x&.
,-F;-l J#r.*t-- glr.r.l-b Lr^L*,- 6 r-rr-l orl:6ji; ,Sld-f a, '..l-f"'6 J<-'r.+ lf - f
dK,riiii ,il+J \ CP €.Jl-* J-t | ,-li ol-r,-l ir.-tJl+ eJjj'"'r t'z'l'4:;A;t #:-'n5.r*or,itJ.(dLj )l ,e.lj rSreJL 2r t-) -r-l!"f qEA;i A tf bV.u; = A'Fi
qEA4 l,i<^" J<-,r.*' .f,",Sl-rr
.$L.r ,j\P ir)t'\t5:t 6!."- CF t;t1..liL"f
"/q fif:..- ,Jt+-"i f="t -- tL; 'l7l (o)
J<-,"+' 6q-& y 6,.tJa:.7b l, r:':""a e :f g-.r,JJ'r-& 1: o'56:L C-b Q)..lj-r,-l Jr.n *it-
Aii SGqU 45 ,F,r i-.,^, et\F*tti : A;i Xi +f /nt,-)<-^ J<-'r.l lO - r
Jr 6!.,t 15rl.^: dt*i;.rt'pite bl;u" ! tlt*,-b.7p 6 q-f ,5lojl-rl a''
L:-e 1r1.r-, 6,Jj;l a;btrtS"-.i; f 1,,l.,, dt5,ro.-- JS. Jlr *f lt-1 J<-'r*p..r-ti.f slo.r-i Jbt t-r;$*,r",'.; 6 di*.
!*,r; .li.i e'--t- ,K-" J<-,r.* 6,ly l2 - |
xt:d(r+BX,xr:-Xt+d*
\=lXv
.r..r: oL.i:.+50.-,t^, f.rE g;;ff'J r3-tir.ri-Jtr*,b esi."^ u.Ult lf , a,;i;: n5
g--.--tl!3s \:+-dqo-o sF.,,i-f'st lt = \ sF = sin|,a = cos0 fl6..r- .r,tt.r-Xr )J* )F 0 o;lsl q :^L"
O?J:4f agl2 a:-941 agl.oL2^" cTlS" lAo
,,-, ]
nn:ifa\ +ft-l o
q,t + f'-t: n.L(-?
.r:*,1 o-t-.i .-+F l) L.lSt l-y .,t<-, fr); lV - f
xt:Xt
xt =Xt * +Xrt\:Xr
.+5''*,iX' : . a;e 2: l.lOABC.r-ll Cr i:it J*rp tr; Gill)+rrL- fffur*; qgLrll.rr-,..1,r,1 4S"t;'Sir.i.,- b d*r', dx(t);1*;l'c,r sgLozbrr (v)
ilrC oL,o y ,Pt t ns-u*:tr.r dKf) = dx(t\ sd.XTt = dX(ttv )t p;.Jrlory,
e ;l4l f U L-_rt; tr xi; to:i;f : +lf e*-l-., lrdx() . dxffi ,glaLi; :* (V)
.-qL; er*; "r-ory ? Jf-,r.*- r,'l.,: +Sb C .ti:.1, dX(r) -ldx(tl.+Sn**,t*,r, N = 0, + l) l{ y &Jt;c.a-;r Bl.rApo.:S1.:)
xr
(:) dx(Y) - 4atr) ey , dx(r' = 41(rt G, + e.') ,
(.7)A0:-fOo(.r)A: g
VY
. eprr
tAt d9 e J:l:.4--t *(+:f JEs
.r.:*,1 oJ-.i o.:l: yj .rLk L-'; ,!S-';1,s I A * f
It-r
x\:X\+AXlxt=XtxY - xY-AX|Y
-ryJ;ri., be tE:i.l-.. ,-Ff,Jir.l:-b u(c,*: &flf1f) ,.*,f .>;EA ei1, nS
-but A et' )l ft t* = X oS ew-&l:-r! .q-ttLa,iLir.ii:r fI S.r*."-: clh j.-rr[-,-r ,-,:-ot5e t5;-f ,t*, ,r_S .t:-E q rJ-ti 3: y' c:j;
l' Axy .l :*q,,j = | 2., -A*,
I
I -Ar- IL' -Ax' J
..(3F kA;i6tr-"rt-j oSx; = Xi i A,ixi€rJrl**r "<^^
)Sa6: Gli.
& rlr-* c,.(-- J) CJt-;) }* ,\-S t+i;r I r.;.f E l), *lef :*!'y-tsi,l s-i! l.,N : (It - It) l'/ \
f e1ry - Att -Ar, - Ar, * Ar, : *q
ls olre\ o"l-; !-.01r*' J<-,t+ 6ly.f " - f
X, X,r\ = ffiI ,xt=ffi , xr=Xr
.r-r.ri (:-r! lt ,-;l-bl-r-rb ;;3^.o -e C ;$-rg"" ,.r-b
C(,,) = Ctrr = 6J + Xr'l', C$r = \ : *q
xr:X\+qxyxt : XY-a)
4=Xr
b-,; '* 4.tlJl JKi.*i," .ll-+" glr f I - f
O?J:4f ag(y a;-921 cSbL=a" 8*K"
,.:f! oyl: n5 +^, .ll-5 ; ,.;5 0..^, l-l e : E ,SL^r.l-L' ,Slr-J.1, or:*,1 c,rL. a [i y E.*t.r;f5;r;,'x't +trt = \ + a\ oy-lrqx,Y *Xrt = \
:*E
DLAB -
.l'l'eij=,.',*4"J
,,-*,1 oJ-.i .+F y-j .:!:L tr *-rg
x\:Xr+ldfiix\=Xt-ld(\Yxr=Xr
l5;-t,r.t t, l+i;l j "r.;5i,*^,'LE,S;ffY rJr-r ,-rf u*E.c,..,t c..rt3 k,si y ^S
li lr.ll- .r-l .+-L; t!;-,t tte *-J rl**r &F l*b c*,t *f Jl; k
4rSrt*r *-F rslt Y si : uii * Uj,i i-f .11 o:Lpi,l .9 clt5"rsr;,tl"r,+ o.*,L
lAt
[1;:]li." c<,- J<-r*+ ff -f
I
al
o
+11
.J*S ,J-)J
-.p-j .:.,)sL ot-re\ ot5.J*:,it .ll+l .Slf ff - f
u\ : fi$,ur = AXrY
UY = AX,Y
.+5C,,r1"- *t €-f .-'L; *,rb ,S--A,.ri1r nf
o .5;r5rlr.,.r s=- r.l-b, e ,5315.rt-r.r-Srr-b 6t^dJ3r (,-il|)
(t , t , ") oLi: ye ).atrrLi" (v)
rAr
:*q€r\ =eyr =€rr= o rtrr : ]A)Q,trr: iA(X,, + YXr),tyr: AX, 1,;:t1
0t\ = @yt = O)fr : o r @tt - -@Vt = jM,
0rr = - @rr = iM, -AX\, aty = - 0y\ -, AXr
tg)=+A,tgl):o,egn1 =-+A (V)
6t-r13,,,-. :l-riol t lt J*,JL^.,!-f3 Jill: ,l/ f 0' qjl.l c,-; ,gl-i ,F 6trf.;tsl rr - r.r;.I.u" 6,f ,;lsl/-j JS-.i ,1A,X't sXv Xt
,/ *i
--xrjl -u,-Jt..e o"r-i r:-j ,Jtr,ri-SP +til 1:
tr\ = Px \.-Y , €yy = fx \o-Y , €'\l : Ax \o-Y
nJ-rti-1": ,f)t' 3 .-uJ.',iia-,'€'tr l-r oln ^ P ^LL f ltTn i:< 'F;YO - V - Y .Jlt r) 6tt+ €yy = e't,* e'v,
Jat : @y\ -@y +5-r..arcll*:; ol @i = \ij*Opi o\q - V- f nJ:l-- {,-it-.- l*1 ! f 6-f
gi:l,-. tr gqu; -\.itr .-("-f J*t ,'l:A fl B Sr$.rLil ;,;*^,t . .r;
x\ = X\ -AXr + AXY
xt = Xt -AXY + AX\
xY - xY -AX\ + AXI
xi
O*J:,<p agl2 o:-,g2; cSbL,:a" *9" lAr
ot;1 ,Sats tt**r F-A r.l-b o o3)t*, .$r../ ,J.iL"J b fE = ") ,--La ,r--a cl!3:
l)oi u>.r-- ,11-. op or--l .rLil *-E )l orti:-,| L, l+i;l j -r,:f J*.*,- c-JL ..,11-1, l.l
.-tsL; tlr'.:",1
ot=A(6,+6r+E) :*q
dt<,,,.*;.,',ll.ql,Stt f , - fu\ : ld(\XY
ur = kX\Xl
uY=tk(x\+x)xYc{ !.c.l n5+^, r.ll..i; _l +SO.o*, Itat u>- r.r-L-l-l ci3-f ,rp ,-rtJ k ,ti y rS
.:;lr (" , , , \) aLi: 6lr;;lel ;i+a> )lri^
0(r): o te)$)=-@()-ikfr, i-y'J :*q
r>.s3,.aat J<-r+ 6,1-r fy - fx\ : X, + AX.X,
xt=Xt*AXrY
xr-Xr*AX,t\OBCD gr elr...r*:!.f 4.i-if 4 f.;) oJ*i or-r tE c4lf ci-,4,.ri;, n5
.{+:;la;l C ^Lh f lt t) oq-L... JK-i;tL.g:L u!;,ty,t
lAd oS9 9 dg'E:tr, *t+:f ,J,os
L06=9 =?oo,L91l=.,ti = \t/VV"(v)
.ri.tt-, L-,i; ,$,)t.r ir<^^ J<*,r.+ 6,11f A - f
xt=fr6**/-txt
xt = -x' + +& + 1/-txr
xy - x, -+ X, * 1/Tx'
r*Sn ,,:,
N = (i,, - i, + I) I ,17 c^{.? .1r 'AVt ;u; '32 y.tti
^5,- Jr-; (ulr)
'-,..t-lA(fu Fi-:.S(./)
6-r+F,Xr = \,Xt: o,X'" = -t oL;p#lolFto6t^."fr51V1
e?-f
6 = y'T 6 + (fT -, 6' + (fT + v) e'
G;rD: *Lf \.f
A(x) : t/rVr (v)
L(^*) = t , l\6;n1 : "/o (:/)
xt=ar(Xr+lXr)xt = atXt
x"r -- arX,
p o.t-*\ -t*r.# ,9,1t;f I - f
O:-J:,F ag(y a:-92g 6L6L,,oc cS*(9
J* r, 0.r",-i orl.: rtl,l; -r--13 6sL .-*(r -;) oJ-i or-r:.1t* ,SL^c4ti ar Jat , a1 bi 2s 6
xr rt
C,/
3o 7rt\ xr
o
X1 rI1
ri5 a,.rOC -El ^iL-JS*,r_+ Jrt t*;trl
J<-,r.+ jl ,r; OB sOA 6ls^1,.* tu-.ltj (V)
ft t;f 55.,* J<-,r-r ^*,_l ,slt -u;f O..t- qtr L"c.rtj S\ ,b-V e)..r-,tr y-*bt'!; ';L (\ )
.JJt-", .,;l,, -; cl-r{ OD sOC ,ti-t.rr *-ltj (y)
/Y = 1 ar' * ar, * arY 1ult1 : *q
cos o: Y 't r-)V Ta,Y + atr
14rY + atr = y arY (y) Arara, = \ (\) (V)
rr=aJ(t+BXlxt = -q)\ + flXt
4 -- 1tx,
lh7
tI,2
tAy
*.;5;.r.r.:.:...-b qilr sF, a<'1i1: a5
,*t SUJ# 6l^c,aa j to';lll (jJt)
R ,=rq.r,l-b '*-i!.r-Lj (:)
'F* a,-3t1 oll$l J 'F-* 1l>y 13-., c.<t (,-r)
:*q(t-ijl)
At(,1 : AtG,) = | Gr , At(r) - AtGO = | 0', At(r) = /vr&): pr
Ri=\ /fi-\/fr
\/fr\/fr
(:)
rr = X\ -AXt + AXY
xt = Xt -A)+ + A){\
xr - Xy -AX\ + A)*
j<J.<,- 6\1, J,J*S.:.-,!;r;L- uiel pi6-r+, tlt-i;.,.:-,1 qV 6.Ar'.ri.;r d
..:-,r N = (i, + i, + fr) laf f rlr:-.1 ;r ,-F*r;-b ),r* J-r-,tr.r..t-b.r,
.+5Cr,.1"' ttA 'F*
a,--91; ;";-'^'r s 'F* 1t:r' ,*-jL
A(rt: \ , Atrl=Afl= :*q
i=Ir,@:fo'(9).t-f1 -h tt lt ,-; )S-rt*;; tlt'r" t'| - f
F-t+YA0 - t -YA
fl+r
\
T[B*,
^o:# ln \+r,4
L f -t-YA
f - t -YA
f+tfl-t+YA
+rA':f
, O = cos-\
O:--ti.F agly a;-921 agla\-" <S*9" IAA
J(."r*+,ll-ry,Sllff-fr\ =/TX' *X,xr:lX1xr=Xr
+scxtt"R ,=- rJ-b e;| d.iL"i (-iJl)
V ,;t*l yrl-i-" a.J.r*--o: .ll.! lV Jg .5.r-r-L- JU ;,r*S r:_,f 1_e*,b (.7)
J::..^-b g.9L- V _l
'Fn a,-j! ollSl S cF* 1r,,.," c.a-7 (-r)
:*q
Ri=
ut=+6-Xr)
ut=+()i,-Xr)
uY:+6-x,).+5"r..^,"
dvlaw,-k- 6t---llrr-t rll:-"1,, irlt ol Jh] ,li--l r]:_i {-JL; a_,-31.1 1r r..._; (._r)
t),-r-r3!"r*--fr, =0, + rir- 6ir)hmrfr, =(ri,-rir-ir,) /f\r
f f+r f-r .l# l-*.r f+r .l (dr)
I " o r/TlLJAr9 = rf7 , A<rl = r[7 , Affr: \ (v)
N = i, , Q = to'(v)slo-r--i o.:b yj .rt!o t_,y; rltir*i; rll_ry ff _ f
lAr <zS9 9 .!S'X)r:;s *(+ :f ,J'oo
xt = y (Xr -Xr)Xr=X\+Xrxr=Xr
.+, eaiL0 - "o (-/) dV ldlP = \r\AV (.-lJl) : *q
'W-. \ - 1 - T JU".,1r o-L^,i 4jbl J6-triiii,- ir,-., rf - v
,J.r,*n''+.K.pY11
V ,jg !r" rJ*,ti (....cJl)
V JLel ,Jg6L^Jl.j c.aa (.-,)
:*q
o
o
\
(.iJl)
,frt=i, (:)
;*,1 o-t-! bV. ,>s-f rt+ JS.,-iJ{*L' cll-t, t'0 - f
xt = F (X, cos BXy * Xt sin BXr)
xt : F G X, sin BX, * Xr cos BXr)
xr = vXr
o
fio
f"rV.an= | '
I
L
frt =i'
lJ
,frr=i'.
.J::..^ Ln..rti v sf , F ii l6
O.rJ:,*4 agly a;-92; 66 L:r," cS*K"
Cr-l Cri .l"ib Stt &<-" -r,*bd$ ai*rJr L.-. c1,- 6lJ J(-,rr..,- e*,1 .1lj Jf f-ilfy.+5Cr.+- b L^..:tj
*fgr*^i or,,J tt{ JS; ;tL 23& +i|,.:-,ttr clt J*-r+ f t <=)
;q *'f ,-1,- v sF ,ft,St"c.iti ta ;,-i.oL +.,--Jl ,,iL-J<-,r+- J:t tf l.r1l: zlri e:-r" 6b.&2r cl_.il .t-t* cill-f .l -r-ll Jp 6lj: 6
)r,$ly l -\-ll $b 6,121: -S;t- q! rSst u,L;,y S,!L *J-rf Jrt (l).lt l[<;r*-.; n&t;t cr..i .:dt ct_.:_,t
:*qL=Lr/!"'+ \ (y)/= W o> (v) trtv: \(.i1r)
r:*'l o-t-j -t-y.';;j .rYsL L-; t:v; J g5t-V),.rt-o:; ti...-J u *t|hia .i- r, - r
/r = Ybc1 sin 4vY=lLqcos4vr=o
+5Cr,r,'tr. +tjf = \ ,U-JJ (\ , -\ , .) ,.ti;,:
F* tYE -l J<-'r*5 1r.r-r-b (-il|)i : (0, + 6.,, + q I fr JLy ca1 :, J*.r-t31: ,-S:-;51.r (v)
t io
Iil eSg 9 cJ9:.t*itr, *l+;f ubs
..il".r ill;t 4t uk s Jp t--31: r5l.r- ,jg 6.-r (v)
-Fto ,rt,rfi-i (.:,)
:*q
(,iJl)p;i =
l'o o \
l.oI -t . o
L
i = 0r (.7)A / A
[;Ii]w=i^* = \r0 (.:,) 1Ir I L1^* = I , { tvr
.S+-& J €t6.r..-& 1r l1i'jt l.r ,SrL u!; 'y ci- c.aa ,strrrgJni sNa JtfV - f
.,t{fi u.a;g611-rfrL; :Sg-irt;/\= x;ANtoA- \" -T al'rL'[''r:Ii ua-'-:-'5'1V
nf rrn, 4l-Li .J,-.ra
A/A-Piin;niGAt)
*.,tii6LruJ f ai,Q;j = |(at; + afl bl1, nf A llv = Qiln;ni + itpl (v)
..ll.i*r!
jl glL* JeJ'',S1"*lf" ,'rL-r- ,,ft- n:,>b ci- 1: lA - |
v, = (xrY + xrx,,\) e'H
v,, = (x,rY -xrlxr) eb
vr=o
t: o"/.1(t, \, \) aLi;y..r-,tr"f n*b...*; tsVlt ,t:*'l c'^rt3 'S'-kil .2t6
r;5;."".; ..:-,1
.-rLi 6buJy (*11)
J<-'r.+ ii t-*Ug,I".t Y'r$" (..-',)
Fto ,tt|5J-* 7iQ)at = y (f -k), h = -f , ar = " (.:Jl):-q
i^*= +y(.7)D(,1 =f ,D<rl=YrDrrl= " (V)
O:* -t'"k o csl2 at*tl 66 4g," .*9" ttr
x\
xY
rt-t'\ 'fr 6,lyf I - f
-v- 1l'1
= X,r€t * X, (r' - l)
xr - X, (r'-e'1 + X,
tSr- ,l-l Oll. *f +^, ,lL$.1 +5i".* l1v; = v; (x) ,.:-er*, rrl-r..
L = F. F*t 1*,att;
t = o +L*J;.r D : i (..)
^k- Jt--i3D Jitu*+ Lt t;L?--'r d (ln d*) /dt 6,J.,j:-L 6la6l&.1)si-r. -f.+Si*+ i -- dx / dx
d(lnd*)ldt=i.D.ff:*E
..-.,,f ol^i orl: yj .fbt, Li,16 O.l:e\ ti-s F-trll+, f l - f
vt = x\xY
vr = xllt
v, = xr4t
.r50.*., Itw us* itJ sW r=- 1-;b (.-J1)
.6 ,_*t-,t (-iJl) rL*i Ctr ,Slt\epqiwi = W* al:t .. (..,)o--!-J c(,- h Ff rr-b."r-lL/ = \ .+-, (\ , " , \) ,Ljl--;: *f J,^5,1[-! (V)
r-11:,r4,i-
w\ : +r/ , wy: *r, , wy:. (.-r]t) : *E
i = {- \ 45 ot_b.W(r) : o, W$) - -W<rt - i l/ I Cv)
Hr
.+'K. -b tt l, y-; '.>-cr-' 1r1.r.,.. f I - f
v\ = {-"t cos at
vY : {-"t sin ott
VY:C
.J::.".4 l^..rtj a sc,.ril, n5
.,:-l c,"t-i o.l! p ooyt n5-t ^, dt^! (.jJl)
.r.;5 i,.*-" 11 a;,.4l':.!' 6t oo"l3, (9)
f 6tV ,se-fu, r 6 Gr ,stt ttA / A - d. (ln /v) I dt e'$ ,fV fq.r*5..*,t*."cr:-,1 x: o .lrff = (6r +
"tl tt/ Y c.*--
A\ = -AVl , At = DVt, Ay = " (V),-!
I r-"t cos at (.t)Y
r>cs-t,ll+" n5.r*r": crl-! ff - f
vr = \lOxr-lxrVt = lXt-Xr
vy : xr - 1/O xr
.rJ Or.l, It Q_if 1t:;) ;4i-,1 )-rH J,.i*,| ,--l-, p*r clf-9: t .lt^:"
*:6,*\106r+re, :*q
O..,,J:'*4 ag(51 a;-9;g CSoLZa+ *K. nr
c,;lJii_ ust,lt..r1,. 13h ff - fvt = xttxr
vY = YrYh
vr=Yx$fr
.+50.+(r 6r - Y6,,) lo ..(i.1l.ry c.4.7 f (l , ", \) y u:.-:s 6:
LIL-_YA:*q
i,r.Lt, fi6r.lqrr.r.t 65'*tJ.'w jd (tnD lat = div v +S4;5,:-l.lro-r
JSu"
rr=Xr+laX,xt:Xr+ktXtx\r-&-h(Xr+Xr)
.r:*,t r:rli ,5- k 1si y 6
j lt -- div v = 'k"=- o't=;7ry,i-r
oJ:t*- dill orli= -"t t+..r,rr,r-. "Jbl b Drj *r.dx\ ,grid;.L \1- \" - f oJ:L f , -f
;l r:-,t .rrk ;t^; ;:* J Dii pJd' (dx') I dtt S ta: clLl,; , U_i aLi: btivgd' (dx') I dlt =, I @,i * v*,r Dol + vui D*) dri fui
o2j.e y-q 0K-^ J<-,r.+ r:*; ;[<-, 4U" -r-lj ,.-'5, JS-, q &J- 4- f* dl f y - futr.r
\ = ltXt
\=L'Xth=QXr
t"..1bit lLt , 1,, tsi ,:6
Ira czss> 9 Lg,Eto' *l+:r J'ca
11rt1
(.jJl)
I2 rt 1
(v)
ft'6 t*)ry tlt , )rt J- 4t ;\lJ).rrt^i .,il'i ,:i., dJ+ OC -E JJt (.jJl)
.rJl-"t .,i! ,;; d.2u.ABFE J*L:"- cr'-L'- (''')
.st-r,-;! -1; AjtACE ,r.b.>t- (V)
)rr' (1r' + 1r') = Y (v) i'Y * )rr' * h' = r (-il|) : *q1r' )rr' + )'lLt' + Lt' lJ : Y (r7)
.r.:-,ly-j etf \ 6f 6lzlr YV - t "L-.ls oJ.-! osb OL! 'r'-lj e-'* f A - f
rr=Xr*+/&Y
Xt=Xr*JtXl|Y
\=Xr
O"-.u.h{ agly a;*9;i 66L!D" <S*& l1?
+Six+ of o.b-J .st
ABFE ,i,.-t,",o F i; C;,ltt
g-'t g:- (V)
v: \ -*t't*l @s) = (-iJr): *q\-\/Y
f
rr=x,+d4+o|x,xy=ofXr*Xr+f'X,xy - ,Y, + Xy + Xy
&-t-fs-,--l J<-,r+ ft +S;*,.; G,:"t'.ll ix u\1, ri:*a L^...,tj f sa.1i'., nS
!iLlt al +ar-
-
(t'+a-\ . ?Ll
,rL--rv = curl rlt LiV .-Lrq tlr,1 4 :l 6yy t>-c,*t clhir.rA tSlt {+, Jli O. - r7'-,-tl.+ gt ;1:1-^,o c:y r,olri 6.-r_Sstl
v\=ox\xY-Y&
vY = -btt,vr = Y,u#Y
A:b I. V.-tlt
t_r
lt)Ar:Yru-,.2 Jil-ri 0
,./l-l ,>Y:1." c dQ, ,>Y:ltu, J:ln ,ilg | - 6c^iri q ,JU.t kl :l.tl* o:-,-l+, 6t"r\2-, .$t<" d/-r!l Jrli ;l ,grl-w
&po 6tr-t1-,.:ly*,rrk n, cJ[el.r,r.. ,J"k #(fQg ^tl, tn.f-l .J.::...A,Fj*
-Ui t ,.-,l!. Vt5"r-t.lr .-ri.rr5;r.rb o1l3 ,r +! 6rr.-5," ,rr)rL.. 4). )4;A 3 :.llr rJJlS
! tttrl ;: .prr,r,c sl-,t ,f sJ ts-*. ls ,S3Jl j c 6la1ull , ju; fts , lF gt* C*l;,
,^ "tl.^i
,S3Jl JrL; ! ."", 6f*r.f l.-'-,,i It AV ufuK" ,S3Jl JrL; 'i- l.lf l ,$Jlo;)\..e.6."Qr;,y JslOiU 0! ,;-*,_ c.ry.:.f ,Jl*, G)lf -l ,r$t<,6La,gjrjt 3r
...:*,1 tif tl-i 9i r;-rt .{"t!- tyj gtr uljiG lb-r, - u"}jytJft cOL
(Jp.,1J {->,-;t-5- ,-b. g '>*cV J+ .Jq {L-y" li ')J^r. Jrb OrlicJ-i o.:l: ef i 6.- J;;, lf.iFt, .:-*'t o.:l,e!: aJrL, f (Jt Kstl ilL b.lss
''--7 ,f--l: t|iJ u"L"i rr 6 oLr.y o(+. c:Atd at-'..1i 61L c;l j u,l, .!lt, .r):L,
&,..11*;l l-: eSf ,rf...:Y.:L. Jli,-/,JE ,gly .rl tE c*,tr-', Oi :-f 6ss S
I o.:L .sV;, ,-eV {tl:0.:y.j u-<':, !,:L,J. J-,r;"l;frlr '*( e.>1: o -;: a-JtJ.., U:*,4!jJ 1.6Ji +S u:-- u{! J::*r! d.,{rqi bi.r:lS.f .4J";i .-; c/Jb .:).:L,
"*l .str,-r Ji,.,r-l .5Lao;l-.i {tl : +:r-,=1*"X1 s c*-ti-,il
.J-::*^ ,Jei 0,'_l
.rts-U 6111 "!1li)
{,^,,i ,€ U* \ V'. Jel;J.L ;t; e.-*,, e(i_l jl #.f,4r.f pte.t .t'r &J y ls
U,." ja ".*l
.r11 J^lJt $. L^JlFl ._rit .gsL
f r Cr, b 6lAJlF:;l a1:tu oU;-Lo I - 6L-ry-V f,--.JSJLL^it r:lr3 i.r 2a-*a 6;-*V !- ,S-llrf 6d;r& c"..."t- rr
.r3J,_t,rl* rj Jlf$l
Oe*JJ,+4 agly o:-94 agbL,2^. cS:JlS" nA
(\-Y-0)
& i' .>l;'*.Sl:1, -l J.as,r-. ufu_Lo-j t, f-O- J>-11 .1: oAf et P*i.. ,Ji J. dd.rt'.- jl o:U:--,| lf ArL s 4l.Li' J-('i -rrJJ ;: uAy* g-,_l 6rL jta..r-,!.-f
;l c*,1 .:l!oA - \ \ - Y
P*ij... (*, P*,i...|x (X , t) , tl ldf
-&5,,1.6;ll5at r 6;1;tjo_r e{.*,t eu dqts* rr c":S f+- 4 tv bfrjt .r_lr! .:eJ tfil Jtfot c.-)\c Ft, bti./ 6 6;Sr;tr.-, -r r.::*^ J{r[ry
,,r-rtr .rLj +r c*i 6;,I;tAr 0.rl: Ol; glr ti ,U; jl cs[,",",.| \? - \ \ - y d:L,
Ptj...(t) = J uP*ii...(*,t)dV
l; I p*u...(x, t) I I dlp = L d'\ ! + p*ii..i) dtp
i''...Q) = *, L t)dv=* !*
Pil..G) =f
l,= Jr d'*t' * vqcP*u"1 JdIv
aV J-rt q drtrly t1i'*4...{x , t) + v41, P*;i... (x , t)l dV (y - y - o)
d/J-.t c*,t O<^. d.rL, grlc 0 - 0 - f d.:t'..lJ cS r-?!l 6:L 6!a rilAl d^Jool-i ,5.l1.u9> t sx ,JJ;* 6ta-r-r:, clrf c.*,rai 11 6:-12 glf) ,f G.ti J.sj.e
.(" *li'i (t) = [ l+*"oV*v4rcP*,j lo"
=[ t+*ftr'P*'i"'),*fo"
lll ur(*aU:6s*(Y:a4's.J,.i $lJt
"r"lj-l-l*, +yii -ry;6b 6
F,i...(t) = I + av * I, v*P*ii-.n1,ds (Y- r -o), rly (
V f4-1r c-t-J str.--l -rhl g+ Js-ra+,|:Pu... ,flr.rlr*g GV: ir tJ't" yl,uj!l :1lr lU cjlil 'l y'i.-L;l 65 sya'., * A-*;l o-ru'3'
"f-l:;l-rl a'itil e''
.Jrr"/
Jl-l.r ar Jlid tr .SrL 6l.ar;,>,-' I drt C-b 6,i26'Jlfst ,lU; '.rt'i:'i'.
J.<.lr l-rS,gtL+ |}_,r5.r!3 Qq..6:*L; c'.nal;4 t'rfl "'-1;i,.:*"t.,. +.,LL
.*.r<.,.h -r, ti otetr Ji.r-r.r,.
Q,i...(t)= I rQ*,,... (x, t) dso = f ,
n-0,. . (x, t) no ds(T - Y - o)
b,ti... (t) =
(f=5.-f ..ii- lst 1x,5t"tri:' i:'r-'.) p'-:l' eJ-ll
l. (Q*u * vvrc Q*ii-..) ''s,
- J, Q*a"'v*o ds,
J, K Q*a. * vv*Q*a..-) 6or-Q*,i..-vooldsu (o - Y - o)
L-,g ol-3 ,.lu-r; 5C ,-y'-L. i u# es:'e-|.rly .rl.l3 ,flf ,SIJ' *'.LL l-rh
,FJtfd
(t-Y-o)Rq... (t) = J. R*i,.. (x , t) dxo
g,-.rb \ - \ \ - f nJrL, jl o.tti'-*l L
O:--u.{d 6(y a:-92q 1gla\^".*9" foo
R;i... (t) :I=[
il*, no * I . ,o,u R*i... tuu
(R*U... 60, * v* R*4) dvq (v-Y-0)
gal-a;l '** d)& .11-Ul ..5 ,t**7 On" ;t ,r:^.-; / t_ I .gsL ,+*> ,,o e.r
,A .oUrl.:*l &Jtl" lsy 6 tr )l jt*; t_ f"-.* fe,l: A S.-.."1 oL..i o.rls
.:lt+; J.lJ stl:.,.-9l.l. I .JiL-, ,rir.r- c.--w,l f* 'f*l c..at- ! .t.,r" (F ,'tp
4 Cr- L^ m f l.-r;L OL; s u'-Vi,5Lar+,. jl ,-*b .:-,1 O<^-. lr .jl-rrl . cSr-.,.:J> 6 .- *ul &J-, 'li)Ln L * € et f l_r ,ri! 6-tV 6qfg s: L V€tsf*-
'fuqeLt u,lF6qJdf -6
A,Mo-_ Ilm -
' AV*o LV (\ -r-0)
rr dJ..:F*, clllr ,'.-3 ,SlLl_s V ,)q 6 S'1,.-;_J- l.s p = p 8, t) 6:J-e,ll{.)_*L-/,lh fj 4t,, L_r,"..+ 4 u.Li * lf ,d/-lJq ..r;1.: pU t aL*J
m: I p rx,t)dvr,V
(Y-r-0)
p.: p. (X,t) ,)q 0h.4.t'J(qj;l) C:r, csq-Sr,r, f*+ pr rJly.d,,.+! 1t-fr../.--J'J *: Jlf,tt L-,g
(X, t) dW (Y-r-o)
,jLJ g-*.i f )l ,j,*3 r^ t- f--? 4 fr- S S'r Vsl p:+ <sl+ oy6aJrt -. c;rL ,iLL. u*r" .$Lu-. .;tf -U dqS:r, y' -lr ,,;*,_ Sq./ t-q 6f
trt = Iu"o'
Io I chJl* <>4.:6 t *lp : S aJas
m= * Iuo$'t)dv-- "
P*PV;,;: o
..:*,1 )i.a \ -\"-0
(Y-r-0)
(t-Y-o)
liJlv-\"-0)t_
(.rv-f-0)
..:-.i1.: f.^lJt r*ii... = P sY - Y - 0 aJrL, ry;6L 5
.tyi1 - | (b*pv;,)dV=. (0-f-0)
JV
;.l c.:;J .ji- Jr-q Jt-Gl t+!-l1r c-t-iL,r,. &J-, L*-jl dJ!;tl .b*i ,S\V ofr J4-r.J
c.1sye d/-Ji p g.J_, jr.:,4 Ll .J-iUu" Sj:l r$,.: .1.r ,&r, aJtl* ,". ijrr. 6.r-t' a;-ii sli.l
b=+ +vi+u 'ili
.g-ii Oli € t) :sb,$i 1r ls ? - Y - 0 oJ.:L.. ,*,,
Eo 0o+ v. _!_ f pv"i; = ou'M.;oP r /n,.1TrlPvil,i="
jt "r,tfQ
.b--S ).-+lrf(r\o:Vit -. *+J;r$!..:,ti.r!: ,)qfl.ttf ,/ Jevr{t;t'l' &,-, L-* ell ? - f - A aJ:L.
Vi,i= " l* divv--o (^-Y-0)
O:-Jr{r. agly u94 6Ll-L't"," *lS" fol
.r-;-ei c.*,-t, 6:V J ef 6ta6.r*rlx, .2:m Ll1.t t*;-lla jllt &t*,.J:L, Jlid.f="r.;lyfs,S;t*,bf -f -O3Y-Y-O rr)rL, Jle,
m= f p$,t)dV= f p.(x,t)dIP (q-r-o).l y Jv.
g,_-rlr x = x fX , t) 6f 6\6 ty qj
f pIx(x,t),tldV= f p(x,t)IdtPJy rlry"
6'.; l.iJf*L; q -y- 0 rJ.rL, !.? s..e-, U-F_V ! rJrL, gr_l .-*,! ftOrSt..:-ib p*l9t OY^?
J*t p (x, t) I -p. (x, t)ldw ='
6.rL uj.na i Sr gt',.:*,l 6.1t:;lIP Ul
N=p. (JJl \._r_0)
;-igr+# crr, b. = " ,OiJ o-l)1,
@)=o (./\o-f-O)
.&U,-f 'fux,rJrL.. drt- b- .JtF) JS..l l, \ o - y - 0 .r).rL,
c^:.i di 6sL J-(..i tl ,-fu-r-, drt*.- aVi JK.i rf J..ar rlti t - f - d dU'Js6'
./ \. -f -0rJrL..1soJ-ioslsOLi.i 6-S,h;lA,dL.t q J-
(N)=iil+N=o
Iof U-JI* c>A.tb s Ul* : A rJ,"e
P-\\-f a.J:LrL-'ycp,J
I = vi,iJ
gulr.l.l5l
@)=J(b+PV;,;)=o
6.:,-;ifrt-, (.:-,t l-4r-r-fur;-b 4lf = det F * ' Ll
P + Pvi,i= "
.-r-,!"f 6'-l;i rfii;r ,-fu*,rJrLr r:*,-.1.r Jf.:;l d)t- j#r 6ly t j-, q+i 6- .rt-l.,rf ,-fun d-rL, -ul6.st
asVA*ij...r.li.r, spA*ij..o7;LaVl.JlJ.P*i...i6;rtr.;3i.r*,.t, \ -Y-Ad'rL,.
.r,Ll C-l I f :+ sl9.1.rl:...aB -f di jl .:-,1
h,i...(t) = (X,t)pJdW* I "n*;i..
8, t) p dv = * I r"n*o-
= f | )*a.. @) + n*ii... @) I dIPvvc
.{["f .rbl5raj Jtte\? \ " - Y - 0 nJrL' rttJ'.i 6
*L
i,ti...Q) = [ )*r.Ndrr =,f uA*,,-pdv
J
A*i...(x,t)pdv= Jr,
)*u 6,t)pdv (\\-r-o)
,r.rl, l.r glE d.rt y-l;l o:ti-r,-l .5ly ,gr-t :. ,g,lau>.att
O,*lrh. <SlX tJ..93 CSbL,:a" cS*K" lof
67 eY:l..,, sL; fp Jpl f - A
rhi S sp p :V CrV fq"- 4 6fl.r ,g:L 4:-,J*',- L-,. -1.: ,.-.-7 d..:*,1 ,of )lf \ - O J<-, aU" pb; otg3rff rSb_1.* s ti6 urJtu,6tiut*it,h f p.-r-,!.f ,Sf )rvi = v; (x , t) .>-er,4 0lJ# -ib .:-; g."4 c o-9)\*,
.:_*,-l +"p t-s tlsl L, f
P; (t) = I pv; dVrrV
6J-p tr .-l JlJ. ju; lts .rlr-,i; "/t"j aJ 6 JSs" 0l* ,b l** ,-kl :
\ - 0 J* g.t a., g*j .jlL ,$i;r r, d/lJli f J.:;l_e a,,a;,
* I" pv;dv= Js r;(D ds +
I uou,on (y - r - a)
(\ -f -0)
t-a$t.Sf JL-.r, LSiL F.-?
Jffd -/-joi;,1 s t-S Jq","3- Jl-S.-rl q l, C! Jlfsl bli,_* t,@ = oii n1 t)j1
C- jl .r-, Y - f - 0 dst -, l-i2. \L4,.,s \\ -Y-O d.:Lrll o.:ti,.,l \.t_Srttgr-ll:.r)\^+
IoA
f @ii - ojrj - pb;) dV = 'Jy
..ii.-- Jr-l-.: JlFl d*.,, \:-,,1 .Stt=:;l V ,i:-tzu..:*,l f'*?
o1i;*Pb;=Pi;
(Y-t-o)
<-b; o(+* i; i st S/
't'-:Sl 'l
r:-'J'r. t'rYi
(r-f-0)
.-r-'!.r c):l"fii .l: o99,,Jc.o c>Als \ it-r 6,u cSt O):L-.v; = o 4+.!1r r:.-,1 c"itJ t- ,it-a rlcy OUf
^S 6^l<$
{t_"r Lri}l5 ,J"b c>u;6
oii;*Pbi:. (0-f-0)
.c*,1 ,r*.. rrl-r,.b u(Jt<" -1, d
6y ,lfY .:Y.tl'a r rJtS;- & u:;i LslAlt;l;6 - 6
af dfil-T,. Asr:w=
,[ii;.r (f - T - 6 .r).:Lr) 6f .:):L, c-ti olt-.il # .:-.i r.r 6Cf L,n
os'ea-t i-rl; - y=r, ,-F :-*ti u,,Ll .1 .:*'l O(^r..r)':L' grl '"r::-"o e):l
€'t ir ot5lf^ nf-t+!'r' t' aYo ,-;-.S,JiJ s*V u .!':- oii i;.St^^i);, s \" - f .3J ;l 6-13i't! b.
&- UU;,5!l-r 6rL gl e:rt,@ ;x sls1.+ili' eli 6'tilo-r-'i 4f el'-f y
...:*,1 ,.r.i 6-Jri 6:V6q,5.1 -r, & JU-; c,1t 6 :y ei .s-t-',-l Jt^tl ff = 6r
-\-.i[ o-t^i o.:1.: l]*i e:'-f 6-urJ5d,l: 15s1'o it, 4.,. 66-,11':y ..{ "{J*t 0K"l O#
e-l ,Sl*l .1-: .-l.iL..r-, gl,.t .lr tL3l 6-ru,.-& Oi ! Itr" uiJ rJit; 4. ta.,.,Ji;t -l
.9q5* 1.4-t r_i-S F :1-,l:.51J f -Y.bJ-rr 6 6laa^,'i: ! qti...-lL'€'l gt Ul
'":^llr 6lta
JJ'.e ar,;e-,1: A So U.>t-,- o? ,u.oJ-.i orl.: .:*i ,,xj ;1.:y ,lltrq poN -.,lrJ
o-t^i c-qrr; \ - Y - Y aJ:L, :r,; fr.t; .,l-ry 5 4iFL"'6 tl = Nnin& .lU; t.\:-i,i 6li r-/ cl:-,|
po(N) (\ -0-0)
O:r-u{d 2g(y o:-92i 66L}rv <S*K" lo?
.r;l1A S" g-.t g;-["JS:,,;.:+fc*.l 6rt eL-, J).tls a:rij.c g.t,jA f Oitr +fo.rl-: ..:*i ea-f dqtsd. e.' 6
",it6c^iJ -J
-\lir.l F p;J^. J-alJu;.i l:..ii l) O-s_rer, $loJ-i
p"& fx , t) ds" +
(Y-A-0)dq-Si-, r-r r.i*i ;q J 13)[, f+> c rgrt Ct-, gil,p. JV , S" .li;r 6r5c.*,1 L!+i'ilJ -t5b,str :F,-r olrE trr;r. st 6 r).a c-i-t!tl .-u,:*^ e_f
tJl:r 6lr'.-.*.1 oJ-i r:-ii e:-f .JqFr, -.1
6b
ai 8, t) : ai IT"(X, t), t | = a"; (X, t)
d / dt Flr!,, :ill,l cu-.o d.rL .:L:; -J
Lac^:J,r,L; u_ra 6 ,y qiJlst.(,- 1-r .-t^! {" -..$ 1., vE b
,.,-lo.rb clu:;t Jl-Sjl ,[-l, q l-r r - y - r eJ.:L-. ;.r
f?II yf .5bc^^*i r1ltlsy-O-0+J:L,c_r,.il9l y-f \:.*;-2,^1jtf qtLn-J.rt;ts1r--,1!(1 JsF.1,..+ 1-l (c.*,lcJ-iu-c,-f ,JJl y - f J63,rr 6cfLn;
.f:-'J Yj dsL.o r, b
I p.N * p"(-N,) ] dSo = . (r-o-o)
,|,,. Iu. r.r' (x , t) dw = Iu. o.ao (x , t) dll
I.r", .-tJtt .tltl tr & 61t":;l ,gLa.:^*i ,JlJ J,_q d:L. yl
po(N) = _ pd-N)
.,.:*,1 l1b-l: ?-l -yqJ:Lr!6e"-f oV;z,t dt^4-'.i., f: Iir" el^i)y Jsg.ett 6li €1, p"& ,i; 1l:y
..:-iiY, r) (0-o-o)
(f-a-0)
po(td =p"iLA)ii , (A= \,
IoY eJl* <>U:l* t U(* : a,J,.s
ei-F It ey r>1..o;;"- 6l'a+>';-c q + n"N) ,i:i;brr 6t'an'eJy aJ:L' yt
N .5.r- -llr-l L-,; ori .+,F 611 >l '2.tp q + g,!bd-r" 0# 6ly eJ-o'r,r'
f.^r-.r,jL?l fl .1-t€tE f-.e )l &-f :lr".. ur:rl*a .5- 6s.sls tr-r.* J'L;
..:-.xb f=rrr ,r/ "1,-J:, a.Lii o J*f l. o*stV
si xi,Sl.^dJ-,, -r-l-b .r-l ..u,!.-r i *-S - \ J-lt ,u ,r;b ,slo^A -rP"n, 611 6.J^r./ 0L:i .t.ib',. fr gt Jt" ,5!b 6it-' 'J-- r:eL"' 'trl'e .l:
ss ,y) )l s 6r.r):t*- oli € i*-S - \ Jrl uls tv\;l ':Lc-'l U
r*.e Y - o - o rtrL, tl Uf! .rrri '.:-'! It e*r!'rL' JSiJ 1: JrL; ":)'rL'.c*il.r frnlf I - 0 - 0 +J.rL',:-r
PorN : P"i6A Ne
Poei = P',61
Po;N = Po,ei Nl.
(2-o-0)..-ar-Jtt ll
(v-0-0)
Pj.* \:'*'l,
(^-0-0)
(q-0-0).|.r.
Fei Nedso + I* o boi dIP = Jo P. a"; dlr
,-#- -,.-: rs;e\b q - O - O drL-' * Jlfdl 'Sss
g,ll3st': '?A )l o':t,,i:-'l b
'€sf,
(Fei,a * P. boi-P. a") dIP = o
P yl-*. JlF ;ljl J-j a-vpt, S! :l-,tlg--+ A-rt+;l 6Lr':^^*i .JtJ J.,-t d:L' yl
Fairq * P. b"i = P. aoi
..:*'l
1,*lJl t.-0-0)
q .r).rL, grl crl;U ,na ,41*,0Ui. ft .u\,f eU jii -2'r 699 c:U:6 6
O:-J:L*4 agly a;*9a <So,LZ^, *K" loh
.{t-.r r,:.tf C:r" rfii;.: rJ"k <zlJ:Q
FeiA t p, bo; = o (:/\.-0-0)
J::., ^ .gsL .rl-ri, *r .L*i .J*_S - \ ,e, ,Sl^ill-f Gf .:ti:.i 6,*,-rlr qi
..tr*,l or.,i orls g*i eU d+.;fu -l.t CL-, q.ts sj.*l: ,l-ll1^)J9 Ln,tL..
C1+ .S.f ./t"q,l l.:,La _* Jr LnOt". o,jly.i- JrU Cr_;n^3.$r,_/ 4+rJ (r.it 4ar.rtJ^r yl.g-;L.**...91, c*,1 O<^, \ _ O rt<-:) f .*
L eii*x1 p"o&) d^to +
I ,.r,,0x1 p. bot dIF = o (\ \ - o - o)
.{t-.f ,rbl5fj Js-t-a.6
fI ttio l( xi Pz*)A + xj p. bo* | dlp = "r, Vo
'if 6-S,h;lArdL?l ti.g-l.r;rt{ b ,-*U)y-rA ..iii r A- o- 0 drt-, oirr6'Pi''l g,,,
I ,"',,olri.o F* + x1 (Fz*va + p. b'r)l dv = o
.*t-f d"tfyjJ<*,qnJ.:t ".C,1l V\"-0-0nJ.rt .. Ly
fJu"Groxi,eFz*)
dlP = o (\y-o-o)
-qt-i JIFJI ti ,-*,,sq isl f lP 6la.:**i u,L^r dlJ J,-ti d.:1,.. .,pl c i.:-r*,
$r../ 4+J c.:y.i jj--eii*xt^ F* = o (\r-0-0)
Iol U-J(*,:Ula t t*(* : $ rJes
\Y-O-O nJ:L, S V-#,; s-:j c-ti 4ibl Y - f &'J-l'dr.Li,',r:a.l. rl:t'J!'!JS6" Ji.u-,| l.t t) o:V
xjrq F* = 4A P .ti (\f - 0 - 0)
d ,Srl.:Sn +.-p btsl fl
Fn : xi,B sM (\0 - 0 - 0)
6 r-l-,,-/oJ-oL.ir \ f - O - A tr g-F-V \ :
NAB - NM (\r - 0 - 0)
- A o)(Sh r.--,i ely ,-A U- ,'r"/ oJ.rl' l.|?t.S- tL f st 'iE )t;l: 6.+-Fe ,syt.S
6 r i &.,L,{' *: i| "l;
:,1*; : r'r:';5}',n r ,.,
df, = oii\dS (\v-0-0)
df; = PAi NA dS" (\n-0-0)
6rL-' o? {.4it- J#J*'J g;J.1;5;'.1, G"L- oF Y - \ \ - f aJ':L-. jt ,5-1-et:U U
;F,f osl.: tLjll $ &t.t L-'1' Cl cs"u:t tr
n, dS - XA,q INA dS'
l.:si,-lg-,! \A - O - 0 -l \V - O - 0'r)rL' q'lr^^ : y-l;l o:t'c-'l U
Otr v Is agl2 o:-9a, aglaL,2^" *K" Ilo
dfi = oy\dS = o1;Xe;JNAdS'= FeiNAdf (\q -O-O)
.S-r,g A;-tj; o-lL:1, t j "$a
u 6
(oyX4J-FAN4dS"= o (Y. - 0 - 0)
J* q -&t*- t{ -i}if - \ JJI u:r J e-S ui.r rf t-rd orrLi,r nJrt-, dr.l jt
fo1; = Fei xjA
.Jj;l.r LVj *(Y\ -A-0)
\:-iJj bli€ \0-0-0d.:L.;l O++^r
foi; = XjA Xi,n Sen (YY-0-0)
.$r./ l1*rl -iff - \ lrr,F q U A-f ,;; saS urtl,#FY s ,slsl G6r-F; cr*t .r-rUi rf
,,,-.:ls ,.lt:j rt*a ,f..oi;.r
6b t5tatr!-; :l.r +5.r1.: ,t+lf ,ll*ii .r.tl,r, Jp,., Jir.f 4Lry" ,V bJ$-*o.c..,a.j ,Sss-re ,-3-r:"-S - \ -l d-f ,;; ,gLa';l.tl ga ,4L:"
r+ J,_ti 'jU;,Stn._Ji; :t J-*_5- \ : ef eLar':; g.;y'torl., ot.:j 61.,l
,Sf ! .rr-ri ..:,*,J., O.:".r t-, .gsl.U j J;lj>ly V - f \:^J 1r | ,r*t +-, 4+"J
uiA Ol5-r:jj 6tnclL-rf 6 6l*'f e'. c*,1 .<"-f ';l-ut Ji;UL.; 5e c"-i. rJ.o 6,.c-ii Oli€ -r..itl € ,;lSl u;.r, trL.o;l
uiA=.(e) , c---.+o (Yr-0-d)
ds.J.:i--".. JrL, dtF) s Slsl ,S1-,;-UfcJ-i, i..i,fv - f c*J-rr d.j-Ft"p,--rb f -V-Y-l\-V-Y.r).:L,d/lJq €+ o
trn\udjn=sii= " (e)
ilI urt,:Usls e Ult:_6 aW.
ls seasFti,ojiF 6t-arllsl YY -O-0-lY\ -O-0 'r):L'''lr.,SOt*:,.t
u'-J;l3,Sff .Sf*tf! .4 -rr-L. ,.rL; c.Jt-- -2r !L;;l ;11 c'*' dlJ 'Jnd lF.lt fe.€,l U-i $,F;: J '-e'1r; t 'Js,J'" € * o 6 u;eJ t);l s*'J"
= i t,,u eenc Fu Fin Frc
pitc-*J., b r.r5.r e6-Vls o* s Fu - uiA * 6u o:V
J = + ew eenc @irq * 6)(ui,n + \iil@kg + 6kc)
sa;lj' LF s 6rt.r$^r ,JLet q
f = | eii* eenc l6in Dp 6*c * Y uhc 6ia Op + " (et)l
5 e;y, e;y6 = ? 6 j4.!loJ*ij l;'ii e-fr-)-. ;!! : e' !'-,1 r:)\^; tli j' 5
.g!,-f eij$iic = Y drc
/ = \ * u*,tc + . (et) (YY-0-0)
6ti./ rss"t?' .iJ,"rJr'uu:* f ;"L"r;:.1au,r
oi; * " G) - F'q;6ei * " (e) (Y0-0-0)
.f-:i,-lg^"! YY - O - 0'rJ.lL,o tSlY. qtl'c"'1 .-<,,' t'
oji=s66n;6s1 6'Pt €+o (Yt-0-0)
,r-i-frii 6ta.;l$l ,Jtt qJ- t, lfJ:'a:,rr Ot*l: Yi - O - 0 3 Y0 - A - A 'r):L'.J::*^ g JsL, 6-* dlL:" i*-f - \ ' 'ir-s- \
O:*J{ro ,g(, a:-9o- 66L!." cS*K" r lr(cSlUU c57 o;l.t;l) 6y o;l,t;l ,)l^, Jelf - 6
6f .;l"ul ;l J-VoL- Orr5 LF.;s ,Slt 6.>l ;-r!" ,Sf ,jl_ul 01"-
';l"t;l ailt ,s;-'.;r g41+*r ls 6sl:L+J U-l .tyr; or;;E g:l&;i a.r ,-:*j ,FjJ: at 6 r5s-',rt*.slq;\ 6r,rl-r:l ;!.al .+fe f^ t'^.7 Sly"s\,SfI u*L-' glo-l-* ,lL. rl; flt'.. 61" La a., c*i r*+ 4 6)? rj[:l 01"- .:1.-;;0lJbt b l+, -St t - o JiS..i -;J o-t; o.:b sLi.J ff 61t.,.:*,1 +!; OT a.r g*j c*
.ry -r.ol-,' $ ,f\:;[5-^: 61.;b ,Slu-Jg 6f ,;[:l Jf^al , *&,Ji -r, (r ^LZ
'=ri )l orli:-,| S r3_tt aJrtr. rJa, r:-o.4,,l 6fu*.^ rt \ \ - y - 0 aJrL.,tr; tE t:
.:Fc* 4" -di ,Ki46t = oqk nq 4;jb ,tr (-# : gy- Jlfslrr cr,lj.rlr
e;tu I xj (pip - o4,,q - pbil - o* | d.V : o
{t-.r..FKrj;[-^l a, n.f
e;ftoq dV = o (Y-2-6)
c-:(-t* -rtoj -,1.: c,-i.e #J c.:aB; (cSr .:)rL.) f - y - 0 +Jst-. )g.6.-r9i jj-- J,_q JIFJI n S:VIV 61r,.'lrJ-,u..:*l
e;i1, opi = o (r-2-o)
;r 6.:y*i; r+ rr .c"*,1 OtE- ,-* -:-*E s otj = ojk ry,g o:lr gL;j Fl:.- L"1 q 6etS f*l'r-5 ot-f e \ - 2 - 0 aJ.:l-. L-,j: cSlqllj ,-Sf ,;lsl ,[-al .r).:L- Jj,jJy\ 6ft^;. dtsclL"- e f 1..:*,1 ,-u,j JLrl f-*-. 6, utr t ,+-, dLa:-*.j-6L^ Ll .c..,,* _:lft &to rJ:.9r1 rl;E:* c...at-- s,x-f ,-* ,# Jlrl:rlj* c.:;J
* Ire;i1,x1Pv*dv: I "rroritoil)ds +
I,,r,,ox1pbrcdv(\ -? -6)
J"
/"
rlr eJ(* c:U:6 s c*ly :6 cJ,os
slst urst:.;3, ta;r-l .1r 6 t>*'l uaVq"-g J*;J 6- ..t-t cr.i o-;t-.il Y J..;t: l5
'ft^'"*
6jjl aJ;l-n,6jjl 6q J.alV - 0
-e.U 'rLB gbgjyl dtil r ,i..;7 6rL ;L ) ,jta +f ,-t ,S3Jl ,SA. J-l
"r.bgjyl tL dLjl nr.r^..- : u^L-' tJLn-lr* L-'y.-r-i ph.il tS'C g"+ U c*,1
.:*,t drS^." Ln,SjyltL JL:" e\;{,g elt,.-*7;l U_ ,rll '',...7
q clLl "r--13 t, 5,riJt<" ,Sl".Jjyl lA Vt5.f.l.lr ..ry! dlr^+ U_
"r*f U. , tj(Jl , ;slf J*tj
gsi )l r5i1 "r;q #* &Jtio L-,-,lyo c*.1 gjY -r t$,yF r, lt Gslr s
.G^# J cF dLn;*6_r .r).:1.- :l oli
'l \ ,S3ll JrL; ,t,yi, o;sfF .l, u<ttS" ,SjJl e -Sl.€,g 3t;;l b lj Jtre+, r)y[i:,. ,-i:: &- E*;_1rr..r-,,3i c*,-r, (f - y - A drL.)
,fi"i ;.r gli ,",.q ;l V 6j4;l g+- .*_ .JSJli^il ;lj' g:;,:t+ <Sji( ,-h_-F ,*;l c*,1 .rtl* \ - 0
pv;v;dV (\ -V-0)
,fiil.: 6 Lrr,L-, J c* gb-l_* L-,j,,s;l+"1 tK erl* ,ftg 6(9:,6;A^a
_f '1,--,-J* J) Cr-V.lnl, ! l:-,1 oJ.i orl.: 0Lj.i
+L+L
= /, /i)
K(t) - pY.vdV--
P (t)
pv; vi dV --
v;dS* f pb;vidV (y-v*o)JV
q.K.F r, L "rJ;t, & 6jJl.9:L ;;. 0r5l
(v; v) dV+t pk=*J"+
l;= I poii)dv=.ly
v; (o1;; + pb) dV
O.-J:4e agly tu9;i 65b 4f.o c4J8"
.*:*,1 o-t.i oJ.p Jq Jl-riit raj .-,,-t+ ,t5; d:J3i g-,! ,JlJ Y - Y - 0 rJ.rL, oi J, 6svgii; = 0; oi); - v;; o;i Vl
k= f Ipbivi* 0ioi);-viioijldVrly
rlr
o.tGif r,; = p,i + WA alp;l-l f.=5"1,-+ ,-,-lr-t'-r q1'i! d^<J \ ,,:L-'J.l^* Jl.r*T**
o;1 D;1 dV (Y-v-O)k: tr6r,ds-lrrvl; pbividv + I,lS
..:b Otii 0ljd lj.os-roq L.)rF d/l Y - v - A +J:L,,-{F jlK*S=P (r-v-o)
Jtf4tdirr r5
t= J' oiD,idv: I r"@.D)dv (o-v-o)
63;l JrL; .eU,-f ,i; 6lylt oU D;i ;.*; Jl_&rl n o:V J d:E/S \ is-rL-,y '-r-i ghjl tS,f rJ.a.r6 Otij 6c-.,1 oJ*i dtJt f - V - 0 a.J:L, L-'; uSJt<,
sS os-t-*-i '$t-j\: 6:A ,S3Jl ",I-lrl
i'^" Oi;l ,;,; 4, .,;.rlr. tJLn::*
.::JCr^U 'Jtl: 6Lr;; L-,g
'-r.-i gta"l
t**J", )dv (2-v-0)
Ll .g:5r.rt-r f*lrr d.:L, ,-/1.r, Jlft;l n blpu.6:;ls )FJ ,I-b et* CU ty-;
6 ,Pt q"V '.$5 g
pit,d,V (v-v-O)l;S=U-ddt I pudv=
vv
ild erl*oU.:lat(y(f ;g !^*
(gg -r-lerr),sUl ,)q\ u"a;. ,-#tt oiit u j ol;lJ 6ii(ltu ul-r'6J-, J^lJt Y - V - 0 nJ:Lr .+.-f
'l
+ u) dV = J' pbiv;dv + I,*tro(]ov;vi
K*U=P
t;6 v; ds
1;Jlr-v-0)+".r)B rA t-
(v^-v-0)
cil{ .rt-:^l; J, Ci-J :-h,llil:*,| o$ o.:ti:-,| tf ts ultlr ,$Jl6ty u :[j)
OtKr."*i;11:y o;lsl t *+; ,-rrl.rs 5.r;1.: r1..J e;S JL">l ttj.t .tP.r:-'l o-t'i
.lSjl ii *-f-tos"rr,;tJ n - V - 0.r)sLr 6g+rb r+.2.(r;i'-.,-F e;l.,u
cJ*3 .,I;l.r ,$Jl J.23.e*t 'rLi! J ,Fc.l & ,gJlr, ut-l;ll *y P ,;+sV
"r*{l,o-t.* JrUrl: Jio-rrl ..J*-,!l sly. eJ.rs fr^tf ,;; ,F-l'6tjFL^ 'ttfi,-f.Jbrr/ OtJ Fl'*te o.1.9,o
-,. ,-/)lf ,$Jl oi ot:.r , 6 ,;t, dAE" €)lF &94 L*" 4 ,str
;l .:-,t .l-l!o ttf ,-l dtL.il f.*?
e: l' prdv-f qin;dsrrv v S
(q-v-0)
ls s{r.l-r.Jr Fl.r6L, L-ry' lra.r-t; s: ctslr> Oi Ol;.1 ,6 ,;tr t9tJe OlJ:f
S +-5,-t €)lF sf; jt1.l.1 q; ;l:L.c*,1 €)(F Ct*. q isr s x5r.r oa,JJ
-t--13 1.r r>L.r J-l; ,s ,.--e ;f--lr, ojf Oi Ol;.1 u. 6 el ';lSl 1l l:*'l r.r.,Le
ss r.f-l:l) t-*rl.:.,-l.t'o ,>-,ln;,-*s*i-h u Jt--i 6dS Lt4L"'? tff .ls clL;
.:^,-l.ta .JltOyE Qi ,;jsly sV tF,-* "bf -J;;l .(,'r1l: I - v - 0 aJrL' ,s t'6:'" c'-)\'"
4;= -KT,; t-
.C*,1 o.:-pj {.:tt l-t at-Sg ,J)}f
Q=- rvT (\.-v-0)
0-il-r;l .rl1- urLi r.r.g Ll ..:..,1 e-slfo-..;r.lL-rlr5T,; s Gif ..:',,-l't'o13 oiJs 5
O4F.x<4 ,g(, a:*g1, <SlaL,:^" *&
.c*t -p gV t-Jo,.c-r5; c^.1;.:"_ha
djl*, Cy*l ,Si;l rl"SJst t ulslf ,$Jl .:E>y.. |)t'd.rljl tt-Sg l: $ )S*'OJS|,-'a *5.:-'l I - v - A aJ'rl-'
K+U-P+e (,.Jt \\-v-o).t:-' s?y '>VF t 5;;-l t-
p ( +v;v; -F u) dV * P (b;v; *
t'#_Li o.:LL-,l 'Sf rri:L, :l : € rt*+ f Jl-&{l q l, d"-, Jlfdl f I
:-#goJ-aL;-. ts J;-req ./ \ \ - V - 0 aJ:L, .tit_ ..,:rKJ*, c (f - y - 0 aJsL.)
/" (pir-oiDi-pr-e;,;) d.v= " r- J @ir-o:D -pr +v. q) d.v= o
rt7
l;dfa Ju r) dv.
I Ir/^)v;-Qin;lds
(9\\-v-0)
S = U-Q
(,--cJl \y-v-0)a.^e)B t-Jo.\
(v\Y-v-0)
aj.:t ,. ! qb st rttfu" 1L'.rl Oi \ as;.s(f 6jrl j:k0lJ-rq.rti3l ,s;aaS
tg tj 6 t-r,g oJ-oLL e*,, .r-,11r .tt*rt GAK" 6jrl ,)tk a., .J.:L.. oi r - v - o
.,,!-ltl ,rbl: ,$;l Oi Ol;.+ u.6 ,it ! .-l ;ll .,l;l, rSb-rr* L-,g,-rJ gl+,1
l.V 6;l;;l f+- 4,sty. JJ,^.. ,* .t-t-'.l f*" J6lr o.)f 6 g;dt{! *t-"r4;jl-,,a UJI ty - V - d nJrl.- JlFt;lJj clrl:rlj !,j!O.:;l_r,k-,| OJ:SI 6;;r-.
;.tsl.l !:-,r b Ol"t l l:W 'j,-a
pit-oiDi -pr + Qi,i= . \pit-o: D-pr * div g = o (\Y-v-0)
6-2-,:.,Y:L, ;f i,gr*,, Y - v - 0 alsLr,r<Jt<r r;il ::T r'r;."n
O
ily
,r.lg&.,-,".1 ,hS S3tt J--pjl &,*i .<r" +S(,rtt ,5y 'jlJ'jl c]l..rl) c*,1y4.;riJ-rl Oit- 01ii 0b \ \ - v - 0 aJrL. JIt, .J.r!.; c-r;q ,-r*l J-t; ,s:lr ,s3Jl 6,4tA aby, q jU tuyt L* ui..b-)J," i ,yS clssq gtlt ..r-,1 .fuQr;.y
.,irt5 6la 6l j u^1, g_1, rlyli .,*f-rlr ,5r..Q:-9*J, fJr dyU ; c Jslr> a-,. -,1r 6Lacll-t r.
,Jttl, ,Sjll C.l-+ J-_* tr 6L:i c-^-,rl .r,.1 tE ./Ll oVf u-b ;(u.ttjt -. ..,15 g--l ?-*-t= jl eV,j$-y" gr+ tl sc*,l ,f ss &J-, cttati*,
.r:t:.r
f blt - ,-/./-j>6 nJ:l.n.t d.tflA - 616 u1sst.!r+ {^:.!U jl jl"i tt',:' i_9,ri v-0 L',r+ st 6 e:_bOL^r csjyl .sl+
rFJ,t dsss.,t-+ ctlE 1-4l,>3tq dL^x{F 6,114'61 rtrJ b.c.*,1 oslf s
,r:;1* ty,;,l1r+ oslf r, ,St(l,olrj'l -,yr ,r<Jt<" -rt561, ,lt.- ca.J-9,.o., 6ly .r-,11S
;t *ra:*.-" t*u.0li.-,,o-j 6,-;'- cr-l q c^-+,5;L,.rt=9;t1U oi ,,k Ll cc*"| oJ.i
.c*,1 4.1:yi f sJ oi6i6l cy_l a*Jl ;Jose,r6!r 61.;,..>;lr-
.{*X5.{3Qr;.;;l .r-9U:, ist.r. l.1 l-t, tlyl! ca:-,_1;- 6ta-t '.-..!t<"o.:ki.l ,S3Jl s oslf clb-ra c^r-rjJ.:r.o JL^.rt cgl; g-tr Oitr jl r,gtE Oi st.S,/0.:Li:-,1
.-)5 J ri5J^c J{. 4.1, 5c;L Oljd ,Slol**_r ,.5i\ - C/_}soQ.r, .c*,1 o-\-i
*3-l t_ '*5.>.2lrz Jrt: c.. 4 q Q; r1lr: ,r<r{<..rl5-rJ-r ,q au..r-jl 6* 6Jb
Oi;t t+ ; JSJ..^r J<* &-.1r 66jl*r gl"l.-,-l .:.-,1 drS-. t' ccrtJ>\S0t 1-r.t
.JJ[S.. .rr .(\ 1P0 c6b $ s $alj.r-r) "r;! j6 ,...7 1 _i'-r.-.,-a ;l o,)f JUJI
o-t-i 3tiil &t, t -" glf ,-l-b*_rJl*,c JLel 6t-l g-1, cl.9;U;l "/l*
,&t4 L*'
}t, eli ,Ss: JL"rl 6bc^r-:3"t*, cuoj.? 6La.t9... .<tt-i ssf .)r b,y' .c*,1
.ty't' ^;iS ,;l.t agi.o yl+ ',1-drlr tLiJl ely. Lr'b cll.;s ,u. ( <g S.*0 ..rs.-.U, r -f ; pbt, trr:e.t c/l -rr
;-l: rljLi r,J:-l-rL, drr-l jl orls*,i L .c,*,1 oJ-i jJrJ u.:lt I .rf;t<" .Jt^f t,
6tar:a-1r d.:L'. d/-ljl ..:*,1 'g clQ fbst- qyJlBaJ.:L.' Jft q .fuQr_r.;.jl:i" :ly,.o .1r
"rp-{ J &;li ,gtag-t, f ,S.t l,. 6b.:r-r-9.t*, rl:tr ,$ ,Sll e 4
..-*,1 ol; orLi,il
ultlr ,$ll | 6 Jj.;b :3,-3 ;il*- 6Lac-""5 &p, L*, ,Sll GV.y' -r:
O/'JJ*4 agly o:-9a 66 Lln" *lS"
.rrt7,-r^.-7 J<-, r.* ,t-rf ,1u,^fJ? t- Ji.ql ,tS ,a-.:-,1 dr<^, Ort .Srt-rf/'b
,53_,.1;t5oF3 t6l) 6^*7.;r "f(..i r-;J d? d 6J'* r*iu clt-sf ss-f f ..t.ib
J t-Fd ^;;S dl*h.-"y, oJbfj,LnPtrt, Orl '^oJ^a- +.r.J.Lar,r.o clt*lj er:-,|
.ri J^tJ+ o:l: 0Lij v,p...2vy1v1 il*-lq
,*yeJ.asrr tl-i -#U cs l, f.-?.-Fb gj-rl ,r(,Qr.1*.1 glbyi 6 GV:,fuJt<-,| J\;t p\U,,4,'lr^, q,*lL-yj +5r;5 ,-if .S,*.r{J*, b Oi Ji"ts
1.r.2,,i o:l: Ji )Sa r, .:.-.,1 O<^" ,-,-r* C/-l ..l.iq ",i6.,lilr
,S3J\,.-,-F Gllyu = f (rl, vu vl, ..rvn) (\ -A-0)
a5ptp(q Fbq u. ;F,g o.rl.: 1t-.;l ,sJb $Jg aJ.rb olp 1 .ri;l :+l ^t1lJ;r d,JFl3 oJ-i A--*'Cps r,.'rLls ,S3Jlrft?Ji )l .trr-l ,c;S u'9,.;".JS,r rl4gl tJ c9t"h---t9; c>Jlovr...rvyrvr Ltil \q *Sj ei-,l cJ,3 cr;3i.Jb.r--l
...:*,1 d,Jjl t r:*j,rbll ,S3Jl.='.' ilp L.r
(Y-^-0)
dL-,-_j 6La;;.- r-, c*j .-,Llr ,s3Jl :l jf 6;,I;lAr cgrlJ o-r)\r
.$r,-/ qrJ b d*Q, i.; 6,1.a;:,-S c,rirQry;
(Y-n-01
:c*ii $ f; f b rrFt, 63J1,-*;t q, J<j Oljd,# ht-rr ll orlh"l U
ilA
0= tuaq
0uL^
-
-No
du=kdrt+}r.ar" (f-^-0)
6L"rr,*5fi J t>2lr-- a->.-;),--r r5 t{,t' c$t--.l-o Y-A-0.1 \-A-0 L-t15, tl.J::.-^ dbr*j.St -rttt, I utstl:l ,*.li eu(.1Qry;
tl yu 6S- &..p, t*y $lc.t.i ,...c.,-F ,-fr. &l -rr 6 e.li i^! l.3-l ,-it \.
.c..-,lf{0t(,l oslf L..1r.:)\^7 *J.,J,Jtl J-,*ltf ol;ri.J.r 4-t -*4 = 4(0,vo) (0-^-0)
.Jr.;J,-: U-f--V dbrj ,St fttt, t Jslfu = u(0, vo) (2-^-0)
.$r./ l, *J!> ,s:lr hlt, 4r- gtS(rtf-A-o aJ-:L, r, e-b drljl o:[i:-'l
to = to(O, vo) (v-A-0)
.$r.-/ +rp t1 dl-"-j,St"r.,tr! A ntS* SYo = vo(0, ro) (^-^-0)
1 r:*,1 ;<-, 6 s-*r; Oh ,r<."b- )yj,f q st,l* ;: .{"Qr -ri J-rl :&l:r-t' 6-f e )s;
(JJlq-A-0)ultlr,sSJl(v1 -^-o):titr9,3;t
(vq-^-o);lsl
(.rl-^-o)r6i;ut
ntr-.,.:;l .,-ti *J.,rl>b 6Jlot"1 ,-S!,1 d,Jt- d{JtSiJrL,.rr .:-'t iF" e-tr .fl
u
rl=u-rl0x = u -2r"r.
E=X-q0=u-q0-2r,r,
u-rp +6-X= o (\" -A-0)
.*J; I j Jtu6tar;. .:i;t 4 r^ tt ep;r .!l O(^- ,S3Jl dt^Ji.-Q i^,
e,Yai 0,Yoi 4,toi 0,to
g.rr(\\ -A-0)
."it.al ..:*,1 )Vxi = xi6* 4 Cb &p, -t -, odv,J$t<" .5y e* s$:,. a'.
s;lr ! 11 ,4.9a.1.0i JrL, \c0 *>sli- r-.1: cl:f dljl .1jt* ,-ri"Q:yr g-t, cl,f.J::,.
^ clLj ; c.i$/ dtJ
l+i Jr -f 6
0 = 0(X* t) t- e = e(Xe, t) (\Y-A-0)
Or*$+4 61, a;-9- 6b L2G"t cS*K" ll o
o.:1.: i drL-. il*-l q JS ,J,::.1't<i cf=5 iJL'- f p u:a, ll,t )F +i f I
.t-frul(\Y-^-0)H- foa,
I,P
r- l*0,UP(\Y-A-0)
cJl*, q 6 6lr" jS a' ,tli g l, g-1.: clyU ..:-,1 uota;.- cgt.F( {1;ryJt<-,| +156cr.2lr> irii-jl-r, edJjl.rr-n:, tf
-l )t ,Jsyl;rr*i; GU:if :rt'tlt* , - 'tq g.^.r ,lp ;l;.4 ,Stf lfultlr Cpil+Ji bLjl +, .:.^-*i 6 \,Strrs ,;isly sV
:tF,_l {.iJi y, ;[<-l
* I:*': I,ffdv- I,trds + I,*0" (\o-^-o)
cSrrlxU J-ftrr lJr OiU O{* q ! t-:JL ylcl:*,| c.-j, d,J}l rJy.lle .n
.sr../ o-r fb.g.r- qyJlE
(\t-^-0)
t-.f .rr5 OV ,$s-rl :J-i u-l:-i 6L-o6;-r U. .rnJ. ,*u.Oli.l l, '# J*, O{l
:t9{.f r-r;j cj-rtf .rb .).l!o e\ ,-*13:y-: a=ai A$:S r.
ffds= [,(#),,0,
c,.bl ,f :e,r#4<i-l u. qi l. 4spl.r-r[* :t e-S,L*jly1.: ,611J o;)toc . ..1:i.*:]Ji oti./ d/ljti,(.-rr* t-r t t-f-O d:t - ry-A rl;)..:*,1 ,_r;;OLi
* I:*"= 16o, I,,a;tas
ril U-rt* r:U:l* g ,*lf :6 J,os
:g;lr cr:-,| .ur;r -ry o, JlFrjl s JtW,*-a;l .l9iJr .:.^*i -f tSlt d/-l 6t+"1 jl -l
pq - o;* (ti),= n - p6. + - fi e,e, = "
..:*,1 lUJlt A-A-O) Jye I gb3.: - dt-j)\SaJrL' L.f y J<'i 'O''ltt!
peq-pr+Qi,i- }.4,0,,=. 1,;JltA - A - 6)
eJ.rL, c.Jt-- c,rtl .lr 6 ,-.J.,-t c.ii 8i : 0,i F u..orlr-'et Ot-rlrf -Jlit:-r,i"f sr JJJ# q 1;JltA-A-0)
pe?t-pr+Qi,i-Jon,f,= ' (9\r-A-0)
rrS.ul.;,-.,*U; .:r!o.r) jd t, ,rbl, ,s3Jl s ,-F oli \f-v-O drL' ! C-b yl .-Ji.J^r.-/ q# lb-1, - Lr,J{jJSdrL';l ,3lcrt- ,:'"5J'(.i J
pery-ptt+Dioi-|n,t,= "
LV,-o'* fr))av= '(\v-A-0)
(\1-^-A)
ejli JJ-"-.Slf VI-A-O iJ.rL, rJ;6! gt^-e.r - .ry-jXf drL';l ,,*U JK,.x .-i-
..:*,1 c/.Li-,:;frU oL;l rf y
,p +nb-Pii"r-*Qigi2. (Y. - A - 0)
tKaQ:3,,j fJ, Jilt ib.t a' ,S;-.Yl :Q" J Glar>-4.ta,. q - d
J*ti eS ;;'-,'l Cl-l- il.*-l*+ Jr-tl ,F&yy 'u''-t*, f ,-r-F t**.&p, L-,- tt.>f ua;.-- elL.tF o4t:s cr:-'l ,rfuL.,-ry' i I fi* rStac"*S
OE*,JJ*4 agl7 a:*9;1 66L*" *8" frfSB'.bH_ L-,-,r*a .*- dlJ ."tjq oli .-o,-f Jr_li d*q syi +(f ,"l.lS rybar.e4lrL'^Lii s oLj q c*,|-l Cti c*i,. tl t b-r -il f,.-l-f cr.ll x 6tv -Luj 6!lr
::y .l;.rl_f il )Sa t Cl-1, y-l .c*,1 o-r-.i ,_.cr-r.r,
xi : Xi(X,t) ,16 c:^i-1, - \qii : o;1(X,t) uiJ JJjtj _y
bi = bi(X,t) lf t-ls J cfr 6Ji _y
u = u(X,t) uera,,, .,l;lr,$rl-yei = Qi6,t) ollr>.;t-i 11.:y -0
r = r(X,t) fr. .lels J. e:lf Zr;:t -?lt = rt(Xl) (f -p cr,_l.r,l-V
0 - O(X,t) (c* d..j:..o) oslf a>.Jr _A
c-tjl6jr;l .Stb-t, : "h; lpy rlr!.; q ,tlr;.r 66b c_:^ grl;t,gl.l.e -r,9.-
-l f -Y-0 L.lss ;: )^-;[ii 1r Jr[u clyti -e-: g.,-l ._ur:r, .F.ner_rj _rj,_lj J:fu$lor.i rlF # ,l .>3liz,,FJ* ! y-j t, -e JloJ-i o.:b \y-V_A
oiij-p;=-pb;pu-oiDi+qi,i:pr (\-1-0)
-rr u^.€ LSJ{ 3 coslr> ir.-;,f.-+ 6ss ,5r.tVi-br,Ji*, C.xl q te drrlJ, C,*,9j;.rS4 sQ; ss to;i cxi cxf oar* t<t 6f*t_ ,g )t3 .:*,1 0.L..! orl: :l:, ,SSb c*,1-,
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0
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" -A-O drl.- Uf -S -b 1.r ! ..:*,1 ;13 .JL^f t,
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ttr 4'J(* aUtb t (*leS :6 cF
.J^it ..Jirb-r;rj dlt.J.(..:, ,*J.J,lt+rf rr. n:*rl-9 ft, Cl-r' 6 c-"1 '# ,f -f t'(S-')l irL 6ly
::,*,-l ,f J rct-llH .rl $lJt oif o'-.-1't ..lL-'lr5 3 'J2lsz 2;:
ori = 6;1(F;a,0,g;); 'l = Q(Fu,0,.g)i
q : ry(Fu,O,g;)i ei: 4i(Fu,0,8;)i (Y-q-o)
L9+B e\- r:..j,e .J.,lo-t^! l:.i2i l*.;i rtri, J Cli e. ,*;*,T,JlJ g-t, Cl-f Cf-l
..:*,1 -i-r.rKq g-t,.rl&;t g! 6tt
ti ,l;t*.gtar.;:. ;f .-rrbF os'e q J''-t It' ifu Y'-A-0 tJ'tL" 1l o':ti:-'l .,':
.rf 0f*
i'= #Frq+ Se * #g, (Y-q -o)
J<- r"+ Ot-rf;t" ),ytu.Stt-rX,t+ )JYbt V-\ o-f d'rL' 1l''rt+-'l U aJsL' y-l
.c.*,1 oJ-.i 6jLo.:L crt* r*a ..lt-rtf J r-of Ot-rF;l ,J)V o22o *t
it'= #L,fin+(f-1-0)ffrt*#s,
:g+-rlr Si-:-dt^d^+ e-S:-f V3 Y'-A-0 .,'r f-l-A iJ:L' tsll-rK-b t''.
'r(# +,t)b. (, -, #'o)",-o#,s, - |4's'>' (o - I - 0)
c:3i a;.3yn,Pri Jst' 1.:4fu-l rSf+ q ,iJ clly O-q-O aJ'rL' Fr;' 6 +S ei
cuf d[..r] i c*,1 d.,ti:r ,-e ltJr,'.:ll iJ:'rKal c\:-'l otui d-aiy o'Fti Js'au'
o.:Li:-,| .tcf .lt-rF.c*,| ,Sflf[ \ -F rlrf rit'al J*l stL;i 'a--rl - olU:" .b*;
,rL-rl:S:rt+lj .str J<-,,"p .rt-rtjtu c.*r tl:l e3tl if ,P lyj cc'*'l o'ui
.{^*J 0JU:, 6 t*r. J<-, -'..*
.:*,1 6i- cJL- d/l 6,-r-ilt .lf t Fk*rJ,"i -u'-lt -f ,Slt {! 9;' ''ryti
,rt.: 6i.,f ool; .-,Jt- Jl-* er. r>slr- qss J rlti ,*l; dta0hi:' Js-t il*-lq
q+ agly a:-921 <SbL,r^" <S*& frr
x; : X(X*t) = Ya + A4Q)IXA _ yel
0 = 0(Xat) = a(t) + [AAi(t)a;(t)llxe_yel
(t-q-o)
(1 -q-0)
6.AuQ) cb -r tY 2X 6-rL tte.i eu r[i-i 1r c^:+J,, -S;u.ya sxe,aJ.:r-, d/.rrs,.+iJ,'.;r.-tr*A tlL;;ldrJr 6l;a(t) -9 cclLl r, a:*,ll slslaiQ) (rJr-u,JS,, )J_1,
:.:f n--,b., Oli€ GU lyei7-s0(Y4t):a(t) (v_q_O)
F^(y^t)=AuQ) (A_q_o)
:.:;i c.r! l) Os:- 1 rJ.rl+ d[ii 1.r t-q-O d.:L, c*,1 g(^, 6_r=Jqy
0(y*) = a(t) + a;(t)[x; -y;l
0,i:8i=aiQ) (\"-q-0)
t*yl:Fu sgi ,0 6Lac6f bri€ 6c*,r ,'-',rsl-l-o 1A-1-0 sv-1-0 L.r3, ,l.l;iJr t*t rtli'*; t, tac.*s el ,-;q 6b.j:.:-. ,illJ o;)\.e .r-5VBu;l Llj:*,sFu s8i 4t Cli'tFJr 0.:'i Ot<,| ,et, 4li ,Ss:,-fu*, rff#, q.sf ._,1*;l
.r1b .r1-3 t+i jU; .rtj:*i.-.L*.1s oJ.i osl.r 6sL U.A .(.t 6ly cf-1-0 iJsLrJ.<r-,yl .5lo.rL t,e+:\: -fu 0l*<r- t-bu. L;i *>er-,,1t-rf 6,*, j]* ,_f.i,..Lr. ,r:Jl- 1 b ,r -Lj ,c-,gy,S # o.j ir+"^r .b = o JLij : . 6c*,1 .rir ot-u gr-l ..:*,1 c1l.i.1p r.1.r
::-j'10-1-0 iJsL, gi = g';.lib c*lr olyiJ.r ctl31tri, ojfr_.-rr clL-rF
ff6
-e "l'(F^#;'s )-g' - + 4i(Fie,'.,s')s'i 2 "
cory ol3J r;-bt}i b3aj.:*,1 ,:rLo 6-ur-tJ-Ptr glr,lyti 6l+i;l o1!j.:
GU: j&' ?--t' d/lrt:1 tS "z;:
lr,Sf!U 6-r.^..iq o"t-.i -'t'-:.it 6t^;f 1.:*'l O<--
.,U\ -p -t-! .:;lf l.-.1.: clL-rlrf
eT(Fu,0.,9';)
-=o
ogi(\\-q-0)
.:J), \ \-q-O aJ.:L. cr:-,| o-l;...,|i:il ,llJ.: oslfa-..;" tlL--rlr5(JlPq g; 6t+i;t
: j:')-cc.ry! ojf a--.-,1r,lL-.rlr5jl .*b c:li 6,i1,6:;lr:JL e-l t
\ip(Fu,0.,gi)ogi
al:t-.Jr lJ-,il^? .-'rlj,.- -=- c,.loc> U-l xP :Q(Fw 0) ty'14"-.':j 4LU)t 5.tpcl 0-q-0
,.rtp>\-tl 60-1-0 a:it- ui,bu aJrt '" s.23+ ss 'JVl oav 6b-Jt-' r-,*6 !
'rL]; +r\ tf ei t-&-.: ,gtll--r".;: ..-*'l Cb:-,| .!U .rt.,l oVt ir!-r, 6.$*.:rti.,,l-r; o-tlf o--.,.t,1L-rlrjll g, = o JL;j = " diolyiJr cl[.;;:-l ,rrl;Jr 6sL
lb-1, - rr-;,,-jy.f e-tllV,.:JL-' irr-l .1r.f$L"f iJL'- o.1l: ! 8; = {i 'lj'J.:-uf, "ulr* a*Ji U os-fu.
- (*'#" ., (F*nl,s*')), - + 4i(F*n0,s*")g;'>" (\ y - q - o)
t--J" 'F.rljl
,5jr;l ,l ,SltV os-re\
tQ(Fw0)a0
q(F*no,gi) :
Oi.-,Jr,ho ag(y o:-9;i SbLz^" *K. fr7
Jr-q -it ,j,s:'l crt,l otlr lo.ssJ J<-,r* rl[.:lrf;l u-b t+- ,l;i 63Jl6l+i;l _l
:ui*,- c-r*iq bi;l u-b J,_t: tFJ
rt = e(Frn9) (\Y-q-0)
,-j-5,-F elt ,J)V t,-rV d-r"-lt dl; gb3, - .,,,tiys el 2i:l -&' 6ry-r6
Lt t 'l-e'Jr lli os\- e..,.t,1[rlr5,-r--li U-l elt.r.i J,6lF -*t tlj ,$Jl -J0-1-0 aJrl*.- .€.*./ts"i| :;v sbl-S'-1._byl_s r>;lr> e.-rr crL_rF.rl.p GU.,
:r.i -t-alJ> J_j Js-teu.
(r<r*anr,) -, # r*) "0
- | a, (Fwl,st')si' 2 o(\r - q - o)
t:yL tiuaLii ?-t' 6 "t:ft,
t.g:j .-r."lr "rlyf L;i e.er-,6t^clt_rlrf Lo ,Stf Jr_q *f.-u,Si; el-t p dst-..1.r 6.rf ./ti:;l ,Sl Lii 6li €
6ii =at@we)=offirit
A.b
a0
(\0 - I - 0)
4-: dp_l jl J* 6rJ,-tOL^ :lj 6j1t +flK; crt q 4?t ! \r-q-A d.:L, t dJ3,;_,1r
:l* 1*lLuac.:--,1 o;lf e.-ls oL-rbf )l Ji;* cry oJ-..i o.rb 0l*ri \ \-q-o .JrL, '4}-,;lJ
!l;, 6 ls .S-rl.r(-t+ ,g1+rlr .)y oJ-i .!i.- \ y-l-O drL, y )b,.,,.61ar.,:, .:..-J
a:;l*.J.;-,)l i:L lgly r.!-.:3J>r.o ,jA O-q-O.,: \ \-q-O; \y-l-6; \0-1-A
48i3..:-*c*
(\P-1-0)
611+U*'i; l' -dc.:,"s.c*,i :;; d-:r.r Ln-l_l-b )J, t) O-f ,:; St CSSt*t r=* -S\-b-fi-t otV I .JUQ .,!tt r*J rl3J., OL,atir 6lay^u;t+" ,Sl^.JUJl .:; S ,SryEoJ-l .rilu:J c,Jt- -SJq rt{,i orl.r 0t-i.i sj. i1-b li +fd;J .iL; ,5toa.iJj" L.o
dt ur:; oJ" C:V t ctJr"-r )Jyb I ,.l; .iLi, 6L^"iJj" pl.irl1.r 6y_{U-;
rrY erl* <>4.:6 s *(st :6 <bS
..:*,1 ./Ll J ff (Tresca criterion) ti,; tl.r t-) {"^'-fl'
7:) .tS,; tql a:-,-9.r, 1gt""-t -, ,Jtt<, -rJ jd It e-fr-t f'€. .F ,Jy-{U.n''
6 jV: e,t, J#i l& t- g'.rn ?*.:4 f -F.1: ai-';r#, L*' cl:y 1du,"*ij
oar,t.i+rI J+ c,.:-,1 ,J.i JLtl 6.:L tte, lk q * p+ oJ-i.-tfl f^ t 6f
./L-,f gLncj.r, J o$ JL-"t r.!-r3J.er,c .:-,1 O<^. 6,b .,*l r/-J4 ;PLf JeV.ty'l ,g)J-g d,.-r. J.rl Uj.i.r;-: 6-U:!
p.rs c*ij r.:.1lr.:-9,-3./Ll ryt, dlr dJ{Ur*-d Ott:,1 6ty. ''t^r ,-}s-, st
f-* g,,q ,Sil urst st .c-,J- f*'oti, rs+: fbn olgcfJr J 6JU ?*)q Eitr'
eu ?*:4;, otfJr-r;l o-t^i c4,*, #;t '*5y t+ -l ,rJt^tl .st^-lr* tu a'i-r}r' L-"'
s $-r,-l c^i,l.: d3 -U ,-Jtrl tJb-l-* J t' c.Jt- c/-l -rr 'r'i!. 0t'*i- "t.,-t -iL;g,.q 6i r;;r 1r * ,,',+ ,5y
"re,t'J s: .s/;,-/ ef oIj*r e:'l ?n:4 W
..:11: ! tn.lf ,l ur^7,5-rt, i!.iir^.Jb i*l .l: '..:*'l 0l''..(.l- oJ;Lr eU ?-*:4.t-r-,r/ o$b-,g co.r-i"raig,o L 6f 'lr^^ ;d 1*.+ *i Jttl ,Jbl;
JL-t elt,ssl-Ll 01,;-e ^-1 :& g..a otJ,uatg^f. .5f ,siss ctP c,r+l tr 'JSs"
.J^i J,6lJt Cbl g_p_tUr*al
./t*,1 q,-p q o[-i "!U
jU; ,-t-14":t? tl .l.r^^ 4'. -*7' 'Sf rref^J
x; = 7;(X4 t)
..:*rl ,57(\-\"-0)
,grL !ta, e.,j-l-f\:*; i^," J.l- q:& p*'+:'r;;g'f cSf tlQ yt;t
q fi ,Jt-j "/-t?tl+
,c*,1 01-j jl .-*b ,322 6 leji ;t O++^'t "5y'.,g o).r ils1
6Lj -l: yx+; 4;tVib c^iii 3,. co'6u4tgt 6f .ry.1611 J*' .'':*'l 'J^i JLtt 'r57,5g45,J{LaC^i,..5 ,Sll'*',--o)U.:j-r J-alf r,.:-,1 CIU A oSt+ = t * 4
On J:Jrlo.:r5o.r[i:-,|"*".]-)\.p;l ttc.l;j iL.C*,loJ-i o.:LL-,| eJrll.r t-lJ,leJ
..:*,1 o-t-i o.:LL-,|"+o.:"r)\.e epl..rr5o.:Li:-,1 Jiel r; 6toc45 ,Slll-: r:r)\c d/-l l-
:t4.l A;.tti J-s Js-te q'I-i..g,'ee I 6f J3 q '-5y
xi* = xi* (X* t) (Y-\.-a)
O-.J{r. 6(y a:-94 3glaL,7^" *& rrA
t+l ;l .s..xi 6:Vjii.1r jX^? osy,a q dliL./ ],.{7JfuJ-_ ,]S ,i, \:7_slsXa = X-\(xt t) 6
xi* : Xi* (X,q, t) = i,* @ t) (Y* t. -0)
.rAul ,4,:"7+, ,l ;:bdb q 6.2*,lXn = [\ (*rt) Srt-K-t? #, \*,;* ;,: -i 6;:.r.!l ol-.i tr,li;-jl u-.Ss 6tL iJrcj co.;3 3s ltia.r- 'L.V rjrl., rlLi 61,
..:*'l g-,lr*.li ir..r,i ,Lf S rf J)
yi* = )(i* (ye, t) - i,+ $t, t) (f-\.-0)
::-*.f di^!i t) Js-f ar.Yn 3Xa,grL c:!J cl!_. 1g:L "^J-lt
(xi - t)@; - y) : plWr,l - i,*0t O)[*:<+ D - i;*1tr)]ro - \ . - o)
O;2eyY1.lXe.g.:L .tid 6l4.li ;l .g-,ly;-e-q.gLrc._".*i y &.-,1by.glrtij ?r- 6iJsl-, tt 6-S,Ft r*, Oi Jl1,, q .sl,Jir--y;jx;.5tac..""5cJ;sy oJ-i ,-.rtjujl .lFr
:! tFc* -*f; ,w sx; !c*j o-\ o-0
aX,* (+ t) Q,,* 0t t) _ ^ax, ty, = oPq
a*,,* @t t) U,* 0t t) _Axp Ayp
y o3)\c .c.*,1 -trLi Q;pQ).r"_jt" 6-to:.r.. lLij t- \ " -0 .,,r V- \ " -0 ij:l-r -yl o.rl-cl-,1
'.rAc/ c-l:-i e', t- ,* 6f JIP t 6o-ri..:'4:g^r cjr ,l ,fV cJL-.,r cyl
f*t! a:.11.: ',:.,-
c"r-.iL ,& "r.Li 4\QpQ).r-_rt d/lJq il*;@tr t) - x; rrrlt.det(Q;) - +\ sQipQ)QiqQ)=60,
c.*,1 g(,^. V- \ o -0 njrt . ,.U,,#e t eSf ,S\ ,fV,r:Jl- n n'1, .1|;:. +,.
(2-t.-0)
Q;p(t) (v - \. - o)
rrt
:g,;lr 6r-/ ,S-SJl-Stit "*6 ,*y
:.:9i a:-i9i,5 J.(; *t a; 1|'s1.:-,1 d,,S^"o 6: ci* (t*) - Q;^Q)c^(t)
- ci* + QuW* - c^l
QaQn=Q^iQni-6^n, J det(Q,,)= \
it* (+ t) = aiQ) + Qi^QY^
xi*:a;+Qg,^
1,...cJll- \" -0)t{
(..-'A- \. -O)
(q-\"-0)l+$ 1.:
(\.-\"-0):6
(\\-\"-0)
aiQ)
xi*
I &txft ,J-j:6 ,>l-.";; -tt (f I li tll_r,_f b '.t.i,lf" t 6f jl qL.i; e,lr--,, j;3 o.,t-i,.-,-F.:La* jl+, JUJI U j c-lb 4l^aU c;(t) ,l:1 L-,j: 6O+x+rx*l*,+J:L.. tt 6,SJtf-sl ,St+ t.Jl-.(\q2q .t.r;JL) r.r5.-Jb -t.iq oJ.i.g Qi^'^]:.,su.
tc>.2.9.,.a q 0li./ b "rpf-lJ
c52> .f$rf #i, r.li;lv-\ .-0P
l-aJK^t.o.r-ir+gAt:& f* il..--l q o+rr*r1*ry+ o1axp1x161r olK;-,, Gr osL;s
p;* =c;(t)+Qa(1p^ (\Y-\.-0)
cb;"-1;r.-ljloJ,i -^ar-p co$ o.rlr Ot.ij Y-A rfS.i1.r 6 "t:_reu.p*
sp+i rl:1 r.5c-;;
O?J;*4 agl2 a:-9;1 agtal2a" *K" flo
.c*,1 O+x+ &*{*, J fufi{r cl* 6[ar5rLa,J,tJ.r"_it" Ft +,.Qi^
\o-\.-0 iJ.rL- L 45 ')4 p,.+ o.r.ilgrf cSf J+r rlL:^;:l -l " - 6 dU..:*,1 rl.aU J A.t-J\iJJ;S L;- eg-,l oJ.i *,-F
'Sf ;t ;ttt - Jsp ,u.Ya yXn g;tJ tf C-r {rti err \ qi \ : r_b
.f_-lls '$grpe-r:@f - yi+)@i+ - yi\ = Qr* @^ - l)Qn@, - l,)
= Q;,nQn@^ - y^)@" - l")
c*,1 -t L;, Qu 66 :l I
@,* -yi\@i+ -y;*) = 6^,@^ -y^)(x, -y,)= @^ - Y^)(x^ - l^)
!.i- coJ..i.-reeJ 6;- tr rLaU d/lJt:1 sr.:-,i oJ.! o.rLii-,| lJr ,g.;ljK t7 c*naB ;l 6..:-,1 oJ-i
;-t...x.lf^ t Sf 1.: b_13; b 0rl, Ot.i; ,Sl-l Z,t, JYa Xa,6rL i.f,n +-, 1l
.JJ\rr* - zi* Jxi* - !i* 6Lall.ry .l"r t--llj 0* 6 ,-if..:*,loJ.i orLii-,[.-J*.o {*-+: p;lr 6lrLa ?t' *l*;l orl;'. .l U
cosg+ =ffi,U .s1l.ry ';lSt ,KL." oJJ) lJj t,J5,1 JSrt Cl? J, e'* .ss n u*2Jtl 6 rr5 r;y.+J ;l '.:li;l s Qa ory JrL:'o c*"at-- )l s-S r*. J A- \ . -0 rJrL, -1, ,srl-r(,-t+
:ptlr .JE if.l JJI .!*i
Qi^@^ -l)Qn@" - z,)wcos g+ =
ril U-rl* ,:Utl* s *(* :5 ,!ai
Qi^Qn@* -Y*)(x" - z,)!:
Ti-r^llx,-znl6,*r(x*-l^)@"-zn)=m@^-l,)@*-z^)=6
=cos0
tvs.,, J* rr.irJ,64)gt 6r -#u 6J& q L'r ot Jb y-l ;l -t-'
g*-i ,-/t-j ,yfu *J..J.i.,.ef^ t F -ssls o'r'r.- 'f*-ll't6 r-i.t'b"' l;ati*t L?*.€,-* '-a4r; t+ a'.
ui* =*,* =# (\Y-\.-o)
o.rl: ;,rj o't' u. uyt (dlcJ.zrj i-rcU ,5J 1 -l A- \ ' -0 d'! e-lS ;t o'rLi:-'l U
.tr,l
ui* = fiIo,1r1 + Q;^QY^|
= *IaiQ) + Q,^QY^l*t)
y-t;l,ghars t)It' ^'..dtldt+ = \ 15:c*'l €lr,.r.l;Q 't* = t + a 6"f-t'S'r;i't* tl
,-/Uj 6J.-il.r-l q :& t':*r* gt $r u>'>'i c-e,"" 6l-,1 ,1t:'V ''Jl":-u-i.l \:r,r ,. -*l d:L' .l' ylt,d3.lr il*7 )l ,f -S
ui* = it;(t) + Qi^QY^ + Qin?)itn
= itil) + aaQW+ Q6Q)u* (\Y- \' -0)..1:r"tc/ 4-f
OlrJ;4f 61, a:-9* 6laL,7-" <*K" rrr
ati?) : Qi*Q)Qi^Q) (\0-\.-0)
: p-rlr Q*'{-'i 1 drf ctrijl &1 d-3_t l.i t_
aij?)gk\) -- e;*(t)ei^Q)e*(t) = e*@ (\p _ \. _ o)
4, *r OU:f 6116laL.b q tf-\ o-0 aJ.rL. d"1-p1s \p-\ "-0 aJ.rL,,Stlj(_F
.:Fg *-.* f*> oriu+i t'l 'F* slri .u-,_;*r, 4* f*
ui* = a; + Q;;Qp1, * Qirrp^ (\v- \. -0)
f, Iooo,,r -- * [\t : .
: g,-:l: o9,:' ,;sf ,j:- t,/ 6 GU:Q;^Qi,*QnQin=.
: t1.,'J1.14*!.r-r!r c./_l.1r S:\-K_V s \?-\.-0 d.rL. tlea -ar-*;l ost;. ..1 U -e
Q;^IQi;*QijlQj^=.
: u's_,S! n- a::f rJ31s rJ.,t e 6.:*,1 C*r,J G-:y', l*J rl.rL, grl
gii =-Q/ (\A-\.-A)
tj q-* L. 6s jz,rllry ,gf ls +5c..*,1 ,1r- Al a.r cl:.,l .*;l - cl.;U:.. A4 rf c^4;- O, t
: c*,1
a)k=-*rrroo (\q-\.-0)
frr uJ(* c>U:6 s cE(y :6 d,4e
:r1S cl! t;Zl: ,f_€*. O;;o r.,.1*,1 0i,.5
Qi = -erjt&* (Y" - \. -0)
oQ r.llyr, g-a 6,lat lls r>-e- Olp +,.lsap 6s7;l.ry t,.-L f..-" .<ifli: .r.1.g,r 1.:
1gtace-fcry.rl-- e,r-,1*,1 djLJr- :l -3t i:tr;-,|-l: .(Jr.":t b ff-O iJ:..-y :r5
t+.r-t:r .r-1l'1 $lJt jl* ,.1-1, * g*+ o-r--i.,gfr ..r5-; c.r.; ,r<-1":,." ll3lc,
e-* J<-,r"; 7t : .,,l:r56ta )t-V J rJ- ,,..q "ree t 6r ,tt l;;-,1 .ri-r.r
.J.i J^tJt
: p,..;lt r-rA- \ " -A d.rL,
ui* =h,+Q,,p,,* -ojl+ eipj (..,Jly\-\.-d)= Q;Pi + Qi,*^+ h;-Q;iai (VY\ - \. -O)
= QiPi + Q;^x,* * c; (:;Y\ - \. -O)
..:*,lori os[i:,*l-iJlf \-\ o-0;t..;y\-\ o-0 aJ-:L, ,FydlJ \0-\ "-0 d.:t--;ld
..:..,1 cJ-i c.:tiil ,-rY\-\.-A 1./Y\-\ o-0 aJlt - jl rl.fr,: ,SlJ.ci - Ai - Arioi jl S
u.J e;;,,fii;: \>.ef .lt-rlrf;.:--1, ?t-." 0t<rl t r, rfS; #-l -1, ,t-eJ-, F_y
or,+ dr,+ ;-'." b 's.'rflf-^ J Js-re
W =QiW+Qii* (TY-\"-o):6+5qi
I&c"
#=6i^0u; 0u; e" Aui A--r,
== a -'.., Ieu(*f _ a*)l&^' a,, fu,*+ e" &**
eb "Xlll c.rl;ul .i)r;\:..cr \x*+ gl; Vn-\.-0 U.:L, ,F )lg-rl-r(-t+ U_F 6
lJ.o.r,r"c {+J YY-\ .-0 aJ.rLr.1.: kJs ,glK-V c...,at-- olr^r r.,
&tt,*
# - g\i* * eqe*,p1,, = ei^ *eije*,IDj" + Wj"l (yf _ \" _O)
O2'JJ'F agly a:-921 <Su,,Lrs" .*K" frr
.:.-,1 ;r5^r 6'rlu4jf'a t cSf;r.-t3.-A C.lt-S S J<-,r.* aJ ?; ,.Wjn tDir S;-F u..qJ 1.l:l - 0tti:" J d.;t-c.. 6la.:^*i 4a tL,cf Ut-rF
^+dU:i"'* -Di^* *Wi^*ffi,n
(YY-\.-0)J
(Y0- \. -0)
L.lssjl .J::-..^^ '-r.i,1t' t 4e ,*4 ?*:1 y u'.ltj J J[5.:, # i; S
6c.*,1 Uls \A-\ o-0 J Yf-\.-0P,i* = QnQi,D*,,
,1r,i* = Q,,r + QaQi,W*,
-9 L-a1lrr-, +-5 c-,1 Jtbt )JJ', lJ) pi g ,rtt UU:t o.8 u,tr*a hll, 0r.rrl ,t- q q
,rit_ JUJI ,,l5,StnJy t e\lt p-r, !.1,," ,SLn-r.rib
Iti* = Qi^U*, uri* = Qi^Qinu^,,
,.iss -'. r-l.lt..S J.?)r'.c"-,+ .its C/l 6J:r.rrr Ot:i t" q,Ji etJ lrl
..f Ll 1l;.., 6:-F ,--u :h 6lrl Ot J,orLi eo$L .,ili ^"Jl L" ..rit-,,f Jta;t c;jo:"g,,',
s ,F -11.:y JLLjI .c.*,1 :& f.^,o.oJ:,6$gf cJf \:"'.t uf.J JElt 6'e O'.{;l 6lqp yl ..r:-; )! ,, oJ,i c.:l.r gf i,,5+t^+, 6Lac45,r:-1, q ui ,56^i)?
6b1l:y dP t L^;* 6 c.-"1 'S ,-?-f .g!,_f c-,1 3; ,5{ d r:; lbrr V.U}.{t_ JUsly, ,fS.: u. ,-/-ru
ti = Qiti
,,rtdt^+" G3tr -l * I AK ! tc*,| CL-, -r-tj J. Ji )lJi' F 6t+i;tdr:!.J -J-c f-? 'ri.le I 6f c.^; .r!lr!-, iL" Jt.ist ,Fe c-,1 ;j)
:syi l;-J, j tj os-n 1 c*,1 O<^- 6:V jii;r C!;l .,._Ulr!, clLJl .i_ .:p
dXndap=I*MxdQ,
fr6
dalr+ =da+n1r+ :J* ffi*- (YP-\.-0)
:*i- Jt"j 2|s1np+ j .:*,1 5a-6rt *l C*,da+ cqU,l r*. eU 6trc.1356g:V
Oat .-fij.1: .JjloJ.i 0k l-L, g--+ cJ.-:leJ .5.2.> dt-' .,1) a,..o -l .- ,,*l C!-,
op-rr.(-r:.* bf "-o 4l:..-).:..,1/+ =J u.';lr,.,'.trtb ilt:;il:jJ,3c-!i.;; ?*:1 .1: r5.1y' oL*,t c.:*,| JS.-.i -,:itJ .lt-rtj ,-f -S*, * +fs&p'
: p*-rb ,5lorsj; i-tsti ,,1.lt5! .-\-i.rt+bl :& f^.? 'Ii.,eeJ
(Yv-\"-0)ilQ ilQ &.
----M.p' Aj M,p
.rr cri ?t' il*-l u.xp 6,ly.0i J- J ?A- \ " -d aJrL. ./LlJ eJrL. Jr-l il^- U-y'+lrl-.- il-+ U-A +-f-tar,r-. Ol-:i q::i tt e$rpll--s ..:*,1 ol-i t;jL Q*
oxx ilQ ^ : a't.lrQ s.:*'l Q;r
's|*.o Yv- \ ' -o
T: *, 'o'clX1, dcj t
(F**)-' -- Fjx'e*t (YA- \" -0)
:r-f,_/ t+r! Y2- \ . -0 rJrL, rr ,}i .-iflrt, ;l eh ,J.rli<-F q
dAp+ = do*no* = Qr/ai = Q6(da)n1 (yq - \ " - O)
*! 1+l 0rl, rtj e g;1.,.., -l !! aJ:L, jl .)Jk d,ifrt*? J ,j*s:O$Lr Y Oti q
tc*,| c..l 44..a.:.-t..r *fti.ji 31 3(da+)t : (da)Y $r,-./
da+ = da (\""-\.-0)
O:-J.{r. aSly a;-9a l6Lb 4^" *K. rw
:6r:*,1 (\ Y"-\ o-0 J Yq-\ "-0 hllr;l o.:li-,| b
nk* = Q*i (ft - \" -0)
g..,7 o;-xuqtgtt 6f c.'.i u.:; iSt- JUst .Jto.iJ-i, 6"e O.# eo$L .,iq l1t|
t-"q ;-u.r4; gt 6f ?*.t4;: ..:*,Lr-c JUJI d/-l g;lt- dlr g-Ls ;L; ..:-,1 :&:\:-ri Jli € -r) os-* \lst;* ,.l:; ;l.ry *-J-.o
ti* : o;i*nj* = o;j* Qj{t* (Yr-\.-0)
: \tFd rt:4 ;;s .2ltL611.rb ,i-f JESi ..:*,1 cJ-i c:Lii-,lf t-\ o-0 iJrL" 6
ti* =Qitj=Q;oflir, (Yr- \" -0)
;l.ry 0.:l.r ;lj rg,j-.-.c*"1 oJ-i ostii-,I ';-f '* J_ft cYY-\.-0 eJ:t-- c.l.r?.^.6 6
:q rFd ,4;'' ot,:'uqtt'f :& fr €f ?-nt41: r.l:;
@,'*i go - Q;o1)nk = o
?t;.-u-!L -fu 9911. *, J"-L: jJlJ d_91: r])\,c7 d/-tJti ec--,| -rlttW -f ell66_7- c".'..i ,,xJ 4it- JL;lt ,Sbrilr,- r5l: -,.-j.r.r!, Q* f .riL+q .r>\^7
o,i* = Qa^Q;no,*, (Yr-\"-0)
JJ-*- t*y u,lJ u-,Ll g.,u, c:3J.u el*l e-t' rlg.,Y9"y oJ'?.r^ ,r:.-=-,\.;s
.J*u l-tr ,F at.c.*.| cJ.i orLii-,| ,:l aJ,;-s.:-,i cJ-t.JJ.;i,-+-ltl Js7o tt
:6 ,1-, C*l 1 r,.t,.lti oJ.aLL
,r,* = QaQino*, + Qi*Qno^,, * QaQlrb*, (fO - \. - O)
rrY U_rl*,>A:6tc*(p:glA
q .:*,1 o-g osls 6tJj YO- \ . -A iJrt', y 6 '-FLA
,Ji.:t ii tn r,'.:*'l f'6r 4+rJ OJ-l
crlb;-.i;i .tyLt' orth*,l ,t1L, db #:..- ,5b;;"'-"'; I t- g-t, Cti OlP
C # elJ 'b .4 ..1b r;l7e .;a+Ur:::ij rhi.r.rr F Ct;t ,rK-' i,t* .styl-r "rJ'ayO-\ "-0 !-:l-11 il o.:[.ei.l tf .lj"f r:4,r yWA t.ltITJJJ*,I';l o.:ift."| ! ,frl-l ,;
C.ltf .r)\^7 fJQii el1. ,F : \ 0- \ " -0
Q,r: W,i*Qio - Q;^W-p (Y2-\.-0)
,7-.,i,; \:r,r q YO- \ " -0 d:L.. I 6it31t+ jl "r",
6
ori* = Wrq* Qr^o^rrQi, * Qno^nQEWrq*
* Qi^fb^, - Wqo^u - W^roElQi,
(Yv- \. -0)
2s t.;;|u. L*r * f...r oriu{j gt 6f q 6,;b+J i^r 0:l.r .llri
:tj,J.,-14+:J fT-\ '-0 iJ:L, jl o.rl-i:-'l S ;SSll.j *
o,j* -W,r*orf -o;r+W1r* =Qi^Iir^n-W^{w-o^qWrqlQin (rn - \ " - 0)
F tr-i-r.., 1l-, t- +f c'*'t atg-A^n - W{^, - W^qos,, +56t:-'l €ls.J56".:r-l-ta (Iaumann stress rate) 6L9q
o,i' =o;i -Wirori-oUWi, (fq- \" -O)
tr.rsl, s-t+s rS./ o.a'b b ,lrl ,t+ b;6 e-1-t rJr;' r: 66at 'J.,--r-
:.:-,1 J-s os-r u (Green - Naghdi,stress rate) 6$; - U-.5 c;;;
orj' : ori * Arroui * o;r{21, (Y. - \. -0)
..:*,1 o-tJ l..oy; \ O- \ . -6 nJ.:L, -ss(2;, 6
ch- t'*-o agly auga 66L:", <*l& rrA
:AJ-+U#J ih,-'19 I abl 6llal',l'u 1. oLJ Jl^,ot 6bqPtJ^. I I - A
f t, Clr ,[ii :l.r l-r
"rLt^.:"-r3J>r,o .fuQr;*r g-1, r,l;iE 6-r-l o! r 1-0 ,.xn .,.r
F '=le l"+ .JJ,-5#e t 6r Q ea.g:1" ./r,rb orF -r*; ..r,o.:.r,o )U3 ,s-,1-,1r:;50{*i- sf .Fl rt 6 :-b OLa ,3r1, yt, Cb ,y;*,gtr;i:, ; ,-r_b*_r_eJ>.,c.t-r.i eJ.,' 4".. .$t<".l.r orLi:-,| 1y 6[ac. _-Jjl .S-l!-f tLit \ . -0 ,.:; -,1.r ._t.i
It.+l g,[ Cl-1, ,-;*-L.;l s ,;rstr., ,ii-, crrltr .ri dbl :& p*-; o-r.ilp^f c.SfJt^.t 6[ac*.r3Ja.,o I l-r ,f -h ']_l .p)bt,-t':& p-.q o.r-ilf$: clf .:*;
r*5"r -3JL'- coJ-i, o.rlr Y-1-A drt ..-2r 6,r-Ll $tu6toai:"y oJ-i
Jli,f -l J$l.',r-'g/! Jrt-t rr-;J.,.rh-el ,.-L g--a o-r.-ilfry clf .:*;: .-:*iJi
(\-\\-0)
6f \:-ii e :X at, 6.>-,1 g*;lJ CI-A-O b -.il11-l-0 !fl-1, .21 os9- c/-l-rsiJ)[.-' ft, Clr ;$Lit: .gtoc-J.rL,-r
"/q J*J.t cl-9! .Ia r..+ .s,,reer;J'lt:& r---7;-r-14;gAf cSf ;t*iJcU,* gb,.ri-l-r tl tii ,l**".,gl,or.r:, -:r
y-l-6tl L-.ly drl jl ,?t) rit-"r JUst rj J.(.i I Clj at 'Fr o+" g-*r .-rjy-!,.r
:(-ril'r.i .2lf; ,>3V;. ""SJ*
q t- J(,; 0t" q \ o_0 ur,.J
u* =u, ry* =q, 0+ =0
F^* = Q,fi"F^* =a,fi"+AqQ*FuLi^* = QdyQ*,, * Q;r,
o,i* = QuP,rQri
Qi* = Qipi
gi* = Qijgj
(qJlT-\\-0)(vY-\\-0)(VY-\\-0)(crf-\\-0)(drY-\\-0)(CY-\\-0)
..:*,1 oJ-i r-cr-rr; \ 0- \ " -0 iJsLr.lr ;l s:-,1 =:l
- O.lk:, ...:-U eu dts o.t-ro t :& ,-*7 ;"r;;g^r c5r u; ft Lij. J ft st;;-..#Ll !l-r.,
:-t f t- Jl;:;t
(.-"Jlf-\\-0),* (q* , Fu*) = u(e , Fu)
rrt
e* Qt* , F^*)
o,j* (rt* , F,**)
qi* Qt* , F** , gk* , Lr*)
= 0(n , Fu)
= Qaga?t , Fr,'e)Qu
= Qirrrq*Qt , F*.q., gk, LH)
(-.Y-\\-0)(Vr- \\ -0)(.)f-\\-0)
1 ?y- \ \ -o b -illr- \ \ -o lr).:L, Jr,*,gtar;. .d d:y .::Jar'o ;;,7^l ,:f
-l O^lf1 .-l ,,If LiJt- 5 o-t-i o.rl: oLi.j J<-, ,+ .rt-rf s (sslF 4-. r'J U-) dJtl
dli *J.,;X^?;.: ,*,Ll dly ;t 6le-e y--.:-'t ;ri' '*'-la; -l: '(\12f eJ;41
ft, Cl-r, ,s: ej t+i q 1-O ,:,;- -rr rf,J::-,)l -rl-9*,JE,,JlJ.-t3q,r.-ltit g-t,
o;i 0 u 1l u-.1:
-J+f .:*,t cre--rr. ,41 tlzs [ .-r:ulls 0i s4 oii alt *J.fL"
.Jr*,r iQi s,$,-/.r3J.*.. \ $tu 6La,*;. f- \ \ -0 iJsL, ':tK7 d:'-l;l dldJ ., Olrr q
5v; r.e s-' ,p 't:-;l: jl L*b oi ":t e-b, cb ,fi '-P-f
6 rs(ilrj, J' Jt"t qry -r.olg r7Y-\ \-O nJ:L, 4;La.:u-:-9-t-, ,ft U-l\.J^t'U.Lii.tef .lt-'F
orj*(p*,uk*,D**,wr*r*) = Qipop(P,uk,D^n,w,*t)Qu (f - \\ -0)
o-r..X q;:; W^, JDn, 4 j t .l--lt - d;U;'o J o;U:.. ,5t4.:-J u Jcf ,lt-'f 6
YO-\ o-0 J Yf-\.-0 cr/Y\-\ o-0 g!:L-. -l Y\-0 '.Jb etS;l o:li,:-,| U .c*'l
:6g*b-"rr:r
6ij(P,ut * c*, Q^F*Q*, Q^pWpqQE, +A*) = Qioeret@,uk, Dmn,Wr*r)Qti
:& fril ,:=r, &*,,.,q :l-,rt+U.Fr-f ,3l,.o-\\-oil'll-l:jlS,-?f )JLbC,/-l {r..:-ri, ,-r[iJl f-\\-0d.rL,./Ll ,ffr:otsg;lr1 ..:*'l O<^'
: p,;l: O-\ \-0 .J-:L.. s 6f yl;l o.rU:-,l\.QA = o ljJ sQ;1 = 5,i S
6rj(p,ut* ck, D^,r,W*n) = 6ij(P,uk, Drr*,W"*) (P - t\ -0)
O..-.t*ho 6(, at*g,1t 66 L*cu *19 lfo
q .,t{Ji .iLi au.c1, jsr1,c.Jt-- itl-1, .,.:*,1 o-r-i orti;-.,| \ O- \ " -0 rJ.rL, *3_l a.t 4?} l.:
d:L,) .:*,1 ,-J-a f...- oriudgr cSr f,stfat ,-f E GU: .;j". 6 : j;u. ity
6_,_l e r.iL !U:*, ).&" Ot.l} j ** ll.ry 61.;1.: F Cb ^?JU" .(++._l l, n_ t "
_0
.:sslcl r7s a'.\?-\ \ -O iJrt -. LF; ot5,l ,g.,1L;l c,Jl--:Jjt .'it 4l *,c.ry.i jj- Jr_q ,JJ 61L
"l;..,.r,gl.oJ*a. ,- *J st ,:,-e-,d/lJti
oii=6;i(p,D^n,W^n) (v-\\-0)
69 cd;.gy_j.,[r;; L; sV-\ \-0 iJ:L, ori Cy.r, g-q Cb.rLlf o.r!1,
:.rf cl! t) ot_f y ,;l j,_S l) ,)p,,*+ i.r-l,ra;ro,
6;j(P, Q^F*Q*, Q^pwpqQ* + Q^,) = Qiptpr(p, D*, w,*)etj (A - \ \ - o)
c'Jt'' c.*l -rr sQii :5r,6lt,Frj, Ora^, .rli'.i;r1+tr 6cir L^,Slr d:p_-rls A- \ \ -0 iJst , 2: L,.ly ,yl A:l';A-* ! .f.+f"f; t,rj,q {l;1*.
6ti(p, D*,W^, * Q^) = 6ij(p, D*,W*) (q _ \\ _o)
\i*; +5c*,1 y Al 'ff -* Sll..:*,1 ,F Cl4U*i; L;y 6rrt OLr.rXU eU:.,- *;A l_ti .-r .lt-rF.*-rl_ c,r[-
.-i .r;L_ uiAl5 Fy st g-t, Cb 6la ,J,.a ,* it:4gf c,Sf ,;-rKtt
,c.Jt-- d/f-rs .*+t C.flf 6-;+ .:Lo>\!l
6ii(p, Q^Fora*) = Qie6ot@, D,*)Qri (\" - \\ -o)
.r;b )lJi. \aU g-F-Q os-r*;r nJ.rl-, y I j c.*,1 ,?ri..r-.l-u. ,r-b e,i ,6 l ,t
6L;;J-p,- J-3 JL- ."uU D^n )l Q,tj:l-l "*b
J,_q .rJ cc*,| :l;t i dT
:s/,! 'J!l ,ts g-t, ot-1.t !D^n;l g9.ir".(*,-l is1-t,s-t--.li
frt U-rl* <>U.:b" g (*lf : a rJ,.s
oii = -p(p)AA + 1@)Db,l;1 + \pQt)D;1 (\\ - \\ -0)
1o o:,_j-6*,_ r ?-_t' -S-U. r ,)q 1 u-ly tt sT p 6
,gl-'l ,>Y:l-u ll - A
1 $t- l-,-,,i Je, 0'-l-r, ii 50i jl kV,iU:. .r)rL, -l .;tL JrL; Otl9i
A:lS:\t- -lr* n.eV rl;, g-t, 15:l, 6rt Ll .rlb:y.ltSa:r-i*,- 6t'o19". u,LiA r€,-* ,tf \ o-Ll <>!!.:(*;t*
"f-l o)JJi g-'! ,JlJ ..-'l 'J.Ji oi< Gjf
F;t .-66" oat:b rltiJ;b.,,BL./LlJ ls,fV :ly, ";:lf
-l ,<=;tt otlf.,Jfut^:ii" 0hif ..r)rl--
"rl !l:-l 6c.*,1 r.li-,1-t,r-'t-,l '.r)rL' c/l rlr:H 4i- 'ue\-t
Jt..,- 0J*:, C-b J-t56!1 .:)rt -. Ji-<5 J rArL./ ef t t"t G-lf -r,r5.;t<"
Ji,4.l :ly {/.fLl .:)rL.. 'fr* -b )l.tjL"f elt l, &3-,6La\*.. 'fut<'
<, a.5.r:-7 * .t:S,-l -L-f t-r€ o),1 tE & rl;*;L$.2 6ly. JJ t.ll-r"q 5L
.S!.rrf,tS{yl *r,Li.2J o.:to { rlrb oL.i:j c.+ti 6c^ii aJ:L. d r1l"r,0t<.1
ot*- rf jl ,.j31;,:- y"tSdl^;Li.; ,S:lS;\ C-b.rlr+ r:e.i rl-9*;l .rL- tl;a
Jre 4,ti lt4- c./-l -r: r,,..rl; "i"-lil
jl ,,r,,U'5.-\ - .1:-,)l ft, Ji'-'.f.f"rq oav slyo l. Cil oav tUt'J+t^j gtJ;rq .-/Ll .:)':L' F.,' s!
otS.,t ,3!.1 eV.lr.rllo-r;tlQ "rLl .:[.ry ',J*3 d/l .-#,5Lar.:;1r
t-,i Oi ! .rl;.g. 4 ,;-r,_/ )tbl 6 u-,t-,l dt+lJ, g* f-r^:-,| .:tJra 01L.,1: c->u
,)ij,-r>\o 3.:-,1 ,J^i Ajbl t+itr I.t-,".+.lb : e-h 4i1.0:.L, (tf fJr.rll$ rJ?J r:iq
,r-: dbl g.,tU, ot-J+Jlr r--- € gy 6.:'l ,*f if ,g)J ts-f ,t'V.Jt- -t+
o$ 4.i C/rt* g.rU-. ,t Cb..#Ll gt-acJ-r. j'l,yJ:l-r*; I t-ei dJ" .J'{L.i.-/J? b
,Sa,->syo \ sy-t g-l .r;lr,-ft*,| .:irl-, J 6.15t''" u,"l-.":;l d+ yl tl:*'l
1l 6ry;L5 tsli;s c-r .-jL;*'. "./i*,1
.5kJJ,"6"ffu,rJJf 6t^J--; 6lr'.61'-ti-
.rf -r,olr; k nf gllr3..:-.rr1 -f tu-1+.,6tat9., .<Jt<"
I ,.:*,1 .J^i 4jLl JJ st isf "fLl
+J:L." J+ jl 13ln-a)\- r:*'i;
,Jt".,:,; o.-p\ ,k $*U( f*" 4,Str ,ir-{- ,i;.:i-:t'-. (.-Jl)
&-f 'Q
O,*,.8.*4 agly aUgaa <Slalza" *& rrr
6;i : CijUn eUn
dl1.srL/ OtJ L tr ,ylr 6.r:.-.o.("LYl .sl^c"rb Cam ol;r a.f
.t-S,g t_t t),rVJft \ - \ y - 0 nJ.:L, :fi:L,* $i,.2
JS6" 0t* b orL 6;-,!l ,-tr; 6.r::*,r ,.*-l-rpt -:I tl;r *5
# oJ:1*9, o$ (*)
t;i -- K;1^n D^n
(\-\Y-0)
(Y-\Y-0)
(\"- tY-0)
'f,e? o/:;6. dtJ .r^rL/
4i =1*6tjDw* \ trt*D;1 (f - \y-O)
(0-\Y-0)
Ot-ii lt ju-.'z oArKij*.sL"ctlj 3 bo.,lJ s: it u:;st;1.JiJ.r nf
J::.,^,6 ;$x,*2 *)* p* s|* di j.r +5
,5+\ ,-;;f - r:; aJrL- (,;)
deqP : Su il
'C*'l
,F ,9-a"A(&-e 6!#l*" ,9;lrl ,>J.rL, 1.r;tPlSi= t {Qlrtii GiJtt -\y-o)
oii = Y Keii (:?-A\Y-0)
trr U-JI* <>U.;b t <Xly : g J,as
JJrsgoq 0l-j r[3*jfir,Sl^.r-l'ty,l { Q } s{ P } dir:6N
{p}= X p,#l= o
{QlM
=X q,#l- o
l..allv-\Y-0)
(vv-\Y-0)
01.; b .**,) I.S*,-J ,flr (r:i-+ .:,tl Lytl Qi s Pi +-lr di r, 6ci? JJlSr-;.i o?i t';! -r-,!"f F JSuK le.i-11: O#.J:.A.:,r.c
lsu" oar'Jb\ ,F : ;l{l,Jt"g-t, ctfl-q 1el, 'J.:L".
ds..,
v" A -f ) ii dt' l..cJtn- \Y-0)
32- aat
0ottrl.., ft - t') i; dt' (VA-\Y-0)
t- +.Y.(pv):. (1 -\Y-o)
,k' <r: . Ul 95.,,- I oJl $': t.r)rL, (e,')
,u = !'.
', = J"
t (f, Lif e(y 'i#iq {, J{, L-'g oJ-'i o':lr Jl'r ualt; dl ir at
.-t-,lr.f (trt c a:r-9*ro 6tr-t 2-, 6Jti Zu g-)*.*,ti #1,-' .1r rJ)lL." .1.<; Atf
, u#.-.fLl .:i:L, di.lll q cJ.,;l: jti *J1> ,.rY.:L, .+f q 'L^, Ol"r., ..r!.:L.
OlJif g<, r:,,):L, , gsf osfu.i:-r-, el4tii 0lJ:rt .gr-.1l-r Ct =l 6-t hlti
C=5"f c*'r6i tr U'U : 6a;V 61l"rf:Lj 3r y' 1r
(V - f - 0 d.:t -),-fu-r,aJrL, (JJl)
(pv*),* = o
OJ-Ji\.4 agly a;-9a <Sl.aLz^" cS+K" rrr
(Y - f - 0 aJ:t--) c.ff .r).:t -. 1.-,,)
o1;;*Pbi=Pi; l{ V.o +pb:pv (\.-A-0)
(\ V - V - 0 a.lrt',),S3JlaJ.:L.. (i;)
pit-o,j Di-pr+Ti,i= " \pit- a:D -pr+ V.q: o (\\-A-O)
teyo->ca r,Jlo.-L,i 'rt.l6ji r *;;3;rlf CV 3 b; u^r,> cSb;X 6f:f &f flu SQ;,oii, vi, p'W-,Jg-,-- ossL.4 &tj J;:*aJ-:L- * bV Oj
.>1lr-aa;: clll-. t t 4tF ccrt..^.or-; '*-lf
J:r;J.(, .lr , o_9)t -, .-r-,!.f! Ilr" nj:l-. o:jL- cclt-,.-.a * ,Jjfq;-,JtJ.$JJ or.l3i,-/L*^., Jr_q f - T $,t)aJrt -, r-, .r-,Ll ajrl-, ui; 6 Layl;l .,-:*,1 jU di.,1r os r:-,raj 0l+a +JlL. g,
.g!.f c.JL- d:l-, _lr; (q.,1t oiE) u;_:lf c^,-l_r.n
-t^:L €:l JL.^-,r d;ls-- \!t J ,h-rs-- C jr; ,ri;t<" .gLaOt4a Jlo) grr--o ,-J-.-*r 4 .rLl njr['- F s ,sf rrlrL, q .lr^, ,_fun, eJrL.
JSL..- bCf_l .J.tu.- ,*"tt ls oij sv;, p Jga- o: rJlJ,r<Jt<",,(;L.r* rJ.rl--
.rf f,"lf g)J,;; ,j--lr 5J:j..^A lglorL ttus*t-Lgta-ti-.
et'''carlson, D. E. (1984), "Linear thermoelasticity," in s. Ftugge's Hnndbuch der physik,
Vol. II (edited by C. Truesdell), Springer-Verlag, pp.1iZ-US.Coleman, B. D. and Noll, W. (1963), "The thermodynamics of elastic materials with
heat conduction and visco6ity," Arch. Rntionat Mech. Anal., Vol. 13, pp.16z-lrg $1,Coleman, B. D. and Mizel, V.l. (1964), "Existence of caloric equations of state in
thermodynamics," l, Clrcm. Phys., Vol. 40, pp. 1116-115 (4).Green, A. E. and Naghdi, P. M., Qgn), "A note on invariance under superposed rigid
body motions," l. of Elasticity,Yol. g, pp. l-8.Malvem, ! , E. (1969),lntrcduction to the Mechanics of o Continuous Medium, Prentice.
Hall, lnc., Englewood Cliffs, NI.Naghdi, P. M. (1984), "The theory of shells and plates," in s. Ftugge's Handbuch der
Physik, Vol. II (edited by C.Truesdell), Springer-Verlag, pp. a?-}aan.Van Wyleh, G.f. and Sonntag, R. E. (1965\, Fundamentals of Classic Thermodymanics,
Wiley, lnc., New York.
tr6 Url* <>A!* s c*l* :6 d,os
lil-
.r-- ;*s-;, * *:;l_";i il ,n; ?,.,:::::: j: ::;, -'
d I e*i ni d.s = f (e*i + e*i vh* - e** vi,k) ni dsdt Jr:'"'-- Js
f.n...-. rJsl*-.li1.t Jful 4*;;.,1.:r.:*,1 c(,-:.t-c \ - Y - 0 oJ:t - ,P*ij...uelr;f -6
c.JL Jr-l 1) J.r.r rl[-.i; .-u:,r" u.:r-L; It V 6,ll;)
i,,i = * I" o, = Iu v;,; dv
.J+S 64J\ t'-; t9,1*Y - 6
eri* ar, = I (wi * wivy -wir;)
1l-i 6:L d;*L 45 -t;.( crtj o t - O +k-.- 4?+ ;/j'.^a .l 6..1L; ;,-l ;t orLn;,t L 3
..t9r rlLi.r ,P- o VS Jt'-,U .j..aj ! ,:*,t l\ ,_ts-,<
* I, wiltids =+ J, tii*ahin;ds
glt*,' 0*+^^ t n-F.flj.ry-, a*2, )l o.:Li:-l ti f - O
dw'
# = i t1o akj - e ij* ero,q @^ vq);
+5 -r*.o: cltJ.;
* I"w;dv= l, Gii*at **iv;-w;v)n1 ds
O:--u.{r* agly a:-y1 6l^ L?n" c&Jl$ fw
;l c.*,1 .rtt* V a:,-J J&l o:L r:;-ra 6rL,;;, n5+1, cll-l; d - O
* I" w;dv = J, ( + eiir,a* r'iv) n1 ds
.r.:*rl o.t-! a3l2l t-; elte\ t>e)-t blJ,, , - 6
vt=
vY=
vr:
ax, - bx,
bx, - ax,
.t.;5 O.** .J.:i.-A Ll*.rb C )b , A tsl st ,,5
g ,-#"r b{V Jfil.* aJrL U-i (-lr)
f r:-,1 "i--Ftil .Sr t-i tv)a = o $ a; <:> p = p. e-rat 6,?s &(.jJt) ; g",q
.r:*,1 o.t-! orl:y-j &b L-i "1.ll
0F JL.1, ooV ';-'J- 14..',,.c(,- 6\V - 6
rij = -P. 6ii
crlgcJL..rl .Stf nJ-r+r,)t-!,r*-,.ry.J:[..-.;l o:li;,[ !.,:-,1 c{U.i-P..li;rn5
.r-5.Jt* t) ctlr-*. i;lig lJ ,i!'
oiDi:p.blp
Ut -:J.-t ;t .c.-,1 c"1U4k,sTp^54,&-hrt\xi = (\ + t lk) Xi cl,- A-A
+SO..l, k St, p. -tl ,*E o4,eet.ltP ,t -- o di*t1.: p = p.4j L-,- Jlfi
o= P'H' (k+tT :*q
Ll)Jttt - I \ - Y aJ.:L 1) o-LJ clrl orp,? - Y - 0 .l : \. - Y - 0 .1).rL :J; t, f - 0
.rg5
frv url* oA:6 s 16,,;les'a J-s
.S;L;11 orl-i,:*,i q l. -0e;ft hhj = I (wi * wi2; -wiv;)
ni ql: .il-5 u<l;i aJrL cr*# J
d (911\ _e;j*a*j*rwjvi.idt \p ) tp
6:L.;.:,.11 o:Uil t, t+i;f .l 150[1 fi *lr .r):l-- I I - O
.futvi=i*v1v;;
n5r*o, r.ll-i; . 'fu* aJsL.1
a (?r,) = @ii - p v; v); + pb;
at
Ooii = -pOiijl urt{.:tl*.F.SL"iJ-rft +5,j,.if, l-t6r.rlsL;t ,rl(-i tr-0
oo"7-'!o':::'""'":;
.fLl aJrL 6!l:,g:L a.l,r^,, tr." 6,- lf - 0
oii=dyDw+lBD;1
U.l S\n \reJ-' olii,i ' -J
lt 6f .rirL .J::"".-r L^*.rb f s a Lri 1: d ,.:-,1
.15;",*r'o:L
pii = pbi * @ + f) vi,ii + fvi;i : *q
oiuf JLrl cl;- JLtr Ai-!J^, t^" c(,- -ttmi f 0"1-3,5LrclL,4511.5 uZY lf - 6
.rsf o;lll ;pl Ll c,*,lJ+'r f - f - O oJ.rL "f5.i
1r )-* 6f .li.rL. 6.r*.o: ,lt-L:
O:-J,{fo ,gly a;-9,g3 ObL't^" cS*8" rrA
clyfl ,51+,--91;
e;11, oi* * m; = "
.r*iL irrt-Ai" $ljr# -&.t '* r.r-b +5.sLr,r,3 c*,1
v; = e;j{Dfr L-; JJI;O us-t,it+," . tr." Jf qlb:: JL.r: :^Lo g--t 4 6lt tO - O
t:SJ--'o;l"ul ;pl c.:r.";s g,-l 6,1r. Jtrr crL:.; .c*,1 6la,-11.1e>-e ,- jtt@j d:j*: rltl
;l c.*.1 etV glq-rlj
M; = 1ai IA)
ff ,?-p-l Jl^'Iij J tr J,r".h-' gb.l.,*,r.L"t lr* Jf bl.* JSMi.ri;, ^5
.c.*,1 ,t.r-l-L' t*,r; r.L-i ,-t-r,'..;
.q5 ;+ JJ ,l-f -J.o,i
Di r;' cll3; ,glr; u!13.2 l, - 6
iF-S - \ J.rt dir Jr*.;b (-iJl)
i:-t; - \ g_,, .',::; rJ*,ti (./)
oi Di : ps.anbnu I Yp. (v) oi Di = p iuPu I p. (i)t) : *qq \ - v - o aJrL .r+- 63jl JlfrJl . I r*" Jr 4*g.-a 4 .rb;r gl; .ror ot*x; lv - d
c*- .*L-cf u:.ot5 *L f* c{at--: ,s oJ,i 4jbt J"fu
K : * ot; ot1 I;i
..f-,tf .f \ O - O +k--., 1: ol-i ,--e.,--1,.., ,-/;-l ,yE I;i bi y oS
L-'g v ,*-- or,'' (Juil :13;.r:..;-t dll c,l*ii 1r,lt* ,slt,Jt.i- ql, dti rA- d
..t-,lf .f y; al.lt-,
h= IuPii,iix*x*-x;x)dV
U = J,, Pbi vi o" - I ,o,i ',i dv + J .',
t,6) ds
rrt
.gr aJlzl 1iJo glrJlFl fi )l ft.r5.r' 6,la 6,t-.-a .e
gtlf ,;,-l.t-o oili;t 3 r:*,t oii : - p. 6ii f|n,-F rri1, n54ir.9. L-.." cf{ lq - O
,$;l d.rLa:-j*,, L{.- J'-l,Sll +^, tlU;.r,-rf,t -h tt JSu" c.i+i Qi = - KT,i
.r:-.il: r.l/ lt r-t )5.:,
pi=-P.v;,i*pr*KT,;;
.r):t --;l .ulr;,r- ,53';l JrU.; d.r9.: rll*i.; of utf -P1: .,5;15" ,53-il & ftl" - a
jV; o,os-t.lr T - T - 0 alrl.-. .l^+.,-! 6)Jt ./J- JS' 1., .* .r-i g-,-q tSf.n-i,g s-,!,5jr;l d:tr.. V n'- 6y ^J^q t el*a O.p,ee:.o.1,. eJ:l.,o i e$rstfat
z-'l ,-; o.t_te\.+-J Jfu ,4 45 # ,-*t-t
ip F. v) + tr (a. D) - pb.v * div (a . v) = .
,/Ll drt ,6!b lu*,3*t -b,-., ci ftf t - a
o;i = - p 6ii + fr4 + pDikDki
.r.,rr crl-l; (Dii = .) . -r-!lry-*Ut'F o:Lftl er:.^..-r lr.rtJf JP r ac1i1.r *5$L
oi; - -Y p - YBIID
.rr-,1 ,|if.,o1, Lt tr.E g.l: UL-! IIp .li 2r *5
-q.o:crl-ij.r$Uy'rft-^r.f$-)l s!)t,Slt Y -A-0 alrt -. ;lff - A
oii:ft)r+tF)eii
O!{.rir. agly auga; cS/a/,zo" c*K. IAo
oij
oi
.r-:*,1 o.t-i a3l2l tl os-t'\..r-Ll .JrL ;; o;lJ .i- 6lr, ff - 0
- p6ij + tij
- p6;i + ),*6ij Dw * I lt* Dij (Y-rt-0d:t -qr-#Lft)
..l"-.l.li r:-,! ! y-; .J:L , c>{s-lr):t .o 2: u-l-,|oJrL i/-l ._i--6.q tl
pii = pbi-p,i + 0* + p*) v6i * lr* viji
..*l'.t d.rl.-.- c(,- Lr r+J :Jr-; lt : ? - A - 0 .1 jJl ? - x- 0 .r):l-.-. fl - O
.tt: ,FV t) o1ye\ ry:.. ,-fu-,YlJS*,-J
oij = dt { R } ew t { S } e;i
jl .u;.rL* .",,'rq { S } r { R } .U; j4 ebtilrll rril: d
{R}=W r ts}=+#? - X - 0 .,.,,Ll .:YrL c.1lSL.:.-;.(;-ylJS--J 4;-rr.' le.-., d,- €C oblf| - O
u5J jl Jel- 6tF,J:5ts1 r.:"ru,' o,rl- 6q jl .i-.,1- 4, 6,- .-Utr.f
o.Jt t-r .*"+t ,Jt--j jl ,.fib f (t) l,:.t-.i 4 o. ,.ti 2, ^J r.:-,1 o, t = o. f (t)
)le.ly Jt-te\l.left-l€rr,8rt.Oyt =OtT =Orr =Oyr =O11 = o *Sn-l
*;s C".n-l (t) : {e} , {p}
:tuL
etr G)= l# o. f (t), eYr = rrr = # o. f (t)
;l cf;b Jrr- .r-tt di:-)trS.*,-, Lp,^ J f V ?- \ y-O s-,Ll crY:L 45 +l5 obf -t O-A
rdt
f(t) : c{Uo. ;-r-\ -h .tt r:11:;lriorr = oflt) ,StF dJ r>"i +5! ,*,q ;,-l
qf.lsrrJ€rrJtrr orr =orr =olr =orr =ory = .l;Srsei.r:-,1 clL;;l ,r^,b
.r50..r,/(t) r {Q} r {P} jt ".*rb
ott'
rKtP) + {Q}:*q
e' '(f)
=
ett(f) =
ffiof(t)
g-eLib.;t.lj: - .fy-.r).SoJ.:t'. iti! "5l5
,F ,-e2* 1::ljl ,fi-il'L-f ;l or[iril L-f ,-A
'.r -i C.,,-p t-) otre \ ostfr5:b -r )-fu-:.r":;;r;.:;I -fv-o
,1, = 0(0, S*, Fu , i'u)
rK{P} + YtO}err(r)=fu of(t)
q = ry(O, 8*, Fu , i'u)
6;i = 6,i(0, 8r, Fu , it^)
Qi = 4i@,8*, Fu , i'u)
.+5Cbl b .,-Lt 6-! Cly J t't-- 61-oc,"-.:j.t-,
&i!(0, g*, Fka, Fkn) _ .ogi
Arlrp , gk, Fw, FkB)
Tw(0, F*n)
q(o,Fw)= --
_ F tT(o, Frn)ori =poirnT
+ 4i(o, Er,, FkB, o)g; ' '
(._ajl
(v
(v
(.r
(.i
OnJ+. agl21 a;-97i 15tbL;ov c*lS rAr
.*-r$,, -h t tt t) 4|'";t ht.r.r -f A-A
u -- tt(C6,r)
0 = 01c6,qy
oii = 6;i(Crqn,rt)
Q; : 1i(Cen,Q,8r)
..t*,or r,rt.l; fb-1, - .r-r-j)\S CA:1E o:ti:-,| t cf*:-,)l i:L cL,- s5lr1
- A{t0-_ery
u = it(C,ca,7)
o;i=i*^(h.&)r,
4gi2 "
aJ:l--.r, :L f-€ .S"#f"l- ,S)--jl n-',F-,-l^+-,6rt ;-.t eLu;f o:|.o, -l U -f t-d.r.,or rrl-.1 lt fi \l,lt o-r,i ,.-.,A- \ . -0
(..qJl&.,4
E=Q*Fin* = QUFiro (v
:r:*,1 y; ctlf q .--l-r |'-q o.r-!rrj;f-rr cJ.r- ,:.^; i,,or5U JU:;t *j +1, Jt-:j -f . -O
J+ =J
p* = p n5+frclb g.r.Sl+ ",yti e,p.rrYo-6 ll.*, i*:j1l o:tp-*,1 L-f l-d
-rJ et:* n-, .-Lr '+-.+
o-r-!,r4;ga-r c.fr g'd uJ ,Jt".iJ.! ,ltrrFni -r+, cJL:i -rr-o:-tiL-,r" JLiil
rar U_rl*,iU:6 I cy(st : g d,.s
oo,,* aomn
q=Qn *"
:.r.-o: r.ll-.iJ lt fi \lss -! ,,* i-"r*f^" '-Sf ..+.rLd 1l o':Lir'l U -ff-0
lr * -7VAB - ,-AB
u.an* -- IrAB
R^* = a,fin
(.rJt
(v
(v
3,i* = QuF*Qrl (.>
.r5o+.t,.or-i.-c,-rly-; +!rt , ett' r.6 6f q:^l- f-"? r.Ll],-) t)y y-fl-6
x(X, r) = aooj)\r t
*=er(rX**-u(r)ld.r*r: rll-5
v* : n(t) + ro(r) x [x+ - a(r)l
..:-,1 ,r-..-t di.lyl! ocs-'@ 3
}ei*Yt 2
,f-.5'Sjil "Sg Ji6 ' a;-"'-;"'Yl I - 2
-pu-b c:tr.1: ;:;: ,s r-#-l (\ ): t-r,l ,i*i b|' 3s U. 'J+-'!lr$J
:2> au+Jr cJl- q J,5c-ity c'..at'-.5!l'r osL 6 6'l<$(Y) 'l ''r'i! ''*f :l t*'
o5L rri €frf,1Jtt.*-,)l o.rl. &-rLirJl .!3U c.ils crllj qgLaii 0J-'t^l ! b
\ ,iLJ.:*,t ir<^-.6+*,)l .rhj-l 6 €rf AJ"ai 6J.'?'s '..:*'l '9-"'t!l 2L
- ;,l-;o.rt*,.gLr,r:--, L-,y r/Jl t-h.lttlJtiJ O',l \ - t J# 'J^:t ,FE'tu-,J;4t 6b:y;l5:&f ,gff .$r../ Ot-r.i 61by 11. 5 4;slS2! ,St^.=n'- U. 'L^'6 ,Ff
,i$-,il -rLi.r ,Jlr \l-r3.J::...^ gr1 J ,)i ,|i- .51.;L 'rl;* ,!'t'i 6 g-VlLt't-'
rt^:el ,frE J.r, d';='-+')l .:1.:ta:"i'"', ^t--b tr)t'- Al elt i t:*'l oJ.! o:ls Lra+'$
"r-'!.r '?l* elt V s
Js Js:,-rr r:Jil U-i '.t't';r 5v'')l l$-l ell ,f!l 'J:L' ' y rL- 1A
.a:.-,1
c = 9@) (\-\-2)
e;rsLa-r;-b {-rl )4 -,r e 3 r>l 0ru:' ,ss*E rl$' Cb -.<',- g 0i lr 6
;.r e,Slr,lt- Lli r..,15u{l -r: I et, Cb gly qJl 'J'rlo-\'i irfu ^S.-'t
,f-fs oe.7^n .f$,gJi;, s!..-r ef )t ,h Cb 4 ,i'; \ - \ - 2 d':L'
-r--t3 ! --U-tr L+ a-c-a.;),lttr*i;.5LaOt--rtJdt- J<,ti; 'rly-r'6 tF.fj c.:-,1 -hu*.JrE ,SJ-rlJ ,JtF) 6,1.a-i,"e3: dr*: ,t.L"j u*'' 'Jtit't 'f &-f ,FaJ:L-. g ,5.79511.+. ,*-J ,*b ..---1; jl f - V - f *JrL' d'-r- o'!l JF' !
C,:-qr" agly a:-9;1 cSl.aL:-" *& r67
WN"W8e
(JJl)
| -, JK"tdi-Yt (V) . ,L; .t*-)l (.,iJt) .str ,-F; - ,-F Cts* ,S; gg.2lsy.1! - grr.rf.rt+ 6l;.6^-.
d;-il -* stri'.t Q) ul;;
: e-tct' oJlJr b of t+r-t;.r ,p5,r, o.rk .lo - v - f
,ii=+ (#.#,)=+ ffi.#) =+ @;;*ui,) (y-\-r)
At' u,L;.<S-,yltli.r ,Jlf L g5*Ll rJ:L, , ,jf ov-f pp ?frs4 s:
'e'r€ O1ycoii=Cijuneun t- a=ge (r- \ -r)
*lt'.nfJ:7;" .r-!,-f \iJJ. yr : A\ 6,1.1: C;1p,,4r)l ?_t' )J.-lj dir:66c*,1 C\ e-S : u:J JJjtJ -es -,;a rl-;ti;
Crjtn=Cjiton=Cii^* (f-\-t)
o.(*,, .i. rr dr 6*-,r, :5trJ;fi ' :-X r);* ".'t-'tr"r 1s J.2ge, 6,b 0t-! Oljd J*iu o$ clt"1 fbply) .$t- .ll-l-1, ,tl*a*,,
o'ri : c'ri* t'* (0- \ -r)
(v)(v)
g,:b Y - \ - 2 d.rt', ti'!r^" ( lJr *r/ .S[^:.lib dlJ JUJI ;l; t L.l
rdy # ,\i2,-.-'.-Al ? J-s
o',i : aiq ai. "*
l_ i;i;iY'] o.^,, o.
.Jor,_/ lr 15 qp A - \ - P d.:t*, t i+:..- +*,-U,.1, 5
C';jp, - ai(I ap apk a*n C*m (?-\-?)
. pU irka {-t t-*tr 'S.,- dlJ JUJI oreU c G;'u-
ia.i-l ;r Ll .-r-U[ ojf e+-rr t aqlg .:-,1 OS--. Ciim V-ls,- , ;S.:Jb .,1r
ojf r.-r:) ,JL;y/l (.rr-U r:-,r jl I ,-rl$ !:.-,! ,-S;1lf 6il SQL-:I \l;ItCUu" eltoil-r -e o.r-1..j ,ru&1, ,-;;f C),->lil #"^A d't o.-f l.1 (.-rll
Og^ ! o:t-. cJ:-ib ,*,13 ,5;-,!l =-lr- ft
.1--Se F t, q'iJ,, jl ,-*b Fl.r-.r-Lt JiU .JSc",i:ci !.:L ,S+,Yl ,flr 6-r:;..^.o jW to*-rtJ grl..ritr.r,
.-r-,lf"f 89 ot; o.t/.1 c-. y* 09;6qt.is-r f - \ - 2 .J:L. L-'9 rJ^i dbl
,SLn*J_f 4;1s u>,-LgJ-j 4,;l o:ti:-,| t: 5J^ tly[i c.*,1 -;;o\ eV ;.."W 6lt'.
;f *--* Oli € t+:.,-l q b .,'*n--rf .<S-,!l gb.+ti g32 o-!,.tpJ) ss : ,*f : F
orr :0r
o\l = ol
orr : oY
ttt : €t
€YY : €Y
€Yr : €r
oYr:oYr:ol
dr\:off:0a
orr:0r\=0t
Ysyr=Ytrr:€v
Ysrt:Ytrr=60
Y€tr:Ytvt:tr
l..JJlv-\-p)
J
(vv- \ -t)
\Y - V - f aJrl-- tr qt-i; L)-- Sl-e, dy. ef gLndJ;. rr Jr ?-V dii.t a5
;t .:*l .ltF .5-9o 0;:tr 0tSl . ,.-cr--,11; C/tjl .c.*,1 ,fi ,jfoa=C"pep t- o=9e (^-\-t)
(J:r-t'.t.s agl2 o;-9;i <SbL,:a. .S!J|S" r6A
Jste\ A - \ - 2 d.rL, e_jV rfi.l c.r::...,r ,i.-i a;.lr 6!b jUy ,gto.:;;rr_;.r_*'ll$ yj
o\
ol
oY
ot
o6
o,
cr rcrYcrrcfic\6cv
ct\cncKcYrc\6cv
cncncrrcYrcY|cY?
cr\ct\cncwcrccv
cd\c$c6Yc(f.c66c6?
c?\crrc?Ycnc?6c??
€l
tY
tr
8Y
€a
e2
(.-Jl \"-\-t)
(9\.-\-t)
(q-\-r)
br.l-b,5 J+<-:J r-,tr.f ,2"13 ,,o:-eY? 6ljs 6 C,p qbi 6 €'* OtJ plt.J-4.:,rJ
,P-,s3;l JrL; $-*,1 -hi* 4tn 0,1 ts u1slf .rl,sti *fu_l ,;i tl
.{t_.f d^lSyj os-r-l \y - V - A .J.:t--
.r.i J^lJt \A - \ o - f d.:L. L-,rs (6,"j5 6t^J<-,-r"r" ,-*r,Slt S
,z-loriD;;p
u=Io;;i;;p
(p* 't-11 fi o;;,25 cSj;.| s.:-,1 pK .ft- 1.1J., .r!rb .fl g u );lr djtljl +-.1 ..r-r!r6tF! rfii.;s 'fu;1*aJsL,
c r5;_l ,S-rlirt* q oJ:Sl ;F,-l,ri..U,ri J^tF Jsl Fy ?_J*' t d,+"^A s p. = p J
r :detr =det(rr .#)= " .# (\\-\-p)
ss hur I il{a << \ "l-t-^bc rllS,r*"i-; 6trOU-rFor1'1 &-f ,ft )l r,d/-lJti
c,,IrL,rs c,.:B 4, -l p = p. Jor.-,/ 4+rl S dg 214;l I = \ .-5--r','J':L'.J-iJ.alJt\.-\-2
jl .*U W &;,53;l rp,-lor;5JLrt 6 GVf cs.fu-,!l ;t:il ,5ty
gr:l: -l -r-,tr.f 'F-S.St^.iJ,ru = u (eii)
.tuu---tiiUeii
1
'd-s:|,-/ s-'! r-/ \ o
\&r1j;; =
-p " ueii
.:-t-,,_l ,jls J-j U.\ L-,y (fo- l-131: ef ,$JDu* tu:;15 ,$Jl ,)ga
tt*=p,u (\0-\-2)
4"+j \f - \ -P d.rLr;l crrt..rf.rt+;l 6 Gvtr.t".rrir,r.!l *.ftJ,Su-p =p, Og.t-f,-;
o;i :p+ = + (\r-\-p)' Ueii Ueii
Cb .51 t37J.r,,LlJ ely ,"ia4 d*-,yl .rLit rL; "/-l
-rr 6c.*,1 t'3 0[U
4.-p -,y i..,! \t - \ - 2 +J:L, tl ,S$r{lrrr, ! t^,JiJ oi;t 6 ,-#-5,S3Jl
,-p..:,*,1 ,9"",A(;-6.:ly. ,".is-r ty,l w;''-ys Ap d/+ 5,$:ly.rPL-/
-rLi-r jl ,-# vi'-y; tf o6jyl ,-iss, OJI {1'*J-1r rbl 'fr-S
)l:}t taz':.' "*ti j;r^
6a-f-r[*f ,gLaJ<-lr* 6 obl r.]-)-9J:tr &>t c a*r,.r.!l n*,-U'. JrU,J::-,!tl+.. Jf a* 0.r1.: .t*i q.f*fr"f as-',j b \P - \ - t +JrL, 'J^i ub J.3 c$bu,
f{-rb
(\Y-\-t)raJ.,1-:
(\r- \ -t)
- \ - P rJrLr ! p+t.- r*,-U." -lr l
(\Y-\-2)
u* (eij) =u*(") . Wrr + + W e;1 epnt "' (\v-\-P)
O?-u,+4 agly a:-92a 66 L!^" {Jlfv. f?o
6ii=+ =*:!") + **,(") em*...'' Ut;i Ueii \e;i\epn
\t-\-2nJ.rl*,;l
(\A-\-t)
l.r .r1l$ rtrJ 'rL ,ffoJ+ gjb ;s o$L.tiL ,3bris 6 ,F,g ,f, J j .. ,g-,ri,f \,Fd^+t.iig:.it:.KUcu*,,.J.iL: r,i= o 6r?rr>lo;i- o 4+rl
.rf 0t*: &f ,S3tl Cti u,,LlJ.rs-n dr+.rlid l,,};.fuyl ..f|*,l drl--
lru*( olo'i=ffi€b'=c;iu"eu' (\1-\-r)
sFJJr Oil d,:'t 1*e. .-5!i.ri 4 Ll cr3*i,r,JAU Y - \ - 2 drt . ! Ot*Sr- d:t - ylg-llr.lr5l 4(L c.:*,1 :l-,tt y - \ - p aJ.:L. L-,y rg cll* 6Jtjj !A +j c.:-;lr
C,j^=Cur;j (y.-\-t)...:*,1 il d+J> *A.6
0\u*(") - ltu*(")de;1 ilep, \epn1e;i
.J,o.:r.-. ,Ji.6tJY\ q y? ,ll:Ciim raL:, ,Slonily rl_u ,;f ,SjJl Cb ,_l--l .r-,"t-r, r:,-.-i;.r osL culi. oalf jl ,I-V,S*-,Y| sVis ,fV {tl Sltd,j-:* uj.K,fflrr tV- \ -PaJ.rl.'...1.r \1- \ -pdrl,-'A&V!n5rr +i.4i.rg*,r
r-tJi Oli.f 09l5l , ,rXif * ,;; *V -ltt, .<,_
u*(ei) = iCr* e;j em = io,j t,j (l-all y\ - \ - t)
.tt-l uas: JJ os-*o A - \ - t oJlt- 6.1ljfiLj tr 5
u*(e) = iC*sa€f : *ooto (VT\-\-t)
..:*,lf{0ltl y? ,l r,1L:, c4l3 y \ -t.A Cof : CBo c;;W L-i \ ,
t7l F '\t2,*.'.Al ? d^s
t<r'*'Yl 6bctti , :rS;t-,,.a ilg,' LSIJa tSg Ji6 I - 2il .rrtr t-rh ,-t..ilf .(*-,il 'Ii dtJl Uri eJ.*,.-,S.,- c^; 0.r1..1ki1 JltiLi, Uti rr a-ev d c"r; f t
'f ,:*.1.(.:-,)lo.:L 1l r>.--a J.ib c:'u.a3 rL.*r.
Ci* '.?-l)e L-,g 4Jf--*? ..4 .5*r!l ,al-r* Jl ., .;>{,', .J-it+ .full ,.r-rf-,
cJ-i orl.: J*;J 4 ;l +tn g )t gt' 66tf e3.ra'a y' tr l:"1 oli o:1.: 6.al9.r !.ii ./Ll d.rL, c J9" * .Slt.Je-F .r,;9/.'n ,+ -U o16 ! rli ,.r;U 0LK,-
drr-l jt ,g.rl.u eJu..U,r.. .lFt ,.^"rp l-l tti -r-lU ,ft*^o 6 6lo:L ..r11: ,5.:-,)t ,.:rb
.rf f,ftf , 'd-rd.g!"f !b gl59 5,5+"!l .,1-rti; a.-.1s r.,Llf lr rl-r"
s-*V 4 f - \ - 2 +JrL'-. g .5.$*,)l JJ*,Li JS6-./1i.-,-l 6.tfL,^^"
e:-*jb q.r-'.;L :f;L *"r JJ*,b .i- .#c.Jt--.r, .slr lrka +y t-&L"^,!
-fl-t g.o- 1l.r2*l 6lasy- te,.eu 4;l -rrt ;" JE:;tJ^ q .r!-1, a-ilJ,,6c^""1
(rl-t--el) *- r-;--. 6La;-p,b urL"t J g.li-;-f -* )l -e 6La;y,*,b..t-iL -r;;01-trt-\ -21*rb .(tolby) JJI$ )trJ.J-rl rl-,,",rKrL,'.-n 6to;9*rb Ll .-r::.,.o .tf;L'**n
,Sl^.r_l-b QJI 6rr, ?:s p J:.21.r;.1.: ^AJ-, OlPq It fJf 6blJ.: 6 I
€:+ rJi;t ..^-n )J*rV Q;.tO - t ,Jz,- q r-t' Lf) J::..A^ g-l: +l,r rFL^^^
f b grk-a €:- rFt*...n )J*'V.fl-ll.r:" .-r-,tf"f c-:(-t- rLi , -p* ly'dJFi o2;eat 0i5l ,".r 5 rl: rrt*i.i Jli € -uq Flt' 6btJ: .r)\q 6!11 6,rt53
r=5.-r ,S-r 9 elt
Ci*=1.6iPn^+ p(6*6p#6oOiil + f @*6i^-6;^6p (\ -Y-2)
,J-l .C;iton = Ciito, = Cri^*i Y - \ - 2 *J.:L, L-' y Ll .-r::..^,o r-v B J P, 1,.1i .1, 6
st j , t "f
hL+ tr .,;a --JU. *. 'ri 0t"1 6Lrcllti; dlJ {q B 6 sVt€ Lt'y,,;ts ,aj o:V
f (6i* 6i^ - 6;^ 6p) = f (6i* 6i^ - 6p 6;r)
\-Y-P.tit- ",,i"tiaJrL,
Jr-,f ,r1l3tr,.*lfQf = . QAP = -B r;.,J.,-loJ.aLL
...-:-il.: f=flJt Y - \ - 2 J.:L, 1.r
6ii = (]' 6ii 6u, * F 6* 6n + F 6n 6p eu"
(J:r-r'qr* agly a;-9;i <SbL':o" *lS. f7r
.{t-.f ,-rb6y-j Os-te\.J.il,dA c..n"t- 6-rt-rK-t- U tl
o,i =)r6;i eps* \ trte;i (y-y-p)
9y -t-3 olt.il >\J 5.iFL^ ..r-,!"f ,j;t-^" .(::-,!l tli-r ,Jtf 89 6yB 66p sl .^rtj J) ! €-r-r Oyu .:"Li Y\ c ,-fu*,)l rft ** -rkir gtr 6p-.1lr
..(*:*,!l o.rL 6lr-, ^5r-# ej .tt-,| *it_ Lri$6r-,tf,.f uA agLa+b ^t t-iJ,*
gg6.2y-,.f:t-..^, F.*-,!l :& f--?.<,- g/16 J*rs,.ll;; I - f -? cJU
re-rte-..1"r !t*"ll ,s!J,sl.lqb 4-9 J::*.4 P, eJ ,JxJfJ u,lJ,5tol.prb ;lel.a:i \:*,J,r t{Ji,rLrt
drr-t l-i t4-t I -r;! ei1 J.al6Laca.7 t, &- ,5bcJL"; i@ g = \, y, f) Jl ,bg.uu_i Y-Y -tdsL,;l ,tSi;ir,.r.iq @ = l, y, f) 6tk)W,rbfrrE,t^JtJ
.f.^r,-f Js-tJ 11Oy,et, t1 6l+Lij
o,i n/d = (), 6;i ew * t pefl nfa)
1 n{d ew * I preii n,@)
'l-L:* tSu- O.rLi cl; slJi: +f *o 6lH l-l .f,rl+: rJ.:t*.... e1r1 ,nfd ;tr.*ii rr t,ffd = o@6ijnjQ)
o,i ,/t) - JL e& n{at * \ 1teg1 n/a)
[ )' tw * \ pegl l n!t)
J^lJt ,>r.lS n{d url^i 6lt c€*,1 ,-;;fAt-t, uJllep, = 0rrl * €111 * €6.1 ofr s
,*J C)d
o,inld = {},[ etrr * €rrr * e1y1l * \ Fee)] n!t)
6blro+"^.r.J.::...^ ,;s,tbl6Lrc-a7 r{d @ = \, Y, Y) 5-r.a:.-/0tJdrL'yt..sltr.f li Js-yet, ubl r.ij y.rtfu
ok)=lIt,rr*errr*e6'1 ]* \Pe@ (q= \,Y,Y)
Jss.0td b,-,*, -t
ef.Sto.iJJ, lfb Y - Y - 2 .rJ.rL, Oli €Jrt*1i = j 0.:lr rp L Y - \ - ? d.rt--. 2l o;i'-",-.r.t ! e,1 l-urlr, t+ .r{ b .rf ,,-&-
6ii=Ylrew*\pe;; =(f ,1 + f ft)e;i (r - Y - t)
,f-S-- Fi 6f - Y - 2 aJrL" tt,;-f--V s eii cAlJ.Y - Y - 2 drL, JF t c o15l
. - g g-,J,r. rrSi:t..^" .-/Ll d:L,
.g-!1j 0liddrL, d/-l Ort'ti#j o;L-e.r !r,11K'.i,i * ,ft-r,3 Jt:-.- 12L'.
7+u,,i = ffi {l \ + ffil"u -ffi a,i "* I
(Y - Y - r)
(0 - Y - 2)
r=5*rr* ftWt:t1-ilt7 - Y - t)
J
(v?-Y-t)
p = tt (Y.j'+ I P)l+tt
1- Y (),+P)
O2-Ji*4 agly a;-9a 61.6 ty^" *K" r?r
.J.r l,rrtr Jj J* atv 3E 'f+
gLnc"rtt yJrf;L*", -lL$; .Slf di9r 0_/E
,ii = * tf '' * v) o;i -v6;1 opl
.'t Ev-:--_1,Y(\ +v) (-iJlr-y-p)
.syul(ir^-Y-t)
(elti =i= \) (,-i1l q- Y-2)
(elti=/=Y) (V1-Y-t)
(,Slti=j=y) (Vq-\-?)
(v-Y-r)
\.+-Sc ovln c;.,,"-rls v s .f,--,rru.Al clsuosL 1A \ &U <Jruf E Li-|1:.4J" ! +', ,f*.)l Ol;4- st -flt {"{..^lb Jt Ali,€, c.lti J) d/l;l ,*tr.J;
.tF€ e--p t) os-roe' <22.,L <JsuL- ,f n Jsut-url .rf
K- Y(t-rv)
.r)l- L-f)J;l c$lor-i, ,:-,* ,3t ,t 6K sG , v , E ,Sl,qU ,Fjj Ji.,Llj
6sF.<; ,iJ cJl-- .,1r ..Ji.i...,a O*S "!6 I - ? JSS -,1r oJ-i o.:l: F-LJ ,F eV
(,-r.ll Y -? F)g..€lu,+ilJ'lL: o1 = +o. l.xrc^{?Js.r:g_0Llitu'i..:.t)
.(oii = * 6. Jfr) $:./ q"t V _ y _ i rJrL,
orr + 6.II
E E
.I vo.tyt=-Uttt=T
.I vo.8tT= -Utu =-Z--
(..tt";) ,5f,or; dp ,:*,tji ?-_t' E p,.c.*,| jei * j ,Sll il.gLa;f ,i++^r6q "i- .rtJ ,-;;;- ,'1..:';.;|.:y-j :-.jt c L,-,J:A J-;f ,-r.i!"f t,lrlf 6sFJ J-rL.:b-r;l
';s g-*r iriJi ;"- E > . .f ,F 4?i.(-ijl \ - p J*).r-,1.i,_/
g*jr, 'r3,!
q - Y - P aJsLr\:.^*.! Vy J,;,Jr 1l ..r.o.:6 l*,-;^SlLLi*:-;
r7d ,# 'r::'i:ulJ rla
s3i 6l,;^Oi r-<* ; ,s!./ ;JS )l u.iU r--lr ;* JV I r-l-r .,.Jt? '$ttt;Jorr : ". ttllr
r,5r, ? Y - ? JSa.;s oJ^i orl.r lLij '.:L ,-3y c,Jl-,Sl;.-r-,t:"f
g,;l: V - Y - 2 d.rL, jl ec*'l *.6'*s y-L ,-f b
\ *rr T.€rY= E orY=Ta (,*Ul \"-Y-2)
\ f - V - Y nJ.rLr;l ':U:-,1 t: o.f.u6,n .St^.15f ,JtJ t
,rr = T--b (.7\.-Y-2)
ott
9z
(v)I-'JK'
ort- r,.ir- (?), Gssu.s,' rJi5 (e,il|) . o.rt- r::--;,.:,YL-'
(.-iJl)
,c;l;S,- 6s5il.5' u'i.i5 (-,) ,
.Oll=Olt=Ofr
O.'JJ* ag(y ru*94 <SbLza" c*& t77
)l .tFC ,ff (tstL$) 6sF ui'*S 61L E 6sllr '#
trL-r o:L 'rt 6lt1G 6
6s3ru r,-' u;,S..:Jt- g1IJ., 'rt)tt ..:*,1 oI3 11.:il G Clt <SFx dl&lL -l-r Orl
,L:;lp ; . t-l oii =tpdijr,?Y -? )Sa(.{rk*,lJr*r-rLtit_)cilr*-
,f..:tr(\-Y/)t):: -"E (\\-Y-P)
L t )1 }*V,,a> rrt .; +,.1s p -rlliK LF J-lJr:Jtl,;",1 0l-rd oi;l +f
.(\A - V - Y drL, 1r_t' er) $r"/ Jattrt e;; u.'SOli r.f l:6-* oiUKJG .gLa.'rl3;lo:$i^l! rfc;t., ol'.iilt.- -u_t ,#+^a.rY.rL. ci? g+,, erJ,.nj 0t i ,_,iryJ ui5 6b1.p,b 6sf s ilFl ,gt".iJ.rr:i..>J
..:-ib f'+lF t, y:(,_iJt \ Y - T - 2)
(v\Y-T-2)
+ltt
--'K\-Ytr
€-': =L'F
,b
.$r"/ 0Lij V - Y - 2 d.:L, qlL.e{ .C=:5"f
Si = f Gqii
oii = Y lfr.ii
tf;t*ap :l.t'61J. Sg liti o KraVl ,rU; f - f)J*t/.;..5- d 9 .fLlf | - ? .:..*;;, .tr(;t-...- -rLi., 6111 59 OjU
.J::*,)l 0tE; g-aLi- ;t l, r4.;; i,..., .J-i +;-ilj.f t:.t s! frt<a {tr, rFl-..^,!
6 ,l-Vl rlL.r ,:-- J"6 <S*il csb,:<r l.url rti;t+t ,St: ..r,-ei c...,-q 6li,lL. Cq,.(f-,Yt ,SLncttJ rf ,gl,r-tn 4 -lr Ox',x',x.', s&rxrxr g3U 6b,.p L-,y
6- .lU.:t jl .€,g,-r-J- l-l lt.f "/tl J*tt rl3{ 14.1.g.. 1cl'il r: dH JElt
6i 1.:*; csss* (fi, lJl ot&6!l-r o.rL €f a(r50Li l2 sj*4 Jr ,-F*6!l.r o.:L €f C o-ruitl 6l+*i.a I # Lale-,;l .,.,6i,;l .i-,lti;tJt .c*,t sF0;W y>,' 4 6 $r"/ 0t-! b rr:Jl- UJt f -? )5;..:.*,1 ,gloo,l'- +-U( Or&Ol;.rt xfi1 taite 5-trr6 Ot.ii b u,-ut 9 f - ? )5,;5Jb tr .$! ,5,;-,)l
A )l 9Y - ? )Sa ls r"lK*jl .Slf JtiJl .S Gil|>..:*,i .SS-,Y| isj.jr o.;-(ritt tf,g.rt-ag.. olK:-,, &- &'rx'p'y 6l-n2g^.6 t-f ,-SJ"!) ,>-,1
"-):;.*;l l, -iJl Y - ? Jii ls.ri sF 6irl,"',
^ ..ts; tb* ! o.rlaL,l.:;y 0.lb ,.rtL->\,
Ot& 4A5;.i- 6c*.1 .,53 r[U oL"^A .,.:*,1 N = Y r I 0 diJ.t atl:..U,-f N nyr,
)l 6leS a,-, Jr y' ,JlJ ,J::-,)l 6La.:rti 6lil o.1l: y.1: l, .tr5;1.,.^.n ;U, (**A(
.sb.f .5!q r*d 09.r, a.tij tti 1: p-?tr 6,ls;'2>u
11 r4
(dJl)
r-r$,.... J2-0 .11.r.;t { .ll.r.lr (e,il1) .5lla,i.a ,}; ,5la
J-(.J ditL, .1r--)l 0tLii .e,;e ,S' 6xpt bJLe jl ,;U .:rlf 0i5l
xi
./)(
.r.rL- eilJ,F. c,JL-- y_l ,Slr JUJI g--iV f*5.-f !f)J b -t-,!._f ?Y - 7
;l c*,1
;]L:llaiil = (\ -\"-t)
,f-JV r$!.ls t"Jti;t .*f t r; .gtariJ -r cSUF .-Uyj ! ,r'*T:;;
11,, 1:; ;: ] f, I,] [; 1l] f, :']
| "-' o' -o' I
=
l; :, ; I (rr Y-r-',
C,:--rL1.,o 6ly o;*94 gbL,:^" csrJg" r7A
o\ = Crrsr + C,re, * Crrer * Crfr * C,oeo * Crre, (f _f _t)
.c-ilr f=ftf ,r-ib.._f _r*o cl.lt olK:-,, crrlt: Ca 6 6ltrjS+,
hi,l I': i::: ] [].,]h:: ll {]
Ii]ilL;::t::i:.ldlf g,;1, .?j c.r,r*.r .rak, A - \ - 2 rJ:L, -ls c"tu yP,r.L;
**I
o'\: Ctte" * Ctre', * Ctrs'r * Ctf'r * C,oe'o * Cve', (f-r-r)
{.o'o - - oa
aJ:L, ra..:j -lr
(o = f, O) Jo'o:oq(a - \, I f, P) -;jl Y -Y-P d:L';l t'l
t'o = - eo (a = Y, 0) J s'o = Ea (a : \, \, Y, ?) JIP,J.^!.J-XJ.6tJtY-Y-t
o'\ = ol = Crtsr * Ctte, * Crrer-Crtsr-Croso * Crre, (0-r-P)
L)uOJ-l rs(i-l 61Lo9Ju" 4L-)\^00 -f - 2 d:L' tf f - Y -? +J:L"*,I-U" tl
tf .Cr, = Cro = . f.-il-t c*il.:4,t-f ,gLf 4;51-".' (-ur.:,r.- OUj b ot ;1J5.p)
Jo'o = oa (q : Y, Y, il \ls: jl *r,/ +F t rsiss 0t^r,.lrrJJ!-;,, Crr, Cro Crn C* Cyr,S-:-,Yl 6ta.-rti,f,,5r*,-U, l) o'o = - oa (a = Y, 0)
Ar;.ax\\ p.;e^3-l .Slf !U"^A CrcJCrr, Cor, Cor, Cor, Cor, Crr, Cr, C1
Al elt..(;-,!l c^:b "r"-yL
c a,Ll C-l l.JjJ'4 ie 6.* J.,-! st ..(t-')l Ortb
.c^""1 J-i,F6!b.:Jt-
l'"',t'rt',| .',t,,,r,...
1",,c',c'.
"a
lt,,rurn
o
o
CttCto
CotCoo
o
" Ctr
"Cvo
"rro
o
"Cn
IC"pl = (? -Y - 7)
n:-.ils t-t+s A; 6;l Ctr 6- Jt ,**^r .{t-,.r ,rf,
O*r!1-o agly a;-921 <SLaLz^" *& fYo
["',t',",,I c,,c,'c,.
I t''t"t'' o
C,,IC"pl = (v-r-P)tCoo o
o oCrr
?-f .i q r-,q n:-ib t1..J ef ,S3Jl eU ft \, ;r-* ?--t' c.rjle.r 6!tr l.f
Js-iJl.rJj aaie -9-r b olr^.o ,-{:-,yl o)w 4r,Je 4 p" xr\ fl'-l +l+ .-u.s€
OtU; .c.;e a*, +S.5lorL d ."ra:,_r^, b C"e Lr"-jt l ,S-fir cli^tt6^ ,J-it:
,-,--i['., 3 o.2-i,ua oJ.lti *s:-y:l os! 4 $q r:..ils e J )J,q-e -l{-lr.<s-,yr.,.:*,1 o-t.i osl.r V - Y - P rJ.rL.;.r 6.:-,1 0i.*-,Yt 6bc":ll
- Jl-p ,g)L L rf;L**,. ,-r--jU f .1,-l;,rrl .fu-,Yl ,*,_;L ui^Kk-i Ua.rl .slr.r--r.t'. dJ:+^-,t J Y - ? JS.i -,1r oJ-x o.:lr 6[.j^i .(::-,)l tt,l]E yz,-,Crr= Cv p.:i,;f \:-,! (,JJl Y-t r$--l)x, sFJf 1. o 0l-llr b.srl,r, JeVrr :f Jf 1. o 0l-llr Slt .Coo = C2 J Crr = Crr, Ct, - C*, Cr, = C.Crr = Crr, Cr,, = Crr, C,r = Crr , Cr, = Cvr 16t_*goJ"aLi.. (:y-? J^(-.x)
f:i'f {,*,J.r (V f - ? ,F) \ uil;J;- fA o,.,lrl, lf , rfYlf .Crv = Coo J.rL.'i! Cop * Crl lllj.f Crr = I sC,,,, = 1t sl:;l,j! rO*lrre.y Crr - C\r -C,,d/J-.: ,ft..*^.rLi-l el:..\ ,Sii*,Yt *-\f ,f-ji3 rl: r.a*i,.-i; u) 6bc,flJ 1.1
.r.l.--iJi OSye
)u+\pl1
o
o
o
1
1+\p1
o
o
o
1
1
)t+rpto
o
o
o
lt
o
p
o
o
lto
I C"n l= (^-Y-r)
rrrri (-iJl)
(.r)
xsrxi
xi,4
x2
(v)r -, JS-,
.y,r Jr "rl-21r
foo (V) :\ Jy Jl15:t"o (./) , x1 Jsz.rflr too (-iJt) .-strr JLiill sdJs Jrl'r+
,g+.St^c.rtJ -tJli€l:,*-iU;r,l'?
- Y - 2 d:L'.,1r o-t-'i Clrl ' ;uL;;l
JJ Js-ree'. u*'-;V jii.1: ,ft-*^,t f cslt 59 OiU {+ri.;': e':r5 bla.v 1E
xf
'ty'r fV
O:-$.F agly o:-9;1 6lbL:a *tg ryf
E=-(t+v)(t-rv)
Voo
Voo
\4oo
" + (t-tr') o
o o ! (vtv)Y.
rI
\ -t)
l)
v
o
o
o
v
\4t
l)
tl
eY
tr
"],^ltrli f,-*f
(q-r-r)
)gf e,-" Jel ,:J3t-,,^a *et *ylt qUl *yl f - p6sll: Jstl.t,-sbf y *b q*-;-,)l .fLt $l* ,F :.rY.,t-. J.f..jJ,r1Lac.=r'-iJ t rsJan .1,1t-,1;-Vt .g*-F.-f 4.1"*lJ( J ,VU(E*l)ll, t+ic.{rQsF")t J .<JU*,1,.:*,!l -lr -/.1.: .J-jlor.i -h;-f ,.f_t -l cSb-l-* dc.*,1
411; &JU- ssf .Sii-"Yl f--?.-J-lr ttei u"L"j,JtJ Jr-q oatls 0lJia.r)rl-,1: ..lrf 0".1i ls j-e 6ss u,a;.-*, t_lri Jr-q Oh.a 6tr-r;:, U^J 6L_a 1.r l rr:fJr-l-i Ol+^i bOtir;;.,9 La.:.er-, 6ss 4sl llt . .{ae.r;"^.,y1 $l* cJt-
'sr cr.'bJ-ib l) Os-ft;. uJ:;. Ohi. .r)sL. l5.ll>.;s .S;L-,I9*,Yi dlJ b e--..t
.€'*U_i.Jrb..rY.rt-. (JJl)
oii;*pbi= o (\-y-p)6t<",#r - ,-;;f eb (:)
T si = u;; * ui,i
6ii=1,6;1 epr*\trte;1
,ii : * t rr * v) o;i -vd;1 op,l
(Y-f-2)59 0,/U (.-;)
(JJlr-Y-t)t_
(vY-r-t)
,+"-7' ", Al ? J-s
.:;,Q jl *5lt F 4" -dj.1r J.L:-^.^A f9t bi 6^-- db-l-* 6 tArf ,f -f YJ,"r.
seii &f ,:-i c oij j;.#;l .:*,1 lr.,Lr €..c P l--i-l.,.: o-t-i c*'gi tJ.:L'
*) ol:V jl F- I rJ;, d,.fti S e)-t" * ett {! lf ,-h-l;.u; clK"..r# u,
'F,-r -*t!.IlloJ.i O*.J t? l-ca.1r LaOlS"r*rU - t
ui : u*i (x) 6s; S (Y-Y-r)
..:*,1 oS g6J q.5.<, FU-* o.;i:-, ,Ji 1r 5.$lo$ O:F t- e^,o.1l tor:.5- Y
t,(i)-_1.*6) d.Js s (o-f-t)
.$lo-t-3 O# S ltSr .:-"J 6;; bclt(rr*.u - f
u; = u;* (x) 6ss 'S\ ('i)l ? - f - t)
S, '"UL.riq C!, gjs o-I; O.F ,5LariJ5t 'lr^"o
6s.t SY (:7 -r -?)
-..,eJt{ji +fc.*,|,lit: f - Y - ? ti \ - Y - ? 0[ir .r)-:L, 1: a;(j r/-J,+
,F csLn-l-x: ,sl-t ,-1-.i-r;(t'rritt' , o,i"' ;lLut".J::*r! ,k tiJJr€+,
,sly. ,-F 4 il,(t) Jr;j(') , orj"' 6.J1-;r c-tliL rr'*(i) 6"J"- 61.or:.53 \b*i
,tKi,.lu-i[ tr'*(i) G"J"-'6Lari59 'b*; un-- cSLn-1.*
o;i : orj$,
-r;1., r,(i) =
+ oitt' , t;j - ,itt' + ritt' J ui = tti(\) + ui$)
tr.x(i) * tr.*(i) J br :'b*; * yb*i 6 ;*: eltl:.Jb 4l.c
O*.J:4f agly a:-92-, 6lb \^" *(& fyr
,F "-^.,1i-YlJ, bJ- at-*-,tj;.: J-i.,o JU*l 6c"r,ib,r, fiI g,> ,!.lOq Al .$C..t-,lf
"for.l orlr f - Y - 2 eJ.:L, L-,9 t^Otirp -J
ej-t Ja,-l,.i 6 ;Bl* 6lJ
f+sr.ii--- OlJi+ rr)rt'-. )ll-2 Jga- r-;;-S J rj;:;:y J.ol3; j.:-b t,Sl; sc*,l
eJ:L, ,S:I-K-E L: , .r-, .SU JX+, OK".=J ,St".il_1.
-J-Ui 4k,... 6 6larf a,
d-, (\ - f - 2 ,J-rt-.) Jrb,.:Y.:L*r;s +a;J,9 (Y - f - P J.rl--;,5J,r O;jU.lr y - y - ?
.Fssl€ i..*J.r lJr rfr, Gf ,Flrn-, aJ.rL,
Itui;j + (), + p) \;; * pbi = o (v-Y-r)
*s-tljLl oii*J rfrli --')-rb.r-l sU'F,s--fl.J::.,.^ u& E|b <>Ut6., ^f
Y - Y - 2 al:t -.1rl:*,lO<^-.4+rt0i, Ju5g.rbb f - y -? drL-. tSjy-t;g++*Abtr.rj;.-rJlr- f -P drl--;.:cl.rrS ef.-V ! rrqtqlt^.""f lr r-r, U-f--q
.-q[ JpVr,J-it-.: oJ-i o.rls (0 - f - P drL.) G-L .5tAui*i5 *J. Sj-t \l-i e:,59 O-;;t-r l.i ..rtr"r, b (rv - v - f aJ:L,) 6a-5r1.*r ebef el16j31L.r)sL.,S*.r$^c .g.rl-u jl ,-*,,.li_i c*,.r, Jrb .:)rL. b rrf .J; (V y - f - p aJ.:Lr)
.V)ut';aj ,:)rL.. a.,
o;1,k7+ s. or*jj*p (b;; +b1) + ! -6;1 pb4u- " (,t -f -?)
JrL;.rY.rl-- tr:=l;rr.-r-.i!"f ,J4- €lr\ li; ,grK;L <>U.tiu*\as-r 6o.:1*, p:q-,1-*- Ll s{.-.1 ,;jj.gl.ar;J -f h 6111 u.;.,.*...{ J-b.ry.rLr;rlp--q c.(;-5rL*r ,-;;-{,StKjL d:1.- tr 6 .:-,1 ll- Oi -u,;L 6.t.i J- a;ty c..*;
.-\.iU o-ti )^",..6, o.rL rA +!.r: (Y - Y - | nJ.Lr) 6yl:YrL, L-,y -rr_! J:[.;.t)rt-rr,.{rQs i.,Yl.;sOhi/ 6tacs5s,L"i OJS| .d/-lJti.-r.iL ,# d-&V ,*l0l+. .tYrL. f."r-rll"t - ,Sl-*, ,-F dq 4+"I ;.: c.tl;Jr.. .;f -b 1s l)L^a:i.o ,y_+ J ot-j jl g,li6-t U-ri $; c Jr-q J- , o_e)\*, .t-t-,cf J,LE, ui = ui (x, t) JS.Sq iti, c cll(,-ro;
lli = Lli* 8, t)
tr6 = tr*6 (x, t)
O;9'eu..,rt-;;t d.li 6s (Jr q -Y-7)
t-
(v1-Y-?)
(.jjl\"-Y-t)
J
(./\"-f-P)
cSJs
6ss
.-r:,-U O..b ru-tsc/ tL:;t rst' a4')-p', 6 4s( L-1.;;,4^r *<t
ui : lt*i (x, .)
iti = it*i (x, .)
.r!.rL, 16t-r.l o.rl.r0l-li 3Li..$U-'l.ry')l .JlJ V- Y -2 dsL'!qL.i;-r'
o!!;b-00U"^-o 5c.'l J) J)-t'r, .{"Q:P!l qJ- 1s Laclt('r*i; Sll fV.S:-'rf oJiat, ,-tb
puijj + (1 + p) ui;i * Pbi = Pili (\\-r-?)
glk^;pd;:"i."Yl 6-2oi,lf5 {. )z&c 6rlS.rl.if s f ,:oV 4*'JiA c u-'JJ4'o 6Lasy;t:i.5slu 1:
'+i=-,r-"-!!l s*y Lacr*i3 #l4rJU, s s--J.,,-r dJr.: -lr J# q q.'.:-')t ..r)lL'
tr -r-r"r &(.i; l2 6l+ti-o JJ; yl u.,..o 6j3L,- JLpl U tt.-r-,!"r a/(a>l''
-J.|9 t+i tq.i.,-tJr ,;r,{,-f 6-* bOtS,r;; 3 torii 6s: uaa'U ')t1jt OLl
4:lr;f y,r-..+ n-,J:.r , 3gla.>in gP,'f. *-:t.€€ r:f ,PV'S-hJ dJlJ#
,.r .:Jl- c,/-lrs ,srt-rfi! .-r-,tr,-r -&t ui.3.r jlr57 -S 'P q ,S- tr 5;U J-r.1 5c'*"t
.J-iU cilJ;<,- .iJl0 - ? )S-,J4U, .>-l-,-) f -f ,:F,€ ,f f -9 c*'l gv o'i'c
P .t*i.,d .5!l: 6tdly,l y.:':- &- osl-1't 4-'J:,6 t 6looiio ,.f;.5 #b r'r{ .rrl q + *.tlii- eit U 6ry r ,tl6! ,3!l: +f l:-'l ,K-' J'i -l' :l F:lc.:-,1 0t--<J- eG. s.Li -r, b-lr 1136 ccJt- crtl.;r .(V O - ? ,F) *31.u" S2a
.1-, fy;.{t! ,rlt-f-,| 6. es: sli'l JJ"
O:-Jr.{f. agly a;-94 ObL:^" ,S*& rv7
6*' J<-t
C:lS:\r u-,^^ .l.ru -i{"1,-.2.r 4t$qd./L.-t ,!V C-* )l,.,6,t.-;v uiir Jfrt^.- (JJt)
"(,- -2.r aJUlcf €Vl ,lV 'l* )|E gluJp 'p; J:b (.f ) .rF.,, JL-et r-j u,-l s s2^c
."Fd cjLct 14.ri rl.r:-.1 y 6,rlS.,L.J l;.^r.6 J-* u:
Jr orr Joil, oft j!;.SLn"iJ-f r, -all 6 - ? )S; c6la-r.i.a rj;s +;s g:fl
.t-i :: xr Jx., )l & ,g)i oltoat.rL",jtf ,Slr.il_1, j JJIoJ-J Jt:.-l ,.c.a e a^_o
,*r, cI!3; € ro-S
(dJl)
oti = oij @t , X), (ij=\,Y) (\-0-2)
rvy F's2,,{-U( ? J.s
.Jit-"f .,b6y-j oaV.r).:1.- q (\ - Y - ? rJ.rLr) JrL; .rYrl-- c r.7.;j 1.:9
r.i r^lJr (Y - f - 2 .J.rL,) oS..* - ,_;;-5 L.ls.t
tr\=Ul,l , tlt:Uy,l , Y€rr=tlt,y*L\; (Y-0-t)
ff - V - f dsL, -uit ul5-.i (fV - V - f drL, ) ,-fr-|,5.$)L.r).rt -r.lt je -r
$lol3 .1tF t+:r-tJ.: 6{.-.il: "url-9;
€rt,yt +€yy,lt = Ysrtrt (f-0-r)
.Sf g di.ii t) Os-ret '.6ln.:.,,it
.a ,F e\ (V Y - Y - P eJ.:t-,; ..5-1o 0-/U ,J)st-,
o6,r *drr,r * pbr= "
oyt,r*oyr,y+pbt:"
err = * ,orr-vott)
tYr = * rrr-Yot)
\ *v orr _T*crY- E urY-yG-T
*or)=#(e,,*err)
(,JJt Y-0-P)
(v Y - 0 - 2)
(.jjl0 -6-?)
(v0-0-2)
(VA-d-?)
t:.lr^^
(?-6-?)- fr {o,,
bQcaya gl-J{ ,-Ff yJb ,i;,SLndJ;, c0 -0 - 2 eJrL,,-11156;5', t
€'t
O:-Ji{d agly o:-92a 66 L,,g." <S*K" rvA
=#ft,y*verr)
E=ff(err*verr)
E€,t= '. ,T*=@* (VV-0-2)r' \ (\ +v)
.rr-lbr .r"_jt- J<-, q 6li,f 'p\
r ! .r)sL, ;y_l
611; d \ 6la->,}-c f Ohi. g)sL, u; (i = \ , y) Otir*; gLndJ-r
-Jlet*b c.rb rr-,j d",tt-,l.9*-,Il ,Slf tt -r_-lU g; .r).rtr; :#
.Euiii+ffiui;i*Pbi: o,
ott 1.-eJlv-0-t)
(yv-0-t)
(i,j= \,Y) (q-o-t)
oYt
EL'lY - \ -v
p] #[ I I,j ]L:l ] ^.,
\ (l +v)
{ilJ,, Jr Jn1 : o l-5t-F.l ,ftQ6-? Jft) 6la-z"i-a,l;f c,*;s,Sll.J::.,^^ xt 4.x\ Jl & ,*.li'$t".,!q t.lt(,r*i;
Ui = tti (\ ,4) , (i=\,Y) (\. - 6-?)
q "/k ,-;;-S e$:L.tYlL, j otir.r..., ,-;;f htl.r..Jrb.rI.:Lrr, cJt> c/.1-rr
\'- 0 -? c Y - 0 -2.r)rLr:iiijq.i-g- c.r,jL€ jV.glllu',.;-o r.iJ r{yaoLr.rr*.fr*.,Slo.^:tJ
-j.(JJlf -Y-P eJ.rt -1,5.9,lOjUe ta.i_ttr.f -d-? s
.r_f +;i' Sl3j.f o2go
rvt F fr,,:'.4( ? dr.s
ott =
oYY =
(\ + v)(t -r t,)
Ew
otr =
[(t-v)ett*zetrl
[(t-v)eur*ze''l
(t +v)(t-tv)
1,JJl r\-0-t)
(9\\-0-2)
(:;\\-0-t)!'lr^'
E ,=1b =@*ow=lf€r, \*y y
(e,r * svr) = v (ou-dvv) (\Y-0-P)
(t +v)(\-\t) (\r-0-2)
L-,; 6l+*i* ,-F; Slt E-* OiU s d:t , r-,.3*r rlrf ,.;5,', t1 ,0iJ oJ)\,
;-r.l,lk fj .rirL,
€r\=+t(f-v)ou-?ovrl
\ *Y r€yy = T t (t - r,) or -vott I
\ *rr Y(t +z) orr drr€rY= E drY: E T =,C
.rJ..j 01"1 Oli € ,*-i\,f5--i 1: l, J;t eJ.:L, u,
t'
1;Jl rf-0-2)
(v\r-0-2)
(v\Y-d-?)
[r]
y
\ -tz
o
rJ.rt -. c ,.5;h*,lJt-,)l 6la->J-.o ,-;;-S elt 6-* 0-tU q Ol.+ .r).:L, tl:-rf 3f; t.l-f ._f \:-,J.r Js-fe C-{ rUti Ls;i-
uyi*pbi=o,(i,i= \,Y)(\0- 6-?)
Y (\ + v)(t - tv)
O!nJ{r. aglz au94 <S/o/1s" *K" fA"
gYrL, l. 6laa..it a'F SU V - 0 - ? tO - O - P .:YsL, 6c-*l.,53 Ot{l-i
srYrL. :t flr"i J:^lJ,t 0t--<r- \ \ - O - ? s\ f - O - P .rlsLr.1.*iq a-.i,," slJf"rr_U;,;;^.o .€,ff v I (\ -v)'l.l:, sE I (l -y') \l)E 6V 6t+2,;.a ui;s.:-ti^o e-f _,1r -iJl 6 - ? ,$i -1r .!-,-r ! J r-rl-l gLr-l:+ fl 16l) Gii> rf pJq;J..*l Cl_f ,-h,c -lr.xq u-ll.: cl;L; 6)J b,;..a t-,-l t g*; Ll r-f-l[ *.tlF-U ';;; s ,F 6tar.l,. c.ll.> iJa,:? Jr .n--Sr-l gop 16la*Se,i:;..Jb' d.,- L
6lao,S'- ,F.S -Jl.o ri- ,Slf ciriir"ri^r6 .s-5'* F st ,.:,,lL.a tSu J-tW L-, j.t,Ssltb.-; A - ? )5,:' Jr €rr
'*f ^;f -r f I ty.l o:ls {-.11 "'*.-i-,!l :,5q €t..c
.-t iL r:.ifr *. )l *
,Ft;?,.{4Ylyj p-Pslls 6s.t,.r$t<. rsLn-l-,+gr-+ s Jslf r--.1r.rlr"itt q L e)J^S s.lg;^ac b.i-I;r.r:-ib f*tr sS: -r E?"fi.A|y;tl^1l.o9.1u" iJL,.-It-+l d*-,Y|L-'g o,li tjay.oslf l-.1.:.rlr.-l; 6,"+r"f 0Lii tJ orL'b*; ,)rf.* +.#J eSS +S t{-l f I dJ1-, ,F p+-;s c !4g,er .c*,1 ,8 Fi*.(::-,!l ,rt;f
.s! v-j g"? -r: ' ,;;ftij=ti/M) *t,,',',', (\-?-?)
{->.-,1r y'l -r, oJ'i t3-3,,.,11.l.'r-S t,i(D , u<Jt<, gLa-l:*;l c5rH, t,/*, ..li 1: +f
,Ej -r, 6-1L:;l ojf r-.1.r r-rl;r ! T s ery ojf .u..1r rll;r r".T. ,31.J:;.,^^ oslfyrtSrft-,..*^ (-.*- 6- .rtlf r>..l.r,*ij :, ,FV ,Jslf ,Jfr.JlJ.JrF ,$;
;l c,*,1 o:V e*t,l'' = q (T -7.) 6ii (\-7-?)
e* r+-rr J p J, p bls 611$ 3 r>-'l ,rL; €jF LL+l ?.,pa il .ts 6,g[l.;;f 6-tos,r. rlLi.j \ - ? - p *J:Lr _,r FF 6,lAs 2jJ- .(mlmfc) _r-,tr,_f
."Ulo-tr! rfrt, e* f--? 4 st Jslr>,e._,1s r*ij 4 t-,f dt,Sl-e,6-* tljE;l orL;-,| -e \ - ? - P aJsL, 1s \ - ? - p rlsL. tlrf r1l9 U
.t)uf .<ri-,YlJ'" t gLl d.rL" o d.:t , rli 1r Ji;tt .St^.r-f
rAt ,rL- ^'.!.,=&, Ul ? <J*;
,ii: +"ti-hdiior* + a (T -7.) dii ff -? -?)
.t1-i,-* €-fu ls os-tet L^.rIf pJ.trr:s r.lt* gLf FL* .Jrl.', y-l
6ii = (\ +v)(t-tv) [v6;pw + (\ - \ v) e;i - (\ + v) a (T -7.) 6iil
(Y-?-?)
.fu*,)l;.; ..r-,t*,| b.lss'r-*-S s F e:f r .-jlFt .St^dJ_r
-J (
ir::r",r-o-a
tF,-/ *V $.rYrL, c^i? OlS.ett
si= ffir,Eoii=ffilei;*q(T-TJl
(.jjr0-?-?)
(vo -? -?)
(\. - v - 0 d-rL,) \s-tt 0-tU l.t.*-,!l * p-a 4 tt ;.lf c.-,_l-r,o -rft
.t-l.iul A*j t) o1req t+!_l-rr 6t,-t.iq '-tt>
Qi= -KT,i (7-7-?)
s>c Uet& ,z ;/p*^iti -51 ; c(c..! c,.itJ 4) .r-,lf .'f g "tslf
.:r-l.to r di ji a5
- Qi,i : Pci'€creJrL'
(v-?-?)
.-ui r^lF ,j,f* 9_h 6\;;lr c^,-l"ta drL,
KT,;; = Pci (^-?-?)
(Y - 2 - ? \Y - ? - 2 dsL,) r-;;-S - uiJ,S-:-,Yl;ry .r):L, q'lr^^., nJ.:t-, ylteyo.>o (Y - f - ? .J:l'-) !tir.;; -,-;;f \lss s (\ - f - P d-rt-,; JrLs .rY.tL,
O:-Jqd aglSl a:-9;1 <Sb Lza" *& rLr
.J-a.:,r,o J*-(.3 1", .<fk-,| ";3 ,h-f*.(+-,)tJr; ,j; dl1 0U.. s).rL. ;tel
..,rr,iJF.i;gb O$^," Jr-q ,rIf e:6.:V.rYrL. I ,S)-f L.F *Jl
LtJl 'P e6v - P
.rr dJ*l-l.r ft-,- Sl-*,,-*Ql .tYsL- cJi-i 0L 6-? 1:;-.1r 6-1;Jo,.lLr
f-O-t iJ.rL, ,,.S,slS)V !; .,f-O-P iJsL"cJrb [lst JFti ,Q;jst-l i:'..ii)l6lA',Le 'p; "fi.i
.1.r t- -l (T-0-t aJ.:L,) 6loJ-c r.i; JK^i -rr a? t,Sp dlti -l
il*l r; $-yk-*,-t, Y-0-P iJ:L, | ,5.q gb-l:* 6jU;..:*,1 (\f-0-t iJrLr)
O.nl # Ab ,a,li € cb; = -Vt 6 A:_bt,Jj-p,-{J* V = V@t,xr) rf*;t;r 6b:8y,-f o.:l.r tj J<-,f C\V 1r Lr'ls c.Jl- .,r_1.r, .f=5*l* lsf=Q@r,x)
o* = Qr * pV; oyr = Q,, * pV; ory = -4,r, (\ -V-t)
.S:-'C dr".b 0t--<r- t* u. J:L; .r):L, tiLs.8 d/ljl o:Li:-,| tf 6-t5.+-":p*-rls Y-0-2 UrLr.l.:0-0-2 rJ-rL, r.l.rl.r r$ L.6la),a rl; cJb 1:
o\t,yr * oyy,ll -v(o|,rl * orr,vr) = y(\ * v)o,r,,, (y -V-t)
d,,r,r * YC,rrrv */,rrrv = -(\ -v) p(V,rt *.Vtr) (f -V-t)
f -0-t d.:L,;r \ f -0-P iJ:L, UrIf -b.rs U c.gl+rlr #f .Jt.- Alf cqLi', .,* u.
:gs,,:fra; Q; r(t - r) (o,,,r, * drt,,) - y(o,\,,\ + ovr,rr) = ydrr,rr (y -V - p)
Q*t\0,,r,r * y/,rrrr *C,rrrr = -(\ - yv) p(V,rr-V,rr)l(\-v) (0-V-t)
€*,! c,\.o.r, cJu.3b ,-r:l3 ,Stn-l:* U- : ,#lf db-l:+;t J*:, Lli gq g;l.a:-i f I
rIr ,*6 fr,,-:-".U( ? LW
.*5C""-lrf-r 4b1,,JJJrrLa d:L, J,-qd 3 r-Li ;* d-V-? sY-Y-? aJrL, J) y'
C,,,,, +\O,,r,t*d,rrr,=$Ygf = o (? -Y -?)
,5lrL- elt ,F tr:t'"1,6 )JYb q +,q b,F J t't> 6y t-l'.i, cJt-.;a 1s ra;Jl
)*,-.sli.i-...r J* J$- et' L.r:.n 6!b 6 5"1*"1 ,Slt .t;F O..'tt oavq 6"*bJti,.rr 6 t-rl !Lax, sxt il,r1b6ltL.+ J+ J<-,1-r ,.i5 g,lt: uata;.-
.I:...^ J-L sc*.| oJ..i o:b Ot^ij .rrj,r" JUI
r, (-iJl)
r1-rc
+ct
,tl
tf +Brc
-BrL
rr (v)
t\-z'8rc
t-v-, JK;
.r,iU t6.s &l ,i.f!,Jii'g)r (V tJ.or,rr JL:; 4rU- rl.r:,olx1 c-.&r.rr 6l:6i ul.L*. i:U (-;tt
l.{r d!-l ,-f .f \ o.t i .:-s q dl t 4.rt,,e ,; 6taa'iJ 'y (V t'.:-l o.t 'i \:rr 4J r:; 'lDv ;"-*.S!Br y-."p .aV
T
I
(iu r'<ra agly ^:-9ai agl.aL'to" *lS" rAr
crt*t; -illt -y-? dti, J<i Jr 0i .rL.t rf ))";* ,51V 6la-r,;-a ,Slt :t - y - ? dU,tf ,f _f ,SJ-I ,F 6b Olr;c q b .-, e.-.1:;l dld^? J+ 4 ,/5J* e.:*,1 oJ-i orl.r
.J;jue Lr.-^,e. rSLnl:+ # ef .-rrSCr+ L t^",S r.li jl-l
Qr = Arxr' * B,. xrr xt * C,,x,,xrr * D,,xr'J# q tsl,rL.+ J+.JJ Cb 4:cJ-,?-l s c*-.i up:La: ,-iJ Cb .Slf .51+L- * ,fV J(.i O{l ./Utjl .-t15..,8d1
..:*,1 ,ci; JlF.jL; "q_r' ,Slo.l.l4L- J+ jl rS-lt *l ,g,la,5ts-t'.rt-lJ yE"jlL oJ'-i r-,i-*-.,-'.3.:-;.JL^.iJj,J dld^+ x" 4;l i-trLl.".Jr O.'4 glly*jl Jn
.gl+t.-.JlJ Jr-tl .r>\^; jl56.rj.Jlcit- lf.;P,.r-r- Oltd s\-V-2 UrL, jl ortr:."|
#l-..- JJ*- t* y t"r irr-a dp-l JL..,.u[J, ,j.^-^ I .r-r.i ti-S -F st ,fV.-u--5'l.r-5',^^;
..:*,1 ,_db-1, 5*-f :tnAt 6(,=5.-f ar-J.at ?-V-? iJrLr -lr gj:.-r d.ll4b tt
:5,*'t c.rl: yj Js;e t LI,JJ eo3)\c r,
dtr = Y Crr, * ? Drxr
otr = ? Ar\ + Y Br.ry
dlr = -YBrrr - Y Crxr
.rr.rF J-.-l o<^-. utsli,;",-,dj", JFL,- ,f-* J -k.iLi'.r.sbc.rt3 ?[iujlti:I! JrlJn qi ,oV l3,t Js I t+!_l
J- -r, bli€ls ss-f yl .-u:.-,o yefu ssDr j+ u.t"*^:tJ lL; t5.,ry (-alt
lr 15 OU"^^ ,.li .r1tat;t jr 6l.ar.rL,. djL JrL, il*-l q ,p .i- lB ,.F gJl-\:*i *W t-b u..f -lb c.{+;: ,l; .-i-S -F1.r t€-,1 cJ.i c.rl: tl[.ii 9\ -V-t Jft,
.JS6..,*ittr sF;t +LaU r,
dyy=Oly=oor\ = 7D1xr;
'> aJ*-i-U( f clr.s
F ,ry sEJ;l ,ie;ArW Jl r+'.1* .,* ,...:*,1 o-:L ,l^; c'Jt- ,5ly 61i" clLr
*f*r1 r:*ix, dfi-,b )t;)l J-pc^{? 6,.:,*,1 ,SE J,'Jr'i JL.el ,F OL'.Sl;
.c*'l oJ.i
.sr,-l *V ,-rt t L!;rJ-.'e d'l;i 3r -,r ,J.ilr ;** (Ct t*)8, t*- ft <*l
.ror"/,ltjj d'- * o 6 Jt- 1r ! r:s J(; d-V-t J.3
f -v - P JA JK.t
*-l a;sJ 2r; OT.5qsr )'P LS)i oS )b;r 5,-;
.rJney $ F e.,,glols r* Cb t:f - Y -? JE'
Q' = Brxrxr * DrxrxrY
./t-#.f ;b .:..; ,ltfr &- e-* 1r Lar:i S- elt iqH ,!,U Cb d/l 5 -t ,tr 1lL-i'j
4t-.^.. drr-l -lr ffi,Jb-1.* a.f15 ,f-f .''*'l oJ-l o.:b bW lP.l):* y-l '..:*'l
.J;.;it-a
,-b yr;6taa.iJ ? S'O' = o +f '>'-'l "5t.tt lU fit;"- e:lS-'-V q .,,Li \iLJo
:Jd-f ,-l r*,1-, \ -V-2 iJ.rL,;t *,a:"-'or = | Drxrx,
oYY=o
otY = -Br- YDrrvY
l.Dr sB, 6bclb 3 .:*,1 ../-t{rl1L U 1l-:fr-, 4, ,Sr.r ! lFlllr ,ii dt'oa;Jj' g'-l
.$lui:t jl b-rt ,tdt, J "/-)q C.b 6ti.Jiill .-r::""o,i::*., ,j.V e:-t J'-F \+i
Cr:-$.1". ,g(, o;-9* 15L6L.,..r" r3*9" rh?
t-t tt9'Xu-".i.i{ D1 .-*18, L_i g'l-l jt orti:-,| l.:.-t.iq Ju q\x\ = tc -,1ro,,,
:r-t'"f c.rlsBy c{lJ 4, t+ .*f dtF y-lr:e Br = -yDrct S,S:-b
otY = -8, *+
d,,-f;t:1 cc*,l oli; J-t,. _r:i ,lji ,Jt{Jl tsJsotr iL F L* Js1e +,. f.,^:. }.PiJ-t Jt^.et ,sJi gLAx' = o rrJ.i,,/-l.6;,lrljiit-, tsss u:;: u-l ,s-SJlStl Jev
:.gldrLr rfS,.: 1: c,.:-,1
P - -f" [-",
* B, IJ*,
6;Jlf.Jl pt+;l ! ..- *,ljt+ ,-r-1, dt ,_F sss.:-)tr .rl:.,lri #, q j., .--)\e 6:s*ii J: os_t- ^.'.bli,g b.-j; ,S1u,"il? C/l1ti .Br = yPlfc
,*,_-rtr
rPo6 = -i;7 rrlr
dYl=o
:6-rl, .Jb e.:*,1/ : Ycr/y ti e';. & ,r*L O; -r 'l
,JlJ t"l
dr= -l*r*ri or-oj oy(=-firrt-rvY)
.c*,lt ul€ GUrb 6:-F )l ,I-V E-tJ q $i.rr 6d jU; sil (JF Cb ,-i::'^b1 +' 6l+J-a i.:i:-i:-,Yl J, f{, J* orr+.ft-l drL OLJ| ,1..1:.. C',l t .L:*^ ,.;$t_ \:*,r ..r;d rl! ,aJl-.rt-.o*
-J,[ixls 6 t-b OLr .0 .9r Jl^.a:z,t:t eW -J],
^L*r,r*, LSlad;J]" q rlr^^
r, 6lr-7"J 1-fW.6b-l-.,+ g"a -l 0LJt 0rl .(5q .f"5"/ j-,- cc-1..i o.:b Jl.:; ?-?
:g--rl: .11j d:b d.rL, Jsge
o'Yr= #('-*t
M, n \ iltre +or-oo *R= o
ar rA0 r
o,'P*or
, -, JsJ
uJS el.a*.t2.r .rii- .rl;lri-r ,lLJl
ljJlv-v-t)
\ aol +fu.+y?*@=or00 ar r(.fv-v-t)
\ ? t -allv-v-P d.rL, ,.1-bsoo;* !/f:i 1) .Ji:"'A # OUrf -ftU@ JR 45
tt{.* g",-tJ yj iJrL. '^b:
o-=r -&-++4vr- r or rr a|r
arooe: F
\ qa \ dYd=r'e :7 N-T arao-
_aAr
1..-cJll-V-P)
(v^-v-2)
(Vn-v-?)(+#)
Ot- t' I o agly ou*92i 6tb L:"* *Lg fAA
L; J,-li d CJ-l F eE rrt- \Q o:;, ):oi )JLb \ 0 = Qe,0) ta;i 1r 6:4i C r:d,r ! $ o.2 ;e a,t ?-y-? iJsL, jl 6 *JJrfii 1.r 6JSO#'iJ Sr,!yt|= "
Y'o = (#.+* + *#X# .+ #*+#r : " (q -v-')
Y'Q= (#ar*)(#1J- #)=.r #r yrr # ,, #i*,ff= "
:tF,g
1,-iJlt"-v-i)q
(./\.-V-P)
(.7\Y -Y -7)
(r7\Y -Y -?)
:.:*,1 l) Js-r- q .Jr*rL4-: dsL, C,1l .Slf ct'-r*,f- 6sl: OH;tllj,-f
Q =A lnr + Brt lnr + Crt + D (\\-v-7)
:s$,g rf I t: t-: ,F r; .gtadJ | $sli;. gJl- S\ 6o,=+ +=!*B(r + yrnr) + rc (,-ilr\y-v-p)rdrr
oo=#=-+*B(r+ylnr)+yc
tre : o
-qt:B 3A A..3L {.il$ r7s (fryc^*i,..fr-*.Yl p-.." cs-91:1gloj> € J-l:, Jle"t i "trl.p JeV.L;i; Oi )5 j+Jsrt*: tgLar; J.2se U_l -* ss lJ: c_r.iL ,i..a6li:s' r, c;l;S_ ,;;S t- -ltJ k- ,-r-,ll c.il$ 6lrF 6,o 6 6l+r,.;-a.Sf-r,: .f-lfe
..r;1.: :_;>_9 o1k:r u!l-6lo-t ^; i ,5'l ,i:-i t, Tr." s: 4;lcjt> q ,lr^n cJt- S ,_eV 6lu,.tu.o
cf)l,J-,-j JU- .l.: .:JL- yl .c*,l r*f )lt rf Ct+Jl .9.r 1.r dLn, -r: grJ tf c*rl
.:-l-'g
r -v -, Ja JK.,
./-t-dl 6LaJrUif q J5; .p -*
0L'.4 c"r;6 "s"l<$oJ.^io.rl.:oLij
rLtl !'.r*; 6i st l, lr,J:J:f -y -? ,JU
.J;5dr.+ ..:1[ lly JrL, c"l.i
ji* ,-r-1. ,$-t hlri .r-*,J ,-f o.:b \ Y-V-P iJ-rL, jl tr.rr t01[i; :l:cJo
.r=bJJJcr:asyclr= o (\ )
. "/l{rl C-b 6r.s ts Iu
o, dr = " (Y )
.,-/-t{tl Cb tsrs st f, ", dr : - M (r)
(Y)J-t 6ss 3-la f t,e = o
C,:-qr. agly a:-921 6LaL2o" *K" Iio
o.rLL-,| C tB ,a s1-qlj ?t-., ,JlJ t#l ;l 61.p, € F--,1j o):1.- q L-lr .i*l(\ ) 6jJ" L-rl:t GW;r 69;....o JU_i uiJ $U &)V-l ,rl,,;b dL"Lb s:.rF,-/
:tF€ l;-.i-;'l tj os-r- u.
A lat + B(\ + y tna) * yC: .
A I bt + B(\ + y ln b) * yC = o
4-.: (Y),Si-t L*:t dbt "rr J rrl$ rFJ oL,ciJd dt{Jl J)rr ct'-f dJrc Gio:r-*,9 c.ii _t) J;-n
dr-.b (\ ) djJ.- L;;l 6h4 dp 1 rJ> I r1,- .rr-r"r- rr JlFt d/_l L-,L..A*Ji (f) djr" L-i:t G1J1r--rr-r ,Jl{Jt e.:y oJ-i JL-el .gbrlL",r,fltit.,l,., ;Fr;
:t-Fgl
L,#dr-l#,1'; f#dr= -M
:p*rlr Jl-S-tl osVll 3 .:*,1 ;"- U',r-l :t esf ,-Fl: il^- (\ ) .b-'l #, f
Qn-Q,c=M
Alnbla +B(bY ln b - at tn a) + c(bt - ar) - M
C sB sA 6,La'>'.U -r J- bliL-f b ttl L_$:t JeV r; iJrl....er ol-r^.o q.r.r!o,_,*l
,f na-,l-, t-l
A - - YM o'6'1n b
Na
B--Yrr,-a,)
Iuo,dr= t:#dr-#T,0"=
c=# [at- aY + r(bYlnb-at lna)]
s:l-G.-Vq b,3",sL^d1-i, oli,-rft?"lr- rN = (bt - o')' - ratbYllnlbla)l\ 6.r-sot l ,taltJ -e oJ-i JLrl tlt^.;l GW YJ \ Y-V-t t:l)rL'.,': Lal:'13
o,: -Yl+t"*+ bYln i + "'r"Ll
oo= #l +t"*+bYln ]-+ a'tnL au'-o'l
tp-- o
.c.*.1 gt+t "t,.U
Cl-' u>*-ella i;.;1.: tt-SU-F ,f1.: .-5 et Qzt L, eb OJ-l ii'.t.:-,1 .f*L $ e; oy, l;sfF -r, 0 1r .r.;*' ls y' jl
"*b 6lP u.Q fl
QQ,O) = fQ) ,nt (\Y-v-7)
.tLr.r"j1 "l-
co--ln = . *f,lt- elt| fJ iL-t .,-ti JJ-"- yL:- a'.
jl JU" d;!..:*,1 olu! o.rl: \ \-V-t iJrL, tr 6l-t .rl- lLi' tsiluf.rtlol 6 t-b
,€A iJL'- l) os-r- yQQ,Q) Cb;l ,lS*i 1l-r,f ei e*'ln = \ 5;i-
f-v-rJtJ,JS-tP ,fVtzL L 6lor,-l.r gt )tfr 5.,-;
O.-Ji+4 ag(y o:-9a 6lh L!.r-," <S+& rlfq, = (Art + Bh * Cr * Dr ln r) sin 0 (\f-v-?)
e) x )- eU \ f -v-t aJ:t'. '^A:1 o$ cl-rl u- Cb 6r+r clbi :F - y -? ,JA.ty,g orliil .t.3-jl.1rP 1b .:; oyls
o,: frn F. ?l sing
oo=fru,*9.+lsing
tns - -fru, #. +] cos g
r=a sro, = tp = . c$-t L_tl, Ort f tf stl:..^.a u::j $U C-b ,-,-.rV I .rLlr Lt-;.5t or; Lfr o13.e t bt i € u .*
5 !;:,;fr;;[ ; :r, J:I:,.1 0:. - -p
.s-rC O"..t D sB sA ,51.l,?U l+i;l 6,-r-i g;- drL- 4r d, L-fF .f_l
yAa-$* D - oa'a
YAb-14* D - ob'b
-A(b' - a') + n Q-- D ln L -- p
's-*ry ,* tJ oy'e q Lnc'lb ,)g'- r*' 1 drtt -./-l J- tl
=*; B:-#; p--P(aY-+bY\
rtr
,Jt".iJj, Urrlr b.:rti yl;l orti:-,| ,gta"lr.ff = at - bt + (o' + bY)h(bla) 6lJ-as..r,c d.+rJ Ur.lj
o,=#(.+ ry) sina
oo=#(".+ ry) sino
P l--atbt ot+b]\cos0t,e = -F [
* lr
o,=os=o J t,e= #tt.+- at + bt'l.
'J"so=+6;r:,S JV )rt,e =
P I . arbr ar +brlo'=rf*T- r I
oo=# [" + d !r, ]
.p*Lr"f Ot-t, f \f-V-2 rJ:L, )tn = f c.Jt- jt JUr.l-r* q b uin OJ-t
cSsF giJ ,5.r- .:* 6 Sg. g-L-'- g:,,.,t;e1-: ,.i; iL,- )- elt:A - Y -? JU
-r.j 'F Cb;t sl:-,| ,-iff 1r .-Q;5,glorrb ;-P &-,gl-rlr -l :.yl.r lljS c;l;i-..:*,1 oJ..i o.rl:OtJ Jft' l6 tb 0L".-t$orLL,l
Q = (Ar' * Brr + Clrt + D) cos y0
:J.::.-.f Js Os-t- q .J<-, .g!ls r;,Jl"*tj. n-V-2 iJ.:L, :l : c.lo
o,= -lrA * 4 * l9l cos yo-' t'--' f ' ttJ*
O:-Ji+. agly a:-9;i aglaL2-" cA*& tlr
oo=
tro =
* \ YBTY
* ?Brt -
\0
sin y0
IY/
IY/
a $'l .o,
?C YDIT-,fJ
:-t:..iq -lf tr,-! -r-j t$-r I-lF e.r;ls;!i i+,;r b ,J-,l; ;: 6 ;$735
or=ScosY0 ;r r--b
tro=-AsinV0 1r r=b
0_v-r.lt:rJg.t
,5loyl.r {r \ S:s*.;[ r:-*; L'^ o
:t-*c/ A;.tJi lj Js_f- r.6.:*,1 ,Jij $U i,;*.rbb p:Or= o 1) f=A
tre = o JJ f =A
(\ )
(Y)
(\" )
(f)
\A+#+F= +\A+?Bb'\-# #=-+rA + #.1= .
\A + ?Bal
:rr5,:,rp : ,F blid ! )U .rY:L, tb * o 6Jb r,
ars(_,=--f
e, = S osy I L^,JS ,0 = o ;: JS &- *fr!.t g-a-, \r ,jJ 6.tJa;1,
o0 =S - YS cos Y0 Jor =t,0 = o { = a ;s .t1i/ ..lr-{ (o.rL- j#)cls--tp-- o
6-u:*roe - S - YS(-\) = fSLaLrrr ,t = YU-g = iO;rt.:*,l,.lijl ;Vso0 = -S ,,0 = n \0 = o S jU:g.+^^ c.:-,1
'?V;r o-LJ n:;t*
'p f-r^, ,hV.,-:*,1 ,f 6fy b 1-r rS ,J-b
ol-r^.a +, t+r-l ;l .ilcJ-i Oat':s tAu:.J l4:s ,ra;;". 'fr.c/l .lr 6 t-f olt il 4!
..r-15orU:-,I -h tr,r #L,-1, Ln"5l-l++,t+ Oi#.t 61Lt:lig,59 OjU
?C YD-7-T="
; D=+n=-*, B=oi
o'=+[' 9'] .+['*Y $l cosro
oo=+ ['.#] + ['.#] cosro
t,o=-+ [' Y.9] sinto
O* r'it* agly a:-9;1 66L1D" *& r1?
'&A- f.<{ ..5,/L- P- ,f.=5.-f U_i l, ,.i.a*, 1r di.-, y jo:L ,F ,b -S ,.rJl 11
,fii 1.: 6 :* OL! 0l dte'l 9 ssM1,JrL, ,j,-f --,t cati .5l,y.lr CEi, C! q )F3 oJ.3, -b+ = o Jr J.r
"t_l#l & # ,iJ.c.*,| d-S:il-p ecri o:lr Ot,ti JJIV-2
&ti-tfc*,l Ol t r**V*,g dl-i sFJf+ = L )r* _&t P&eS 6,lr-t_31; .:.-; ll-l..5ro S SV'l GL. a;Je ,jx.:9', r>tJ t1_g*t J.5f 6l+*b
& 1r,5lrtii cr.Ll Alt.Sg,ll-r-1, sr.ll c^rb.Jt.g:,l;l oJ-a itrU !.*b&* * CF" ss P 'IL;;,-JU, dl; .E*€ 0+'r-j: i;l.ul r h st fl-r gFtr[ix l: 6 t-b 01.^ 6ort \ s-t*;lR iJ-au lr .*t- 6 $'r ,-P ]t&hR git*,
J(J).:.-,1x, s\ .>l-.ari. eJ J.IJP iLfi tt {rU $f .r:-,1 cJ-! o.rl.r rlLii c-rV-p
-r*JJ F-p : aU B L-jlt nt:J J cr3+.4J* 6drU d/tjt orLi,:-,| lf .tgV-f.d/-llti.cos(l ,-P):*;sin(l ,-fl) = *'"-, r..:-,1 osL0 yJut Jutq5tacrt*
ut=-Axr4i \-Axr4i uy=o (\ -^-t).€,-,1 sF ;,e.l: J* dljl q
'F*ia5lr9 6
d.rl,- jl b t^.F-S Jli€ 6Ye;i - uij * ui,i ed-V-y iJrL,-rr 6 a:j rU tiJ:t'- r, tr.rl;-rS6-1li ,rl; ,Su,nlJ? Jt:j cd,r t Cll tF,r-5+t-, \-A-t
.(p = G,-cJln-v-p drt-- jl lrj) s$'tr$ V-f -f
otr = Gfu, ; orr : -Gfu, i otr = dyy = otr = olr = o (y -A-p)
6ly c-tisy .J-i cLi:-,| $ uis:.lLr q t^r:;-SdJ?..^ tui;; ,s6^i)? yl 6b;i;lt. .tF'l O.n't \"Y-V-Y rJ:L, ,StKjL .r)rL. cJ.tj.,^.o ;"- \ s cil-,a:z,t )l *b(ls t* u. \ -f -t rJ.rL, .gtc,, ..r)rl-- ,;e u* J,-s;r c.Jt- 61,11 ,u.LJu yb,;6LadJ!. r1 ,.-lL ,J-l ,rsJ.cg!.c-,,1 .r'l5 $u tP;t+ it-,.rp € vV
rty aL; o:t,,*Al f ,bs
0xtR
P (v)
v -, JS;
.rU;+ (V r s"t' .|i.r, P' u.P i-L;.rLttlt+ (V tM1 l.>1.:' t2> 4;615V'S t clr;-l (-il|
f 'e';' * F*
O.*J:,*4 agly ou9;9 <Sh.Lza" *lS" nA
^-, J6;)-t I ,fV -t rSs-t' ii,- 6l'r2lrra U olliJs P P
{?t ./V-? )S:;r oJ-i C} '*-t & b-rl* 6j,6,ltLi; yft 6,sj--o €.{+ rJ
!..:,-,1 Jj,kV & C*' jl glJr,ily'-rs ,-F.*1./; eclL;^a .,* y
t=tlffi =6g\M=GoR (f -^-p)
..:*,1 1t'-, 0t J rJ.re 6 6lr&fi y' tr Lf q 'it
bt1..-*,t:i. Ft* .S
:6g.it-.fJ, f sF./-t{rl 'r.;e tSss }l6l^-rrd 6ta0L, g.7 tt
Mt= JJ fr,"r. -xro,r)dxttut = ", il Rtdxixt = GuIp (o-A-p)
.c.,.1 &*,r*l uJei b]'^-lp16
C* 1r r,7ji;.lt- e0.,1-r-l: )l -* ,.'. u,L;. C! r" 1.6,.2j.:,;. sF -f csll
upL. 6lla;-a .|a.,, \:.*.i 6loJe CrU, ec*,i cJ-X o.:l: OLfi A-t J<-, S;t;J# q toOti r*.:.; t1 \ -A-2 iJ:L, J{q ,lt.- * cSll.$r"/ C, J.r.rU 3 _rujL,r^,
.rya 6>va ,jur = -Ax,,4 i W = hx\xy i ur = t{@t, x) (? - h-?)
J g-,1-rr jl #:..-. J+tt a.fl$ qi .A-Fr; ,t9;i,(:2' +b eb'l'(x'' , ,rv) a1 6
af-t s74t +f ,",|'.rf obf c1t4^a..:*,| OL*i- ct'-f CrU" fk alr cr-lrQ
.r:*,* fjY ;JL, t-b u.lr Cf-t 6c*,1
il:L.;l o.rtiil !. .,llrcl ottil, 6loyb .rL-i ssf ): df ot-r- JJ- dJ?.!.^
d/-lJti s:f r*,tr., ?-^-? iJrL, jl b ",:s
U3" Y-Y-2 aJ-rt - o1)\-e ar 6-V-f
ou =dyr=orr =otr = oiorr =Gq(r!r,, -x)i dyr= Gq(rlt,, -xr) (V-A-t)
..r11$ tFJ c^it-i J*,SbOUlC,"t.Jl-t .r:; 6,o +f .:*,1 €ls r;6tar;J ? ,ll:l,* -'. ur.-.o. 6b-U ./1,:i ;: V-A-P il*; r, \ -f -2 iJrL, .Jrb J3l iJsL- j:
lJe:,rr ra;i J.rb lr iJ:Lr.1., 6lljKr-b -l sr € E:U JL-Sr-
orr,r * o1y,y * or,r = G0(p1 * 9,rr) = o (A-A- t)
.-r-,tf ..(*-fJLa .:it-i ut'-f & 6lt !-ll r/ 6.r;l: cJ)r 3
Yrtp = o (1 -^-t)
.JiiJ.i 0-.b Jr-t ;j $t^i C-&-,. cSu G)-t },-ltt
.l;b ftl- 4q A-f -Y iJrL, .r"LlJ $ \lr.t srrl ui; $U 5;t C!-,y
ot{lr*o;nr-- o; ottllt*osnr: "1 ortllt*or1n1= (\ " - A - P)
g..-'tJ oli- ,-* y s):t .-. cp-l jl Jll iJrL, 1.r .r:-,1 fly = ' b.i-l 6 4;5 qy: n, *,1 -t:"jt J Lf y 6 ,)V .2: c.b1i6
G0(1,,r-xr)n, + G0(1P,\ *.r,)ny = o (.-iJl\\ -A-t)
:.rrj6;.r ti JK.i +, +LU$ 5
O.-.L*4 6ly oa9;1 <S1'alza" *& l'o o
rl),tflt * {)Jflt = # = xtflt - xtflr (9\\ -A-?)
S! 6*-r"-rLa A-t ;$..X 1.r oJ-i olr OIJ c^il-! Lt'_f &.rr -lr_! {({, , .rr) ."rrlf QC O-rl;* L-.- rJ-r-r ../\ \-A-t iJrl.- J,-q .;b & I JUj 1 c-; 6i;s, -1
.-^5Cr.rtt l:'if g!i.,il ,P-c.r::.,..l JJe, $U
'** *U" i^"..o; q *t+5.+ i ,,J-ly o5)tr
L-t'-f Ct|.i, di .53., orL
!l ",*ttur = II ",*rtur = II "*r&t =. (\ y - ^
- r)
0ljd .JlFl Or:*,;.J t'. *! ! ..-t ;"- U&_lrr JIFJI ,Vy s drr = . 6ta"i jl:c-!i
o e I ! rt,,
*x,,px dx, = " { ilfil*, t t,,-r, ) I * * V, (e,t * x,)l} a*,a*,
(\f- ^-7)dtr JlFl u.l: Jt-fu| ylol.r;6 u-J'#;l ..-*,1 or-i o:th-,lytE = , L; S
,-5.1.,_.t+ q.) rstf C L*,- Jf
Ce !"x,{(rl,,r-xr)nr*
(r1,,, *x1)n1}ds = o (\f -n-p)
6fOt-.f;: lLL.Slq-,t-, tf .rr"f g.n'f, CpJ,l ./\ \-A-2 iJrt', il*-l +, rj.r_F,-r;6"b ;.; \ y-A-p rJ.rL, ,JlFl d,i.rr
:r.5".1 irr'ti .rr-tf yj L-lr! ,Jcr = L sh = o s..:^it-i "dtpl C-bJr Jr
I ! *t*rtut = I I r"*t&r = ", il (x,o','- - x,,o,,)dx,tut = Mt
(\0- x-?):t:,j.,t q# JF*i ,Er )l .s;"-.J-rl JlFlr)ow = . 6tdi;l o1b3.r
Mt- ", ff
(r,t + x,,t * xfl,t -x,,tp,,,)dxrdx, (\p- tt-?)
J.t-te ! ,.l?+, c=l.a.i_J^t t
K = G JJ O" + ryY * xil,r - xrtlt,,)dxrdx, (\v - n - 2)
,lt* -r-j Jy'e I ,:?d, L--llj tt:,-'l n -'t:.," ,l.,U c'r'i!. or'i uih^?'9@' ' x') is 6itF,g
0Mt:-K
(\n- x-7)
doLrry
- -
" 6[y
aoi otr = -A
ao&r - - G0(lt,
(\q-x-7):.rlr'rQ
*xr) (Y" - x-7)
(YY - x- ?)
:t--J'10 '**J 6
# = Go(.l,,, - x) i
:t-Fr.l Jdi s4J)L. $+ U-l )lq) 'jJ- ll
vY<D= -\G0 (Y\-A-P)
$U +,q er:.-,h:,- sF\ A:l-f6cJt-i ;t? g!-, ..:*,1 ..ri.i aryj gtSt; 6JJ"L 0L^
!.rr O.n'lt Jr-U. \ o-h-? aJ.rLr;l d:t , (yy 6.:-,1 ,:.r 0t-rr d/-l .$t^1 "5j!glJ
,rt ub h 45"-Jt -i rgta q jr,. rl-r:,1 ,.t,3 n<r-rl, J ft Lt: ,y-l FI ,41 :izs
6 t-f ,f 4+:il n-,J.t-a jl f:i:*. ,* y..:*,1 Jj cJ^i o:b 0LiJ A-t Ji<-i.l:-6
1.: ,.1J .5Ln.iJ-!, C L7.:l.u,l :, J*&;f.y1, dS '--'.-1 ,r:-k ,rr=#:J-, .l;.6lJ* o.rl:;*j aJrL, L-'y Gt9^-e c^{?
dh dx\orr6 -ovr6= "
da3_3 q (Yr_n_t)ds
I
I
o.rl: ,,aL^a:;l r.li o1 .i,a )lJ;t Lr;r-1r, 3 c,,.,1 qV gt P L-u rl-rirl 9Q ut-l.rtb.
.J.i falJ,;
AO &r ,. OQ tut
Arr ds Axy ds
O:- t' I o agly ^a.92a 6La L:t-" *K" Yol
,prl: Q
-t \ Y-A-2 aJ:L,
JJ# tu,&,= JJ [f *,J tu, = J.], tu, =.(yy -^-?)
ilrtr-.;l allt'-. dr:lr 6dl--s.r- .#, tu coti,c t-b y..:*,1 c{t3 0JrLd, ,sr:Q \)-l: J-ll LF ,t .t-fr,g g.n'ft;j .J.rL, ,yyorr = o ofr s :F,go.r'tr \ y-A-t
:t{,g t-j rste 1 t+i (.f y s Jd--J.,g O.n'lr i \ O-A-p iJ.rL,,.
JJ t '' #l -* #ll tu,tu,- M, (Y0-n-2)
..rl ,fi oo-f jeC t -, ,SssQ +f c*;t, d/_ljl o:LL,l s ,F {. e g e}lt$St U.
Mt=tJJtuxdxl:t-F.* {+:J
(Yr-n-2)
,-*-r gLL I.., cSss g,l: yl6c*,10 ,-* 6b d,.:- fJt,- ,jst alq J-.d/_l;tliJrL-. ie ,;1g../ r,)tr Gb .rLl1^.t{.l O.n'b ;., y\ -A-? aJrL, j *-,1 j.a
.f=so.* \ \?-h-?
f-^-rJaJ6;.s,"V.23^n cSy+ & *"
b ,ri o.rb ot.ii ,ql q 6A.J,t? .iLi ..! ,Ji"+, \s\ s LauiJ :l - A ' ? dU
..r;S,:".*, c.:*,t6:jlJ) J) Mt ,56r!,25 t>; 6 iU:
l..:,-,1 l) J-s-t' u ula.Yl iJsL' : a.,jo
rrY,rrt-rT-T_ \
:e-f ,i J. JJ J* q t)0 ,P Cb otlrQ
o=r[4*4-tllot bY 'J
y \-A-t drL-. ;l ..:-,1 *- rt'-f & luf- ,SssQ.ollJq '":*'l ctt'r dl 6:g+tlr
"[,+. ]] = -\Go
atbt Ge
^--ffi
:Orlfe
t\?-h-? rJrL, jt J;
:6rJr--5;
II,,r*,tut = I"y: LnbaY
II*'*,tut=I*t=|nab'
II *,*r : nab
.J
:.:lr gta,l M,,Sltlt,y Oli€(..:-,1 Lt'-f P ':--L-)
naYbrGoMt=ffi; qri ;l-ljl
C,:-.u,1,+ 6ly a:-921 agla L--x <S*9" lof
- (at + bY)u- ' vt" naYbYG t
iJ.o.r,r". n-r;j c.r3,i gtljK_t+ O ,*I.:t!o r: 6;L;
e=-#[#.#-'1
do \M, -- do \M, :.,,'F tr u;
otr=A: --\i oyr= - ",
= -ffi*,
:U c*,1 i-t s Sl.-f ..jU;l .q-f .t=.sqr;|.r, ,,.^,_fL ,;IM,
4-r''tmax - J- .TftaD',
:U c*,1 ,11i5 t|,€ +;, ,-fu, -=.I.a ,-lLi q
x = L: raYbYT = G(A\r
0 a, + bt yntlo
.:-,lrrly ..H; ra ,t kV ri; c(.gloy l: u.,orc Ctt) a = b 6,,:Jb ,Slt S +i
c=tffiQ__+,
.c.*,1 jt-rL 6:-F tl ,ilt rr 6c:f r.r! o.:L Ltll; q 0l-f.'# b l+i .t*" 6 s,-la)Jr- ss-f ss6r-1'.i ,tt.il .r,_tl
il-tol :l ;l-p LAJ"LJ g,l ..t-;-ei r:-,r I c6r J* q F Cb;l orU:..,1 lr .,|-r"f l, ,y..:-,1 GVJb u:in, OJ_l
g4 o-. i;*,,;,.Ylq - ftl-i -F u \*- ,S* J-r-p Jr JJ 13Jn tu i'. ,**,Yl ,1:t*" ,F ,*,E -F :t
:b
Y-? jb))rJJU,.rYsL, J-,.rLtf 6.:-'1 6l& 2t;'; cS*cJg'fcJ-ll '.:*'l ^iia*S ,fr 6+ c|yfcg-l: rJli J-rS.tU-"f A-,-'ti i \ai c/l.r, o.lL.;'r I s ,l!Y-? jb) 1.r ,-"lA - Ji* ,.:,; .r)rL, 'L^r 1 Jrb O):h," F :t 6 tP€
1.l ..r):L^ Ul .tf 'l 'J#. lr.,.iJf *J e:6)V '.r)'rL' .rLlJ ; J'S f.;it-'t
.J.,lo-\.3 .S.r;irt+ ;4 .5rn r-iflrt,
s orf U-f .l-jloJ-i '-*,?l Y-? it;.).t, 6J(*tt-,ll-)l u'"l.ll lrllL' I,€,;rlF +.'+ 6lao-lt^'i q h!-lt'l-rl+i o-,!;'
c\ -f -P nJ:L, cJrb .r!rL,
oijj+pb;= " (r-1-i)Y-T-2 nJ:L,,crt^(, # ,,-Pf ,.r).:L..
Yei=u;; *ui,t (Y-q-P)
uY-Y-2 t- Jjlf-r-2 d.rL, c,5.l,a OjU
6ii:)'6iiew* \pe;1
r,; = *t(\ -v)o;1 -v6;,owl
(.-Jlf-q-2)
(vr-q-t)
iJ.:L. r: Oi qp drl.t rLi -r ,..,J|y-q-P ilrL, ,5.,^ rlyf lr t-r-j drL' drti(-F t1
:7-SSl€ \:-'J +r (\-1-P
l.4uijj + ()v + p)ui;i * pb; = " (f-q-r)
.c*,1 .:s* ,-9b <>AsbblP o 3.:*,1 ss4-t.:t jr Ji.*lli, iJ-rL. u, rl'tJ l5
iJrL-. 4-.: tJ V-f .Jir*.:-1) o-t'..! C-y ,-;-f A:6)V,.r!-:t-, ";.J
.S-r-f J-rt ,Sl-t
t€rf-'tFt, Oi to:,-t ss 3 osf ,t,-r=,YY-v-f
eij,m * ebn,ij - eW^ - Eim,ik = o (c - q - r)
\ -1-t iJrl-- ! 0i :-5t J ._cJtY-q- ? 'dsl'-;l ..:U:-,1 q b..rij yJ.'F-t.l.l!o q
:7:i,-f r:rr 4J
Cr- \*t o ag(y a;-9;1 agLa L=x <SrJ& Yof
o;jJ,* * * o4;i * p@;; + bp) + 1fr 6;fbq, = . (t - q - t)
ev:.J_f;t) a? Ln1,,tf 6JzJtL,.?1.:.-tr,.-.o ,g(*- ,Jfu <S:E)L c:4ilt 6,# gln-t.* 6 €'* r+-r
"/-bc.Jt- I t{.:j cr.iJ S\ J-r-p -rr a? -r clt* r*;;
S i),y,f^^ J-,lr* ,sfy:t lf \ &tf:&t a.fdb-r.* o* )r;> .r;ri-c.:-*'t tb ej-r t"lts./LlJ 61,!J Jlfdl J# q JJ-*-,_b r.
q.f=s..i 0K"r*if, 6Ji J-f-plr 6lciL- e*'t!.Jt"J- t lql l.f44r.;bl
.Jb,Fq *J b Ot<" J*iii 0Ui. cgj:.,- t* u.y_1tj ..r)rt" J- ,JlJ .ri)b .Jt?
cJi-ar,_r,. r2,"t t, jt* ,-r-r, 6t",Fq $i J- 6 | ;)rL, 3 o.:r5..1! 6:ro s ,SslrJ.
€'.,y';-'t.rY.:t-.-,y-:U.r)sL, 1) oJ-i C-b.jt^.,L*jb,
-J.ui 6lJ. rJ)+ 0t"i l.i
5l;- .-/t^,j,.jk, J+-".-lj,l .e:ig '-;rr q l-r.*t- .Jlrr[tlr., ,JlJ rtt-d-tbl tJU .rY:L, .F y rJJ::"-l 6\l-rv' ,jss ^-,.-r;13-b-9s .fuJrJLA tr .fu_f-lb
''f r,rlrIJ{^l^ :S& 6:_F J. & sr.:-,| o-t-i orLii-,t 6i;l ,-is-t 6;;;.11 6,9l,r;ctyS' iuo 6a-frL*r-rs -l s! :-eJ.*,o .11 ej,,6 6sltL db ro 6 S'l rltl .e .*,1
li;r-TJ*b tlr-l 6J1t s F*;lsy ,r(, cr:*,tauJj,. jl &i'..i_ al *,-jal "!U
1 c.*,lQ -r.;t 6Jti,l Ot-rFa .t(:i c-r;b ,;.a a dJ?-$.6 olgiJr -rlrrt .i- J-S f ld/lJq
,5t cOU;^-o t_J y.:Fr.l ,;;S ,1ltr:tdi a.:.rE-rl ,it U_ c.*,1 o:;;q,o; a
..:*,1 &*tlrf 4, srtt tt.2a^.a e-flt.it1 Jf a.tKi,-r;U ,;-aa;by l$ly:o.rl.: yj Jy.;,e I 0tS";:iU 0h:. S *t,_l ,f -fIr3*t S.:_F q ,.plt 1r cr*Ll yly
Ui=Qj*tioqrls,o
ir:-,| oJ.i
(v-q-r)
rrtir*;-;l.ry grl ,glK.-l+.,.:*1 '#* ,rb\ J-S t,-F** .p.-fiU.Q,, S: Lrr!_ co.$ ,;ifF t, j.. {l .2: t6b; y f -1-p ilrt - 1.r
1n-1-p).i
:,*-_1.!lcr, r:d,r 4.,
puijj + (1 + F.)ujii -- o
lQ,rii + Peipsl\s,pii + @ + t4f,iii + Q'+ t4tin'l'wii=' (q-1-P):-rricf f J) c>y,e +' 6
Qr+ fp)O,iii+ Pe;o4tPq,pji= " (\'-q-2)
:,.:-,1 J-s Js-f 4., \ " -1-2 aJ:L, d,1rk! 611-rf:Lj s: slpq\q,pii = o aS
@ + ty)YV'O + ttY xYttP = " (\\-q-t)
V-q-t d:t -.031r +5 u"t9u7 rJO.-L l-, t.-1-P rJrt". 6{ s0'^eJ4- }.t-r-,,go..'b 1l-\-P) y-3t, iJ.:L, 6 S,geli l, ;t<" .*ii OlJi.r,(''-t' ,g\JK-V
: -,-*\:-,) n.tlt SQ rStt,SrJ,"Jt^ 1-lJ tf 6l^'ca t't *'(U t* y
YtQ= "
YtE=o1,.Jlr y - q - 2)
(v\Y - q - t)
.rY.rt - cf -* rf-,,:*,1 l-1-2 ilrL, Jp \Y-1-t iJ.:L" 6 CV::;i ';l-il .t-U
,O )ful-b,f=SVlxjl \I-q-2 iJ:L, ;:ls{ = o JVtC = c4b t-r5l .c*+rajti
Or-l dr3i.tt{ 3 \ . -1-2 aJ.:L, )l el3:t-, Oti-Sq '>-,1 t.ia':- *) ,lf .fb' clttt q
:6.t.,-.:,'.*t;t s.ll *- ,)f .r"ljtl-, 6.-1LJ
YrQ= "(\r-q-t)
ui 6lJ. )- er. iQ ttjA'* "ib,l,
...(iJ,,Jta dlt'4 )JYb u.,PV aJ:L' )l 'F S
:6.t1t-,-/ ;r \ o -q-2 iJ-'L, )l J-S ,f J \ caJLi' t* u. ;v6 fi-
VYr/ = . (\f - q - t)
.Jrr,-/ ail2lu; elt 'bjd d/-l 6
*1.0L<^ ;;-J 0l+- Ot* +#i .rLlJ Jll., .r!'t' J-' g.0. 1s: ois.)
O?JJ*4 agly a:*9;i <SbL':a, *K" YoA
C) F = 46; o-23-a a-, to.:--l ri c*"1 ()S)lS )ltJ u. slp 6slrJ fJr dLa#.j-oev Jfu q ,#,rJ& /Jx*-tu; JV,^r..*
^Sf.:S.f a.f ,_issd.r{l.1r .:yg
ui= \(\*v)F;;t -Fi;,
lc*,1 oJ.i o.lls yj
(\6-1-2)
-1 oJ-i o,L'il Jl*ilt{, ,ff ! .rr"r.srllK_t+ A-q-t il.:t , yr p=c.,- ,-b y Sd)y-rrt-lu.lY-rLr drot-.r sslv = ffi. csL;,S^So J-l-.rY.rt .1,,i,65
:6 ;dr;n o.n'b
VYF=o (\2-1-r),tLrrl 4jn 6 ^'f clL-o..:*,1 .*L, ;rSjK j:y ,Sll G-Es.: "(t;.-zLrtlr; y' d/l,1i:1
OLL; Ol-r;"/ .r;l: ry>3 F \rp sQ ,t.glaJ:l-- s.J;:..^A ;l--i- iJ:t - J- l+i rr;s .r:_,1
:nf rlr
(,--4Jl\V - q - t)
J
(V\v-q-2)
q:/\v-l-t iJ:L. otKi,-r;U r*.fu;,rtn 4,Jdq, "dblr.<JJ,.rin
L;;,Ii Jl:{1,_.f
"p|Sfj Jy,e
(,JJ|\A-q-t):t:-,€ F sQ .,rl tl*_rl -t
(v\A-q-P)
.'Fe cif o-! ,_;;-S ,h;VA ..cJt- c/trr
c.*.1 oLa:ir jl ,_*bFrrSF - FrQJ(,.:;q OSlg)lrJ.u.qi lr:t _ 1_? JU"T+'. r, fr- d-Ji,Jirr 6f .51+L..- j-- .11.:y ,ll 6-5:61 -rFr:Fr @$#r) ;,-,-u c:1;-a cJLrt 31L1l**r..<S-,)l f.-? 4 ,u.xr -l-l*,J*o c^{?
-1s 3 ol..a:L,,
;F'* uK O.-S ^ F""nCf_l q .4:j r:4,r
Q : -Fi,i
tij*rlrUj : y(\-v)F;r,
t;j*!hj : "
Q,ii : -Fijji
fol
.c*,1 or.i, orb \|-q-t iJrt*- st S s* dL.6u; = Y(l-v)F;;1-Fi;i -t a.: o;t-71 :gjo
cU.Lt Orlt
ut = -F1',rr
ur = -F1,rr
uy : y(t-v)(rr,1 * Fr,vr + Fr,t ) -Ftttr:.,tj .u-e ,5.,-B 6Fy = BR'^\ ': 1r. J F"-SCF .1, Tr., ;' 4LrU Cr" L: .#L:, bFr
:ilj Jst' o brlt(,, J*-., dt,l;l;1 t€r; e-*RY = l,t * xrt * \' s
B+x,ut = -Rr-
B\x,lut = -Rr
*:,[+'#tbli,J oly.$lr*,t-",!U 59 O;;U jl o.rti;.i q L^,.rS col(,.-r*iu,gtra;J? 1;^l:l
:6.:lr ol.i.j
orr = *1,'-',(# . # . #)- #]":r{.f u:,D\^?:=f; 3 oJ-io! 6fplrn-r gta;l ! 6
o* = _rt-Y(Y Ih- _ r(xrY + ,r,Ylhl"tt - -"[ Rr Ra
J
6.1-t t 1+_1, ! -,|5g,1 .c.-t./ 6;J\&tl 61L 6;'--'[. Jft q rll-rcr I nJsL. OJI
:..:,*,1 li J.s-n 1 oJ.i Cy.rl d:L, .t-$,-l Ost' (c*,1 r'--xr'+xrt ;Rt=rY*rtY
6tr=_Br\#)4.#1
Ssi yJ.\B 6 sr"/ Otil L q+ = c,1b '^-Je Gss sxr c.{? 1: to3l g.7:c*,1 l) os-r- q jU :1y, Jlf;jt .,..5g..+ P JJI-el
O:-&..{f. agly o;-9;1 CbJ.:"," cS+lS. t"l o
(-on)\nr dr
:O*lrQ stdr = RdR l,t
: d,*1b_,rr 1r c;ljl3D_ P,,_EG-,)
- F*q Ji.F 6ori '-rb gjr ,is.t st.*tf o3)\-o y;U .r)rt Ji- Clt Sr ,-iss
4;l .(*;,1b 6;li *t j)rt .. a'.6:yXu" dS(Paptrovich - Neuber) ,y,4 ti 6 6l^.'i! ,'.0t<" Jeii.zl:y .:.Jl- C*l glf .ry€r+r ,,t *.,b, rlrf 6- 3 Ji<-,|,pfeF -r, p $*' q 6rF .ll*x; c..JJ, .):1JV ,'.jU', s|ry.8 Jt<-,| ,J="Jq
ui=Vr-8i- (4{i,i (\q-q-2)Y( \ -v)
"= J-
P = \nB[fr-t,x'[#+ yx/[+l
, $-f 6j L o: Li = ffi .sr L; j t o: [ii I ! -r,srl -ri- t+ rr-r 1'; .f^^ .rY r L, 1.r i : 6:1.ti,; s-'r
''r
tV,;i - A + \p)B,ijj-{tl + p.)(Vaii1h,*Vi,i) =. (y. -q-t)
.s--J.,-t O.,tt t.lr^i! tltt -t: \l-.J 6 jU: Cf r r, ..r!:L, ylVYV= o
VYB= o
1-iJlrr - t - p)
J
(vY\-q-2)oti r;;1l.ry fut1l eltB Ji = \, Y,f L.V; ,'$s,,rb Cb :V -* dc,*,| A-l _flQ :llry 6taji jl .ri rnlJt *ijrtiq ,5,g0..,'1, tr -rr-lU .t)rL, 6\ q-l-t .J.rLr;lz;
J $i-+ t' jl #:.,- ,!S t-b {.8 svi }l(*,'l Cb tka erll: rnJj, u, Qu rlt* rogrJrL, ,* ,flg-,lrr ! b,fq yl6sl.r Ot-i; oljL/.$Jt<-,lCb.-,q.rb6rliE
:-ljlc*r.1r,
v= Y(t -v)vYF
B=v.F- Y'*Y( \ -v)
A,Qt)=A,+T
1-iJlry-q-p)J
(9YY-q-t)
1 ol-2;J5 L.r:.n lf "*."e "!.U
Jit - 6lJ Ctu t* u.y-ll, .l)lt" J-' 6 ;s;- st
J'- .5{ ,Stf cl.;Li; e.oB ..r)L .JlJ t{J 3 ci f{01(,l JJ " t} u.oat'^bU
11 67sV Lk -lr' ,-rLls Llj q J;i'-54, 6\ GlV..:,-,| c.,.,i-,r.1r iL-G'Li-l ,r ,l.rlj; l--lr .rLlf .4.K.,E -r, e.:*,lp', ,l)V -1VJ 3p1.rl-l'r lLi;i .-:.:;
,€C e-f il rst' I ;lS, r.P; Ol-t..
u; = QQYI (YY-q-2)
iJ.rt -,;s yy-l-?'eJsl'-,ftr.-,Stl-r(,l+ tf .-l c*,11/ q Ql Q Sr' : x{;6
:p-*sc/ $ J:* J#L4'-'iJtL' +', A-1-P
(YY-1-r)
:c.*ii bli € li Js-re o l.; 6ld:l-- ffi €f S- aS
dto+Ydo-.4r'r r dr
(Yo-q-2)
-J.f-1-t iJ:L, ..t lc*,lj 6-t l'-lri +r 1 J!.*A 6-SJl-5.;l ,9Lac,'.l3'4, sA, 6
:r.:*,lJi Ji5-'i .Sl.rts -l oJ.i dJi 0t<" ,=*i; 6b,;*'
6ii = )v6;fl4* + p(uii * \,i) (Yr-q-r)
:p,;lr YO-q-t .l YY-l-P ..r).rL,;l .rli-'l U n5
6ii = yiA,6ii. *[("' . +)r, - ryl (yv - q - 2)
l5c*,t op=otfi{tj(.J+*11 tJ \-V-Y d:Lr) €el,*J-c.{? 2t,}.,:S.zlrra 64;;irtt- f:s -*,g Y O-q -t oJ.:L, e:l -€-.V.rLtJ
Ory*a agly a;-9;a 6tbL:"," *tS" rlr
oN=(fi.+ \pY,.ry (T^-1-P)
jlo.rtiil tf ..-,L".JfU.ff ,.XScrLL t* y..:*,lcl.i o.:tiilxi = m;4yV 6:c*,[ Js Js-f- 4J ra.1 ..:*,1 ni 1 sge K_ _rlry v; ScJ.ib €oo = o;iV;vi
as-(fl+ \py,.+ (yq-q-r).V;Xi = . lyj
r,6.:*,1 Als ru--J.€O.r'J j# J.ftl- ,Sj-t \l-r3 rlA, 5Ar,SLnc+tj
ON=-Pt -1J f =lt
ON=-Pl 1J l=tl
:g,;lr l, 13 ,tgU glaJ;rJ cou o.;t il 6-* crqLe gt+t tf
_ prr,Y - p{tY r,rrrr poN="h T# (-iJlr"-q-p)
_ prr,Y - p,,r{_ * t,,trrt pt - pt ,os=- f*=f;1T (yY"-q-t)
"!f-r,-t..it-i Pr = " 6fl> b_p, = " r5,rrJb ,ff*t 6ly. u.-\r, rlYrL, yl
Al .1-l,f .lq,t, q dJn a-,,Li 6lf ,t.1l" ,6)'a:t' e.'1 tt b.r:+ y-l ..Ula>tal
q,d/-lJ o-l)to .S-*J.,_l g,,l,Jslr;.rYrLr,Stf ,-+bJ- tJsl3i\:#,r t'.-13L;-r1 Cl-1;
1s J:b .:)rL, 11 .-u! r15jL ./H - J-i* .r).:L,. tl Oi -qt{ J- ,rf J1i6 )JLb
€g A-i ufr db-r*./liioijj= o (r\-q-t)
:r-fr,! 0l* -$ J1j,e t uJ Ol+,
6ii = eip&t*rQq4pn (Yv-q-2)\
'p,JLndJj-. ,drL, 0,,-l f.i!.,- !*.t q ..:*,1 .rLa;;, jl ,S-l-l-b 0-ll"- 6b
<D4 6eJB, 6ly .r-50k Or& J**;q
-J.6li,tdr\ = epftUnQqt,pr, (rr-q-P),->sf ..1 c-iK-t- rL;
"r*t+lt+ c"..-i- t i+f s g-)-fi,gLar-_-Ul g.s: g+1.6
:.tul,rr-lt .;'rj
rtr ,]6 E?",i.-A( ? J.s
or\ = YOvr,lr - Qrr,r, - O6,rt
Oyy = Y@rr,rr - @1t,r, - Oyr,rr
Ofi = YOrr,rt - Q,,,,rt - Orr,1,
oly = Q1r,r, * 06rr - Orirt - Q,rr,rl
oyr = @ra,rr * @g,yr - O,,r,,,1- - O,a,rt
Or\ = Qrr,r, * Orr,r, - @,r,r, - Qrr,rt
(.-Jlfr-q-P)crLL )* u.
(:Yf-q-2)(r;fr-1-t)(.rrr-q-t)(drYr-q-2)(6,Yf-q-t)
0..-'b ,rl; .5trdJ -r ,y-l tr J:L; ,.:.r)rL-. 6.rb OLij f:ii:.-. .Stt-r(-ta ! ..ll-r"f.J-r=/
J* ,O,t = Or. = Qrr = " Jl ,-P- rQq* 6Fjj dLad^? o-rl.: jlj je U.
,A4, e-b dbd^? 0:l: ,$ t"- \ .f-:i,g g-,) u \ (Maxwelt) )'.'SL 6:t<;*i".'.,
t I o.t.l r ,i:* (Morera) ltt-r Li 6 ,.o.tr_l,F q,Orr = Qrr = Qrr = . F-u.Qq*.Jtr.iJj" i^r -St 6.:,*,1 1Jt Lt'y C*l q qi .t{,r/ *ttti -'* gU oL^n
q .,; r:.-,1 o&,Jf V-? j:h.-lr 5 oty,g4lrS Cblil3,udl cJ-.i,U tn a rhr. grr!.rL- el-,.. & q b t O4 J:;Q qf l..-:*,1 d)J. JrU ff -q-2 rJrL" il*-r
\lr! *^: f. c..4 ,.7|A- ,fu crirL, l.t,ilt.r, J- Jl ,.r^16 ir Jrb'oov
O:-J;.F ag(y o;-93 aglaL2o" calJ& rlr
$t*
.,:*,1 r.rL "!U
J# '"u* oxf ,S3jl Jq , ,c,* r:JL 2r I - f
tl* = C*ofsa€f @, f = \,..., ?)
er rJ.*.r :-5; tr t'lly,.f t1 aJ:L i/-l ^5 -gar rlt-! .iitt"# arl;:."Lt1
u* = IC*sa€F
g,-.llr cl15l rai 1.r er.:*,| 6;tiu Cop tsl y oS
Au*
de1 - CoP to = oF
."t-,lr"f A - \ - P d:t, [ .;ilylr 6
4.St-t o.t--i r.,-;.--,' &y.-f 6# J .-jlFl ,Slo.ilr, u r-Ff J ,5,St .r.l-[- ftf - f
cl,- 1u*11) LLrt ,Sj;l ,)q 6,- C.- o13,ee.tu* ,-*f 6jjl Jq 6,lr;6,1^L,.11
.+5.F u*$r ti'u; ,$j Jq
u*= u*r,,1 *lt*(r)= *"uti,+iS,irtU tfQ
jt ..*:b { ott e-p rS ;t- i,/-+.r-q or.i o.:1.: q--t'u* u*S 6j} Jq ftf - fu* = u* (Fd ft ,-*-.-L.iq cl.7-f ,Ff ,Slnil-r gto'1 J<-,r**.rtrrtjg[n.iJr.nJ:l-l-. c,Jt-- cl--l.lr d.u.rr Sl*:J,rK:-l:, oJ:l.--n I Cj-rl d:1.-. 1l o:i-it-",| l.i -t.!tl
'y' af--Vy; drL, tt \t - \ - t
,oii = #^r",S3j Jq er.:-,| o-L! i-ys Y - Y - 2 d.:L Lj S rfL*-n 6*:-,)l o.:L ci-,9,lr1f - f
.-t;5,ik t):sl-r -Jlt
'*_S
C*op,.1i ;r *5
.rtjsrr-j JSj
e4 c;briJe" (iJt)
o;; C;t'l-Jf (V)
e4 g;t"t-! (V)
u* = { (l et e;; * Y Feii ei) (iJt) : f q
,* = ft ), + t p) lloii-)r.o;iolrl.tft),+tl't)
u* = ( +, * ,)rrs' -t t'ttr"
(v)
(v)
6 6,lr;f *.r-iti l-1, -;r" ,ft--^" ,.1-b -*TU f t a - f
,'ri = 4;,^ ain T^r, = Tij
6bJtiul Jt* rJ:lSla;1 u;;2, r,l*:" JU:]t/ ,Jh
l'oo
Iaiil= I -t c
l. \L ]r'Ia'1=lI ;l]
-\c
0
..r-,tr,.r dA Ftf 6lJr 1l ,S)e uP;.r: +*l ,f L**^ rr-b I 6 "r..rr ,.ltl;
.r).r1,, ;l orL,:-l lr -l V \Y - Y - ? -titl \Y - Y - I.r!:L, #;;t ^tr$ c/)J., -,.tirt+1.P V-Y-2'J:l-... ?J.-e!l A-l -?
lJ Y - f ^Ju;.: oJ,.i(i-r! +ap \Y - Y - t drl-- ;l ortiil U .d+-Yl orL c(,- 6,lr;Y - f
..r4SAt* G .lK ,rW 6.:-,Yl ,Slr.,itj -J
u* = ikt; eij * G (eiit,l-* e;;e1) : *E
-t bli,/ b ;t4;c(.:*,)l o.rL .(,- ,Sl-,t U*<rl i-r-; S3;t JQ ni +1, bl-, A - ,
O,*.);<'a agly a:*9;1 6laL!", .*& rl7
'r-5.rt* t) r:setu,J,rl cslA,rr-S(:) t ckl,gLau; (Jlt)
u*(r) =(or-or)t + (or-or)t + (or-or)t
\YG
* (q -er)Y * (er-e,/Yl Gu*(r)={[fr,-sv)Y
lr:I^\-rt -ffi't':t
I tt (\ * v) - rK(\ -yt,) (.r)
(-aJl)
(v)
'S ;b; d*:-,Yl o:L...{ glr +", ilb.r+tUr (-ilt y\ - \ - t aJ:L) u* j-r,.r ,tt - fAu* lUt,i = o;i c-.Il;f-rl.r otK:i.r;Uo.rr.-! ctll V- y -t L-y - y -f drt-... -b-y
(ile;i I 0t*, = 6i^6io nSr-y' o--r') Ou* I Mrj = eii
*:- p- ; .r-SJ o:r ).cl 61,t1r.-, ..tFl"..* 'h;
,S-;"*,il d^, g-* 4 6,111 l, - f;sli,g tl a4; i,r--l -S-r*.rr crtJ;.!! 4L..)\" \ - y - ? Jg 2;6 o;jS clLr.u;-.r
.r-5,Jt i l)a1f\v SE ,/46t"c.rtj .-q,1
(\ + v)o@ -v lo<rr * orrr * o' I (q=\,Y,r)
,; crl-|1" r:-; d e* Sltlt,J.rl 6La,J.:Jfy = "tld 3E = \oi pSi 61" u--..t*:5A.* r-,!.f tt oty,\ (bi ?...-J)
Ioiil =
8(t) = -fr0 x , €(r) = o7f x \o-t , €(r) = \Yt \o-t, fq*5.r*^, tlLi ,f;t*" ,sii-Yt orL c(,- ,g,lr, I I - f
e(q)
::]L'\ o-t
\ =r{l+t4 1jr1\ *rz f 1,+l1t
I ltt' Y Kt'-
-
t9,\-Yt/ \ *v't'
-tox\x\xy 6,GyaJ JtA;l .r-rV 6 Slks*,| d.rt'a;".(::-'il '-";4 6.xrxr A'"iP lf - '
;l r:*,1 olV ox' 1x' 1x'1
r 6 0lIt ' 'lIa,il: | -\
I
lo\lLJr:-:*.t rbl-f-* 6iL CVI tllll,'.,.:*,1 o.l*i.:Lril Vt5.f-l y oS ';jK;Ls 'r*rr cll-l;
(V - f - 2 nJrl,..;--) r.;ls 4;-,il gL^c-rtt ,-r-iV Sst
Cqp .<S-,il .5Lo,.:,t3 .r--il, . r.lLN : Y qy &il r.r.rUt ,r''' 'S'-xt tF f t tf - f
..r{,j-.a" .>-'l Cop = Cpo *-t ,-rt\tt:*q
Cttoo
Crroo
Crroo
Crroo
.coo co,
ocv c2
Crr C,t
Crt Ctt
C* CY(
C* Ctt
Crr
Crt
c,,
a,:IC"pl =
L
o-i rfL..^^ f tSll,.i.:-,)l ,SL^crti 'f-;U.c(S-'!l cl;t'i;6Lre"*' jt .t;5 o"t lf -
'.r:-,1 a:iL- d:^lsJ-j J.Ct
o0
0o
oo
oo
'Ctt o
o "Cy1
rlc,, c,, c,, o
I
lc,, c,, c,, o
Ilc,, c,, cl, o
I
"Crt
ICql =
L
O?.L.4f 612 a:-921 6lh L*r," *9" ilA
(,:*,1 ,g1t;;1fi 6l+n J.- jl d+-,)l <s1W y-,^ blits+,x1 ,:* fl a5r*.o: ot-!.)bq-- i,r,-k-rlr).r.p r-ol.r-;Crr = C,, * y Crr.L_*ier.l_-i ^:$.,L;2s
.(Sr \l)8'yr = arq ar^ tqr, J o,ty = atq ar* o*nnf .(+*,)l
"rrh, lf rj[-,r-. .>l,ay,-,gl-a.u-,. ti gll*.,St-l-rj eS 64.' ft la _ f,s+*,Yl
'r-jJj:;1.: rJ+J J.&*" d::-,)l c.rti orjlj: nj+1, r-r[.ij or_iLN = \"
..r-:!.f sg oy,,.ae:
IC"pl =
c', c,, c,, c,r c,o cvc,, c, r c,, cv c,, c,o
c,r c,\. c,, c,o cv cl,cfl cn cff cfi cro cvcw cil cfi cv cw cro
G, Cw Cn cro cv cYl
tlLij.l:*,1 N = ?.*-r {n*,Yl urla )J-- t*- AT+ :-t* 45d$_yl f*_ ,_<,- 6\, lf _ P
, Coo = Crr , Cn, Cr, = Cvr jl .u.rt+- AT;"-r; d+-,)l tSt c.rt-j oj_r.^,
.Crr = Cw tC* = + (C,, -C,r)
cfu*,)l o:L a-5,r:Jt- )ru* ,.Jf 4.j;;l Jq,Sl.;,,rf,Slr*t.r, t-....-J J-rS lv _,..rr r.: rr_-,y .r_,tr 4;jjtl
:*qu* = + fc,t€r * Y Crve, * Y Crr €r) sr + { (Crrev * Y Cyrer) er
+ { (crrrt + crrert + coor ot + crre ,t 1
,-fu*,Yl Ulii )J-,. tsli;oa.x1 y^- 4;lj:45c(+.yl &:{ J4.-.r .(,- 6h +fr bl; lA_,t.." cl,- el-,t6.:y.r-olr; JS-: rlLr,g!tr
'#f 4.j-it Jq o,-.,.t N = y n*..r,.
.!tt c5,S-,jl ,lrli,- a>.;eh$r 4^ip s 4j-J +.r
jl ,:*,1 olV &J-, l..'." ,Ji.Sl-l u;'tll'ql
I x\+xt drr "lloiil: | "', xt-xt
.lL ' ' xtl
iJ-ti ,tr6lr3r vl*.rr J.:U.; e):l-, ft'r;,\r-lxr Jxtll ,r-bot1 t1i2' 6
1Gt) - (\ + xr) E, + (r -xr) 6r !-,;rr = \ a^ie 6s:,-F 2ltv fl I sli
.-r.;5r.ll*1xy l\ )l eE ot-t'q l.rorr .rti djtJl
olt:xr-xr*6:*q
: *L=
\tt*: er(CtL€r * YCrrer + ICrp,' + tCrfr) f e, (Crret * YCvpr t Y C#r)
+er(Cnpr * Y Cwe) * CrPrY * eo(Cooeo * Y Care) * Crrerl
t1-,
.ry' .rtdy-j J# q 6:^ d3jti t' r5.r-5- b..:f - T - 2 eJrL' r o - I
vG6ii = ffi Gii *vep,6;1)
y)ot-raqy-lti oJst-- \P \ - T -t lY -Y-2 g).rt"!..r-,tf U-l)1 '#-;
G (uiji + u14) pbi = "
''-''-..
\ -Itl
.:JL--cl--l .l:nJ.u-o:dtiUl ..:-.,l ,,'"b | = otd 6!ltnJrl-'i/-l ntc>''l->lX
lrrc,/ .,7p J.:k .r):l-. 3 ,59 tl;E
Guiji + +oii,i * pb; = .
O-,,'J:,,,tc4 agly a;-921 66L*q *& Ylo
.,'*.1 o$ ,Sltl ,; al:l-- L-3; Q; ol_*!,r,_b1l:r,
-J. dt5.f,i, olt*" f I _ p
"r=WQ; -lbi = " -tS"r-" vt+*t s t--j.ei,fb (v - f - P aJ;L)
-r,-3b alrt-- dl.l, .lt*1.r a5.r--iLQt = -\ I r sQt = xt /r fl .Q;11t* = , ;.1--r_iL,r,-b3.r.!-rtb
.+SCr,.t, l) AJ'<l.- r; rll.r*" .c.-lrY : xfi; tsi
:*qor\=-oyy=2@rX, lf , ofr=o, oly:yQG(Xrr-xrr)tf
orr=-or\ = f@*rlf , oty: o, Q=t (l-r) tC
eJrt'-)Jr-JLc<.4t!-r,.:*,Yl al:t--lJ-t--arrrl :;.u*itl -Ftllup.a>6Ll,* ftff -ftFr-l..ryL- y.; dt<",.*ir,ll-ry l.r (\ \ - y - t
tti=Q,i*eij*ltkj
'r:scl" dlytr b 6-1 64 a- .:Yri', rlt,* sQ.j1; q Cli nS
\ f - 0 - ?.1 0 - O - ?.:Y:l--.,p- 6*.riU,gla-xit-a u:.s 6lan5+o, 65 ff _,.r-Sdt* t) ot-tc\ p sA a-") 6L"c,.lE .--r, blie
./fj rr L*ii jt I -rli i,."t; b Y'Q = . ,r,-ij3.r .to.r.rL^ oJrL .r,-L / +f "r*r.r clt-:;
-St .u.+;,rL- ,.1; 6Laa-cJJ- ir,-l L-,;"\*J- JrL; .rlrt - ('uee dt :.r
el-n B sA ,t ,SfQt sJi.i-...r tr*.rLj B lA.li.;, n5 -t.il'i d - '4xrYxtr - Bxro
.J*t,i*-- .l-.iL 6.'^:'.o .r' gb n*'-t
/-\"8:*qd::r (jJl) cJL 1.: "<S-,il o.rL +5 r+: +*-,j u* uxf ,S3jl Jq dt, d.l.l: 6,, r A - r
.tJ'l 6,1^-^i-a 'Ff (:\ Olc^)-
u* i I o,,t * orrt -\ votr dvr * r (t + v) o\ll I I E(._oJl): g"-q
tij : [(r + v) oij-r\4o*ll E + 6ii (f -f ) q:*q
(t" j,k = \, y)(,-ijl)
u* = (lt + i[Gut + trrt,) * I' eu svv * Yl€rrY (v)
.;*1 .r:*,1 ;.*.,g,r--lui;Cb,5i Q:xr'xt + f xrYxrr -xto aJi-t.b)dl iif,-,
,g,la',iuc ,-F;,il*1 .i- .'S:--l obt q .:Jl.- ,l-l el-,t r'l; rr*,b .l -r-ili.f V'0 = o
.+5o,r"t*, r>-'ltt = 'lY0
:*q
fYrrYlI tt xr\xt - Y" rrr -t rrr - Yf x,xrY .
I
lo;il= l-rr,t-yfx,xrY \YxrYrv+Axrr I
L o o ltrYrr-ttttJ
Jr d.t r.r rrL.i; di"--b .6'*ttb f - ? - ? d.:L 1.: Y - ? - P d:l,. rr-fu fV -,..r,9*irt-1,11 Y-?-ta);t ,U':,.5;p,trr.ll1,.f lJ O- ?-? AJ,- f )ffa)rl--
e; (q) Slr.^;'* ri!- (-iJl) cJt- t) l1r.*:*,)lyr 6)J ,?;ia ,-r';l.rl:L f A - ,.JaA) Ar-rj 12 6'la-J-a
(i=\,1)trr= -loiilE+a(T-7.),
O4tJhe agly a;-9;i AbL':a" c&JlS" rrlo;i=),6;1ep* | pe;1-6tift,t + r lt)o(T-T.), 6 j,k = \, y) (:)
orr = uoii-aE (f -7.) - Itii-ft1'+ \p,)a(T-7.), (i: \, t)
t-.t tS:tl F- Cb q 4-3; L rt - tQa=D&rtxtY+Forvo
.r:*,1r.,^, F6 - + F- Cb .r-l .Sl; n5+^, dt*! (,-.,Jl
o.:ls,-f,K,j"' lt il 6-r F' Cb (V
Q=Qo+Qr+Qt
Q = Do(xJx',Y + + B,x,'x, + + A{,txt
on = Do(?xrtx, - fxvr)
oyr=YDoryr*Brxr*A,
ory = -?Doxrxl -F.fi,
J^-., J5; &tJ- b efr --t5"- ori:S fi ,lL J-*l J-t.,r ,-f-f L. ,-JJ,l*-; c<.,- f , - P
^.5l) -* ,t-1 ,Sl-,t,S:y b-i rj*:..r:1tr,r.o ,Ktt * kr;l .lr srV ,Jt glr3x.r5.,"
.+f, .--rg ..r5 r.,L J,-q b,-;;*k--.1.: b"S n5.r+, clki .Y"-i +k-,ls oS r:*,r4i ljY3\ 6-t b-l_i;l o.:li,il l.f l - ,
- *r,',
l*t. L
_:s-.Yc'
o-.L.
Y
a.f l:-,1 gt) Y1-f
Do:#At= BY-
or\ =
0t\ =
orY =
i,,',_ crxt_ i u,
ft {*,", -
+ (+"'
ft {*,'*,' + cYr,)
.r:*,1 ; e-f P * e-r.-lirl-,./ = + cY oS
t -'fr 6.lf gjf L-l-i.f '\-t +k*n 1.: ol*i " uL' .SL^uS ;l o':t'c';*'l lr n5 +f' Jl'ljf l - '
'Fe ,3,lV t e:r b-,- ls'sr,t,$ Cb * jVl iJ^- ..(,- ? 6). b-n ,yL ,P a''.e oYY-2 3 Y\-2 JL- J1+' q rr - '
,;*'- r:3-1,U" i}l
Qr = Drxr'
? 6y b; nt*'tr. a5.r.r: clt-l;
Dr=#,* -#r,:g;l: c"-l+.1:
dr\= ftW: -i"','++"-)L'F,
OzpJJ,k4 agly a:-97, <Sl.a Lz-" cS+& rlr* J+tt Cb C,L,ll63l-..:. d^L" J.fu \,-t'-f C;t 1., sjL: d,- sl.^6.r*.os cli-l; rr -,
:s.*,1 ,-1 ol.f
{@,, xt) = 1@,,' - \:r'Yxv)
:"t=f ;*.n'.siL-t rLl l-*-y,tr .:,t: (.-all
K ,->rr.*I- (.7
.r-*t. F g'-t't (6\- lTr
7A
1y'-faY
o'
)JX\=A, Jy= o
1- (JJl)
(v)
(d
l{-
YMpovr]r"r =YK
xt
ljr,,- c.' -,1 ,ieli. 6Ji - C1; ^S 0sjl3 )l)-r. {. ej Lf A _ ,
F=B(''e,+ac,+ta\- -\r-r r r-)
:u-1-.ie o:1.:y-i otf q.Jt<"-xl,,SLn-Jj,6+^, ttt-:; f = X*i; c..:B ,5,-B
tti=-ryr ,r--y$)u, uel-o.\ dtt ,.*i; 45t;5oJ-oLL (A:-5.:t-.:;; or.i orl: ,tb 7-h;l o;Lp-*,1 L
.(.tlf.r o.rt--il clt(^ .# ,** r;l? *ta- 3.:-; g* ,*,-sl
rrA ,.b',i?",7a-Al ? 4-s
u*iUr=T
i/-lJq ilV = tte = o d#?.cl il, = -YB(\-Yv)rY nS.r*^rob-l
\tt, fB( t -rv) yB( \ -yv)er =T =- JEV=EE= -T
59 cl;l; 1l .e-,t e, * er1, * eg = . u{S- 1L;t .9., y-l ;l
6;i = )udqu4r + \trt(ui; + \,;)
d/-lJq ,+L-.f p o, = \laeija, L.i-l 1r ef
tr _ nB(t-rv)rr
rB( t - rv)ow=ooo= T
,S*.>KcShoj-i
d9 tl,F*t cabosLi e ok ,.,*:-*8 I - V
,F r5.:-,1 ..1ie, jKU- c*.1 Ct S.f dr*" 6o.1l.i -* g',-rt1:1 4^di4 ,Sl
rA ort-! 6 tFC 3y(-u-i! t.(a-f*tOS^, :4 6c*4 tr) gt6t".rr0J<-, JL- -rr otLi 6_ cd/tJti .raJ., J<-,r=o sr'l JL".et Laru S jV b u-,j-,
6 "l:-f J^ I J:SJ, -t lt ,_*". F * c".,ai;.rl'i (.-La f $f c.Jt- -l: U_)
o1[^i;l 6],,Lfiy' -rr c!;l ,5.1l.rl 'F 6.- 6,51 ui;1lsy +f sL l! Lyy crl .-t-,tr
"rr, .S!.-f jt+ jl Ji;* Ll g.:..,1 'p ,sl n; JV-l tf .*b 0J<-, Jt> -rr
.g-tc-/(\ - \ -v)
)Lis tr Oi -Jjl +5.:*.t ,(=; .Ulgets / is (c..-.)p. .*,,t:; *e.1i;.t 15
6,'11b er \ - v - 2 aJsL,;t ..til;6 c$zb-192;=.a
6ii =-P.6ii (Y-\-v)
G+ -f (f- e.:-,1 d-rL:i t+ a-r 1.r u.i:; 0F Jt- tr o-rLi 4 6lJ -tlr.,, OtJ 6Jt -i "lJ
d"3l..Jl; $k-,|-l-rJ+ ./'ii 5,ri!./ +ti;g 2s ).al u:J c..{+ oLA
"-''wl(Y- \ -v)
tl*rl fAqtjJS,qdl-d.rL,4t-,-" p ,)q j T,->jr> e.-r:11rLli.*l
.:_1ii o.:1.:
F(p.,p,T)=o (y-\-V)
v
trd=oii ni=-p.ni
P, = -*oo
6,4-J:',,te agly a:-9;i 6lAL*. ,S*9" rfA
{JL- drr_l -lr 3 c.Jri:*-l ;-- YJ*r-. yJ.,JL^..JJ tr5r- Jt- .rJ .JLi 6lJ
6ri =-p6;1 +t;1't-t€
(0- \ -v)
;.: o-;Lj 6 s.9Ja3.:*,1 .Syjl .-*b 6cJj..1-,6 6 r'yn? c,F ll*bl:tU diJ:.5\ : y-t -SA <52"Q:9s.Lo ,r L;jl, p;l-i.i c eJ.rt-- d,*l -rr ,tf g .3i- J-iq OJi, Jf-c,Jt--l.rp. .(;L-,1.lLli .j/JSa:J<iL" r-t'..f ,Jl2l p sT +r.c*i u-b !l+rt Ot*
,;u_*l,.llsJ:L;(t-\-v)
,i <+ JL"-i ,-F tlt p ,6f Jb.1r o-rlJ .i- ell6 ,:-, qi 6 - v - 2 nJrL.;l
.t1.{,* oll.r r.)-1j.,a ,t-4.e*-,1r Ll cc....aj
P - -|(oa-tr) (v- \ -v)
.tFe it P. L.p ,(tij = o) 0J<-, Jb rr ..rb 4, cslJ. e,dp-l ^5 pig *Vu.l ql c o-L..-; lgLao.;t^i Slt ,3,Jl .:Y.rL, ar-,j 1.r
JIJJ OrJi Jt -. tf I cr-t' .-ii- 0J<-, Jt- -lr s5[ao.;l-.i dlJ -qq 0J.i...,^2, ,j:;s )r;l;,'-,1.rL L .JJb rDU J<-,-+ Lt:y-b jl ,_*:b 6q 6f-5"r L-fJ c JJ, r'.' '{'' 'r v J" eJ Jr J (5-'- -
,*,'.,,';"]'J.,,r'q *-lLtii = f;i (D) (A- \ -v)
Itfaru ur: .n--5,; oJ9;*l ,,,tJ 1-r oltJ cJ^iU F* aJrt'. #-l-rr u*b Ll;)l flL-'; tt ttt t c-*,1 O;;S: olh \ asf o.,,t^i cJ5,r, 4rrJ Dtj jU;6b rll;rn,
.t r^r"f OH lj aJ.rl--
t.. = K;1o, Dou (q-\-v)
.Jss",.r-<i.l_l r-ll.i j-t;...2 atlf K,io, *-lf di j, d.r:*.A,ft,..-" Laolt-i dLi c.*,1 o.rr-.t.JrU a_n" -r-b.ldgtrbb
F(p,p,T)=o
rrt +lJS 6tao2h ;tl ,J.as
J$t ualt; q rlr^, grl ..:*,1 ,fjt,.^, J'-ti.5{ q - V - p aJ:L. s, K;jpq,d/lJq
olL3.i- ,Sl116 *f,g 4++.!U- di,bts \ a'Kqou?--t'A\ 6r;l.r,r, -114- r;1 3D;1
1l c.*,1 otV "/Ll
eJ:L.t, O<^r rfL** '/i
oij = - p6ii + 1* 6,i D* * \ trt* D;1 (\" - \ -V)
#-tjl .JSu" i,tl l,.;l;,r5-u...7 cflf +f .r:.*,n ,f+ ?(rpx ,)v*.JiJ,6..-*.1 l) J.t-req d-11..,1..i .Sl: OF+ JU-j r* 6ry.:,,r,. oJ-aLL aJrL,-
* on =. -p + + (\" ]'* + \ p*) Dii = - p + R* D,, (\\-\-v)
L; .c."-t .rfq- g, cfq- *p t is-rK * = + fy ),* + \ p*) r.li;, a.f
K*=+(Y)'*+\lt*)=o
]r* = _trlr*Y
.l.lsi4\:*,r ,€ C-o \ o - V - 2 oJrL..rrl-rl+ifl .J*q
S, +*du okk= -pdii * *do (V1* + \ tt*) Dw* \ p*\ii (\r- \ -v)
1,-all tY-\-V)JrL.. rA t_
(v\Y-\-v)
L; dfl 6 t-r.t oraLl \\ -V- P d.:l--11 -ec.r..:U,r, ,y'1l*l L.;y.j,s-r
.':*'l
'F elt -ilFl 61,^rr;U ft
Si="u-*d;j ow
f,i =P;i-}.6aD*
(,-AJl \f-\-V)
J<-,r.* 7t ,Sl-t s
(v\Y-\-v)
O?-t4-o aglSl a;-921 151.6 L,"g",".Al9 Yfo
si= \P*fii
oii = -f @ + K * Di)
(,-iJl \0-\-v)
(y\0-\-v)
Jti ,-pL.li;rr.3r: $rc/ tt+rt ,-p it-o'jl *. l-r cSy et il 4+ yl l-rl.rjLL/ 1-r tr ;$x.-.2 -l .r5r..Qr yj tW l{ l, JSt .
,-{*l - JJU .:l.tL- ( c&! Jl'.f ,rlel eYsL." I - ve t,J53 !,i- orl JLrt dtrl.* cl:f La-, ! b *r"I cJbr bo;b *<i . b
st ,S)11J:€t JJ '-;A .r1l.r I'F-rt \>efOl.r.r ., i^;- j*..ztJbl:+ rl1l7
,-*,.:F,f o.l,1 -rE oJ:.r+ .-rt^i qF i Gil slJu", JiL-,JlJ ftt- .r):L, F,.JUr.i O,-b -t,-tr fj 0tJi. g):L, ,C 3a-'.rLi 6- u<Jt<rJ, j sVis elt
(P - f - A .JrLr),-fu-r-,dsL, (-iJl)
P + Pvi,i= o (\-T-v)
(f - f - 0 rJrLr) cSf .l)rL, (.-r)
oiii+pbi=pi; (Y-Y-v)
(\ " - Y - ? dsLr) ./Ll cr):L, (.-,)
6;i = - plii + 1* 6;i Dw * \ trt* D;1 (Y-r-v)
(\f - V - 0 rJrL,),S3JldrL, (.r)
pit = ori Dri - Qi,i * pr (T-Y-v)
(2 - v - 2 drt',) di- ,* aJ.rL, (.:,')
P=P(P,T) (0-Y-v)
rrl 82-W 6bo2h ;tl d-,s
u:tt(p,T)c,Jl--,4JKdJrt (e)
(2-r-v)
(\ " - V - A d.:Lr) d)lf .:r-l-tr al.:t - (6)
Qi= -KT,; (V-Y-V)
V - A - ? V \ - A - 2 .r)rl.-,-r.*,- alrL-o olK;-,, y-l
\Dt-vij*vpl.#, *i;
Q J'-
(^-Y-v)
.::,'ljlJl ..r;r:r".Jbl b T:P,Qi,tt,Dij,vr,P,o;1 }S44,,YYUaJ.rL, 11 -t
^;cr!:[r.- {-.: jU Lii orj.:k;-*:{ odV ,r<Jt<" aJtu 6- ; .rJ+ p*i -t' ,J )lf0-A-2al.rl--.J-fu4,-l A-A-2 aJrL,.O#J Y-
^-7 ti\ - A-?
..ry -trlf ltJs-n*'.O.slF.{'.Jr jl fuP=P(p) (q-Y-v)
.ll^ru/ JJ"iJ p sDii , vi, P , o;i lgze \V -9 aJ:L, \V otK:-,.r.5{ *f
! ,.rrij .r)rL, Js^lJ J *Si fs lr .:*,t O<^." iy 0l't=, .:).rL, ;t ,,;-'.rj:l*-Jrf -
^ -? +JsL.-..S-rl.ri-L+ !,.,-,.Sely oJi*+ 6Lao1Li #b-r,
;-;'Sl,-f\:*,J, cA -A - P aJrL..4;., jl o:t-i:-,|9 Y - h-?
pii = pbi-p,i + (1* + p*) v1;i I lt*vijj (\. - y-V)
!'lr,*" rrI:t'-. Cr-l .r.ltru- tAoJl*i ,SD ,#*(- ,* rrirL, \isr 6, p, vic,^i^ dl1 drL. c.i^ ot<:-,r.i- ? - X - ? sA - A - ? eY - A - 2 .r)rL,
,.r.*Ll rJrL, 4r - A - P d.rL, 6 6t2+s! rr a?j .Jrr./ J5$ l.s T iu, p
,*r.rlr ,SJ-ll ..r)rL, Js-iJ s:lJ-s.tu.^.^ ,-b* 'f *t- JJU rrirL, c\:-,1 *-
ii=-j+ vjvij
O2uJJ-g ggly az,9;i <SoLza. <S:J& rrt
tU", J#q \' -A - 2 eJrL,,J.,!,8,-?-f ( ),* = - * r* )
pi; = pbi-p,i * *p* (vy; + y r,,j)
'6-t*,1 L; -St
.{t-,-r(\\-Y-v)
- :e-l[. .r!.rt - ,"t;! 1 - n - P a.l:L- J;;ott c,Jt- Lf],:+ d.rL. ft r,i*r n
ls p sv; )142,-rQa t drLrrka J-S la 3-*ata.4,$-yr, drL.. !.1,,"" ,-{-t*,t.$r"/ JS.:r
t^Ji- ' ji -;r o-tj cl! .Ju-.*;
'JLx Jt.- g;l; ,.:YrL-" &53 Lfl 6L*, -r,
t F,Sl^dlrJry' ess4sl s 6)-* !l-i q.lr^, l) 6;i,0hi..r!.:L, -r,[
a4 rOi J)sls Jt--j I t{J d r.;fr;t* *-ftl Ct-,.i- 6ss t$_f \lr|.lusg.,b &erJil LA.Jt-i jS..+ tSS.t atS)-f r,-)l rf 116 ss j.J4r-!i- F^ #f u-,L, ^AJ.0+"^,!cS6OL_:+\L-r.-r l.g:;.:^*; c,r_l-;: ,J:, eLr,:,,)rL, Js-U *fc.*,|-nfi q pj).r;l.r
..r;l.r 6.;*X*.rG->t n1 jU d!L:,nl C6OL-*.-r."tr"f cSl+A
ojta;botUf -vx - 7 .:**i.,1r oJ-i dl-1| .:Y.rL, lJg -,1J cJi-.-7 t5Lao.;t i &JU" 6J+
-,^
)l o?-s 6tac-.r-.;-9 6.rl-t*;.<, gr.+^J,-r"i!.f llr to'.,11-.i.qK, y\ j,al ,j:,ii.trjc r-t+t ls 6,ty.s6 T *ti:;l +t? JSt.- S ,-;-,,-l
6L', )l*a:il +r 3 o.r;+j i*,s4| It n-i g,r.X+, Jjt-r:, ,_b.6 ov-rt;l ,grlu , lq.:,_lJ.)
€,-l g)J oav 6too.;t.i a-r c*-; l, t+i
J^iL '>2lr-- d.->.;r ) "l;- cJL-- nJ:t -. fl , *=liS/, agLaosh (*iJl)
b.JL-.rlr;; c\bl cJ-i gt:: I -n - 2 rJ:L- L-,y, dCft"-oqg[.ao1L-l JSs- c.*J L)r; d/l jl 4t jLa'JLi s ,t:r-_F,-f 6*,-lrr-lr!a.->.1J c;iJr {5) ,J6;y/olr*;; -l: y' trs.a;l;;! ..U;l.r fU d*,-lrr-t-,1f
o;l-i :l eV t- ,-rl; ,t:lr 6P 6) 9stJa5:: ol-r.,{ -l (c*,1 c.it3 oJlf. f..,,r-
S .-f .S*,-ll-t-, t : ..rlr*,u bl p t l.1 ( rr.*$lJt c.Jt- aJsL, ,$q crtj o.l!j .i- JK" f | , :r+b f U agLo osLi e.')
tL
p=c1B (\ -r-v)
&W agtaosLi;1f !'as
,5^5. ,b : o r..fsLtc/ Lry yl .-ta-:,r" eii tr,Sr-{Ut'li a'
,,-*-;-F;f .vi,i : . r{Udp .gb..lt-i Sl-*. ,-fu*, aJ:t'-'
1|'.:j.rlr.; ef g ^S ups,rb 4 Jr.l6,>-,1 ;,'. y-{ dy-tufli
qgLaoll-i el-r'..*l{.-,^-; (f*- t) Jt<".1.: b e-r*-; €'rJ,f,. tl-i.r.i $tJt F*t- ,l-rU .r).rL, , y-+Ufli
pi; = pbi - p,i * lt* viji (Y-Y-v)
€f ,-h. rc-t-L, o^,. .1t , 6iss -t ql "r-'!,.r vi,i = ' L; rtE 5y-{dli i+* tn;t5.-Ut 6 JV;J cJ'ir'r"a v-t[fti 6 s-F.t ,ft
.gtf,-f
.5tarl.; -UlJ, 661,1t'.i 6.-' ( 8l<1.l Ot+ ) 6 v:-qp's 6laosLi; (u)
L- o r': .? :s o-lt-i .i- Js,f *-'l ':Sr> Jb r) 6 'Pt e \ it
ss F* s]r* *)-* f t, eatt t-h,;r,u/ .J:rti aJ.8 ol34 i't ,?.JS6. J:ai t., ff.tS.rt-l ,51 aJ-rL, rli (J.ib -k es;' \ o - V - 2 d'rL'
.{t-.-f "'i^6f-j
os3'eu #*l- J{JU "l)lL,. -e
pii = pb; -P,r (Y-Y-v)
o-,t; 6- ll o v-( 113 6- .rr"r Utl r:l-t'' ,,Je( oSp '>U:6 fb. :&t r5
tso^ ct'i t'";60;U;t 6.:-,1 ;[..15
P:PRT (f-Y-v)
rf Ol..i; pV+,!..r-,!,-f {..Jl}, t>,; uaV i6 elt b;lSc-ti R tli.rr 5
.,i ii L u..tr d r. J::*r oJi...q : t-4fli ,J+ 6LaolL'i "/L"j
*5
J*q rtL.f ( 6"r;frri oU,f ( 'S|;* ol'-.,? f - V -
cz;19{ OL-.nls $f ,-r-i! crL;;l ,fu,.;U '5{ \tef ,glaa;'Jt' fltteJ't 6.rL .;..i-. c ..f)b dri? Jr .eU./
+ + viv;i
O.nJ{r-. 6(y a;-9a1 66L*" cS*K" rrr
.*t_"r pS $,:L j-ii r,
i; = v1 v;;
.t-r-'ct' Ft Jsye.f { J-lt aJrL, e c;lrSl ,lb-..,a "i_ gtf ,."r,
Pvjvij = pbi-p,i (\-Y-v)
-9 .:*,1 0J<-, Jb tr o.lb , s[. f- tl]^ s {,1tj t? a-e-a;r .r-of0hia f t , Oty o3)\c
F*t - tJU .t)rL'. . .:Jt-- ..,r1 .r, .-u-_5,-* <$,lLlil+ b Lrt Al ,Sll y-h
Pbi-P,i = "
jl .r;.r!,." Ft*1(Y-f-v)
.g!"f l) Js-reu. sbi eb&_.-,-r*r cltilp sp ,&.(*,-lrr:r! LrS ,ftl,.
P (p) = f o h
Jp. P(\"-f-v)
,,fu," Ctr 4 .:^.rJ l, tii bli€,.r;q ctr5+luh, ,,q*rst,o_r.* -St,oi t 4y.rf 0[: *; &): .L-,-" A
bi = S2,i (T-Y-v)
P,i : * 0,,
6 r--J ,-1e2..! c Y - \ o - feJrL. -b-* )l
t_ VP=Yplp (O-r-v)
.\:*!i Js;o if_{,llf"f l-r f - \. - p drt -.lf:Sl .r-,(fl+P),i:. (p-y_V)
u.*.--> rJta-l-x- l-r .!,lrr-l! o-lb .i- c;lr;i- ,ll*-,+ ,JtJ f5l- ,JrL, Olirt 6..t-,!,-f Clrl5.dlt-,
rra SW astaasLi; :f Jos
&- rsssq3t? jl c.rr ..;lF- crL--r .5- r*tUt't; ';i,.rLi 6- | - f -Y J&'
* Srr-,t -Li -9 J:,U.h azJp J. ),f olt i .:,ti^i f | 't5.l f^lt l, 0 -$ U. 'e'}e
O.rg.lt+:a ,t-l ,Sl'.ls,>,cr--l;S .3t^,il+, tF ^i-5 -h:lr (cll'i &-)p : p' .llii
.+5
6V;Kbe4 d.rL, L-,y.s!r, = \ (\, v-) , vr : vr = ' l5 ef d-
7r:*V cr-,.vy = vr (4) Jvy,y = o t)ct.,-l jl 'vi,i = ' tt-+U,*Slt tlt'.,a
d.:L, .J::*,r .,t-o 1la!{ -l Drr = Dr = + (AvY I W) ,SLn"iJ-f 6!b "$'lr"v;
;l .:-,1 otV c,Ji- drtl.r, di..fLloii=-p6ii*\p*D;i
.rf ^t-1
., bli.t' ta"ti;t
x2
loul-p
o
o
-plt* (\vr
L
p. (;, t uy)
ler) -P
.-r-,!,-f uf.+ esi k; q3t? oJ"
b = g gin 0 6' -cos 0 E)
..:*,1 l) Js-fq ,F .51.,111 6 '37 ..r!lL' 3
o;jj*Pbi=Pvjvij
O*J:4f agly a:-,911 66L:". *tS" rrc
(611i = \)
(,slt i = Y)
(slt i = Y)
.")u l,rr-r: il ,-eV \lst osg-! cilr<r- ot--f tSLt
-P,\ = o
-p,t * lt* (o'vt l&'r) + p gsin0: o
-P,r-Pgcos0= o
6fl al.:t-, )l 6,;Jl$,sl\p:(pgcos0)xr+f(xt)
(xr= h),p= p. rljiCt-,-;)..-,",1 ,s-SJl-Sr;l;l ;[.at-61$l eU {f@r) oijrn5
f (xr) =p.-pghcos0
p=p.+(pgcos0)(+-h)
.uv t r:*j (,slt i = Y/ frJl;j;h*,r drt-- ,,.;tET: rt:;.3* "
vy: -Pgs!n0 rYr* axr*b' \lr*
-F-tslO c*,!
.f{.rl: ,s.t-t hl-i;l Ll .la.a ,s;JtftJl ,gLac"ltl b sa
b = o ..*ljQxr. = . 6ci", vy = o (\)
a -- # sin o.r-lrQtr, = h 6,-rtJ otr = o (y)
..:*ilr f=alf a = (pgh / p*) sin O,Stl-rK_t+ ! u, t Ly.yaJrl-.-1.,..;yl1
,'*Lrt-o.A rL/- ? J
f,
rry ,Sa-US ag(ao2h :f Jos
V\= W$ h-x)xr
.-r-,!"f ,fS-l i,{Jr. ,?tq,J!b a'
x2
,rL--f €f ,g. )-t' -!i-Wr;-b +f't.3q ,i^;f ^'.'lb'i- ier 0Ui.'ft
J'Frt WU.YY - \o - f drL, L-2' 6tw u'):f tl:y.c.Jt'-c,*l-r:(.rl ,t;;8.*F**t.lL-ra 6ty +4i.1r c''' "l ;"tV a-o-a.1r dLa'^'6 ril:
wi= it4ovhj= o w=JVxv-\curlY=oiY(v-f-v)
aJrL-, ^S A\* 4 ,Sty. S 7-$C o.;j curl ( grad Q) = " .5-eLj jl c'-,fYU
-eJ os-yett tlstt,fq yJ.6l1: ,-f ls c-e ,-,Obif ,J5;6"b ! v - \ ' - I..:-ii
vi=Q,i t_ v=YQ (A-f-v)
t::y. u,;r7r! Sll.,JS t gt) tr dcurl Y = o tri 6rl.1 0t ii dlyd c 6it-e ls
,>-,1 <J:-:4 O|*fU glf .-rU: l5.:-'l'-t; 01il A - \ o - 2 aJ':L';r cll ta;; 3..:*'l
.SL..-f r*;Jd/l q,L'.':.Dl rf
6li,-/\ ;tar:;nJ.:L-. s ):l d:L, ct-ddti F** O!,.r 4 .str
Or-JJ,,F ,gl, o:-9- 6LbL!.rt <S*& rrA
6 - t' Q,,, (q-f-v)
.ry; el; g tf .ltr.f l);*-,y *JrL.. s )sl nJ:i.-.r-J,"dlt.(*"_lrr_l.,q
(t' 6,j - v; v) vj,i : o (\.-f-v)
oJ.:l-. ,. &-t:.aJrt-, y-+Ut'lt ,f 1." .tb-.r .5ly .-:1lr f U t"j g ,S 1: altb.6.{t_._f ,_.^5.r!)
Q,;i:. b. YY4:" (\\ -r-v)
rr.:-fc./orJtt{ A- \ o -f aJrL. 1l o:U;..l !.:-r-,Jhi.rt+"-l .Slf ui,SbJ*
J(^- LaF All- ,Vl g.- .!-l t*;!.1r cr:-,| ,p.r.,\) +JrL.. 6c.*l,5i ;t_i^i t*!-l
."r-l!.r
U-{4#3, Jit aJ.tbu 6 - v\? - \. - 2 al:t*.-.st-.:
"(*,_ly-l-rll ,b_b, tr SVs,; !,rl; ,4fl.t-S AF_t+ Jll +Jrt-- .:-1, ..:r{*, -r: tJl_f c/
Sti;=(Q+P),i (\ -0-v)
O|;+ J.-;.i- ."f*-;lr.c,r alrl-- C,1l .Sl-l J- ,-<r_ or.,13T cd U. t, l$ "f-,- ,llpq
Iit, (11-rr1)'tS t>ef c.67 u.L"- llry ca_L.^; p ,l ^Li; r^ tr 6 €g qr-f,S-SJl-fu| ! .-r-ltr"f clL_-r Lp,;.:l-u,l.,1: o.;Li ,>lrs ,*, c crlJ;<,_ .lb--r 4 gfri
6:11OL-rll.l,.;," GyUFtYY-t 4[..-t)Ot-j-t;51s,1rr \ - \ \ -t alr{-,11
I"*'r,+*, *{l+e+ P-G(t:)Y
(Y-0-v)
rrt &l.E as6osh: f ,Jl"*;
aJ.rb -r..sqgr-l ..:*,1 Ot-r L;.rl-r:'l lr r,,L-.,11''y. rltili'5.l- fut '''l 'l'6
4 q 6;J1#l ,l ).aV G (t) iv:6b c-r'l;U tilF- ':57 ft 'c*"t dry
-51 ,,-l\ ..r:5 ".;
,s-&-, q ,lL-.r tt ,.Jy 1l .:-,1 o<^" +5 {t-.-/ .prt5 G t'rl"
..:*,1 r:g 6U-- JSr' G. '* -P c4U 6- ''r*:'! 'Pf* tlt-.,a
c.1i -,1, +5 A - gh g-t '" cr"i;L t' -r't ).rl-l ,ssi l+- qit+ 6 s"l<a
..r-,!.r olb 1: e-fg! u)! L6rt ';lst 51 h i :'*'113q c^lli g : \rA \ mls'
crli,:r-F, t>-e flbrl glJ;rq hu : vY / Y g s sB t&.rttllJ:'q hp = P I g'Sl
-r*:r -r,olr; rtUftj 6Lao.1t-,i ,Slt J-ytoJrL.,,lK'i
h+hp*hu:h+ -G. (Y-o-v)
Gy\ u.Sy-,| '=r; jt )'aitt: 0 - A - Y nJ':L' A-b;l a'3-l g''-li'U- !
.jt(i &- ,ss, * ltFtt q l, 4r*). G,cr_- e- Jr tV Jlfsl a:;' o'-l'9,-'J,*,
l.r ,rLi jl n:*l * 6" JF t" O*!Jf.r- <Jsr !-; ./l L-'9 'J-as.r" lFrl
€,Y'-L-fJ(Y-0-v)
C ;;-,..q.fL.. 'f "l*;lA, fu; sC Li tP S it-' & UU; n; $;t 6
,ly=.,i-f.r* -g curl v = o c .:-,1 '#**oL-.f
6 dt ..:-il': 4?i J.,-! '"t^l!.f
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:r1-J ,-l,rlr f .r*^b tl:l 6tl
f J L.r 1.r J* ,t.r*; 5 sl*.(: d/-l Ft1 V -t0 gLn.r,-yy- ) : 'J-b L 'siF S
4-J d,Jpl s );b ,SiJl *J.l::i,-,1-A-0 iJ-:L, e:l-K-V.J^'./ ? t'1l3
iJ-o.:,r'"
(1 -\-A)
..:*,1 o-U.i .l;-ii f J [..: 0:y c"ftj .Si^.r"- lJ) iP\ Slt S
a-il-,11 Ji,ql 6tr6.:*,i I t^rU g^, o.t.:.lf,,l4 6l+*,-U" q-\-A iJrL, ,g)J.
jL ,.:1,fj \:!r cJ.a.:,r-o J<-, r"nt o:L 5 jV: JU*'-f -rBL ,tni[ 1': .-tS"r"
,gtc.iLil 6si b,.sl f r t s ossV;l cylJ ol)\o t sFCt4r' Laa;l .st,r"
il 1-\-A rJ:L.'l'q &fl.sL"f gLS,- ta,-l clL. ;1u! +r" Ul ,t3''6 4i...(,- ,Sfr ..rrlti d-Jp f6*, 6l cJ,L,r,
"/q r*.r ur{ b1-t ;Kutt 4-t 6 4t
;: ,j:,i-. 1l^- Od....-J #-tJli oll$ ultl) ,$Jl J,Sil 6 ,5-p lr r*ii ,Ji,-r.,-l .fu-')
";*ltf tJ t€Jl tJ l.,il ,/KV'^J-13 4iJ,.i r,+5 rgl tlzs 1..,'*,1 :i,;'\-\-A iJ-:L,
LaOL"r*-7 ;r ,:,.a5 g-r-l ."tl.:r, ,Atsl, La'rg;; .JlJ O<^- ,5bclL"tg gllyb cosl'r
.:y,g J* ,-;--lyl -il t &stl )r LrtLo Jeir, .*tlrrf ,S&o-t-* q t6:,l q ts;l,t .:,<t . J-b ,-rLlJ tcJ ig't 6-,Str d,stlPQ) ,f- vlsJ Jtd-l ! .*b Ln,S-ri.:-r ,e :ll'; +(i-l 1 4?i q 'r.:-il: J^lJt
:g-ii JJ os-re '". rsli rlls ,J,stl U|:'. iJ':L' 11 o:Li:-'l U. -9 c'*'l
F=#-e#
+ r [un
({"p)ry:ry.*klnPQ):rt.
O:-J.,h. ,gl, o:,-9- agl.oL=a" *K" l"Ao
rSs* (JloJr_l .*-,Y 4 ,St : q-\-A iJ.:L. )r Jill o:Ln;-,| ..:*l ,yly.cttiK 6!J.o.:.r.. t; o_s_re 1 b t{:r o,,.9.;) 4 .Sh 6*f J o.ql )l )-V
r:4tqy (*.-,1 Ir s\ A, Jir?S 6Lac*; jl ,jev JL"rl .5Ln:,,1 6fts *5tS-eI,, 1a;Lnc.a- jl .i- -p t: '.Li gl+"1 JKc"- ;l|y .orL ;l or"v;; -f 6ss,:; jJ
s7G) - In". r* = # I"'r* = # fq,rtt1, - \h, (c- dr{)
,5tac.a-7 jl 4, _ri -r:,JS)LScLa.r.,n.j; Ll,g3;..r.i gt+"| .;K,1.:.r3i._,t*:-l o I:*-l't \:-,J r., .fLa:;
l*,', = f f6oy - rl ),, (g"a,r-r*) (\y - \ -^)
c-{? .r, qj-rl t.5t6Jl tJ L{Jl c"** iL-u ;*-r & cu,osgil :l-u ;'t ,t,i, g.-a .rrLc 6:1 rFcf r4& c*.1 &,sjsl-l t-r:,!: .<.i-,) i.S,i-l ,-r-S ,->-,lx;
Itt -klx;t)i= L<rt>i (g.+rll{)
4*,;r r5.rlrt r e? J \ Y-\-rf iJ.tl-. ss?-\-l iJ.:t -;tp ,!"-n 1 d/l .5Jh<_t? l.a
lJ-ar,r,o r-*;j *^a:i.o
w :n&_o*F,, + lr, +ir, _ rl (\r_ \ _A)Y <r'>.
:.i..iJi ls Os-f I .Jli"./ ! aJrL, .yt c.:J,X-, Sll.6
(\\-\-A)
l{-+F,t+ Ar, +trt_rl (\f-\-^)
A
Ao
rL.
J-
r-^JS;
.r.>f oJ*i.r:5A ,t'-f * * :L J*q afA. e.f P *:L. qJ-rlJrU q 5''-! c;
:,l-r,g o:lr .,1j b)ssit*3 t 5.:-"1 L-f -t JSbG J f-V 6
N=+
<rt >,G=Nkq-,:<r'>
,.,;#,gi.l..:*r (f-A Jii) d1^, ..5',-.;r 6t-r .-ft ,Ji$ Uf f -b -t' L. '..",o.,
'-!t-"f .',a"6rj J;j'eIr=)', Ir:1r=i
,-t.i gt+,|f.rl5..:*,1 oJ; o:Li:-,lltllry = \ Jiii ,.,-f V r[Ur." ?--P r.' c^'=L 6l.:*,1 p OS-fe \
tr=+[r'*+-tl (\t-\-A)
dla-l:er il.=.,-l ar o-\,-..i ft+l JK U. t-l ..:*,1 gr*1, 6La'r3:; ,,|ss ofi fta"l 1l5 .J*l
dsLr;.: o-t-i o:1.: 6ki rlsi.Xl:Q c'.>l ;t-J(',r- t'oJ; ob;i Ji''r.l-l s(::-') tsg tz:V
J-(; r::'J )l kG ,sti.sr-lirL- U .c*,1 l9€^l^:l;i 6jr''l tr -,*'t'lJ \t-\-A
orlr.;r-j Jsp ,-,.1 u;-':S.i-;l Jel- 6J* cA-\-A iJrh' '^b: q o$ 4"*3_r-'c/
(\0 - \ - ^)
61* r,.t.o ag(y a:-y <SbLz^" <S*K" rAr
J* t c,.-l-,.- I " dlJ-) 6V = A.L. = AL pr--llr cc*,lf.-- "rJ-S y;-J<-, -;r5.:*l e;lJcg-l ), -- +c.JL-- c.*l t: ,_l#S!,.*j nft?.jill ..:;b c.Jyr qJ3lL.
:ri,-t' 4+*i \t-\-A iJrL" ,t S-$,-,;1l;;1-, sq.p O,lll o:lg,*l t,.+:+.dL L.
r.i w dw dl4/ dJ,,-:-:_:__aL dL d"l, dL
r=+- Gv-iI
F =T t'-+l =z.Gv-+r
(tv - \ - A)
o:;s (neo-Hookean material) <is* *'osL r't:sun 6s*,aJrL. irr r;r 66rr_r t'tFct'
PrJ 'ui*=i,, Yl aglr1 ,-rr.SrSjjl &gr l - A
OtJ b lr: ,slr.r- f , *lF 6Lao:1.: G.r t .Ji ui'1r ciL- {i-,Jj ,S:-Fjl .*b S
'*f ,S3tl ttrJ o.f 1l .1a.fu-) ft, cl,rf J-r. AU -*,,f .-r.or;.t-Sr-*.r'bj ..:-,1 3,i = F;rqFjna JS,:, ,*is * l*b O.j",a+, ,[S-.1*iij ,Jl*rL<
(Rivlin \qYA) ,|y-, L-,g Oi;l -r1 3(Moonqt \lf . ) j_r L-,y l.r:,1 6 €rs s,l,g_F.fLl ..:-,1 ,cif .-r!,, c-r.i d-:.r \-A u$., -l:6 J;j| -f Gs: )lcrir-L;.\'*V L5Ladjt"i;l -r,_tr,.(*,_ty-lll l"r:,1;t !:L6c,*,1 d,{l 6ijl Jn ,S6C;F s jy
.-u,5.- =,f &f ,Sjtl Cb ury JKj ,r*,tr q 4?'
tJtrt-t .;-.^1J ,-;;-5:t.!,-rj-r)-l.f:b ,+ae -ul3 t,-if ,Sjil +f+S o.f.Stf /, s I, ,1, J<-, _+ :r? t_r,b
W = W(Ir,Ir,\)
I, = Bii
(\-Y-A)4'
--,*'1LY-Y-A iJ:L, ,Jt^t-t, ,tKir,-rj-9"i ./tsil B,i ,*l6b1.e"'-' fl 6 -rJa+-1
:rjl.,-t 1*'J) ls o.t-ye u.h slq A\ ;:J:S,5l,oc:*;
/,=*IB$ii-B,Ful
Ir = eiiFr4rp* = det{Bii}
1r=lrY *)Lr'+LrtIt = )rJ )rtt + )rr\ ]rr\ + A\t 1r\
Ir = )rrr )rtt )rr\
Q(Fu) = "
6(c*) = .
(Y-Y-^)
(Y - r - n)
.r-tgst,- "[Ur.r
:,1;l 6 -r.l.,l-r ,r(;t<, ;-,q c4;r'"]-Yl J Lr<':-') :19" jl A-rl':*r
J.U* f -f \ J3l* ,l uelr;l cc^*a ,j(^" dF ,!Ur." ,hS t* a., i:L e- qfl
J.(; rg ,rt_rFJ .3r.; Otp a., o.:L f t; ,tBEi fQ.sp€ ,y,rrri dtiu-b ;(u.tt.i- il.."-l r.r osL y-",Ut'lt':..rL 6c.*,1 ,-rl', Ol-t., Al .ty.f sFql: Jt<"y-r,Udlt i.rL.5- elt .'y.t J.r, f$::;;,rt-rf ,51u,^;l? e. CVIN =p. t- (A-f-O iJ:Lr) ui,i = o al.:l-- il*; q f-0 ,.rb )t Lyy O.,l .Jjlu,
e-, .;Jlt "-f-A eJ.rL-. s *Vrl.^1t] ,Jt<" ,5t3.:i jt .-u clt* (..-e.Jtt "-Y-0 iJ-rL,)
:6.:-,1 €ls, Jfu y-iJ.rL. rl! rll-i;c
I=det{Fu}'=\ (f-Y-^)
,rl-^;l:6# ! *rr, ;h*- ;'5 c.Jl- q ,sy-{U f l; "'V t-tri ;t OFI
4rK.F -r, ls t) rs-r- q 'J5
e* .7-:f 6)l(0-Y-^)
Can = FieF;a 6ta;i;l .t-
(?-r-r)
'-t.3.;t' -t sls6r c"';6.:;l: ! .$.>raJ.a> ylcrrLl: -ti,str g-lr,$i y-l
: g;lr O tt e-S,,|*;ltrrr, ! .sb.r "lq
r*! o;+ :& r^.*
hiea=h|rrF,a+Fuit,u)(v-Y-^)
O:-Js.{r. 61, a:-9- 6LbL}*" cS*& r6r
OrEr 1J*,b.i- al d.;E:, rr-b .i_ q {.-**j jf 6-5,};l*-, tr ,r,-/ n_:,;i 3
:6 G;q ol"t, .,r*-,-1,r,
(^-Y-A)
drr-lJq c.:-,1 oJ-.i orlr \V-\.-f iJrL-. L 6o:y Er16 a.; kl :tCan 6 -r+50+-1,
,J;-2'e a, .:-,1 O<^- V-Y-A iJ-tL.
(q-Y-^)il*-r t0;1 6(\"-Y-A)
.t_r-c/,-,"F
3 c,r-t*,| yL, jt Clil*l ,jG eU,Jj.J .gl,^dJ_F g.7 q oi:J +5 6 'pt.{[.r J(J,,,J-|, -ti )l kv-r;1 JjAt F.9t^dJ-]
- 6;i + d,i (\\-Y-^)
6;1Dq : " (\Y-Y-A)
-9 J::*r DU t JrLi, o;j S0;i Jr y' 5"tor,r, cll*il t y-y-A j l-y-A .r)rL.. i*.,-U-
6;j : lhij (\r-Y-A)
6;iD;1 : "
d0 -\ (,)+ *,ld\ocen r \ ac# acu )
Q;p;1 = "
Qij :;rr(oc*L * ffi\r,,
(\Y-Y-^)
iJrt-- r.: r-li.s;t-r(_L+ U .q-y-A ilrL-. .pfi;j - 5ii ?t;:;t i!*,-l ! JrL- .,_b ,.\
Trii : -pdii
'.r1;,g d.++ \f-Y-A(\0 - Y -
^).,.:*,1 ,!i; gt! .:tF .,) 3 o)lr c,^.'ie r*; ,Lii J*b .r|J(;il .Slf Jt<-'l 45
::.y.:; a*:ry l) o:.se q .,1Jr,Ut'$ irL 4 t' ,;J .:-'l O5^- Jl-'
,tij=-p6;1 +&;1
,st^n;l"-f e-f ;x slt6 r-r,-/ or.*,.fLl 3-t, jl \2-T-A iJrt-'rr fJr il^?
:>-f ,g na;: JLjr O.tf I .JJ
(\2-Y-^)
(\v-Y-^),rii = -p6;i * Fu aWaFin
.r, 6 '*f..,iJ
tt..:*,1 ,-.-- -i-13 J. Af ,SiJlW s ltl"t't ;'U ,SlJ.l : \ a'
"^.-,g r'Ff ,SiJl.:tF j.l-9ilr',J^i Ajlujl .,Li,i jl GltV *J. \V-Y-A d':L'
J - r r!,Ud$.rL Glt.ry€ ^if.r'Ul W ,sltsi,-eV F *;;l oi ;*i:\!ii /j JsJ a1 oli./ t, t-f -n iJ:L' JIr = \ ,l-lfQ
W : W(lr,I,) (\A-Y-A)
.t-r-,€ J^V,F.5tar;j ? r,S*; i.rr[i./LtJ 'F;'Sill tt,Sf ,.|*jti'!
,,ij : _p6;i * Fi,tl# #; . # *o . # #;l (\ q _ Y _ ^)
;l ,r-,bt.r *il, SIr 6.:*,1 .l;L.$loJ-3 orlr Y-Y-A iJrL' srltJ /t 6t'ot-t1 6
::y:; a;^l-9,r - ;.Ji-j, q oa;l olyo I JsJ r*; .lt-'lr(
It : FuFu
/, = * IFuFu\rFp - FaFplF;n\nl
Ir - detlFu\
(Y.-Y-A)
O4'JJ# agly a:-92i <SbLz^" cS+lS" r67
&L5 c!lrLi:--..- u{trc.i*s .llv.J? 4*--l a'. ei ! /, -l /, .Sbt-t, Jf 6tr,;;.d/_lJli ,.tp€
:6ipdtp (y\-y-A)
Jlte u. d-f ,; ,gLrr;Ji. c\l-Y-A-lr yo-y-A 3 y\-y-A ..:)rL, 1l o.:ti:-,| U
:l:r_-l,g c*,r \ l-)
oFu
w
6,i =
J# q G ayt ey ::s:;#": fi ,""T,'J
6ti =y.6;j + yrBij * y,,BnBp (yf - y-A)
: TJrBil' + (y. - yl)dij + (T, - y{,)Bij
:+f-u-r,rr 4"J
6,i = W
I, - r, ffiuu
. K p,i * r # tr, (yr - y -
^)
rLli .:-rt o 6fqii;l ,c.r*lJ oy)\c *r# = o Jfr = \ n'.:-,1 gjy ,drr{UaSf; Af-r:
.J; -rLi,i .:r-l!" ! rr-r-n iJ.rt -;l .lt* 0+-*i sr:-,| ,r3,J: ,rbb ,il .rj.,1.r
ut A-f ,-:;6\ il o:Voi i*1 ;Fc
+ \BtiW * r,ffi+ rn*Bit,ff (yy-y-^),,#uu
oii =-pdii +#tr*#tn,
oii = -pdii + #tu * #',
(Y0-Y-^)t_
(Yt-Y-^)
.s*,1 oJ.i.*-F \?-?-y iJsL, 2sC;1 6c,*.{.ii, "*b J'lj tl cc*,| ,E o>tI, slrjl .r,tb ef ,s:tl c[ grlls
rdv ,#*'a4,,,=z-U( : I J-s
r:a osye 1r,-q ,-#-S ,S3Jl 6 tf n; ;;t:' t* r. (\lfA) ,|y..:-,1 or;i
.r1,- 0t l I, sI, yJr56t.l^+
w=2c"p(1,-y)"(/,-rf (Yv-Y-A)
b o-t-i A* jI,, sI, +1 c..j/r-Y l1'-Y jl ,jtJqr yJ. ,-#-5 ,$Jl 6 :y'"5t azi.:y ;rr^.a: ,* ,r:;f ,$Jl s p ,,;-f 0d &.U.i
tsf dti.,L-? .rl.r; t4.z4 rlt-;i o:l: .-,:-l- cJ5..x F J ':L p; ,./LlJ
?-lr4 -fl-, tCt.=G:/tsil !t,JE" elt.tA.t !/ti;jt YV-Y-A iJrt ,.l: o.:[i:-,l
iJsL, id.r-r q L^"rJ..:*,1 it J;e G 6 r{.l JeV ,t'-*--F t1t,4 ,r"-.s$,g1fi o:tiil ts-f (Y-Y-O nJrLr) '3;-.J!:L, , s tFc-f r*,1 ., Y2-Y-A
.*,,L, 6l \lrJ ! rlr^^ Jitl* 6,la *Slr.c,-s rrY.:W r'ey--. os.f 1g-tS f t
.- .'r u. G)-f },-lt:" 0..'1r.,1) 3 o-t-i :J; tSy..:i.rL, t; O.*"U -,S .sr d ,-F
.q-tc
,#.5 csiil 'Pb agla':JbY - A
or,-j n:il u -<+!,.;.r5lo.:L ;l 6 uoV6l+L; gl* q o-uJJ1JJ dLj
L-,L." 6ly r-iK t/j d-l-rr t{ J.."tl ;ret-' i.rL."; ;l 6i}l t-.,.*,* -\-:jL s!:*,1
Ite,lu,:{l-ji 43-ljl # Ll ,rf rsf J,-q .rlt-ji ,*-S e:tl Ct't Jl:;" 6Lac.".tJ
;t ,grll-; q *3-l U c*,1 JrL-. J r-p, ,./tsrit Jr-! .rtt 6Jl tl ,-*V J'(..t, r,r,5
je 6 S:-b t.ri-t'./U"jl .r,-b (rv-r-A d:L,) ,|y-t- G-r'#;,S3Jl.)t)q't7r
'#f ,s3Jl :l s-fi-' JJ'^,, 6LaJ(*i .-r5Ot r ! o.:L U\ it t* ,.r or'11
l, tfvf ,.rJr:) # Cb C-r il*-r n, a.S.r;l,rlil Jrt-,61.11"u1 b +"r 6.t.1l:
6tnJr-r.- dJ+-L.a 6-* 6tac-a5YV-Y-A il.:l.- :lC"p gLn,*tJ 6l.Ji jl .(+l..!i...,6 .fU, iSU- ;;tf p gtoJrrbt, ,-Fl tt tr-tJ Jus.-r.o.j OtJ L "-'-,+".Y1
5c^."1 6p\\,4, J.r, 'rif .*-,).5lorL Slt 'if ,S3Jl r[ii OJ;':Lt tF€./L;;jl G ,,,*l.Jl! Jl-;l- q tsr t,bt, ti .t-*J.,g A ,>S*fi os!
t.:*,1 e:al1B;i J<-, r.* 1*V,gU-U, g+..-.J q W '#-S
63Jl
O:-.u.{..o agly u-92i <SbLz-" *lS" r6A
W=G(1,-y) (\ -\"-A)
.gl.;lr +5 6-r sl-e. Eu JS,.l; -':.+r .-ra .u-L; )JLb \, ,kl 6l-asjr,u ,f -,t \a:,..ii l) Otf a-' \-Y-A iJrL-"6c.*,1 ui', dlJ., tosj'.Ay s\ &, J.^,l6tt./5
W:G()',t*irt+,lrY-Y):t-Fd
(Y-Y-^)
j.:U.:*.r ]rr' = ]rr' : ]r-' ,lr' = ]tr osJe a,, Lar:,,.t,S qS)F,-*,J:Sc,Jt- dlJ
d/lJq ,JL\tltYlLyY = \ -r*i!J{Udlt ,:V fl ,t-l ,-o-e)\.e
w - G(r, + * - tl
"r-t3 6l;l I .,J s4isjzzt tJ u:-fscJt- 6ly \?-\-A U.:t'r t: .)t!r yl;l ,:li;-,l I
f =p,,=#: yGfr-il (Y - Y - n)
..:*,1 -i,*if-\ '* :l \ \ diJ3'"P,, 5.}i, ,Jt^J- y .:*,1 A; ,Si;l Cb .Slrl JS,.i yjort*, O5;^--r- 'rL
- 6iy,6l*L' 3r L ,o.rL Jtn tu"1^-" FJ"tr.i- .rf.f ,a;-lt 6;gr>-,lt-
;:-.* )J*'U lJ, J J;l ,St^t-! o c*i A-S 6Jl db dt-- c.rtl-rr..:*,1 c-rrJyl
ss:-,l;a{Udlt -r Q;y:yl o:L d,i-t ,f -e L ,o.113.: ;Fe t;f _F1s ,rb "$.1,
:t-..:€ {^ii tJ os_*e I d; ,S3Jl
w=ct(tr,'+ 1,, +1,, -y) +.,(,!.7!.+-, (f -y-^)
,5.)-G--V.J::.*^ 1r' : )rr' :1-' s)rr' : I' ;!-',l 6taui5 *isyz.c,ss r4U;i6;-t--ly l6):il 5, .;l; )v a-'..2.i 6-S,L;lA.r; f-Y-A iJsLrJ.: br::*ifor|
.$r"/ l-,1 *iU J<-,r.* ceL,-
(0-r-^)
Jr rLsl ui+-, J dV tr .,|i.-f b *it- r*1, J#.l.: ..>L,.-c rfu-l n1 4?y q ut>lt al
g!.{t""f f$-l e; -f ,-pcr.,,3i.:-,, u.JLr s\ "aJ*,J d,..iiit \xr s\ 6tnc'a;
er.L'4=".3 q .r;q/ r*; r:>L,... J Ji ?t' *f c*,1 ;r, rll"r, 4l $,;iz'-.* dt>i:F€-*df ,Sp.<;.;;
f=P,,=Y(r-iX.,
d,, = Y(i'- +X., .' +)
.+)
(2-Y-^)
il:t*-, -rri 6L.n drL, di j, A n3-l q .lrf a+j'tJ^! 1A-A-f iJ-:L.;l orUul b
: _ 'l \:4rr ,.,_F-S )l.,;trtr gJ'F ,slt Jy-e cl:-,| t-r-A
(v - r - ^)
*-.E11 0:y *-f g-e J* \ tSr o.tt' I V-t-A iJ:L. A'^J^? o*, L.".1
or\=?(Cr+Ct)E":t rJir/
(^-Y-^)
;,*tf {.ii E : ?(C, * Cr) Jsj,e tu A:i*##,Yl J-1"t .<"-5,*f Sll,d/lJt*t
dlJ,. Orl .*:.-,1 .16 jt Jylp\:*,i o.& u|;tdUt'li orL 5t+i;t ,c,1lJ o3)\r
G :Y(Cr+Cv) il*,-l y dl J;& c-alh-Y-2 iJ:L. c^.at- il*; ar c+fc.*,[ ;,-.:yc/ 'rl.: .(3-95 ,*-5 Sll
U 6a-6,Stor:,56\ G.r o I o.:L g-t, ,!r:- i-r dld^? Jr corL Jr'
Ci-l jl ,?t..:*,1 jt+ ,-r-1, y)! .ir" .r)\^-7 327 u;'-S,Jlt t-l ,$r-/.rQ l-,-l-o.rL ..5- cJk .gly .J.:-:,,.^ spv6tnf l 61.21;- u;;f ,S3Jl Cb glf .iL; 6tajS;
:J^1!. a:.il.: ,4j J<-, u. ,Ff ,S;yl 6r.r; ,; ,-et c(\ lVY ,O.rfl; OSI
w = }Zft" * ),,,an + ),ro' - Yl (q-r-^)
O.-,):,.,tf 6ly o:-9;i aglaL2a" cS*8" Y?O
,71-Slp, = -YCv Jltt : \C, A1 : -t Qt = \ jtt = \,Y 4turlt- Alf JKi OJI
.*t".f "b6OJr:
- Gy o.:L a; c.r;..i
,F-ro- -r' .t .r+Uf ,; iil' 'K- cSl:t ,3j: h f - A
iJ.:L, ',>5.,>.:!:L. jl ort"iil U.
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6j;l6b..:*,1 G ,ft J-l-tr.*,L; c4tj."t'i 0! \-A cJ$-, st 6:-b oL.a cg ,,l
x+ i* r-l-b 6t-t dr+..ij I, 6.:*,1fl - Y q .#Ur. ,f-* - J.ij .Jt-- :t '*f:t{.l -p il.>st' q F ,JL""iJ-F ,.,F-5,S3Jl dr.+ 4 .c*,1 ,fS-:,
6ii =p6;i +GB;i (\-f-A)
.€*,1 J<t ,.* :ra -l-l-b Bt t gt JsbG cr.r.U -,,L:.i .r-rt .o P 6u.cff tsLn-l:a: :/Ui -r, 615o.n'lt b JrU .r!.:L, J,-! ,'l; ,su,^A?
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b Jrb,r.-[*jlj,-, ,*tt- iJ.:L, Y-Y-A iJ.'L,.1: \-Y-A j\?-?-Y..:YsL,. e:l-K-V
y,j-=-y-a ,r-:-;uj &;
"l@,ffi#)#;.(#V)#;l :"
w-=G*- Gytx;dx; AXAAXA
ryr_ _y_ _ 6enuj oxe
:t-r-id orL yj O;-r- ! 6
(f-f-^)
6
r7l #*'ni=,=i".lJl ;; 'Jos
0'*j LXu _ "axAaxB uj
u*,ti ..,..,..>J tLli .r.rF ;; -f ! .Si.*L, O1y.a r f -Y-A iJ:L.. ..:*loJ-i o:Li:-,I
:qi"f r:-,) t tS-*: o.rcG;l orLi:-l J et';tS; ;t
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..:*l t :Xe;l "r*,bP
6
x=x6,\; y=y(X,Y); z=Z (2-r-n;
Fie = (v-f-^)
'-rrrcb &;-, L^r*.p.r g.+ rl:;lj )l ,fV ^Jtu d/-l J- f 7ps Sl ^S q5or-1,
iJ-rL, jl o:tru-'l ; j5* * .* ,-l-b .4F 1-r ,f5,.: ,-.., ..lt-rf g.rl-tK,-t? .,',-lor.:i
lJ-o:,r,o 4+J \ -f -A
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(A\-r-^)
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uuoAX AY
NfioAX AY
oo\
oo = _p + "[(#),.(#),]ovv = -p + "[(#)'.(#)']ory="[##.##)
O:nJ{f. 6ly o:-9a aglaL2^" *8" r?r
AcAYUcAXNAYNAX:::=r-:--=-..rr-r-rrlN- ay' aY ay' aX &' aY ac \
o;2e 4).1t = ry ,+s 6;)jtu li ..ll-r,-r b ,J:; .JL^dJ-F irit- r*.; J*:!*Ji fi-Vr-h
F^-t =
oo:fI
ow=fl
AX AX il(AxAyUAY AY AY
Ax dy Az
az az azM, dy Az
'au uc
AY AY
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oo\
ax L_ ax 9L:,AXAY AYAX I (1 -f*^)
(\.-f-A)
(sJl\y-f-A)
(9\T-r-A)
(r7\Y-T-A)
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:rF,-l :1;t
o"y=_"1##.##l
afIa,
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(9\Y-r-^)
(\t-f-^)
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5x(ty) u-ti J<-, ;t ,8 ,rbt o'J>..:*l e ,j- 1: oJ-ic-:r.r df*. u* cr-l.r
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(\0-f-A)
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f dY = \'0y
rJ:t -. :l q(x) = V(x)l-\ r,'e*r': L..:*,1.r q .:.*, jts -flU.f ,*,-,1 c".>\e 6
:5.t.1-ei \:rr ,. : Jlftl oti./ \ O-Y-A
Y: q(x)y + g(r) (\t-Y-^)
o .-l 6;Jl$tl o.,.U.l_e,J: Cb ,<tg(,r) 6tr 6
JrL; ..r)rL-. t :-*d O*.b .JrJ,,Uf l; L* ,Y 1X o"t-.i o.rl: Cli ;t o.rtc-,| U
O=PJ41 612." a;-91 6lbL:." cS*8" r7r
:s*ii J) O)_re y Jl/ ua It
Glff'+ 0q'+g')(yq" + g)l (.-4Jl\v-Y-A)
#=Glqoq'+g)l(.f \V - f - A)
caJ:L-. t4s:..:-,lx 4J \:*j J=*jlF-, "fl+.rL..j y' jl J*1 gbf-y, o;!9.: 6.:-r'l ,c;f ,ltSilfI ,-Slf grV or)Jl \:d,r ,". ell €>l)q 6t/\V-Y-A
*= |w'T'+qg'y+M(x) (\A-f-A)
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:r/,-f *tJ,)r
AII
-=a,
+*(qq")=q'q"
* rnn'r = s'q" * q'('
M'(x)=ff'*g'd'
(-ail\q-f-A)
(.7\1-f-A)
(r7\1-r-A)
.rLlf q(x) eU 6q" = ktq ea"t st oj.3ue,S-SJl-S-tl-e o-\^i jti -iJl\q-f-A iJ.rL,
.rf ,/ 'i=.*;B -9A 6la,->''.13
q(x)=Aek+Bek (Y" -f -A)
g(x) Cb
q"=k'q 6c:"it-, gl ucta;:: s4j Jl-r,lls? J..-dltq-t-l .r!.rL,,.1;i.:*,.t r, l-,1
g(x)=cek+Dek (Y\ -f -A)
,Y@y) :X(Uy) eli ,F-i,-ri-r,i{p,r qg(x) :f@) 6ly +f Jl-.-rlc".rtjD sC 6!r:.."1 y--t.l,0til lj O s -f-
x:l#y=1A&+Bek)y+cetu+Dek
s F 0t-*<.,- ,* u.D sC,SLnc^:ti cor-i o:rJi,J$tu -qi-,) ,F e&,:td,1l;t.:1 'r._*,-t tif e 63L'-B sA.Sbcttj
y = (lrA1-t[tan-\ 1&1 + c'1 lUJtrY-f-^)
(vYY-f-^)Y= YAycnshla
6,1j),5;-,i ! !+-r" orfu J<-,r*+ .J;t:.;"r'J Jr-q 6sl,rr-tac.itjC' JA K 6
JK..x) .rt db- J<-,r"''; J et' ,->l-.o*, ft..* Jr y' i+ Jf rJ-,1t'L" yL e' )S;drL.;l C' cf t-i JJ-- ,gly. tx : " $ uisX : o !:l.r ,ly Ulr:Q ,(.-/ -l ,--cJlO-n
cliloJ-i, c.1tt ,,j,^j ar. Oie",)-,o SlqJi;l .-r,a:,r, q+S t-l "/lO n ,l$' 6,.;JltY-f-A
2l - .}. Icosntru)l-t 6Jri,-/ \:*,r u.A '>u.13
si;l 6.:*,lx : +\ .rb..e"p 6rY:y:t--J1d r*,1.^, L sl,"i!.r*i; J {,rt-r#J ,Jt^JJt tl gltV fJK -tJ cft4jl,."
kL : y cosh (kD[tan-'6- +l (YY-f-^)
.rltY: !.H 1s l.t:,|1.r *fo-t-3 dle(-l., iL-,5rJ^i o:b 0tii.t-!.ii.r )- a'.r-l-!:"t dlf
,r cc...qj y--rr,!til +f,,*o ,ll el-t 4;U-,L J J::.-.4 F :l La; cu,iL- r*{ J3;: cdl.:
rSt+ q ,,i,a g1,.. j-f ,Ss-s JeV CJi ;Kutt ! ,ri o6 ,$:.I;4 tef.)y{-f O-rb ,..t.i; y ;s ,it-." ;;S df,il.)
2Hi
I
iJta - t J6;
(...aJlYY-T-^)
(vYT-r-^)
*lI
i2L
I
-L
,;i g:r,-|5,.:.1r of,s:j 5.:-) cr-:
Or-Jiqe 61, o:-9- 6lhL:r" *K" r77
u'A-AJK;r.r, .rLl-i 1., ,,tit r--- J,C, Oi?E .Oi c.rU-"+ ,_!5J 2-, af,S;).1-:-y cr;
A* j l) Js-re I F ,Sl.l.n;f ? cYf -f -A iJ-'L,,St"c^:b Lr O"r; oarh,U:J,-l-,1
(JlYo-f-^)oo: -p +-tcffiiffi
oyy = -p + "iu'fffi * Y ##ffiJ,"Yo- Y-^)
ory:-|ckyffi (vYO-r-^)
yp 9E,sl b ..l:;*"r or.iuiL..i p oE dUl .rL:i ..:.1t"" 0_1.,.l .,r Lari; crlK, ..t-l .rr
r-r-{,f ,J-KJ- ,r:iq ,iuo 6lrr;4 Ss, L-,1.. ,Ss-i S 6luzlg! ,S-tt -n ,j-V F ,Stf L; 6*,;.11 6lr.:,r, dt<,| J<-, -*o rr rr.,t,l;
ta"l; ,t S-SJ|fu| tr.(f-,r J(,.l;) .;--fu.y2" .dyi.3jai I l.6r coJ-i or^-- 6)U
:p'--ll:
[' ,1,=!: I orl,:!
e = + ryffi #coshYlk/;tanh(/</)
(Yt-f-^)
t_
(Yv*f-^)
orx
lt l
,-^J6;{.it- J,(^i, J*-t' S.r-Y 4;t; Ou F
:t-fr,-f ,lt1r; ti r2f t d.rLti 6Ji..;L;.i o;V,i*jlr U
F- (YA-f-A)
crw
i*i*t*tj*c'rx
Io o*l=,*
€tv
Carrolt, M.M. (1988), "Finite strain solutions in compressible isotropic elasticity," /.
, Elas.,pp.65'92.Kao, B.G. iria U Razgunas (1986), "On the Determination of Strain Energy-Functions
of Rubbers," Proieedings of the Sl'rthJnfl. Conference on Vehiclc Structurnl Dynnruics
P-L7l,Society of AutomotiveEngineering, Warrendale, PA'
McCrum, N. G., Buckley, C. P., and Bucknall, C. B. (1997\, Principles of Polymer Engi-
neering, second Edition, oxford university Pr9ss, Oxford, u.K.
Mooney, li. (tgAO), "A Theory of Large.plastic Deformatiott," I. Appl.Phys', Vol' 11,
pp.582-592.nivlrn, R.S. (1948), "Large Elastic Deformations of lsotropic Materials: IV. Further
Developmenis of thi General Theory, " Plril. Trqns. Roy. Soc', A24l-, pp' 37-9'397 '
Rivlin, R.S. (1949), "Large Elastic Deformations of lsotropic Materials: VI. Further
Results in the Theor-y of Torsion, Shear and Flexure," Phil. Trans. Roy. Soc., A242'
pp.t73-195.Rivlin, R.S. and Sawyers, K.N. (f976),'"The Strain-Energy Function for Elastomers,"
Tians. of the Society of Rheology,20z4, pp. 545-557'
Sperling, L. H. (lgg2),-lnirocluctiiti to Phviicnl Polvmer Science, Second Edition, Wiley
& Sons, lnc., New York.Ward, I. M. ancl Hadley, D. W. (1993), An lntroduction to the Mechnnicnl Properties of
Solid Polynlers, Wiley & Sons, Inc., Chichester, U'K'
O?J:,.*4 ,gl, a;*9- ObL=-" .S+K" r7A
J;u'.J.y-) oy..c yC.4s U"-S ,F r*; r-;t- .Jt^t-q ,,.t-l6to.;_p... o.l-,lJ.I - A
Ir:Lrt*lrt *lrtIt = ),r\ )rrt * l,tirt + )rr' )rr'
Ir - )'J ).tY lrl
:qSr.-6: dt*! .,1ltjdt, , 4rr.,,-t o:l- 6,1y.
1,=lrYrlr'++n, trr'
,,=*.#+1\Y1,1
:.t Jnh-lCen Jf" *n- ..t!rf : I, sIr -rl, ,Sb!! o*lt fi L:tj1f - A
aL+: \6'asowAB
AL
d=I,6.en-CnaI*
* = Irc,qn-'owAB
(._4Jl
(.-,
(6
:.r.-o: rll*Ilr slr J,, ;t t 6Loot*i5 *J.1, sI, d-yu.11 o:li*..| \f _,t
#=*,r,' -^;\(#*tl #) (v
J 03.''*.l o:L c(, 6,lyW(IyIr,\) ot4 q fo.- -*;ts -t: ,,-;-S ,fijl U;rf -b y! r - rr
:c-:i y) o:to u.lt u:-,f-\ ,.l!- ,JL^*JJ. ,;li,-16+1, rJL5.,cS-J -rst-l
P,a = # = \ X *, #(/,d, - B)Fi,o+ y/r # r^^
w1 ,#* ".+;=-; -U( : t, ,Jas
:r:*,1 oli o:lr yj sst' q '# F A - A
oii : i r,u^
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o,i : + lX p,i * # r,,uo - B;)By, * + # urf
:+r,( _h -t' b.tj J<-, *. &f 6ilt, - A
w/ = u@,) + ut()tr). -(#)
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^ aYw ^ aYwA'm:n'M
:-,) o.t-* y '*-S G,3Jl oL.r'.fK: .fu*,) ,S;l: ,,."Le ,K- 6,y,- jt 6,116rl. ,g,ly.Y - A
'.tF,-l l;3,-y
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-JIt sI, Jr gbt-! *'-r 1l o:l,u-,| tt (.ill)
(/, - r)' = *. +. #. y/, - r/y + q
:.r,-.r-ri s*i) tu fLi ./-S 63tl t jJl;l Jel-. eb grliK,-h ! (v)
u@,) = (C, + ycr)irY + (C, - ?Cr[tt * Cy',-t
-(c,*ct-rc,
OlrsJ.1,.o agfs., a;*921 obL,r-" *& fYo
:+5
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ni +o, ,lL: t - a
- Y(l,-t - 1rt7,,Y1 {il,
- Y(lr-t - )rr')r,,'1 Uil,
s Q1ts,,t .O(^^ orL.;r 6tryj: .,.-5 e:1:f.,\ Slt u--f ,*-* ;;.St^*J.1..orr : . af c--c^i.- U_l )lp &.-,L tLi.i ,glrl J-- 4-, {, ..!.i*A c_rf .<a-) .r_*Lt'F
1Ft;tSr 3L *.,5,*.) .Slt r^Su" otat*l-" JLy, q {tJKi -# - rJrL! Ssi 6Loortr t. - A
P & s "l[\"in qj_el a-.;-- Jp 6,1jt.fu.-j yl .c.*t oJ,i.- -5 ! ;rt-, ;l:.-l \inY rit*; Jfu,-+"
Displ. (in) Force (lb)
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4.62E+02-5.35E+02-6.728+O2-7.40E+O2-8.13E+02-9.458+02-1.018+03-1. l5E+03
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:.ry5 g-t l) ;t-l,S - ,':; .1try"; (iJl
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q r! 3 .:-'l ("ir - ,-pf at dr*lt) )-fu-: oYl.." r Q-F-|-.rs otl') .5$-,)l
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02*.l;'k4 a5(y a:-92.= 6lbL:r." cS+K. rvr
,F J*;lA: tilAl ,15; p,K;..'YQfup '/Ll
c>Y.tl'e f - t,r<Jt-ft'" ,$tlZ-r;i.+rt s g\lt c.(s-,it }4;)el r>Vrt'a; )l ,-.5 -
crl orf ,tji ,-F t* ar 3 J;-a:,r-o J<-, r* s:lS.:\il*-l t, 6,r,tK:"n c.:-,1 :,,2, 6.L
d trl{ Ft<" ,SiJl r:-fu-s,lL--f ;t 'gSeL.ty€ a:^ilry tn-r!,-;"-l ,S3Jl
6 6"6a clL;^,t t* y d+ ff tlyll .5.r1-u .r4,l -ib c-,y-, t-b y.gl'r.-:
,dl, 1: ..t::5r, .il J or;3 l, ,-$tt 6itl rsf ,g rly ,Jkf ,St^-l-r* rff sJ
, ;L; L- 3 o-;>.i eLslryjb s s:15,fi "o4a' y \ 6i1l c sli:- t* ,. A'J-:ly i^,4
J<^." .(t-,Yl1fu,- s slis tr-# t* t .c.*.1 ?y.<.:*,)lJ(*,--l q -lli-., O{l .JSu,
! : jF--; ,lL--f -fi-t e-jl: .(+-,)l JS-.i rn'r l.: r 4, )l 't rsf r, .:.*,1
,.:- u-*;l.r!.:t--t-tJq esj;,;-g.,ait+iOtni.:3.t*..;tc,JL;,c/l .5La.l;."*1.,,
;fu-:,lL-ra J .<*-,)l J<-,r.* 6-V.r)t- 61t' o 6.{:-,)iJS*,,-J;Lit 6Lt
fli l, J.:i...^ 3r ,-ri jl -.5-; dbtlir f .r*-,U- .Jlr-fJl clL;^r t* y3 o-,11r Jst ls
rr y' GU: 66L; JiLx g : *f s .* Jitj f^ .t)rU g;; c;13 t-b y.rsct"
.ry -urlj; * ef : ,_r*'
.,S, Sl-e. eLl .:YrL- ,F !r-;lrS--t 29,-lt-( SSa "o--'; s:
.rY:l-- tl *;7 { c.:*,1 o"t; o.:b Ol,:;?- \ Y-0 iJ.rL. st 6.t* JLr (,5$-,Y1.jK*,"J
.,.:*,1 'JJ JIF t+!"1 6(\ Y-Y-t iJrL.) ,rr-rt-r
s-o;j 5::-,)l sVis ,Slt..ft*,l
Si : 1G4;i
oii = Y Keii
si : llt.\ij
uii :-y(p-**D;i)
1.-,oJlt-y-q)(.f\-Y-q)
(._cJlr - Y - q)
(vY-Y-q)
c(\ 0- \ -v iJrL',) i |fu-; rrL-ra 6ly ,o;)\c *,
L* 6lr. V; ;.:)rL.' yl .slo..r; oV, e:f ; .ilFl ,Jt^yt, -J.i_
ro 6F 6tn.r[rt ( 6 €'* git ,F.*-,Y|Jfu,. s 6s-F 6l; .-r;.+*, &,:i:tlltt :1;ue ,a,:', sl; orb gL;J \ A* \ " -Y ijrL.o -lt S l* rlL.r I J::...^ 6,7tSu;ra ,:lS,^
t;?*-J-U(tS*-9:1
./\f-\-V: Y.-V-Y ,J):t , jl Jt- d/lJq
4a=fu (f - Y - q)
t -.rLf^i l. -Sl U- e -r^iL. cl.:fr.h.-r t- ,!V3 .fur'f **; q ./Y-Y-l iJ-:L. .:sp ;V3 f I
-# aJ 6J-5 -;l q.:-,1 o<^" 1s) otr,;, 6 6J-S..tlii- '* ,sJi blP
T-Y-q iJrL-..l Y-Y-l ijrl--. 1J- Y-Y-q iJrL, '{ +i c:el; +itjlDii )Sa:.:y,..i tl:';*.1 tJ os-y- 1-l 3)\.al c.*,1 J(,o-
iij = Dij
iii = Dii
si : \P* qij
oii = y** i;i
{P}S,r - r {Q}eii
{M}oa - Y{N}e;;
(.-.cJtr - Y - q)
:6 syiu" naa 611s,li jl r(vY-Y-q)
(.-.ajlo - Y - q)
(v0-Y-q)
3 oJ-.i c.,r"6j ,F-*.Strctb t, t+i 6.tt:,-r, JtJj \-Y-q ;0*Y-1 .r)rL, L-{Lt
crill.: .i*tl c$loJ-i O!.Ff ,S6aJ tl ,lltV yJr; ,5La.;J*Jtt 6ci:L'O,l6ul*' at
",L;.I;ti, ,SLn-l-l,ly,l ,Sr b ,.:YrL, '^eJ^--' Jr J^;l .r^.^; d/lJtl
)JLb 4r..rFcf s..,:.rc* Jp' K C 6-*,sLnc^rtJ rst+ q oJ-'l o:lr v-\ Y-O Urt - .,:
b p-5"f dt..,l aJrL, gia,\b q b gtqE, "!1.il,,l1r,SLn-r-l,ly,lcg:+;
Cr,lOrf J,tS
'f{.1;l csrr tu
1-illt-Y-q)(*7-Y-q)
^9; tfu"; ,lb-.f 3 c.,,^.:-,)l ,J$ ,.r):L, ! lFrl r, ,-5rL 61L 6:J-e .5to1;5U 6{M} '{Q} r,{P} r !L"j ,}t }-t,r1r ,Sl^t-l,lr*l cr.i f3 6 t* tlL-a ..:*,1 or,j; a;-ilr
.>l*ota; -&U.6o.ry V-\Y-O iJ.:L, strti Jm; Qi Qi ,4-ln tt JS..i g {N} 3
j Ls jl e)i,ul5cJt-.,;r .--^.1 O<^,. =-l* Ul q-Sl.c,*'l ,g)J.:.'; irl- 'fi-tJ
J.1S-e I t{Ji s-r.l e-* t+!-l 6 'W
irL 61i;;: t-1.':..iL ,S-fr-t gt^fttU, U-
el-r. 6 €,f cf4 6J-i 0L,. l.u.,,l .1: 6 :-b rlLr .-r:r; € t;f -b;r ,r1Lac."Li
.:)rL, {N} : K J {M} = \ c {Q} : G, {Pl = \ glr:-l,ly.l tl "-aV
6La.-,L;Jl
,{Ql =t; } rt t: \ dlJ 16r;;sf.,yscJ56 i-ys b.{:-,)lr$rt-Y-q.r-pL/ o:ls oti; )fu-:
'!-1Li2 ,{N} :
"* } : {M} -- I
Osyeq tql-lr c/r€, .rl;x i^.o "!.."
1.: +fc.*,| o.:lr 0[,i.i ,:;-J6Lnrl-t"ji
O:-J.6. ,gl, a:*9- <Sb 3-s" *K, rY7
d 9*; "+li ry 4t-.- q-f 6J+ rlrl: cd,r jl cl;{ #-lJq cl:.Ii.r, JtiJ ..(.:-,Yl
I .'-t*ilrta: ;j-l;l JK.j ;: up;.<S-,Y|J<*,-; ;tti.r ,St: ._! .,., Ll ,.:YrL-. 6i J2j,.a
fnt
t - r !iK-i
.$r./ .')lJ L uav ,ft..t\.=-- u.^(, or1.. e5o.rL ;, crtJl
{P}Sr = t {Qlqii
oii - Y Keii
1,-Jlv - y - q)
(:v - Y - q)
t-: r-! {Q} : {P} dt rJ,ltl (,-(i*rr-r,al rli.r 4.Str.r.iL Q:istl L.,..,61L
P ti \ tl 6i,_r"_$l l.{Q) J{Pi]. 6r-b oL^cJiiL u:,;3,s)l-erll-r-'L;brrlyl g"-.t3,J ,-1.,t_r+ l;t, iJrt ,-,*,q ,gJlV-f -1 iJ:L, d/l;tl rJ-a.:r.o clt-il r,"r5.rr r"ri;
j*l* glal* e4ib,4,St-Ff - t,p;l g-t-: 4 ,lL- 6F;) &-i-,YlJ<*,_J ,Jrtrl 1g1-oo.r.,ljt gt!.*:*ilFl g...,q q 4-r f=i- t* yorL.(-illJK*,-J ryt,.1;a s ;.y,f ,;rq b .rJ ;l orL dJc,Jt- L sJ-i o;l-.il V-Y-l iJ:t - t cl{--r st 6 s-*r-lLr cc.-,1
orL;l .-'<, 4 .rt :Q '^:5,g../tijl ,6rLi 6Lro-r,l ef i 6ll telj -;&- d_*6oJ-.i o)lr OtJ \ -1 J(i st 6 s*..lL^ .p*J.f Ji .r, ,o;tf ,t; orL ;y ..:; +fl,
.tt-.f ,j,,65d.:L, 4 q V-y-q iJ:L, 6 €e 4?j d,Jt- Crt ,Sl:
{Plorv = Y{Q}q,, = r{O}e,, = {Q}7,, (\ - Y - q)
e\ Jt..:*,| o!.i orl.r 0L.ji \ -1 JK.i st S _s* oL.o sg*,| "r-,"rg '*t ,-*-Syr, S
iJ.rL-, ;, {QI : {P} .SLn-rr-lf,l 6J^iL d+"Yl P ,* ,.,r'i i:L ;t5, r*3;:rgcf oath ljo.t-f o aJ.:L, t:t:(Q: G, P = \) r:lcqt-i \-f-q
{-
rYY
orY : GY* (Y - r - q)
Q*,-,;-,otst-) J9t .:t-:;l 6c,*,1;t- ,r..i. aJ.:t .o y-|..:*l .<i:-,)l tfl J-l& G 6:r-r,_/ o.:l: y j iJrL, J;f a-r j o*,1 4 -*F ori JL-t 6Jt ,,'
"!;,-$.jt<" ,i d
F-k6 (r-Y-q)Jl,;l..' u!; -p c$L-i- rf5--i.,rY-Y-q I Y-Y-l .:).:L,6y,.7 e.:-lri c.tljk *5
yl.:.,11.: \G Fk t:-,,_l o.-f 6,.:-l ol;3tiil o:L.{:-)l u.t,-$"j15.. qL:-
..1*,1 ol; g*; Y-q Jfr,.1:;l:3j il*-l ^'. Jtilf c>;_2,o q djLqLl..e:.ii \-f-q iJrL.criq )-fu-_: clL-f c6*,5-;rt- --t, ft '+1*,jsrU.
:t -t-t '/
onl 4_or2l/" lllqrL
Fl / "^AM/'J I
LLd-:r-1J6:
..rL;r, dlrl .,illS, d-t.i
rt,drY : F Ttt (..."aJly - r - q)
! -.f;-5*," J ?-t' lf.:*,| ljJt.<i:-,YlJS*r-J G:-F;r.r:*,lcr-j-$*,--l ?--t'lt 6
?--t' ,SltI rJt<-,| jl L "rr,
d/l jl s t-"'J ,-l L;- iy grl3 c:yl.i.r.o orl: clti4:l^;I O22.o a,, ejjlf -f-1 iJrL, d/-ltq t€,; or[l;-.| a:l--,-5*,--l
ott = ITtr (vr * Y - q)
,-*-) Jq )*; L.tl-- o--l FtS.. a-,LL .t{,-r ,Jei y'-ljl .s-r,r .5t"rra i^-o rr(f-p tr 6'.ty (r;*J,.51 ;f,*'-1;! 1l o.t-i a 6;l:J=. rlj-;: 6l glrt, 0F-+,
i CtS j.,a clil -lr c.:-,1 F 6,53^, ,SJi
hz
O:aJ:,.4e ,g(, a:-9- <Sl.aL'=o" *lS" rvA
F=rtd (o-r-q)
.,,-4> on,
I[]tctz {-
L,
l,l/,Vr
f'n
6tt
r-1 J4;.:fu-: uL-rr.rF
gLnJ.t-. iftt*, L, cJ-i ':b 6jF )tf t, +5.5.r$-, J-al9.: ,,,t*,lr-'i.:t-, :# 3r..-.'"1 ort*, J-1,-l cci Cfl -ll -.*,1:.. d; il*-l *., 6.*,!|JS*,-J
jii .,c.*,1 6l-r Jsf u. -S ry : -* 4i O*...'.J .-vs,€;*3 q Gsf o."V\-f-1 iJrt -. c,JL-- drr-l .r, 1;,:Su" J--p 11 Jve?af lr.X.rKJrl+ 4te..-eJlf-q
dry= {G+I\)Ts
:t-,'J'€
(2 - r - 1)
osf u. -S y-eJrre (lJr.c*,l r.r; Ot"1 A, = *il*-l q 0t j 1 r:*l Gf i* 6q
"fLl i:t-- 6!1.: : J13U Jqr Ft"i c,-rf-l ,fS..i c.:-,1 ori ,)^ f t t5r
l.:*,1 p )S:(v-r-q)
,-r; 'Jj gLnJ-r, .(rr ol.,:J,ft;i elr:s r.r;b .StrniJj" ! l-ti" Jt5-,1)z = ft Su
"!; -t' oi.*- tt C--# -r-l-r 4, ;Ss .tr,/ ,ti!,-l .iLi t ;j OUl 3:;l ,+;lF -,"t- Ft* 6 t_r'l *.K kV J"r, s._oJlo-q J6j.:;* .Jj 6_r rs-re
to * *)"', = {G o,l T*
rvt #'^j="=a-UlrS*r: I &6
J* cr-1-i 4; 6r ctj,re y -Sy* "i- q dirsJ--tj tlL-a ft..>t rrl-r,b-,|
.i-.,Jt." 4;l .rl-r,. ''f t* I .$)L/ Ot..ii b,J;"bq .-,Jq- "i- J-r-. ;,1,-,0-l
J'tU ;t-,i y i -frt Sp\|,,r- 6t-aJ-r. ..:11: ,!La 'F )l,F ut-r- f l, ,l;i -# y*.uU. -S yr- -5-- t- ,li .i- \ dly 6 J t*SV -r-lr ,-i- ,Jtj- .Jlr d..t;y.ai
o*- " orr-1JK-',
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g JySU t A-F g;l-aJ"t- clrl
6-&11r .$r--i, ,u;L .:-,1 irS^,. j,ll tY! csLo+r"r : 6r\l,ltaa ,5LaJ"r.
f,.- JJ. q 6rL;J.r;lr:;-,e ,)-*t t* lr)! *.,r" gLnJ-r. €+.Slf .fLlJ-ti,gt:.(-alt2-q J*,) ,-r,Jrq.:,. o.1l: 6r Js-re I d//sJ-'\n 6 A3S ^:s\
J.(;) u--5'* J-.SU isU- -;
J-r. "ri
t 6.:*,1 ,$l-r rst' I JrJt- J->l3n
.(vp_l
a-1,1s;
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;.-) Os-re *'.,3si:lr',l rfS.i.1: (\-f-1 eJrL")./Ll .r)-:L, taJ-u y-l glf1,.:*,1
,iJl, - I ,$;Cr_rl5".rU- gp^,- J-r,
ctf - I J<J
.r)y.SU a:i[- r-.-,- Jr,
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fAoO:-JJfr. 6ly 6J-t:? <Sta,Lra".S+8"
rrr = # - #+ ... + # (,"-j,n -r - \ a-{
d*= Gil-tr tg++...+
Gfhl (vA-y-q)Jn,st
ar* tr_
u,*, a,+;
t- ..F ! o?-J 6lo.:L ;kil 0.:b OLJj Jli:, r, .:-,1 J<^" U[ 9.,or; J-r, ,.rY.:L.. y l.;s
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"t t+ glf .fLl iJrL, s.rit-a 6jl.,,.. t1.i.1,illn-f-l iJ:L, 'roJ,*^tr..liJl*: q 6.-u.3 rrlr; orl: s..all0-q J*, e.:.;l_r.j[:.,1
aaa
G,
'a:=""=:'- U(t5.,,- 9 : 1
-s -l o. js|. ! or, .rL ,-3y ,tJ 4 c.''t ,J^'<' irL ti:b ,li J*ts G? c*;
,-;;fo ..:*,1 rlL;;t u-ti bl-* ,. dt ,;;f 6f 4l,rl 6s-f ,* t j .'-ri ;;':'t' K;
\Ts.glaLJ ,gJ r;;f ,4 .r:; ;1atlt r;i.1: ;r,-t'A ;f yrr(t) JeV
JpV rs Ot;* ,* ,.s :-*,-f al[ oK: 1* oi -rr -l '-r-i JL'cl .-'<' a'.y. slti'
nl ,r"{Lji .ll-c '} 'F tl& 1r ,:.,oK .:-rc/ q 0t"j jl .-*b Jl-*l- q orr(t)
'^U: y el) os2e q tr.S-ll-rf.lL .:; d/-l ,i! .:-i,g o:b .:*i 1:;s ;kttti: -*,g i-A; y; il.:L. o s t' o' oS t rS u' J s i'e u(t - /')'r'-l-e d, Cb 1l o:ti:-'l
u(t - t,)
c:yi; Jt^elr - " Ot-j st Gf e-tlS.t\ft.:*,t oJ-i o:b gL:;V-q ji.i.,b_2-j;r3
:t-J€ A*'Ji t) Js-re I ,JlJ
U(t - r,\
I
drr : o. u(t)
v-1JK;
,-Stnlt Ct- .21:.p, ;t-;
(Y - f - q)
iJ:t - c(2-Y-1 il.rt-,),rL.-,*rlf J"r.4, elL,fLl iJ:L.. 1':,'xJ yldrb ':Jt--s U
kv "Jr*;li,
:{:t2t,
t < t,(\-f-q)
o.u(t)=GT,t*4'ir,t (r - Y - q)
O:-.L.1"o 6(y o:-9;1 <SbL,:^".9*& rhr
yrr(t) : o.(\ - ,t/') + (f - f - q)
g, &.5 d + @ dJ 6 t-r-, +i.-l +t? ..:*,1 ft^".* ,*.L ,"*,-rtKl 6V t 6)r df ft 3 '>,tT ,"i,,, ,yf j tt : o 6 rlu:or;3^, .:^-,su.c} oWrh;,.a'U rsirt q.#, Cf-lq .*s€t =r,.lLj tr.fl..Al+ )l$.a., ,.:,;l:.r^ a-.l:lg;r ef-l
&:JJ €-p O;,2-,,-a t *.Q-o-a t,-f ? 3-t, ,A dJ,+^,b cf -f -1 iJ.rt-, .tj; ,-l 6
y,rQ) - J(t)o.u(t)
:t-*c*
(0 - f - q)
Ctt..,K ;.r."U.i- blp !.tyct' I ,-"c-1JU" G ,ftJ_l! ,r.k / c"ti 15
gr 11 csr os;e *r 6.:*lr:"La-t-l-9 df Cl-1, Cl-.<S-,)lJ<.--l J-r, -f j Gf.jJb-q ,[K,.] rrl-r,U-,| ,]r -r.l+ .Str i/lrq .ll;f^.o:..
I(t1 :/,(\-ett"1 +J,, (v - f - q)
JIP-q J* cxrl5 ,u;u- a,^; J-v elt s
N
r(t1 =I+f\-r'/',),- I
iJ:i-.-) .fl*,| ijrt--.lr y-y-1 iJ.tt-, 'JSV J-v ,Sll ,-if ,S:IS , .U !L;;l .,1r
:.:*,1 $ ,I;l*.: iJ.:L-. t>-2y.o rr q;i _9 oJ-.i g.;l-r(.t+ (V-Y-q
l(t)-+-,/(\ -r"/")
:t-r./ df -b(2 - r - 1)
(^ - Y - q)
d(r) + o 4LGqTtt = o. (q - Y - 1)
'^J:,t*Al9U9:1 ,J,as
,r-F ,* o ..:.-,1 -r-lr d, Cb .,,t'j J,:.i;' j t..:*'bJ: Cb d(t) 45
d u(t - tr)d(r-r,):i (\"-f-q)
:rPcf 4"-p -1j .r)sL, b 5
d(r-1,):", t*t\ 1-;Jlr\ -r-q)
-t)dt=\ (v\\ -Y-q)
6rl:otJ Ollpd r
- t,)dt' - f(t,)u(t - t,) ,Slt t)tt (\ Y - f - q)
1 ilrL.. 1l cr.Ll y-ty .c.*,1 :l-,tfi(t) &Po Cb ,o .Slf
:f-r-'l J-rSU 'i? Y\
{: ,-
u' b ol; q5
I *,r'ru,',K:tfat q-r-
y,r(t) : ".f(f
I(t1 =(t .;)
\.ry,-14++ ,-if .:;;l .:.i, 1 ,;" &;l-e r:it
* n)"u' (\Y-f-q)
J-rSu d? 6b oi;l 6
(\f-f-1).,.:.-"1
! Iir" oVf '^t-"i:
- o dLJ;r ,rJL".el ef ,t(\0-T-q)y* = T. u(t)
iJ.rL, rrlrv : y.6(t) e:l-€--V il*,-l t f-rr,*. t* ,..f-rK.r$t ! I+r" kV .p: Oi ia"p 6 t-r,g o:1.: 2-Y-1
o,r(r) : y.lGu(t) + ry 6(t)l (\t-f-q)
O?J:4c4 agly a;-921 15L6L!^. cS+K" rAr
'pS:t .--l 6l1t : " rrLj.,1.: 4i.:jJ.*.o uiJ a.5,-*,1 6i -€"qrJ:L. c.r_l t: tJr 5b
;r e \0-f -1 iJ-rt-, .rlL;^.a st_S 6 jL.: rJ-*JL.,8.., 6lt.rr-J^lF )Vy..rL;^,r
orr * G y. 6(t) (\v-Y-q)
:6"ta:,r" 4*ij \ Y-f -q iJrt - 1l o-rlo;-,1; )! ;l ,S-SJl-flll 6c*.1
o'r(/) =y.Get/'u1t'1 (\A-f-q)
;, c-iL-.f ,*l:l Jl6..r-lb,r, +,JiJ OJI , )r.,,;U;l .lt; ii 6:*,r, oJ.aLL
8 JrsL J-r. 61r,. ;!;sllj rrLl t ,t-lt\ rr-{,-f l.r* ,irs _p ! F t=t o.V.s
.r.Fqf6lorL -f ,Slt\G(t) ,rtt ,r<.,,1_r Cb ,,_r;;&,i t: W-tI(t) GoCb qtj,
€e -q-* o,tQ) JS 4r, ut't& il*-r q
o,r(t) = G(t)y.u(t) (rq _ r_q)
J-,SV J-v slt r-it ,&.,,t_, Cb c\A*l-f il:t-,;t
G(t1 -- G e'k
.J";trq, iJ-:t'." JF 6LriJ ;(artr.:yi 6.,1UKta v-f-l iJsL-
(Y. - Y - q)
:..:*,1 l) Jlr- q cJ9Jt" a5b- n*.r; J-v Sll3 c*,1
N
c(/): I G;eak;i=\
Io,r:t
(Y\ -f -1)
cj-l"tr -,1r orL ,5ii-YlJ<*,-J 6LaJJ, tl ,-rl.611 ;!;s.'ls s ,_ir {-,/L draJ_l-r,o.:l: J3J>
".,r1 -1, jl ,-iLi, 6LnJ-r, J db .-,. -L*jF_, .ry:L. ..:*,1 oJ\, orl.r \ _f _q
..' - -.1 oJ..i
SJr't -Ar$-$-l-Yolo-16 + sl+
Bl: E lo
]o _le -lL Ol+\r luo
"t\. r,
9o*,,u il " e L {Il lv r r' dl , a; *'l o 1" *. ; l"- i.;-tu -io ;i-u ,tlo j;
el-1.lo
S \A r,.L6e ss; FiI 's '$ I I I ; -j * 6 Btt rr !r SA I o'-; + Ub b b t1" I I [ .i,:.o* o -lo tt I
30R-
+o
st\avtq vtneEo
_i'Il
J,
\5
q.5f3
{r, q-j-f,\v i :.,1, 'l J'5 '-\' 3
$ui
*"++.i ai ci
#<,
j-f,
-f':r
\eo
1'{
I
5I
3"1b{3l;9,\\"{ '\1A-1, Y_59i-:)A
5J.\41' ri
"(la\.J
I:J;dJ
Jq-
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90
#I
#
)
.*'
T,
rrp
$tll
^oJo++ q.
!{t
-q) 9++
C-qa.
JI*l-sl<l
{3
Ot
I
IQ
"-l tI Gr-
q't!=?]++J
-rlq jl-<. 1 Jlq+il:1? olJ X' -A--' Sl+ a ++ -l-o -6 ait: q s vV++ -<r $ +j -il" s rlet: - | O
O
sl-t*l* ol.sl ol- - lo< | d_ :,1_ * 1."1
i' oL 9'l- tt | -Fl
'rl
|3$ $$31"1--j^s'!I Gt-
(\v
r' Jlo+lsr: I-<l
o,sjs+9' "t- sl -9 -93 il 1lo + +I ls il' a oq sL{ -i *i *lsl'.
sl-ls+ lisl'<l
(\"{
q:+
-l lq'il
Qlq''l++
.sl1 l--* ;-q+
ol-lrFtl<l
+{3 l-\_/q.
Ql--l
+is l*
+ol-*t-la2'
I$
Ln
v
+13
s.*:
r-+9:O -\'a_+z4-
{9txqrv- eq
k?*-J ^!J(tS.-9:1
A- I JK-t
olp:.r- (./ t cSldli F (,-tt :o't*i' JL.-sl ,; C;Loa-:-l
jLli ltAJtfr;t 1"eft kld - I-,1r ..:*,1 -p "ref^f Jel "-=,*i*,'Yl tl;r- ; t;A; a:.-.t-)lf*'- S 6lt
USr:;) ,f ,-;;-S 6 S,-l t":l ,!-.rl d,,rll-.i-|.,1J r:r t;-f ,P-f - ,r;:j \ls-, 6F
:t311 (6l.a.r:J) ,JL^.i"-5 g"a .g-lL,- (t^,jif ) Lar.l:; ;l 6l+e y'- Jt^'"| )l kV.rL r;y.r, c5l4,rl; i;;13".Slr.rlrQ .i:.-l ,l; (61^'Pfits1^.ri:; il.-'-l q os
c.:*,1 .l(t) G? Ctt "i-,51.,111 6.glorL ,. fl,'.:*'l o.t; o:lrgtJ iJln-q JS'; J) 6
lry J.al-g ,kV ef o,y.:; JL-el
Y
y,r(r) = A(r. J(t) + Nr, J(t- l') * Nr, l(r- tr) = I Lo; J(t-t;) (r - O - q)
i:o
ef'!*t ct;F a- eJ uilFr .ri:; ,s-l:6:\ 6l; ;t(:;i .,^:*-; "oJ=-3 o'. j
I r;3.a..,-e-Sc(,;A-l J<;,) Q-; € df -F 9 &7iS .q*,Jl"t",Stn,S-'l'r3-t!
y,r(t) = It*rO
-r"lfflar' (Y - 0 - q)
.r*,ti osfr,/ oLl :r,-*-5 -&Q rr; oF€ ^*3;-l; Jl-S-tl dia., 6rric/ orlr
Ora.J:4d ag(y a:-92i <S/oLza" cS+& rAA
tfrJt : o r-rLj -,1.: ,lJ.,: 4Jl j*-'*,8 f 1.,.:*,1 t = -q dt j jl ,rr;'+,L^ -rl
:J-i r^lJr..rt-s,(q-q J(i) .^j,L. -i- Q : ) O[ Oi b r::; ft r rt;U.a.:..il.r
r-qdKt.d)l J**rtu.i_ q rii :;,.,t
,Jlfsl;J^i iB.il :-eJ> JL"t J eF I oF oy.,a t )q iJ:L, JlF.;l:t ,S$4tfatU
-5.-' ,)<a
(f - 0 - q)
..:*,1.I. = I(.) 6J_l"f \:*,r 4J
-t .,:; -5";. 6-i)l_r 6taJllll "t,li,g c.trl:- &g ,* +t u,3js +t
i.I;-,..a )V; .>;ye 1 .-f Ll lgtaj5* .f..a) a-'-,i l, .l-r*1, ,JL^.Jijfjl .g[-r;;
v,r(t): o.J(t)* l"t
J(t -r'rl#fo,
y,t4) =J.ot(/) + I'"o,r4)ffi1",
o'r(/)= J:
G(t-qwfo,,
(Y-o-1)
(._;jro - 0 - q)
o,rQ) = y.G(t) *1. G(t-qlrylo, (v6-o-q)
o'r(t) = y*(t)G.
fAl ,*; "a:=,.,*Utss,-l : I cJr.s
..:*,1 Q. = G(") lf,J)rL, )l
'fV c,Jl-- c-r-J Cl::-,| o.:L ,-il. rfVlt S cY-0-1 iJ.:L,. C)li Jl-fljl
:;+ il-,-l y,6 :-b dLo.A c**,1 j:li Jlfsl "$!.,1:
,lf 5::-,)lJ<*,-J .-,+Lll.-:*,1 o-ru.i o:1.:,rIU+t : E!yl,Jl"-t,
-J fi ol:V
\ryiil) = f*l,(t -qlrylo, (dtr-o-q)
ret*Q)= J: I.(t-qllffilr, (:?-o-q)
'it,-F djl.-.-q U:h,O#c,.:-,1 tF {.*JU"rrt
I:;rfrlr:su(r) =
J_ yGs(/ -,'llfflta,' 1.tv - o - q)
or*Q) = J: yGv(t -,'ll!@)a,' lurv - o - q)
.r, b.rLi-r +fl+i;l .c-,1 lql ,t "$:irt
j J:& Gv ,s er:r -r: ;*;rtt J-r! G5 6,.:*,1
s??-d-1 iJ.rl*-,-rry' Jf, = * = Kr5lo.rr5,ff $Jt .:lyaqtql.*t-"f
"p.rS,;V-Y-l iJrL, ;tl, ot& = YK ew J* q r-,V-0-1
JIJE ,'l,.gb eJaJ;ia Jlh r$y2l^ cSlocStlJftq, - 1
o.:l.r 1p .<JJTJL-I 'F;1+ .J.:;.; c,-'Lt 6 GV:.*-,)lJ<*,_l gt..7l ,$,
tGs-F;l u> ,t-l ,fsl t3l; .,'*,1 *"*-.*,Yl;S*',-s 6s-F -&-t g j4 r*F,gsfJ. ';;f
,-(.,- .:g 6 t-q ,$.i 1.r o-l..i o:ls clLiJ ,grL .-r<" g*t, q ls:jz gi; t)s1.a u.
'*-S ,ll .6'l:J. c*i-S sl-,t r ,J6t ! 6*-f.rLn
liJlr -r-q)il*-r q t_
y,r(t) - y. sin ott
O:rJt,(r. 6ly a;-92g GlbL:o"" *9" flo
yrrQ) - y. cos at (v\ -P-1).rF"f .lt*
y*(t) - /.(cos rttt * I sin arr) = y. ,''' (Y - t - q)
c,r-J J 6^Lb JeV F *A,Jia,F-j.l -b )l6.:*.t Lre-{.c.*,li - fi 5
,Jl; .JJI$J. f.o q f.it*. t* y4 ,.a €P-f 6l,or:; jc J .,Jbl ,tXJf ",+e; o_6,*a ! or Ot-*i- ltSj Y-2-q iJ.:L, i!-,-r q o$ t* &-* )l ]-V
o-tg q,J C/lJq cdl: J,aJ+ Jt^rto,r(t) -- o'ei'' (Y - t - q)
l.t t-t, cJt- \!;i c..:*,,1 ,J.i J,$ii c,"r*-! J) jl eq .c,..,1 -tlt" o* 6 rF"f .-lt*
st .r4,-t' !r.rlL; o r:*j ur-t^j J.t.F q r5b$fJr J rj JrlJt , d6-fjt ,-*b.7)b*€.ri +, 6c.*,1,-t, *Jb eq kt ,4 ytLbl
qt- .-(.:-,)|JS*,-J ./Ll iJ:L, tr Y-P-l iJ.rL, 3 Y-P-l U.tL, 15JU<-i? tl
oL.a {O} s{Pl gl".r-r,t;l J* f'-'.Jijr) ro-frct' o:l.r \-Y-1 iJrL, il.-,-r q e.f
:y-:i|,.5t.- - 'r 4r soJ..i c*'yi V- \ Y-0 iJ-:tr- :t 6 s-b
N
o'pp(ir,t)k r''' =2 y.qo1ialkei'tk=o
p*(ir)k
(o - r - q)
qnQr)k
N
xk:o
(Y - r - 1)
clj .3lt
N.xO k=o
,F : ,Ji-h )lri't a-,Li., ,sLn-r-rfU 'ri- q
v.Nx
k:o
:VJ-t€ J-j )S,S \ J s)--p t, Ol G.1tot) r,!L; J;& Oli.o q 6
rt I
.5t -I ss'.(., 1r ol-.i oJ+3 ,S3Jl JlJr,. o u*rll .J-l! Al .G'(a) cu/2i> c*;
1-l cG"(ot) t'fP-f .bJ .s-fr-l ,;;S rr;i,)sb; .:*,1 S:lS.:\ J-t5J4.",!Li, J-l&
-J..ty.t c,lf ;>Ut J-l& J |[51*, .sljl q ori .jl 63Jl b .:*,1
:\:*ii -fi Js-n 1 dliL/ b .$ u'f Y-?-1 iJrL,.r, 5 ol, ,i'l;
orr = G'(iot) y.rt'' = lG'(a) + iG'(a)ly.u''' (v - t - q)
iJ;j,e orG'1ia1.;tst gl-rr yrrrr,ilL .JlJ,. ,--r", tr -t
G@) = 1t 1c' 1o)l' + IG' (r)l' (^-r-1)
:r{,-lo.rl.r yj ost' a,.G'(a) tG@) O-,.d ,i)til L--rlj -jlttt o-rXo I
L - G'(iot) =G'(ot) *iG"(a)v.
tand =#Ef
(?-?-\)
(q - r - q)
tJy,..a q (V-2-1 UlLr) o't ,.iJ JL-
dry = G1o4ei6y.ei'' = G1ot1y.si(''+a) (\" -t-q);Fr-* +;-ii
F -flv.llJi, 6 ty'l oJ.oLt". rJ:L. d/ljl
o. : G@)y. (\\-t-q)
"- *'l
J.*J \ "-1 J-fu.c-..1 .'; q s.i-,id .i)bl ia.2lt tlti. q g;jE r;b 6\l: utsf s
,*f :r:.;6to.;tryc.ftJJl$..;.rc' ill\.-1 ,!5-.i.,1-:.$r,-/.Jbl boJr-+yl,51lrl..ts;7u-co c.1tj ,Slor--rlj ref ! .,ltj ,l-i+ JJ> eJjlo-t-i t"rr d -tf t--ll; ! 6
,fii1.:OLj I c.*i c.:-,1 ,*-5: ,;; ,Jr';{_JrU, -frU.6La.ll-ry 4l 6,:i s J--r.tu
O*J:<p agly u92i <SbL:e., *& r1t
o. cos 6 ,-*_5,r ,lJ f{* ,S;U Oye q ?\ "-q0t* yj O4e r orai3 r)sb (1-P-1 iJ:L, oJ>U t
o cosdG'=v.
-i,, o. sin dLt =-
v.
;t<-,;t .c,*,1 o-l.i g-,,:r ,-,\ o-1
\ \ -2-q iJrL, '(.n5 t 3 .:*,1
:r-du;(djl\Y-r-q)
(v\Y-2-1)
(a)
(r)
l. - I c,ilg
,;;.5 - 'p jV) 6,Vua-' (y r r_* -t sh 6La;ls; rllj.r (..,il| :,5.:-Ylrt--1 orL .1r ,-it -*iU
)r 'p; J jLLi 6tor;.; 6.(IL*YIJS*,_ s ,Ss-F L.o -lr )Fy 'S6rt
U !L;;l 1.:
.gLa.:-J (f -p rtl'1ira1,!hi.. c*JL..,-F )Jyb{, ulju d/l ,# 6La.:^*i:etf ,.tt-p l, JL"tl g'b 4 cs,a:>,a t-b y.r- t+qt+ &1,-l-*-r : uprj>
(\r-r-1)JeV u.ijf '3Yr a.,
oo sin ar I
sn(ot-0
olY = o.ei''
t ialT;=T e (rr-t-q)y:y,-r f,.r.4.);,.s)lJ<J.t+ \-r-1 ij:l.'-. s" up5 6
= I'(iot) = J'(c,t) - iJ"(rtt) (\0-r-1)
5.:*,ld -i)l61 t_-rl; ilr*-r q "rJ
q + *-5;V ;G {."LL Fq,ri^.--)\." 6.t-j C..c.,-,*, l) Js-r- I gJt-
"t-l .,1:
I
v_o.
..iyil c?1J.t..-.f'(ot) t o)-ti c*/L J'(o\ etLi,. Jsb ,Sl.lail')^ U .5la*,_[L .;.r
I(ttt) =
J,-*€ oJ.rLi
(\v-t-q)
:t-*€ or[.r y j Js2e *, .:*,1 *f -{to slby.,)i ), 5 .;lL c*/L
Y. = J(ot)o. (\^-t-q)
tan d _ J"(or)J'@t)
G'(iotSl.(irtt) = \
(\2 - t - q)
(\q-P-q)
.I.-tr.-.l ,ill,l6c.*l Al ,-eV i.:L [ .J.r.. .i- s, (J' t*) C' i*,1-.-. .gly ,.rL d-["_lidttnt.r
-=o; {..*! ,el tlt JeV 6-* iJrL, : U-fi_Viot L.osV..f Ll il-tL, ;slsl, jf
T* ' vz
it 6t"2-f-q iJ:L..21 .r,,Lld/lJ .{*5J- (.rjq )V ,.t-rc-"s0i f lL = ,/. b-)orr
4+t'i 6or, = (G + irla)yrr:-*'t'ory = {G + e 0}y,r.r-Ll iJ:L, ,,y-#Jt *, ,Jli V-f-l iJ:L.- jl cr,,l-i-. ,-* y .oslT* - G(\ * ira) = G' t--; ,_*
:,.1i;l +5(i, + L)o* = (Gia)y* r-Fr.t'{d, * I}o,., = {GL,Iyrr,J-rJt...,t' "tt J L' '' d=GG'or'+iat)l(\*ttotYl=6'Trr
p'@)l' l.l'(ot)lY
O2-J:,*4 agly o;-92i 66L** *lS" rlr
"J=*!lts-, :iltl f$-,;t ctL;*, c-1JL 3 tLi, Jsu6tr l""l;, y'e'-,'ii ;t
6Lac.1..5C,,-l .Slf JrL .)bl* .,*l.r.f ortL,l U., 5..:-,)1J5,*r-J 'fLl
.r!rl--
)JYb y-l q .t{.t -tF-,| j., ".fLl
.r!:L, ;i)li Jl-€al J<-,;t o.rli:-,l t LL;",41 slzs ! .C=5"t 6rl6,F .."Jlo-o-q iJ:L, r: l, t-P-l iJ.rL,
-t) J1n o .bi 3r rr ,.L; ;3"ittt Ct, 6.:-,1 gj) ..g-;ljK-[ C/l jt #
G(t1 -_ c.[ r - o(t)l
:t_* rt-F
(Y. - t - q)
co.tif3.S-rl-r(-t+ s\F.XI .Jtl, 1.c.-,t t: o rlL;1:G(t) )l$.G. - G(") 6it3'J,g Jl0-0-q iJ.rL.
o,v(t) : ittyy. G.
et s E= o t'=t st 7 2ls .9 dt' =
- t')ldt' (Y\ - 2 - q)I-'-*'*'o't\ - c(t
aE ;-D\ (t - t' - f' Lor.;lj .rJ:L. c/-l -rs
:p.;l:Jt-6 = @ {t:@
o,,r(r) -- iat " {#- f etuEei''Q6)rf} (yy - r - q)
:{t-"f .r"15y-j Oye et S
o,,(r) = r.{o.- o. l.-(far
cos @E +ar sin .aoilO(ildE]'r^' (yy- p - q)
:Yf-t-q il:L, jt A-lfh orr(t) = y.lG'(a) + iG'(a)lt''' (V-P-l ilrL,;l Ll
G' (ot) - Q. - a G.J-rio oE O(E)dE (.-4Jlyy - t - q)
n00G'(r) = -e) G.
J. *t aE 0G)dE (9YT-P-q)
.0@ = -r6(t) tG.= G 6e:* qc(t) - G[\ * zd(r)] .y-rlSi:td6ly
t^J= "r' . Ul gS* t ; I tl"a6
lJ:.or,rr {-'.i YY-t-l .:):L, ty--J5 J-.l; .i- Alf ."f-lf Q
G'(!,i*G-", I" z(6)sin otEclE- G
G"(tt) = -G ot f -Al cos arf dE = Gott - o)4, J"
4Q):\-e4" rG. = G 6 s-t* 1 .cO - Gl\ - ( \ - e't")l,Jt'v!:L d 6ty
iJ.:,,ar,re 4--i YY-t-t .:)rL.. 'J f V J! 4 glf +-tf Q
G'(ttt) = G - G ttt f-(t -'E/") sin of dE, J"
- e - c o, lL - o'"'= =f = Golt-'
--v lo, \ tzYorYl t *ttr,tl
G"(ttt) - -G ,,, f "* (l - ,+k)
"os rn€ dE = #
1.: r5l:-...^ aQ-l q ;ilt :t J-fV J U-# gbJ.r,. ,SIJ.G" sG' ,Slt rr.rte,. g',1
Y-6-q il:t-- ;.)i J* -51 cr,L'i.,o bS t* I */tr.):.riloJ-! r*,t-'.,,J+ .-jlflrt,
*r r,,Srl(t-" b o*=o.ei'' .JrL, ,-li .rr -l (,.:5 SB.;l b ,-r-t-,t iJrL. ./t dl Jl.;;t 1
:g-:i,t,-.'r q,fis +'-V;J(t) = /.1\ + ?(t)l rtf '".ltJ(t) 0.1!.r
T* = lt *t[\ + v(t - t')liot o.ri'''rtt' (Y0-2-q)
r* = ".U * r.ctt J-rin ollp(€) dE + u., I:*s otfE(gdffr''' $r- t - q)
:gll: gi it-,t, . 6
O:--lig. ,g(2 o;-927 16Lb L:t" *t9" r1?
I'(r) = J. + sin rof g(il d.E (lijlyv-t-q)
I(r) = -J.ot cos oo Erl.,G) dE (vYv-P-q)
\ j*.J(t) ,if Cb q o."V !rL. ,,o 6,1L.f'(a) sJ'(a) b:.s1i cd,r ,u. 6ll..rbt r .r-l
.l:1t*rf gcl.,.;l
JilLt Jel s4ib. - ,jl*V - 1
.,rY.rL- il*l u.,]'*J1l*; sgly,l r$.-i;s tj-, .<.:-,)|JK-_J .-.fLl .rYrL,
! l+r" c-,ti aa-;r "$i;.:-r*i;) ,-fVfl ft, O* o-t-Jy_L^:, llg-,* u.t-y-qrJJ! f.+' 1r,.,i;) rrfqt e\: (JJI2-Y-1 iJrL, eo-t-i JL*el ,::; ,:t*t ,-P.
'lr5.ll.-r; ..lt* (Vt-Y-l iJ.rt',.'-t-i JL-pl u:ij 6J_5 'fr. ! I+r" J(j;:,r;O1.9,o t ,rIql cJt-,, ft, ,-r-,+ tly o?-t dr;lr sly cJY.r ot;S 6tar:"_U;iiJ.rL. Jsge rr dP-Y-l iJ.:L, u;t_ r:,o6Jft, #r Cf_l q j r.6u"J!'iJ.<+-Ylo .r).:L, ,r-l ccib- l*f .St^.;J t, ,F ss .s--J r.t' o.r[i:.,i .lt:fy-l-,1r co-\.i o:lr V-y-l
:$r'l,it-L.j lj Js-re
{P}(o,,-+ ori) = y{O}(e,,-*',) f-iJlr -v-q)
{P}(or' -*", = v{O}(e,, -}e;;; (.',\ -v-q)
{P}(or,-*", = y{O}(e*-*',, (v\ -v-q)
Lrf
I
{P}o,r = y{O}e,,
{P}ow = Y{O}ev'-
{P}or, = y {O}err
(.rt-v-q)
(dr\-V-q)
(C\ - v - q)
1.=i ,, U(95,-.-9 : 1
oii : YKeii (C\ - v - q)
Jl.L5 )-J ti.*,i O<^-. \-V-q iJrL, tl ,ft so-u JLtl u::t oov c,Jt-,.rt.,l;
g.,t[- '^-^', j- el F-lSl ;r 5 Oc\\ 'd",,4b-;: orL .ff ,*tb cJB" ,Slf .,f Crr"L
s..)\-V-1 iJ.rLr.* a,, \-V-1 iJ.:L, ui,otsq ,S G-r.5::-,!1Jfu,--l tli-l ../LlcJL--.Slf .qb t-b y.ry€ .,7;a t-i,01* \-f-q iJ-rt''" il*-l *,*f;;l tlLr
"rrJ 4.JtJ t-)oty : oyr: or\ : o JOt\ : oyy : or( - p.\.r$.*tt-,lltJ+
c(c.*,| orr : otr : drl : . ! orr = oyr - ofr = o. Gl-sls 6c;lr;i- 6)JrA lt4t
oQ-+ )l ,-S-;: 1t-i'.i t,- A-r'f .(1 lrL ,;JS elt -fl-t.:t!t 1 ..:-,1 6::-,)l Jti.)
r;l-9 cJ-i J.lrl,i e-*-r LSJn .:iflrt, .r, 6 6tia" r \ -V-q iJsL, )l ,it sr.)L.arir..
.s,*c JJ"c.:.-,1 c'^{-? r.J f tJ# 6rU P C!-, rgltr 6 c*,1 cS:.;s rSJ-s\ c.{?
oii - o.u(t) 0ii":J 6 a:J;tS r1,- ag,5L,'.'lt -fi-t JeW i^.4 lorr = o.U(t) g,-lJt:1
jl J;l iJrL,. u,.:Jt- Cf-l gl;..:*,1 tii = o,u(t)lfK c6\-V-1 iJ:L, jl n*U" cl:*,i
:t-fr,-l \ -V-q iJ:L..
{Pt!o.u(t) - + o.u(t)l
{P}t- I o.u(r)l :
{P}t- } o.u(r)1 =
= \{e}[r,, -rc; (,Jry-v-q)
\{Q}[r,,- #l (vY-v-q)
(VY - v - q)
(r-v-r1
J ./Y-V-l il.rL.;l ,lrtS;i .Jij'a ,it ,Sbuu-S -fr! cn-"-1 --, :l -fi-t iJ.:L, u, e
+, .:*,1 O<^.o .-eJlf -V-1 iJ-rL.o ,.,rrl J oy)te ..:*,1€yr = err sF,l oJ.aLi,o 6Y-V-1.t*,Ferr rJlJ p:-h
il<{P} + {Ql o.u(t)rr\ : o.u\t) -- qK1g) = W
jlcoLi. t* u...:*,lJ3b- J;-r, E S:iltl J'*' {E} = \K{Q\\(VK{P} + {QD 6
O:-J,-h. 61, a;92i 16L6L,.o. <S*& r1A
:6p1t-.f .,lJr:yy : firr 1l 'rtL-l U j Y-V-l iJ.tL-
r:rr yK{PI - \lQI- {v}
f:Y tf:t I
(f - v - q)r* zK{Pl + y{O}
J:fu, i,,.:'J |tu ,sr-,ly + S-till,l J,.(-:, *5
6loy-J J* f ,L.fV.glorL Cly.oVf q.lr^^ : )-.;.,-J- +t*," 11L:- r"
.r-r-€ o-:lr 3-al ,o-t-i o:;-ei,Jtj. ! 6.'*,.lb jt*i.:L i slt {Ql : IP}svtilll )l
,g.li Jl;l. t bnr, 3r:11 .J.+.;,-r JL"."t ,rc;f ,l;xt )F cS[l.r t: I r:;L;,,yKi-:L..r;5;".- ttL;;l
s {P} - t 6 gFC -h r' c2-Y-l iJ:L- 1y--# !:L,51 S\ ,,,Jl Urt-- ;t:<-!-..>u^.e ,S-tlJbjl Jn) r +.; orl: rl; Jlf -V-1 iJ:t--. r.rj-,r .,:jj S rle\ - {G + e0}
trfc/ 4+* $ "t;ti, iJ:t-- O;ya I ,(cSr+
i'r *+= ".,<,tffi].# (o-v-q)
',ii t1-ri 6,:-,-/ r!- -:rt-uL-,| 6LaurJj ll "[3jlF"-r
U.:t -. ryl
_! _la"(,) - o.u(t){# (' -,' ) *k} (r -v- q)
4 + a 6OU*a..:*,1 (*7- )r.-*-,)l ,l-r.kVdc,, = u. #,1 = " ,;ps.c,*l cgl> ,-t-*)l --t, r)r\ + rr.t(yK + Gy\KGl -- ?
rrQ ;rrt-uu-,| J(j, 1, .,.,;) syr clt.J slt f v Fli-rijrl-- ,-,y-v-l il:[,- 1r
lo;3^o t (rf
i,, * +=o.uo[&-+1 +.# (v-v-q)
',u , r:*ul9S.-g: 1
tr..1,-/ ra"i6iLorL 3 ,S-SJl-&il 'P 6
a',',(r) - -o.u(t)i# (t - "" ) * o.u(l frV (^ - v - q)
efs..>*,lt:,1 .uJ1l rl& r.,LL,.<f-)l lql jt .rLlr 5t4, - !+ a - " is.nyr + (YG - YA)/\ hGK,J+@
;s-l-tl JSJ; 4-t .,..Ll .:)rL. -b )l dJn o-,rFt- Jr4c./-l .1: Jt- f b
J1ful J-fu ./L:"Li df ,.lLn q -r,l-r"f L J-tJ Ll c'.:*l oJ-'i l:':-r uJ.*lli'a*.*,i coJ-.i o.:l: V-O-q ilrL, t* 2-O-q il:L.. \ 6 t-b JLA u-'l.,l .:j:L' G)li
t6;,f i).tjjlt-O-q iJrL..)t,1:,J^i J'l*i 6sY5t,J':J e.:lS-,\.;t ":,9'i; o:l:
!(t:. OL.j f rt .a r:r .-e)\r ! t:;i - tt.u(t)lYK Joxx - fKes1 ! lt*it .1: '(;-')l:r{.f -p J).:)rL-
Y[,,,-#l -'"' (" -?) u,,'uQ-/)dt' (-;rtq-v-q)
Y[,''-t#] =
r[ *-"''(t\ =
-t')dt' (.fl-V-q)l.' (- ?) uo'*u
l"'(-?) ,o'*o -t')dt' (Vq-v-q)
.5lo-rL ,Sll rF oVf "r,.-'3: )-$b t J d;yr : trr 6c,*'l )t<^Xl o.,1Lj': lr):L' d/-ljl
dS fj J[i..r) 6 S-t-b,,rb.,"*.,.," orL.r.j;y c*/L J" 6l-f, GtV -u.jt+ ,,fVl.:*,1 oIi orl.:
.li:t-- ;jy Jl-S-tl JS:.:;t o:LL-'l ! tr f -v-1 JU.. iJ:"- ,F:l - y - 1 JU.J.-o: a-.-"i cc*,| .5::-,)l lL1jl f s 4-F eVylt'6tr.Sttl.r.Sll .fLl
/"= ( 1 - e"t"1lc c2-f-1 iJrL-.,(.rj-) d-e, {-1-/t, ,,}F i:L.i elt:c\o:ty':,gorr = o.u(t) 6lt (-jJlq-v-l iJrL' 6 a:* c":-'l
O:-Ji.. aglsl o:-9,31 <SbL*. <+& foo
T[,,, - #]= l"#2 (' -,-+ ) 0,, (\" -v- q)
3-r,g 4-'.*i \ Y-Y-q !J:t-- 1l orLa-,| ! : ,r S-SJ\fu| {,+*" ,* yc*,l;(^. 6
c,r(/):"""<,>14.+J (\\-v-q)
_ !_ :':L ,!lri +r." ! t- -r
err(r):"'"Q)W #l (\Y-v-q)
c*,) \,3.)S.t! ..*l .Slt Vq-v-l iJ:t ..1l ce,L:." ,-b y.t-V-q iJrt-, ! ,.;;11 ,,:6:sl cl
,[,,, #]= I'" #(,-,-+)0,,
evr(/) =-o.u(t)W.+] (\Y-v-q)
f=a..- srsl1*.r tsr4J 4:^.--;,)rF*,-r rr 6il r: jli:li " s
,jJ .gL-ooll-- J*lJ, jYL- .Sl4 .c*.1 r_l-rr, cr-S J* ,llrl )! cSLoJti dJ*^^J:i- Jsyo a, c*,1 OS^" 4U.; ,[.al c;.-t .*^,, L-r:.n ! "f
L-l .St_r: U_ s cj u" j -c
J *'..+-Yl 1r -t--13 .9l4k*, .ffl .:):L. Od lL:j I & ,Js: gl.ry*i; or[*-"1
Jl*" c,Jl- 6l; ..:*,1 rf )J.c.rd r(-:-J)lJ<*r-J "AJfu,St-., .:)rlr, .r\y "l*+;.;[ LaOli -r-J q \-t ,ry).SL^-lr* c,*,1 ;(- 5,li:-,)l_fu-:.*L-,l g.it &Gj ciss '-\.jr jL.i- L-r:a 6,4:-)l;5*,,-3 J .q:-,)l lL*?l 6lt s r,-\:-i[
;L.:;.<+")l f"*t 41J..-*rl .:-,1, f oF€ rrl-: gjf $ st 6jJt OLA
"Jfu.;r +5 G;y.1J crl,-r-r-6..Jl--j )l ,$;* Laol* * J<,L^,J:j-SLa"p,c^:$
Fo t
"b 1, {..U(tS*9 :1 tbs
,-rS-; rlt ; ;l u-ti r.2ye ,. ,;li q ! -t-r e:lS:\ q-51cd,L:, 4;-,Y|J<*,-J
&r+ c,.-e.,-yu ;*J .1p./ ,lr-{ .f\) 4r J J::..^^ Ol-j I n:*,|-l ft-, .:).rL.'^b: u. oii(\t) u,l.; Jtj, g;l; ,.lL; t +:*tl: 'l9J: &p, Cb .i- ./\)
a;;(rs) = I"* orrx,t)e'tdt (\f-v-q)
.J1.,!,-+ -3U.r.,1[ .:.")\.o ,5!lr ,5toe.:,.5-l jq+ r.':"S 6i.l-t 6ly16 orb
u*,1; Clj ./\) 6L^,Iry J;l-t+ ,.r\i .J*+;: :lllb-,| ,r-,t: ,St^Vl:S
Ctt .f.,- ,.rU:.i "r\Y 'l*.+
c;a-; ri^:.-r .dlql c.r^"ol .$lo:rSc*'-#\ 'f P OLj q
:r'*; J3l J"*. Jr SlJ. d/ltq cJ,l"r-,Ll l-u orl.r
f ao{!t)
e''dt = -oij("; + so;;(rs) (,_oJl\o - v - 1)J" dt
(v\A*v-q)
a:.,,^.:-,)l .fL-l .r):L, Cr*i iqbl il*,-t or. b &U.J kl uVf t;ly"r" Jt-.Q-t-',-l tlFl+:r-l-,1) oS c--,-ras? J-.3 :t 6:-b 01"^) gr5Ot*-t
o;i,i$)*b;(x)=. J:b1\-r-P)
ve;i(x) = u;,i(x) + u1,;(x) otir# - .r-ftt - Y - ? )
Sr(x) = \Gqii6)
o;i6) = fl(e;(x)
.fLl LtlJr(,_ijl\ Y - Y - 2)
ry e'tdt = -oij(.; *so'(.; + sYo;(rs)
(9\Y-Y-t)
O:-JJ,F ,gl, a:-9- 6b L:* <St& f"f
o4;(ns) *b;(x,s)- " JrL,.;(iJl\t - v - q)
Ye;i(ar) : u;j(ry,v) + u1,;(Xs) trt<".*=J-.*f (:\t - V - q)
P1"; Sa(r,v) - r0(";rla(r.r) '#Ll Uyr(v\p - v - q)
-r;;(rys) : YK.;;(rys) (()\t-v-q)
3-L-o.gl+L : -r:q @1.r1 , P(r) rti rr -r J:-:-,.-a br!,-r+ ,! *t ,gl;b 6Lac*5 6+5(\2-v-1 .:iri.- 1l .gla*,-U," \O-V-q iJ:l-- ,*l.r .J.,--r+ r.ry 1^.,.-,1
$l5 6la,l*il FU. 'J^: r5! l'*,,+" Yl .:):t .. L r,J-::.--a clL; ;l ,fu s 6-F ;L,.lJ).-.Q@lP(.r) c^l JrL,. ft,ut r! *t d_rt _r .,! ** .5fl: .91a.:-,nt- 0t","
)V: ,>sye t b &U.J ;[-al 6"tar./ U 1 b .jl+l ;,1 .:.ry o:1.: ,.aLaSlG et J:*j_,1 f^ ^J!*,,n J-.r\),f,"+,rj;! oJ; asL; a;--,;-)1.2:6ll[*" ,tr -Sl,f=:fOto
t_rlnl -ri,ril-o.91+L-r
+ -t}(s)/P(,r) + g,l.r(,-t+ t "{i-)l fu-:.r{") .'Jtr,lr-r* ! ,-*l: ..rLj q c-rl: SGS|S;\ s,G LrJ Jsu6t* q sl}+c*i JE '&\ Jsb
-1 q+-+-.)l .JtrJ- jl .S.,[* ^tl+i )l .ry€ rt+"-l t{ji
Cf l JUst ,SL^Cf"-6,t+ y-V-q I y-V-t .-.,!rt-,. 1l 6.;,1 lL c-r.ilol; +;^ii v ,b *l r.:Jl_r o;p 1655!;:-t l, [ae+l,J
Er.r) - \KO -\/ fKP+p
rKP - r0Ft'r) -+\'/ ?KP+\Q
(\v-v-q)
(\^-v-q)
t ,.a-.-Ulg9.-9 : 1
r -v - I JA rS:.fuf ,L-.1 d". c5L;i \tn , "'i- JL.-r'l t'. f ,* s:i
(SJ*.i- 6Jt l,Ss-er;;' cefuu)l .gLa; c.! 1r rt* uel-;Lr:J :f - y - 1,JU
Jtp +.r oJ-^i o:l: 6L;i ,[5--:, 1: j -i-'="
1: F. c.1Li f *
F.oo(r,z) : ; l( \ - \v)A(r,z) - B(r,z)l
o c*,lr .fu-!l1<*,-J ,,J .-rrl,*,t-a:; gi g F 1l t; B(r,z) j A(r'z) 6.:-,1
ft.tS,t* c,.--,1"1:-)l lql ;: s ct,# oi-r.,, ,S\ G-l;3 '"+
Alr oo(r,z,t) oV)
.r.q.; orl: F(tS - F.u(t).st{,s-,|-rfi! i!..-, q IJ.. ;s J*
\-Yv ,.:-,!- ./\) ;|.-r; f-t't,.- t* y -,li,gVtrr\ 6tr. \A-v-q iJ-,L- :l:clo
-,! 6b./>[,) ,,t1r. .f.5*U r1ttxP+Q) ,-,r-*.1 lr.(+)l ,p r: o-r-J;9!,t
o.1 jrlfi .*-it.15*,- : "F./U .!"r+ ..,r.lrQ ,:-F€ o:l:1 ,-,tp ," F.u(t)
O; j,o
rr,,(r,z,s):# lffi-Bg)l;i;l .,5 lQll{PI-lG*q0,} garl, cP-Y-1 iJrt---,d/.'1J5 Jr-".,#!l iJrL. 1l .":-,1
6,s:*-ffi -G*qs
- B(r,z)loo(r,z,s): fr tY(G + qs)A(r,z)
YK+G+rt
OlpJr.l agly a:-921 66L*" *K" fof
,JL^"lr_Jtj 'q-: -2.r 6.:-,,hjk-,1 C,., ,^ ;l taJ[,Jl J.l.ra .<^5 +, .:-,1 O<^" .rt!o
"f-t:rt; )eV cclLl r q*,1; .(ii-,YlJ<*,_: ,F b o-r..i o.J<r-.f\y
#t"+qK;ryoo(r,z,t): + t[ yK+c
F t = o
'Fs6+5'troo(r,2,") = * fyA(r,z) - B(r,z)l
t+ais:oo(r,z,t* -) =
F lY =q/A'4 - BQ,z)l
Yz 'YK+ G..1*,1 ..(ii-,)l j- S
fn,'o - u(''41
etb
Ferry, I. D. (19(tl), viscoelnstic properties of polynrcrs, wilev and sons, New york.Findley, w. N., LaL l. s., and onanran,'o. liszoy, creep and Relaxation of Nonlinear
viscoelastic Materials, North-Holland publishing company, London.Flugge, w. (1967), vis-coelnsticity, Braisdell publishirig comp"ny, w.ttt am, MA.Fried, J.R. (1995'1, polymer science and Technology, pientice'Haii prR, upper saddle
River, NI.Mc€rum, N. G., Buckley, c. p., and Bucknall, c. B. (lggz), principlcs of polymer Engi-_ neering, second Edition, oxford University press, New york. 'Pipkin, A.c. (1972), Lectures on vir^coernsticitv il,rory,springer-verlag, New york.
B -t ,JKJ
J3t*rl dit-firV-Y-l iJ.,L,1: Urj =tt - + 6,it*-rSi = "t - +
6,f*s,liK-h t1 t - r'.Clt,gsiltl JS-r q dbt* o.:irL
,1. .-! r:,ti (,--all
E &\J.r-r, (v
v il:-,1A.-; (C
.-!5.',..-,'' v.. :fuLgr
lll-K-t{Q)lvtp} (.jI
lEl = qK{O} /ffK{P} + {O}) (v
lrl = (rIqP) - \{Q})/(?K{PI + Y{O}) (6
= + o+-t+!t 6tft, .r-l;t*r .slr,r.(,-. gbJ;.r, bc*,JL r - t
.r.:*rl
:*5,r,-.rl: dJn.i. i.rL r:5.i- "#-5- ,_,li ..:YrL 1l +5.r.a: tll-:;
D-
J* J1JL. -r-l: lr 6-r- oV q i,rJS.r-lj ci- 1l o$ o.:l: dl-jj ,Jrrl.,q tkt J.r, f - to):L) ,Ssjltl,.r!:t .o i;1" q VJr- = T;,--# * f ;r-su di-l J:;l: [ .r:*,1 o.t..i
'+5 J*.' b J-r" .r-l ,Slr ..r'L| iJ:t,. ,(V-f-l -l 2-Y-1
eqGflrT+G\Gyy=eg +(G, *G,,*rt, lt)o+(G, ltr)o
=[a
Jr+ \ Brq
.1,-6) 4r-lj oJ-.i o:l: .ll.i; gpbL a*,,5lr,oJ.t" ,Sll,ll '/;l
s):L f - t
O - t't.o agly a;-921 <SlaL,:^" <*lS" fo7
v
o
o
+ i, ttt : l(ryt + ry) I rlrhrlo + (\lr,qr)o
* o I tu = (Gt + Gr)i, + (G, I "r)y
* o l\ = eti + (Gt + 11, lt)y
tel',
(-iJl
(.:
Q
.r.-:-,1 o-LL Jf$ JFt -t--lj ci- \,Stl_r ott' q drJS-r-l.l 4 jl .5rk..,* J"r",51d - I.qf ar*.,- It eLl il:L, .J-r, ,t-l Slt
'PL,o + o I rt = Uv + (Gr + Gt * rtt lt)i, + (G, I tr)y
.+5 Cr1rr,' ,,or.i o.rl.r ,ll-J; gpb! .,LlA )t' 6|112 - 1
,.r-L-l iJ:t -. (jJl
4-.t or;,-|..":,6ta.r-lj 6taG(f) C.+l-.l.- G(t) ,S +5orl G(t) ,Ji$.rl-r CL'(v.[.:*,1 Stl-f o)f
,el.'"
o + o I tt = rtyy + (G, + G, * qrl t)it + (G' I t)y (.-e!l
G(t1 = G', * G,F"h + /t&Q) (v
,Sl-tlty@ ,-;;f .r:*,1 or-3 ort, oi olr*,o 1l:y'; i!-,J q ,..,S t"'+ o!.! o:l: trL:; J'r.o 6laY - I
:r*5;*a y-.r.Sto.jlr u,b 6,tt6tL. y-t
"<tksY(.jjl" =tlt S Y(V
.+,.r.li r:-r) tu tt (V) Vl,tt "ref^J;l o:Lp:,-,i I
y(t) = o.I(\ -e4")u(t) - o.I(t -e4t:t)tc1uQ-tt)
y(t) = o.I(t-e"k1u1t1"rq(,ill
(v
:rr5;.*r.r' o-t-i o.rl: ,-lt-:; J.t 6! - q
..#Ll iJ:t .. (iJl
.G(t) ,jfdrt.r CL'(v
OEf,Jr.,h. aglz a:-9;g <SLaLz-" *lS" FoA
.r:*,1 o$ o:l: a.o.-n ir;1.r.9,^, il*l ". ,ff 6 el* a., ro S t
= tt 6trO(t),"r-- (C
G
0t
:eL,
o + o lt =rA + yGi + (G lt)y (..-Jt
G(t1 =y6(t) + c(\ + e-'t") (vo(t) = )'(trt - rye-tl' + Gt)u(t) (c
1) oJ.i o.:1.: dl-.:i.1bjo, otf *y(t) nt jU; l.)o(t) tn-1 *!-....,1.: or.,i o:l.r tlt-ij J-r" ,5ly r - r.'4f ,r*-:" rr-
Fo t
Y
fe
0
F.q;..1;/=fl, (6.r:l=Y/r (.-,,:l=/, (jJl .,1r tro(t) oj;'oJr-i o:l: t'rl-! J.u,Jl.r I I - q
.-qf orte.:-l tr' I V ,Sl:t,*t't 11 r.uiL or-! orb;b1o, il-.1 q.J-: Jl-rl ,f f
7
7o
0
o(t,) = Q.ry lr,)[y - ,",k1
o(rr,) = (y.ry /r\)[-T + \ea,k - r-rtJrl
o(rr,) = (y.ry I t,'11-e4r/" * \e-ttrt" - e-'trhl
'&!(.-Jl
(:(c
(.it^ ./(t) =(y,l\G)I\-e'th"lu(t) ir*+, tt ,.'.!,S.:- t-*.1r o-r.J ortr 11t tj Jb fl I r - t
.+S.:..t" tto(t) u.i;; ,trS.,lt .,- *,1 o-r-l o.r*5;l:.rj .,1r 6 .,*
o(t) = y.u(t)
tgl.,
O*^.,J,.{:o aglsa a:-9;1 6U Ln-," .S+& flo
L.2G
0
y-O-q iJrL ,Gji Jt <;;t 61.,1 tr - r
y(t) : [_tU -
t,)(do(t,)tdt,)dt,
,t b';*, or, o::,il 1", ffi; : ; : rr;;,? ::,Jr;
..,JlO-O-l il:l-. jji Jlf$lijL-l t.\f-1 *L.rl3,a-,r o(t1 : e"t;l o.:L''r",| L lr- t
o(t) -- {_oU - t')(dy(t')tdt')dt'
,;iy,c>-,tc(r) .r\Y ,.i,-+G1.t) -sG1sF(r) = * o*, rJt-! \r-q *J:-, iT+ r.e-'l ulri> J $5
f-O-l it:t -.-$-:1: c{*,)l;S-- s td)t,Sl-t, j:li,StrJrFf Aff -b.,: U Id - I
y(t) -- I.o(t) * J' oQ)ldl(t - t')ld(t - t')ldt'
C0-0-q iJ:|" 3
o(t) = G.y(t) + f' y4)ldG(t - /)ld(t - t')ldt'.t"
s.tVT: t -t'\:o(t) = e" FeiS:!,S!6-r..r.r,.lt-i;
G.Z(q +/.^a(s; +Z1s;r1s) = o
rt I
A(s)
L;l 1r S "F't J"-L
-''t'7clt ldD(tr
^;r'plG t (tT)(tr
.,.:.-l
qU m sb a 6$! or-..1orb G(t1 - a(hlt1'" oty- d.r u:!- ;Kzttt Ct'.5 S et lt - tm1\ U0) = * sin mr (*f'orL ;l stt, .)r gb 6"r..o: r.rl-i; ..:,*l JV;t ,,Jt^"!,-.i1; ..rLr,:^.).--c ,5l1lr 6Lrc*J6+SorLir:-lG1.r;4,r1 = \/.rY :1.-,-t 1l ..:-l
.J.:i*i\.ftY
tt,r."Lt iJr[,-.J-r" *l Sll.r.,1l: oJ-i orl: ol-j; J-r- cr-p.--n Ju,.Apl2! * J-t-,-5,- ly - I
.+5ir.+ Ju st.t {s't )t s +Sf;r-t;9artt Cb
(.-dl
'rF 6U 1-',
= I.-,
41.,; = I"*
,
G(t1 =Gu*Grerh'J(t1 = (\/(Gr + Gr))erh" + (\/GyX \ - e'k,"')
tr': (G, + Gr)t, lG,
:fuL
(*Jl
(v
l)G(t) ;f;tttt CL'J.r, ,l-l ,Sl-r(*iJl .c,*,1 o-r-i J.r-. so$ orl: ab 7.b dJ,raA ,5lorL tA - q
.-t*S 0..+
.ry' JLrl J-r" q r.:-l oLl o.rl.: cll-ji.,1l:1 , cly.*,o dtl:\-:,1 gL- ,5!b S '*-S f t Q
.+f 0..n, lr .r.r .r-tt C(t) ,J"t-i .S '*V
jiy Jl.€Jl ll o:1,'*.1 [
OE-Jt{rr gS(2 ot-g7 <SbLzo," *lS" rtrGtr
"---l 4l_-*"@G 1't
v(r) Il-zt
G(t1 = G * lGe-tttc * 6r-t/tt
o(t) = G)'[t + fr -rr-"'" -\rc4t"1u1t1
:tubgt
(.-rJl
(v
iJ:t-, y \ 6,ts.ia 6,rttK-lq.1l o:te;.t ! o-r.i ort.: rll-t; Ju elt.\G'(iat) l"L:r." J3_r, t I - t.+S 0..*, .f Ll
4'*-J
J' tr"
c2
:gqG'(ira) =[G, + (G, *G,,)t,tat +iG,t,all(l +t,tott)
,sltJ' JG' 45+5 d)J.d.+^^,t>*,lJt = + (r + rSYd),,F r*q n5.r+, dtJ r. - I.({s (!rG'l' = \ q:,.r-L-b) rrss, dr..L't,.rL;t ,,-t cr-rlsJ-r,
y-; il:t . ,rt (..t*,f or-.! o.rl: alsrl'Qot) q f+!-l.r: *j) Lj.b.-e-j-f---l n5*-r( F ,rf t - t. . tl . t.Jr/tJ o-lJ !e)-P
o.ei'' = t7'1io4r.ei''1
rrt-:; J.u elt.l.try.(iot); +iCr*,- (..rr*r bt-f-q ilrt ,) G.(ira)
-Jtrrt'Qa).fJ oa-,i-" t-l .l> ol-i oll:
teL"
ry.Qro): lry(f + o\tt) -,ttrU (t + ot\r\1
G" : o'sind .1i1rn5.r\Y-l-l iJ:t-, j G' :ry.1i2:.,S*;Jt\Y-t-l iJrl--;lrr- tv. y.
.-r{o:ta, lG*J* = \ ;l J": ''tin' JJ' =ry a5-u-o: rtt*ro."o
nJ-r.^rtrbo(f) :o. sin ot oatt, l.J"KJ<.-..S,- S:r[ody Jlf'll-g..Lf," -t.c*,1 L j-rl' .i)Ail c*.,_/L q f*tu ,* ,. J(.*- y' l: or--l -;l: Sjtl
.r*5A; trc*1ittt1 oi )l s,/;lil:t'.".L;-1.,1.: oJ-i o.:1.: rlt-I i+=+ s*i q J-r-. ,g,laf I - I
..rSg-r ln(rtn) j,b r, G" ;>.:-.1r I tE;;l y'rl-r,;
rlr ,#'^J*,i*A(&Lj ,F
a + (o I tla+ (\ / rt)o = yci +(r rc I rqi+ (G lr')y 'eq
G. (ii.a) : G' + iG' where
G' = G[\ + (ro I YTrYr',at + Yttatl / [ t + (tv / Y)tYt'tt + trtttrl
G" : GlYttt + (q I \){arU [r + (rv / Y)tttttt + ttt'tr\
,Sll. ln(on)=o, G"= t/fG
CD ln (an): \ , G' = \/\fG,Slt ln (on) : Y, G" = ./0VYG
elt ln(tot):@,G"=o
SQ *. GP cjy ,1-11...t ,--..S" r.5i J(.-,e, c{-!t3<--j 6lo:L )l 6*.6,- fd - t
O.*J:,,te ,glS1 a;-9a 6lbL&* cSlJ& rlr
JLel .-,S.6Yq C! JP = -P.u(t) c;tr:i- ,* *...,r-.i 6)r rt:4 ..>-t,ctf
1l orfp-.-l! l.r,r,t(r) "*
..1-F.-U!.-(*:-,il lL+l y j $31- e-r.rt ":Lrfl .t_yr,l
.-rJ*.-,t*. t,rr,,(oo) _rdrr(") .r+5ir*^,- \-V-q iJ:t -
tgL"
o'r(r) : -p.l\ -eG /YK + Yc))le-tKt(rK+$)""1u(r)
orr(o) = -p.l(rK - \G) l(rK + fc)lrr''(o) - *p'
utf )l-pory(t)=6.u(t) 'p L orL jS 6tS) [ c-..-.,- c<-:-,)tts-: 4,,tr Ui. Li- it - I
orl: cll-j.i J-t^ ,,ri-t.,1: 6.rj,-3 J-r. rlll[:-,t ,,L; -uV ot-f a, .:*,1 Jfu orL .,- *ul
iJrl-- 'G:li,SL"J| K"t 1l orlc-,| tr .rF J-r" -tL, il 1r ci:-)t o.tf r, I r.;l: b o-t-!
.+5.,..-- lrrrr(t) dt- ,r-S:c,r(t) 6tF 'pf .i-0-1
(;
rr.. -
e ,r(t) -- o.{\QK + G) I vx - e"kllyc}u(t1
syr(r) = o.[(\ / qK)-( t - t"h1l?G)lu(t)
,gq
8 et* + ,*f )lf -.b Gp b3y ,;>l1a, {i-)tts*,-3 .5torL ;t ct*-t ,-(i fv - f
6blr I L{:*,Yl lL-+l .r: p-..r .(;1ls :3- j ue\,.^I .r*, Gi^) .c*,le ,1=ryy:€o= o
(rrrO "Ki in:tr (-ri:) ft.c--t.4tj./. ,ri;, n5ory J"=1.(\ + ,) --?.:*tL._gst).,y; t,
(rrr0 - {AlKt + r(\ - e"11lrt.}u(t1'PL,
o;U;l CV-*t )) J o)J,r.fu-il tL+t ..:JL 1: -5-1J1 J(*i t cfu-'!le('-- j u^""'7f A'
cSjj s.-lJ,Ji- ,* *..t5or, : -p.6(t) Sly-ru;L:i ..:--*; S-*.5't ,Ss-* Jr.5totK;icyr:rrrr : " ,3 ct*.r-:[ r; SA fl *>-l u;-tf i],or-!, gti \ 7!-
*5 Cr..l,- t, rrt t (l ) -, r: r t (t )
tgl'"
rrrr(r) -p"l(yG-yK)do l(rx + rc)-(tG l(YK+ YG))et-rKt(rK+tG)tttVO
<r'r(r) - p.lydg / (rr + rc)-(yG I (YK + YG)et-YKt(YK+tG)tt'VQ)
.,.:.-l or-i o:b .;lri *J-- .rF .i- Jl- g5[i tr-91: ,Jf ! Lij-,)lF*i3 a,l+il 'iS:
f l - f
c(,-.1:;,.-L c,*-.r, j oI-X o:l.:1lri.r,l-;-lJr : " * 6S-t sl-.-u S ib J+\ 4
or.-X orlr bV 6r-\! "-.t"1- .>tt'.r: orL ft .SC:.ll-l J..r+titt:t. c-rU ;it'Cf
q F)t1 ,Sr u! tr<trr(t) -l (tr t(/) .rri J-r, tL" :'rl v: c(v-')l elt' a' i 'oi)a P
40.o,,
rt?
@'- ffT"G rr
.rl
rr',(r) - -r.l(tcz / f)( \-t-"") + (rK
rr,r(r) - 8"1(r& / fX\ -t't') + (-/(
.B;(rr : A - - .>st' qt^r,L1t. );l>;Lis c_,_; .(+-)l f) rl_r- cl_r:-t4 otif,. _ {
orlr4 : # F(f - \v)r + Blrl.';1,-u,.ucV;dlS,r::.r 1osy.{rs: n * #Itu <16 (tr.c--nl Js-laqy r.J.t JsbE A. hA ab-rU B sA S rt-,g
f t -*t.S+-)l tL+l tr _r c.r:,rK Jt y +512 .r,[- rtnl ti c(;-jlts- j a,li-l 61,
'eqo, = p.u(l) '-*ep" Y; 6.:eL;- ;,_t t .*.-yt J- .rL^
o11 - p"u(l) 'y-,yP" Yt-- 6oeLa J,_l t "(;-il J* JL.^
u(t) - ryAr I (rX + yC))( 1-u-tK+t;1"'''',)r(,) + (B I tcr11t-c"h7u1r1
It+l .1: &i)l s'tt t, .r_rK ,>st' {, c<+-)lF--_, c.(aF.,!-: "+ crL;; 4 f t - tJl*"tl - " Jl-jlrrp;tlj.r,irL\t-arr1t: p.e" 6lr.t6 S:ifl.,-,.,i or.jJr.
.r*(.'**- .'-L .:*- u-- u__ ,
rl
'e\+
+
rGlY)tlu(t)
\Gly)tlu(t)
t^.:d*A(t5.--9 :1
\>)t' q,,eU;,,Ji:J cii-il 4-f .rt*, n-i/ ../-La;.JlJ ^<"-l d,;;tr L.ltoo(t) ,:tJ'
oft - P.[(t -rv)A-Blltn
.$J..a, sJ:-:.-.^ .>l-o:;". q di*tt., t+ .*ly B JA 45l)
'eqoo = (p./rzX(r(G - q)e" + 1yr-(rK+G)trny l(rK + G - q) - e'B
, P.LYft *>t-;fi ,lr.x:L f F.,r1ia*;l lL .:-; tltfr.i .fu-,)t; 4 6,ly.ff - I
JJ- Giil,-rt ! rt rl J[" L',Jii-ll6-"s E Gl:tltu(L,t) ii.-Ut: P=P"u(t)
d+"il s,jt..l, c.xrKirL L6; ! gr^r o-, ,r$; (V -1..r--rl5o:L r"Lly L5r^i 4.{i5 JF tt-(L,o) ,.:lL .p .r, .-F5 J*d b !l .+l 1r
u(L,t) : (P.LY ED(\ - e"kn)uQ), fi = r/E
o(L,o) - (P.LY I yED, ,5;-!r;.--
tel'"
(,-Jl
(vo(L,t) = (P.LY I yEDI(\ - ,'/") + (\ i \IQea/"1u7t1
@@,@) = (P.LY I yEI), ,g-yr;.-
O*.u.1,o ,gl, a;-921C5(o .6"r", cS*K" rtA
c^iU 4 q. 6t@'t) -- q.t,lL.r t n:*rl,.r ,siKtL..:-; cL:-,YlJS-J ,p p J, 6;rr - 1
.-!.S;ap-.rO .-fu-Yt ; j5; *Jtru(x,t) i F*>-t o;tf 1t,; r>-,1 ,ti;t ,'Jl P.r-rK-lg;Jn 4-- (y; .cS-r*r 4 .1.,*rK(.jJt 45 t:-' ,_etf orL 6 ,Jrr- ,,
.+5.*-rU, l, ^,t:j
.Sti tt*+|1: c(;"Yl'L'
u(vtt): X(x)[r -rn(, -r-tttllu(t):;.L(_t
(.jJl
(.-,o(yvt) = X(x)[r - (yl<r El(r/( + c)x \ _ e'/"71u1r1
6A-,*\tE=rtlE
l^bb O-.Jge
gYPA cY2' eYO\ eY0A IYOV sf0P eY00
Ytq
e \AY e\PA c\0Y c\0\ cu.ltJ-?.rlr5i
Yf \ tY'P s\lY c\lY c\AV
r\Y\ cPA c|Y t0. cff tr>-.32-, )l)J-t
Y\ ' s\Y0 t\Yf t\ YY c\ YY
YA eo!-.l;l:;1
(\tY tYl tYA sYY s\Y t\ ' .+-i- tl'.t
cY.V c\lf t\A\ c\24,\0f c\0Y c\0\
Y\Y
,lY .1\ tlt2 cltf (vf c2f r0A (u::; 0li'l
YA\ e \ \t c\ \Y s\ \ \ c\'2 slA.tV.10
YYY cY\V c;l5,Jl'^g-t,
\ \ 2 cll sVl cVA ..J:J dtlt-q
\ f Y s\ YY c,.;!f .Jt^t-t,
\YY s\ YV c\ TY t\ Y' ta'Jll 6q,f4e\Yl c\YY c\Y\ c\Y' r,,J.rt+6q-fo
Y.0 cY. . s\lY c\1\ e\20 c\0A c\00
s\ YY c\ YT s\ Y \ c\ Y' cgf../n 6Ji-5#,
s\f0 s\Y0 c\YY t\Yl e\YV c\YP c\YY
s\Vl c\VY c\V\ r,\V' c\11 s\0A c\YP
cY.A cY.l cT.0 sYr. c\11 s\1\ c\l'
YYO r'\ YV c.r^'* LL;lY \ Y c6;yl .sQ J-lY' ' cg.ra dq J.-l
f . T cY. \ sY' ' (Yqt .&U"' J--l
YYA sYVf cYVY t.ztii Cl* ,l'-lYYY .J:rl^A )t;- Jel
cYVl cYVA cYV0 cYVY c.(;k*'l-9;*')t
r'0YY' cYV0 .YVY .'(.-"Q:1*')l
YAP cYVO t6la*4 ^+*f*')ls\2\ cTA cYV sYA r,f 1 cJ-'t':-' JUir ;l
Y\0 'Y2\
Y \ f cY \ . sYAA ctAV c,,5jtlr; 6t^JtJSl
YY' tYY' tr:J ._!lFl
cY \ Y cPY s0\ (0A ,;a; cSr-- '11't.,l
Y\V
YYV cY \ Y c6l+r-311 cSr.- 'llSlf \0 sY\Y cu'nir2; ,Sifl
YYA rY\A cY\Y sY\Y r6i.:ry 6JlY\V cY\2 eTt0,"5i:tf ,S3fl
r,Y\A r,Y\V tY\2 cY\O sulj'b,5jtjl
Yfl IYOA tYYY cYY\ cY\1
eY2. cY01 I,YOA cY00 ,,#; ,g-/lc\"OY eYY0 stY\ sY\A ct\f cYV' cY21
Y.1
Yqq ,u*.i|.:r,,. .-rb 6b
O?rr4.+ 6(y a:-921 66Ls" *K" ffo
\VA r,\VP c\ YV s\\? s6jlfY -;rryc\Y\ c\Y. c\Yl c\YV cq5rl.-. , ):€ts
Y.Y c\VP e\YA
eY2\ IYAV cYf Y e.(;*,)l .5ta*-ft-t\"\A cY\V sY\0 cYV. r,Y21 c\?Y c\??
YY. cYV. cy2y sa-.) .tLnc,1tJ
\"YA cYYV c\'\"\" (J[*jb clL--f
eYfA eY\"V eYY\" t,_#f-* db_J-?
rrqcYYP gYYO cYYf cY\"Y ec;lyi_ .tb_-f
ff \ gYYA
1 cP cLalb J fs\"1 c\"V e\"0 cYf ,\"Y ,s!-l 6Laca-;c\OA r,\02 c\fV cA, sy? eVy s0. cyA
YP\" sYPY s\2V
cV. e0. cfV cYY s\". sYl e \. cr.tir->s\PA (\01 c\0n c\0Y (\0\ r,1A cV\
eYf P cYYY c\ 1Y (\AA s\AV e \Af c\Ay
YY \ cY.2 cYlP
Y. . e00 ,lr JqYf 2 s!*-fr 21 6r
YYA c"rg 6yYf \ s\"11 6cilF_ 6'>
(YAY c\"AY c\"A\ cYA. (yfy cri.-_jf \ \ cYlP cYAV c\"AY
AV eAA cr:; c1l . sy i5Lroyl:
\ fV su5r5 ,Sll s-f LgLaoyls
\vf s\ yy ca;Stj Otyi,sll cVY cYY c\A c\ . cl cljt 5c5u,
cYAA cYAf cYAf sYAY q9l-l,JiJ Ctt
\"YY sYT \ cY\Y
Y\"f c;L:.i 6b\ YV cr$i #s )J;b
Y2A cY0v (Ji# *, )r;ljtYA eYYV eoJ-i,*-q' ,F :--E
Yq 6f*?U )pV\AY c\Vl c\YV e:9Lu ,-*f ,_*V
\0 e-,/^,a €-t s-*E\2. e \04 6uJ.i l,-j+
\"Y \ rIA r c{+*::.,)t_frr,
(\A\ c\Vl c\P. c\OO,O<^,!J<-,-*'j
\11 c\lf r\AP e\A0
c\0\ c\YA c\ \1 s,--l,.a p^"? olSr,.:i;
Yf . cY\"A cYYV rY l? t6,y-*t:sYPl eYPV s\?7 s\? \ s.(S*,)l ,.l.rlr;
\"\A rY\V cYv.
slA s10 cAA r,A0 cAf cpf 1-iy. d.Jc\ \2 c\ \A c\ \f e\ \Y e\ \ \ s\.y s\. \
gYYV e\"Yf eYlA sYAP c,Yf y eyTl s\ \V
YA \ gYVA
r,lv s12 e10 clf sly r,6l+r.,,i-"a d;Je YA. cYVA I,YVV e YV2 r,YV0 e
ypV e \ f A
\"Y\ s\"Y'eYAY
0vr'r|,,t',:;,sAA 6AY eVl eVP rVO eVy ,Jlel,gtrr*,
l\2 c\ \V s\ \ \ slf ely
Yf . c\Vl s\V2 c\ YV e6J-el . +6r,
ril l;^^l) Q*)'1P
2 ceJi 6t1;
YYl ,953*,1 a;3
fYq cJlf:l,l ,=ri
cY . V c\ 11 cPA ,AY ,gr"li ):'-t *; ;
YYA sYY' cY\f cY\Y cY'A
YYl cYYA cU-|S a*3
\ lV ,JrLi drpli
c\1\ c\02 c\fV c\Y0 cy';-,. ,-p;YAl c\"A\ cYV2 cYAV
cYVO c\0Y c\fl c\fA cdl+*r.., "l-s
cYP \ cYY \ cYAY cYA' cYVl I,YVA cYVP
Y2Y
Y\2 '\f Y 6ul."l gt'.""-5
YYY cYtL,Ji.-rrljtj
r,\AA c\Yl c\YP cr$ir::+JoLa,lrJ
Y\f c\Vl c\VY c\2f
\0\ c\f y c\f \ cgtK-.,.:+,lt_:F
c\VY c\21 c\2Y s\ ?\ ce>-er*'tlLr.'F
Yf . cYYl cYYf gYYY cYYY cYYY c\VY
yV c\n,cllrfy ,f_iV\\ c-1iua.r"-il-
\ 01 cV. cf V cYY cJ..L;. .r"-il-rY,.rF e-iv
YV cf 2 c\"V cYY sJ>l:r ,-p-JV
V cJ-"L:"' dq,
c0. cYl c\"V cYO c\"f c.rL",al 6ls;j^-"
cll clv c10 clY cAO cAf sA\ cVA lVV
c\21 c\2A c\2\ c\01 c\0V r,\fY c\ \'
Y2A cYOA ,YOY ,Y\ 2 cTPY
t\0 cYA' c\Y2
YA c\Y c\ Y c.:L:
cY01 cYA2 cY00 sYf I c.i:*,)l JtlircYlV cYVO gYVf cYVY cYA. cY2Y cY2.
Y\l cYlA
Y \ n c,-r(.-.b-r-l-.y db"j
cOv cYl c\1c\2 c\0 c\f cA ,.? cua;V
c\ fA c\ YY ,\ YA c\ Y \ cAA c/t? c? 1 cPA
YV0 c\ 0Y
YYA ..rJg-,| o;L;
cYY\ cYYl cYYA cYYf '=(2"i; jt!'rl-J;
rfY
Yf \ cyl-Ugflf .ti;
ff f co Ji:.-,.--- ,j, o;l-i;
Yff crf-15.1t^:;
Yl cJrL:" \lri
Yf ' cYYY 'F*l Li\"Yf c\"' Y cYr I ct';ini :=,b
YYl cYf Y c.r('r:"*; =-lf
\ 0 c.'1L- ?-t'cf \ rYY sYV c\1c\P c\0,.rLl: ?-t'
\ 20 cVP
\ 1 cta,r*,-;V *-eYVl cYVA cYYY tr$ a'.r""
\"Y1 c,r,"-.- ?--t'yy. c.ri-Qr-fy ,Lli
YYV cY ?? ,.,5;l*]3.lJ.-a.lLLi
Y \ V c\ lV c,-fuQr-ry J:l ,iiE
YOV c59 oJ; o.:lr {t,.J'^e diU
O?Ji,.p 6lso a:-9a1 66L*." .S*tg rrfcyfy cyy\ c\lV cy,tll*.:)rt-_.cY\". cYAY r,YVl cYVA sYVf cYVY cYVY
IYY cY\"\
cY. 2 cY.0 r,YV0 cYVf cy--lL, ,.rYsL.-
cY\"Y cYY\ sYY. cf \ \ cY\ r sf ry's 6f r!
Yf \ cYf . ,YYf gYYY
Yf Y cYY \ sYYl cYYA c,rJgiy aJ.rL.
cY.Y 6Y. Y cY. \ cY,, r*S;-.t+, al:tr.o
cY\f cY01 cYOA cYfV cYf2 cYff cYYY
cYf \ cYfA c\"YV cYY0 cYYi cYYY cYY.
YOY
YYA cLa;15.-<iat!-r eJst--
Yf \ cYYA cu"\) aJ.:L-
YYV c6f oJ:['-
YY \ cYAY r,",l-b3r .<iJrJtb +J:L-.
sfl cYl c\"V cY2 cYO cYY ';)eltrV,c\2V c\01 c\ \0 c\. \ c\,. cVY c0.
\?Y c\1\ c\AA c\AV c\AY c\A\
YPY c\AA c\ . . cAY cVY c\"y $?-J )IJ;A
Y\Yc.rJirl_,e*
Y01 ('<.L"Yl;-La:1,e,
f Y c6;lsy 0l+.\ \ V cf \ cf . cfl c6;-l*jti 0l+.
c\YY c\YY c\ Yf c\ Y. c01 *ref !lJi.c\lf c\lY c\lY c\1. c\VP c\2Y c\2Y
cYY. cYYA,rtr,;;l
I;: :;J :;;\ 2V c \?? cu^XL /1
c\2V c\20 c\PY c\2Y cr$i;*.*, a;
Y0 c,_rlJ:.,,- giG 6la;y*,c\VY c\ YV c\ YY c\ YY c,r1Vj .>le*A
\1.c\ Yf c\ YY c\ YY c\ Y \ c6:L .>l-a*-
Y.A cY.2 c\Vf c\fY c\Y\ c\Yl
YI Y cYl \ c-ilul J-rr-
cY0V cYO\ t\?? c\?Y cu;;+. JjL-'f . Y cYAY c\"VV cY2. c\"01
\7? c\7 f sYfY 16.* JsbYI Y cYl \ cor1i3 J;-r.
Y2f cc!.a J;!f \ 2 r,f .0 rf . Y cYlV ,YPY ,..K[ J:!c\20 c\2f c\YY c\Y\ c\\". cdsL Je,c\1Y c\VA c\VV c\VY c\V\ c\21 c\PV
cY\Y cY\Y cY.Y cY. \ cY.. c\1A s\lfYf Y cYf \ cYYl c\"YY cYfV cYf 2 cYf0
Ytl rYAl cYAY cy'J c.1y'L
YY\ cJyl ,rY:L,c\ .V c\ ,2 cPA cPV eJrt'i .rY:t----.
cYVV cYVf cYVY r,Y.A cY.V sY.A c\ \V
cY\1 cY\Y c\"\Y cY.0 gYAY cYA\ cYVA
Y?Y cY?. sYY \
YfY r,cJt-- .J)rl--c\ Yl c\ YA s\ YV c\ YP ,.:5r.- rr)rl*.cY\Y cY\Y cYrV cY.A cY.f c\yy c\yy
(YYY cYY. cYVY cYO. cYYV rYff rY\P
Y2. gYOV cYY0
cYVV cYVf c\0Y c6;t511*, .r!.:1*..-
Y'2 sY'0 cY12 sYAY cYVA
rrr L&l) o*)+s
cYY\ c\"Yl ,YYl cYfl s\1T c\1\ c\1'
YYA
\?? Lj:-:S ai\ 1\ c\ 1' c\ tO ';!45 5;
cf . Y cYlA cY01 cY ?f cb j*'ly, +f \2 cf '0
YOY c\2Y cf OYc..r(.r:S+cfO sY\ c\2 c\ \ c\' si^.i.<J-L-7 -:[^-i
Y\Y cY2\ cY\Y
cY\ c\2 c\0 c\f cd c6z;V 6.1l-rfiL:
c\\"\ s\Y\ cOY r'0\ cf \ cf ' cYY rYf
YfY c\ 1\
cY \ Y cY \ Y sOV c00 c..r*l-, iJta-l:r-YfA
cYAO cYAf cYAY cYA\ ,YA, 'u!{tljf \ \ sf .V cYlf sYAl
YfY co-ry,FK-r
configuration 6q,5.1"
confiquration dq-s*"discipation function i)til Cbwarping lunction
"!u)tg vb Cb
airy stren function 6tl ,* 6bollJ.c-" ,-a Cb
multiple - valued function
.tlr>l-h*al +rUojlj
adi.abatic
lanair
stagnation
dilatation
e!!J r-.(;LL:lJJ/ i_>.
,Slq) - flrienthalpy .t-tt*,_l .,,l[:Ji
?l;l\-*,t ql;l(). L
.l- cl)-Pv Jl
Conespondence principle &U., J-lmapping function
spin tensor
orthogonal tensor
polar decomposition
distortion
deformation
Invariance
multiplicity
deniatoric stress
vorticity
ratotional
a.t.r..i;6b
F*sybf"l-.i. -,f-b
cffr, <."> z
cubuical dilatation
trace of a tensor
deviator
contraction
stationary
image vector
vorticity vector
dual vector
dual vector
preuds vector
traction vector
eigen vector
tnvanance
dissipation
-, L:**,1u-
1j-a: ;l:1
;;;t> ;l:1,tt5y.r;lry
..tt53r llry
;K*t*, ttt"
,li511:yoAj;l:1
gtl-t_t,
6stt,
second - order tensor lJr +.;J/ ,,,y8Li +,'.*-;t>. -J.
t-U)2?r
li: *;U J..
.5y-!Ur*;
-;*;lFl tJlJ
spatial description Gl;3'-i:€i6i\Sl' ):-i
lagrangian description
material dexcription 6:V Ju,a j:reference description e-t',i:-iupsfteam ,r*,:)Ur.l[-:aprincipal direction ;tel C;Lnc<-
K:^t*^ )J;> Jel
.9lFl
"tqlisothermal ;1,,,.-a ,>tl-r - JG; jSl
principle of
equipresence
Hereditary Integrals Gi )li d;LaJlFt
transpore of a tensor;-*;V 4. Otr;f,
conservation of mars fJ? dqConsfimtive responses ;li .gtrry!
rrd ub o j(e
adjoint matrix
idencity matix
arthogonal basis
orthonormal
, ql>.-o . *l iL"JJ.l)-J
l-l , *, iLJl)'J
&Li. dq,Jrlr:..
stretch 6vminor of determinent t-rl;;.r :taS
postfactor matrix ?t' rp",f-JVprefactor matrbc ?t' cr^d. er"-yL
viscosiry "!u"atetrahedron u*stE
conservative ,.j!64Lit--
control volume J-{ fnull motion (6_r+) ,* 6fnontrivial solution *-* Fcreep LP?stream line ot-ra Epath line y L;divergence
't'lr_f-:thermodynamic,ri.Q:;^y -Jbf-,substate
measure
free index
dummy indices
fiwfluid
pseudo
designated plane
torsional rigtdity
dyadic product
dashpot
';l-trl - "i+rttit
".;t;./i6 ua-l:
l:.)w,
1:O)t4)
4"r.4
oJ*'J a2tJre
:-l'rr.r4) LJ-JJJA
Q ?.?
4.,[, ?t'-5,.-e
thermal abfusivity ,J )lf ,.rj;., ?-t',rJ ?--t' - ;!u.2 ?-t'
coefficient of vesosiry
oJ;.y',-7 ,*c
q$br?
Lt,-5+;s
materail coordinate,s 6:L crld"--,
Loss modulus
bulk modules
bulk modules
storage modulus
materail derivative
multipler
j)lJl ,)sb
,F J:!
tr J;!oJ1>3 Jsb
6rl- Jtu?.P
Shear compliance L-fJ.c-;.9I^Lconstitution equations,-fl-,l o)rl-.
fundamental equatiolxs y rLi .:)rL.
e :K)6.:):t--"c omp atib ility equations
equation of continuity &y, oJ:L-
characterstic equation aet^;.aJ:L-.
principal values ;Ipl JrU-.
approperate ,s.;t;"eigen value oj; ;l-&,
body moment u^-.- 0L"-
proper ,*,L"pfopef .*,$unique $u. ra>,;-
inviscil
antisymmetric
curl
stretch
O:eJi*4 ctly d,:*t:i 15(oL,.c.e <*lS" Ff,
symbolic notation
uncrement
nonion
body fore
relacation
Vulcanization
thermal conductiviry
actahedral
homogeneous
steady
d/-rL*j 6;ljfiL-t;
Il- ..JL4rUt-
u# 6ri;kttt
,l,11-lr*ti-l
G-slf ..:'-l-t'o
t5g j':'!'a4
.j\.o-a
,l+,t, - c;lt<,_
orientation +J,*";b, -t^t aa-U
multtply - connected region
imperoper .=*,L-U
anisotropic ,ft ,,*o[:
rate of change o|*; aJrate of deformation J<-,_r.* -;rate of extension ,F t;rate of extension ;S a;rate of stretching ,j6 t;indicial notation ua;V 6;l_r(.rLj
