~ \\{ TO

N') THROUGH:

SUBJE;CT:

l

OPTIONAL FORM NO. 10 MAY IQ62 EDITION GSA GEN. REG. NO. 27

(0/P 4-65) 5010-107

UNITED STATES GOVERNMENT

Memorandum ~~m~~a~;; ~~ H, T. Falvey ~ ;\ · \~~

K. D, Mahnken

(

Conformal Transformation

HYDRAULICS BRANCH OFF ICIAL FILE COPY

Denver, Colorado DATE: November 5, 1975

Conformal transformation is an application of mathematics by which the flow around one body may be transformed into the flow around a second body of a different shape. The circulation as well as the stagnation points remain unchanged in the mapping. The mapping also maintains all geometric similarities between old and transformed configurations. Practical application of such a transformation evolves from the fact that the new configuration may be easier to work with. Program TRAN was written to replace the radial arrange­ment of wicket gates in a turbine with a linear arrangement. This new configuration makes the problem of studying the components of swirl in the flow, which causes surging, an easier one. In figure.89 a form has been plotted on rectangular coordinates with constant spacing. If we now replot the same form on ~olar coordinates, we may effect a desirable change. The elements in figure 89 are squares, and the elements of figure 90 have the same characteristic; that is, a is equal to b for some average radial distance r. Infinitely small elements in the cartesian system have dimensions dx and dy, and the corresponding polar elements are dr and rd¢.

E1. _ rd¢ dx - dr

choosing the scale for cartesian coordinates such that

y = ¢

and by differentiating

therefore

dy = d¢

dr dx =

r

and by integration

(¢ in radians)

x = log~r)+ const e

Buy U.S. Savings Bonds Regularly on the Payroll Savings Plan

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• - _, " ., -e .. 3 f'4 ~ "6

Im. 11.-Carteliaa 1Y11tem.

1! z

r ··r-

I , .. .. \• $ ,i,,l,,'t· •"' . .... ' .... ,. ,;.

,

..

\. ·~., ........

Assuming that the number of coordinate lines in both systems are increased until the elements of ABCD (figure 90) are infinitely small compared to the radius r, we can assume such elements to be squares. The only difference between the "squares" of the polar system and cartesian system is the scale factor. Therefore, any two geometric figures in the system will be geometrically similar.

In program TRAN the following transformations are used

y = s<ji x = slnr - sln(g-JI))

The following is a listing of program TRAN and an example which was run. The microfilm plots at the end are the transformed gates in various angles. Note the now linear arrangement of the gates from the inputted radial arrangement.

2

i

I .j

\

\r

\

Pr. /:1 ra,roa9ulor / .x ( y p<J la r, · t?f { <J

I

l e

,.,&=oqle of' 1nc/1no-!1on

r/'10175-forme,d Coord1nol~. y~6<p x = 6 Ln (r') -5Ln ( £-d)

(a:-oco I e ~oc f c:Jt")

74/74 uP T = 1 TKACE

~~O G~AM TRAN( WG;O~ , OUTFUT,TA PEJ =WGCOR ,TA Pt5 =0UT ~~;,TA ~Eu ) C C ----- ------------ -- - .

C C C C C C C C C C C G C G

C C

TRAN:..i FC f~ MAl I ON

THI S PROG~AM T~ANSF ORM3 CO MTOU ~S OF ~ ~ICKi T GAT~ TO X ANO Y GOO{O I ~A TE~ USING GONFO ~M~ L TRANSFQ~~M Tl O~S.

I NPU T F l l f::: : •"1 l,_; 0 I'(

CON TOU RS M1\Y J ::. ::1H.:: RC: u IN ~ lHE~ POLI-l e< CR ~C:C f AiiGl.J L~ R COJ ~ LlINATE S A3 R~F~RENGtD FROM Trl~ PIJUT POINT. CO\ITROL CAr~O c..;AKO 1)

COLS 1-3 i =NJ . OF ~TS . COLS ~-o N=NJ . OF I,,C Lil\ ,HICiJ ANG L=:S CO L 7 Li= .; IF Pul14R

L1= 1 IF Kc.C TA NGULAR

CON T~OL CA~S <CAR Q 2> COLS 1-3 RA JIUS JF WI G~ET GAT~S

..>CALt. FAC TO R C C C

CO LS 3 -1 ~ GOL S 11-1 3 J I STANC~ F~CN PIVCT FOI ~T

10 T~~I LING EJG~ CF G~T~ r u

C C

CuLS 16-:7 1 5-1 -3 ZC-21

X Y (X HJ = Y FT) ,W !1B c. K UF GA T r.:S c---------------------------------

c C VALUES OF I NC LINATION ANGL~ CF G~T ~

F O R i·iA T C r- 8 • 2.}

------·--·-

C C C

Be.G I N Wi frl TtE. ANG L:: WH IC ii R i:FR::.S i__ ;n~ TH c CLOScQ POS lTIJ N

C C C

DA TA INPUT !FCLLOW i l~ G CA r< JS) OI S TA,\J Cf Ai J MIG L'.: (P OL AR CO-

. 0 :::ro I N ~ T c. S ) , F J r(f1 A 1 ( f o • Z )

X ANO Y ( ~::G TANG UL A~ CCO~lJ i t~~ l ES ), FO R1-1AT (F d . ~ )

C C OUTPUT: ~ ---~-- ----·-------------------·~-- ---c T~ Af'-1 3F ORMEJ X ... N;J Y COOROi iJAT~S C MI ~F-,O FIL 11 uF P~O TT EU Ti<ANSFO~r-lEQ POI~HS C C C

-c----7r-n- nT-l1~r A 0 I · 1 E t J S I fJ 1~ X ( 1 ., J I , ( ( 1 ~ : ) , X T ( 1 1J : ) , YT C 1 (; : ) , 0 ( 1 .:. ~ ) ,

+ 13 :. Ti-\ ( 1 0 J ) , PH I l 1 j .. ) , C ( 1 0 0 ) , 1) ( 1 IJ O ) , " ;. :1 M A ( 1 Q O ) - - --+ , \~ C i : - j > ~ ff < i J ~-,--, ::;-i X l iG .... > , G 1 X 2 l 1 3 CT;l:";JYlT~~ ~~ ~,-, ~G~v~v~2~<~1~,:~D~> -, ~J.-T""".2....,<...,...l T} ) - - ·-

+,YT 2 <10 ~ > CALL :1 0u t: '.:iG( 6 ,1)

----CA LL - ·_:; U L J E G < - · 1 -~: , - 1 ~ • , 15 • , 1 ::> .

CA LL lJG J t., TG(.t :5 ,. ZS ,. tS ,. EiS J ~L AJC3,1> M,N,L1

---1- FO R Mlf°Tl I 3 ;r3-; I 1 >,----·-----·

~~ AD(S,2 J ) ~ ,~,J,IX,IY,IG 2 0 F O 1~ "i 1..l ( f ~ • ~ _ ,_F_ 5!. ~ -'-' F_ S_._z_,,_I_ 2--=--,_I_2"""",_J_2 ___ J ----------- - --

.; ~· 4 D { S , 2 1 ) C J ..__ T .~ C J ) , J = 1 , N I 2 1 F0 ~:-1j l Cl JFo . 2 )

____ _ ~,=- A J _( J , b I < X C l l , Y C I J , I= 1 , :-1 >

6 F J R :1 A r < 1 U f '3 • i J ,W I T ::. < :.i , ~ cl

__ ?_o FO R:,1AT_t1H1,,,._.J U·1-IA RY OF INFUT UAfi.'1- ) \·l k. IT c.. C 3 , c:. 7 l C 1·1 , ~ , ( J :. T A < J l , J = 1 , ; J > , J E f A < l ) , lG >

27 FG~M ATl///J X ,• i~Ll tl dC:R OF P OI.'JTS - •,1X,IS, + / J X,•N~~~~~ OF ANG LES - •,lX, I J, t /JX,1 :JF >3 . 2 , + / 3 X,~CL OSL J P J~ITIO N ~~G L~•,1 X,fj.Z,l x,• o ~G~c =s·

___ +_ _ I 3 X ,~NJ 11 3 ::. -~ 0 F W IC t<.:: T GA T :.: S ._ , 1 X , I 2 ) .,; f< I T :. C 5 , c:. .-3 ) R , S , J , I X , l Y

28 FO~M Afl///3~,• ~~U IUS CF h iGK i T GA T:.:j - •,1X,F 5 . 2 , + /3 x ,.v- s ~ AL :: FACT J~ - -v-,1X,F:5.2, + /3 X,•0I S TA NCE F~OM PIVOT PJINT~ • / 3X ,-v-T J TRA ILI NG EOG~ OF GOTE - •,1x,F s . 2 , + / 3 X , ,;.. A L L O i M::. i-. .; I u 1'l S I N I., · ~ rl := S - :.:; CAL ~ ; • , I 2 , • 1. J\ = ,,. , .i: :2 ~ 4 F' T .(< l -----· ------

IF { lG. L ~ . 1) Gu TO ~~ PI =o . 2'>32 /I G X0=3•ALOG ( RJ-S• ALCG(~-Ol -----YJ =G. G XZ =~• S I N( PI> vz=-~+ '<-··ccs , iJU ---- --------- - - - -------------------·-------x: = S..,. :~ L J t, < SJ R T ( l ;( 2 > u 2 + ( ~ +Y 2 ) -1< -IL 2 ) l - :::> .v- ALO G CR - D > Yl=ATA N2 (X2, ( :..C H2))-l-S WrUT::<S ,3 0 ) XO ,Y O ,X1,Y1 ----·-· -~ ---------·-.

3: fJ~MA T(///3X,• J~ I GIN OF T~ANSFJ R~~~ GAT~-IL1X,F3.2,•,•, + F 3 • 2, / J X, -1< O,~ I G hi CF ;,i EXT GA TC: ..i;. , lX, F:;,. 2 , -1<-,"'", F 5. 2)

4~ Q·J i"J J -=1,N iJJ 1S I=~,M IFCL1> S ,5, ·t -c- -----· ·--------------------

c C

POL AR

3 Ri:. u= ,L 1--+l-[ i l 3 O:.:--­ ·------------------- ---- -

G/.l. ,1i·1.!\ (U=':30- (d i::fi'.dJHYCI)) GAM :-1A '1 >= GA r1 MA li> "'"i-(AO - - ----------------------c ( I l = X ( I l ... S I t~ l ~ .... M M A l I ) ) t"J C II = x C I> .,. CJ S < G..i. M i1 A <I > > Z=R+-C{l) o < r i = s u,< r < a < .i: > ... ,,. 2 +- z .. """ 2 > PHI{U = A fAN 2 ( :3<IJ ,Z>

GJ T J 7

Kc.CT A ,\I,'., UL A R

!+ W(U=X<I> u <I)=Y(l)

·--- - - r F c J ~ I' :1 ~-1r -r, 1r,---~----------------·--------- - ·----- ·-· - -· - --TH ~ TA=~ c TA< J > - J ~ TA< 1 > U=Trl ~TA•(3.1416/1S 0 .0)

----·,i, n -~xTr > •CJ s-·< LJl-:::-vrrr•~s-rNTu· .... ..-,-----------·--------- ------- -------- -

1./l I > = X ( I ) • S I t ~ ( U l + Y ( l > •GO S < U ) ~ l= R•·Q <I> ----- --· - - ---·-------

C C C

0 ~ Ill)= AT ANZ (~(i),l) O(Il=$J~T(Wll)••2•z••2)

GO : lF u I{ MA L T RAN :.; F Li< MA f ION

7 YT<I>= S•r.>hI(U -- ----- ------ ----------------x T ( U = 5 • ,.:. LO ,; ( U {I ) ) - S • ALO G ( ~ -0 l YT2(I) =YT (I) Hl XT2lll=XT(U ---· ·-----------------·------

15 t_; QrH l 1iU.= C~ LL PAG~G ( J , J ,1) CA LL S~TS MG C1~ J ,3.~)

-------------- --- -------------------------------c.:.L 1... G.{1J,:, c..:...,1.., c .,o., , CALL flTL ~G (l,l rlX ,1,1 ri r,11,11rlW ICKi T GAT~) C~LL S: TSMG ( 5] , J .J) ---- -----------·------- -----------~-CA LL l 1fr.SG ( ;-1 ,xr,YT) CA LL LIN~ SG( M, XI2 ,YT2) IF(L.:.l 19,1'-3,22 ------------------------------------- ------- ---c

C C

OUTPUT

1~ ~2IT ~ <~,1) J::: l ~ {J)

3 FOR. :1A T(1 H1 ,• KAJlA L I NCLI /JA TI ON - JF ,H Ci<c.T GA TES •, FE . 2 +,;;,. GcG~i:.:::S •> --- - ---------------------------------------

9 wr< Ir:.C;-,1 ~, 1 0 F O :< :1 A T ( / / / 1 -+ X , Jf. ., I V ::. r~ • , 1 J X , Jf. T R A W> _F J KM E i.i ,... ,

+ ----- vi ?.. IT i: ( :5 , 11 )

I': .. ; ,; , • J IS T ... , 3 ,'( , .:;:. .A \ SL C: • , ·.) X , .:;:. ..< ,... , 7 X , '" Y '" ) ---(X( < ; ,Y ( K ) ,XT( i< l ,Yf{ Kl, K=l,M)

C

11 FO~MAT(~X,Fo.J,3X,F6.Z,~X,F3.2,3X,Fj.2) GO TO 16

22 W~ITi:.l3,2J) B.:: T~ (J) 23 FO~MA T(1H1,· ~~or,L I ~CLI NA Tl3 N OF ~ICKiT GA T~S •,F 6 .2

t , -\< D:: GR.::::: S • 1 12 ~ik'.Il '..: (5,1. 3 ) 1 3 F L ~ I A f ( / / / 1 -t X , ,,. ;; I ii ::: l J • , 1 J X , 4 T R I. N SF _; --<. i·E O • ,

+ /1 2( ,~x~,7x,~-y~,.-,-3-x- ,·~~-x~~-,~7~x- ,-. -t-•_J _________ _ vJF. IT t: { S ,1~) ( W (L) , J (Ll,Xf<L>,YT(U, L=1,Ml

1+ FOi·UIAT{lCX,FS.2, 3X,F3.l,5X,F5.2,3;(,FS.2,SX)

-------·--·

--1- o~·-i_~OtHi l·U -=: --------------

CALL PAG~G(C,~,1) Cf.IL L ~Xllu

---2~- CA LL-rx~ T ---·---------------------------~NJ

--- -------- - -------·-------------------------

--··---- ·-·---- ·--··------------------------·

~EF EREtiC .:: 1AP <t-< =3)

0 .: F L l.'E: i< i:. F c K : : J C.:: :; - ----·- - ------------·----- -- ·----------·-·

l

-- - · ~ - - _ _._.. • .._......_..._ ~.... • - - ____ .,,_: ~,. .. f

f SUH~ARY JF INPUT OATA

__ __._.NUMB E..L..O_f_e_o L'lI s ______ s 5.

NJ~BE~ OF ANG-~S - 10 77.)l 75.JO 72~03

• 67. 00 62.QQ

CLOSED ~OSITIO~ ~NGLE 77.0J OEG~EES NU~BE~ OF WIC~ET G"TES 32

' - - - · :) -- - . - · !- - .J - -

SCALE !='ACTOR - i.o.oo -OISTA~'.:E FROH PIJOT POINT IO t~"ILI1iGEJ:;E OF G~ TE - ;.LO

ALL OIMENSIO~S I~ INCHES - St:ALEZ 3 IN

ORIGI~ OF TRANSF0~11EO GATE f+.&O, o.ao OB,IGI~ OF ~E~T !:iATE ~-601 Z!32

57.10 52. 0 0

= 1 FT

r

'+7.0Q 1+2.oc 37.0J

" "-MU&.14~ l 1'1 u '- &. I .. M I &. IJ fl "' ".L V " '- I u .. ' l. ..J f I •"UU L U"'- '- J I

I

I . . ~

__ GILE..N T~A~Sl:O~HEO .:_ )( '( )( '(

5.70 J. J3 f+. 8 3 t+ • 3 3 J I _5.70 _ .31_~ __ 5.33 __ 1+.JO

• 5. f> 5 .35 5.C3 t+.26 · t+. 9 5 • It 0 s.o, 3.73 It• 5 C _. _It 1 s.c, __ 3.39

• 3.95 • t+ 5 5.1:; 2.98 3.f+5 • 53 5.0, 2.60 3.oo • . 55_ 5. 06 __ 2.26

• 2.so .5:J 5.12 1.66 2. 0 0 • 75 5.19 1.so 1.so . _~f, 5 __ ._2_:, __ 1_ ,_1

• 1.JO 1.no c;.Jf, .75 .50 1.15 5. 1+7 .37

O __ ._O O 1._2~ 5 ._53 0. O_O -.so 1.29 5.57 -.37 • -1.00 1. 30 5.5B -.11+

-1_. _5_p 1_._iq 5. s_, -1.1_2

• -1.95 1.25 5.So -1.45 -2.47 1.15 5.51 -1.64 -2~38 1.) 5 5-__. 4; -2.~2

• -3.1+5 • 35 5.32 -2.58 -3.98 .65 5.2) -2.99 -4~ ~5 .35 5. IJ 3 -3.36 -t+.96. .02 4.73 -3.77

0 -5.20 -.2J 4.6!+ -3.96 -5.1+5 - • I+ 5 r... 47 -i..11 -5.60 -.7J 1+.2g -1+.31

0 -5.62 -.95 4.1) -r+.34 -5.56 -1.15 J.q; -4.33 -5.1+5 -1.1+~ 3.7. -l+.25 • -5.2c -1.60 3.57 -lt.07 -4.95 -1.7) 3. l+7 -3. 69 -r+.1+8 -1.6) 3.35 -3.53 • -4.30 -1.82 3.33 -3.39 -3 _ _. 97 -1.67 3.27 -3.13 -3. 45 -1.64 3.25 -2.12 • -3.00 -1. 82 3.25 · -2. 37 -2.46 -1.AO 3.2; -1.96

• -2.ou -1.7? 3.3J -1.57 -1.so -1.r,2 3.35 -1.16 - ~_._ o_o -1.53 3 _.__!+; -.16 -.so -1.1+~ 3.sz -.39 • o.oo -1.23 3 • 6) o.oo

_ ._5_0 -1.1, J.Ei3 .39 1.00 •lel8 3.1, .78 • 1.so •• 95 3.68 1.1& 2.QO -.ss 3.97 1.55 2,ft8 •• ,2 I+, 0 1 1.91 • 2.98 - • f, 1 4.2~ 2.29 3 •.?_C •,SJ l+. 3 J 2.69 4.00 · -.40 4 • 41 3.07 • 4.50 -.21 .... 51+ 3 ... ,. S,00 -.15 .. _._r,; 3.81 5 .--r,-5 -.os ... 11 lt,29 • s.,o ,.a, It, e J lt,33

I

~ ~AOIAL tNCLINAT(ON

I OF HICl(ET GATES 75.000EGREES

• .GIVEN TRA:-tSFO~HEO I l( y X y D'- 5.70 -.20 ft. 6 6 '+. 3 4 I J

_5.71 .• 13 I+. 9 3 __ '+ • 3

• 5.6& • 1; I+. en . ft• 26 4.96 .?3 I+. 9; 3.75 '+, 51 ,25 I+. 9 !t __ 3. '+.1 3.g& • J 1 ft.9, 3.00 • 3. I+ 7 .33 lt.97 2,62 .3 • . J.2 • 4, 5.CJ 2_. _2._6

• 2.52 • 5 ft 5.05 1, 90 2.02 • 66 5.1+ 1. 52 1, 53. ,_8_1 5,2.~ 1_,15

• 1, 0 3 .96 5,33 ,17 • 5 I+ 1.13 5.43 .40 ._c_4 1,_, _4 5.53 .Q3

t -.45 1. 31 5,56 - • 31+ -.95 1.33 5.61 -.11

-1 •. '+ 5 1_,_J ft 5 • !;>2 -1,~_o s -1.91 1.32 5.&t -1.r+2 -2.1+3 1.24 5.57 -1.81 - 2 !...9 _4 1.15 5.53 "!'2 .19 -3.1+2 .97 5.41 -2.55 -3.95 .79 5.3J -2.96 -4.lt_l+ .51 5,12 -3.34 -4.96 ,19 r+. 92 -3. 75

) -5.20 -.02 r+. 7 6 -3.95 - 5__!..._~_6 -.26 !+. 6 2 -r+.17 -5.&2 - e =:; n 4,4!+ ,•f+.31

'"\ -5.65 -.75 4. 2:, -4.35 ~

-5.62 -.95 ft .1 J -4.3'+ -5.50 -1.21 3.8~ -1+.21 -5. 2 5 -1.1+2 J.71 -4.10 -s.u1 -1.53 3.61 -3.92 -4.54 -1.64 3.4! -3.56 -4.36 -1.&7 3.45 -3.f+.2 -4.()3 -1.73 3,36 -3.17 -3.51 -1.12 3.3& -2.76 -3.06 -1. 71 3. 31t -2.41 -~~54 -1.71 3,32 -2.QJ ---------2. 0 6 -1.65 3,35 -1. 62 -1.56 -1.57 3. t+ l -1.22 -1.05 -1.46 3.47 -.82 -.55 -1,38 3.53 - • '+3 -. D 5 -1. 21:J 3.6J -.04

• !+6 - ~.!..? 0 3.66 ,36 .96 -1.11 3.75 .75 • 1,47 0 1. 0 0 3. 6 If i .1 I+

1_. 97 -.CJ2 3 • 9_2 1.53 2.45 -.31 '+. 0 2 1,90 2,9& -,71 4.11 . 2.28 3,48 -. Q_2 ~_!_21 z_~_6_i, 3,98 -.54 ft. 3 0 3 • 0 & 4, 49 -.43 1+,'+2 3.41+ ,._~ g 9 -.3~ 4,51 3,82 ,. &It •,2§ lt,6!t ... 30 ,,,1 .. ,~ "'~" .. ,,,

~ ~AOIAL I~:LINAT[ON OF' ~IC<~T GI\ TES 72,000EGREES

I t

_GIVEN TRA~Sf_ORHEO J )( y )( y

··- 5.&8 -.5Q f+,45 4 • 3 5 . J I .5 • 71 -,17 It• .71 __ 4. 35

5.&& - • 1 l+ l+.72 4. 31 t f+,97 -.JJ 4 • 7 ; 3,77

4,52 .• J 2 '+ • 7 => ___ 3 • !+ 3 3. 9 7 ,1J f+. 7 3 1.02 3, t+ 8 .21 It. 8 3 2. 61+ 3. J f+ . ,29 It. 8 3 _ _ · 2 . _3 (j_

2,55 , It 1 l+,95 1,92 2. Q 6 .57 5. Q & 1,55

. 1,.5.7 .]3 - .1_, 1,_13 1. J 6 • 91 S,23 .a1

,60 1.10 5,43 ,45 ._11 1, .. 2 !t 5 t..!:U 0~

-. 39 1,33 5,6] -.29 -.86 1. 36 5,6+ -.66

-1_._3 6 1.-..lt 2 5_. 6l -1_. a -1,63 1. I+ 2 5,66 -1.3& -2.36 1. 3 6 5. 6:, -1.75 - 2._, 6 8 1 I 3_1 5 ._6..!t -2.._1_ -3. 3 6 1,15 5,5!+ -2.5ij -3.91 • 99 5.45 -2.92 - 4 ,.!+ 0 I 7 !t 5._._2 3 -3..J -4,9f+ .45 5,11 -3. 72

) -5.20 ,25 I+. g 3 -3.93 - 5_._!+ 7 .J3 It . _6 3 -~_.__1.2

-, -5.64 -.21 l+,67 -4.30 .) -5,68 -.It& ft.,1+6 -r... 35

- .?..!_6J~ -.E,E, ~. 33 -lf.,35 -5. 55 -.92 4,12 -ft.• 29 • -5.32 -1.11+ 3.93 -4.13 - 5_. __ C_6 __ -_1_._2_6 3_.__a_2 •3L -4.&2 -1. r..o 3,66 -3.61 • -l+,44 -1,l+ft 3. 6!t- -3.47 -4"_1Z -l,_5_2 3_, _~; -3., -3.&u -1.53 3,51 -2.82 • -3.15 -1.55 3.47 -2.47 -2.&3 -1.,s 3 ,..!t J -~..1.0_Q -2.1 ft. -1.54 3,4+ -1.68 -1.64 -1.1+6 3,47 -1.28 -1, 1.3 -1.Jt1 3_.-5., -. -,62 -1.35 J,55 - • l+ 8

t -.11 -1,29 3,61 -.09 • I+ 0 -1'--22 l_._q2 ,90 -1. u; 3,71 ,70 • 1. 41 -1.)'i 3.7, 1.10 ,92 -1~.o_z 3_,li_ !t 1_._ft

2. I+ 1 - .. 93 3,92 1,67 I 2,92 -.67 3.9~ 2,26

~.!tit ·....t..60 lt_,_o r z__ , _ _f> f> 3,95 -.75 '+,lit 3,05

I 4,lt6 -.&6 4,24 3.43 lt,97 - .~ _'-_tJ .. _._:u !,82 t;&e .. , .. .. ... 1 •u 3i

I . •' •,iii \1\t tu '~~ IL, ~

c .. ~,AO[ AL t~CLINATION OF' H[CIC ET GAT ES 67.QGOEGREES

I • .GII_E.N T RANSF:OR.HE.O.

' )( y )( y .• ,_ 5.61 -.99 It. 0 7 It.JI+ , I J

____ 5. 6 7 ___ ~ • ~6 ____ 4. 3 l __ It. 3 6

• 5.62 -.61+ 4.3!+ It. 32 l+.91+ - • I+ 7 1+.1+? 3.79 i..;;o - .• 3 6 4. It :; __ 3. '+5

• 3.97 -.24 4.53 3.03 3. It 8 -.11 It. 6 J 2.66 .3. J 5 • J.2 r+ . _63 ___ 2_._3_

• 2.57 .19 '+.79 1.95 2.10 .39 It. 9 3 1.59

._6_3 ,..5.9 5._c &_ 1._2.J

• 1.16 .fH 5.2? .87 .69 1.05 5.3g .52 •. 22 1._2i 5. 5' ._16

-. 2 7 1.36 5.6? -.20 • -.76 1.45 5.69 - .56 - _1. __ 2_5 1~ 53 5.7, -.93 -1.70 1.57 5.8) -1.26 • -2.23 1.56 5.8) -1.65 - 2 .• _r__c; L55 5.81 -~!.ll

• -3.25 1.1+4 5.7; -2.41 -3. 81 1.33 5.7) -2.82 - 1+._3_2 1 ~J._2 5_. 57 -L_2

0 -4.66 .66 5. l+ 3 -3.65 -5.16 • 71 5.32 -3.66 -5.1+5 • 50 ;.19 -r+.10 -5.6r+ .26 5. 0 !+ -lt.26

0 -5.7~ • 0 '+ '+. 8 ~ -4.32 -5.69 -.1, I+ ._Li -!+.34 -5.61 -.43 I+ • 5 J -4.29 • -5.40 -.67 I+. 3 J -'+.15 - s.~ .. 11 -.81 !f. .17 -3.99 -4.72 -.99 '+.!JO -3.66 • -4.55 -1.as 3.g; -3.53 -4 .• 2 3 -1.15 3.85 -3.29 •3. 72 -1.21 3.77 -2.89 • -3. 27 -1.?7 3. 70 -2.55 -~-· 75 -1_. 31+ 3_. 62 -2.15 -2.21 -1.35 J.6C -1.11 • -1.76 -1.33 3.53 -1.37 • _1_._ _?5 __ -1_.i o 3 . _f, J - .9.7

-.74 -1.29 J.6J -.57 • -.22 -1.27 3. E, 2 -.17

·--~ 9 -1_! __ ~5 3.Q3 .22 .so -1.21+ 3.&; .62 • 1.31 -1.20 3.69 1.02 • 8 2 -11..1, 3,71 1,

2.32 -1.14 3. 7:, 1.so • 2.83 -1.12 3.6J 2.20 3.36 -1!..1.0 3 ·--"- !t '~-'~ 3.87 -1.03 3.67 1.01 • ... 38 -1.a; 3.9ft 3 • 40 ~_,_ g 0~ 1-~ __ o_~ 1t. o_o 3 • 7() 5,5& • t , 33 ... o~ It, JO &. &1

, 1,, '!:! fj~ tu§ , \%3 \ -

~AOIAL INCLTNAT[ON 0~ WIC<~T GATES 62.000EGREES

.. _ _ GliEJl T ~A~ sr:_o_~ '1 EJJ.

~;-)( ( )( y

5.51 - 1. 1+8 3.6~ t+. 30 _ 5. 5 9 _ -1 • 16 3 • . 9 !+ __ ft • 3 4 5.55 -1.12 3. g:; I+ • 3 0 I+. 8 6 -.69 I+ • 0 9 3.76 4 • 4 5 ---~ • 7 7 1+.15 ___ 3. !+3 3.93 -.59 4.2:, 3.02 3.46 -.4t 4.37 2.65 3 • J It - .• 25 I+•'+ 3 __ 2_._3 2.56 -.J4 I+• 6 2 1.96

• 2 .13 .21 4.73 1.61 -1, 67.' .. ft_4 !+ . _93. 1 .•. 2&

• 1.22 .71 5 • 1 I+ .92

.76 .98 5.3!+ ,56 , 3.2 1,_2_J 5,5J ,2'-t .;

-.15 1.38 5,63 -.11 I

-.63 1.51 5.74 · -.47 ~ 1. 1 2 _ _ 1._5_3. 5._._, 3 - ._8_ -1.56 1.71 5.9) -1.15 -2.09 1.75 5. 9 !+ -1,51+ -2,&1 __ 1,L9 ~~-'U -1 __ ._9_2 -3.11 1.71 5.95 -2.29 -3.66 1,66 5,q3 -2.71 -4~ ~J, 1.1+9 5 • 6_!_ __ - ~_!_11 -l+.79 1. 3 J 5. 7!t -3.55

) -5.07 1.15 5.6!t -3.77 -5.36 .gs 5, 5 !+ -4.01 -5.59 • 77 5.4) -f+.18

• -5. 6 7 .. 51+ 5.23 -4.26 ~.5-!l> 9 .33 5.03 -!+.29 -5. f,J ,J6 l+,87 -1+.21

t . -5.44 - • 2J l+,6:, -f+.14 - ~.!'_2_~ - .. 36 4.52 -3. 99 ·---, -!+.79 -.58 f+,32 -3.68

• -4.62 -.o; 4.2f> -3.56 -4.J_.2 -.1, f+.1~ -3.33 -3.81 -.68 ,. • ;:; 3 -2.95

• -3.37 -,98 3.93 -2. 61 -2.66 - _1 _ _.10 3,81 -2.22 -2.36 -1.11+ 3,7; -1.135 -1.87 -1,18 3.72 -1.46 -1_._3_5 -1..L.1_9 J .~1>9 -1.06 -. 8 5 -1. 22 3,65 -.66 -. 3 3 -1,25 3,64 -.2&

~18 -1,,27 3.62 • 11+ .69 -1. 3J 3,6) .. 5 '+

- 1.20 -1.31 3,6J • 91+ 1_, .7 1 ___ J, ·-~ I+ 3_!_5 3 1.!..~.!+ 2.21 -1. 31+ 3,6J 1,73

· 2. 72 -1.3:; 3,Gl 2.13 3, ~ 5 ~ 1.!_39 3 _! f> 1 _ _ 2_ • .5 .. 3,76 -1.42 3.61 2,94 • 4,28 •1,ltl 3,G!t J,34 lt,79 -1 ..... . ]~z{i_:, _ _1 e 74 ,·.· .. -~ .. i'a ,i J, s, ... 2, ;. i; l . -•1""' l,t;1 -..Ja

,;

' I ~AO[AL [~~LINAT[ON OF W[C~cT GATES 57.00lEG~EES

• __ GI\l~N TRA~SE.0Rr1EO _ ___ ______ _ X Y X Y

5.36 -1.q5 3.31 4.22 _ _______ s. 1+ 7 --~ 1. &1+ ___ 3. s 0 _ _ 1+. 2 s. ____________ _

I .• , ·-·· 1 I

• 5.43 -1.oC 3.53 l+.25 1+.19 -1.32 J.7o 3.73

_______ 4. 3 7 ___ ~ 1. 15. 3. 8 , __ 3 • .1+0 _____ _________ _ 3.87 -.gJ 4.0J 2.99 3.41 -.71 !+.1!+ · 2.63 • _______ 3 •; 1 - _. 51 _ _ _ 4. 2_7 __ ,2_._3 .__ ________ _ 2.56 -.26 4.4~ 1.96 2.14 .J2 4.6, 1.63 • _______ 1._7 _0 .$_0 4_. _8_!+ 1_._2_9,c___ _____ ______ _

.&1 5.o; .97 .86 .91 s.2g .&s • 1.28

_______ !_fti. _ ____,1,. _ _1_7 ____ 5_. __ !46 ._3 __ ,,._2 ____________ _ _ -.03 1.38 5.6!+ -.02 -.so 1.56 5.77 -.37 • -------- ~ _9_7 t.. 7 _3,,:_ ___ 5. 9 J - • 71.__ _________ _

-1.40 1.84 5.99 -1.03 -1.93 1.93 G.o; -1.1+2 • -------- 2. 41+ 2. Q 1 &_. 1 !+ -1~...c..7..:.g _________ _ -2.35 1.98 6.1!+ -2.1& -3.52 1.97 6.1; -2.58 • - 4_~ 0 6 1-~~ 5 6. C 9~ --~2,_.,,.--=9~8~--------- - -----4.65 1.72 6.Cl -3.43 -4.35 1.53 s.9; -3.65 0

________ -_.:.5:...:, • .?~ 1. I+ f+ 5 • ~'...!.7 __ -~3~•....::.9~0 __________ _ ____ _ -5.50 1.2& 5.7, -4.08 -5.61 1.J3 5.59 -4.17 • -5.64 .63 5.45 -4.21 ---------5.60 .55 5.23 -4.21 -5.43 .28 s.02 -4.10 • ---- - - -~~~ ~~ - .1~0~--~4~·~8~7 __ -~3~·--=9,~7 ________ _ _____ _ -4.83 -.16 4.6!+ -3.68 -4.66 -.24 4.57 -3.56 • -4.37 -.40 4.43 -3.35 --------3.~7 -.55 4.23 -2.96 -3.41+ -.&, 4.16 -2.65 • _______ -2.!_9-_s - • s"-1+-'----- ~ !_o_~ _ _ -2. 2,:__:s~--------------

• - 2. 41 • • g3 3.93 -1.91 -1.96 -1.01 3.85 -1.52 -1.45 -~_.07 3.7'3 -1. l 3 _________ _ __ _ ----- - -

• -.95 -1.11+ J.72 -.7!+ -.4~ -1.21 J.60 -.34

------- • 0 7 -1 • 2~6'.__ __ ~3~·-6~~1 __ __:_• :-0_ .:::...5 ____________ _

• a57 -1.36 3a55 e45 1.J8 -1.41 3.52 .85

_ _ ___ 1. 59 ___ -1_ •.. ~ -'--8 ___ 3. 4 7 _ __;1. 2,_!'.5:...._ _____________ _

• 2.08 -1.52 3.4, 1.63 2.59 -1.59 3.42 2.03

_____ _ _ 3. 12 __ -1._~! ___ l. 3 3 2_!_45-:_ _____________ _

• 3.&2 -1.74 3.35 2.85 4.14 -1.1g 3.3~ 3.26

• It• & 5 -1. 8 ,;_$ _..........__3 • J 1 . 3 • & &'.':-_______ ______ _

- --------, ieCJ .. 1iti11 t , ·~s-~ -.-. i:, ~ _,c.. · -• rii; '.I_'-'• I.. ~~

~M>IAL l'-'CLINAT[r>N OF' H(C(C:T GATES 52.0Jr>EGREES

I • __ GI~_EN • T RA '-'Sf.._OR'1EO t )( 'f X y

··- 5.17 -2.1+1 2.93 4. 11 I J 5. 3 1__ - 2. 11_ 3.13 __ 1+.20

• 5.27 -2.J7 3.2) · 4 • 1 & 4. && -1.73 3.42 3.6& 4.25 _ ..:- 1. 53 3.5, __ 3.33 3.77 -1.26 J.73 2.94 • 3. 3 4 -1.00 3.91 2.59 2,95 - • 7_ 7 I+ , .'J 1 2.2s

• 2.53 -.+g '+. 2 7 1,95 2.13 - .11 It. 5 l 1. 63 1_._7_2 ._ 1_5 ft. 7 3 1 •.. 31

• 1.33 .43 4,C38 1.:) J .34 • 63 5,23 ,70 .-5_~ __ 1_._1_2 ~-~~s ,39

• .Jg 1.38 5.6'+ • 0 7 - • 3 f, 1,60 5.6) -,26 -:_ _. _8_1 1~_tO ?•9~ - ~J:,_O

-1.24 1.36 6.07 -.91 0 -1.75 2.Jq &.13 -1,26

-2~ 26 2.21 6.26 -1 • 65 -2.77 2,23 6.31 -2.02 • -3.33 2.21 6,37 -2.43 -3.89 ~.20 &.3!+ -2.84 -4.49 2.11 6.31 -3 .2 8

0 -4,80 2,J2 6. 2:, -J.51 -5.13 1. 9:l 6,13 -3.76 -5.37 1,73 6.09 -3 .95

C -5.49 1,51 5,9+ -l+.06 -5.54 1,32 5,8) -4.11 -5.53 1.J 3 5,53 -1+.12 • -5.39 ,75 5,37 -4.03 - ?_.!_~_o ,55 5,21 -3.91 -4.82 ,25 4,95 -3.64 • -4,67 ,17 4. 8 8 -3,53 -'+,39 -.02 4,72 -3!34 ------3.90 -.21 4.55 -2.qs • -3,49 -.36 4.33 -2. 67 -3 .. '.l 1 -.56 1+. 2 2 -2.32 -- - --·--2,54 -.71 1t.1J -1,96 • -2.04 -.63 3.99 -1. 56 ~-1._5._4 -.~4 3~~-~ -1.1_9 ------ -- ---1. 0 4 -1.a& 3,73 -.61 • -,55 -1.11 3. 7) -.42

- .. 0_5 ___ -1_._2 e 3.61 -.04 ,45 -1.ft.J 3.52 .35 • .96 -1,49 3.45 .,75

1,45 __ -_1. 6_2 3,_Jj J __ ! 1_1+ -1.94 -1.7) 3.31 1,53 • 2.1t1+ -1.81 3,2!t 1,93 2,96 ___ - _1_._3J 3,17 ?_._} _4 3,4& •2,J5 3.13 2,74 • 3,96 -2.15 J.o; 3,14 ... 4 7 - ~-~ t? ~~-0) 3,~5 S,10 -2. ltl 2,CJl ft,06 'h!f •I, ftt ,., •H lt sll

" ~ ~AOIAL INCLINATION OF WICKET GATES 47.0CDEGREES

• ~'-'

)

"\ .,

•

GI/EN _TRA~S[O~~EO~----------->< y . x y

4.q4 -2.65 2.55 J.q6 5.10 -2. 56 ___ 2. Bl ___ 4_. _0_l ___________ _ 5.07 -2.52 2.63 4.04 4.49 -2.11 3.1) -3.55

_ _______ ,... 1 o _ ~ 1. I\ 9 3. 2 5 __ 3. 21t _________ ___ _ 3.65 -1.59 3.47 2.86 3.24 -1.29 3.66 2.53

_ _______ 2. 8 7 -1. J 2 _ _ ~ 3_._8[ _ __,2_~_2_~·- --------------2. 48 -.r, 4.1J 1.91 2.11 -.35 4.35 1.62

_______ 1.73 -. t i 4._61 1_._J_2,__ ____________ _ 1.37 .37 4.83 1.03 1.01 .75 5.17 .76

_ ______ !_6 2. __ 1._o~7 _ __ 5_._~1 • ~&,<._ ______ _ __ _

.21 1.37 5.63 .16 -.22 1.63 5.82 -.16

------ - -. E;, 5 1_!.._.6__7. _ _ _ __,6"--'."--'0,._,)..__ _ _ -.....,.,_,4,....,8,:._ ___________ __ _ -1.06 2.J6 6.'1!+ -.76 -1.56 2.23 6.23 -1.14 - 2_._J_6 2. I+ J 6. 41 -1 • 5 -2.56 2.46 6.47 -1.86 -3.12 2.55 6.50 -2.27

___ ___ _ -;J_._68 ~.53 6.57 -2....,,_,6"-.!7 ______________ _ -4.29 2.50 6.56 -3.11 -4.60 2.1+3 6.5+ -3.31+

_______ -_ ft._._94 2. 34 6. 5J -3 .60 -s.20 2.19 6.41 -3.79 -5.34 1.99 6.27 -3.91

--------- ~._ft._1 1 • 7 9 6. 1.,,_1+.,__-....,3'-'.'-'9~7 ___________ _ _ _ -s.42 1.s1 s.91 -4.oo -5.30 1.21 5.71 -3.94

_c ______ •<-"5 • 1 ft 1 e O O 5 • 5 !t - l • 8,_,,J'------------- ---~ 4.76 .66 5.27 -3.59 -4.63 .57 5.16 -3.46

______ -_4_._;37 .37 s.01 -3.,..:,3:_,,0'-----------------3.91 .13 4.81 -2.96 -3.51 -.JS 4.63 -2.67

_______ - _3_. _0_5 -. 32 4. 42 -2. 33 -2.59 -.49 4.27 -1.99 -2.11 · -.os 1+.13 -t.63

-------·-=1 .• _J, _2 -. 63 i..:JJ -1.25 -1.13 -.96 3.87 -.68 -.65 -1.12 3.7~ -.so

------- - <! 1_6 -1_. _~7 3 • 62 - .12~--------------.33 -1.44 3.49 .26 .62 -1.57 3.39 .65

------~1 . J t __ -i. 7 4 3 • .z_ r 1"--'!.~0,....,3..__ ____________ _ 1.79 -t.66 3.1~ 1.41 2.2s -2.02 3.31 1.ao

______ _ i.ra -2 .. 1.s 2_.9~ 2.2~1 ___________ _ 3.26 -2.35 2.a; 2.60 3.76 -2.46 2.11 3.00 4. 2 6 -2. 6 3 2.~ ~-~ J • ftO. ______________ _

--- ----.. I ft ,--. ·2 ;-., , 2 I 5 ft JI 91 ,i~~ .,,~, e, g& a, ,a

() ~J\OIAL t~CLINAT(ON OF 14IC'<ET GATES 42.000EGREES

I • GI 'I E.tf _ _T RA NSf_ORH ED I l( y )( y ·e:- 4.&7 -3.~7 2.20 3.76 J

4 • 3 6 ----- 3 • ~ )_ _____ 2 • 4 '*--3 • 91 4. 8 3 -2.95 2.47 3.8<) • 4.28 -2.51 2.1e 3 .1+2

. J • 9 Z - ~ 2 • 2 5 _2. 9 7 ___ 3 •. 11 3.1+9 -t.9J 3.22 2.76 • 3.11 -1.57 3. 4:, 2.1+1+ 2. 77 __ ~.1. 2l 3. 68 __ 2._.16

• 2.41 -.~2 3. q !+ 1.67 2. 0 7 -.53 4.22 1.59

_1. 7 2. -,16 _4 • 5 J __ 1. 3 - ---1.39 .25 1+. ,n 1.06

0 1. ~ 7 .&6 5.1J • BJ ._7_1 1 .• Q_~ 5_. _3 7 .5_3 -- -----

• .JJ 1. 3 4 5. f> t .25 - • () 7 1. 6 4 5. ,n -.os ~. !+ .9 1 .•. 9:, S_._C_3. - ,..3_6

• -.88 2.14 6.2J -.64 -1.36 2.36 6.37 -.gg ~ 1._8_4 i _._5 7 6.53 -1.3 -2.34 2.&6 6.62 -1. 69 • -2.89 2.62 6. 7 !+ -2.09 -3.!+'+ ?.84 6.78 -2.49 -4. J 5 2.65 6.83 -2.92

0 -l+.37 2.82 6.81 -3.16 -'+,_ti . 2.1& 6_.--1..9 -3~_!t.J. -4.99 2.64 6.72 -3.61

C -5.15 2. 1+5 6.59 -3.71+ - 5 __ . _2 3 0___2 6 ~~45_ -3.61 -5.27 1.98 6. 2.3 -3.86 • -5.18 1.67 6. (I 3 -3.61 - _5_._0_J 1 ~.!t-5 5. 8:, -3.72 -4.70 1.10 5.58 -3.50 • -4. 5 7 • 91} 5. 4 8 -3.41 - 4._3_2 • 75 5. 23 -3 • - - -·----3.88 .. I+ 7 5.05 -2 • 93 • -3.50 .23 r+. 8 5 -2.65 -3 ~o--9 -.05 f+. 63 -2.33 -2 .. 62 - .. 26 1+.4; -2.01 • -2.16 -.1+7 4.27 -1.66 -1 • P 6 _ ___.::__e_Q Q 4 _,__1 ~ -1.•-~--1 .. 21 - • 8 E, 3.9; -.94 • -.,~ -t.06 3. 71 -.58 - .. z 7 -1,_25 J__, _{,_J . -,

• 2 Ci -1.46 3.47 .16 • • E, 6 -1.64 3.33 • 5 I+ J. •.1 5 __ -:.1 ~ _ ~ '+ 3 ,..1_6 • ..9 1.&2 -2.01 3.J5 1.28 • · 2. 0 9 -2.21 2,<H 1,6E, 2,56 - Z., .4' 2._J _7 Z ,_O 3.05 -2,62 2. 62 2.i.1+ • 3.53 -2.8l 2.5] 2 • 8ft a._._o 1 -2!_9_9 2,3! l.,ZJ . 1+,&0 -s.211 2,19 3,73 .,~, . • J~ ~, lh ~i l 1f8

e . UOUL [NCLINATION

I OF' lfIC'<ET GATES 37.00JEG~EES

• .. ......... GI II:: N T RA NSf OIU1EO . ' 1 l( y )( y ·J- 4. 3 7 -3.5~ 1.86 3.57

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•. 8 0 .35 5 .. .. 3.2 • 6 .. 0 • 45 1.31 s.sa .33 • .01 1.54 5.81 • 0 5

- ._3 2 1_._95 6 ._j 5 ~ • ..2 .. J -.69 2.21 6.2, -.so • -1.15 2. i.7 &. 4!t- - • 81+

-1._51 2_. 7 2 6_._Q__I -1_._15 -2.10 2.a1 6.7, -1.51 • -2.63 3. J 6 6. 131 -1.89 -3 .• 18 3 .. .. 1 .3 6~~ -2....L2-9. -3.79 3.20 7. fJ 5 -2 • 72

0 -4.11 3.19 7.05 -2.95 - 4-.._1+_6 3_!1_1.6 1. i:~ -3-'_2_ . -1+. 7'+ 3.06 1.01 -3.t+O

C -t+.92 2.86 &. 83 -3.54 - ~ .. .. 0_1 2_,.L.1 Q_,J7 -3_. -5.G7 2.43 6.53 -3.68 • -5.01 2.12 6.3, -3.66 - !t., __ 6_ 6 1,-6 . .6 Q_,_i_j -3 -'+.59 1.50 5.87 -3.39 • -4.46 1.37 5.75 -3.31 -~--2_1t- 1.12 5.57 -3_. & -3.63 "61 5.31 -2 .. 87 • -3.47 a53 5.09 -2,61 - _3_._ __ 0_6 "~2 ~_._a J -2.32 -2 .. 64 - .. JJ '+. & 2 -2 .. 01 • -2.19 -.28 4.42 -1.&6

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