Wave exciting forces and moments on an ocean platform

19 JUNI 1973

ARCHIEF.

OFFSHORE TECHNOLOGY CONFERCE

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Technische Hogeschool

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THIS, IS A PBEPRT - -- SUBJECT TO CORRECTION.

Wave-Exciting Forces and Moments

on an

Ocean Platform

By

e-. ii. Kim and F. Chou, Stevens Institute of Technology

© Còpyright 1970

I.

.2

j.

77.

Offshore Technology Conference on behalf of American Institute of Mining, Metallurgical, and petroleum Engineers,, Inc., The American Association of Petroleum Geologists, American Institutè of ChemicalEngineers,America,n Society of Civil Engineers, The American Society of Mechanical

Engineers, The Institute of Electrical and Electronics Engineers, Inc., Marine Technology Society, SoQiety of Exploration Geophysicists, and SOciety of Naval Architects &.Marine Engineers.

This paper was preared for presentation at the Second Annual Offshore Technology Conference

to be held in Houston, Tex.., April

_{22_2L,..l97O.}

_{Permission to copy is restricted to an abstract}

of not morè.than 300 words. _{Illustrations may not be copied.}

Such use of an abstract should contain conspicuous acknowledgment of where and by whom the pape is p'esented.

ABS ['RACT

The authors describe a new method of predicting the wave-exciting forces and moments acting on an ocean platform restrained in

oblique seas. Procedures are based on strip

theory.

The wave-exciting forces and ments on restrained cylinders have been investigated before, in beam seas as well as in oblique _seas, but the cylinders were always Lewis cylindrical

forms. In this study, the two-dimensional method developed by Frank is extended to the

calculation of wave, forces and. moments, and the

strip method devised by Grim is applied to obtain the three-dimensional forces and moments.

The results of the numerical calculatios for _a

Series

60

model are coared with the results of the latest experimental work by Lalangas. The results of an experiment conducted for the present study, on a rectangular light-drafted barge restrained in quartering seas, are

compared with theoretical predictions.. Theory and experiment are in generally good agreement

in both cases..

INTRODUCTION .

The vertical wave-exciting forces on Lewis cylindrical forms restrained in beam seas were

calculated by Grim in

1960.1

His method was later applied by Ternura2 to the calculation _of References and illustrations at end of paper.

sway- and roll-exciting forces and moments on Lewis cylinders restrained in beam _{seas. Tasai} then performed approximate calculations of the lateral wave-exciting forces and moments in oblique waves according to Watanabe's strip method.3 _{Grim and Schenzle, in 1968, further} refined the cálculation of lateral exciting forcés and moments on a ship restrained _in oblique seas, by applyipg the strip method

[crossflow. hypothesis].

Grim's method is based onthe assumption that the disturbance of an incident wave caused by the ship's body is represented by the

potential used in describing the water flow around the body when the body is oscillating harmonically in the calm water surface. _This potential, together with the incident wave potential, constitutes the potential that

describes the flow around the body under restraint in waves. . the present study, we

select for disturbance the potential used by Frank.5 _{His method enables us to compute}

hydrodynamjc foròes and _{oments not only for the} non-Lewis cylindrical form, but also for the widely varying configuratioñs of ocean platforms

of the senisubinersjb1e type.

To determine the degree of reliability of the prediction method, theoretical values are compared with Lalangast lates _experimental results for a Series 60.model9 and with addi-tional experimental results for a rectangular barge obtained in the present study.

COORDINATE SYSTEMS

Let o-XYZ

and

o-xyz be the riglithnd

rectangular coordinate systems, as illustrated

in Fig. l

Coordinate planes

o-XZ and

o-xz lie

on the calm watêr surface, and the Y- and y.

axes point vertically upward.

incidence of the wave be desinateë the wave progress in the positive

Then the wave profile is

h = a cos

(vX -

wt)

i-n the space doordinates, and

Let

the

by

and

let

X-direction.

in the body coordinates,

wave amplitude

wave number [2/g]

circular frequency of the wave.

Now, suppose two vertical control planes cut the body at z and z dz, and observe the wave motion within the fictitiouslr confined

domain. The wave equation [Eq. lI can be

interpreted by noting that the teim .vx sTin I.L

determines

the wave Íorm in the two-dimensional

domain

and

that the term vz cos

.i represents the phase shift of -the wave at x = O and z = z from

the crest of the incident wave at the origin. Cutting the whole body by many vertical control planes such as are demonstrated above, we ban

apply the strip method, or the crossflow hypothesis. That is, the three-dimensional

forces on the restrained body, induced by the oblique waves, can be determined approximately by summation of the two-dimensional elementary forces induced in the waves of the strip domain at the subdivided elémentary sections over the length of the body.

WAVE POTENTIAL

-Let us suppose that there is no obstacle in the strip domain defined by the two control

planes. Then the potential of the flow is only that of the wave [Eq. 1] and is represented by

=

_{e' sn (vx sin}

+ vz cos - - wt) or, conveniently, by the wave potential per unit amplitude of the incident wave,

-1.

p

_{(x,y,z ;}

t,)

_{= e' sin (vx sin} + vz cos -

wt)

-.

[2]

The terms Vx sin j and vz cos t have physical

where a

(0

h = a cos (vx sn

meanings interpreted as in the preceding

section.

The wave potential is broken down into two component potentials, of the form

p0 =

e' sin

(vx

sin p.) CO OS 4.

;i;

..

[s']

ò - . - r p = e

cos(vxsinp.

e w .- . -F

sir-(-vz-»s'-'wt)

...-a

[l/a]e, is applied to the symmetric motion of water about the y-axis. Therefore-, the odd

function will not be employed as the potential causing the sway-, and roll-eciting force and moment on a cylinder section in the beam sea, while the even one will be used-as the potential causing the heave-exciting on the section.

DISTTJEBANcE TENTIAL

-The non-existence of the sectional body in the strip domain, assumed in the preceding section, does not, in fact, hold; and this

creates the disturbance to the wave potential flow represented by Eqs. 2, and 1i-. The disturbance is mathematically expressed by a distribution of pulsating sources over the

section surface. Frank5 has reported on the source-distribution method as applied 'to the

calculation

of the- hydrodynamic forces and

moments on oscillating cylinders in, or below,

the free surface of deep water. The potential used by Frank is employed as our d-isturbanöe

potential.

Since the disturbance is supposed to

respond to the incident wave and the position and form of the sectional body, it is written

y , z ;, p.,t) = Re E c m)(S)

G(x,y

; vz cos p. .

wt) ds]

[5]

In this equation, m designates the- mOde of excitation [m - 2,3,1 sway, heave, roll]. The

expression Q[m][s] designates the unknown com-plex source intensities distributed along the section contour C. The expression depends on the mode of excitation, the geometry of the

section,. and the incident wave. The expression G[x,y is the pulsating source potential of unit intensity at the point

,ij

in the

lower-half xy-plane of Fig.. 2. The term e1"Z cos p.

+

vz, cos wt)

-[1]

Since the potntials [l/a]0 ad [i/a]è -are, respectively, the odd and even functions ofx, the odd function [l/a]0 is applied to the asymmetric motion of water about the y-axis in the xy-plane, while-the even function,

WAVE-EXCITING FÓRCES -AID MONTS ON

r OTC 1180 _Ç.

_{H. }1 and}

_{F. CHOU I-333}

represents the position of the strip section z where the disturbace occurs in response to the oblique incident wave of wave numbér

_and

incidence _{L. This term takes into aouflt the} phase difference of rna.ima of the disturbance pressure

_and

the forôé acting on the section

contou. from the crest of the Incident jqaré at the coordinate orig-in.

Since

QJm]

_{and G áre- the complex source}

intensity and

functIon,

respectively, let them

be

Q(rn) q(m)

. ¡Q(m)

G = G.

- ¡G.

--r

i

where i

añd where m3

_and

Q]

are

real

and imaginary parts of

["],

and Gr -a -Gj'

are

real and imaginary parts of G.

Eq.

5

is thë

changed to

Cm) (m)

f

(Q

G +Q.

G.)ds'

-.

r

r- i (m)

cos ('z cos

-

wt)

J.

(_ Q(rn) +

Q(m)G)

ds

1 _{The deepwater condition.}

he fifth condition to be satisfied is the kinematical boundary condition on -the body

surface [nô flow through the -surface of the

restrained body]. t'or the thrèe modes of excitation, wé w-rite

= _;

--(

_{) = Ö}

_{) = O}

òri S

--

t8]

where

/n designates the normal derivative of

the ±unction at a point on. the body surface.

The equations are reduced

_{to-simultaneous}

-li-near algebraic equation systems [see Append-ix

for explicit expressions].

HYDRODYNAMIC PRESSURE - AND FORCE

-The linear hydrodynamic pressures induced

at a point on the body surface by the incident

wave of unit amplitude are,

û S S

[for sway]

--a

_-

-[for heave]-

-ò-

-a

at

[for roll], ..[9]

The -integration of these pressures over the

submezged surface of the body gies us the wave exciting forces and moments on the body per

tth-i-t amplitude of- the Icidét wave. The

föllowing symbols

are

_{introduced for the forces}

and moments.- - - --

-surge-exciting force sway-exciting force

F heave-exciting force

-ÌvI roll-exciting oent aboit the origin

= itch-exciting moment about the origin yaw-exciting moment about the origin

We formula-te them as -F p

.1

f

_{$_.!dxdz}

a

(L)(c)

a = -

ÇÇ_..! dydz

a M

-1f

a

(L)-(c) r

z dy dz

sn (vz cos h -

wt-.[.6]

BOUNDARY CONDITION

-The ultimately required potential wh-ich -describes the water flow around the restrained body in the strip region Is therefore obtained y superimposing the wave potential [Eq.

_2]

_and

the corresponding disturbance potential [Eq. 6-].

Specifically, the potentials are written accord-ing-to the mode of excitation [i.e., _sway,

heave

_and

_{roll], as}

-=

ke

--A

rr,

r

= -R

ao

The. potentials

, -and- satisfy the

fol-lowing four conditions, out of the five

required.7 -.

-The- continuity of the liquid -in the

wolè domain

-The-linearized free-surface coMitió

I334

WAVE-EXCITING FORCES AND MOMENTS ON

AN OCEAN PIAO FD

I OBI_QU SEAS

OTCl18Q

where [L] and [C,] mean that the integration is executed over the length of' the body and over the contbür of a section, respectively. As to the surge-exciting force Fi/a, it may, if the restrained body is geometrically syametrical about the x-axis, be calculated by taking the

strip -in thé direction parallel to the z-axis.

The positive signs of' the forces and moments take the positive directions of the coordinate

systems [Fig. 1].

EDCPRThIENTS AND CALCULATIONS

To establish the applicability of the theoretical method to the calculation of wave forces and moments on ocean platforms floating in oblique seas, two very different models were

chosen: the Series 60 model of' CB = 0.6 and a light-drafted rectangular barge model. The model particulars are iven in Table 1.

The experiment with he. Series 60 model

was reported by Lalangas, but the rectangular barge model was tested under the present study

program in Davidson Laboratory' s a'ank 2. The

forces and moments were measured by a three-component dynamometer at a 45-degree inclina-tion to the longitudinal axi of' the barge,

whiàh corresponds to 135 and 225 degrees,,

according to our definition of At 135-degrees, the heave-, surge- and pitch-exciti.g forces and moments were measured; and at =

225 degrees, the sway- and roll-exciting forces and moments were measured. This was done by rotating the model 90 degrees while the dyna-mometer remained fixed. The origin of the body

coordinates lay in the calm-water level, and tha moments were measured about the origin. The

wave recorder was located 40.1 in. ahead of the origin [Fig.

3].

The phase difference of the wave force and moment from the crest of the

incident wave at the origin was determined by

correcting the location of the wave recorder..

The numerical calculation. of the exciting

forces and moments for the- two models was

carried dut on the I3v 360 computer at Stevens Institute of' Technology, and the results were compared with experimental results [see Figs. 4

to 18]. It was found that the roll.exciting moment is largely dependent upon the number of sources and the position of the moment center. In calculating the surge-exciting force on the barge model, an attempt was made to take longitudinal strip, and the side forcé in the direction of' the z-axis was taken as the surge force [Fig.

3].

End effects were not corrected

slendernes of the body, às expected.

The agreement between theoretical and experimental results is fairly good for the barge. The agreement is attributable to thé large end effects.

The agreement betweén theoretical and experimental results for both ship model and barge model is generally better for the

verti-cal forces than for the lateral forces.

ACKJOWIEDENTS

The authors would like to express their - appreciation to J. P. Breslin, E. Numata and C. J. Henry for their expressions of interest and their advice during the period of research.

NOMENCLATURE = water-plane area a = wave amplitude B breadth of model -F force -G = source potential g gravitational constant h = wave elevation L = length of model

M

moment

m = number indicating mode of excitation o = origin of' th body coordinate system p hydrodynamic pressure Q sburce intensity s = contour length t tithe- -X, Y, Z space coordinates -x, y, z = body coordiflates phase difference X = wave length -= incident angle -V wave number

circular frequency of the wave

= x-coordina.te of position of source

= y-coordinate of position of source

REFERENCES

Grim, O.: "Eine Methode ftír eine genauere

Berechnung der Tauch-und Stanipfbeweging in glattem Wasser und in Wellen", HSVA-Bericht

Nr.

_1217,

Hamburg [Juni, 19601.

Ternura, K.: "The Calculation of Hydrody-namical Forces and Moments Acting on the Two-Dimensional Body -- According to the Grim Theory", J. SZK [Sept., 1963] No. 26. Tasai, F.: "On the Swaying, Yawing and

M X = a M z

₌

-a - (L)(C), a (L)(C) à z

dx dz

(x dx + y dy) dz' .[1O]

in any òf the calculations.

CONCLUSIONS -

-1. The agreement between theoretical and

experimental results is very good for the ship model. This agreement is attributable to the

OTC 1180 _{C -H. I<tM and F. CHOU}

_I335

APENDDC

The Kinematical Bouhdary Conditions

The kinematical boundary conditions of Eq. 8 are presented below as formulated by

Frank,5 with Frank's. symbols retained.

The normal velocity components of the

wave potentials [Ecjs.

₃

and 1] are, explicitly,

I

-

we

_LSifl

sin

(VX.

sin

.&)

sin

+ cos

(Vx.

sin ii.) cos cv1]

sin (vz cos

.&

wt)

- Sin (vx. sin

.&)

COS

(vz cos

.

-cos cv.]

wt)

where _{is the slope of the contoir at a} point x1, yj [Fig. 2]. Thus the kinematical

conditions at the point. Xi,Yi on the body

sui-face are, for sway,

N _,

, N , ,

' 'J'.+

j

J=1 J=1

-+ cos (vx

.

sin p) cos

where I and are the "influence

coefficients" given in thé Appendix of Frank's

report. For roll, the formula is the same as for sway, where, only the mode number takes

4.

I N sources are taken [i.e., i _1,2,... N], we obtain a [N][2N] euatïon system_with

2N unli,owns, - the real and imaginary päits of

the unknown complex sou'ce intensities _Q. The Hydrodynamic Forces an Moments The hydrodynamic forces and moments on a section of the body are the pressure integrals over the section surface. They are expressed as noted below.

For heave,

.

J - .

cos (vz cos

-(C) a

+ -. sin (vz cos

Ij. - wt),

For sway, PS

T-(C) -Sc

= r.co

(vz cos

L

-+ -

_{sin (vz cos}

ji.

- wt

For roll,

(C) a

-x d-x +

y

dy)

Rc

(VZ cos ji.

Integrating heave., sway forces and roll moment

4.

Rolling Motions.oShp

im

db1iuê

Waves",

Intl. Shibuildthg Progress [May,

_{1967] 14,}

-

sin (vx;

.

sin

Q(2)

) cos -cvi J

Q(2 I)

No.153.

-

-.

Grim, 0. and Schenzle, P.

_"Bereumg

der Torsions-be1astg einès Schiffes im Seegang", Foré chungzentrum des Deutschen Schiffbaus Bericht Nr.

_5,

Hanbug an der

+

j=1 '

j=1 N+J IJ

5.

Alster 1

[1968].

Frank, W.: _{"On thé Oscillation of}

Cylin-ders in or Below the Free-Su±face of Deep

and are, for heave,

Fluids", NSRDC Rept. _.2375 [Oct.,

_1967].

Q3)

1Ç)

+

.

o

6.

Lalangas., P. A.: "Lateral and Veitical _j=1 J 1

j=1 J

'J

Forces and Moments on a Restrained Ser±es

60 Ship Model in Oblique Regular Waves", Davidson Laboratory Rept. 920, Stevens Institute of Technology [Oct.,

_1963].

Q3)

(3)

+

j=1 J '

I)

j=1 'J

7.

Wehausen, J. V. and Laitoe, . V.:

"Sur-face Waves", Handbth der Physik, Band. IX,

Vj

= - w e

_[sin

.

sin (vx.

.

sin

) sin

cv Springer Verlag

_[1960].

cpo

=

ueV[s

COS (vx. sin ¡.i.)

sin cv1

- wt)

_{sin (vz Gos}

_-

_wt)

E-336

WAVE-ED(ÖITING FORCES MID MO}'NTS ON

AN OCEAN PLATFORM FIXED -IN OBLIQUE :SEAS OTC fl.O

Each force and moment has two components which deteiine the. maitude añd the piase differ-over the length of the body [Eq. 10], we obtain

the total forces and moments. Heave- and roll-exciting moment, for example, become

F y a

a

p

=

f ....!i

cos (vz cos .i.. - wt) dz

(L) a E +

f

sin.(vz. ços - wt) dz (L) . -. F F cos wt +_!. a a sin wt VZCOS

-

wt) dz. sin (vz cos pt) dz M M _.!i

wt +sin wt.

IM = (M

+ M5)

- F -1.. ys - 6fyh-= tan M .l zs = tan

().

mzh -

Mzc;

-Tab1e ]

ence; that is,

IF I =

(F2 +F2)

y yc ys LB? (L) Breádth (B) Draft- (H) Displacement (Fw)

LCG (abaft midship section)

'VCG (blow waterline)

Rudder area

Waterplane area

Load watèrline coefficient

- Section coefficient

Series 60 Model Barge Model

5.00 ft 15 in. 0.667 ft 10 in.

0.267

f-t . i i 33.27 lb 5.+l lb 0.Ó75.ft o 0.022 ft 0.03Ó ft2 Nil .2.355 ft2 150 in.2 0.706. - 1.0 - 0.988 throughout

o o I I I 0.2 0.4 0.6 o A 0.8 1.0

uB

2 TEST THEORY

O -

A -

AMPLITUDE PHASE ANGLE X LATERAL FORCE (D 4 -I (I, LI Ui ¡80 w o 120 ui -J A z 60 4 w 'n

I

1.2 1.4 1.6 ¡.8 o. X WAy E

FIG. I. COORDINATE SYSTEMS

Y,Y VERTICAL FORCE 0.14 0.12 WAVE RECORDER 40.1 TEST THEORY

O - AMPLITUDE

A

-PHASE ANGLE WAVE 0.10-J 4

008-

_o aOG 0 U. o 0.04 0.02-A A o I 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 I 8 uB

--o o 0 ..___.____._o 4_-j n LI w 1 180 I o

-Il2Oj

I (D A

-I60

I A I A I I A I I IJ _o u,

FIG.4. SURGE-EXCITING FORCE FOR BARGE FIG.5. SWAY-EXCITING FORCE FOR BARGE

(I35)

('225)

FIG. 2. STRIP SECTION A-A FIG. 3. BARGE IN AN OBLIQUE SEA

WAVE MOM E N T 0.10 J 0.08 d o

06

0.04 0.02 o O

1.0 0.8 TEST THEORY

O - AMPLITUDE

A - PHASE ANGLE

2 TEST ThEORY

¿L-+ - -30°

Ö 600

FIG. 8. PITCH-EXCITING MOMENT FOR BARGE (joI35°)

WAVE LENGTH TO SHIP. LENGTH RATIO. AIL

FIG. 9. AMPLITUDE OF SWAY-EXCITING FORCE FOR SERIES

60 MODEL

O 0.2 0.4 0.6. Ò.8 1.0 1.2 1.4. 1.6 I 8

V.B 2

FIG.6. HEAVE-EXCITING FORCE FOR BARGE

(o350)

0 0.2 0.4 ö.6 0.8 1.0 1.2 1.4 1.6 I 8 o u-B AthA _A I 1- - I. 0.2 0.4 Ò:6 Ô.8

uB

0.6 0.5 TEST THEORY

O - AMPLITUDE

A - PHASE ANGLE

FiG. 7. ROLL-EXCITING MOMENT FOR BARGE

-

(aa5°)

A TEST THEORY L X

----

9Q0 O 120° D

---

1500 1.2 1.4 1.6 Io o 4 w -j CI) w w 80 w o IZO ,j_--j CD z 60

4

w (n D-, O -. -'0. s .

-

,.--.-. -a..-'

- . --- O _-I_.--- I ---I 0.5 10 1.5 2.0 O 0.5 - 1.0 1.5 2.0

WAVE LENGTH.TO SHIP LENGTH RATIO1 XIL

FIG.IO. AMPLITUDE ÓFSWAY-EXCITING FORCE FOR SÌES

60 MODEL -1.0 TEST THEORY

O - AMPLITUDE

PHASE ANGLE o 4 w

L

--j O.e 4 0 u, W O.6 . I8O U- _0.4 0 : -

oi

0.2

-.

A A 60 .Jcsj O.6 000 0.4- .

A-o.: A AAA A A

-.1

,

C, I.0 4 -j (n 0.8 w w 180 0.6 w

-J-o _s 1201j O.4 60

a:

a4 o -I 0.3

0.5 160 0.4

t

₄

.1 _0.3

-/a

/7:,

..

-,

0.2

i ,

0.1-

. £

I,

/

,

-- /

o I I I 0 0.5 1.0 1.5 2.0

WAVE LENGTH TO SHIP LENGTH RATIO, A/L

IG.II. _{AMPLITUDE OF HEAVE-EXCITING FORCE FOR SERIES}

60 MODEL TEST ThEORY

X - SWAY-WAVE

O - - - HEAVE- WAVE 160 80 -6--

--- -

---o-80 WAVE-TO-HEADING ANGLE, DEGREES (°) 30 60 90 120 150 180

.13. SWAY-WAVE AND HEAVE-WAVE PHASE ANGLES, 1.0 L

WAVES FOR SERIES GO MODEL

O-TEST THEORY /L 90°

O - 20°

D _150°

--V 80° o

/

.

/

f,

7,,

i,v

//

\'.,

/

0.5 1.0 1.5

WAVE LENGTH TO SHIP LENGTH RATIO, A/L

1

FlG.12. _{AMPLITUDE OF HEAVE-EXCITING FORCE FOR SERIES}

60 MODEL TEST ThEORY

----

30°

+ - 600

I I 0 0.5 I 1.0 1.5

WAVE LENGTH TO SHIP LENGTH RATIO, A/L

FIG.14. _{AMPLITUDE 0F YAW-EXCITING MOMENT FOR}

SERIES 60 MODEL 2.0 2.0 TEST THEORY /.L £

-

---

0° 30° 600 + 0.6 0.5 0.4 a -j D' _0.3 u. 0.2 0.1 O O 0.10 0.08 0.06 0.04 0.02

0.10 0.08 0.06 0.04 0.02 A - ir---I ¡ I-

I--'

I J 0.5

-- IO

1.5

wAvE LÑGTH TO SkIP LENGTH RATIO, AIL

FIG 15 AMPLITUDE 0F YAW-EXCITING MOMENT FOR

SÉRIES GO MODEL

O - 0.5 1.0 1.5 2.0

WAVE' LENGTH TO SHIP LENGTH RATIO, AIL

FIG.I7. AMPLITUDE 0F PITCH-EXCITING MOMENT FOR sERIEs 60 MODEL

-I'

TEST THEORY FL

WAVE LENGTH TO SHIP LENGTH 'RATIO, AIL

FIG.I6. AMPLITUDE OF PITCH-EXCITING MOMENT FOR

SERIES 60' MODEL

TEST THEORY

X _YAW-WAVE

O PITCH':- WAVE 9OWUP 240 ¡60

o_J'

_c,;

' o-g-w 80

o

w -J 30 60' 90 !2Ö 150 ¡80 4 80

- WAVE-TO-HEADING ANGLE, DEGREES

¡60 _o\

240

-FIG. f8. YAW-WAVE AND pIrCH-WAVE PHASE ANGLES, LO L WAVES FOR SERIES 60 MODEL

-0

o D Ai o, 0.10 0.08 0.06

04

0.02 O £

--

0° '30° 60° I + -'

I--I-- I--I--.

+

+

-

--7.-'X.

1'

I'

I --I

1' /

"I

I, /

I 2.0 _{O 0.5} 1.0 1.5 2.0 TEST THEORY FL X

----

90° 120°

O -

_o

_iso°

A

----

¡80° TEST THEORY FL X

---

90° 120°

O -

_o

_-.

_150° 0.10 0.08 D A_ 0.06 o, 'p.;, ü04 02 o