The distribution of the hydrodynamic forces on a heaving and pitching shipmodel, with zero forward speed in still water

PUBLICATION No 124

FEBRUARY 1965

SHIPBUILDING LABORATORY

TECHNOLOGICAL UNIVERSITY

-

DELFT

THE DISTRIBUTION OF THE HYDRODYNAMIC FORCES

ON A HEAVING AND PITCHING SHIPMODEL,

WITH ZERO FORWARD SPEED IN STILL WATR

BY

PROF. IR J. GERRITSMA AND W BEUKELMAN

SPONSORED BY THE. OFFICE OF NAVAL RESEARCH

THE DISTRIBUTION OF THE HYDRODYNAMIC FORCES ON A

HE!VING AND PITCHING SHIPMODEL, WITH ZEROFORWARD SPEED IN STILL WATERO

by Prof0ir J. Gerrita and W. Beukelman,

Summary0

As an exteneion of earlier work on the distribution of the hydrodynamic forces, acting on a heaving and pitching shipmodel9 this report deals with the case of zero forward speed0

Forced oscillation tests were carried out with a segmented shipmodel to investigate the distribution of the forces along the hull for heaving and pitching motions. The in-phase and quadrature components of the vertical forces on each of the seven sections were measured as a function of frequency and an analy8is was made of their distr4bution along the length of the shipmodel.

The experimental results are compared with Grim's values for damping and added mass. The cohiparison shows a satisfactory agree-ment between theory and experiagree-ment0

-2-l Introduction0

According to strip-theory the damping coefficients, the added masa and the added mass-moment of inertia of a ship, per-, forming harmonic heaving and pitching motions in still water, can be calculated by integrating the corresponding twa-dimen-sional quantities for each cross-section over the lengthlof the

ship0

-Tiis very simple procedure does not include three-dimen-aona1 or forward speed effects on the total damping and added ma580 Forced oscillation tests with a 2.3 meter model of a nor-mal cargoship have shown a very snor-mall speed dejendency of these quantities, at least for frequencies of interest for the usual seakeeping problemsi]0 The opeed range was Fn

=l5

to Fn=030. The tests were c:rried out in the 1f.3 meter wide towing tank of the Deift Shipbi1dir.g Laboratory. Teats at zero forward speed were not considered in thts tank because a large wall influence was expected from the reflection of waves, generated by the forced

model motio'is.0 Without zero speed experiments form effects and speed effects could not be separated0

In a later stage it was decided to carry out zero speed tests in a wide basin to minimize wall interference. The results could than be.used to study. form effect in damping and added mass by comparing the resulta with calculations from strip theory. (The

.ero speed experiments could also double the range of speedi con-sidered).

This is of interest for the analysis of the damping cross-coupling coefficients of the motion equations. These cross-coup-ling coefficients depend quite strongly on thé speed of the ship and this i rportant for the calculation of ship motions i,n waves0 From 6trip_theory1J it follows that:

e =

JN'

xdx - Vm

E =

J

N'x dx + Vm

where:

e _{is the coefficient of the pitch velocity term in the motion}

equation for heave,

E is the coefficient of the heave velocity, term in the equation Thr pitch,

N' is the: sectional damping coefficent, m is the total added mass,

V is the forward speed of the ship0

By plotting e and E on a base of forward speed two straight lines cor-respond to each frequecyof motion. The two lines have a common point

for V 0, where: e = E

=

¿ N'

dx and their slope is given by ±m. The experiments have shown tile validity of equation (.1) in the speed range Fn = 0.15- .30 Th zero forward speed, tests could giv,e

ad-ditional support to the theory because of the considerable extension of the speed range.

Funds became available from the Ofilce of Naval Research by,whch forced oscillation tests with a segmented model at zero speed could be carried out in the Seakeeping Laboratory of the Netherlands Ship Modèl

Basin at Wageningen. i

'In this report the results of the experiménts are given. The re-suits are compared with calculated values, using Grim's two-dimensional results for dámping and added mass for ship like sectiortsE2jo

2 Experiments0

The oscillation tests were carried out with a ₀₃ meter modél of the Sixty Series1 having a blockcoefflcient CB=0o?G The main dimen-sions are given in Table I. The model is made of polyeáter, re-inforced with fibreglass, and consists of seven separate' sections of equal length.

Each of the sections has two end-bulkheads. The width of thé gapbetween two sections 15 one millimeter. The sections are not connected to each other, but they arekept in position by means of stiff strain-gauge dy-namometers, which are connected to a longitadinal steel box girder above

the model0

The dynamometers are sensitive only for forces perpendicular to the baseline of the model9

By means of a Scoth-Yoke mechanism a harmonic heaving or pitch-ing motion can be given to the combination of the seven sections, which

form the shipmodel. The total foròes on each section could be measured ae a function of frequency.

A non segmented model of the same form Was also teetedin the same conditions of frequency to compare the forces on the whole model wJ.th

the sum of the section results0 A poesible effect of the gaps between the sections could- be detected in this way. The arrangement of the tests

with the segmented odelandwiththe whole model is given in

Fig0

l.

The mechanical oscillator and the measuring system is shown

in

Fig0 2.

In

principle the measuring system is similar to the one des-cribed by 000dman[3]: the meaeured force signal is multiplied by cosWt and aint.i)tand after irttegraticn the first harmonics of the in-phaseand

quádrature components can be found without distort-ion due to vibration noise0 In some details the electronic circuit differs somewhat from the description in[3J. In particular aynchro resolvers aroused Instead of

sine-cosIrL potentiometer,, because they allow higher rotational speedS0 The accuracy f ti Instrumentation proved to be satisfactory which is important for the determination of the quadrature components, which are small in comparison with the in-phase components of the measured

forces0 Throughout the experiments only first harmonics were deter-mined. It should be noted -that non-linear effects may be important for the sections at he bow and thest-ern where the ship is not wall--sided. The forced oscillation tests were carried out for frequencies up to

W= 11+ rad/sec.

-- The motion amplitudes of the shipmodel covered a sufficiently large

range to study the linearity of the measured values.. An example of the measured forces on section 2, when the combination of the seven-sections

performs a pitching motioñ, is given in Fig.

3.

- -

-- -- The position o-f the model in the tank is given in Figure .k0-There

are beaches or. two adjacent sidos of the tank and a wire mesh along a third side as indicated in the figure0

Each oscillation test was completed within the time that observabLe

-Blockcoefficient

Coefficient of mid length section Prismatic coefficient

iaterplane area

Jaterplane coefficient

Longitudinal moment of inertia of waterplane L.C.B., forward of L /2

pp.

Centre of effort of' waterplane after L/2 Froudeilumber of service speed

0.700 0.986 0.710 0.572 ni2 0.785

01685

rn3 0.011 m

0038

020

Presentation of the results0

3.1 iJhole model0

It is assumed that the force F and the moment N acting on a forued heaving and pitching shipmodel in still water can be described by the following equations:

Heave:

TABLE 1. Main particulars of the shipmodel0

(a

+b

+

D

+E

+

o o

Pi t ch:

(A+k2P'

nj.

,i.

- 'I

.J_tI.1. ,.-\

yy

/

cz

F sin(Wt+)

o z Gz= _Msin((.4.)t+J) cl

+e+ gQ

=-Fsin(()t+J)

(3)

where: z is the heave displacement and Q is the pitch ang1e

Length between perpendiculars

2258 m

Length on the waterline

2.296

m

Breadth 0.322 m

Draught 0,129 m

Volume of displacement 0.0657 m3

}

For a given heaving motion: z=ZsinLt, it fol].0w8 that; F sinc(

b-zC4) a

cz -FcosO

a z

a=

zW

a Z

W

gZ+

McoaJ

ZW2

'I

Similar expressions are valid for the pitchin motIon. The determination of the damping coefficients b and B and the damping

cross-coupling coefficients e and E, is straight forward: for a

given frequency these coefficients are proportional to the quadra-ture components of the forces or moments for unit amplitude of

mo-tion. For the determination of the added mass, the sdded nass mo-ment of inertia, a and A, and the added mass cross-coupling coeffI-cients d and D it Is necessary to know the restoring force and

mo-ment coefficients c and C, and the staticál cross-cou1ing coeffi-cients g and G0

For zero speed of advance, these coefficients can be calcxlated easily from the known values of the waterplane area, the longitudinal moment of inertia of he waterplane area and the position of the

cen-tre of effort of the waterpiane area, as given in Table 1

The results for thé whole model are given in Figures ₅ and 6. These figures also give the sum of the results from each section to

show that the ôomblned effect of the slits between the sections and possible inaccuracies in the measurements is very small0

3.2e PeSulta for the sectiona1

The componénts of the forces ori èach of the seven sections were determined in the same way as for the whole model. As only the forces and moments on the sections weremeasured two

equa-tions remain ior each section: Heave

= F*

_s1n((t+..)

Pitch:

(d*

+Ç.Xj)

_{+ i Q + g Q - F sin«.Ot +}

J

)

where _çS7 is the mass moment of the section .1 with respect to the pitching axis. The ,star () indicates the coefficients of the sections.0 The section coeîfiçients divided by the length of the sec-tions give the mean cro3s-section coefficients, thus:

*

a -,

1/7L

and so on0 Assuming tht the distributions of the cross-sectional values of the coefficients: a', b' etcetera, are continuous curves, these distributions can be determined from the seven mean cross-section values0

In Figure 7 the distributions of the added mass a, the damping coefficient b and the cross-couplïng coefficiente cl and e are given aa a function of frequency.

Numerical values of the section results, a*, b etcetera, are suiunarized in Tablè 2. In thïs Table the z-eBults for the whole model are also given.

The sui of the section resulte, as used in Figures 5 and 6 for comparison with t}ìe whole model results, are determined according

t the. following equations:

(5) : a* = a b = b = d e = e

fd'xdx=A

fe'xdx

= B L Ja' x dx = D

L1

ib' xdx E L

The values for A, B, D and E are given in Table 3.

TABLE.2

Added masa, damping

coefficient

and

cross-coupling coefficients for

the

sections and for

the whole model0

F0

.4)

rade

sec.

1 2 3 9 5 o 7 sum o. p sections wuoa.e

model

a' kg sec2/m

2

11

2Q58 .L+o9

365

2,90 2q78

o68

1799

i88o

k 0.1+8 _{1.38 1.78} 1.93 _{1.75 11'7} 0.21 8.70 8,61 6 0.23 1Q02 1.18 1.29 1.26 0.88 0.18 6.01+ 5.81+ 8 0.20 0.81 1.12 1.20 1.22 0.85 0.11 5.51 5.1+3 10 0.20 0.7k 1.28 1.1+7 1.36 o085 0.12 6.02 5.86 12 0.21 0.78 1.35 1.57 1,63

090

0.1k

₆₅₈

6.'+8 1k

₀₁₉

0.85 1.1+8 1.68 _1.73 1.01+ _0.17

71k

6.82

b'

kg sec/rn

_b 2 'aOO 1.1+0 3095 3.90 3.20

230

0.90 17.65 10.25 1+ 2.08 k.k8 5.18 _{5.93 5.75} 5.25 _{1005 29.72} 29,17 6 1.98 4q23 5.10 _5.15 k,62 3.82 0.90 25.80 25.03 8 2015 3.80 2Q93 _2.13 2.81 2.53. 0.71+

17.09

18.00 10 2.18 3.06 1.72 1.28 1.31 1,78 ₀₉₅₆ 11,89 _12,31 12 1.98 2q22 _1.32

_0.97

0.61+ 1.1+1 0.1+2 8,96 9.1+3. 1k 1.5k _1.78 1.51 1.03 0,69 1.1+8 0,30 8.33 7.87 2

kgsec

_d 2 -0.55 -108k _-1038 -0.06 +1.12 _{i1.33 4031. -1.07 -1,15.} 1+ -0.1+3 -0,99 -0.69 _-0.03 0.68 o.8o 0.21+ -0.1+2 -o,k8 6 -0.31 -0.68 -o,ki 0.01 0,1+1+ _0.56 0.16 _-0.23 -0.17 8 _-0.25 -0.50 _-0.33 0000 0.36 0,1+5 _0.13 -0.1k -0.09 10 -0,20 -091+3 _-0.38 0,00

039

0,1+5 0.12 -0.05 -0,02 12 _-0.17 -0.1+5 -0.1+1 0,00 0.1+1 0.51 0.12 _*0.01 +0.01+ 1k -0.19 -0.1+9 -0,1+2 0.00 40.1+2 ±0.53 +0.15 0 +0,03 e'

kgaec

_. e 2 - -0.07. -0,18 - - -

+013

k -.1.21 -1.28

_-O53

0.02 0,33 1.19 0.52 -0.1+6 -0.69 6 -1.99 -2.80 -1.82 0.00 1.61

210

o.81 _-2,09 -1,81 8 -1.98 -2,60 -1,09 -0.01 0.81 1.62 0.72 -2.53 -2.1+6 10 -1.81+

-.96

-0.66 -0.01 0,1+1 _0.95 0.51+ -2.57 -2.56 12 _-1.67 -1.1+5 _-0.52 -0.01 0.30 0.69 0.1+5 -2.21 -2.08 1k -1.55 -1.19 -0.60 +0.09

0.30

±0,63 0.38 -.1.94+ _-1.95

TABLE 3 Added. mase moment of inerti, damping coeeficient and cross-coupling coefficients ior the whole model.

For A and B the sum of the section results were obtained ly using equation (6)

k. Analysjg of the re8ults.

The experimental valueE of the coefficients òf the _{motion equa.,i} tions will now be analysed by using the strip-theory. For _a descrip-tion of the strip-theory reference ja made to

_{iJ .}

_{in the ,eptia1 case} of zero forward speed the coefficients of the motion equations, accor-ding to the strip-heory, have a very simple form, as shown in Table k, where:

Coefficients of the motion eauatjons (hydromechartical part)..

Added mass of a cross-section (zero speed)

Damp:Ing coefficient of a cross-section (zero speed).

iVaterplane area.

Statical moment of waterplane.

Longitudinaluioment of inertia of waterpiane.

a..

A.

s N Aw SW

I

lo rad/ sec0

.2

Akgmse.c B kgiisec D kg sec2 .1 kg eec

sum of sections whole model sum of séctions whole model sum of sections whole model sum o sections. whole model 2 k 6 8 10 12 1k

226

l5k

ii8

113

ii8

3o77

203k

i+9,

113

1.09 1.17 1.22

372

609k 5.83 .

k.50

3°53 O0kk 3q72 6.90

565

k.k7 3.29 2.8k 0Q58 -0.19 -o,o8

+0.06

.o.O8 -.

s128

. s.Oçkl -0.18 -0.06 +0.02 +0.03 +o0ok 0Q52

-165

_21.2

-258

-2.k6 +0.55

o63

-i83

-20k9 -2.6k

-232

-2.07.

lo

-TABLE Lf _Coefficients

_oi

_{the motion equations according}

to

strip-theory for zero forviad speed0

Table ¡4 _{shows tìat each}

_of

the coefficients can be calculated by integrating the _{corresponding cr035-sectional, values over the} length of the ship.

This. procedure has been carried out for che

ocol

under consi-deration. Grim's cross-sectional _{values for the damping.coeffja4ents} N' and the added mass in' were used.

In Figure 7 the calculated _{distributior.s of the added mass and} the damping coefficients are shown in óomparlson with the experimen-tal values. In general there is satisfactGry agreement between theò-ry and experiment. The calculated added

_mea

at

_{the lowest} frequen-cies is smaller than indiated _by

_the

_{experiments, rind some of the} distributions

_{ehow a}

_{slight shift in longitudinal direction when} compared

with

the measured values.

Also the values uf the tàtal added mass, the damping coeffi-cients

arid

_{the cross-coupling coefficients are also compared in the}

Figures 8 and 9.

The numerical valuee are summarized in Table'50 Fn = 0 a =

f'

dL in' x dx b =

fN'

dx e

.'xdx

c _g A ₌

f'

X2 dx D. ' .x dx

B=fN9x2dx

_.EL!XdX

CÇgIw

.

Gzfgsw

A

kgmsec2

B kgmsec

2

a kgsec ¡w b kg sec/rn

TABLE 5, Comparison of calculated and measured coefficients.

rad/

sec.

mea-sured

cal-c:ula ted sur ed

cal-culated mea-sured cal-culated mea-sured cal-culated

measured alcïL-ated measured calculated

kg seca kg sec

2 3.77 3.03 .o.kk. 6.12 18.80 i2.k5

1O25

2537

-1.28 -1.15 -0064 +0.55 +0013 -1038

4 2.34 1.60 3.72 8.29 8.61.

6.3

29.17 31.66. -o.ki -0.48 -0.33. -0063 -006.9 -2.12

6 1.49 1.15 6.90 7.64 5.84 5.25 25.03 26.22 -0.18 -0.17 -0.19 -1.83. -1 .81 -2.54

8

113

1.09 5065 5.81 5.43 5.41 18.00 16.91 -o.o6 -0.09 -0.10 -2.49 -2.46 -2.66

lo 1.09 1.18 4047 4.01 5.86 6.04 12.31 9,46 +0.02 -0.02

-0.05 -2064 -2. 6 -2.47

12 1.17 i.a8 3.29 :2.69 6.48 6.61 9,43 501k +O.03 +o.ok -o.ok, -2.32 -2.08

-2.08

1k 1q22 1.37 2.84 1.84 6.82 7.00 7.87 2.95 +o.ok +0.03 .0o04 -2.07 -1095

-1.65 E/e

12

-In the region of resonance (for heaveC4) = 6.9 rad/eec,

for pitch (AJ = .7.0 rad/sóc) the agreement between theory and experiment is excellent.

For frequencies lower than 5 rad/8ec the cali.ulated pitch damping coefficient andthe added mass moment of inertia show a

rather largo difference between calculation and experiment0 The. heave and pitch damping coefficients for

W>

10 ead/aec are smaller than the measured values0

Although only first harmonic componen:ts of. the hydromechani.... cal forces were measured, there is no garantee that the 90 degrees out of phase components correspond exclusively to wave damping0 Particularly at high frequencies damping of viscouS origin could be present and the first harmonic content of such e non wave making damping could influence the final result. Because the calculation only considers pure wave danping this

could

be á reason for the ob-served differences0

For practical purposes, such as the calculation of ship mo-tions in waves, the differences as shown in the Figures 8 and _{9 are} not of much interest.

As a final cosparison between experiment and strip-theory, the calculated and measured damping cross-coupling coefficients e and E at zero forward'speed, are plotted In Figure 10, which is taken from referenceEl]. In Figuro 10 it is shown 'that e and E vary linearly with. forward speed at constant' frequency, of motion.. The range of

for-ward speed were this is valid now inòlude zero speed and the calcu-lated damping' cross-coupling coefficients are in agreement with the measurements0

5e Acknowledgethent.

The work described in this report Was sponsored by the Office of Naval Research under Contract No. N 62S58LlO9?0

The Authors are indebted to Mr. TSR. Dyerfor his assistance in carrying out the experiments0

-6 References0

/

L J. Gerritema and W. Beukelman.

"The distribution of the hydrodynamic forces on a heaving and pitching ship model in still water"0

Paper p'-eaented at the Fifth Symposium of Naval Hydrodynamics, Bergen, Norway - 196k0

International. Shipbuilding. Progress

l96k

2 O. Grim.

"A method fcr a more precise computation of heaving and pitch-. ing motions both in smoot water and in waves".

Third Symposium of Naval Hydrodynamics, Scheveningen -

1960.

3o A. Goodman0

"Experimental te'Ghniq.ues and methods of analysis used in sub-iiarged body research".

Third Sympos&um of Naval Hydrodynaniics. Scheveningen .- l96O.

7 List of symbols0

a . . . gi Coefficients of the, motion equations.

A . . . GJ (hydromechanical part).

S *

a

000

The same for. a section of the

ship0

AS f1S3

£% 0 0 0 i. -a'0 . .

.The same for a cross-èection" of the ship0 AI

SI Q S '.4

Waterplane area.

'CB Blockcoefficient0

Fu Froude number.

F, F

Amplitude of vertical force on e heaving or pitching ship0

g Accelèration of gravity.

-Longitudinal moment of inertia of waterplare Longitudinal radius of inertia of the ship. Length between perpendiculars0

M, M

Amplitude of moment on a heaving or pitching

ship0

m' .Added mass of across-section (zero speed)0

m Added masa of a ship (heave).

Damping coefficient of a cross-section (zero speed).

S,

Statical moment of waterpiane0

t Time0

V Forward speed of ship.

x, y9 z Right hand coordinate system, fixed to the ship.

X,y, Z0

_{Right hand coordinate system, fixed in space0}

z Vertical despiacemerit of ship.

z Heave amplitude0

a

xi Horizontal distance of centre of gravity of a section to the

pitching axs0

O9ft99

Q. Q a.

w

.(W&S 5632) Phase angles, Pitch. angle0 Pitch amplitude0 Density of water0 Circular frequency. 1k

-Ç

Volume of displacement of ship. Volume of displacemen.t of section0

r

«h1

z=rsinwt

-JLJ

IL

HEAVING TEST WITH SEGMENTED MODEL

w

V.

z=r sinwt

F2sin(wta2)

1

F1 sin(wt+a1)

HEAVING TEST WITH WHOLE MODEL

F2 sin(wt+y2) ]

ARRANGEMENT OF OSCILLATION TESTS

FIGURE 1

1 2 3 4 5 6 7 1 2 3 4 5 6 7

PITCHING TEST WITH SEGMENTED MODEL

PITCHING TEST WITH WHOLE MODEL

D

_.11_t_ sin wt

:i L

GEARED MOTOR

ELECTRONIC STRAIN INDICATOR CARRIER AMPLIFIER

f

]

r

]E

ITI

L1L

II -u i11 STEEL

STRAIN GAUGE DYNAMOMETER

MODULATED CARRIER SCOTCH YOKE '\PHASE SHIFTER CARRIER BOX GIRDER Ii iii -H III iI

-H---H

L RESOLVER

V

o

PRINCIPLE OF MECHANICAL OSCILLATOR AND ELECTRONIC CIRCUIT

FIGURE 2

fr-ml

A M PLI FIER

DE MODULATOR

INTEGRATOR

f

_100-E

-x

+100-.200

PITCH

SECTION.2 (AFT)

Fn=0

I I Ï

123456

w - CIRCULAR FREQUENCY.

rad/sec

2r

a L

r =1 cm

r =2cm

A

r =3cm

IN PHASE COMPONENT

QUADRATURE COMPONENT.

10

11

12

13 14

FIG.3

COMPONENTS OF FORCE ON SECTION 2. PITCHING MOTION

-60

_in

_'iV

'Ola

Cloe w Cd

*

X

-

LL

f

-20

E

WIRE. MESH WAVE MAKER MODEL _iijjj

WJ'JZ# YWJ7FjW# WZWY Wf 7L'

W#J VW WFW

7I 7#F7W7WJ7)7#j!JWF FF#Fm7W#M

012345

liii,,

10m i

FIG. 4

POSITION OF MODEL IN SEAKEEPING

BASIN

E XI TATO R RI

INSTRUMENTS BEACH

I

I

z

w C-) IL IL w

o

L) (D

z

- Q-D

O-2

L) T ...

o

5.

w

10

HEAVING MOTION

0 ai U) D) 15 Fn = O 40 20 CD

z

û-o

-a

I

z

4 L). Li IL w

o

L) O (D

z

J

û-D

a-4

46

U) u) FIG. 5

EXPERIMENTAL RESULTS FOR WHOLE MODEL

5 5 10 15 SUM OF SECTIONS

o WHOLE MODEL

.

I

¡o

I

/

I

/

/

5 10 2 5 lo (J) 15 3 2 O 10 8 o

a,

s

-ri

o

I

o

u' I-rn

Io.

O

rnc

I-rn

Ics

orn

oc)

rnf

r-;;

_a

zu,

u, 2

d.-COUPLING COEFFICIENT kgsec

a

(s)

e...COUPLING COEFFICIENT

kgsec

I I I N,) D N.J o q'

r

(J, D u, u, D (n

D

2

A.-ADDED MASS MOMENT OF INERTIA kgm

_sec

Ca)

C

B.DAMPING COEFFICIENT

kgm sec D

F o o _0,5 1,0 0,

iL

IPPIL

111k

hulk

U!i1U

Fn nO - EXPERiMENT CALCULATION Fn n O

FIG 7 DISTRIBUTION OF o,b,d AND e OVER THE LENGTH OF THE

SHIP M OD EL 20 2 a

2

L

o

4

2 o E _4 o 2

4

o

a

iu,' p..

uhu!

ÌPJiJ;

UJn1O

uiiííi

!i

¡I

r

djsac

1L 11

R__il

Ii

_u

l'lì"

IÌIìl

IL-31415

'hl2

20 10 o 20 10 20 10 E o 20

13 1415 [---'

I

s[17I

1,0 O O - aß UI o 0,5 1,0 0$ o 1,0 0,5

g 8 7 E 6 3 2 1 o 10 40 30 E lo 8 6 4 o -2 -4 -6 , -s-s

/

I

III,

/

/

'NATURALFOR HEAVE

FREQUENCY, NATURAL FOR HEAVE FREQUENCY 4 3 2 N L) G) U) D) 2 -3 w (AJ 5 w _o EXPERIMENT

Fn0

_w

-..- CALCULATION

FIG. 8

COMPARISION OF THE CALCULATED VALUES OF a,b, D AND E

WITH THE EXPERIMENTS

15 _{5 10 15}

10 15

4 3 2

i

o 3 2 i U) a)

-2

-3

5 w lo 1F; u a) U) a) w

lo

9 8 7

FnO

o _EXPERIMENT

-. --CALCULATION

3 w w

FIG.9

COMPARISION OF THE CALCULATED VALUES OF A, B ,d AND e

WITH THE EXPERIMENTS

/1' 1

Ii'

/

III,

J

I

II

,p" NATURAL FOR PITC FREQUENCY .kyy/1o.25

____

'NATURAL FOR PITC FREQUENCY /L = 0.25 2 D 2 4 6 è o 2 5 10 5 io 15 10 15

F1010 DAMPING CROSS COUPLINGCOEFFICIENT AS A FUNCTION OF FORWARD SPEED

-I I I I o s -O _{k6 rad/sec} D s radsec -s ()10 rad/sec -s s W12 rad/sec

iiiii:

_-r P0,1 0.2 0.3

Fn..

JEXPERIMENT E,e CALCULATED 8 6 1. 2 o -2 -4 -6 -8 6 4 2 o _2 -4 -6 -8 -lo 6 4 2 o -2 w -4 w -6 -8 -10 6 4 2 o. -21 -4 -'6 -8 -lo