Pitch- and heave characteristics of a destroyer

I

824825

TECHNISCHE HOGESCHOOL DELFT

AFDELING DER MARITIEME TECHNIEK

LABORATORIUM VOOR SCHEEPSHYDROMECHANICA

PITCH- AND HEAVE CHARACTERISTICS OF A DESTROYER

w. Beukelman

International Shipbuilding Progress

Volume 17, No. 192, August 1970

Reportno. 257..,P Shipbuilding

Labora-tory Deift.

Deift University of Technology

Ship Hydromechanics Laboratory Mekelweg 2

2628 CD DELFT The Netherlands Phone 015 -786882

PITCH- AND HEAVE CHARACTERISTICS OF A DESTROYER)

1. Introduction.

Asknownthe pitch- and heave motions may be

represented by two coupled linear equations of

motions including the important speed effect. For the determination of the motion-coefficients. the wave-forces and -moments it is required to know the two-dimensional damping and added mass of

the ship's sections. A solution for a circular

cylinder oscillating at the free surface was given by Ursell [1].

By conformal transformation of the

cross-sections tothe unit circle it is possible to define the damping and added mass for each section.

Such a transformation, in which a three

co-efficient or Lewis-form transformation was

introduced, is given by Tasai [2] and Grim [3]. These methods are useful for cross-sections of conventional ships, but fail for sections with

extreme shapes. In this case it is necessary to

extend the Lewis form transformation to a multi -coefficient transformation [4] as given by Smith

[5]anddeJong [61. For the destroyer model was

made use of such a multi coefficient transforma

-tion because of the extremely bulbous

cross-section in the fore-body.

Taking into account the speed effect it is now

possible to determine the hydrodynamic

co-efficients for each section in different ways: according to Gerritsma and Beukelman [7].

Ogilvie-Tuck [8] and Vugts [9],

Semenof-Tjan-Tsanksij, Blagowetsjenskij. Golodilin [101. which

) Report no. 257 Shipsiild1ng Laboratory Deift, University of Technology.

by W. Beukelman.

Sum mary.

The coefficients of the pitch - and heave equations have been experimentally determined for the model

of the Davidson A-destroyer with a bulbous bow. Moreover the wave-forces and -moments on the restrained model have been measured in different head waves. These measurements are compared with the results of computations based on two modes of strip theory and on a 'rational' strip theory for slender bodies.

The motions have been calculated with both the computed- and experimentally determined co-efficients, forces and moments and are compared with each other. A comparison for one speed has

also been made with the measured motions of the Naval Ship Research and Development Center (NSRDC)

and the measurements of Breslin and Eng.

henceforth are respectively called version I, II

and III.

After integration over the ship's length the total hydrodynamic coefficients for the ship are known

[7] [11].

Toverifythese theories the hydrodvnamic co-efficients have also been determined in an

ex-perimental way by means of oscillation tests with

the destroyer model in still water. The wave

forces and -moments have been measured with the model restrained in waves.

The computation of the hydrodynamic

co-efficients, -forces, -moments and the motions

for the destroyer has also been carried out by

Smith [5] according to Gerritsma '5 version, while

the motions have been measured by Breslin and

Eng [12] and Smith and Salvesen [13].

Finally the motion equations have been solved

both with the computed and measured coefficients.

forces and moments.

In the past the hydrodynamic coefficients had

been determined in an experimental way for some other models [7] [11] [14].

2. Calculations.

The right hand coordinate system xyz. connect -edto the ship's centre of gravity is travelling with

the ship speed V. It is supposed that the centre of gravity in height is situated in the still water

surface. The wave surface. -force. -moments

236

S

S

cos

(ct

kx) wave g

2pg Jvxdx

F' _{Fa CS (oet - F} wave force L

M : Ma cos (oet + EM) wave moment (eq. 1)

Z + Ez) heave _{A Jm'x2dx}

+ v

_fN'xdx V

fxdx

8 8 _{et + Eec)} pitch

L 2 L 2 L dx

e e

in which the wave number

2 B

fN'x2dx -2V fm'xdx -V fjx2dx

k

2/ -

(eq. 2) L L L

and the circular frequency of encounter

+ _kV while

=wave length

g = acceleration due to gravity

equations of motion [7J for

(eq. 3)

heave: (a+p)z+b+cz -do -eO -gO=F

9 .. (eq.4)

pitch: (A+k pV) 8+B 8 -D -E - G z=M The hydrodynamic coefficients in these equa-tions are according to Gerritsma and Beukelman

[71 (version I): C - 2pg

_f

L D= fm'xdx L

E=JN'xdx-V JJixdx

Other symbols to be used in this analysis are:

_{G-2pg fyxdx}

_(eq.5)

p = density of the water L

y = the half width of the cross-section at

W the wave force:

the waterline

m' = sectional added mass F CO5

kT COS

N' sectional damping E 2pg

f ye

(kx) dx +

V = volume of displacement a sin _L

k =radius of inertia for pitch

yy * sin

sectionaladdedmassatthestern

J(N'-V-)e

T (kx)dx +

L dx

N0 = sectional damping at the stern

= length ordinate of the stern * COS

fme

(kx)dx (eq.6)

It is now possible to derive, the well-known _L

and the wave moment:

M cos_a _kT* COS E -2pg fy Xe (kx)dx + ç M 3 w a sin sin L sin (kx)dx +

f(N'

- V -)xe

dm _kT*

a= fm'dx

L dx L 2 -kT + fm Xe _(kx)dx b

fN'dx -V f-- dx

_cos L L L (eq. 7) c

2pg fydx

L Inthe case of a ship with a transom stern it is

d fm'xdx+_Y_ fN'dx

--4

f - dx

dm possibletofindbymeansofpartialintegration for

L L L dx the hydrodvnamic coefficients according to

e e version I:

C _{pg fYx2dx}

c = 2pg fydx

L L D_{= f m}'xdx d _{= f m}'xdx + V f N 'dx L L L e E = fNxdx + V f m'dx + Vm'x e f N'xdx - V fm'dx L L L L G

2pg fyxdx

_{(eq. 8)} g = 2pg fy xdx w L L

For each section holds as described in [7]:

A_-= f m x dx 0 L * 1 k kz T

-- ln(1 -

Je dz)

(eq.9) B= JN'x2dx k _y -T L in which:

_{C2pg fyx2dx}

T draught of the section.

Forthedeterminationof the sectional damping D

fm'xdx -

-Y- fN'dx

and added mass has been used a multi -coefficient L c L

C

transformation of the cross-sections to the unit

_{E= fN'xdx+V fm'dx}

circle. The transformation formula looks like: _L

L N -(2n-1) a 2n-1 nO

G-2pg fy xdx

(eq.1O)

Lw

The integral terms in the above mentioned

where a maximum number of N - 32 was used for coupling terms are ignored.

b- fN'dx+Vm

the sections 17 and 18.

L The transformation coefficients for the sections

c

2pg jvdx

are given in Table 1.

L

d fm'xdx + V

fN'dx+m'

2.1. Other striptheories.

e e

A so-called 'rational' strip theory for slender

e = fN'xdx - V fm'dx + Vm 'x ships and high frequencies has been developed

L L by Ogilvie and Tuck in [81 (version II. In the case

g - 2pg f Yxdx _{destroyer it is not entirely correct to make use}of a ship with a transom stern like the Davidson -A

L

of this theory: the influence of the transom stern

2 has tobe neglected to some extent. To show this

,2

V , V , .

A fm x dx +

2 JN xdx + 2 fm dx + influence the hydrodynamic coefficients

accord-L o L L ing to version II are plotted in Figures 2 - 8.

e e

These coefficients are given in [8] as follows:

+

m'x

2 00

_a=Jmdx

e L B=

Nxdx+Vmx

, 2 L

00

b= fN'dx

L

Table 1.

Transformation coefficients - Davidson type A.

section 0 1 2 3 4 5 6 7 8 9 10 y 0. 1096 0. 1322 0. 1508 0. 1653 0. 1706 0. 1756 0. 1788 0. 1820 0. 1848 0. 1854 0. 1854

a'

+0. 634763 +0. 538712 +0. 475447 +0. 405580 +0. 337851 +0. 276044 +0. 227559 +0. 204095 +0. 196954 +0. 194498 +0. 193795 a3 _{+0. 008824 +0. 008272 +0. 016456 +0. 033001 +0. 040889 +0. 049993 +0. 061934 +0. 048229 +0. 030584 +0. 016564 +0. 004721} a -0. 018587 -0. 032341 -0. 029574 -0. 028604 -0. 029224 -0. 031944 -0. 031540 -0. 016625 -0. 006254 -0. 003277 -0. 002115 a _{-0. 014569 +0. 000787 +0. 006433 +0. 004568 +0. 009562 +0. 012645 +0. 008042 -0. 001761 -0. 002119 -0. 004268 -0. 004379} a9 0 0 0 0 0 0 -0.004427 0 0 0 0 section 11 12 13 14 15 16 17 18 19 20 y 0. 1826 0. 1718 0. 1550 0. 1332 0. 1120 0. 0870 0. 0630 0. 0422 0. 0240 0. 0001 +0. 182833 +0. 159812 +0. 129027 +0. 087717 +0. 044207 -0. 008312 -0. 069047 -0. 156982 -0. 321085 0 a -0. 000003 -0. 013634 -0. 037071 -0. 063870 -0. 092086 -0. 115450 -0. 133027 -0. 153040 -0. 167356 +0. 849541 -0. 002151 -0. 006390 -0. 023765 -0. 038991 -0. 053038 -0. 062931 -0. 065924 -0. 077415 -0. 079463 0 -0. 002419 -0. 004420 -0. 015333 -0. 024243 -0. 029453 -0. 039177 -0. 044839 -0. 051034 -0. 053505 0 a9 0 0 0 -0. 016527 -0. 017523 -0. 029847 -0. 035147 -0. 036673 -0. 036656 0 a11 0 0 0 _{-0.007892 -0.011381 -0.021379 -0.024859 -0.027850 -0.028859} 0 a13 0 0 0 0 -0. 011643 -0. 018640 -0. 021567 -0. 023534 -0. 029944 0 a₁₅ 0 0 0 0 -0. 008125 -0. 012810 -0. 014849 -0. 016615 -0. 016122 0 a 0 0 0 0 -0.006435 -0.011540 -0.014018 -0.016084 -0.016587 0

a'7

0 0 0 0 0 -0.008154 -0.010711 -0.011900 -0.009329 0 0 0 0 0 0 0 -0. 008382 -0. 009449 0 0 a 0 0 0 0 0 0 -0.008175 -0.008663 0 0 a 0 0 0 0 0 0 -0. 007344 -0. 007466 0 0 a 0 0 0 0 0 0 -0. 004600 -0. 005942 0 0 a27₂₉ 0 0 0 0 0 0 -0. 004671 -0. 005488 0 0 0 0 0 0 0 0 -0. 004484 -0. 004837 0 0 '31

For slender ships the differences between

Gerritsma- Beukelman and Ogilvie-Tuck only

remain in A and D.

In Figures 2-8 also are plotted the

hydro-dynamic coefficients according to Vugts [9] and

Semenof-Tjan-Tsanskij e. a. [101 (version III).

Their sectional mass coefficients contain a term with the derivative of the damping to the length

and as a consequence also other coefficients

include this term.

These coefficients can be written according to

version III for a general ship in the next way:

( , V

a = j m dx +

-L

b = JN'dx -V fLdx

c

2pg fycix

L

d= Jm'xdx+

2V L V

+-.

2 JN'dx co L e 2

f-_xdx

A = fm'x2 dx + coe fN 'xdx + 2 V idm V dN 2

- - j

xcix + - j - x dx

L co L

BJN'x2dx

-2V fm'xdx -V

f'2ci

+ v2

Jxdx

dN' 2 dx co L e C _{2pg f}

yx2dx

L V2 rdm'

- J-dx

_{2 dx} co L e C) e dm' e f N 'xdx - 2V f m 'dx - V f xcix + dx L L L v2 1dN'

-- Jdx

_{2 dx} C) e

g=2pg fyxdx

L L e

E= fN'xdx -V j -- xdx

cdm' L L G = 2pg

And again in the special case of a ship with a

transom stern the hydrodynamic coefficients

according to version III can be derived:

a = Im'dx V

2o

N L e b = JN'dx + Vm' c

2pg fydx

L B

fN'x2dx+2

+-N x

₂₀₀

e

C2pg

D fm'dxdx -L ci = fm'xdx + JN'dx + e e e = f N'xdx - V fm'dx +

₂₀

N '+ Vm 'x

₀₀

L L e

g=2pg fyxdx

L A fm'x2dx + V fm'cix

_2oo

N 'x 2 L L e e 2

+i-m 'x

_2oo

e

jfN'dx

fN'dx+Vm'x2 +

L V

N x

₂₀₀

c) e E f N 'xdx + V f m 'xdx + V m

00

L L

D=fm'xdx+ V

2

fdN'xdx

(eq. 11)

S

240

G2pg fyxdx

L (eq. 12) e E pv\gL pV .. gL

For slender ships the hydrodynamic

co-efficients of version IlIand version II are similar the wave force the wave moment

F M

except for A and B. In the last named theory the a a

forward speed correction lack for A and B, while pg A pGI kç (eq. 13)

version I does not have a speed correction in B

andD. 2

where A = fy dx and I

_{= fy x dx}

w w L w

L L

2.2. Calculations results.

The calculated hydrodynamic coefficients (eq. 3. Experiments.

5, 10, 11), wave force (eq. 6) and wave moment

(eq. 7) are given in Figures 2 - 11 for the

The experiments have been carried out with a different speeds in a non-dimensional form as wooden motlel of the Davidson Type A-destroyer

follows: (Maruo-design). The model was constructed by

the National Physical Laboratory and was kindly

for heave for pitch

offered to the Deift Shipbuilding Laboratory to

the coefficients: the coefficients:

carry out the forced motion experiments. This

a a

1 + -

1 + destroyer form hasa conventional afterbody, but

pV

pVk 2 _{a strongly bulged forebody. The hull form (see} yy

Figure 1) has earlier been tested for pitch and

b\'gL B _{heave motions as reported by Breslin and Eng}

AP P

pgV

VL'L

[12] and has recently been tested by

Smith-p g

Salvesen [121.

ci 1) The main particulars

of the model are

pVL pVL _{summarized in Table 2.}

6

DAVIDSON TYPE A DESTROYER

Table 2.

Main particulars of Davidson A-destroyer

model built to a linear ratio of 33. 545. Length between perpendiculars

Wetted surface area

Beam Draught

Volume of displacement Blockcoefficient

Midship area coefficient

Waterplane area

Longitudinal moment of inertia of waterplane

L.C.B. forwardofL /2

pp Centre of effort of waterplane

aftofL

/2

pp

Radius of gyration for pitch

3.481 m 1. 505 m2 0.371 m 0. 127 m 0. 08845 0. 536 0. 778 0. 954 m2 0.7519 m4 0. 058 m 0.266 m 0.25

The experiments consisted of three parts:

a)forced oscillation test for heave in still water b)forced cscillation test for pitch in still water c)measurement of wave forces and -moments

on the restrained model

For all experiments three speeds were

considered: F_n

.15. .35. .55.

3.1. Forced oscillation - Heave.

The model was forced to oscillate in heave by means of the forced motion oscillator of the Delft

Shipbuilding Laboratory [15]. The model was

connected to the oscillator by two vertical struts,

fore and aft. The connection consisted of two

force-transducers. By means of a Scotch-Yoke

mechanism the model received a sinusoidal

motion for a certain frequency.

The frequency range for the forced oscillation

test was between: 2 and 15,

or in non-dimensional form between

e

L/g1.2and9.0.

The oscillation amplitudes for heave were

0. 01 m and 0. 02 m. The resulting equations for

the heave oscillations are:

(a+pV)+b+cz-F cos(t+

z e z

-D-E-Gz=M cos(ct+

z e z

If the forced heaving motion is supposed to be

z z COSc.) t

a e

the hydrodynamic coefficients may be soluted

with the following results:

cz -F cos

a z z

a-2 - pV Zc

_ae

F sin E b= Zc

_ae

Gz +McosE

a z z 2 ZCO

ae

M SinE z z

E=-Zc)

_ae

in which:

F cos E F cos E + F cos

z z zA zA zV zV

F sin = F sin E + F sin E

z z zA zA zV zV

M cos E = (F CO5 E - F cos E )t/d

z z zA zA zV zV

M sin = (F sin - F sin )'/l

z z zA zA zV zV

(eq. 16)

while:

F

zA = force amplitude on the aft strut with

phase angle EA

F force amplitude on the fore strut with

zV

phase angle

1 distance between the struts

After measuring _{FAcosE zA' FAsin E zA'}

F cosE and F

sin

it is possible to find

zV zV zV zV

the hydrodynamic coefficients for heave with

eq. 15. These resultsare given in the Figures 2, 3. 5and 8 in a non-dimensional form (eq. 13 for the different speeds.

(eq. 15) (eq. 14)

'242 L 6

wVt._

L 6 Fn..15 Fr.35 -EXPERIMENT EXPERIMENT £ 6 o r.OlOm o r..020m 0 0 0 0

\

0 0 Fn..35 8 0 VERSN I -CALCULATION VERSIONI -yE P St ON

Figure 2. Comparison of the experimental and calculated added mass coefficient a.

0

L 6 0

wVt7i_{.-_--.} VERSION I

o r..OlO m CALCULATION VERSION

-o r. 020 rn

VERSION m

Figure 3. Comparison of the experimental and calculated damping coefficient b.

£ 6 0 0 0 oo 0 00 0 0 Fn..55

0.'. A 1. 6 wVt7 Fr'.15 -\ A Ae OO o r..OlOm EXPERIMENT _{o r..020m} A .005m

Figure 4. Comparison of the experimental and calculated mass coupling coefficient d. wVE7 Fn..35 -VERSION CALCULATION VERSION 5 VERSION Fn.35 -_0. .1.5 (VERSION I CALCULATI0N VERSION 0 VERSI0N_..__ 1. 6 0 6

Figure 5. Comparison of the experimental and calculated coupling coefficients for damping e and E.

Fn..55 0 2 0 wVt 6 EXPERIMENT E J r..010 m J r..010 m pvVt r..020m 0r..020m A r. .005 m 0.5 0.5

\

.0 - ₀ 0 A 0

-

7-244

3.2. Forced oscillation Pitch.

The forced oscillation for pitch took place in

the same way as described for heave and for the

same frequency range while the oscillation

amplitudes were 0. 005, 0. 01 and 0. 02 m.

The equations for the pitch oscillations are:

-do - eÔ - go = F cos(c t + E

2

0 e 0

(A+k pV)O+BO+COM cos(wt+E

YY 0 e e

(eq. 17)

and the pitching motion is supposed to be:

0 = e coso t

a e

in which:

o 2z /1 with z = excitation amplitude.

a a a

After solution the next hydrodynamic

efficients are derived:

CO -M cos

a 0 8 2

A-

k pV 2 _yy 8 0)

ae

B= M sino 0 0 0c

_ae

0 8 Fn..35 -EXPERIMENT o r..OlOm o r..O2Xm A r. DORm go + F cosE a 0 0 2 Oc

ae

F sine 0 0 e 00)

_ae

in which: M cos =(F coso o 0 OA OA M sin E =(F sinE -o e eA OA F cos E = F cose + o 0 eA OA

Fsin

=F sinE + o e OA OA while:

-F

coso ev ov F sinE )'/21 ev ev F cosE ov ev F sino ov ev (eq. 18)

FOA = force amplitude on the aft strut with

phase angle

OA

co- F0 = force amplitude on the fore strut with

phase angle

00V

After measuring FOAc05ROA. FOAsinEOA,

F

cos

and F sino it is possible to find

ev ev ev ev

the hydrodynamic coefficients for pitch with eq. 18. These results are given in the Figures 6, 7, 4and5alsoina non-dimensional form (eq. 13) for the different speeds.

A 1VERSJON I CALCULATION VERSION VERSION I I £ S

0.

05

Figure 7. Comparison of the experimental and calculated damping moment coefficient_B.

L S * 0. Fn.15 EXPERIMENT o r..020m r..005m I. 8 CALCULATION VERSION VERSION Fn..35 wVt7j VERSION I EXPERIMENT o r..OlOm CAI.CULATION VERSION

o r..020m LVERSION

Figure 8. Comparison of the experimental and calculated mass coupling coefficient D.

I I 0.5_ I _0. -0.X 0.

\

\ \ -

\

_N \\ A 'N_N 0. a

\

0 N

/

D 0 02 0 _DO

0-

0.2k...

/

/ N N N 0. N Fn.35 _Fn.55 -L S wVt7 0 r..OlOm 0

246

3.3. Restrained model in waves.

The destroyer model was motionless kept in

the zero amplitude position of the forced oscilla

-tion test for heave. The generated waves were regular and long crested with approximately a

height 2 L/40, L/50. L/75, while the wave

lengths were 7s 0. 6L, 0. 8L, 1. OL, 1. 2L, 1. 6L and 2. OL.

The wave force and -moment on the restrained

model are:

F=F cos(O t+E

)andM=M cos(c t+

)

a e F a e M

They were recorded on an ultra-violet strip

chart recorder, together with the wave-height.

Amplitudes and phases were manually analyzed.

The results are given in Figures 9, 10 and 11 in

a non-dimensional form (eq. 13) for the different speeds. 1.0 Figure 9. (eq. 1) Fn.15 180 LI) Ui Ui C, Ui _180 _270 4. Motions.

By solving eq. 4 for the different theories the motions are known. For the strip theory accord-ing to version I and III the motions are given in

Figures 12, 13, 15, 16, 17 and 19. while the

motion results calculated with the hydrodynamic

coefficients according to version II for speed

Fn = . 35 are shown in Figures 14 and 18.

The motions are also calculated for the three speeds and plotted in Figures 12, 13, 15. 16, 17 and 19 for the strip theory according to version I, first with 1.0 .8 .6 .2 D =fm'xdx - -Y b. 0 L 0) e

to show the influence on the motion of the speed

correction in this term and second for speed

F = . 35 with the same speed correction for D_n but without speed influence in A. These results

are given in Figures 14 and 18.

In all above mentioned cases the right hand part of the equation has been taken equal as described

Comparison of the experimental and calculated wave exciting forces and moments for F = . 15.

1.2 1.2 .1. .8 .8 vT7r 1020/L=1/50 0 2/L=1/80 IL 2 c, /L 1/35 EXPERIMENT CALCULATION

1.0 1.0 .8 .6 .2 0 .8 .6 .2 0 0 0 .4 .8 .4 .8 vT7r 1.2 EXPERIMENT 1.2 Fn..35 180 _1SO _270 fO 20/L1/50 0 2/L.1/80 2 L /L1135

Figure 10. Comparison of the experimental and calculated wave exciting forces and moments for F = . 35.

Fn.. 55 180 U) Ui Ui Ui C) - 90 w _1 80 _270 U, Ui 90 Ui 1.0 .8 1.0 a. .2 .4 .8 CALCULATION I I I I .1. .8 1.2 1.2 180 180 fO 2,/I.1/50 EXPERIMENT Q 2/L1/80 CALCULATION 2/L.1/35

Figure 11. Comparison of the experimental and calculated wave exciting forces and moments for = . 55.

U, Ui Ui 90 C, Ui _90 .180 _270 U) Ui Ui C, Ui C)

248 2.0 1.8 1.6 1L Za/ 0,8 0.6 0/. 02 0/. 0.2 DAVIDSON-A Fn. 15 VERSION 1 - VERSIONS

VERSION 1 WITH D.Jmdx_-Y_b

-A CALCULATION

Figure 12. Heave- and pitch amplitudes for F = . 15.

in eq. 6 and 7.

The amplitudes and phases of the heaving and

pitching motion have also been computed by

solving eq. 4 for the measured coefficients,

forces and moments.

The results are shown in Figures 12 - 19

together with the experimental motion amplitudes and phases for speed Fn = .35 of Smith-Salvesen

[131 and the motion amplitudes of Breslin-Eng [121 for three speeds.

5. Discussion.

Concerning the coefficients of the motion

equa-tions it is apparent that the strip theory accord-ingtoversionl for a and b is in close agreement with the experimental values; the deviation in b

for version II in the case of high speeds proceeds

from the supposition that the ship has to be

slender and so the influence of the stern has to be

neglected. The differences in d are rather small

just as the deviation from the measurements.

For D

it is evident that the forward speed

correction is a good improvement. The calcula-tion results with respect to the measured values

are,

regarding e and E,

different for the

considered speeds and frequencies.

For the

highest speed version land III are in good agree-ment with the measureagree-ments. Considering A the

deflection from the experimental values for

version III is striking and in this case the results

of version II are correct. For B it has to be

confirmed again that the discrepancies between the measurements and all version remain rather

large. The non-linearity with respect to the

oscillation amplitude is the most evident in this term.

The motions calculated with the measured

co-efficients, forces and moments generally agree

well with the calculated and measured motions.

Two exceptions have to be ascertained: first the discrepance between the measured motions of Breslin-Eng[12] for Fn .15 and the calculated

motions and second for Fn = . 55 in the case of the

longest considered wave the difference between

18 16' 1/. 1.2 - fo-Za/ç 08- 06- 0/.- 02-1.L 12 06 0/. 02 DAVIDSON A 2/.-VERSION 1 22 - --- VERSeN

- VERSION 1 _{WITH D.frr(dx--Yb}

20- CALCULATED WITH L EXPERIMENTAL COEFFICIENTS, FORCES AND MOMENTS

o o EXPERIMENT NSRDC i EXPERIMENT BRESLIN.ENG /0 I/I oi/ 'I / / 02 0/. 06 0.2

II

0.6 0.8 1.0 1.2

. CALCULATED WITH EXPERIMENTAL COEFFICIENTS. FORCES AND MOMENTS EXPERIMENT BRESLIN - ENG

10

08 12

2.6 2.4 22 2.0 1.8 1.5 1.4 12 1.0 Za/ 08 0.6 0.4 02 14 1.2 11.0 08 ea/bca 06 0,4 02

correct for the motion results according to

version I.

Afterwards it must be ascertained that for

comparison with the measured motions it shoujd have been more correct to carry out the

oscilla-tion tests about the waterline belonging to the

considered speed; the calculations too should

then have to be adapted accordingly.

6. Conclusions.

For the lowest speed F = . 15 the results of

the several versions show less mutual differences and a good agreement with the measured

coeffi-cients, wave forces and -moments, while the motions too are in good agreement with the

measured values.

The mutual differences between the versions

and the deflection from the measured values,

particularly for b, eand B, increase with speed.

It is however probable that the mutual differences

2.4 2.2 2.0 1.8 1.6 1.4 1.2 Za/ca 0.8 06 0.4 0.2 1.2 - 0.8as EL - 0.2-I I DAVIDSON-A Fn..55 VERSION I 'I }CALCUIATION I.... VERSION I WITH D..Jrrdx_ -b S S CALCULATED WITH EXP COEFFICIENTS, II FORCES AND MOMENTS. I.

EXP BRESLINENG II \ Ij \.

/!

\ I / / /

/ !

-,,,

S Vt1

The figures show that this tendency is only Figure 15. Heave- and pitch amplitudes for . 55.

0.2 04 0.6 08 1.0 1.2

VEA

Figure 14. Heave- and pitch amplitudes for F = . 35.

the pure calculated pitch amplitude according to

version

I or

III on the one side and the pitch

amplitude calculated with the experimental co-efficients, forces and moments on the other side.

It is remarkable that a speed correction of D

results into a closer agreement with the

ex-periments for the coefficients, but at least for

F = . 35 shows too high heave amplitudes with

an exception for version III and for Fn . 55 in

the case of long waves gives improbably high

pitch amplitudes. The lack of measured motions

for the highest speed to refer to is regrettable. It is also noteworthy that in the limit case of

infinite long waves (frequency of encounter zero)

the motion amplitudes z / and 0 /k should

a a a a

tend to the value 1 and the phase angle for heave

5zç and pitch should respectively tend to 0

and -90 degrees.

I

250 - 20 - £0 60 _180 - 216I.. 20ho - 6080 - -133- -120- 1h0-Czç Eg -160- -180- -203- -220- -260- -280- -300- -320-DAVIDSON -A Fn 35 VERSION I VERSION V I I -,

Figure 16. Heave- and pitch phases for F = . 15.

Figure 18. Heave- and pitch phases for F = . 35.

-- S

E6

- - VERSION I WITH DsJmdx_.b S Ezç

ICALCULATED WITH EXPERIMENTAL COEFFICIENTS,

E9 FORCES AND MOMENTS.

o o EXPERIMENT NSRDC

02 Oh

vtx 06 08 10 1.2

S

Figure 17. Heave- and pitch phases for F =. 35.

_20 _h 0 60 _180-_80 U 0 -EzçEeç 160 22 o EXPERIMENT NSRDC DAVIDSON -A Fa. 35 VERSION 1 WITH 1 0-.Jrridx_ -b andAfrxdxjCALTION VERSION U C _20 .60 _180 2h0 280 .320 3-DAVIDSON-A 55

\

5---___5_ \. \ \ \ teç \\S \ VERSION I VERSION CALCULATION - - - VERSION I WITH 0- Jmdx_ -0-2b E

CALCULATED WITH EXPERIMENTAL J COEFFICIENTS, FORCES AND MOMENTS

S S

02 04 06 08 10 12

Figure 19. Heave- and pitch phases for F = . 55.

_200_ 260 300 _320_ _3h0_S I' \\ DAVIDSON_A VERSION I 220-200 .250 _280 31.0-VERSION CALCULATION. VERSION I WITH D.Jmdx_ _Lbj

S I _{CALCULATED WITH EXPERIMENTAL COEFFICIENTS,}

FORCES AND MOMENTS 0

02 Oh 06 0.8 1.0 1.2

between the versions are insignificant with

respect to the neglects in the strip theory. A

correction of the speed influence in one of the

components (e. g. D, b) necessarily does not lead

to a more correct motion. The speed influence

as a total complex of both the right- and left hand

side of the motion equation is evidently important.

A pronounced preference for one of the versions

is hardly to give. On an average version III

perhaps procures somewhat better results, in

every case for the heave amplitudes and the phase

angles, however, the deflection from the

measurement for A and the pitch amplitudes

remain unsatisfactory.

References.

Ursell, F., 'On the heaving motion of a circular

cylinder on the surface of a fluid', Quarterly

JournalMech. and Applied Math. Vol. II Pt, 2, 1949.

Tasai, F., 'On the damping force and added mass of ships heaving and pitching', Report of Research Institute for Applied Mech., Kyushu University, 1960.

Grim, 0., 'A method fora more precise computation of heaving and pitching motions, both in smooth water and in waves', Third Symposium on Naval Hydrodyanmics 1960.

Porter, W.R. , 'Pressure distribution, added mass and damping coefficients for cylinders oscillating

in a free surface', University of California

Institute of Engineering Research, Series 82 1960.

Smith, W. E. , 'Computation of pitch and heave

motions for arbitrary ship forms', Neth. Ship

Research Centre, report no. 90 5, april 1967. Jong, B. de, (a) 'Berekeningvandehydrodynamische

coefficienten van oscillerende cylinders', report no. _{174, maart 1967. (b) 'Computation of the} hydrodynamic coefficients of oscillating cylinders', report no. 174A in preparation (translation of report no. 174) Delft Shipbuilding Laboratory.

Gerritsma, J., Beukelman, W., 'Analysis of the

modified strip theory for the calculation of ship motions and wave bending moments',

Neth. Ship Research Centre, report no. 96 5, june 1967.

Ogilvie, T. Francis. , Tuck, Ernest 0., 'A rational strip theory of ship motions', part 1 , Department of Naval Architecture and Marine Engineering of the University of Michigan, no. 013, maart 1969. Vugts, J., 'The hydrodynamic forces and ship

motions in waves', Deift Shipbuilding Laboratory,

report in preparation.

Semenof-Tjan -Tsanskij, W. W., Blagowetsjenskij, S. N., Golodilin, A. N., Book: 'Motions of ships: Publishing office 'Shipbuilding' 1969, Leningrad. Gerritsma, J., Beukelman, W., 'The distribution of

the hydrodynamic forces ona heaving and pitch-ing ship model in still water', Fifth Symposium Naval Hydrodynamics 1964.

Breslin, J. P., Eng, K., 'Resistance and seakeeping

performance of a new high speed destroyer

design', Davidson Laboratory Report no. 1082, 1965.

Smith, W. E., Salvesen, N., 'Comparsion of ship

motion theory and experiment for Davidson A

destroyer form', Naval Ship Research and

Deve'opment Center, Hydromechanics Labora-tory, Technical Note 102, February 1969. Smith, W. E., 'Equation of motion coefficients for a

pitching and heaving destroyer model', Deift

Shipbuilding Laboratory, report no. 154, September 1966.

Zunderdorp, H.J. , Buitenhek, M. , 'Oscillator techniques at the shipbuilding laboratory,Delft Shipbuilding Laboratory, report no. 111, 1963.

Nomenclature.

ab cd

ABCD

A w a n F F zA

Fv

F z F a F OA F0 g k 2Tr/A k yy L M

eg

EG

coefficients of the equations of

motion for heave and pitch area of waterplane

transformation coefficient wave force

force amplitude on the aft strut

for heave oscillation

forceamplitudeonthe fore strut

for heave oscillation

total force for heave oscillation

wave force amplitude

force amplitude on the aft strut for pitch oscillation

force amplitude on the fore strut for pitch oscillation

total force for pitch oscillation

Froude number

acceleration of gravity

longitudinal moment of inertia of waterplane area with respect

to the Y-axis wave number

longitudinal radius of inerita of

the model

length between perpendiculars

252 M z Ma Me m

m'

₀ N N'

N'

0 T t V xyz yw

moment for heave oscillation wave moment amplitude moment for pitch oscillation sectional added mass

sectional added mass at the stern

number of transformation

co-efficients

8

sectional damping

sectional damping at the stern

draught of the model

time P

forward speed of the model V

0)

right hand coordinate system

0)

half width of waterline e

z Za

heave displacement heave amplitude

phase angle between the motions (forces, moments) and the waves (oscillator)

instantaneous wave elevation

wave amplitude pitch angle pitch amplitude wave length density of water volume of displacement circular frequency