Equation of motion coefficients for a pitching and heaving destroyer model

Report No. 13h.

September 1966.

LABORATORIUM VOOR

SCHEEPSBOUWKUNDE

TECHNISCHE HOGESCHOOL DELFT

EQUATION OP NOTION COE1ICINTß 1OR A

PflCflItO A}D HEAVING DESTROYER MODEL.

E4ûation of Motion Coefficients for a

Pitching and HeavinDestror Model,

W.E. Smith

Abstract.

The equation of motion coefficients for a pitching and heaving destroyer model are measured using forced oscillation techniques. The forces due to waves on a constrained model are also measured. The pitch and heave motions in regular long created head waves are

measured. Al]. coefficients forces anti motions are compared with

reault obtained from modified strip theory computation. Agreement in all casas is excellent.

o

Physicist, David taylor Model Basin, Washington, D.C. at Shipbuilding Laboratory, Deift on research assignment.

-2-Intro due tion.

Formulation of the problem of calculating pitc) and heave motions

of a ship in head waves is now well estab3.jsbed. The motions may be ra-presented by a pair of coupled difSerential equations as developed by I(orvin-Kroukov8ky [i]. The validity o' cuch a representation waß

estab-lished in a series ot experinents by Qerritama [2][3] [i+] , in which each tei'*n of the equation waa meauured. The experiments emp1oed a forced

oßcjllation technique which permitted the meaßurement of the individual coefficients for a particular 8bip. The wave exciting forces on a

con-strained model were al8o measured. The experlmentakly derived coeffi-cienta and forces were then inserted into the equations and the

_motions

computed. These

computed motions were then compared with the results of a motion

experiment

in waves. Such resulta not only verified the

vali-dity of the differential equation representation _{but in addition} es-tablished the super position principle and the linearity of the motions.

Subsequent to this, methodswe developed which permitted the computer l.

calculation of the coefficients and thue the motions. In the process

of developing such a computer program, it was recognized that addition-al experimentaddition-al information would be of considerable vaddition-alue in estab-liahing the accuracy of ouch a calculation. Also, since the Gerritama experiments were confined to one parent form of modela, the 60

aerea,

block .60, .70, and .80, with the major empbaae on the block .70,

it was decided thàt experiments on an entirely different form would

be desirable,

The computer computation of the motions employs a modified form of strip theory which has inherent assumptions that are in some ways similar to the assumptions for elender body theorr. However, it should be observed that a ship moving with forward speed in the free surface

is not approximated by either a two-dimensional slender body _{or an} elongated slender body of revolution. In view of these somewhat tenuous similarities with slender body theory, and the posibjlity of futuro

analytical relationships which may be based on slenderness assumptio, a destroyer form was selected.

This form waS much more slender than the relatively broad ship

used in the Qorritama [2] experiments.. Also, since there were no large

longitudinal slopes in the forward section of the ship, it was

anti-cipated that experimental results from euch a form when considered along with the 60 Series data, would provide information _{as to the} im-portance of slenderness or longitudinal elope variations in strip theo-ry computation.

-3-o

3

Model Tested.

The model used for all testa was a conventional frigate hull of the Friesland class and waa one for which the motion characteristics

had been extensively investigated in ful), scale sea worthiness trials1

(Bledeoe, Buisemker and cunimins _[51

).

This model was constructed of fibreglass and was 2.&tain length. For all testing the mode]. was

bal-lasted to th. design load water line and Wga operated with a radius of giration of .25 Joa .259 L1,». This radius of gyration was selected

to coincide with previous full scale trial conditions.

Table 1.

Main Particulars of Ship Model.

L. centre of

_{mess M}

_{.0293 AP?}

Radiva of gyration pitch

_{.259 L»}

Scale ratio

Length L1, L

2.810 Bea !t. _.2935 Draft (DWL) M.

_.0975

o Displacement KG 44.55 Block coeff. _.554 o

Mjdship area coeff.

.815

Prismatic coetf. .679

Waterplane area coeff.

_.798

k

S1.

Force Ocillatjon Test - Heave.

The model was torce oscillated in heave using the Deift Shipbuild-ing Laboratory mechanical oscillator [io]. The model was attached to the oscillator b -means of two force transducers, as Bhown in Figur's i.

The oscillator employed

_{a Scotch-yoke mechanism to impart a constant}

frequency, &inusojdal motion to the model. The frequency _capabilities

of the oscilldtor' were euch that oscillation tests _{could be performed}

at any dj.aczeet

frequency between W=a and

W=

15. It was also possible to vary the

oscillation amplitude

_{and for this test amplitudes of .01 s,}

.02m, and .0km

were used. Heave test conditions are sumrized in Table 2.

Table 2.

Heave Oscillation

Test

Conditions.

Speed F _{.15, .25, .35, .45, .55}

Frequency range

_(J)OVIE

= I to ₈

Amplitude Z5 = .01, .02, .0k .

For this experiment the model was

oscillated vertically and

the

vertical force required to siatain teady

state oscillation was

mese-ured with a transducer in

the bow and stern of the model. See Iig. 1.

The sum of the forces of the forward and aft

_{transducers is the total}

heave

force, and the

_{difference represents the heave into pitch coupling} terms which is due to asymmetries in the hydrostatic and hydrodyn%n%ic

forces on the model. The force transducer outputs were connected direct-ly to an analog Fourier anadirect-lyzer which provided a direct indication of

the in phase and out of phase cómponeflt of

the first or fundamental

harmonic of the forces.

The higher harmonic content of the signal, it

5

Force Oscillation Experiment Pitch;

The ship zodel was force oøcillated In pitch only at a number of frèquenoiea and

ainpUtudea, as shown in Table

_3.

Table 3.

Pitch Oscillation Teat Conditjon.

The measuring apparatus was idential with that for heave.

Wave Excitation Force Experiment.

The ship model was rigidly attached to the carriage by two fore and aft mounted force transducers which permitted the measurement of

the forces exerted qn the

stationary model by the

incident, waves.

The waves were regular long-crested and were approximately L,/fO in height. Wave lengths were varied from Lpp/X ₌

_.5

to

Lpp/X = 2.0..

The forces and momenta

on the model

due to the waves were recorded on an ultra-violet strip chart recorder. Simultaneously,

the wave height

was measured using a resistance wire probe, mounted four meters for-ward of the model's

centre of

gravity and directly ahe*d of the mo-del. This data was also recorded on the ultra-violet strip chart

re-corder.

The information recorded was analyzed manually by averaging the

value for ten consecutive Cycles of motion, F'or the wave, height

meas-urement,

the phases relative to the forces

were adjusted to compensate

for the distance between the wave probe location and the model ceñtre

of gravity. Test conditions are as

hOwn in Table k.

Speed

Frequency range

=

.15, .25, .35, .45, .55

I to

_{W0V±1E = B}

k. Notion experiments.

The unpowered model was connected to a towing apparatum which was

so arranged as to reatrict all

modes of motion except pitch and heave. Al]. testing was done in regular long crested head waveS with a peak height of approximately L/k0. The wave heights were reduced at fre-quencies nears resonance to prevent the model froa shipping water. The wayS lengths were varied from .5 to 2.0.

Pitch, heave and wave displacements were recorded, for each test

condition. The pitch and

heave displacements were sensed by micro-torque rotary potentiometers mounted as part of the towing

apparatus.

The towing strut and motion transducers were arranged so that the re-straint forces in heave and pitch were negligible. The wave height was aenaed by a resistance wire probe located four meters

forward of the

model's centre of gravity

and directly ahead

of the model. All data was recorded simultaneously on a multi channel ultra-.violet strip chart

re-corder. }Iotiou information was recorded

only after the carriage and

mo-del bad

been running at a constant speed for a sufficient length of time to insure steady state conditions.

The information recorded was analyzed

mMually

by averaging the values of ten consecutive cicles of

motion.

For the wave height

meas-urement, the phases relative to

the

motions were adjusted to compensate for the distance between the wave probe location and the mode]. centre of gravity

Model teat conditions are ahowfl in Table

5.

6

Table k.

Wave Excitation Force Test Conditions.

Speed

Iave length ratio

Wave height ratio

.833,

.500,

.555, .625,

1.000, 1.250, 1.670, .71k, 2.000

-7 a

s

7

Thble 5.

Motion. Test Conditions.

_It.*

-8peed

Fn

.15, .25, .35,

k5, .55.

llav

length rutio L,/X

.500,

.555, .625,

.714e .833, i.000, 1.250, 1.670, 2.000

Wave height ratio 2 a/Lpp = 1/40

Each of the above tests were performed at the

speeds Fn 15, .25, .35,

.i5 and

o55i

5. Anal$ia Forced 0scillatio.

Selecting a atandard right handed coordinate system as shown4n Fig. 21, the equations of motion for pitch and heave in head Waves ara

(A+PVk2)+B+ço..]Z.E±.iGZ=Mcoe(wt+

_EM)

Inherent in auch a representation are the iaual assumptions of

super-position and that coupling from other modes of motion is sml1. For head waves euch a coupling assumption is apparently justified.

To expeiimentally evaluate the coefficients it is necessary to perform two linearily independent experiments at each frequenày' and measure the exctting force, moment and diepacoments.

Por simplicity of computation

the

two experiments can be designed

so that only one mode of motion is present in each experiment. The resulting euatioris for the heave

experimente are:

(5+pV)+b+cz

_{'ZCOS(Wet1fl E)}

D.+ E± +

= coB

(Wet + E

For a forced heaving

motion:

ZpW

GÇ

MZCQIEMZ - 2

a'o

-M

ainE

ZaWe

The pitch .xperiaent equations are;

d*+e+g9

-T0coa(Wt+

¡*+ßò+c9

_{M9coe(Wt+ E)}

For a forced pitching motion:

Q

*

Q 008W t

the

remaining coefficients are.:

¡

CMaO0eEMQ

2

QW

_se

B =

I4ain ENQ

gO+FQcoaEQ

d=

2 QaWe FQBiD

E79

aWe

8

the coefficients may- be expressed as:

a CZa - F cosE

-pv

Z

aWe F

sinE

-9

9

6.

Ana].sie wave forces.

The force and moment on the totally restrained modo].;

Fw

FaC08( W5t +

a00

Wet s.

This measurement then provided the relationship between the wave 8hape and the force and moment exerted on the model.

Discussion.

The oscillator experiment provided measured values for all eight of the dynamic coefficients (a, b, , e, ¡, 8, D,

E)

of tbe'equation of motion. The coefficients were measured for several amplitudes of

the motion; for heave, 10, 20 and ¿40 percent of the designed draft,

and for pitch, the vertical motion of the bow was 1+, 28 and 56 per cent of the designed draft. The coefficient values obtained for the

different amp].itude8 showed only minor differences nd would, to a certain extent, indicate good linearity.

It muet be remembered, however, that a Fourier analysis was

per-formed on all test information and only the first harmonio component was retained. Under such circumstances, the Fourier anaLyzer can in

itself act as a. linearizing device which could mask certain types of non linearity Therefore, it cannot be said that such an experiment is a complete verification of linearityv Such a final verification of linearity must of necessity await the completion of the analysis of a transient or similar oscillator experiment in which higher

harmo-nics are considered. The experimental resulta for the different am-plitudea are shown in the Figures 1 through lo.

The coeffioiets were aleo calculated using a computer program which employs a modified form of strip theory. This program uses the

tlreell [6] solution for a circular cylinder and the conforma].

trans-formation of the circular cylinder into shiplike form, Tasai [7] Porter [8] . For the computer computations two methods were tried:

(i) using a Lewis forni or three coefficient transformation of the

cylinder and, (2) a so-called close fit program involving an arbitra-ry number of transformation coefficiénts.

-lo

The Lewis form for three coefficient tranaformationa ia one which approximates the shape of the ship oectione with an elliptic curve which matchee tb. beam, the draft artd the area of the shiplike sectIon

ex.-aot1. While this is in general not a good approximation tor

ebip

sec-tions, it fitB the pazticuisr destroyer considered here very well. Therefore, auy differences between a Lewie form and cloae fit computa-tion should be sma3l,

The Lewis form computed values for the coefficients ¿uo eho in the figures along with the experimental reBulta. In every caOe

experi-meútal results

nd

computation agree quite well, with the beat agree-rient for the main added mase and damping term. The cross coupling tera generally chow good agreement with the 8jeed dependency clearly evident in the damping

terms (e,

E). The speed dependenol normaUy associated

with the

restoring force terms (g, C)!

which for

ease of analysis has been

arbitrarily included in the added mase term (i, ¡), la also

clear-].y evident in both

computation and experiment. The absence of speed dependency for the terms (a, b, B, D) is also clearly demonstrated.

The agreement over all appears to be considerably

better than that for th Series 60 block 70 data as reported by Gerritema [9].

Thisis eonEistentwith.s1ender body assufllption8 and indicates that such assumptions may indeed be applicable to surface ship computations using nodified strip theory. Assuming that such a relationship exists,

the satisfactory agreement between computation ad experiment for both

the Series 60 block 70 and the Freoland destroyer is an indication of the large deviation from a true slender body which are possible while still maintaining satisfactory computational accuracy.

The close fit or multi transformation coefficient program was also

need to compute the equation

of motion coefficient terms. The differences if any, from the Lewis form computations were small. Thin is not sur-prising since for this ship the Lewis form transformation is a good fit. While the differences from a ship deign standpoint are insignificant, it is interesting

to note that in every case where a difference occurred

the close

fit data showed

improved agreement with experiments. The close fit computation aluea, where different from the Lewis form computations, are also shown in the figures with the experimental coefficients.

The wave exciting forces and moments were also measured.

Theae are

shown in Figures 11 through 15.

Agreement

between the measured exciting forces and moments and computation is excellent, with only small devia.. t±on at

the higher frequency. A oomariaon of

the phase angles shows good agreement between computation

and experiment at low and medium

tre-uencies only, i.e. below L /

1.0.

-11-Agreement between experiment and computation for the motions,

amplitudes and

phaee angles is excellent. The only difference of any agnifioance occurs in the heaving motion at the higher wave frequen. cies, that is near 1.5. The motions, however, at this

frequen-cy are so small that this difference is not considered to be important.

Conclusions.

The ability of modified strip theory to account for forward speed eftects even to the relatively high Froude number of .5 is

demonstra-t ed

When the results from this experiment and the Gerritema Series 60 experiments are compared an estimate of the importance of deviations from the slender body assumptions is possible.

¶Fhe capabilities of a Computer program based on

modified strip theory for the computation. of pitch and hevo motions in head wavei is demonstrated0

The agreement between computed and experimental motions provides still another demonstration of the linearity of this

problen.

11

-Acknow1edement.

This work was ma-de possible through the cooperation and support

of the 8tudieoentrui T.N.O. voor Sobeepsbouw en Navigatie.

Partioulaz- appreciation

iB

expressed for an objective evaluation of the reeearch ¿dma of this project to Mr. W. Spuyzan

The exceUent computation assistance provided by the Wiskundige Dienst (Computer Department) i greatfilly acknowledged.

Timely completion of this projeoj was made possible by the enthusiatic assistance of the Shipbuilding Laboratory Staff.

12

Nomenclature.

a b o d e g

- Coefficients of the equations of motion

for heave and pitch.

A BODEG

- Block coefficient.

F Porce on mode]. due to forced heave motion.

Pg - Porce and model due to orced pitch motion.

Fa Vtave force amplitude on restrained model.

Fri - Fronde number.

g - Acceleration due to gravity.

Radina of gyration of model in pitch.

L - Length over all.

0e

-

Length between perpendiculars.

Ma - Total moment amplitude on model.

Wave moment amplitude on restrained ship. - Moment on model due to forced heave motion - Moment on model due to forced 4tch motion.

t - Time.

- Right-handed body axis system.

z - Heave displacement.

za - Heave amplitude

C

Phase angle between the motions (forces, monenth) and the

waveS. 13

X

-

Wave

length.

f3

-

Density of water.

V

-

Displacement of volume. W

-

Circular frequency.

W0 Circular frequency of encounter.

Q - Pitch angle. _{- 14}

- Instantaneous wave elevation.

L

- Wave amplitude.

R.feiencea.

I E.V. Koz'vin-Kroukovsky, W.fl. Jacobs.

"Pitching and Heaving Notions of a Ship in Regular Wave&', Transactions Soc ety of Naval Architecte and Narine Engineers, 1957.

2 J. Gerritema,

"Ship Motions in Longitudinal Waves", International Shipbuilding ?rogrese,

1960.

3 J. Gerritama W. Beukelman.

"Distribution of the HydrodyrLamic Forces on a fleaving and Pitching Ship

Model in 3till Water", Xnternational Chipbuildin.g Progress, 1964.

4

J. Qerritata, W.E. Smith,

"Full Scale Destroyer Motion Neasuremente" Laboratorium voor Scheeps-bouwkunde, Technische

Ifogesohoola-Deift,

Report No. 142,

_1966.

3 M.D. Eledeoe, O. Buseer,aker, W.E. Cummins.

"Seakeeping Trials on Three Dutch Destroyers", Transactions Society of Naval Architects and Narine Engineers. 1960.

6 F. Ursell.

"On the Heaving Notion of a Circular Cylinder on the Surface of a Fluid", quarterly Journal Meob. and Applied Math. Vol. II PT2 1949.

7 F. Tasai,

"On the Damping Force and Added Mase of Ships Heaving and Pitching",

eport of Research Institute for Applied Meohanics,Kyuehu University,

_1960.

8.

W.R. Porter,

"Pressure Distribution, Added Mass and Damping Coefficients for Cylinders Oscillating in a Free Surface", University of California, Institute of Engineering Research, Series 82, 1960.

9 J.

Gerritema,

«Distribution of flydrodynamic Forcesalong the Length of a Ship Model in Waves", Laboratorium voor Scheepsbouwkunde, Technische Hogesohool-Delft Report No

144, 1966.

10 H.J. Zunderdorp, M. Btxitenhek,

Oscillator-Techniques at the Shipbuilding Laboratory", Laboratorium voor Scbeesbouwkunde, Technische Hogeechool-Deift, Report No. 111,

_1963.

14

6

o5

4

3

2

i

o

FO.15

r

o

.

-.

FriesLand

cotcuated

o exp.:

.4

calculated

cLass

ampL. = (101

O

cLose

Lewis

fit

forn

m

I2

,,

f334

b 1

o

-i

2 3

4

5

6

WE'L

= 0.15

p

5

C L

3

i

-/

/

/

/-/

r

-fi

s o

/

fi

cl,

o

¼ - TI -ci 'I

2

3

WEV

6

o

u

0. 0.2

L/4

0.1

o

-soi

0.2

0.3

o 1

2

3

F,

= 0,15

WE"LL

s

r

u

-Friestcnd

caLcuLated

SPt ciii

exp.: a

cLass

ntd I pwi.s

"npL=O.O.l

0.02.

ccse fit

form

rad

o

_______ H

0.04

's

'o

u

o

10.2

e

VL

-0.4

-0.6

-0.8

-1.0

A-a

e

/

/

/

/

/

s--F0.15

2

D

:

D

2

3

4

5

6

Eg

1

₇

i

s

6

5

a. p"

i

L

2

o

o

I

FO.25

L

2

5

Eg

o

s

0.3

f

0.2

vg

L/L

0.1 O

0.1

0.2

- 0.3e

/

A

0

e

u'

's

EJ

u

A

a

u

A

a

flO.25

P14-I 2

hA

4 5

6

1

ei

10.2

e

0h

s

06

0.8

1.0

1.2

jO

/

I

I

I

I

I

¿

I

I

¡

/

i"

-w

D

s---.

- s

i

z

F

0.25

.

-Ç/: 2

P

L.

6

s

o

3

2

i

t

t

o

F= 0.35

WEV

6

Q

5

b

4

2

i

o

o

D D

/

I

I

/

I

/

/

K

'o

o

s,, s

3

= Q.35

s

n

0.3

¶OE2

av9

L/4

0.1

o

01

0.2

o

Q

u

C

s

u

-n

F =0.35

.-u

G n Q

s

u

2

3

4

56

O

i

7

o

f_02

VL

-0.4

-0.6

-0.8

-1.0

-4.

wV

_Eg

F.

=

0.35

I

5

3

i

o

o

FO.45

u t U Sb A A

i

4

WE\L

._!._,...

a

s

b

V

4

2

i

= 0.45

A

I

o

A

A

'4.

7

D

u

o

o

i

0.3

0.2 0.1

-0.2

o

n

A

2

5

Eg

0.45

I

u

-0.6

-0.8

.

-1.0

o

.

/

/

/

/

/

/

j

/

-p-.--

-/

u

_u

3

=0.45

u

(.

i

p

4

3

o

o

& t

3

5

wV

_Eg

O.55

I

Ó

b4

F= 0.55

I.

s

o-5

L

I

i

L

A

A

/

/

/

'-J 'I

/

I

I

I

I.

/

\

V.

A I

'

t

-

--o

i

2

3

4

5

6

7

Î02

VL

-0.4

-0.6

-02

-1.0

-1.2

o

/

/

/

/

/

¿

I

/

/

/

I

-o

u

Eg

F0.55

o

a

6

.

3

2

i

o

o

Friesland class

calculated close fit

calcuqted

_{Lewis_form}

o exp.: arp1.= 0.01

rQd,

'I I

002

ft

0.04

i

F

=

0.15

t

5

0.6

BVL2

0h

0.3

0.2

0.1

F0.15

L

/

L

L

D a a a

As

R

flr

1

2

3

5

7

Eg

4,

t

o

DV9

1/4

2

3

4

F

=0.15

3

6

c. 1F

-

i,

_-D

---.

FriesLand

catcutated

calcuLated

xp

class

ampi =

O

close

Lewisform

fit

fi m

0.0

ûOL

2

FI-I

0. 25

u

I.

F4-

YB

5

2 '4 :3 2

i

n

-.

o,

-A

-

Oo

-n -

-i

2

3

4

5

6

7

o

1;;

L/4

-2

-3

4

s

6

w,

o

F =0.25

6 7

2

3

i

o

4

0/.

VL

VL

0.2

01

-0.1

-0.2

0 1 2 3

4

5

i

Eg

F

0.25

I

u

t

t

t o

u?

w

:

j

u

6

5

Iu

lo-+4

'o

3

i

o

a b

o

-FO.35

WEV

o

i

2

3

4

5

6

7

2

e

0.6

0.5

VL

BVL2

0.4

)) B

a..

o

o a

s

wV

_Eg

FO.35

6

£

a

Çic:

88

a O

3

0.3 0.2 0.1

o

t

-2

-3

-6g

o

=

0.35

/

n

4 5

wV

_Eg

s

9e

01.

01

o

-01

-0.2

o

.

s

t p'

I

p' t t t t p'

t

t

t

t

t

I,

t

t

_a

s

/

D a

2

45

6

w'LE

_Eg

=

0.35

a

s

s

f

2

o

5

3

Q o o

F

=

0.45

5

6

I

o

i

2

A

s

i

L/4

-6

o

=0.45

6

3

4

5

Eg

I

2

3

4

5

Eg

F

0.45

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