48(昭 和 40 年 11 月 造 船 協 会 秋 季 講 演 会 に お い て 講 演)

On the Measurementmeans of Forcedof theYawing

Stability
Derivatives
Technique
By Seizo

byMotora,

Masataka

Member *Fujino,MemberSummary
In thisthepaper,

yawingtheauthors

oscillator

describe

theand measuringforcedsystem,yawingtechnique,especiallythestability

derivativesof a shipby whichtheby thismethodwiththe ones obtained

in

low

frequency

mechanism

ofmodelaremeasured.
In orderarm

to compare

technique,of thesethetests,

theto getderivatives

authors

carrieda fairlyIn addition,
order

the

well

authors

data

more

obtainedout

agreement
discuss

accurately

thethatwhatandeasily.

1
In dealing
it becomes
forcedwiththenecessary

yawingin regular

towingof depth

controlof a surface

be stated
was

later

of oscillator5)
been

A shipauthors

intendoftechnique
to show
and thea co-ordinateastechniquevaluesbasedtheones

at theseries

obtained

Principle

fixed

at thefollows

model inor disturbancecausedby waves,
purpose, adependents.

For

thistechniquealso has a meritthat

it is applicable

has

used

been

atDTMB1)

to measure

60,Leeuwen4)

derivatives

one which
University

wastheIn thisof Tokyoof Technology
betweenby theofthethebyandhasstability

wastestedstabilityarm

forcedyawingas shown

Motoratherotating

C. G. of a ship,

in moreas functions

used

Cb=0. 60 of which

Institutecorrelation

2and Vanof the stabilityon theat Stevens
theto a ship

for thestability

case,problems

derivatives

only

a combined

used.

by Paulling3)

constructed

model

thewastested

As a result

to be given

as frequency

Thisof Michigan2).

has

expressed

yawing.

was applied

arm

Using

derivativessuitableUniversity

rotating

ismovement

techniquecan give reliable

techniquerudderThistheyawingareat thewill

type

stability

forcedof motions

waterstechniquehis techniqueTherange.

recognized.

Introduction

to periodical

mostwasbodies.

whichof 1964.

theis thought

kinds

data

in restrictedsameof Michigan

to evaluate

bothof submerged

ship

The

A refinedof ships

tanks.

motiontresponse

techniqueexperiments

betweenby the rotatingrefinedways

of thecircularand

been

thatfrequency.

Couch at the University

working

derivatives

since
wereby this oscillator.

derivatives

showingobtained

the middle
obtainedby theforcedtechnique.

techniquein Fig. 1, a linearized

equationof motion:(1 )whereU : speedof advanceβ:driftangle

原稿受付*

昭 和 40 年 7 月 10 日

東京大学工学部

byIn this paper,

Fig. 1

Co-ordinates

SystemOn theγ:yaw

Measurement

angularof thevirtualmass

toIz+Jz:

virtualmass

momentY : hydrodynamicforce

longitudinal

In thismotion

periodically.
as Yo and No,

anddistanceswayingy-axisofforcemomentYo, No ; external

PurethelateralN : hydrodynamicα:Derivatives

by meansof Forced

Yawing49

Techniquevelocity

m+my:(i )Stability

direction
inertiaon the

about

ship,

z-axis

∂Y/

Yβ=

∂β

on the

ship

about

z-axis,momentdeliveredto thebetween

the

∂Y

Yγ=,

/∂γ

∂N/

Nβ=

ship

∂N/

Nγ=,

∂β
by the∂γ

oscillator

centreof addedmass,

my,and C. G.

to sway

without

yawingmotion.

of theship

motiona shipmodel

Denoting

theand thephase

is forced

force

and moment

shift

which

areof Yo and No to thenecessary
motionTherefore

fit onlychangestheon thismotion

to confine

as e1 andship£2 respectively,

we get :(2 )Substituting(2) into (1), we get :(3 )Equating (2) and (3), we can get the in-phaseand out-phasecomponents

of Yo and N0.

(4 )

From
to(4),

wecangetm+my,YƒÀ, NƒÀ, myƒ¿

for

the

specified

frequency

at

which

the

model

wasforced

isalways

kept

tosway.
( ii )
In

Pure

thisyawing

motionmotionthemodelisforcedtoyaw

andswaysothatthedrift

angle

8be

zero.
Therefore
γ=7cosωt,

Substituting

β=(5)0(5) into (1) we get :(6 )Similarto thepreviouscase,

we get :(7 )(iii ) Combined motion
All the unknown terms in the equation
found to be quite delicate
Therefore(1) can be obtained

and contain many causes of scatterby (i) and (ii).

motion.

In the combined

which runs on a straighti. e. :
γ=γcosωt,β=βcos(ωt一Substituting(8) into (1) we getπ/2),γ=βωisof data.

a combined motion is tried to use in place of the pure yawing

motion used in this case, a model is forced to yaw around its center of gravity
course.

However the latter(8 )造 船 協 会 論 文 集

50

第 118 号

(9 )If

m+my,myα,Yβand

set-upforcedSwaying

MotionPure

YawingMotionCombined

MotionDefinition

ofof
toand

theyaw

350

mm

and
give

theyawingtheperiodicalasthebed (1)slidesandswaythemaxa" purezero.

If

theTo

give

a"pureTherefore

astrokes ƒÅsand ƒÅy

motionsyaw

yawingmotion",adjuster (8)
theis

as

given
is

is

scotchintheFig.

and‑(m+mx)

in Fig. 2.

3.

Apparatus

InFig.

3,

shaft (11).

wholeset-upyokes (7)isandaship

TheisPhase

model

model

mountedand (6).

adjustedthe ƒÀ

necessaryand (7)intoforof (12)in Fig. 3.

the eccentricity

Iz+Jz,Yγ,Nγ

get

Analyseronattached
istheMaximumto•@(3)

forcedtowingswayingtoswaycarriage,
amplitudeisto40•K.

providedyokes (6)canYawingverticalscotchonly

isForcedThedenoted ƒÅs,itweoscillator

ofbytest,

are shownshown

transversely.

wasWhere a is the arm length
stroke while

Yawingmotion", (6)of (6)sway3

rotation

amplitudeswayingofpureFig. 3

arestrokephase

the(ii) and (iii)oscillator

bybespecifiedbyMotionsyaw

simultaneously

To

given

PureFig. 2

TheareShip motions given by (i),×U+Yγ.

isNβtotheaspecified

equation(2)

adjustthisshould

thepurpose.
satisfy

phase
In
the

To give a combinedof (6) is kept zero.

strokeisbetweenthewhile (7)isadjustedasfollows:

expressedcasescotchofyokes (7)and•@(6).

yawingmotion,pure

following

relationmotion,only (7):is adjusted

to beOn theMeasurementof theStability

4
To eliminate

the high frequency

harmonic analysistechniqueby means

Measuringnoise whichsimilar

Side forces Y1 and Y2 acting

Derivativesof Forced

YawingTechnique51

systemmay be containedin the measured

to the one used by Van Leeuwen

to the model at the points

forwardforce or moment, aand Tanaka6)

is employed.

and after the C. G. of the model by

distance l are picked up by linear transformers.
ThenSide

force

Momentare produced

through

Y0=

Y1+

N0=l

(Y1‑Y2Y2)

an analog

computor.

TheseYo and No are fed as the input

Then

weY0getto sine-cosine-potentiometerssynchronizedwiththe ship motion.

cosωt

Y0 sinωt
N0 cosωt
N0 sinωt

asoutputs.
Integratingthesevalues

byananalogcomputor,

wegetforinstance;(10)(10) is a straightline of tangent -Y0COSε1

superposed/2

bya sinusoidal

the term we like to obtain is Yocos ei as shown in equationof (10) gives the answer.

In a similarway,

we can get the stability

N0sin

derivatives
5of ship's(4), twice of the tangent

2co.

Sinceof the mean curveε2.

by equation

Results

A 2 m model of series 60, Cb=0. 60 was used.
The radius of gyrationof frequency

we getY0sinε1,N0cosε2,andTherefore,

curve

K of the model is adjusted(4),(7) and (9).

obtained
Dimensionsof the model are shown in Table 1.

to 0.20 of ship length

to get more accurate

value

added moment of inertia.
Table

Stability

derivativesof Technology7).

areobtained

1

Characteristicsof thismodelby theof therotating

Modelarmtechniqueat Stevens

Institute52

造 船 協 会 論 文 集

Fig.

Fig.

46

my'一

ω' curve

Nβ'一 ω'curve

第 118 号Fig.

5

Y'β 一 ω' curve

Fig.

7α'一 ω' curveOn theMeasurement

Fig.

8Fig.

10of theJz'‑ω'Nγ'‑ω'Stability

Derivatives

by meansof Forced

curve

curve

Fig.

Fig.

119(my‑mx)

YawingYγ'‑ω'U+Yγ

Techniquecurve〜 ωcurve53

造 船 協 会 論 文 集54

一 ｍyαU‑Nr(kg・m・sec)

Fig.

Foris

Fig. 13(Iz+Jz)

conditionofthewithrudderandFor

therevolution

FortheshipAstothespeed,magnitudes

12‑

myαU‑

Nγ〜 ω curveFig.

ω+Nβ/ω 〜 ω curve

themodel,twopropeller-wereof

第 118 号

conditions-one

iswith14rudder

myαandω+Yβ/ω 〜 ω curve

without

propeller

derivatives

propeller,

aFn=0.

10

and 0.

20

andr,

the

followingareasshown

inof ƒÀobtained

theotherchosen.

the

self

propelling

weretested

conditionforwaseachchosen.

condition.

rangeswerechosen4•`14,

where

symbols

to

keep

the

mean:

range.

Stability

andFigs.

motioninalinerOn theMeasurementof theStability

Derivatives

bymeansof Forced

Yawing55

Technique(11)As it is expected

thatvalue of thederivativesame

thevalueof a stability
fromfrequency,

frequency

wereextrapolated

tocomparedwithones

wereas shownin Table

which

zero

From

Table

model

that
pure

from

theandIn

examiningmostapparent

detectthepurebilityofandarebination
Infor

view

This

tests

derivatives

Cb=0.
pure

coincidewith

theof thederivatives

vs.

thus

obtained

werefrequency

at Stevens

Instituteof Technology7)

scatter60

ofDerivatives

obtainedbyagreement
model

yawingyawing

is

data

also

fromveryrotating
therecognized.
show

ontainedisthebetweenmotions

motionAeffectof ƒÀerror

causedmaycorrectsmall
bebyof Iz+Jz

and2,valueswith

difficultchangesscatterpureyawingrealizearm

techniquederivatives

In

more

toin

contained

lowU

for

Figs

4•`14,

thane.

it

and
Cb=0.

70

can

be

it

the

ones

motion

;i.

will

the

data

as

it

will

be

easilyfrequecy

measured

or ƒÖ

in

existing ƒÀintheofderivatives

ofrange.

valuesobtained

mainly

is

cause

very

the

the

comes
difficulttomy,fromadeviation

fromerror.
seen

that

the

Therefore,

using

kinds

in

adjustingdifficulties
motionandswaying

fromarerecognizedthreeYƒÀ anditNƒÀeffect

will

which

the discussion,

obtainedmotion,andofswayingofvery

set

ofcombined

twosetpureobtained

bythosefigures.

combinedThis

derivativesofToof ƒÀ

be

wiser

are

obtainedandis
tomotion

as

shown

that

we

yawingincankinds
theofrelia-myƒ¿ƒÖ+ YƒÀ/ƒÖ•@ arepureshows

three
check

andmotion

motion,Figs.

pick

11•`14.

upany

com-motions.
for

the

pure

motionlarge

yawingexceptcomparedmotion,
for

with

very
inertia

it

will
low

bewiserfrequencies

tochoosea combinationwherehydrodynamicmoment.

Conclusions

it can be stated

It was found that the forced yawing technique combined with harmonic
data of stability derivatives even in low frequency range.
Stabilityout

motion.

(Iz+Jz)ƒÖ+ain

combinedisbeyawing(my -mx) -myaU-Nr,

which

bewillpure

obtainedthosewillout

bythemotion,

derivatives

agreementtwoallswayingtheSummarizing(2)

at zeroStability

reasonable6(1)reliablevaluesshouldcurves

motion.

comparedcausedforfrom

pureof ƒÀ

Chaptertechnique,momentfrequency

technique,armof theofA

measurement

value

fromofvalues

andperfectthethisswayingwell.

thepurepurethatpossiblei. e.,

oftheby rotating

perfectly.

theinReasonablesaid

obtainedconsequently,calculated

and

arm

ηsω/U=ηy/arelationswaying

at zerorotating

Comparison

fairlyLeeuwenaactualobtained

2motions.

that

statedmotions,

bepoints

swaying

theAs

Van

data

fact

β=0,

by

can
agree

by

thefromsatisfyitmethodobtainedseenof2,present

the2.
Table

the

derivative

obtained

obtainedby the forced yawing

techniqueanalysisas extrapolated

can produce

to zero frequency

造 船 協 会 論 文 集

56

agree very well with the ones obtained

第 118 号by the rotating

arm technique.

( 3 ) The added mass, added moment of inertia and the stability
combination of two out of three kinds of motions ; i. e., pure swaying

derivatives are obtained from amotion, pure yawing motion, andcombined moton.

motion and combined motion is

And it was found that a combinationeasy to conduct and also produce more accurateof pure swaying

data,except at very low frequency.

AcknowledgementThe

authors

experiment,

likeandtoacknowledge

usefulladvicethegiven

usefullbyDr.

helps

givenby

TanakaofShip

Prof.
ResearchInuiandhisstaffin

conducting

Institute.

References

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

Goodman A: " Experimental
Technique and Methods of Analysis Used in Submerged Body Research." Third Symposium on Naval Hydrodynamics,
Scheveningen,
Holland, 1960
Motora S. & R. B. Couch : " Maneuverabilityof Full Bodied Ships in Restricted Waters." Presented
before the Fall Meeting of the Great Lakes and Great River Section of the SNAME, Oct., 1961
Paulling R : " The Dynamic Problem of Two Ships Operating on Parallel Courses in Close Proximity." University of California,
July 1962
Van Leeuwen G. : "The Lateral Damping and Added Mass of an Oscillating Ship Model" Technological University-DelftPub. No. 23, June 1964
Motora S. : "Forced Yawing Technique to obtain the Stability Derivatives as Frequency Dependents" Appendix 3 of the Comm. Report. 10 th ITTC Sept. 1963
Tanaka H. & Kitagawa H. : "On the Study of Characteristicsof Ship Motions by a Forced Oscillation Method." Journal of Zosen Kiokai, Vol. 111, June 1962
Eda H. & C. L. Crane : " Steering Characteristicsof the Series 60(Cb=60) " Davidson Laboratory
Report 923, Oct. 1962