The Society
The
SocietyofofNavalNaval Architects
Architects ofof JapanJapan

9. 0n the ModifiedZigzag Maneuver and

its Application

andSeizo MoToRA*, Member
Masataka FuJiNo",A4izinber

JaPan, Vot. 128 Dec, 1970)(From I,S.IV,A,

Summary
In caselessstableofin conducting
to examine

zigzagordinary

the

ships

ships suchor unstable

at amaneuverkeeping qualities

courseofsmall

rudderangle,

the authors

ships,

such

fu!ibodied ship,

as

say

there are diMculties
5 degrees. In orger

proposed a modified

zig-

Zn.8.he,,2e,spll[g,SS,i/rPg,gi?h,:d,:;{O,:l,,2r,di,"e:,gdhit"he,rtd:,E: {:,;nM,: i;ml.SZa:2.:,:,1 ;;
liZt,yl £ilc,Szi
ieiillairlel
l,i,i,i:ln,ldli.to,
11:////cd
b:TS,/[.8alp,3,,M.Z",2",Y8.rk,:'ttZh,
'

M

it,
Z

,es

,

Introduction
In case ships
quality, thewithstablemaneuverzigzag

with

couse

keeping

proposed byease

It has been aangle.

rudderra e "a.>i,g

e

Or

even

at

usefulof ships
the maneuverability
systemof the firstorderby rnakingusemethodis notthisanalysis2)3).
However,
fordeterminingthe maneuverability
suitableships, because theor unstableof lessstableofbecomes rernarkabledivergesin spite of the corn-

such
or

ships

frequency

to determmation

functionofresponse

steering.

1. Modified Zigzag Maneuyer
1.1 Dithcultiesin executing
in casezigzagmaneuverto searth

trajectorye,wlilt:.

neuver

Kempfi) can be executed
method

ac" k

"t,hi

functionof maneuverability.

response

small

O

or

unstable

theofnormallessstableships

As stated in the introduction,there are
the norrnal
some
diMculties in executing
zigzagrnaneuver

at

a

small

rudderangle,

to say 5 degrees,in case of less stable ortankers,
as full super
ships
such
unstable

because the trajectory of ship in process ofdiverges to infinityor
maneuverthe zigzag
if it were not so, the differenceof the course
anglefrom the original eourse grows so large
is unable to be
maneuverthat the zigzag
reFig. 1 shows the qualitative
isdis- continued.
which
zjgzagmaneuver)the modifiedb'angle
lationshipbetween the rudder
maneuverzigzagtinguished from the normal
andtheamrnaneuverusedat the zigzag
angleti"is notin the repect that the rudder
as theaswellasymmetricmanever
pensatory steering so that the zigzag
of
the
sake
For
is unable to be continued.
have
the dificulty, the authors
renioving
type of zigto
executea different
proposedbe called asitwill
zag maneuver(hereafter

angle
plitude & of courseT'(=distancerunlship's length) of
period'is reyersedO.
the rudder¢ at which
limit cycle in both cases of stable and unIn this paper, the authors shall again in- stable ships. As shownin fig.1, itis impostroduce the tenor of the article 4)with somezigzagthe normal
to executedescribe sible at all small rudder angle in case of
together, and moreover
examplesatmaneuverzigzagmaof the modified
the applicationandeven
in case of lessships,
unstable
it is almost impossiblefrom theships
stable' EngineeringFaculty of Tokyo
University
alwaysequal

to theswitchingcourseangle

133

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Masataka FuJINo andSeizo

134

t.
i'l2
i;aj
I,,.
t

'

S

O

i,/il

t

L

MoToRA.

':"'.Ill

,

5ti

f"ot

.e

4

-1

OO

10.tlg!"3i-e.

:Odtstanec eetwee41t"m ee res

practical point ofca}s-s1"104-Et-tgntO

Fig. 1 Amplitude of head angle
d (=TUIL)

e and

l3S
.y

periods'+8fZ-s'-td-tsYte'Ss:

modifiedm-s"to' 4

zigzagzigzagunstalbe

maneuver,

ships

can

even

be steered

thelessstableondithculty. The

without

a straightreasondo not develope to create unstablehydrodynamic
moment.
Hence, the ships
can quickly responseto even small rudder
angle.

As an example,

maneuvertests onthe prescribed unstable

ship whose

equationofthe results
motionofofcourseisreprsented

angle

grows

gradually larger with the Iapseof time incaseof 1 degree switchingcourseangle,
while
it reaches to a steady state, that islimitcycle, in case of that lessthan O.8e.
The ship motion
atthe moditied
zigzagseems
motion

the

with

small

switchingnorrnalzigzagmaneuver.

have proposed

' .]L

/

t6

2e

Lg 4j
±

--(d)5-Q6-yt,ess.
T-s'-to'

8

20

-

T'

--Ce)s-o.4"he'S5'

.

-5
-to'

T'

i216

(t)5LQ2'

Fig. 2

Simulationof zigzag

manettverofequationusing

non-linear

in case(11)(Ti'

7le,=O.485, Th,=O.895,=-17.75,K,-9.15, pf==-8.6)
suitable

to execute

maneuverin orderthe modified
zigzagto examine
the course･
ships.

keeping qualities ofIn the followingparagraphs, we shall comkeeping qualities obtainedpare the courseby the first ordersysternanalysisof thenormalandthe modified
zigzagmaneuvers
with

each

other.

1.3 The airnof the firstordersystemanalysis

Comparing with the results obtained
frorn
the firstorder systemanalysisof 150-150
thannormal,and5e-50 normal
5e-le modified
Therefore, zigzag rnaneuvers(theformer of the numthat itis morebers connectedby the symboi-means
thecourseto be more similar to the actual
at the course
keepingrnaneuver

the authorss:-s'-td 4

zigzagby a second order nonlinear equation (11),
are
shown
fig,2 (a)-(f)at whichthe rudder angle
courseis kept 5 degrees while the switchingangle
is varied from 50 to O.20. In this exthe change@-an2IiFfr

will

velocity

maneuver7'

Vt1O.Sorthebe that at the course keeping maneuver
direcrudder
isreversed to the compensatorytion before the deviationof course angle
grows larger than 1 degree even in case ofmanualsteering;
muchless the deviation is
in case of automatic
steering.
Therefore,
the angularvelocity
as well as the drifting

ample,

4e(c)s-o.ein executing
the dificulties

of

u-･

--

maneuverIn spite

i6
.(b)

5-1

view.

1.2 Proposition of

cour$e

20

16

Iiii[

10

--

Oe

normal

v

s

1

angle

ship

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135

its Application

On the Moclified Zigzag Maneuver andissmaller than the latterby the dotted linein fig.3.
one
as shown
andthe latterdoes the switching
maneuvers
wecandraw the inference The less stable a ship is, the larger thecourseang!e),
differencebetween the solutions of eqs. (1)that K, T indicesfrom 50-50 normal zigzag
largerthan those and (2)is.
maneuverare remarkableNow we assume that there exist two ships,
from 15n-15e normal one, while K, T indiceskeeping qualities areareof whichthe course
zigzag maneuver
from 50-le modified
andby eqs. (1)and (2)respectively.
represented
smallerthan those from 150-150 normal
angle
with
the records
of course
This
seems
Comparing
zigzagmaneuvers.
5o-5enormal
amountofto coincide

the rudder

withexactlyofresponses

lessstable

modified

such

angle

one, is virtuallystable

and

ships

such

as

the fact that theships
toor unstable

fullsupertankers

without

50-leto thosesirnilar

therefore evenareso muchlinearequation
T, Th

as

forrnersolution

andruddersuch

able

at the

angle

of

?

5

to

v !.eg.")-ss/eg,C2)

..

x--

Ns

XN

e'

ti

x

it

1
t

Nl

N,

L..t

N

representativeZ3,g+ -E,ldil. (z + Tli)-ZllZ-

Fig. 4Comparison
first order
order

these

two

whose

equation

betweensystemzigzagmaneuverof

andadvances

ships.

and

that

of

secondsystemNamely, in case of ships
isdescribedby eq.

of motion

the rudder

the tirning of reversing
(1),

(1)

=K6+KT,ft

maneuver,in fig.4 by thedifferencesas shown
dottedlines may be found between

follows.

dozigzagsolid and

unstable

keeping quality. For the sakeof the courseofthat the equationof brevity,we assumebythesecondmaybedescribed
motionorder

thedithculties.
the aim of thein order to decidebeThen,'we shall examine
analysisordersystemfirst
are wellT
indiceswhich K,
steeredasmaneuverzigzagmodifiedships,

at zigzag

used

the overshooting

coursethose of

angle
angle

of
decreases
Tindices of
eq. (2).Therefore, the K andhere follow thethe syrnbols
used
where
than
described by eq. (1)are smaller
ships
of
order
approximation
article 2). The first
and
hence
the
forrnerbyeq.
those of ship(2),
this equation
proposed by Nomoto2) iscoursekeepstableships
seern to be of more¢-e,(e,ing quality than the latter ship.
Kb
T'Zll
,- +
exaggirated
This tendency will be muchcourseangle
decreases.In
T== T,+Tle-A(2) as the switching
to distinguish the K, Tindicesobtainedorder
The solution ¢ (t)of eq. (2)will agree exact!yanalysisof thefrom the firstorder systemwiththat of eq. (1)after the lapse of inof ships
diescribedby eq.
zigzagmaneuverfinitely
long time, while at the initialstage
let'suse the symbols(1)from those of eq. (2),7' values
of
These
K'
and
K" and T'.
iftheanaloguefromshipobtainedstablezigkinds of modified
simulationof various
figs.
5
and
6.
shown
in
maneuvers
are
zag
K'
Tb'andThe prescribed values of Tt',7'li',
O,893
are
12.68,
O.420,
atthese
simulation
used
t
of
The
K
and
Tindicesand 5.82respectively.
Fig, 3 Comparison between the response of firstexactly
to the K' andthe eq. (2)correspondandsteering
order
systern to stepwise
modifiedzigzagfrem
theindices
obtainedT'
sy$temerder
that of second
compared

with

ships

=

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136

Masataka FuJINO

maneuvers

infinitesimal
rudderoflarger theangle

rudderis, themaneuvervaluesare,caseof smallsmallerThe

angle.

the zigzag
K'k andT*oftheThis tendency is remarkable

1,O

in

Q'/6'

ratio.

The K" and T' indicesobtained from thevarious kinds of zigzag
maneuvers
areindependent of the amountrudderofangle o""and are constant, only ifthe ratio ¢ *f6* remains
unchanged.
But the K* and T" values
do not coincide with the K and T values of
eq, (2).
This differencebetween the Kts',T"valuesandthe K, T values increasesas thecoursekeeping quality becomes worse.
The
K; T values of unstableships are always
negative,
but there may exist the positiveK*andT' values in case of certain small ¢ '/6"
ratie.
Therefore such modified
zigzagmaneuvers
are able to be executedevenin caseof unstable

Seizo MoToRA

and

to.2s
#vp-.
o.s

r

eXl!?C'oO5

10

Fig. 6 K" of a stablekind ofig the purpose

the

of

zigzag?

maneuyerof the maneuvering
teristicsof ships?
representative

Principal modesof steering

motioncharac-ef ships

be classified in the followingitems,whicho{prhvsc.

.5-rs

:

'

O.4;k

!COURSaKEEPINGee--ptIS"-O...xb"t5'o------r]--e,5I2e72S'>D'le1.0

D.2
2'NORALz''/3iL-2.o--7-Ttt'EMERGENCY'

le11T'
O.1

L-h'COUR5E--CHANGE1

o

o510

.

T* of a stablekind offrom various

obtainedzigzagmaneuvers

6 qualitatively.

coursekeepingmaneuverb)

coursechangernaneuverc)emergency

maneuverofmodified

2D

30

40

S\ (deg)ship

obtainedzigzagnormalzigzagand35e respectively,ofsteering

150 or 350 are

c)

are

of fi*
because in these
the rudder

motionfrequentlyusedcoursegrow to the similar extent
of the used
rudderangle.
Accordingly,the K' and

as

the

obtained

to sayfrom such
l50-150and

be well

representative

normalangles

kept

and

of

until

well

maneuvers

the changes

unchanged

angle
amountsT" values
zigzagthus

maneuvers,

350-350 normal
ones,of the maneuvering
the items b) and c).

will

of
On
hand, in the mode a) the helmsman
endeavors
to keep the changeof course
angle
withinlessthan 1 degree by steering
the rudderof more or lessthan 5 degrees.
Therefore, the modified
zigzagmaneuversofsmall
e"fO"ratios, for instance se-lo
characteristics

the

other

modified

O,3Fig. 5

=150

40

a)

by

modes

As stated above, the K* and T' values
determined from the modified
ornormalzigzagrnaneuvers
are usuallydifferent
fromthe K and T indicesof the first order system analysis. Then, whichindicesare best

will

30

Among these items, the b) andships.

What

ship

modifiedin figs.5 andare shown

EMERGENCY'

CHnNGE

20s' Cdeg)-

covered

1.4'

COURSE

one,

are

more

adequate

and

more

zigzagpractical than the normal
maneuverto examine
the course
keeping quality of･The K* and T' values obtained
ships.
fromthese modified
zigzagmaneuvers,
as statedalready,aresmallerthan those from thenormalzigzagIn other words,
T' values suggest more
keeping quality than the lattermaneuvers.

the former K' andfrom various
maneuvers

stableones

coursedo.

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On the ModifiedZigzag Maneuver

137

itsApplication

and

:,o

20't'(deg)

1
io

10

o

o

10

S

?O

15

O5

-oft!g!g!Cd)b)a)Fig. 7

Amplitude of head angle

di

kind

maneuvers

modifiedofperiodandzigzagcarried1.5 Conditions for the existance of stable}imitcycles.
Even in case of unstable ships, whose norzigzag

rnaneuvers

not be executed

cahbecause of divergence of their course angle,
if we choose an adequate small switching

di'comparedangle

course

with

angle

rudderbeO", the modified
so that on the phaseable
to be executed,
trajectoryof steering motion
theportrait
zigzagwillof course

amplitudes

withrudder

angle

angle

di*in casestab!e limit cycles

fi"

of

unstableanis nonlinear.
shall

ship

whose
our

as the equation

the switchingwith

ip*are able

to be executed,

case,

small

to sayofwith

Therefore,itwillthat there

maneuverangle

in the re-e"*.compared

they are unrealizable.
ip',
value

coursewhile

existsabeyondip'1fi"of

Vt{:}.
--th-

(71'

ship

rnaneuver,e[e)the symbols

whereK{1+liS)5{oe[orifio'ptM.

i

vs(t)

s"+Is)"+ns)

1'S'

L-

sh,p

ManeuLer

M9zag

Fig. 8 Block diagram

ip'andHereafter ing

concentratecases

comparison

zigzagthe various

S(t)IE.

ofis linear.From the studies statedabove,
it may be deduced that the zigzag
ti' inlarge rudderanglemaneuvers
with

the

at

unstableofzigzagmaneuvercourseof rnotion

criticalanzigzagmodifiedarecombinations

the switchingof motion

be suspected

with

will

dimension-

againstpresent, we
attention to only such

verse

out

of limitcycles

asone
kind ofbe consideredkeeping
for judgingthe course
criterion
in
this
we
shall
Then
paragraph,quality.
which
the
under
investigatethe conditions
bystablelimit cycle is able to be reachedzigzagrnaneuver.
executionofinterminable
In fig.8 is shown the block diagram of

ratio

the

exampTe,ipand

to somerespect

shown

an

angle

lessperiods T' of such

equationfor thelimitcycle at last.

(b),asIn figs.7(a) andwill

maneuvera certain stable

reach(=rU/L)r'-17.75, Tb'=O.485,7le'=O.895, Kt=-9.ls, pr=-s,6)

=

mal

20

15

10

"EttS-gggL(d)unstablecertain
which

ships

o"* areof the switch-

representative

angle respece(t) and fi(t)meantively. Besides,dii(t),dio(t),
some
time-dependent function,to say thecourseangle

desired course
error

of

angle

angle,

coursespectively.

and

angle

rudderangle,
actual course
andrudder angle re-However,rnay
ipi(t)

bemadethe desired course
lostnull withQut

generality.
angle
In case of stable ships, the rudder
will beti(t)andcour$eangle(hereafter
ipo(t)as o(t)for the aim of brevity)will
writtenoscillatory
state,
to sayreachthe steadyoflimit cycle, after interminableexecutionofThe
zigzag
can
not
reach
the stable limit cycle.
cyclesof the ip'/t;'amountof such critical value

The existance of such limit
be assured by the followingre-

steering.
can

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Masataka FUJINO and138

The frequency characteristics

Seizo MoToRA

hysteresis block representing

the zigzag
b(t) respectively,
itlarlyagainst the time, can be well described is able to be easily deduced from fig.9 thatby the so-called describingfunctionK}q as
the relationof eq. (4)is a suficientcondifollows:5)
tion for steady
oscillatory
state.
asoning.

the rudder

6(t)whichangle/-ie',
7,
%
cos

:=

}

oi

e,

,,13

,1.

Assumingthat the error

steering

rectangu-`nfiip-* -j-lo:-'-sin

`'

'

=

variesof

signale(t),, ,-ipo(t)andare

G(1'a))'KLq=-l
Transforming

this equation,

El,
)
7;),
)

Re(G(1'to))==Re(-is sinu-

soidal functionof time, this equationthat theof rudderamplitude(4)

impliesangle 6(t) is

1inCG(io)=1}"(-(5)

times of that of error signal e(t) and:K}ql
the phase angle of 6(t) is aheadof e(t) by
In other words, a cross point of theNyquist
G(jo)Besides,z(K}g.
the fact that the describing curve and -11Kb4 curve on the
diafunctionis independent of frequencyw but gram corresponds
to a steadyosci 11atory
only dependent of the arnplitude diof inputstate,
to say steady limit cycle.
signalmust
be kept in mind.
By the way,
The functionG(iw) shownin fig.8 is the- ¢ COs ¢ ifrequencytransfer function of ship with rudRe(.
ii:l,
4"a
der angle regarded
as the input and course(6)angle as the output.
Taking account of the== -itrLn(fact that the input and the output of the'Accordingly, the -lfKk, curve IS a halfline

)=
k,)

E(O

parallel to the real axis of the Nyquist diaThe distancefrom the half Iine to
gram.

oect)

the real
combinatione(11--%tt)

and

S(r)

of

determined by theswitchingangle

coursefi", to say

op'

angle

-xdi"/(46"). On

hand, the behaviors of the frequency transfer function G(]'tu)K(1 + Tijw)1
TiJ'w)･(1+Tbjo)) on Nyquist diagram(Liw･(1+are differentaccording
to the course
keeping qualities of ships. In fig.10, are quan-

G(5)K"+T3S)

ofrudderthe-SCI+TISXI+T?S)Fig, 9 Block diagramis uniquelyaxis

limit cycle

1

other

-'

;J-

Jm
S')i
oRe

Re-i
t

-1/keq,):tsb,.TY't
oQ-ut-IE`(a･,,s4(e3ri4E`

sVl81v).scLVttLVGiy'Lv)a)i'1,'r[lkr+stabTeshipb) unstab]e $hip
Fig. 10 Exsistanceof limitcycle

c)ofzigzagT* andK* of anunstableship

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On the ModifiedZigzag Maneuver and139

itsApplication

so that it canby II in the figureis unstableFrom
the reasonnot be realized
physically.
it rnay be presumed that
(1) In case of stable ship. The G(1'w) ing stated above,zig-,20e) normal10e) and (200,eyen
if (100,
curve
always stays in the third quadrant of
nurnbers
wheretheformerrnaneuvers,
zag
regardto vanathe coordinate
plane without
courseintheinfiniteparentheses mean the switchingtion of frequency a). Namely, forof Gijw) angle
cb*and the latterones do the ruddervaluesimallysmall to the absolute
to be executedin caseis infinitely
large and vice versa. There- angle O", are unablernodifiedzigzagships, suchofunstableintersectsthe
fore,the G(fo) curve always200)may beas (le,10e) or (20,maneuvers-11K}, at only one
half line representing
executed.
in fig. 10 a). Hence, thepoint as shown
case
Similarlyas in the figs,5and6intrajectory on the phase plane approachesof stable ships, in fig.10 (c)arequantitagradually to the stable limit cycle correthat will
the
K*
and
T'
valuestively shown
to the cross point. As the fresponding
systemto be deterniinedfrom the first ordercorrespondquency bl and the ratio ip1fi"
inzigzagmaneuvers-IYK}qmodifiedanalysisofeachpoint of the G(jw)curve and the
the
The
value
of
ships,
ip of case of unstablehalf line respectively, the amplitude

the behaviors of Gijw) intitativelyshown
two cases of stable and unstable ships.

period Tof the limit
from thedetermined
cycle are automaticallyatof these two curves
pararneters, (u and ipli*,course

angle

and

the

.

parameter

¢

'/o"*

corresponding

to

the

scissaof this figureis its critical value
it is possible to executedecideswhether
zigzag

ab-

that

the

maneuver.

the cross point. Accordinglyif the ratios of
'/b",
other
are equal to eachzigzagof the modified
di"to 6',to say ¢1.6 A few exarnples
zigzagkinds
ofdifferent
for
the
even
maneuyer.
for instance 200.v20",100-.10o
Maneuver,that
the K* and T' values
Let's show
their limit
maneuvers,and 5e.-J50 zigzag
useofobtainedby makingrecently
werecycleshave the same frequency tu. In this two actual largeships. The principaldimencase, the amplitude
ipis strictly proportional sions of these ships are shown in Table 1.
course angle ip'.
to the switching
The G(;'w) Table 1 Principal dimensions of ships and theship.
ofunstable(2) In caseandmodifiedzigzagresultsof normal
shiftsfrom the third quadrant into
curvetu
maneuvers
the second
quadrant as the frequency
value,
acertaincritical`'B''ship
lessthandecreases
ship-11KL,
in
the
line
stays
stM
the
half
while
in this case. ThereLength between
third quadrant even
313.0
307.0
perpendiculars, Lpp Cm)interdonotalwaysthese
twocurves
fore,
48.2
48.2
B
Moulded breadth, (rn)other. There is no cross point in
se:'t each
19.39250,75019.40250,2511166.711
Draft,d (rn)wherethe switching course angle e"case
Displacement (ton)withthe rudderis larger in comparison
lf694.562.381.13O.927
Rudder arealLpp × dcourseangle i". However, if the switchingangle
ip" is diminished while the rudder
nsr*K6s'*100) normal(loO,
the Gijw)curveo'"is kept unchanged,
angle
maneuvers
zrgzagintersect the -1/K}, half lineas shownmay
7ks'*K6st*in fig.10 (b).In this case, thereusuallyao,sO) modified
maneuvers
zlgzagexist
two cross points. This irnpliesthat
states rnay exist at
two steady oscillatory
r'-6'
There are fullsuper tankers whose
the sametime. After the inspectionofby reverse
spiralobtainedcharactemsticsof these two steadystates, however,stability
hysteresis
loops
representing
have
the
tests
state rnarked
it is deduced that the steady
modified

1"A"

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Masataka FUJINO and140

keeping qualities as shown
12. In these figures,the time
historiesof (10D,
100)normal zigzag maneuver
courseunstablein figs.11 and-O,5qNsuuva

Seizo MoToRA50) modified(10,

to sayt- ¢ , 6 curves,
are also shown.
The K* andT' values obtained
from the firstorder system analysis
ofthese time histories aretabulated in table 1 for comparisonof eachother. From the results, it may
be deduced
andthat theK' andzigzagT' values

from (10o,10o)
larger thannormalzigzagrnaneuver
2o4e
those
from
50)
(le, modified
S(deg)starb,'reverseSPIratest-o.s

40-20

Port

ships

are

of

more

the normalattptrs*).ao:lo")

iigTagmaneuver51O'

o'-1O'r2o`-30
200

400

600

800

are

one,

stablequality at the modified
30' sdi'Starb.
2ct
P

maneuver,zigzagzigzagcoursekeepingmaneuverthanmaneuver.

2.1 Determination

frequencyoffunctionby Fourier
Analysisof limit cyeles

Port

10' Starb,
sV'
sv
5'
o'

6oo

aoo

POrt11Results of reverseofspiral'`A''

test andzigzagship

atresponse

the modi-

maneuyer.

response

to obtain the frequency
functionis Fourier analysis of the

response

to the sinusoidal

The direct method

800
t Csec)-5'-10'Fig.

maneuverfied zigzag

tigzag m)neuver

mod;fiedthat the2. Applicationof ModifiedZigzag Maneuyer
to Examinationof the Frequency Regpone
Functiont (sec){VI8b-(1:5)

sosteeringof various

frequenciesw.The modified zigzag maneuveris also able to be used to obtain the frefunction,because the steadyquency response
oscillatory
stateswithvariousfrequenciesw
are able to be realized by interminableexecution ofmodifiedzigzagThe

maneuverE)'lattermethod

is applicable even forunstablewhile
ships,
the former methed,to say thesinusoidal
steering
is not applicable.
This
is the main reasen that the modified
zigzagmaneuverhave advantage
overthe sinusoidal

Lsteenng.

40'

However, the time historiesof the rudder
angle

20'

b(t) andcourseangle¢ (t)arenotal-

the zigzag maneuvers.
Therefore, it is necessary
to pick up theharmonic component
withthe fundamental
ways

o'

-?o-4cr
CVti')=":50
1O' v8starb,
v
5'
200
o`

modif;ed

zig!agmlneuyer

56oa-5--1e'Fig.

t Csec)/800

PDrt12

Resultsef reverse
rnaneuyer

of

`'B''

spiral test andship

zlgzagsinusoidal

ones

at

frequency from these deformed signals of
the rudder
angle
and
courseangle
in orderto obtain the frequedcy response
function.
The rudder angle of rectangular
shape
may
be substituted by the equivalentsinusoidal
function(7),of which the arnplitude isequal
to 4o"*ITand the period is the samea3 tha:ofthe rectangularruddersignal,NII-Electronic
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On the Modified ZigzagManeuver46'
sin wt
6(t)=

(7)

rt

141

its Application

andD,G-s=-hO.4

caLcv[atTen.S".lo".}.'.?,lg)O.2`.gOi,gr...,of
As the fundamental harmonic componentis (40"/=)iG(j'w)isin(ofthe course angle ip(t)Nrshand, the absolute
on
the
other
+Z:G(fo)),-1.D-O.5ag6"o.-2o..,SO.5
.D
of
angle
z:Gijca)
and
the
value
phase
IG(ju)lLegtu'
arefunction G(J'ca)
the frequency responseNoable
to be determined from the amplitude-OA-40'-s
at the steadycaLcu{atlon
andthe phase angle of ip(t).h--O.6-60'

1.

N"""

Hence, the frequency£ ...
Tij'tu)1((1responsefunction H(ito)l"Vca)-08u80'by using the+lljm)(1+T2fo)) is calculated
function of a stable
Fig. 13 Frequency response
followingrelationship.
state.

oscillatory=K<1+

IH<jw)I=utIG(ju)I,In order to examine
statedmethod,the validityof

the abovefunc-

frequency response

theof thedetermined from the analysis
kindsof the' rnodified zigzag maneuver
varioustion

simuthe analogue
withthat fromlator will be comparedby the relationshiptheoretical calculation
Htr'w)= K<1+ Ty-to)X(Cl+ Til'w)(1+ Tito))
which

were

on

executed

ship

(8)

.(H<ju)=LGijw)+n12Fig. 13 showsthetions thus obtained
maneuvers

culations.

comrnonquencyoffrequencyresponse funcfrom the various modified
andThe abscissa
logarithm ofm'(=toLIU),

kind

rnaneuvers

zigzagmodifiedzigzagfrom various

obtained

the theoretical calof this figureisthedimensionless freto say logca',and the

Iegarithrn of the absoluteis common
functionof the frequency response
Ti', value to say loglH<jo)1,or the phase angle(1) Stableship. ThestabilityindicesHijw),[Tl',Ts' and K' used forthe analogue simulathe cirThe dotted line connectingz<:Miw).
werein table 2, whichtion are shown
funcresponse
cled
points is the frequency
by rneans of forced yawing tech- tion ebtained
measuredfrom the modified zigzag
tanker').
niquewith a fullsuperwhile
the solid line is that fromrnaneuvers,
These two linesagree withthe calculations.
stability
well.
This fact means that
Table 2 Hydrodynamic coeMcientsand
other
very
each
is useful forindices of the ships used in the analysis
rnaneuver
zigzagthe modified
ordinate.

of frequency

responsecharacteristics

i

ship

unstablem'+mv'

}3'
Nst
Nfi,

stable

26.2

xlO-S

20.0

20.0

"

O,09

n

5.03

"

-10.78Ki- O.437
Y},M'1}t+1}'- 3.15
1.79

-

ship

26.2 xlO--3
"

"
"
"
#

O.09

"

5.03

"

-10.78- O.437- 2.3e!.79

Tlt7h'7le,K,12.68-11.78

O.420

O.484

.-.L

O.893

O.895

5.82

- 6.20function.
the frequency response
Inspectingthese two linesin detail,however,
it is found that the gain characteristics
frornthe modified zigzag maneuver
IH<ju)1
Even
is slightly less than its true value.
themeasured
betweenthese differences
negligiblyarethe true values
andvalueserrorswiththe experimental
srnall compared
the
sea.
at
maneuverof the zigzag
As stated already inship.
Unstable(2)
obtaining

"

"

"
n

function can
1.5, the frequency response
less
for
the
frequencies
net be determinedfrequency, at which
criticalthan a certain
the function G(jtu)posses itsleastimaginary.

value.
ofthe

However,
response

artificial deformation
thefunction G(fo) makes
the

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142

andSeizoMoToRA

This circuit posses the maximum
possible to bephase adfrequency response
vance
equal
to tan-i((1-r)1(2.vX7T))
at the frefunctioncan be obtained even for the fre- quency we(=1/Tt)v'r).
The response
funcquency less than the critical one. For thetion of this circuit is calculated
by eq. (9).8)
artificial
deformation of the frequency response
function, we shall rnake use of theGD(s)=r'11++r7:lrbPSs
modified
executed,

zigzagsomaneuverthat thecircuit which

rnakesinput signal

advance.

1

the phase angle of its
It is most desirable(9)

r== R,R+iR, (o<r<i)ithe circuit whose
gain characteristicsdecreasesproportionally to the freTb=R,･C,
J
the phase angle increases
quency tu, while
as the frequency to decreases. As a matter
By insertingthis circuit after the block G(s)of fact,however, thecircuitin as shown in fig.16, that is to say the deas shown
fig.15 will be used in the followingstudy.
to chooseeFig. 16Blockcasediagram of;nmaneuverof adoptingzlgzag
phase-shifting

circuitformed course

angle
is fed back to thedi'(t)
hysteresis block instead of the course angle
e(t),the frequency response functioncan be
deformed as shownin fig. 14.
The resistances
Ri, R2 and the capacitanceC2 must be predetermined by adequately
presuming the frequency wo and the maximum
In fig.17 is shownphase advance.
the analogue
simulator
withthe phase shifting circuit, where
the part enclosed
by adotted lineis the phase shifting circuit. The
frequency response
functionGDijto)which isFig. 14 Modificationof frequency
responsefunctionq,
R2RlDUtPUt
ifiputFig. 15Phase-shifting

circuitintegrator

Fig. 17

Analogueaddee
simulatoroflncaseof

adopting

phase-shifting circuitzigzagrneneuver

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On the ModifiedZigzagManeuveris able4O'?o'-1.5-1.0

Fig. 18

Frequency

response

shifting circuit
shown

in fig.18 wasby theanalysis

varioussinusoidal

to be executed(20e,50) modified

evenzigzagmaneuvers

where

the former numbersin the parentheses mean
the switchingcourseanglesin terms of the-O,1-Q2
artlficial
courseangle
andthe latter
ip'(t)
angles.
-Q3-O.4-05Li[Gptsal)1ones do the rudder
From the frequency response
functionC'(J'e))determined by uslng the artificial
courseangle,
the frequency response
function U(ju) whichwewant
to know is calfunction of phaseculated
with
ease
by eq. CIO)
e.s1,tolvlfibCjN)1

143

for the remarkablesmall
frequency. For instance,it isevenpossible to execute(5e,50),(10,5o)and

eo'zopv)60'(CbeleJ

itsApplieation

andIoglH(ju)l==Iogbl+1og1G'(jto)I-1og1GDijev)1)
determined previously

the responses
inputs.

to

of

)
(10)

ztH(jo)=ztlGiijw)-t(:GD(jto)+n12

the

The modified
zigzagmaneuverat which
In order to examine
the validity of thethe artificial course angle ip'(t)is fed back new modified
zigzagmaneuver,the stable
instead of the real course angle
isshipalreadyinthis
ip(t),
quotedparagraph will
turnedto be realizable even for small fre- be turned to be unstable by changing
only
the yaw damping moment
coeficient M' fromquency. Namely in case of the modified zig-3.15 × 10-3to -2.30 × 10-S. The Ti',Tle',Tla'zag maneuverat whichthe real course angle
and K' indicesof this artificiar unstable
ship¢ (t)is fed back as hown in fig.8, even thein table 2 as well as the hydroare shown(le,50) modified zigzag maneuver is unable
to be executed, while the new
The circled points andmodifiedzigdynamic coeMcients.
zag maneuver
the points enclosed
by triangles in fig.19
at which the artificial course
angle
is
fed
back
as
shown
in
fig.
16,
representthefrequency
respensefunctionsdi'(t)
e.4N's"lo4eeuF"'
ie.P/5'r･""Ye"//J'02g.hLtl.Jm.e'Lagcve"..xtsN

-10
.o

o

-0.5

C.5

1,

e'Lv･thphase-sh;ftingNc/peuit).-let.'b-1fo"･
vwTthoutcircuitphase-shlf{ing.xi-40-o.za-NK"
-;
e/.x
sb-y
-i:aLculat;on'd-oo-O.6sc"･
NK'V'vL-'cEr.eutation
L?ILI.Vjhox'.".-1oo-:.e'F.x-:20.-T,?
[HCJ'W}

Fig. 19

Frequencyfrom various

responsefunction ofkind of rnodified

anzigzagunstableship

maneuversobtained

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144

Masataka FuJINo'

andSeizo MoToRA

by the new

obtained

modifiedzigzagmaneuistanceof Iirnitcycle, it must
be discttssed
the original modified
ones
respecin detail which nonlinear
equationof motion
tively. These points agree well with the is best representative of the steering motion'true response
function shownby the solidof unstableships. However in this study,linesin the figure.
the following nonlinear
equation
with･
theAt the region hatched by oblique lines so-called cubic type nonlinearity
which
has
where
the logarithmic frequency, to say
been proposed by Nomoto9) and NorrbiniO),
logw, is less than -O.125, theoriginal modiwill be used
for the sake of brevity
fiedzigzag maneuvers
is not realizable.
As
71Zil+(Tl+Th)i+r+Pr3=:Kh+KThS(11)
stated already,however, it is possible to
The rudder
angle
executeb(t) at the steadythe new modified zigzag
oscilmaneuver
latorystate,
that
is
to
say
even
the limitcycle,
at the frequency within
the region, so
canbe described by the errorthat the frequencyresponse functionis able
signale(t)(Eto be obtained at the frequency whoseof
course
angle
(t))
as
follows;¢
logarithm
is lessthan -O.125. The circled points
i(t)=ae(t)+Be(t)--crgb(t)-Psb(t)in this fiure wereobtainedby using the
4a*
4b* .
artificial
courseangle ¢ '(t), to say the newa:=
COS9bi, P
ncs(12)
modifiedzigzagmaneuverandpoints enclosed
by triangles were obtainedby thedi,-sin-i(e-')
original
modifiedzigzagmaneuvers.
They

and

vers

)

I

:-Tcs(vSM9bi,(de- Substitutingeq. (12)into eq. (11),finitelyin the Figure. However inspecting
TlThip+(71+Th+KTleP)dithe points in detail,it will be found that+(1+KTlia+K)9)di+PdiS+Ktr ¢the gain characteristicsof the response func(13)tion from the modified
zigzagmaneuversare
The sinusoidal solution of this autonomousslightly
smaller than the true values,
andequatlon, to say ipsinwt, gives the amplitude
that the measured
phase angles lag behined
l and frequency bl of the limit cycle
at the'the true ones. Nevertheless,this method
to zigzag
maneuver.
SubstitutingQ sm' a,t lnto
obtainthe frequencyresponse
functionseems
eq. (12)and equating
the coethcientsof theto be very useful
because the differences fundamental harmonic functions,sm' of andbetween the measured
valuesandthe true
costut, to zeros,
agree

well

with

each

other

as

shown

=O

ones

are

negligibly

small.

2.2 Investigation of existance･ofcycles in case of taking accountnonlinearityToestimate

of

the

of

the

of

un-

motion.

steering

motionitisnecessary
to takethe nonlinearityof motion.
Thenin this paragraph, the conditions
under
which
the existanceof the limit cycle at the zigzag
stableaccount

ships correctly,of(-2pm2-TITb)w2+

limlt-(Tl+7le+KTsp)w2+Klx=O

J

The graphicalsolutionthat will be statedbelow seems
to be useful for solving theseequations,because this metho

to

the

revealamplitude
cycles,

diand
and

the

d isappropriate
between thethe frequencyto of the limit

close

relation

switchingcourseangle

is assured, will be investigated. and the rudder angle ti'of the zigzag
Before begining the investigationof exing. Rearranging eq. (14),maneuver-aa2+Pt.2

(14)

¢

'

steer--K<1+1le2.2)[T,+Ti-Tb+(T,1)T,--2-pTlaip2).!]T?2toi)(1+
Tb2w2)+F(e,te)<1+Ti'w2)(1+

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a2+P2tu2F(di,.)=

T2+-2/pip2)to2]-K<1+Tla2.2)[1+(T,T,+TiT3-Ti

ptu

145

itsApplication

and

(15)

w)]
Tlica2)(1+lh2to2)(1+T3'w')+F(ip,('iiltTp2ip')w, +196p2T3Sop,--;pT,T,ft2ip2).6+(gP(Th2-T,Th)di2+

gpip2.i

The lefthand side term of the firstequais -ndicos ¢ if(4fi'), and that sects the curves representing the right handtion of eq. (15)equationisequal to -nip'1(4o"") side term. A pair of opand w correspondingof the second
re$pectively.
Each of them is independent to each of these cross points satisfy theequationof eq. (15).One solid Iine
secondThe right hand sideof the frequency tu.
for one prein fig.21 is obtained
equationis graphically as shownterm of the second
'/o"'
of
by
plottinga lot of
by a curve as shown in fig.20 scribed value¢
represented
satisfying the second equaif the parameter ¢ is presumed
previeusly. the pairs, (ip,ca)
the left hand side term andtion.
Similarly,
left
hand
side
On the other hand, as theterm of the firstequahandsideequationis constant in- the rightterm of the second
by a half lineparalleltion are represented
dependently of the frequency, it is reprecurve
respectively
abscissa
andcertaintothesentedby a halfIineparallel to the abscissa.
thattheamplitudeis
ip previously
provided
The distancefrom the half lineto the abfig.
22).
The
cross
determinedtheequapoint at
determinedby(seescissa is uniquely
the
curve
re-zip"1(4b').
which
the
half
line
intersects
intertion
This half linealways
right
hand
side
term
deterpresenting theaaftCpv)t
bl

o

- !!E
4si

the first
equation
dotted line in fig.21 is obw)
tained by potting a lot of the pairs, (op,corresponding
to the cross points of fig.22.
stated above
It isobvious from the reasoning
of
the
solid
Iineandthat the cross pointofthe dotted linein fig.21 is representativeat the zigzag
maneuverwitha lirnit
cycle
of di'/6'.
The ordinate
the prescribed value

mines a solutionTheof eq. (15).

satisfyingofthe crossandabscissa

point arethe am--atd"(PelFig. 20
o

-tw

F･

.

secondeg.Df.

eq,"51w-t

F"Ft
i"t''''':

7s''`t.

Xxfirsteq, of eg.tlS)
.

× Nx

%

-

-

N.

Py'""'lt"}ks-ofaf'vlS"'NN-s--.-eveFig, 21Wo

Fig. 22

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146

MasatakaFUJINO

plitude & of course
w

of

and

angle

the limitcy ¢ le respectively.

the frequency

As anexampleof the graphical solution,
the modified zigzag
maneuversof the unstable

ship

be studied

will

type

so-calledcubic

eMcient

P' of which

23 shows
unstabletheby assurningthenonlinearity,

simulationofthe modified

zigzag

The graphical solutions
do not well agree
withthe results from the analogue
simulation. However, itisevident
from this figurethat both of the circred points and the triangular points locateon the same
dotted line
representing
the first
equation.
Consequently,
if the solid line whichis representativeof
the second
equationwere
brought down up
to the broken line, the graphical solution
shouldagree
withthe results of the analogue
simulation.
In spite of the inaccuracy, thegraphicalsolutions seem to useful for studying the limit cycle quantitatively. By theway,comparing
the results obtainedby taking accountof the nonlinearityof steering
motionwiththose from the Iinear analysis,v

theunstablefollowingfactsmay

(1)

at

the nonlinear

analysiszigzagmaneuver-11.78,(71'==

metionp,

cycle

71i,=O.484,T5,=O.895, K,=-6.20,

the second
The points

maneuver.

-O.6
used

first andtheThe
above,

themean
analogue

D4

modifiedstatedby triangles, on the other hand,
limit cycies
obtainedby the-O.2

shipgraphical
Iinesare, as

enclosed

O,2

an

solid

the circled

solutions.

co-

'

of

dotted andti',where(15)respectively.

o.4

characteristics

the
of

r'e.6

r'Nony'areof eq.

ipand the frequency w, to say (ip,tu), of thelimitcycles are shownin fig.24 for the

Fig. 23

porntsequations

Some pairs of the amplitude-o--2eo20LOE,Cdeg

kinds of ¢ " and

the

the

of

variousrepresentative

is equal to -20,O. Fig.

r'Ne"' characteristics

ship.

Seizo MeToRA

and

of

=-20,O)¢*!oh',

be deduced;

In the case where the equation ofis linear, the frequency of the lirnit

is uniquelydetermined by the ratio
independentlyof the values of ¢ ' and

1
1

O
I'SL

O.

･,i:.,

O,w'

Fig. 24

Amplitude of head angle
the various
anunstablekind of modified
ship

diand
zigzagin consideration

frequency

to' ofmaneuversoflimitcycles at

carriedout

withnonlinearity

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On the ModifiedZigzag Maneuvero"*, andthe amplitude

moreoveripisstrictly

Howangle ip'.
proportional to the switching
is not true in case of
ever,this statement
The frethe nonlinear
equation of motion.
as theincreases
quency of the limit cycle
fi* increase,though thevaluesof ip*andThe ampliis kept unchanged.
ratio di'fti'

dipossesses its minimum

tude

certain pair of ¢

"

value

b' as shown

and

at

in fig.

24.

(2) In caseof

there exist

motion,equation

nonlinear
more

than

one

of

stable147

its Application

andif the principal

Howeverthe nonlinearity.
rnodesof thesteering

motioninto a)coursekeepingrnaneuver,b) coursechangemaneuverc) emergency

maneu-

andareclassified

the
is
still
usefirst
ful because of simpli ¢ ity. In this case, thedifferent K and T indicesmust be used pro-if they arever, andorder

treated,separately

approximation

systemto the relevant
steermg
perly accordingc)rnodes,the K
mode.
Narnely,for b) and150) and
T indicesobtained from (15e,and350) normal(350,

zigzagmaneuvers

respec-for a)

mode
kinds of ip"and O", tively are appropriate, while50) those from (le,50)modified zigzag maneuver
equal
to O.16. The CO.80,
to say ip'/6*
forinstance,has are suitable.
modifiedzigzagmaneuver,(3) Even in case of unstable ships, the100)modified
three limitcycles while the (1.6",
has only one limitcycle., stable limit cycles are realized if the ratio
zigzagmaneuverlimit cycles

atHowever,of these three limit cycles areto')
The limit cycle whose Cip',
andunrealiO.52)is unstable

not

all

stable.

certainis (O.23,

value
zable,

the rernaining

while

two limit cycles

1.002)andareco') values(O.073,(ip',
are stable and reO.28)respectively,(O.78,
whose

alizable.

former limit cycle of theseif the zigrealized
ones isusuallyThe

two stablezagmaneuveris started

from the straightcourse.

Conclusion
Sumrning up the discussions stated above,
will be deduced;
the followingconclusions
or
unstableoflessstable
In
case
(1)
te
executeimpossibleorships, it is ditheult
the norrnal

zigzagrudderat smallmaneuvermazigzagat the modified

angle
coursethe switchinge" to the rudder angle 6* is less than a certain value.
Nevertheless,there exists no stable lirnit
cycle with the frequency lessthan a certain
If the artificial course angle
criticalvalue.
ofthe modified

leads in advance of theis
fed back insteadof theangle
angle,
it is possible to executewiththe large
maneuverzigzagswitchingcourse

angle

phase

whose
real

coursereal

courseangle,

sothat thestablelimitcycle is able to be realized even at thefrequencyIess than the above stated criticalvalue.

(4) By analysing

the responseto the zig-

cycle

is reached,withenoughsteering
after the limitof maneuverresponse function
the frequencyzag

ing motion

canbe obtainedac-methodisby far betterthan
curacy.･This
angle. However
steering,
because it is applithe switching course angle is the sinusoidal
neuver whereships.
such less cable even for the unstable
angle, evensmallthan the rudder
The authors greatly appreciate the kindstable
or unstable ships can realize the stable
by Mr. M. SuGiTA and othernessofferedlimitcycles. Therefore,the modified zigzag
thestaffs of Seakeeping Laboratory, University
is very useful for examining
maneuveris of Tokyo, who helped the authors to drawa shipcourse keeping quality, whether
They
the figuresinserted in this article,
stableor not.
ShiptoHitachi(2) It is not appropriate to describeall also express their gratitude
building& Engineering Co. Ltd.,and Sasebomodes
by only one
kinds of the steering
Heavy IndustriesCo. Ltd. which tendered
As proposed byfirst order approximation.
with
full
to
kindly the results of experiments
Nomoto and Karasunoii),it is necessary

describe the steering motion with the second
taking account ofequationorderdifferential

scale

super

tankers.

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The Society
The
SocietyofofNavalArchitects
Naval Architects ofof Japan
Japan

148

Masataka FuJINo andReferenee
1) G. KEMPF:
Maneuvering StandardsforShips,
Deut$che Schiffarkts Zeitschrift,
Hansa, No.

27X28 (1944)

SeizoMoTonA
andControl Engineers,Vol, 5, No. 11,
PP. 24-33, Nov. (1966)(inJapanese)
7) M.FuJINo:StudiesonManouvrabilityofSh{psin RestrictedWaters, J.S.N.A.Japan. Vol. 124,
PP. 51-72, Dec. (1968)(inJapanese)

ment2) K. NoMoTo:
Analysis of Kempf's Standard
Maneuver Test and Proposed SteeringQuality
8) K, IcHIKAWA; S),stematic
71Fzeo,ly
qf'Automatic
Indices,DTMB
Rep. No. 1461 (1969)
Control,(lnJapanese)
Asakura Shobo
3) K.NoMoTo:
OntheSteeringQualitiesofShips, 9) K. NoMoTO:
Approximate NonlinearAnalysis
I.S.P. Vel. 4 (1957)
on
Steering Motion, Contribution to 12th4) S. MOTORA, and M. FuJINo:
On the Modified
I.T.T.C.(1969)
Zigzag Maneuver to Obtain the Course'Keep10)
Nils. H. NORRBIN:'ing
On theReducedNonlinearofLess Stable Ships, ContribuQualities
Equation of Steering and Its Applicationto thetion te 12th I.T.T.C.
(1969)
Analysisof the Pnll-OutManeuver, Contribu5) J.E. GIBsoN:
IVbnliner xtlutomatieControl,tion to 12th I.T.T,C.

(1969)
McGraw-Hill
11) K, NOMoTO, and K. KARAsuNo:
A New Pro6) K. FvJII, H. ITo, and T. KITA: Measurement
cedureof
Manoeuvring Model Experiment,of Control SysternDynamics Using Limit Cycle
J,S.N.A. Japan. Vol. 126, PP. 131-140,Dec.
Information, Journal of the Society of lnstru(1969)(inJapanese)

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