Analyses on low cycle resonance of ships in irregular astern seas

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Analyses

on Low' Cycle .esor6r5f5

in Astern Seas

by Masami Harnamoto*, Member

Naoya Umeda**, Member..

-Akihiko Matsuda***,. Member and Wataru Sera***, Student Member

Summary

Low cycle resonance occurs due to the dynamic variation of righting arm with respect to the relâtive

position of ship to waves. The dynamic variation depends on the encounter period of a ship in astern

seas. This paper discusses on the following items. (1) An analyticál expression of GZ curves varying

with respect to the relative position of ship to waves. (2) Non-linear equation of motion describing

low cycle resonance. (3) The effects of righting arm, stability range and encounter period on low

cycle resonance.

1.

Introduction

To investigate the dangerous situations leading up to

capsize in severe astern seas, free running model

experiment1' have been carried Out for a container ship

and a purse seiner at the seakeeping and maneuvering basin of the National Research Institute of Fisheries

Engineering. During these, the low cycle resonance was

observed for the container ship model but was not

observed for the purse seiner modél in scvere astern seas. Details of the experiment will be reported by the subject of separate publications7'.

It might be very

interesting for ship designers and operators to consider

why the difference happened. The problem, here is to

investigate what leads to the low cycle resOnance.

When a ship is traveling, in severe astern seas, the

rolling motionwill be developed to large amplitude or

capsized in a very short time if the natural roll period is

close to a specific encounter period. Such a rolling motion is so called low cycle or parametric resonance

which is a dangerous situation for ships of poor stability satisfying 1MO gesolutions A 167 and A 562 only criti-cally. Although ships are usually designed with consid-eration of safety margin with respect to capsizing under

most conditions, the. combination of severe waves loading condition, ship speed. and heading angle may lead to poor stability;

Analytical approaches to this problem were

origi-nally made by Watanabe1',

Grim2', Kerwj&' and Paulling".. They pointed out that low cycle resonance occurs due to the dynamic variation of righting arm with rEspect to the relative position of ship to waves and the unstable regions of low cycle resonance are specified as the solution of linear differential equation

of the so called Mathieu's type. However by solving the linear equation of motion mentioned above, it is

impos-sible to simulate the rolling motion óf large amplitude

leading up to capsizing because the righting arm is

described in a linear function. S that it is difficult to

know the rolling motion of large amplitude in detail. Kan and Taguchi5' carried out an analysis of a mathe-matical model which takes into account the non-linear GZ curve and made an interesting discussion on the phenomena of chaos and fractals. In their study, the non-linear GZ curve was obtained somewhat simpler

manner.

This paper focuses on making an analytical approach

tO this problem by taking into account a non-linear

righting arm GZ more rigorously than the previous

studies. Since this equation of motion is described in an analytical expression, itwill be possible to investigate the features of low cycle resonance in detail.

2.

Equation of Low Cycle Resonance

The righting arm GZ varies with respect to the

rela-tive position ¿G of ship to a wave when a ship is running in longitudinal waves.The low cycle resonance occurs when the GZ varies with a period in a specific encounter

period of ship to waves. The characteristics of this

probleimare usually described mathematically as

(1)

where I. is the moment of inertia of a ship, J the added

moment of inertia, K the damping coefficiént. W the

t

_{Osaka University}

** _{National Research Institute of' Fisheries}

Engi-neering

Graduate School, Osaka University Received 10th Jan. 1995

198 _{Journal of The Society of Naval Architects of Japan, Vol. 177}

ship weight and roll angle.

And then Eq.( 1) leads tÓ the following form

(2)

'

\TΫ

T)

GM

-where a is the effective extinction coefficient, T the natural period of roll.

The GZ(çb, _{) in Eq. ( 2) is calculated, based on the} assumption of quasi-static equilibrium, by integrating. the quasi-hydrostatic pressure

over the momentary

submerged hull under the wave surface.

In order to

solve Eq.( 2 )riurnerically, it is necessary to make an

analytical expression of GZ as function of and Ec. This is the problem here to be considered because the

characteristics of low cycle resonance can be described depending ori the type of GZ curve.

Property of GZ in still water

In general, the GZ curve is mainly specified by the metacentric height GM, the integrated

area A of GZ

curve and the vanishing point of stability q5,.. They are given by

cvr_(th

'

\

d 1.o

A=f'GZdcb

(3)

GZ=O at 5q5r

Êor simple expression, the GZ curve including three parameters at least may be described in the following

form

GZ=(GM C)-'-sin (T-)±

cft

sin (22r--.)

(4)

Here GM and 5,. are immediately obtained from the GZ

curve calculated by numerical method, C is

the coefficient describiñg concave and convex configuration of GZ curve at the small heel angle and determined by integrating the GZ curve as

Figure 1 shows the GZ curves of two type which can be

described by Eq.( 4). The type lis the family of GZ

curve in case of negative C and type II is the family of

GZ curve in case of positive C. The GZ curve of the container ship used for our experiment belongs to the type I and the GZ curve of the purse seiner belongs to

the type II.

0.4

02

o

Variation of GZ in a wave

As mentioned early, the GZ curve in a wave varies with respect to the relative position of shipto a wave. According to Eq.( 4), it is possible to consider that the

variation of GZ in a wave is based on the variationof

three parameters GM, A and r

with respect to the

relative position of ship to a wave; The variätiôns of them can be expressed in the Fourier series as

GM(ec) .

A()

=ao+E(acos nkEc+b sin nkc)

_nL

(6)

where k is the wave number, ao, C

and b are the

coefficients determined by mean of Fourier analysis of

the GZ(,

).

In Eq.( 6) the relative position¿cis a function of ship

speed U, phase velOcity c of wave at t.=0, and the

relative position EG of ship to a wave at any time t is.

given by

ÇG=Eo(c U)t

3.

Calculations of GZ-variation

To investigate the regions where the low cycle reso-nance occurs, the calculations of GZ-variation in a wave

have been carried out for a container ship and a purse seiner as shown in Fig. 2 and Fig. 3. In addition, the

principal dimensions of them are shown in Tables i and

2.

GZ-variàtions of container ship

The GZ variation was calculated in cases of GM

0.15 m and 0.318 m which satisfy the 1MO resolutions A 167 and A 562. Fig. 4 indicates the coefficients as a function of wave height to length ratio H/A in cases of

GM =0.15 m, Fig. 5 the three parameters of GM(Ea),

A(Ec) and as a function of the relative position Ec/A and Fig. 6 the GZ() in still water, GZ(, ¿) at the

0.6

0.4

0.2

o

Fig. 2 Body plan and GZ curve of container ship GZ [m] G=0318m G=OE15ni 40 _[deg.]

'(Gl)

r

Y4\

GZ/(GÌfPr)

''ner

GÏ755; o

r

o 0.4 0.8 / 0.4 0.8 Ø/,

Fig. i _{Types of GZ curve of container ship and purse} seiner

4

2

(-0.08

0.04

Analyses on Low. Cycle Resonance of Ship in Astern Seas ₁₉₉

Fi'. Ai'.

Fig. 3 Body plan and GZ curve of purse seiner

Table i

Principal dimensions of container ship

Table 2 Principal dimensions of purse seiner

wave crest and trough amidships

¡ri comparison

..'

between the numerical calculation and approximate

expreiöii Figures 7, 8 and 9 indicate the coefficients

the three parameters and the GZ-variationz respectively in case of GM=O.318 rn.

' GZ-variation of purse seiner

The GZ-variátion was cakulated in case of GMO.75

m which critici1y satisfies 1MO A.685 weather criteria at the wind velocity equal to 19 rn/s. Figures iO, 11 and

12 show the coefficients, the three parameters and the GZ-variation in the same way as the container ship.

4.

Stable and unstable regions

of low cycle resonance

As well known, the low cycle resonance has been traditionally considered as the ' problem of linear

.Mathieu's equation'4 given in the form

.

±2(--) ±(2-)2[i+

COS(ko_Wet)]O

(8)

where 4GM is the variation of metacentric height, a the

'

effective extinction coefficient, and the lowest and

widest unstable region at an encounter period of half

the natural roll period as shown in Fig. 13.

The problem here is to investigate the characteristics

of Mathieu's equation described with non-linear GZ

curve as shown in Eq. ( 2). For this purpose, a number

of the time domain simulations are carried out for the

container ship of GM =0.15 m and 0.318m, and the purse seiner of GM=0.755 m.

Stable and unstable regions for container ship

The stable and unstable regions for the wave height

to length ratio H/A versus the encounter period to

natural roll period ratio T/T are shown in Fig. 14 for GMO.15 ni and Fig. 15 for' GM=0.'318m in cases of wave to ship length ratio A/L=1.0, 1.5 and 2.0

respec-tively. Then the experimental results are shown for AIL

= 1.5 in comparison with computed results. An initial heel angle is five degrees and the relative position of

ship to a wave at the wave, trough amidships at the

beginning of a time domain simulation to initiate the roll motion. The region. bounded by H/A and T/T is

divided into 10.000 elements. The dot area indicates the

stable region and the white area the unstable region.

The stable and unstable regions simulated by Eq. ( 2)

described with the non-linear GZ are significantly

different frOm that simUlated by Eq. ( 8) described with the linear GZ. For Figs. 14 and 15. the unstable region become biggr as the wave to ship length ratiO A/L and

the metacentric height GM is smaller.

This is an

understandable trend.

Although, in the solution of

linear Mathieu's equation, the first resonance occurs at

T1/T=1/2 and'the second resonance occurs at Te/T i as shown in Fig. 13. both the first and second reso-nances shift to the direction of the shorter encounter period due to the effective natural roll period which is

smaller than in still water. The boundaries between the

- Items Ship Model

Length L(rn) 150.0 2.5

Breadth

B(m) 27.2 0.453 Depth D(m) 13.5 0.225 Draft

dj(m)

8.50 0.142 da(m) 8.50 0.142 Block Coef. C6 0.667 0.667

l.c.b. (aft)

-

-1.010 -0.0168

Gyro. radius

in, pitch

k/L

0.244 0.244 Metacentric GM(rn) 0.15 0.0025 height

-Natural roll

period

T(sec)

43.37 5.59

ModeiScale

-. .

-

1/60

Items Ship. Model

Length ' L(in) 34.5 - 2.3

Breadth

B(m) . 7.6 0.507

Depth

D(m) 3.07 0.205 Draft dj(xn) 2.84 0.189

d(m)

3.14 0.209 Block Coef. C6 0.652' 0.643

l.c.b. (aft)

-

1.742 0.116

Gyro. radius

in pitch

k/L

0.332 0.332 Metacentric height GM(m) 0.755 0.0503

Natural roll

period

T(sec)

7.47 1.93 Model Scale

-

-

1/15

GZ[]

G=0.755ai

20: ...4Ö

t.

'[deg.]

200 Journal of The Society of Naval Architects of Japan, Vol.177

'.2

₌

.

=

-A 20

-s-

-o X 0' , GÏ=0.15[iu]

I fL1

GÌ(c)

A( ÈG) o

X 0

, GÏ=0.15[

IJL=1

]

approximate expression

..

numerical computation.

AJLL5_

I S. L 0.04 0 HI I

I JL= Ls

I /L2

Fig. 4 Coefficients of GZ-variation of container ship

with GM= 0.15 m 0.03 HI I

approximate expression

numerial computation

AJ =2

H=8in i H=4m

Fig. S Three. parameters of GZ.variation of container

ship with GM =0.15 m

.ao

ai A_bi o a2 t. 0.05 0

H/I

o i 10 O 10

E

E

O

O

Analyses on Low Cycle Resonance of Ship in Astern Seas 201

GÏ=O.318[m] r(ÈG) E à) 0.05

H/ I

Fig. 7 Coefficients with GM =0.313 rn H=8m ugh

_4troLigh

Ik1 VA1

crst

+

crest'

t

0

5O0

- 50 Cdegj ç5 [dègj

Fig. 6 GZ-variations of container Ehip with GM0.15

m

apprôxiniate expression

s n.

numerical computation

I /L=L5

tiough

H=Sm 50

[degj

A L2

of GZ-variation of container ship

0.03 HI X an o aj A bi o a2

, Gl=O.l5in

approximate expression

numerical computation

202 _{Journal of The Society of Naval Architects of Japan, VoI. 177} E E O o o X 0 , G1=O.318Cin]

¡,IIIIIII

I'II1Iii

¡,IlIi

ILWdI

£/L=1 I

UUJIUR

O

c/ A

x=0

, GÏ=0318in

crest

trough-Illr8m i

togh

.,-VAWI

lo

50 0 Ø [degj IÌL=L5

¿ c/ I

approximate expression

nunierical computation

Ì!tRP

10

Fig. S Three parameters of GZ-variation of container

ship with GM=O.318m

- approximate exptessiôn

numerical computation

-H=8m

tro ugh

Fig. 9 GZvariations of container ship with GMO.318

m

50

[degj

H=4ni

=

z

z

Q '-I Q o z.=O _{!: G10.75S[m]} 'A /L=l

GX()

r( ¿ c) od. o HI A

X 0

GÏ=O.755[iJ A /L=1 O

lo

Analyses on LOW Cycle Resonaice of Ship in Astern Seas ₂₀₃

A/L=L5

Fig. 10 Coefficients of Glvariation of purse seiner

A/L=15 I

RRRRR

s

approximate expression

numerical Cowputation A/L=2

- approximate expression

. A

numerical computation

H3.45m K=2.3in

10

Fig. 11

Three parameters of GZ-variation of

_{purse seiñer}

.. ao

a a a2

° bi.

0.05 0 H/A 0.05 H/ A

204 Journal of The Society of Naval Architects Of Japan, Vol. 177 02 02 O z. 0 , G=O.755m R=2.3in

Fig.. 12 GZ-variation of purse seiner

approximate expression numerical computation

roll period ratio Te/T are shown in Fig. 17 for GM=

0.755 rn in cases of wave to ship length ratio A/L=1.0, 1.5 and 2.0 respectively. The unstable region becomes

bigger as the wave to ship length ratio AIL is smaller.

This is the same trend as the container ship. However,

the feature of unstable region is quite different from that of container ship. There is no first resonance but the second resonance at TeIT 1 which is not so wide and deep. This might be the reason why the low cycle

resonance did not occur in. the free

running model

experiments of the purse seiner. Figure 18 shows the

time histories" obtained from the experiment in

cóm-parison with the results simulated. 5. Conclusions

An analytical approach was carried out to investigate the effect .of non-linear GZ on the stable and unstable regions of low cycle reonance summarized as follows

The mathematical model describing a single

rolling motion was presented for obtainiñg the

stable and unstable regions of low cycle

ro-nance.

A practical method was presented to describe the GZ-variation with respect to the relative

position of ship to a wave.

(3). To investigate what leads to low cycle

reso-nance, the. stable and unstable regions were numerically explored for both the container

ship and purse seiner.

(4)

Finally it was concluded that the occurrence of

low cycle resonance is one of the different

features of unstable region between container

ship and purse seiner.

o 0.5. _Te/'r 1.0

Fig. 13 Stable and unstable regions of ship with linear

GZ curve

stable and unstablé regions are not distinct as pointed out by Kan and Taguchi..

On the other hand, free running model experiments7' were carried out for the container ship in cases of TeIT =0.4-0.7, A/L=1.5 and H/A=1/25. Fig. 16 shows the time histories7' of pitch and roll obtained from experi-ment of container ship model in comparison with results simulated.

Stable. and unstable regions for purse seiner

The stable and unstable regions for the wave height

Hl A 0 1/20 o H/A ¡ff25 O 0.5

Te/Il

O

Fig. 14 Stable and unstable regions of container ship with non-linear GZ curve of GMO.15 m and

the effective extinction coefficient a=O.271

Unstable

,, A 1/22.5 - 1/45 è.=0271.: --I G=O.318: A7LL0: 0.5 Te/T 1 0

Unstable

_H/A 1/15 O

o Not capsized in experiients

Capsized in experinents

---H/A

1/30 0.5

Te/Il

1/60 O o

Unstable

H/ A 1/30

Fig. 15 Stable and unstable regions of container ship

with non-linear GZ curve of GM =0.318 m and the effective extinction coefficient a=0.271

Unstable

H/A

1/20

0.5 1

a=O.2l85

i55

- --= GÏO755

able

-1i

!Stable_LL=L5

StableLÀLi0

0 1

Te/I 2

00 1

T,i 2

0 1

Te/I 2

Fig. 17

Stable and unstable regions of purse seiner

with non-linear GZ curve of GM =0.755 m and the effective extinction coefficient a=0.2185

Analyses on Low Cycle Resonance of Ship in Astern Seas 205

T/T

1

0.5

Hl A

0:

io

-4e: Experimenf'

Pitch

25: Roll Angle(deg)

lo -25

Simulation

Journal of The Society of Naval Architectsof.Japan, Vol.1717

20 0

(sec)

First Resonance at T/T.36 and. 1/LL5

Pitch

25: Roll Angle(deg)

O

-25:

Simulation

Second Resonance at TdT=O.71 and ?/L=L5

Fig. 16 _{Tire histories of pitch and roll for container}

ship model in comparison with results

simulat-ed.

This study was carried out under the panel RR 24 of

Ship Building Research Association of Japan.The

authors wish to express their gratitude to members of

the RR 24, chaired by Prof.

Fujino for productive

discussions.

Ref èrences

1) Wátariabe, Y.: On the Dynamical Properties of

the Transverse Instability of a Ship due to

Pitch-25: Roll Angle(deg)

O

-10-40: Roll Angle(deg)

oj\

-4O

Experiment

\

10 20 30

(sec)

3'O

(sec)

0

_C

(sec)

-At TdT=L21 and l/L=L5

Fig. 18 Time histories of pitch and roll for purseseiner model in comparison with results simulated

ing, Journal of the Society of Naval Architects of Japan, 53, pp. 51-70, (1934).

Grim, O.: Rollschwingungen, . Stabilitat und Si-cherheit im Seegang, Schiffstechnik, (1952). 1

Kerwin J E

Notes on Roll in Longi Waves

mt. Ship. Prog:,. VOl. 2(16), (1955).

Paulling, J. R. The Transverse Stability of a

Ship in a Longitudinal Seaway, Jour. of Ship

Research, SNAME, Vol. 4, No. 4, March, (1961).

Kan, M., Taguchi, H.

Capsizing of a Ship in

Quartering Seas(Patt 4. Chaos and Fractals in

Forced Mathieu Type Capsize Equation), Journal of the Society of Naval Architects of Japan Vol

171, pp. 83-98, June, (1992).

Haniainoto, M., Fujino, M.: Capsize of Ships in Following Seas, Safety and Stability of Ships and

Offshore Strucbires Third Marine Dynamics

Symposium -, pp. 125-158,

(1986)-Umeda, N., Hamarnoto, M.,Takaishi, Y., Chiba. H., Matsuda, Sera, W., Suzuki, S.C, Spyrou, K., Watanabe, K.,: Model Experiments of Ship Cap. size in Astern Seas, Spring Meetings of the

Soci-ety of Naval Architects of Japan, (1995), (tò be

submitted.)