Stochastic Predictions of Parametric Roll Motions of Ships

CeSOS, Trondheim , March 2006

Stochastic Predictions of Parametric Roll Motions of Ships

J. Juncher JensenDepartment of Mechanical Engineering, Technical

University of Denmark, Denmark


Parametric Roll


Container Ship in Head Sea

Length L

BreadthB

Draft D Block coeff. Cb

β1 β2 β3 GMsw Radius of gyr. rx

Speed V

284 m 32.2 m 10.5 m 0.61 0.012 0.40 0.42 0.89 m 0.4B 6 m/s

h=0.05L, Lw=L

0

1

2

3

0 10 20 30 40 50Roll (deg)

GZ

(m

)

AP L/4 L/2 3L/4

3

31 2 2

( )2 0

x

g w GZ

r


Parametric Roll in Regular Waves

-20

-15

-10

-5

0

5

10

15

20

0 100 200 300 400 500 600

time (s)

roll

(d

eg

)

h=3.7m h=3.65m


Linear Stochastic Waves

1

( , ) ( , ) ( , )n

i i i ii

H x t u c x t u c x t

2

( , ) cos( )

( , ) sin( )

( )

i i i i

i i i i

i i i

c x t t k x

c x t t k x

S d


Equivalent Wave for GZ

H(X,t): Stochastic wave profileh(t): Equivalent wave height used in calculation of GZxc(t): Wave crest position

0 0

2 2

2 2 2 2( ) , , cos ; ( ) , , sin

, cos

2 ( )arccos if ( ) 0

2 ( )( )

2 ( )arccos if ( ) 0

2 ( )

( ) ( ) ( )

e eL L

e e e e

e

c

ee

x xa t H X x t t dx b t H X x t t dx

L L L L

X x t x Vt

L a tb t

h tx t

L a tL b t

h t

h t a t b t


FORM Analysis

The design point D (*) and the associated value of βHL determine accurately the probability that the response exceed the prescribed value. They can be calculated by standard reliability programs.

,i iu u

3

31 2 2

( )2 0

x

g w GZ

r

0 0

0

( )

Linear response:

standard deviation

HL

HL

P t

1 1 2 2 0 0 1 2 21( , , , , ..., , ) ( , , , , ..., , ) 0n n nnG u u u u u u t u u u u u u


Conditional Most Probable Roll Motion

• Roll period in waves must be in the range of twice the wave encounter peak period. • Simulation time t0 longer than the hydrodynamic memory.• Results to the right:

Design point wave elevation Roll response for this wave

for Hs=6m, 12m and prescribed roll angle of 17 deg. (0.3 rad.)• A minimum wave height (3.7m) is needed to trigger parametric roll.

-20

-15

-10

-5

0

5

10

15

20

0 100 200 300 400 500 600

time (s)

roll

h=3.7m h=3.65m

-20

-15

-10

-5

0

5

10

15

20

0 50 100 150 200 250 300

time (s)

Ro

ll (

de

g)

Hs=6m Hs=12m

-3

-2

-1

0

1

2

3

0 50 100 150 200 250 300

time (s)

Wa

ve

ele

va

tio

n (

m)

at

am

ids

hip

s

. Hs=6m Hs=12m


Conditional ResponsesConditional mean response = Most probable response

0 50 100 150 200 250 300


Reliability Index βHL

Linear response: βHL = constant*Roll/Hs

0

2

4

6

8

10

12

10 20 30 40 50roll (deg)

Bet

a

Hs=12m

Hs=6m

MC(12m)

(Hs=6m)/2


Variation with V,Tz and Heave

0

2

4

6

8

10

12

20 30 40 50roll (deg)

Bet

a

V=6m/s

V=3m/s

V=9m/s

0

2

4

6

8

10

12

20 30 40 50roll (deg)

Bet

a

Tz=11.7s

Tz=13s

Tz=11s

0

2

4

6

8

10

12

20 30 40 50roll (deg)

Bet

a

Hs=12m

No Heave


Mean Outcrossing Rates

The mean outcrossing rate is given by

21

2 2 220

1

1( )

2

HLn

i i iiHL

e u u

0 01 / 0F

/0 0{max } 1 ( )eT T

TP F


Probability of Exceedance

0

0.25

0.5

0.75

1

10 20 30 40roll (deg)

P{M

ax

(ro

ll)>

Pre

sc

rib

ed

|15

min

ute

s}

.

Hs=6m Hs=12m

Given ship, speed, heading and zero-crossing wave period


Decision Support Systems

{max 15 deg 15 minutes} 0.5 P


Conclusions

• The First Order Reliability Method (FORM) provides a fast and accurate tool to predict parametric roll in stationary seaways:• The design point and reliability index follows from FORM• The mean outcrossing rates and probability of

exceedance of a given roll angle within a given period is given analytically in terms of the design point and the reliability index.

• It requires a realistic time-domain formulation of roll.

• The procedure is well suited for generation of a operational polar diagram in an on-board decision support system.

Stochastic Predictions of Parametric Roll Motions of Ships

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