A modified approach for the application of fokker-plank equation to the non-linear ship motions in random waves

A MODIFIED APPROACH FOR THE APPLICATION

OF FOKKER-PLANK EQUATION TO THE NONLINEAR

SHIP MOTIONS IN RANDOM WAVES

by

MR. HADDARA..

Abstract.

An analytical method.for the study of transient behaviour as well ás the stationary solution of

non-linear ship motion in random waves is presented The problem of rolling motion of a ship travelling in oblique random waves is considered for the purpose of illustrating the méthod. The presence of nonlinear damping and restoration terms of general form is allowed Differential equations which

govern the propagation of the expected välúe and thé v.riance of the motion in this case are derived,

using the Fäkker-Plank equation.

The results are specialized to the case of a ship rolling in beam waves. Results are compared

with similar results obtained by other methods., This method gives the exact stationary solution when

the damping is linear.

1. Introduction.

Considering the ship as a rigid body, one can describe its motion in a seaway by a set of six

second-ordeí differential equations.

These.equa-tions are coupled and nonlinear. The problem is further complicated by the fact that waves are

random in nature. Thus, ship motions-are

actual-ly stochastic processes. Onactual-ly special cases of

this general problem have been discussed in the

literature.

The author (1971)**) investigated the stability

of nonlinear rolling in random seas,- using a

modification of the method of slowly varying

para-meters. A brief review óf the different methods

used in studying nonlinear ship mot-ions in random

seas has been alsogiven. We would like, how-ever, to mention very briefly thesé methods.

Hasselman (1966) and Vassilopoulos (1966)

used the

functional representation method.

Hasselman showed that the nonlinear transfer functions are relatedto the higher-order statist-ical moments of ship motions in the same way as the linear transfer functions are related to the spectrum. While Vassilopoulos considered the

problem of a ship rolling with nonlinear ±'estoring

mo ment.

The - perturbation méthod was used by

)Port-Said Shlpyard,Suez - Canal Authority. Egypt.

Seethe list of references at the end of this paper. -j

Deift University of Tec!rnolcy Ship Hydromachanics Laboratory

library

Mekelweg 2- 2620 CD Deift

The Netherlands

Phone: 3115 786873. Fax: 3115781836

Yamanouchi (1969) to study the effect of quadratic damping on the spectrum of rolling motion.

St. Denis (1 967) developed an equivalent linear

-ization technique to study in a general way the coupled pitch-heave equations. Vassilopoulos

(1971) considered the problem of a rolling ship in beam -seas using an equivalent linearization technique. He allowed the ship to have a cubic

restoring moment term and a square damping

moment term.

--The method based on the Fokker -Plank equation

did not receive any attention from naval archi-tects. This may has been partly due to the diffi-culties-encountered in solving the partial differ-ential equation which describes the conditional

probability density function of the stochastic pro

-cess (i. e. the Fokker -Plank equation). Besides, the number of nonlinear problems which can be dealtwithusingthis approach is limited. For

in-stance, one àan not use this method when the

damping is nonlinear. The main advantage of this

method, however, is that exáct stationary solu-tions may be constructed in some cases.

This approach may, however, be mdified in

sucha way that the actual solution of the Fokker

-Plank equation is no longer necessary. This is

achieved by converting the Fokker -Plank equation

into a stochastic differential equation (Saga and

can obtain stationary solutions as well as transient behaviour for a. wide class of nonlinear

stochastic processes. For instance, problems

with nonlinear damping can be tackled using this -approach.

In the present work the problem of a ship rolling

in random seas is considered for the purpose of illustrati.ng..the modified method. The presence

of nonlinear damping and restoration ofgeneral

formis allowed. Besides, the restoring moment

is considered to. be a function ftime. Thus, the

case of a ship excited by oblique or following

waves is covered.

The stationary solutionis then, derived for the

case of a ship rolling in beam seas. It is com-pared with the solutions given in the literature.

2. Equation of motion.

The equation of motiön of a ship travelling with

a constant average forward velocity in random oblique waves can be written in the following form (Comstock (1967)):

+N()+F(, t)=K(t)

(1)

where

the angle of roll

the derivative of with respect to time

N()

the damping moment

F(, t) the restoring moment

K(t) the wave exciting moment

The restoring moment acting on the ship is

al-lowed to be a function of time. Thus, variations

inthè righting arm of the ship caused by oblique. or following waves can be included. These

varia-tions are extremely important when stability

study is our prime aim.

The excitation K(t), should be Gaussian..

Besides,it shouldsatisfy the following conditions: <K(t1) K(t2)> =p0 6(t1 -t2) (2) where < > means the ensemble average, a is the

Dirac delta function and p0 is the variance of the excitation.

Using the following change of variables:

Y

Y2D,

one can rewrite equation (.1) as:

"i '2

'2=-N(i2) -F(y1, t)+K(t)

We also Write equation (4) in a matrix form

as follows:

-i'=D(Y,t)+E(t) (5)

where -. .

1(Y2) - F(y1;

t)]' E

=

3. Fokker.1'lank equation.

A stochastic process Y(t), is called Markôv if

the conditionäl probahilitythat Y lies in the inter

-val(y, yy)attÎme t, given that Y is equal

toy1 at time t1. y2 at time t2, ..., and at

time tn_i (Where tn->

te_1 ...

> t1) depends only

the value of Y at the time t.4. Thus, for a Markov

process, we have

Pn(yltj,.Y2t2,... ,Yn..1tn4 Yntn)

=P2(y_1t_1I yt)

(6) where P2 (y, t1 y2 t2) dy2 is the probability that Y

will lie in the.interval (y2+dy2).at time t2given

that Y =y1 at timé t1. Then, the. conditional

prob-ability density function P2, describes a Märkov process completely.

A Markovprocess may also be associated with

a first-order differentialequation of the form of

equatioxi(5), (Papoulis (1965)). Then the two di-mensional stochastic process (Yi. y2) of equation (4) is Markov. The process may be described by conditional probability density function,

P2(y10, Y2Ô y2, t) where y10 and y20 are the initial values of the angle and velocity of rolling motion. It canbe shown that P2 satisfies the

fol-lowing partial differential equation (see Appendix

A for details):

ap a a

ap

2 (8)

ay2

where we have used the short hand notation P

in-stead of P2(y10, y20j y1, y2, t). The solutionof

equation (8) subject tó the initial ondition

P(y10, y201 y1, y2, t)=a(y1-y10)6(y2-y20) as

t -, 0, yields the conditional probability density

function which describes the process (y1, y2)

completely.

Instead of actually solving equation (8), which

is impossible any way, we are going to convert

it into a stochastic differential equation. Rewrite equation (8) as follows:

kil

,D4

dP(y10, y20 1y1, y2, t) =P(y10, y20 y1, y2, t

+dt) -P(y10, YoIi

y2,t)

la

a

=I---(y P)+-

2 _ay2 I (N + F)P}

32J

(9)

Equation (9) can be used to deriVe the differential equations which govern the propagation of the

ex-pected values andváriances of y1 and y2.

Expected 'values propagation:

We multiply the two sides of equation(9) by y1 and integrate the equation with respect to y1 and

y2from-to. Wethènrepéat the same process

using y2 and get (details may be found in

Appen-dixB)

M2=-<N(y2)+F(y1, t)> (10) wherep1 and are the expected values of y1 and Y2 respectively.

Variance propagation:

We multiply the two sides of equation (9) by

(Yi M1)2 and integrate with respect to Yi and y2

from -

to . We then repeat the same process

using (Y2 -M2)2 and (y1 -'1) (Y2 2) arid after some manipulations we. get (details may be found in Appendix B)

'1=-2M1M2 ₍₁₁₎

V2=<y1F(y1, t)> -i.i1<N+F>+i22 (12) where V1 and V2 are the variances of y1 and respectively.

4. Transient behaviour.

The transient behaviour of the mean arid

variance caribe obtained from equations (10) and

(11). The main problem here is how to evaluate the ensemble averages on the right hand side of the equations. An approximate value for these

averages maybe obtained by replacing the quan

tities whose averages are sought by their Taylor series expansion about the expected values M1,

and M2. If the nonlinearities are small one can use terms up to first order only.

The following approximate equations can be

used to study the transient behaviour of the

motion:

M1 = I1

= -N(M2) -F(i1, t)

.Ç7 =_,2aM2)V242

a2

Equations (13-a) are two coupled first-order non-.

linear ordinary differential equations. They may

be replaced by one second order. differential

equa-tion in the expected value of the moequa-tion. The

stability of the solution of this equation may be

investigated after we have specified the shapes of the. nonlinear. damping and restoring moments.

Equations (13 -ib) are. two uncoupled linear

or-dinary. equations. The coefficient of V2 in the secoñd equation is always positive. Thus, these. equations possess stable solutions as long as the

solutions óf equations (13 -a) are stable. .5. St'tionary solution.

If the process is stable it will reach a station-ary state after some time. The stationstation-ary

solu-tionis obtained fromequations (10), (11) and (12) by substituting M1-I12T1=V2=0. Then we get <Ñ(y2)+F(y1, t)>=0, and

<y2N(y2)>=2.

(14)

Special case: A ship rolling in beam seas.

We. would like to consider the case of a ship

movingwith zero forward velocity and excited by

random, beam waves. The object of this section

is to compare the results obtained by other

methods with the results obtained using the

method presented in this work.

I. Quadratic damping and cubic restoring

mo-ments:

In this case we have

N(y2)

_{2c9(y2+1y2lY2I)}

F(y1, t)=F(y1)= °21

_{+ 2} (15)

(13-a)

Stibstituting(15) into (12) and (14) andevaluatiig the averages. We get

./!3]

4c,[c4

+Ej p0

where we have used the familiar notation

2.

2-v

a,T

1a4

₂

In deriving equations (16) and (17) wehave.used the following assumptions:

2

1.3

IY2!>-%I

,

<y14>=3

:

The roll angle and rofl velocity variances in the linear case can be obtained fromequations (16)

and (17) as

at2p/4c,, ¶pL2tJ2/C)P2

wherewe have used the subscript L to denote the

variances when the nonlinearities are absent.

Substituting (17) and (18) into (16), we get

3E2+1

13

E2a,2»+a,

(19) This is the general relation between the linear

and nonlinear variances of rolling motion.

Equation (19) is similar to the expression

ob-tained by. VassilopouloS using an equivalent

jnearizationteChfltqUe. We are going derive his

result from equation (19). Let us rewrite

equa-tions (16) and. (17) in the following approximate form:

2

fl

= (

+32a

,

Substituting (18) and (21) into (20)we get

a2(1 3

E2 _{PL2) +}

Lp

4.5

2

+3E2a) _°pL =0

which is the expression obtained by Vas silopoulos.

II. Linear damping and cubic restoration:

SubstitutingE1 0; in equation (19) we get:

4 2 2_

32a,

+0 _pL 0.

From which we get

2_

_'1-3E

p

çL'

2°pL

which is the exact solution as has been obtained by Caughey (1963) using the original

Fokker-Plank equation approach.

Ill. Cubic damping and linear restoration:

In this case we have

N(y2)=2c(y2+E1Y23)

F(y1) =c2y1

Substituting into ecivatins (12) and (14) and evaluating the verages, we get

4.Ç(&.2+3e1a,4)=t40 , (23)

and

(24)

Substituting (18) and (24) iñto (23), we get

2.442 :ï2.

(25).

The positive root of this equation is given by

2 2 2 2

OEc =1 (1 3E1c _pL) (26)

This is the same result obtained by Crandall

(1963) úsing the perturbation method. 6. Discusions and conclusions.

In the previous sections, we have presented an analytical

method for the study of both the

transient behaviour and the stationary solution of

nonlinear ship motion in random seas.

The application of this methOd to the problem of rolling motion shows the following:

(22)

I. Not only the stationary values of the expected.

value and the 'variance of the motion can be ob-tamed but also their transient behaviour can be

studied.

Time variatjons in the righting arm caused by oblique or following waves can be allowed. Such.

variations are extremely important when the

question of motion stability is being considered.

The accuracy ofthe method is best

illustrat-edby noting that the stationary solution obtained

for the case of a ship rolling in beam seas with and

linear damping and cubic restoration is exact.

unfortunately, this is the only case whosè exact solution can be found in the :1jterare.

IV. Coupled motions can be easily dealt with using this method.

List of symbols.

angle of roll N damping moment

F . restoring moment

K wave exciting moment

t.

time

5 Dirac deltafunction

variance of wave exciting moment angle of roll velocity of roll expected value of yj expected value of y2 variance of Yi V2, . variance of y2 .

Y,D,E column vectors

P conditional probability density function

natural frequency in roll.

nondimensional damping coefficient

ratio

of the

coefficient of nonlinear

damping moment to the coefficient of

linear damping moment

ratio between the coefficients of the

non-linear and non-linear parts of the restoring

moment

variance of roll angle variance of roll velocity

A dot over the variable denotes differentiation with respect to time.

Appendix A.

It can be easily shown that the conditional prob-ability density function which describes the

Markov process (y1, y2) satisfies the following

partial différential équation (Caughey (1963)):

= - (a P) +

_yy

JP) i=1 i 1,3=1 (A-1) where <Ay.>

a.=lim

1 i _{At-IO At} b-= 11m <AYAYJ> At-.O At (A-2) 7. References .

.Caughêy, T.K. (1963), 'Derivation and application of

Fokker -Plank equation to discrete nonlinear dynamic

systems subjected to white random noise excitation', Journal of the Acoustical Society of America, Vol... 35, No. 11, (Nov.1963),. p. 1683.

Cornstock, J. P., (1967), 'Piinciples of naval architect-urè', SNAME Publications.

Crandall, S. H. (i963), 'Perturbation techniques for

random vibration of nonlinear systems'. Journal of

the AcOustical Soéiet of America, Vol. .35, Ño. 11,

(NOv. 1963), p. 1700.

Haddara, M. R. (1971), 'On nonlinear rolling of ships in

random seas', Report to U.S. Coast Guard, (May

1971). AlsoInternational Shipbuilding Progress, Vol. 20, No. 230, (Oct. 1973), p. 377.

Hasselman, K. (1966), 'On.nonlinear ship motions in

ir-regular waves', Journal of Ship Research, VOl. 9,

NO. 3, (Dec. 1966).

Papoulis, A. (1965), 'Probability, randomvarlables, and stochastic processes', McGraw-Hill Book Company, (1965).

Sage, A..P. and Melsa, J. L. (1971), 'Estimation theory

with applications to communications and còñtröl', McGraw-Hill Book Company, (1971).

St. Deals, M.(1967), 'Onapröblem in the theory of non-linear oscillations of ships', Schiffstechhik, Bd. 14, Heft 70, (1967), p. 11.

Vassllopoúlos,. L. (1966), 'The application of statistical theory friorilinear systems to ship motion

performanòe in random seas', Proceedings Ship ContrOl, Sys

-teñisSmpÓsium, Annapolis, Maryland, Vol. I, (Nov. 1966), p. iX-A-1.

Vàssilopóulos, L. (1971), 'Ship rolling at zero speed in

random beam seas with nonlinear damping and

restoration', Journal of Ship Research, Vol. 15, NO.

4, (Dec. 1971), p.289.

Yamanouchi, Y. (1969), 'On the considerations on the statisticalanalysis cf ship response in waves',

Selected papers from the Journal of Society of Naval Archi

-tects of Japan, Vol. 3, (1969).

We can evaluate the averages in (A -2) as follows:

a1=lim

.

At-.0

At t+dt <-(N+F)At+

_f

K(u)du> t a

= um

2

At-.0

<Ay1 b11= hrn

_At-.0

At b

_{22 At-'O}

= hm

=0 At =- <N +F> t+dt

<i-(N+F)At+ j K(u)du>

t At

=Var(K(t)) 'V0

Substitutinginto (A-1) we get equation (8). Appendix B.

Heré,we are going to evaluate some integrals

whôh are needed In tie derivätion of equations

(10) -(12).. 00 : . ffy2-(N+F)PdY1dY2 1f oo -00 00

_2°°

00 = f dy1(y.2(N+F)P

- f

(N+F)Pdy2) j-f y1P(y0, Y201Yj.' y2, t)dy1dy2=I.11(t), .

-

y.2=-oo

=._<(N+F)>+fdy1(Y2(N+F)P ) Inderivingeql1atiOflS(I.OK2) we have assumed

= 00 thé following boundary. conditions:

= _fdy2(y1y2P I -fy2Pdy2)

Y2=°° Yi=° y1=-00 .

y1 y2 P =(N+.F)P I I

y1=00 . y1=-00 Y2 Yl=°°

= '2 - f

dy2(y1 Y2 P

372=00 .

- y1=-00 j =y(N+F)P

y=oo

00

is yi(--;-1(Y2 P))dy1 dy2

00

f5

y1ì(N+F)PdY1dY2

00 .y2=00

= _f00y1dy1((N + F) P

-fy1

'j9 dy1 dy2 r2 7 y1

dy1(.j2)

fi Y2((Y2

P) dy1 dy2

= f y2.dy2(y2P I y1=-00 00 = 00

=0, i=1,2.

a Y2