Roll response of a ship under the action of a sudden excitation

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1986

t\RCH1EF

ROLL RESPONSE OF A SHIP UNDER THE ACTION OF A SUDDEN EXCITATION

by A. Yucel Odaba0. and J. Vince

Lab. v. Scheepsbouwkunde

Technische Hogeschool

'Delft

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SUMMARY

This paper examines the

_impOrtanee

_{of roll damping on the gust} _responsE of a ship. As 4 Mathematical model an

ad

hoc roll equation

is

used and the. action of-the wind gust

is

_{represented by means of a}

_step

_functict, A solution of

the

_{equation of motion is developed by} _{using the asymptotic.} method of Bogoliuhov and MitropolSki[3]. _{Computational results on the}

influence of roll damping, initial conditions and the gust magnitude are given

for a

stern trawler. _{Results cleatly indicates the importance of} toll damping. In

the

_{light of this investigation it is} _{suggested that}

in determining the gust response of

_a

ship

in 4

weather

criterion

the_

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INTRODUCTION

Jr. the assessment of intact ship _{stability by using various weather} criteria the final

_-step

_{is the determination of the maXimut} _roll angle under the action of a sudden wind _{gust. Which is assumed to Lake} place when the ship reaches a maximum angle of roll towards the wind.

In the estimation of tht gust _{response the effect of}

_roll

damping is neglected and the solution is reduced _{to an energy balance problem} which is determined graphically (see. _{Fig,1).- Apart from the differences}

in

the evaluation Of the gust exciting _{moment, various 'national} criteria also differ in their ways

an the

_{determination of the roll} angles towards the wind,

of,

_Kefs,[12]. _{Therefore the initial angle} at which the wind gust starts to _{act differs betWeen different national}

'criteria,

The aim of this work is _{to study the effects of the initial angle.} variations and roll damping on the

Loll

_{responEe of a ship under the} sudden action of a wind gust which iS represented by a step function To represent the roll motion dynamics an ad hoc equation with _non-Linear damping and restoring _{moment is used,} _{The method of solution is}

a

modified version of the asymptotic method of Krylov and Bogoliubov[3] which is presented in the Appendix, _{In oroler to make the explitir} scittion simple the restoring moment in the examples is aNpressed as.

a sum of a linear and a cubic term although _{the method can comfortably'} tccommcdste any kind of non-linearity.

The resultsindicate that the

linear damping coefficient has

_{a pronounced}

effect on the maximum r.:,11 angle _{espeaially when the initial angl. (at} which the gust starts to act) is large. _{Numerical experimentation has} indicated that the difference _{between the computed maximum}

response and

the

_{value obtained by area balancing increases with incTeasing gust}

moment, It' is therefore advisable _{rict to neglect dynamic. effec'ts}

In

such calcula,:icris,

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2 _{PROBLEM FORMULATION AND METHOD OF SOLUTION}

Following the common assumption in uazious national weather criteria let us consider a ship which under the action of waves heeled to an angle 00 (see Fig.1) and than subjected to a sudden wind _{gust with the} wave action ceasing to exist*. If the restoring moment of the ship

is expressed as a function of the angle heel, i.e.

M ° g(0)

the usual procedure reveals that the settling heel eagle (i. . the angle that will be approached as the time tends: to infinity) is deter-mined trom the solution of the equation

g(0s) Mg (1)

where M is the gust heeling moment and

0s is the settling angle.

Determination of step response to a, sudden gust is determined _from the principle of balance of energy, excluding the damping effects:

Om

j[g(0)-M

] do = 0

where Om _{is the maximum roll angle following the action of} _{the wind gust.}

' To reduce the degree- of Complication let

us assume that the righting moment be expressed as

g(0)

Bcp3

₍₃₎

(2)

* The reader can assess for himself the logic in _{the Sequence of events} and their relationship with reality,

I.

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then the settling and the maximum

_responee

_{angles are obtained from} the solution of the following equations

3 B s -Mg- .= 0 .13 4 A 2 _A 2 B ,L4

u

_A

-

6 4' m

m

g m

2 .o 'fo 't *O1 =

If, however, one would like to consider the ship's reaction as a dynaMic system it would be more appropriate to study the response of a ship by trying to solve the equation of motion to a step function excitation. To a first approximation rolling motion of _{a ship can be} described by the equation** (where the restoring moment is simplified):

where I is the virtual mass moment of inertia, _{N1 and N2 are linear}

0

and non-linear damping coefficients to be derived from forced rolling tests, U(t) is the Heaviside step function, and

₍₀₎

_{denotes the} differen-tiation with respect to time,

Dividing both sides of the equation with I _{one obtains} 0

02); 0;20 to3

wu(t)

where 2y111/I0, co2A/I,

c=B/Io

and W=Mg/Io.

. .

To determine the roll response to a gust action equation (7) needs to

, be solved with the initial congtions

) 410, ;(D) = 0 ₍₈₎

(.2y

(41-+ N 02-)

0 +

4

863

= M U(t)

_

** See Ref [4) for a discussion _on

_the

_{equation of motion.}

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Following the modified asymptotic method of Krylav-Bogoliubov, Murty(51, the solution is expressed in

rhe

following form:

4

t) + k a e (a,W) + 0(e2)

0( 0-6

7

where a--=a(t) and 11)=*(t) varies with time, k Lakes values 41 depending on

the initial and settling conditions, and

Ne2)

indicates that the terms neglected are 'at least in second order in e.

In accordance with the procedure outlined in the Appendix the non-linear part of the equation is

f(094)

=

Y10 0 - 0

and by virtue of (4..8) and (A.10)

3 1

. f Lf1

+s

--k

20s (34YT )4.e. {(62PY

-0 c=0 -s

( 9 )

-y)-3] (a/2)

=IP 2

+ _{k[A1-3(1+yyl)1(a/2)3 1+e} t-kclys[34112+IMe/2)

[3(14-1Y)tAY1](a/2)31+e2*0 (a/2)212y1 (A-y).-.3]:

e-405012)2[3+2yi(X+10l+e34ifyi(X+*)-11(a/2)3 -31,b 3 ke (1+i1(A+y))(e/2) f(0,,O) = 0: (11) (10)

I.

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1

F (a) = - ---4cO a2

(34-2yY1 )

o 2 s

F1(a) - 2[-s,-( _ )-3)(4/2)-4-kfty -3(14.1,Y

Ma/2)3

F2(a) = 45(an)2[2y ( -y)-3]

F3(a) = (a/2)3[Y1(X-/)-11 (12)

G1(a) = -k0:[3.411(.+Y)](4/2)43(.1+YY )tAy ](a/2)3

G (a) = -0s(612)[3+2y (X-py)]

G (4) = Ic(a/2)3(1+Y10.+Y)]

22

-Where A, = (y- to ) , and for n:!..4 On(a) =

Fu(e) =

O.

Consequently, equations (A,12) and (6,13) become

yaAl-(y+.2A)A1-1-ya2Bi-2X01=20:(yi(A-y),3](e/2)

2k[Ay 3(1+y,(1)](a/2)3. (13)

-(2A)A3.--ya2W.,2AaB11 =-202[34-Ys (X+Y)](a/2)

- 21([_{3(1+YY1)t)0(1](a/2)3} '(14)

'Solving for Al(a) and

B1(a) one

obtains

A1 (a) = =-Y 0 (a/2)-(kJw2)13y+), (w 21(2)](012)3 (15)

s

2

B ( ) = _{s(3,iry1)/2X-(kA/2w2)(3+2YY )(a/2)2} .

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By using these results one can construct a first approximation as

da .

dt * (a),

A,

Which after substitution takes the form

da_3

2 dt ='- pia - kp2a

d*

'

cl7 = P3 -

191a-where 2 P1 = 1""14)s/2 p c[3y+y

_{(w2+2y2)]/(81.02)9}

, ₂ ₁ = l+E0-2 (3+YY1)/2A, P4 a.(3+2yy )/(8w2). s a°P1i 2 2P1 2

[(P1*kP2ao)e

-42ao] a(t) -(17)

Donating

a(0)=a0

and *(0)=*o, the solutions for the first order approxi-mations are obtained as

(18) 2 131+192a P3t P4 kn [ ₂ (19)

*(t) =

₁₁₎₀ 2P2 131+

kP2a

₀

Having obtained a(t) and *(t), the roll angle and the roll angular velocity are expressed as

1 *( 1

0(t) = Os +

le(t)et)

+

Tka(t)e-*(t)

(20a)

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The initial conditions (8) demands that

a(0)e")(°).+ ka(0)e-*(0) =

2(00,0s)

- (21)

.(A-Y)a(0)e0j-k(X+y)a(0)e7*(°) = 0

whence

a(0)e*(°)

(00)(x+y)/A,

ka(0)a-0(0)

=

0 -0a)(X-Y)/X.

Eliminating the exponential factor, a(o) is _{obtained as}

[ (0)]2 = -k[(00-013)w/X]2.

The sign of k can now be determined from the requirement that a(0) must be real. _{Since for the rolling motion A2<0, k}

needs to be +1. Hence

ao = a(0) =

(0 -0

_o _s)w/a A=ia (22)

Equations can further be handled to yield (with _k=+1)

1 [e

*(0)

a-*(0)] = (d,0 s )/a =

7 [ego)

e-11)(0)] y(00-05)/(Aao) = -iy/w

indicating that *(t) is

a

purely imaginary

function

_{of time, i.e.} tp(t)=16(t),,such that

cos 6(0) = a/w , sin

e(o)

= - y w resulting in

6(0) = - tan-1 (y/a)

(10)

and

- -1 P1.41)2

2.

e(t) = pt - tan

(y/a) t r

Ln [- -2]

P2 p +p a 1 .2 o where 2 P; = a 4' 08(34-YY1)/(2a) and. pi = ea(3+2yy1)/(8w2).

Consequently, the first order approximate solution

= Os

+ a(t) cos 6(t)

where a(t) and e(t) are defined by equations (18) and (24) respectively.

Further improvement on the solution can be obtained by computing the function u1(a,0) which, following the procedure outlined in the Appendix, is obtained as follows: O 20 a2 s

u14

(a

6) = s 3+Y Y)a + -f(a2 2 -82n2-) cos 28 + 2 -2co (Z2+11 where +0 3 ' ) + 2a2 (E3+n3

(3+2y Y)/4, 82= -y1a/2, E2 = w2_4a2

'{(a .34-83n3)

cos 36

(c33-933)

sin 381 (26)

a3

= (1+yy )/8 8

a/8

E w2.4.

2 .2

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The improved solution can thus be expressed as

(1)(t) = (1)

s

+ a(t) cos

e(t)'+

_eu1(a,e)

-which will be used to compute the roll response of a ship to a step function excitation under varying initial conditions, damping charac-teristics and excitation magnitude.

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3 APPLICATION OF THE METHOD

To analyse the effect of roll damping on the peak roll response of _a ship under the action of _{a sudden wind gust, the method described in} the previous part has been applied for the calculation of gust _response of a stern trawler. _{Principal particulars of the ship and} _the

coefficients of the equation of motion are given in Tables 1 and 2 respeCtively. _{Computations have been carried out up to first order} accuracy, including the correction function su1(a,0). _{Variation of} the functions a(t), 0(t) and eu1(a,0) _{of the solution}

s + a(t) cos8(t) + 00-1(4iP)

with time for a sample calculation CMg=4965, _{00=-0.4) are illustrated} in Fig.2.

To make calculations indicative of the effects of the variation of roll damping and of gust magnitude four values of Ni, N2 and M have been used while keeping the rest of the coefficients _constant. _Computations indicated that for this particular ship and the loading condition

variation of N2 had only a marginal effect. _{Variation of N1, however,} indicated the strong dependence of the peak roll response on the value of roll damping moment. _{Figure 3 illustrates these results for} _{M =198.6.}

In this figure 0101 indicates the fesponse _{angle obtained by the energy} balance method (see Fig.1). _{An examination of the response} _{curves and} the comparison of peak _{response values with corresponding 0102 shows that} ,the maximum difference reaches to 0.193 _{radians (-11 degrees) sufficient}

to make a ship stable or unstable in a weather criterion evaluation. An important point to notice here is the increase in the role of roll

damping with increasing initial roll angle.

Computations carried out with different wind gust moment magnitude confirmed the above trend and indicate

_a

_{difference of .5}

_to

_{10 degfees} it the peak response Values compared. to the corresponding Om values. Furthermore the differences increased with _{increasing gust moment} magnitude.

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4 CONCLUDING REMARKS

From the limited study presented here one can clearly notice the impor-tant effect of roll damping- on the peak response of a .ship to the sudden action of a. wind gust. Since one Li interested in the response peak gust duration has not been included in the study. _{If however gust} duration is less than the.half of the natural roll period the over prediction by the energy.balance Method will become more pronounced.

In the light of this investigation it is suggested that if the gust response has to be included in a-weather criterion the tole of damping

should be taken into account.

ACKNOWLEDGEMENT.

The authors express their thanks to the Council of the British Ship Research Association for permission to publish this paper and to the Department of Trade (UK), Marine Division as the financial sponsor of the work on which it is based.

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REFERENCES

Intact Stability, Including Analysis of Intact Stability -Casualty Records - Weather Criterion. IMCO Document STAB

27/5/4, Submitted by Japan,

1982.

Intact Stability, Including Analysis of Intact Stability

. .Casualty Records,.- Weather Criterion. IMCO Document STAB

27/5/3, Submitted by USSR, 1982.

Asymptotic Methods in the Theory of Non-Linear Oscillations. BOGOLIUBOV, N.N. and MITROPOLSKY, Y.A. Hindustan Publ. Corp. Delhi, 1961.

Hydrodynamic Reaction to Large Amplitude Rolling Motion.

ODABASI, A.Y. Int. Shipbuilding Progress, Vol.28, p.74, 1981.

T[5] A Unified Krylov-Bogoliubov Method for Solving Second-Order Non-Linear Systems. MURTY, I.S.N. Int. j. Non-Linear Mech., VO1.6, p.45, 1971.

[6] _{On Perturbation Methods. Involving Expansions in Terms of} _a

Parameter. BELLMAN, R. Quart..

of Appl.

Math. Vo1,13, p.195,

(15)

Table 1

Principal Particulars of the Sample Ship

* Displacement

mass

Table 2

Coefficients of the Equation of Motion

Ship Stern Trawler

Lpp (m) 56.85 B (m) 12.20 D (m) 6.8 T (m) _4.10 Trim (m0-aft) 1.20 CB 0.515 A (tonnes) 1567

Ship Stern Trawler

63555 0 Ni 6172 8 N2 10735 A 10545 1316 N. 198.6

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FIG. 1

DEFINITION OF GUJASI-DYNAMIC ROLL ANGLE

(17)

0 8

r

0.4

0.7

I-O. 3 Ox 1O

RI

, A', EU x10 2

0.6-0.2

.

0.5-0.1

0.4 -

0. 3

-- 0. 1

/

0.2 --0.2

. 0.

-O. 1

- 0.4

./

= 63555.0

NI = 6172.0

=10735.0

A =10545.0

B

=1316.84

Mg = 496.5

\ 00 = -0.4

I. \

12 \

14 16

/

--

0. 5

FIG. 2

VARIATION OF THE SOLUTION

COMPONENTS WITH TIME

-e

A

EU

/

18 20

time

(secs)

(18)

0-3

0 dians)

0.2

01

O3

-0.1

-0.2

-0:3

-0.4

0

diant)

0 .2 0.1 0.1 0 .2 12

N1= 51440

N1 = 77160 time 20 (secs) time 20(secs)

0-3

(radians)

0.2 0'1 0

-02

0.3

0.4

03 (radians)

0.2 0-2 = 6172.0

FIG. 3

_{EFFECTS OF ROLL DAMPING}

_{AND INITIAL ANGLE}

ON GUST RESPONSE

12 N1 = 9259-0

time r

0 (secs) I time r 20 (secs' sfro

(radians)

orn

-0-1

01376

-0-2

0.2386

-03

0.3380

-0.4

0.4389

01 -0-1

.0'3

-0.4

(19)

03

0

'ad ions) 0.2 Mg = 397.2

04

0.3

0 (radians)

0.2

O!1 0.1

02 -03

_ - 0-4

AMrLJMITI AI

t mni

P z_

ern(radians)

,.jvg 198.6

297. 9 397. 2 496. 5

-0.1"

0 .1376 0-1561

0 -1760 0 .1942

-0.2

0. 2386

0- 2564 0.2757 0-2949

-0.3

0-3380 0.3568

O. 3770 0.3950

-0-4

0.4389 0-4571

0 .4767 0 -4962

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APPENDIX

THE METHOD OF SOLUTION

Let the equation of motion of a single-degree-of-freedom non-linear system subject to step function excitation _be

X + 2yic + w2x + ef(x) _{= Mgt)}

(A:1)

where x.=x(t) is the system _{response, y, w and W are positive}

constants. e is a small parameter, an over dot denotes _{differentiation with}

respect to time, U(t) is the Heaviside step function defined by

0 for t 0,

U(t) = (

1 for t O.

(A,2)

and f(x,k) is any analytical non-linear function of x and

As the time tends to

infinity

_{any stable motion will tend approach to} stable position, i,e. x(t)-oc.s as t4co, where acceleration and velocity

vanish, i.e. X4.0, 3140 as t-).,=. _{The value of xs can be determined from}

2

xs + ef(xs, ) = W.

The'solution is sought in the following _form

x(t) x +

e*

+ k --e-* +

s 2 .2

(A.3)

l(a,0) + e u2(a,*) _(A.4

where a(t) and

OW

_{are time varying amplitude and}

exponent, respectively, ank k can take values +1 or -1 depending on the conditions. _{The functions} a(t) and *(t) are expressible as

(21)

da dt la

e41(4)

+ ek2(0.) !=1, = A + eB (a) + e.2B2(a) + . dt where .2 2 = w )

Substitution of (A04) and (A05) into the terms

M+2r142x

yields R+.2y11+w2x=w2x0+lee/q-yaAi+(y+2X)A1-ya2B1+2XaB1l 2 1 u -ikee-/qyaA1,-(y-2X)Acya2B1-2iay+el(yar" 2 aa BMa* 2u 2u .,2Aya

1 +2

D1 +

2)ty

4 + w-u1} +

0(e.2)

2 .

IP2

au

Where 0(e2) denotes terms of second and higher

order

powers of e.' The non-linear term can be expanded into a series in terms of e as

2raf dx af clic]

.

ef(x,i) = E[f]

0(2)

(A7)

e 'ax de

a

de _c=0

iC0fi = f0 = f(x + ee*

+ kae,

(A-y)ae* - x

= (A.5)

(A.6)

ik(X+y)ae-*). (A.8)

au

Y a aa

(22)

Substituting (A.7) and A.8) into (A.1) and equating the _{coefficients of} E we obtain , 2 2 2-

aul

3u1

2 a u1

aui

2 (ya) -3a2 aa

y a 2Aya isT-p- + X + 2Xy - +

w ul

alp

42

= - ya2

-.A2XaB1

+ iitel[yaAl (y-2X)

-

ya2B1 - 2XaB1] + f(xs,0) - fa.

(A.9).

By using the definition Of thethe difference

fo-if(Xs,-0) can

_{be expanded}

as

fo - f(x8,0) = F(a) + E [E.

-We"

+ (a)e-1141

n=1 n

and substitution of this expansion in (A.9) Yields

F (a) + elqyaAi-(y+2X)A1+ _{-'2XaB1 -2F} (a)1+ike'llyaA!

1

(y-2X)A1-ya2B;72XaB12kG1(a)]- E [Ft(a)en4)+Gt(a)e 4=2

In order to avoid the appearance of secular terms in u1(a0P) the coefficients of e in (A.11) are equated to zero

ya.A! - y+2X)A1 + 1a2Bi - 2XaB1 = 21' (a)

1 IP) (A,10) 2 2 a2u

3u

i

2

3u

- Y2a 1 3u1 2 .1

(ya) 2Xya - + 2Xy + w u ₌

.aa .2 Ba Da34, 1 r! tr.

ft

I

It

(23)

from which the unknown functions Al(a) and

B/(a)

are determined. The rest of the equation (A:11) yields a partial differential equation for the determination of the unknown function u/(a.,*)

2 ,2 ,2

2 a u1 2 au1 ° ui 2 _ 1.02u

(ya

2 a aa -2Ala Ma* + 2Al alp +

aa.

= -F0(a) - E (F(e)e' + 011(4)e-"].

n=2

One may assume that u1(a.,0 can be expressed as

u1(a'V) = v

_{0(a) +} 1.

Iv

(a)en* + w (a)e-1141

.

n=2

and the equations to yield the unknown functions vo(e), 7(a), %(a) are obtained by substituting the expansion (A.15) into (A.14).

vn(a)=Fn(a),(n=0,2,3,..),

- (ya)2v:;(a)-y(y+2nA)av;( ) +(n2A2+2nyX+w2

_ 2 " 2 2 2

(ya) wri(a)-y(y-2nA)aw;(4)+(n A =2,nyX+w )wl'a(a)=-Gn(a),(n=2,3,...);

Although the method presented may be thought of as a small patameter expansion technique and hence only

valid

for small values of

e,

this in fact does not pose any serious drawback. Following Beilmen[6] anOther .perturbation parameter

v may

be defined as

v.= c/(1+e)

which for all positive c takes values between 0 and 1, i.e. 0.1p<1.

Therefore all the previous analysis can be repeated with c replaced by p .since

2 3,

E =

mu+ .11

+ o(p3)

(24)

which implies that the first order problem in p coincides with the first order problem in e (e being replaced with p) and differences _{starts to} exist in higher order approximations, i.e.

gf(x, "o+p2+....)[(41,0 + p(df/dp)11.0 +