Comparison between simulation results and theoretical predictions for a ship rolling in random beam waves

COMPARISON BETWEEN SIMULATION RESULTS AND TlffiORETICAL PREDICTIONS

FOR A SHIP ROLLING IN RANDOM BEAM wAttefR University Of Technology by Ship Hvdroineclsanlcs Laboratory J.B. Roberts'

Mekelweg 2 - 2628 CD Oelft Tha Netherlands

Summary Prrone. 3 l is 786373 - F a x : 31 1$ 7S1836 A simulation study has been undertaken of the roUing motion of a ship in iMidom beain waves. For this pur-pose it was assumed that a single degree of freedom equation of motion was appropriate, with non-linearities in both the doping and restqriiig moment terms. By generating long realisations of a suitable random input process, and numerically integrating the equation of motion to obtain corresponding realisations of the roll response process, simulation estimates of the roll amplitude cumulative distribution, and the standard deviation of roU, were obtained. These estirhates were coinpared with theoretical predictions, for a realistic range of parameters. I. Introduction

Under certain conditions it is possible for a ship, when operating in a random sea state, to exhibit severe rolling motion, which may lead to capsize. This is a phenomenon of the resonance type, which arises when the natural frequency of roll motion matches the fre-quency at which wave energy in the sea is dominant; it is exaceibated by the relatively low hydrodynamic roll damping which is usually present.

The important problem of predicting the statistics of roll motion, in a random sea state, has received con-siderable attention in the literature over the last two decades (e.g. see References [1 — 8] . Most effort so far has been directed to the simplified case of un-coupled, single degree of freedom, roE motion in random beam waves. Even here the analyi^ is diffi-cult due to the non-linearities whidi are present in both the damping and restorii^ mom»it terms, and to the distinctly non-wlüte character of the wave exci-tation process.

Recently the author has proposed a new approach to tiie analysis of uncoupled roll motion, which is based on a cpmbiiiatipn of stochastic averaging with Markov process theory 191. This method allows one to cope with arbitrary non-linearities and results in ex-pressions for the probability dhtributipn of the rbU amplitude, the standerd deviation of roll motion, and various other statistical parameters. It was shown in Reference [9] that the theory gives results in good agreement with some simulation results due to DalzeU [10,11]. It has also been shown recently [12] that the theory leads to predictions which are in good agreement with some experimratal results from a 1:20 scale model of the fishery protection vessel the 'Sulisker*, rolling in beam waves in a wave tank.

To furtbsr test the validity of the theory desaibed in Reference [9], a digital simulation study of ship rolling in beam waves was undertaken by the authoi.

*) School of Engineering uid Applied Scienen, Univenity of Sussex, Falmer, Brighton, UJC.

For this purpose a sirigle degree óf freedom roll motion equation was assumed, Pf the same type as that con-sidered by Dalzell [10,11]. Realisations of an ap-propriate wave moment input process was generated digitally by a simple but very efficient algorithm, and the equation of motion was integrated numericaUy to obtain corresponding realisations of the roll motion process^ Due to the rapid method of generating the input process it was possible to obtain very long reahs-ations of the output process, and hence accmate statistical estimates of the roll response.

In this paper the details ofthe simulatiPn prooediu^ are discussed and some typical results are presented; these are compared with corresponding tluoretical estimates obtamed by the method of Reference [9J. A simple modification to this original theory is pro-posed which leads to improved accuracy in the theoret-ical prédictions.

2. The simulation procedure 2.1. The equation of motion

It is assumed here that an appropriate equation of motion for a ship rolling in random beam waves is of the form :

li + ßci4>) + K(»t>) = M{t) (1) v/hen ! is the roll inertia (including tfae added, hydros

dynamic, inertia), ^ is the roll angle, c(i) is the non-linear damping moment, K(<l>) is the non^ear restor-ing moment and is the roll excitation moment. Some justification for the adoption of equation (1) may be found in Reference [121, where it is shown that, in a linear, t^reatrnent, the roll equation may be uncoupled from the sway motion, by an appropriate choice of origin for the coordinates. Moreover results derived from equation (1) have been found to be üi good agreement with experimentally detennined roll response statistics [ 121,

Numerous authors, dâtiiig back to Froude [131 have shown that the roll damping can be well re-presented, in the majority of cases, by the following linear^lus-qiiadratic forin:

c i i ) - c ^ i + C2i[i\ (2) Here and are: constants, which can be estimated

from free-decay experimental data [14J. Accordingly, this form wül be adopted here for the purposes of the simulation study.

An. analytic form is also required for the non-linear restoring moment iC(^). The simplest form which can approximate r&al KQt») vs 0 chaiacteristics is the linear plus cubic form

where

(3) This is the form adopted by Dalzell [10, 11] in his simulation work, and wül also be used here. It is noted that X,C^) = 0 when a critical roll angle, is reached, such that

(4) Figure 1 shows a comparison between equation (1) îuid some typical, real restoring motnent characteris-tics, normalised so that the maximum value of K(<l>) is unity (after Figure 8 of Reference [ 10] ).

On combining equations (1), (2) and (3), and dividing throughout by /, one obtains the foUowing specific equation of motion, which is suitable for a simulation study:

0 + + Cjil^l -J- A"^0 ^ Ä:^«^ = Xit) (5)

• • • • PASSENGER VESSEL, FULL UWO RCaANGULAR 8ARGE OLD WARSHIP

OLD SHElTEfi DECK CARGO SHIP Figure 1. The restoring moment characteristic.

C ^1 ƒ and (6) (7) (8) It is convenient at this stage to non-dimension-alisè the equation of motion (equation (5)). A suit-able npn-dimensiohal roll angle, 1^, may be defined by

0* (9)

i.e, ^ = 1 when the restoring moment reduces to zero. Dividing equation (5) throughout by 0* we obtain

* (10) Further, complete non-dimensionless results by replac-ing time, /, by the npn-dimensipnal time

where

(11)

(12)

(13) is the natmal frequency of undamped roll. Hence we obtain the equation of motion

r i + äiif + bhh + if - * ^ = ( / ) where a and (14)

Here ä and b are now non-dimensional damping coef-ficients arid differentiation is now with respect to T . x(t) is a non-dimensional input process.

2.2. The input process

It is possible to generate reali^tions of the process, x(t), witfa any desired power spectnnn, by means of a digital filter operating on a sequence of independent random numbers (e.g. see References [10] arid [15]). However, for spectral shapes wltich typically arise in practice it turns out that the required digital filtering operation leads to fairly lengthy corapiitations [10, 11*

151.

Since the objective of the present study is to test the validity of the theoretical method of Reference

[9], it is not essential to generate realisations with a spectrum which has a specific, precise shape. It is suf-ficient that x(t) should have a spectnim which has rtiugfaly the correct overall character - i.e. a single peak and a bandwidth whicfa is representative of real excitation processes. Accordingly, in the present study, we choose, for the input process jr(r), a process which has roughly the right character and for which reaHs-ations can be generated in a very effident, rapid man-ner, with the minimum of cpriipûtational effort. This approach enables bng realisations of the output rPU response process to be subsequently generated, with the advantage that accurate estimates of the statistics of the roU mption can be found with a reasonable computational effort,

A suitable input process, x(t), with a single-^eak spectrurh, can be generated by using the relatipn^p

x(t) - h [aiO coswpf - bit) smcj^t] (15)

where ait) and bit) are two independent random processes with zero mean and identical statistits, and A is a height scale. If ait) and b(t) art slowly varying processes (on a macroscopic scale) in comparison with sinw^r and cosw^f, then x(t) will be a pro<»^ with a single peak, at frequency w^. Moreover^ if ait) and bit) are Gausäan then xit) wiU also be Gaussian, by virtue pf the linear character of the operation in equat-ion (15).

For ait) and bit) the simplest processes which are useful are first-order processes, obtained by passing white noise through a first order linear system. Thus ait) (and &(f)>can be obt^ed from

a+ßa = ßn(f) (16) where7i(/) is a white noise process, such that

E{nit)nit + r)}=SiT), (17)

and 0 is a bandwidth parameter. If the power spectrum of a(0 is defmed by I " y(w) = — ƒ w(f)coswrrf/ " 2ff (18) wfaere w(T) = £ { ö ( f ) a ( f + r)} (19) is the correlatiori function of Jc(/), then it is easily

found from equation (16) that

72

(20)

2 j r ( ß 2 + t j 2 j

Regarded as a two-sized spectrum (—«• < u < *), Sgiüj), as given by equation (20) has a single peak at w =5 0. j3 is such that as |S «», 0(0 -* nit), i.e. a(r) ap-proaches a white noise. As ß reduces, thé width ofthe spectrum of ait) reduces, as shown in Figure 2.

On combining ait) and bit), according to equation

1-0 H 3 o«r 07 ^ 0« 0-5 03 03 1 1 1 1 1 1 1 —

\ \ "

J

Figure 2. Power spectnan ofthe process a(f). (15), one obtains a process x(r) with the spectrum

ç , x _ A V r 1_ ^ 1

4ir L 0 ^ + ( c j - W p ) 2 ß^ + iu+Li^)^}

(21) litis spectrum is obtained, graphically, by first ap-plying a frequency shift of to the spectnmi of fl(r), to the left, foUowed by a frequency shift of to the spe:Ctrum of i>(r), to the right (see Figuré 3). A

weight-• \ ^ ^ a t C T R l W OF o l t ) w LEFI-SHintO a l t l SPECTRUM

\

Figure 3. Derivation of the input spectrum, 5;f(u), from the spectmm offf(r).

ed sum of these shifted spectra will then produce the spectrum of x it), as given by equation (21 ). From this graphical construction it is evident that, when ß is smaU, the overlapping effect of the two shifted spectta isïsmaÜ and, to a good approximation

4,r L ß 2 - h ( a , - u j 2 j (22) for w > 0. Figure 4 shows the variation of the shape pf

(u) with changes of ß, according to the exact result given by equation (21) (here = 1).

nfCORtncii. iNMt SPKTU

I « » it I k

Figure 4, Tî» hiput, spectrum, Sj.(ij), for various values of the bandwidth parameter, e.

Treating x (f) as a fimction of non-dimenaonal time, r, then the corresponding non-dimensional frequericy n is defmed by

n = (23)

and the spectrum oixir) becomes

where e =• and "p L *2 + (n -n^)^ + (n (24) (25) (26) and both non-dimensional parameters.

The non-dimensional standard deviation of x(r), ff,, is given by

az = ƒ 5 , ( a ) £ / n (27)

On substituting equatipn (24) for 5^ (n) into equation (27) and integrating one finds that

2 ^'<-o^

(28)

The appropriate non-dimensional form of equation (15) is then

ffV2

x ( r ) ~ [a(r) cosßpT - bir) sinfî^i-I (29) where a ir) (and similarly bir)) is generated from

a + ea = « n ( T ) (30) and

£ { « ( r ) n ( r + u ) } = 5 ( ù ) (31) where differentiation with respect to time.

In practice, to generate x (r) at eqm-spaced mtervals, spaced Ar apart, it is necessary to first generatea(/Ar) and bifàt), frpm equatipn (30). Here it is not neces-sary to approximate the white noise process «(r), and to integrate equation (30) numerically. Instead, the foUowing exact recursive relationship can be uséd [16]

a(r + Ar) = a(r) exp(— eAi") "**

+ 1^^^ -exp(-2eAT)}]**W. (32) (and similarly for Ô(T)). Hère w. is a Gaussian raridPm number, with unit standard deviation and zero meari, with a fresh random number generated for each time step. To ensure stationarity it is best to start with an initial condition such as a(Q) - 0 and to ignore the transient portion ofthe realisation generated according to equation (32).

2.3. The ou tput process

With the input prbcess defined in the previous sec-tion, and the equation of motion as given by equation (13), the roU response process, li'(r), depends on the foUowing five parameters:

(i) a - linear damping factor (ii) b — quadratic damping factor

(iü) Hp — non-dimensionalfrequericyratio(=o;p/wg) (iv) e — non-dimensional bandwidth parameter

(=)3/a;„)

(v) (7^ - non-dimensional input strength parameter. To generate ^ digitaUy, the input process ;C(T) was first generated at closely spaced equi-spaced time stants, with ;e(T) assumed constant during these in-stants. Generally 50 samples per *cycle* were;gerierated (i.e. A T = 2ir/5Ö). A fourth order Runge-^Kutta in-tegration routine was tfaen used to compute \>(T) at the same time instants that were used for the input process (i.e. r^=»;Ar).

To obtain accurate roll statistics it is necessary to compute a large number, :V. of successive sample valu^ of the roll response (/ = 1.2 , iV), for any given set of parameters. This is partly due to the amaU time intervals used m thè computatiori arid partly due to the

riarrpw-band nature of the response process, which results in a high degree of correlation between succsive sample values of the réponse. The statistical es-ftmates presented later in this paper were obtained by using 1 ntUlion successive samples (A^ = 10*), for each parameter set .

2.4, Output statistics

From the numerical mtegration of the equation of ihotipn, yaiùès of roll displacemerit, \t'(T), and roU velpàty, ii'(T),,are computed at times T^. = / A T (ƒ =1,2,

. . . , A 0 .

For the purpose of comparing the simulation results with the theory of Reference [9], two statistical features ofthe rbU réponse.were examined.

1. The starid!Ü'd deviation of the roU response, . This is sÊmpIy estimated from a simulation cal-culation, throt^ therelation

N, f l (33)

2. The probability distribution of the rbU amphtüde, Air). Since ^ ( T ) is a narrow band process, Aif)

can be r^axded as the envelope of the peaks in the roU respond, as shown in Figure 5.

ROLL %w ENVELOPE PBOCESS / \ / \ -s

\ \ \ \

< \ \

/ \ / \ /

Figure 5. Variation of roll angle with time and a compwison with the ainplitude enyelpp_e process,

^(0-To find values of A(r), the eneigy envelope (sum of kinetic plus potential energy) is first cajlculated at every time instant (/Ar). Thus, from equation (13),

(34) This is then cpnverted to the amplitude envelope, A(y}. through the relationship

A^ A^

^ - ^ ^ V i r ) (35)

With this definition pf Air), it is evident that Air) = ^ ( T ) w'heriever = 0, i.e. the amplitude prPœss ^ ( T )

wiU 'touch' aU the peaks in the roll motion process

(//(T). It is noted that other defuutions of ^ ( T ) are

pos-sible, but the one given here accords with the theoret-ical metfaod of Reference [9], which is based on a study of the eneigy envelope process K ( T ) .

Solving equation (35) as a quadratic in A' one obtains:

^Cr)= {1 -[1 -AV]^f' (36) From the sample values of ^ ( T ) at - /Ar it is a

sim-ple matter to form a hBtogram and hence estimates for the cumulative probabiUty,?(.4),of>4(r).

Finally, the facifaty of forming spectral estimâtes of both the input process, X ( T ) , and the output process, ^ (r), was incorporated into, the simulation programme. Spectral estimates y/ere generated by performing FFFs on contiguous blocks, and subseqûâritly aver^ aging over the blocks.

3. The theoretical method

Implicit in the foregoing discussion is the assump-tiori that, for the rPU equation given by equatipn (13), the roll response i>(f), is a station^ process provided that xit) is also stationary. However, due to the pre-sence of the negative cubic restoring terrh, this is not, in fact, correct. It can be shown theoretically that tfae solution ^(r) wiU never achieve stationarity, since the roU response wil eventuaUy exit from the regime of stable motipn in the phase plane 19]. Thus strictly, it is improper to discuss stationary statistics such as ff^ and?(X).

Despite this difficulty, it is possible, frdiri a prac-tical viewpoint, to treat the output proMSS as station-ary provided that the input level is suffidently weak. In these c^cumstances the probabihty of \£'(i') exceed-ing the regime of stablè operation (i.e. a câpazè oc-curring) is negligible smaU; then both the simulation method discussed here and the theoretical method of

Reference [9] cari be used to determine estimates of cr^ andPiA).

This tlieorètical method is fuUy described in Ref-erence [9] and the details wül not be given here. It is worth pointing out, however, that the method is essentiaUy an asymptotic orie, which wiU become in-creasingly accurate as the roU damping approaches zero. More precisely, the ratio

^ - bandwidth of output process

bandwidth of input process (37) is the parameter of importance here; wheri iî is small one can expect the theory to give good results. The error incvored when R is finite is not easy to estimate theoreticaUy and this is one of the main reasons ^yhy a comparison between theory and smtulatiori is valuable;

it provides a ^ide to the magnitude of the error in-curred through the approximations in the theory. 3.1. Tbe modified theory

Althoi^ some account of the shape of the input spectnmi is taken in the original theory, described in Reference [9], it is limited in this respect. Here propose a simple modiflcatiori to the thePry which enables further accoimt to be tak^ of the shape of the mput spectrurh and can be expected to give im-proved agreement with the simulation estimates. Both the caigînal and modify t î n m e s wiU be used here, forcomparispn ynth the simulation study results.

to apprecäte the limitation to the original theory, referred to above, it is sufficient to consider the ^edal case of a linear restoring moment. In this case it turns out that only the value of the input spectrum, S^i<*i), at w = is required [9] ; i.e. the basic as-sumptwns in the theory are equivalent to making a white noise approximation for xit) with a constant spectral level, as ^pwn in Figure 6. This observation is finther supported by conadering the even mtïre par-ticular case of linear damping and linear stiffness. The original theory tfaeri gives a result for the staridard deviatiPn of the roU motipn which agrees with the exact result for white noise exàtation, with spectral levelS(w^)[9]. S f u l WHITE NOISE APPROXIftATION OUTPUT SPECTRUM INPUT SPECTBUW

Figuré 6. Irtpùt and output spectra and the white noise approx-imation for rfie iiip'ut.

For the linear case the error in approximating the mput as :a white noise can be readily calculated from the standard theory of linear systems. Thus,

a | =2ir ƒ |û(aj)!-5^(£j)dw (38) where aiu), the frequency response function of the system is given by

1

a(w)=-— + 2 / f w c j

(39)

and f is the usual critical damping factor (f • = a/2). For a non-white input spectnmi, S^iu), cm be. eiräluäted nimericaUy, from equation (38). . .

Let ff^^ be tiie white noise approximation for a^^, obtained by setting 5(w) = 5<Wg), a constant, iri equation (38). A ratio, r(f ) will be defined as

(40)

'RW

where is the exact result, for a linear s^ystem, ob-tained from equation (38) by using the correct, non-, white, input spectrum. As indicated, this ratio r will depend on the damping factor, f, and also on the shape of the input spectrum.

The function r(i-), cPmputed fro'm linear theory, can be used to correct, approximately, the theory given in Reference [9], for the general case of nonlinear stiffness ^.d dairiping. The idea is to replace 5^(£J), in the original theory, by a modified spectnim

5i(w) = r(f)5,(u) . (41) Of course, in the Éneai case, this will have the effect

that ff^ is now given exactiy. In the non-üiwar case one must chpose an effective value of f, which will depend Pn tite energy level, V. This has the effect that 5^(cj) becomes a function of K, in the Fokker-Planck eijtiation forTgiven in Reference [9].

A method of dioosing an effective f value can be found by considering the' case of free decay. Setting xir) = 0 m the equatipn of rnption (equation (13)) pne has

i +ai +bi\i\ + ^ -.^^ =0 (42) If this equation is averaged, by the method of

Refer-ence [9], one obtains an approximate equation fpr the free decay of the energy envelope, V{T) (see equation (34)). The result may be written in theiform

1

2V dr ^ where

QiV) = —- [ai +bi\i\]i är

(4.3)

(44) and f(V) is the periodic time of undamped oscillat-ions, at energy level V. The function QiV) ^ directly related to the fractional energy löss per cycle due to damping.

In the linear case b ~ 0, TiV) ^ 2v and one finds, from equation (44), that

ö ( n = t . (45) In the more general, non-linear râse, QiV) can 6e

regarded as an appropriate, amplitude dependent, effective f value and one can replace f in equation (41 ) by <2(K)-i.e.

S'iu) = r[QiV)] 5,(w) (46) may be used to oairect the spectral level, at each value qf V. More details on the method of determining the function QiV) may be found in Reference [9].

This proposed modification is a rough method of accounting, to spme extent, for effects of the shape of the input spectrum which are not accounted for in the original theory.. Its value as a means of improving the theory wiU be assessed throu^ a comparispri of predictipris with simulation results.

4. Comparison between simulation and theory

A number of cases were chosen for comparison pur-poses, each case prescribed by a particülar choice of values of the five parameters, a, b, e and a„. We first describe the rationale behind the choice of values for these parameters.

4.1. Choice of parameter values 4.1.1. Damping parameters a and b

For most ships the roU damping is principally quadratic, with the result that the Unear parametej a is ùsuàUy srriaU (typicaUy in the range 0.01 to 0.1) and ,the quadratic parameter b is usuaUy relatively large (typically in the; range 0.1 to 5). For example, from the comprehensive set óf model tests on the SuUsker, which had a bare hull, the foUowing values were de-duced [ 141 :

a = 0.0214 b = 0.853

(note that a « 2a*, b = 2b*<l>*. where a* and b* are dèfiriéd in Reference [14]; also it is ^sumed that 0 * s 90* for the Suli^er). This resist is shown in Figure 7, where a is plotted against b, together with simiUff results obtained from a variety of ships, with and withoiit bHge^eels (see also Reference [ 10] ).

In the Ught of these results, values ofa = 0.01,0.04 or 0.07 were chosen for thè cases studied here, b values \yere chosen from amongst the foUowing: 0. 0.2, 0.6,

1.0,2.0:and 3.0 (see Figure 7). 4.1.2. ß values

p

iri practice severe rpUing is usuaUy encountered in resonant conditions — i.e. when the roU natural fre-quency is close to the frefre-quency, at which l:he input spectrum is a maxiraurn. This implies that fl^ is close to unity. In the am.idation study reported here, n^. isi in most of the cases, either 0.9 or 1.1.

4.1.3. Bandwidth parameter, e

To iUustrate some typical shapes fpr input spectra.

0-1

0 01

0-001

t I I 1 I I 1 1 ] 1

O WITH BLßE KEELS • WITHOUT BItGE IŒEIS M SULISKER

X SIMULATION

- T — I I I t I I I - I 1 I I I I t U

r

O'OI O'l

Figuré 7. Typical value» of Ûie linear damping factor, a, and the quadratic damping factor, b.

we show, in Figure 8(a), some wave moment spectra derived in Reference [12]. The wave morawit spectra were dwivcd from the wave elevation spectra through a Unearised analysis [12] and here are presented in a normalised form, such that the peak value is unity arid the frequency at which this peak occurs is also unity. For coniparison purposes, Figure 8(b) shows the theoretical input spectrum used in the preserit simulat-ion study, for several « values, normaJised in the same way as in Figure 8(a). Comparison of Figures 8(a) and 8(b) shows that a value of e = 0.3 gives a theoretical

— ' • ' '1 1 1 1 / / \ \ D*«SET * - 3 l i : i • •

/ / / ^ ^ ^ •

F^ure 8(a). Typical wave moment ^eclia.

S S i Z M S 1 1 1 r / t-toi c . 5 7 •

.spectrum which has-a bandwidth amUar tp the spectra shown in Figures 8(a). e = 0.3 was the bandwidth value chosen for the majority of the simulation work reported here.

4.1.4. Iriput strength parameter,

As already mentioned, if is too large then the response process will exit the stable regime in the p h a » and it cannot be treated as a stationmy process. In the present study a^ was adjusted, in each case, so that a re^priably large aniplitude respÔTÙe was achiev-ed, but not luge enough to precipitate instability. Gener^y speaking, it was found that the response prpcess remamed bounded (i.e. stable) provided that ff^ < 0.2, where is the standard deviation of the non-dimensionai roU response process, ^ (r).

4.1.5, Ùîosen parameter values

In aU, 14 cases were chosen for the simulation work, each case being defined by a set o f values for a, b. . e and ff^. The appropriate parameter values for eadi case are tabulated in Table 1.

Table 1

Parameter values for the various cases

Cass No. a b e _°x I d.oi ~o.6 09^ 03 0.012 2 0.01 0.2 0.9 0.3 0.03 3 0.01 1.0 0.9 03 0.06 4 0.01 2.0 0.9 0.3 0.10 5 6.01 3.0 0.9 OJ 0.13 6 0.01 1.0 0.7 0.3 0.075 7 0.01 3.0 0.7 OJ 0.10 8 O.OI 1.0 1.1 0.3 0.06 9 0.01 3.0 1.1 0.3 0.13 10 O.OI 1.0 03 OJ 0.08 11 0.01 3.0 0,9 0.7 0.15 12 0.04 0.2 0.9 0.3 0.04 13 0.04 1.0 0.9 0.3 0.07 14 0.07 0.6 0.9 0.3 0.07

4,2. Spectral analysis of input arid ôûtpiit processes

TP check that the simulatiPn prpgrariime was genör-ating the correct input process, and to examine the bandwidth of the output process, in typical cases, a spectral analysis was c^riied out on both the input and output processes. Figure 9 .shPws some typical results of such an analysis for Cases 2 and 3 (here the vertical scale is arbitrary). It is observed that the: spectral es-timates for the input process agree very closely with the t;heoretical spectrum. Moreover, the bandwidt:h of the output process giectrum is seen to increase with in-creasing damping, as one would expect. Even for relatively high dariiping (b ~ 1.0), however, the

band-• Hwûr

•loufwr

rwofltriui.

IMKIT s p t c n u H

Figuré 9. Spectra of the input.and output processes, fl «'0.01, n = "09, e •= p J . 6 = Ö.2 aàd'ï .0.

widtii of the output prpcess is appreciably less than the bandwidth of the iriput process. Thus the bsysic require--ment of the theory is satisfied for these values of darnping (see Section 3).

4.3. Roll anplitude distributions

F o r each case, the simulation estimates of the cûmuiative probabiUty distribution, PiA), of the roB amplit^ide procew, X<r), were plptted against A on Rayleigh probabiUty paper. This paper is scïüed such that a Rayleigh distribution appears as a strai^t Une, Also, for each case, the corresponding theoretical predictions were plotted, as computed from the oi^igin-al theory of Reference [9], and the rnpdified theory of Reference [ 12]. The appropriate plots for Ceases 1 to 14 are shown in Figures 10—23, respectively.

Figures 10 to l 4 show the effect o f ^adually m-creasing the magnitude of the quadratic darapirig fac-.

P(AI M9W M 9 M 0-M93 0999 049« 0 9 » 8^99 i<9t M 7 M S 09 0-8 07 0-5 oi. 03 02 O'l SIMUUnON ESTmATES THEOm-- « 0 01 02 03 O i OS A Figure 10. The cumulative probability datribution fot Ait), a = O.OI, b = 0.Ó, fï^ = Ö.9, e = Û:3,\ = "Ö.'Ó 12;.

P U I 09999 «99B «Î99S 0999 099S 0995 099 0 « 097 O » M M M »6 05 « 0? Ol —1 1 • " T ' 'SinUUTION ESTIMATES 0 1 »01 a«>0'9 c . 0 3 MODIFIED THEOltY 02 O'L 05

Figure 11. The cumulative probabiUty distribution for Ait).

a = QDl,b = Q2,ap = 0,9,e= 03,0^ = 0.03. PIÄI (h9999 »99S 09995 09999 0990 099S 0 « 0 > 041 D9t &> » 097 ftp'09 0-95 t > K 09 0« 07 os O i 03 03 01 -w

SIHULATION ESTIMATES CASE V

MODIFIED THEORY

ORIGINAL THEORY

02 03 05

Figure 13. The cumulative probabilit>' distribution for.4(r). a « 0.01,6 = 2.0, fip = 0.9, e = 0.3, o^ = 0.10. P(A1 09999 M995 0499 099B 0495 099 090 04T 0-95 09 OB 0-7 0^6 OS O'L 03 02 0-1 -1 r 1 1 1 • SIMULATION ESTIMATES jf^^^^} MODIFIED THEORY . / / / -a > 001 bm 1-0 0 • O'Oa ORIGINAL THEOffr 02 OS

Figure 12..The cumulative probabiUty distribution forj4(r).

a = 0.01, !> = 1.0, Öp =• 0.9, e = 0 J , = 0.06.

tor b from 0.0 to 3.0, with a = 0.01, = 0.9 and e = 0.3, For very Ught damping ib = 0.0), Figure 10 shpws that the theory gives excellent agreement with the simulation results; this can be expected since here the bandwidth ratio R (see equation (37)) is very smaU. Moreover, the modification to the theory, proposed in Reference [12], here has a negligible effect. As b increases (Figures 11 to 14) there is a tendency for the original theory to increasingly underestimate P(A) but Üie modified theory gives, in each case, an improv-ed agreement with the simulation estimates.

P U I 09999 0 9 9 « 09995 pW9 0>99B 0-995 0 « 0-9B 0'97 095 09 D l 07 06 OS 01. 0-3 02 01 _j 1 1 ^ • SIMUUTION ESTIMATES T P MODIFIED THEORY a * 0-01 b* » t . 03 0x10-13 V ORIGINAL THEORY

Figure 14. The cumulative probability distribution fôr^(f). a = 0.1, i = 3.0, iïp = Ó.9, « = 03, a, = 0.13.

Figures 15 and 16 give some indication of the effect of decreasing the frequency ratio from 0.9 to 0.7. Comparing Figure 15 with Figure 12 and Figure 16 with Figure 13 (same values of a, b and e) it is evident that there is now closer agreement betwen the simul-ation estimates and the modified theory; moreover, the difference between the two theoretical curves is smaUer. It is noted that for = 0.7, the peak in the response spectrum wiU no longer be near the peak in the input spectrum but wiU, in fact, be to the right of the input peak, where the input spectrum is not chang-ing so rapidly with frequency.

P I A ) 0'9999 a99BS l^999S ^ 9 » 0-9« 099S 099 »91 O-W «95 04 O l 0-7 »6 »5 0'; 0-3 07 0^1 —I—•- 1 1 T r

• SIMULATION ESTIMATES CASE i

ORIGINAL THtVtr . a> 001 o - « a 0 7 5 MOQIRED THEOmr 0-2 04 06 A

Figure iS. The cumulative probabiUty distribuiioa forvl(r). a a 0.01, i> « 1.0, iïp =• 0.7, e = OJ, a^ = 0,075.

PtA) 09999 -IH99S vm*> 0499 049S « 9 5 0-99 « * O'OI fr9B b> 10 0-97 ap>it 09Î e . »3 09 08 0-7 04 05 »!• 03 02 0-1 —rrr—. 1 1— — r

• SIMUUriOH ESTIMATES CASE ^

MOmFIED THEORY / / '

' (SlIGIKAL THEORT

O-I 03 0'3 05

Figure 17. The cumulative prob^ility distribution for.4(r). a = 0.01. h = 1.0, Üp ^ 1.1, e = 0 J , ff, = 0.6. PIA) 0-9999 0-9990 0-9995 0-H9 0998 a - O'OI 0995 b» 3-0 0995 b» 3-0 0 99 fl^.0 T 0- 98 « • 0 3 0- 98 « • 0 3 0-97 ffjj'OlO Û-9S 0 9 0-8 0-7 0-i 0:5 0 t 0-3 0 2 0-1 -0-1 • SIMUUTION ESTIMATES ORIGINAL THEÇRY /J /A NODtFlEO THEORY 0-1 02 0-3 0-1. 0-5

Figure 16. The cumulative probabiUty distribution for.^(0-a = 0.01,* = 3.0, ßp « 0.7, « « OJ, for.^(0-a, = 0.10, P (A) 04999 0-9998 09995 0 999 0-998 0;995 0 99 0 98 0-97 0 95 09 o a 07 0-« 05 Ob 03 0 2 0-1 -»1-s m u u n o N ESTIMATES _^1 MODIFIEO THEORT , a 1 O-Ot b > 30 0, . 013

-/ /

Figure 18. The cumulative probabiUty distribution for-4(f). a = 0.1, ô « 3.D, r^ = i.l, c = OJ, ff^ = 0,13.

Figures 17 arid 18 show results for = 1.1, - L e . the peak in the response spectrum now Ues a Uttie tp the left of the peak in the input spectrum. Comparing Figitfe 17 with Figure 13 and Figure 18 with Figure 14 (same a, b arid e v^ucs) it is evident that the agreerrierit achieved between the simulation estimates and the modified theory is not significantiy infiuenced by the relative position of the response spectrum peak, prov-ided that it is close to the input spectruiri peak.

Figures 19 and 20 show the effect of increasing the bandwidtii of the input process. On comparing Figure

19 with Figure 13 and Figure 20 with Figure 14 (same a, b and values) it is clear that the tendency is for the agreement between the simulation ^tîmatès smd the thepretical curves to improve as e increases, in-deed, in Figures 19 and 20 the agreement between the modified theory and the simulation estimates is excel-lent, despite the fairly high level of damping present in

P IA) 09999 0999B • 0 « 9 S 0999' 0 « 6 099S 899 098 0-97 0 « 09 OB 07 • 0.6 frS 0-4 » 3 02 0-1 -81 a • aoi b m 10 dp • 09 c > 07 0, > 0 « CASE 10 MODIFIED THEORY / / ^ CRfljINAL THEORY 0-2 03 0* 05

Figure 19. The cimiûlàtive probability distribution for^(r)

a « 0.01, è = 1.0, flp s 05, c = 0.?; o, " * p lAI 09999 89998 89995 0^999 0998 0995 099 098 097 09S 09 h 0 8 0-7 06 0-5 03 0 2 0-1 -0-1 - I 1 1— • SIMULATION ESTIMATES CASE 12

0 • 0-04 b * 02 MOOFIED THEORY 0,> O04 ORIGINAL THEOW 03 83 M. OS

0.08. aFigure 21. The cumulative probabiUi>' distribution for .4(f). =" Ö.04, b - 02, Sïp - 0.9, e = 0 J . a, = 0.04.

p (A) 64999 0 9 9 « 89995 0999 0998 099S 049 098 047 045 09 oe 0-7 0-« o's 0:4 0-3 02 0-1 -I r C A a tl MOOIFÎD THEORY b 3 34 y ^ ORIGINAL THEORY J U 84

Figure 20. The cumulative probabiUty distribution foi Ait).

a~O.Ol,b- 3.0, Üp = 0.9, e « 0.7, = 0.15.

these two cases. Of course, tfae effect of increasing c is to decrease the bandwidth ratio R (see equation (37)) and the theory is knowri to improve in accuracy as R decreases.

Figures 21 and 22 show the effect of increasing the linear damping factora from 0,01 to 0.04. Again fairly good agreement is achieved between the modified theory and the.simulation estimates. FinaUy, Figure 23 shows a result for a = 0.07 and b = 0.60;here the damp-ing is more Hnwr than in the previous cases which have been discussed. Once again the modified theory agrees fairly weU witfa the simulation estimates.

P(A1 04999 wm 04995 0999 0998 8995 0 ^ 098 097 »9S M OS 07 0 t 05 0-4 0-3 0-2 -I 1 1 r • SIMUUTION ESTIMATES CASE 13 Qi 0-04 b> t o _{MODIFIED THEORT • / /} / / / / 0. • 0'OT ORlGJMAL THEORY 02 »3 OS

Figure 22. The cumulative probability distribution for^(r). a = 0i04, b ° i.0, flp = 05, € a OJ, = 0.07.

4.4. Standard deviation of roll motion

For each of the 14 cases shown in Table 1 the stan-dard deviation of the roU response, , was estimated by the following three methods:

a. from the simulation.study;

b. from the original theory of Reference [9] ; c. from the present modified theory.

The values obtained are tabidated in Table 2. A study pf these results reveals that, in iripst cases where the modification to the theory is significant, the

P(AI 09999 »9998 • o m -0499 0998 0995 099 s * 087 898 b * 840 097 Bp-04 895 C <0-3 09 Ot^O-OT 09 88 87 86 05 04 03 02 91 I I SIMUUTION ESTIMATES CASE-14

KOOIFIHl THSJRif • /

THEORf

-01 M 83 84 OS

Figure 23. The cumulative probabiUty distribution forvi(f). a » 0X17, i a 0,60, Op « 0.9, e = 0 J . - 0.07.

Table 2

Standaid deviations of the roU response: Comparison between theoretical and stmulation

results

CaseNo. amularion original theory modified

1 0.154 0.158 0.157 2 0.161 0.164 0.161 3 O.ISI 0.159 0.151 4 0.165 0,182 0.165 5 0.169 0.191 0.167 6 0.168 0.154 0.159 7 0.145 0.129 0.133 8 0.142 0.156 0.146 9 0.1S2 0.184 0.157 10 0,153 0.153 0.150 U 0.166 0.163 0.157 12 0.170 0.174 0.169 13 0.160 0.171 0.161 14 0.179 0.192 0.180

modified theory gives a much improved agreement with the simulatiPn estimates of . In particular, when is close to unity the modified theory general-ly gives very good agreement with the simulation es-timates.

Figure 24 shows the variation of a^ with for the case where a = 0.01, fip = 0.9 and e = 0.3, and for four different values of b (0.0, 0.2, 1.0 and 3.0). This clearly shows that .the modified theory gives exceUent agreement with the simulation estimates, whereas the original theory becomes progressively less accurate as the damping is increased.

b > M SlMUUnoN SmÄTES • b « 0 . i !) • 0-2 • b a l » 0 - 3 a > 0-01 ( > 83 MOOFIEO THEtHY ORIGINAL -mm 008 810 812 » 4 016 Ox

Figure .24. Variation of the roll standaid deviation, a^ , with the wave input standard deviation a..

a = 0J)l,np « 0 5 , e = 0 J . ô a 0 . 02,1 and 3.

5. Conclusions

The comparisons between simulation estimates of the roU response, ixid the thepretical predictions, shown here, demonstrate that the modified theory proposed here generaUy gives more accurate predic-tions than the original theory of Reference [9]. CDver a reaUstic range of .damping parameters, and for a reaUstic input bandwidth, the modified theory gives good agreement with the simulation estimates of the cumulative probability distribution of the roU pUtude and the standard deviation of tlw roU am-pUtude. As the assiiinptipris in the theory would lead one to expect, the agreement is best when the band-width ratio R (see equation (37)) is small; this con-dition is achieved whèn the damping is Ught and alsp when the input bandwidth is large.

Ackno wledgemen ts

This work was supported by the Marine Division, Department of Transport, as part pf the- SAFESHIP project, in coUaboration with the National Marithne Institute (NMI) Ltd., Feltham, Middlesex, * Thè author would Uke to thank Drs. N. Hogben,

R. Standing, N. Dacunha and A. MonaÜ for help and advice.

References

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