Non-linear dynamic behaviour of moored platforms driven by stochastic seas

BEHOORI BIJ BRIEF

a..

van

ABSTRACT

An overview of the currently available methods for predicting the response of nonlinear systems to general random excitations is presented, together with their relative advantages and limitations, with particular reference to ocean engineering applications. A new technique is described, and its results compared

to those

of various existing procedures for the particular case of a simple system governed by the D.iffing equation.

An application to the prediction of the surging motion

of a tension leg platform in heavy

seas is then prOvidedL _{The surging, motion probability density} functions of the deviation with

respect to its mean

t1ue and of the maxima are both obtained.

The method may apply to the stochastic description of the dynamic response of any floating body to a general rzndom excitation taking free surface memory effects and various non-linearities into account.

NOMENCLATURE

e exponential function

EL I mathematical expectancy operator

h1. h3 first, third order impulse response functions

NONLINEAR DYNAMIC BEHAVIOR OF MOORED PLATFORMS DRIVEN BY STOCHASTIC SEAS

C. Duthot, Research Assistant and J. .L. Armand, Professor Department of Mechanical and Environmental Engineering

UniversiW of California Santa Barbara. California

Pm(a)

e()

S. S1 so Z( u) k, Vk Ci C lii, U1 V0, Va I N GE KO ME N

3 0 JiJifi

1987 Be antw. .Th

&eomuca

MsIal

_{2. 2B}

_Dofft 1k., OI5.. 78 _{F oie}

linear harmonic response or transfer function logarithmic function of the same order of magnitude as probability density function of the aeviation of y(t) with respect to its mean value probability density function of maxima real part of original system. "equivalent" linear systems input white noise frequency spectrum intensity (double sided) time variable system excitation

system response normalized Gaussian probability density function constants 2i-th order moment error integration variable i-th order cumulant, moment 0-level, a-level up-crossing expected frequencies system linear damping coefficient linear system response variance moment, cu m ulant generating functions

I

INTRODUCTION

The design of ocean structures, similar to that of ships, is for the most part based on empirical _{rules and codes,} and relies heavily on past experience. _{Although such} experience may be extensive in the case of ships, the

design of which is the result of centuries of mostly

careful and slow evolution, it is necessarily limited for

ocean structures, as exemplified by various _recent failures. The need for a more _{rationally-based design}

procedure for ocean structures is now_{well recognized.} Ocean structures operate. in _{an environment which is}

random in nature. The _{development and}

implementation of probabilistic models for predicting the loads acting on an ocean _{structure have been the} subject of extensive research in _{recent years., On the}

other hand, it

is

_{the stochastic prediction of}

_the resoons of an ocean structure tq environmental loads

which provides the significant _{information necessary} for its rational design; this

_{area has yet to be fully}

investigated.

Whereas linear system theory

_{is a well developed}

body of

_{knowledge, the application of which is}

relatively straightforward, the severe limitations of

linear models are now recognized _{in many situations} involving ocean structures.

Nonlinearitjes play an iinpottant rOle in the design Of moored floating structures. In particular, the response

to loads in unusual or extreme

_{conditions, which}

constitutes an important part of the _{design process, is}

essentially governed by nonlinear effects.

The aim of this paper is to describe a novel method for predicting the response

of nonlinear systems

_to

general random excitation, susceptible _{to apply to the} stochastic description of the dynamic behavior of any floating body such _{as a ship or offshore platform in} random seas, when nonlinearities prevail. The method

.-iU be tested here

_{on the simplified case of the}

prediction of the surging motion of a tension-leg

platform.

I- REVIEW OF EXISTING _TECHNIQUES

Since the pioneering work _{of St. Denis & Pierson [18)}

who applied linear

_{spectral theory to describe the}

motions of a ship _{among irregular waves, numerous}

attempts have been made to take in _{account various}

nonlinearities in the original model by

_{means of}

existing nonlinear system theories.

A general theory for nonlinear dynamic response to

stochastic processes, of the same _{scope as the linear} theory, is not yet available, and thus progress toward a

satisfactory stochastic theory of ship and platform motions has been rather Slow.

Nonlinear stochastic modelling is a relatively _{new and}

difficult field, drawing on the latest

_{advances in}

nonlinear system theory and stochastic _{processes. All} the existing approaches are, in some way, limited in

scope, by assuming particular properties of both _the excitation and the system; moreover, most of these

techniques yield a limited_{or incomplete description of} the response.

1L1_ Time _{Domain Simulation}

-Time domain simulation of _{the equations of motion} remains the foremost way of predicting the response of a nonlinear system to_{some prescribed input.}

Apart from its systematic aspect, such a method

exhibits many well-known drawbacks, mostly linked

to the prohibitive amount of' _{calculations necessary.}

Nevertheless, it has been applied to the description of f-ship motions. _{DaIzell 17) showed, through a}

time-stepping procedure, that for

_{most of the practical}

dynamic range, the distribution of roll _{maxima does} not correspond to the distribution of the maxima ofa random Gaussian process predicted by theory, i.e. the Rayleigh distribution.

In the case of a stochastic

_{excitation, a spectral or}

probabilistic description of the _{response becomes}

rather cumbersome using a time domain simulation'. For this reason, _{stochastic frequency domain} techniques

_{are generally preferred,}

_to

_{the more}

expensive time domain techniques.

1-2- Equivalent Linearization

-Linearizing the system comes next among the available techniques. Basically, an "equivalent" linear system is substituted to the original nonlinear one. The price to

be paid for such a drastic _{simplification Of the model} lies in the choice of _{a linearization procedure, which} does not follow any specific guidelines, as well as in an 'The nature ol which is deterministic,

ir,complete description of the System which ignores the speCifiC features2 of nonlinear Systems: the system is thus globally assumed to behave as a linear one.

One approach involves replacing the original nonlinear equations describing the system, by equivalent linear equations which minimize the root mean square error,

and was described by Caughey [31. The resulting

system thus depends upon the

statistics of the

response, which is not really inconvenient provided

that stationarity is preserved.

The method has been applied to rolling of a ship in

random waves by Kaplan [13), who assumed viscous

damping and by Vassilopoulos [21) who assumed

viscous damping and nonlinear restoring moment.

among others.

13 FokkerPlanck Equation

-The appealing aspect of an approach based on the

Fokker-Planck equation is that the derived solution is an exact one. However, the assumptionsunderlying the

application of this technique are quite restrictive: the

nonlinearity is of static nature only, and the excitation is a Gaussian process the frequency spectrum of which. is that of a white noise3 [4).

Essentially, the method is based on the fact that the

response of a discrete. dynamic system subjected to a. Gaussian white noise behaves as a continuous Markov

process. _{Then, it is possible to show [41. that the,} response joint probability density function satisfies a partial differential equation: the Fokker-Planck equation.

This method has been applied to the nonlinear rolling

of a Ship in random waves by Haddai-a [111 and by Roberts [171.

1-4- Perturbation Techniques

-The general feature of perturbation methods is to substitute an infinite number of linear systems to a

2Nonlinearities usually have two dillerent effects. The first of these Ieas to a response which differs only quantitatively from the linear response, while the second one induces phenomena which are not

predictable within the framework of a linear approach, such as certain

types of dynamic instabilities, sub- or sUperharmonjc responses,

bit urcations,..

3We may however emphasize that the Fokker-Planck_{equation does exist}

in the more general case of dynamic nonhinearities and non-white

excitation spectrum, but an analytic solution is not generally possible in this case 1151. and the corresponding solution is rather numerically iavolved.

287

nonlinear one through an expansion

_{in terms of a}

"small" parameter describing the magnitude of the nonhinearities, In this way, the nonlinear features _of

the system disappear, On the other hand, the system

response is now expressed in terms of _{a. generally}

infinite, series [5). The difficulty of evaluation of each term increases geometrically as its order in the _series.

Two fundamental questions then arise; namely the convergence of the series, as well as the. number of

terms necessary to get an accurate description of the

solution. The answer to the former is generally not easy although one may get'a good approximation of the solution from the knowledge of the first few terms only, even when the series diverges. Once _{again, the}

nonlinearities must remain weak in order to insure both convergence and accurate prediction

of the

solution with a limited number of terms.

Such an expansion procedure has been applied to the nonlinear rolling of ships in random seas, first in the case of viscous damping [241 and later in the case of a static nonlinearity 1101.

_{Both papers discuss the}

influence of the nonlinearity on the response frequency spectrum,

1-5- Functional Series Representation

Methods

In most derivations of the equations of

_{motion of}

floating bodies such as ships and offshore. Structures, the. system is conveniently reduced _{to a set of} second-order differential equations with frequency-dependent coefficients, whether linearity is assumed _{or not. The}

simplicity of such a description is only apparent; it

actually represents integral equations in the

time domain, the physical interpretation of which lies in the fact that a structure freely floating in _{waves is a} space-time system. Various approximations and integrations are made to reduce it to a time system [191. The price to be paid to allow such a simplification resides in the

memory effect which appears as we get rid of the

space dimensions and is mathematically described by

the integral equations over the

_{past history of the}

motion.

When the further assumption of linearity is made, the system is compactly described by its impulse response

matrix in the 'time domain and by its

_harmonic

response matrix in the frequency domain.

The fundamental importance of these concepts in ship hydrodynamics have been stressed by Cummins [61 and later by Bishop, Burcher & Price 12).

represent a natural generalization of this procedure

handle the non-linearitjes,

_{It may be intuitive}

thought not as a regular expansion in power series, b

rather as a "power series

_{with memory", namely.}

fuctjonal series_{representation technique.}

The functional series representation of differentj integral and _{integro-djfferential equations originate} with the work of _{Volterra by the end of last centu} (see e.g. 122]). Essentially, the _{input-output relation 0}

a given system is

_{expanded in a functional}

powe series, Wiener [231 later applied Volterra's _descriptio of _{general functional}

relationships to nonhinea communication problems.

Two major applications of Volterra _{series4 to nonhinea} systems are analysis _{andidentification The}

nonlinear

ship-wave system contains several unknowns, and

both _{aspects have been}

considered in ship hydrodynamics For _{analysis, the Volterra series}

representation of a given system is determined from

the system itself (e.g. _{a set of differential or} integro-differential equations), whereas for identification _the Volterra series is _{determined from simultaneous} measurements of the input and the output functions.

1-5-1- AiJy/s

-Vassjlopoulos [201 discussed _{the applicability of the} Vofterra and .Wiener _{series to the motions of a ship in} irregular seas modeled

_{as a nonlinear}

autonomous system, while, Dalzell _{[8,9] investigated}

the applicability of the Volterra series model to _nonlinear ship rolling, and especially of the third degree _Volterra

funcjonal

1-5-2- Identffjc,zt0

-Identification _{techniques for linear}

autonomous systems are well _{established: essentially,} cross-spectrum techniques_{are used in order to}

evaluate the

linear transfer _function.

Hasselnian [12J showed

_{that the nonlinear}

_transfer

functjs can be obtained

_{from high order}

moments of the ship motions.

2- THE PROPOSED_TECHNIQUE

In most

_{applications to the}

determination of the

40ther luncona1 series represeaCjon techniques are possible: Wiener zxd Hermite functional series,

to ly Ut a al, d ry 1'

r

xi r

response of nonlinear dynamic systems to Stochastic excitation _{perturbation methods are}

not _carried

further than the first

_{nonlinear term because of the}

resulting excessive _{complexity. Moreover, it will} be shown that the _{improvement provided}

by a two- or a

three-term perturbation _{method only covers a region} where the

perturbation parameter 15 so "small" that

the linear and the equivalent linear _{approximations} are already quite _{accurate enough from a practical} point of view.

The main idea _{underlying the following}

develope5

is to provide _{an improved theory}

over the Volterra

functional series_{model in deriving the output statistics} (e.g. _{moments and cumulants)}

of an

_autonomOus nonlinear system driven _{by a Gaussian noise. Rather} than expressing these statistics by _{a truncated series}

the convergence _{and accuracy of which}

remain questiona

closed-form approximate solutions are developed.

These statistics may eventually be Used to evaluatean Edgeworth..serie5yp output _{probability density} function.

In order to illustrate this _{technique without going into} an excessive _{amount of algebra, let us consider the} particular case of the _{Dutfing equation.}

2-1- Application

_{to the Duffing Equatjo}

-Let us consider the nonlinear _{system (S) described by} the following

differential equation (Duffing equation), written in dimensionless_form:

g+2.y.y3_x(t)

(2.1)

in which the excitation x(t) is _{assumed to be a} zero-and Gaussign _Stochastic thite 'i'se with double_{sided frequency spectrum s0}

The output variance of the linearisedsys is easily shown to be, by_{residue integration:}

TTS0/2 _(2.2)

At this point,_{we may note that the}

only two quantities necessary to describe the

system and the excitation are S0 and

. The response should thus depend upon these two parameters alone.

1though our method is

rather general as it is applicable to non-static nonhinearities and non-white

j0put frequency spectra,

the above example will

enale

US to compare the results obtained using this

method with

the exact response statistics and probabilitY density function derived from the Fokker-planck equation technique.

The following "equivalent" linear systems (S1) are

defined next:

2g + y1(l + E[y12J)

-

x(t) (i *) (2.3)

where successive parameters cz are chosen to minimize the following, quantity:

In other words, the system (S1) should provide an,

accurate linear approximation to the output 2ith-order. momentE[y2'Jof the original nonlinear system (S). Yet

this minimizing procedure needS to be clarified further.

Qearly, fOr i - I, (S1) corresponds to the equivalent

linear system (equivalent in the mean square sense) as defined by Caughey [31.

The actual forms of the equivahent" linear systems (S1)

(equations (2.3)) are suggested by the form of the

original differential equation describing the

input-output relationship of system (S) (equation (2.1)). Once again, it is' important to emphasize at this point

that the proposed method is not restricted to systems of the type (2.1). In fact, a much more general class of

differential equations and even integral or

integro-differential equations can be handled within

the framework of this technique.

2-2-- Output Mean Square of System (S1) -Since each system (5k) is linear, the corresponding

Output mean square (or variance)E(y12J _{is simply given}

by residue integration of the output energy spectrum

over the whole frequency domain:

E[y121 - 02 /(1 + a1E[y121) Ci E f'l') , (2.5)

289

Solving forE1y121 _{in the above equation, we obtain:}

E[y2J- ((1 + 4ca12 Wa - l)/2o: (1 e f*)(26)

where a,,,2 is the output variance of the linear part of the original system (S).

2-3- Output Moments of System (S)

-Since system (S) only involves terms of odd order5, all output moments of odd order are identically _zero, while output moments J121 of even order are

approxiniately obtained from the systems (S1)

considered separately:

Keeping in mind that y is the output of_{an autonomous}

linear system driven by a Gaussian, stationary _and ergodic random process, it is possible to prove' the following set of equalities:

E[y12iJ - ((2i)!/2 ii)Ei1y121 Ci E

f*)

_(2.8)

2-4- Output Cumulants of System (S)

-Throughout the following, we will be more concerned with the cumulants 1<21 of the response of system S.

than with its moments jj2. Cumulants are simpler and appear directly in the probability density function of the output y(t) discussed below.

The moments_{Jk correspond to the k-th derivative of} the moment generating function [161:

q,(it) - J,p(C) e itt dC (2.9)

while, by definition, the cumulants _{1<k are the k-lu}

derivatives of the cumulant generating functioir

w(it) - lnc,(it) (2.10)

5me proposed technique is, in its present form, restricted to analytic

systems involving odd order terms only.

Expanding functions 4(it) and _{s(it) in Taylor series} and equating the coefficients of same power in (It)k using

_{equation, (2.10), we obtain}

_the

_{first few}

Cumulants:

K2 - J-2 K4 - 114 - 31.122

K6_{-116+ 301.123- 151121.14}

(2.11)

Substituting equations (2.7) and (2.8) into (2.11), we Obtain:

E[y121

K4 - 3 E2[y221 _{- 3}E2[y2J

1<6 - 15E3[y321 + 30 E31y121 - 45 E[y2JE2[y221

(2.12) where the only unknowns are the _{cr coefficients, the}

Ety12J being expressed in terms of 'these _coefficients through equation (2.6).

6The Vofterra functional series model essentially consists of the Taylor

series expansion, with memory. of an input-output relationship. Therefore, it does not provide anything more than what can otherwise be obtained through a regular expansion technique. The real _{advantage of} the Volterra series model lies in the rather advanced body _{of results} related to its application,

2-5- Determination of the a1

_Coefficients

-So far, expressions (2.8) and (2.12)

_{for the output}

moments and

cumulants would _{not provide a} significant improvement over other _{existing techniques}

if we were unable to determine the

_{coefficients a1.}

Assuming that equations (2.8) and (2.12) represent

uniformly valid approximate expressions of the output

statistics of system (S) over some practical _dynamic

domain, some additional information, not provided by

the linear theory, is needed in order

_{to completely}

determine these approximate statistics. _{The idea is}

thus to match the solutions '(2.12) with _{some 'inner"} solutions valid as the parameter

G2

_{becomes small,}

i.e. solutions obtained through an_{expansion procedure}

in terms of

a12. _{Volterra functional series will be}

used, though a regular perturbation "with _memory"

should yield identical results6.

One reason to describe _{the output through its} cumulants rather than its

_{moments lies in a quite}

interesting

property of

pure Volterra _functional polynomials of order n, namely that all cumulants K2k. k > (n + 1)/2, do vanish. Such_{a, property can be better}

understood, keeping in mind that7:

K2t - O(a,2(2k-l)) _(2.13)

and that a pure nth order Volterra

_{polynomial will}

yield only statistics of order 2n.

Taking now advantage of this property, all coefficients i 3, can be determined through equation _(2,13). Finally, a 3rd order Volterra functional series will yield cumulants 1<2k' k

2, which, in turn, lead to the

determination of the two remaining unknowns a1.and a2.

In order to assess the validity of the procedure, an

application to the case of

_{a nonlinear system, the}

input-output relationship of which is described by

equation (2.1), is provided. _{The response y(t) is} assumed to be written as the Volterra_{series El):}

y(t) - Jh1(t-t)x('t)dt

+

I1h3(t-rl,t-r2.t_t3)x(tl)x(t2)x(T3)dt1

dt2dr3

+

(2.14) Clearly, for Gaussian white noise excitation, the

response 2nd order cumulant

K2 _{is obtained by} integrating the output frequency

_{spectrum at this}

order:

- Oyl2(1 - 6S0J_{IHi(w)J2C(Hl()) dw +}

(2.15)

where H1(w) is the linear frequency response function. Carrying out the integration, we obtain:

K2 - Gyl2 - 3yI4 + O(u16) _(2.16)

7WhiJe such a property has been proved for the first few_cumulants

only, its generalizatjo _{Is based on the conjecture that it can be}

inrerred from the memoryless case by computing the Vblterra _system oitput cumulant of order 2k (III.

bere O2, the output variance of the linear system.

atUr3lly appears as the expansion parameter.

We now evaluate the 4th order moment of the output

and u equation (2.11) in order to obtain the lowest

order term Of the 4th order cumulant:

K4 - -24S03J _{J J IH1(wi)121H1(w2)121H1(u3)12}

- -

-+ ... _(2.17)

Carrying Out the integration, we obtain:

K4 -6a16+O(cr18) (2.18)

Coefficients tx are obtained identifying equations (2.8) and (2.12) with equations (2.16) and (2.18):

- 3 0:2 - 4

If we now take advantage of the cumulant property mentioned above, the remaining x can be evaluated

much more easily:

0:3 - 5 0:4-6

It is interesting to mention that all remaining c (i - 5. 6, ...) n be evaluated in the same way, thus avoiding the troublesome task to compute the moments through the corresponding Volterra series.

Lin:a\ Volterra 6th., 'Fokker-Planck Volterra 4th Theory 1 (eq. linearisation) 291

Linear Output Variance,

6th

Figure 2 - Output Fourth Order Cumulant

Finally, we obtain the following expressions _{for the}

first few output cuinulants, through _{equations (2.6),} and (2.12):

1<2 - (-I + -Ii + l2a12)/6

1<4 - 3(-I + /l + l6a12)2/64 -(-1 + .11 + l2cJ12)2/12 1<6 - 3(-1 + .11 + 2Oa12)3/2O0

+ 5(-1 + 11 + l2a12)3/36

- 45(-1 + /l

I6a12)2(-1 + /l + l2o2)/384

(2.19)

Figures 1,

2 & 3 show cumulants

_1<2, 1<4 and _1<6

respectively (labelled Theory 1, 2 & 3) obtained using

equations (2.19). as functions of à12, and compared with the results of several other available methods, In

particular, the cuinulants are given exactly by the Fokker-Planck equation, while Volterra 4th & Volterra 6th respectively correspond to a 2-term and a 3-term Volterra functional series.

As should be expected, the

linear theory yields

vanishing 4th & 6th order cumulants, and Volterra 4th

yields a vanishing 6th order cumulant, while the 6th

order cumulant using Volterra 6th is rather algebraically involved and has not been computed. As emphasized earlier, the improvement provided by

Volterra 4th and 6th (2 and 3 term Volterra series)

over the linear theory is only discernible for very

small values of This is particularly true in the

case of the 4th order cumulant K.

It is seen that equations (2.19) do provide very good approximations of the cumulants over a rather extensive dynamic range, - 0.15 being a relatively

OI5

Fokker-Plarick

-

'

Volterra 4th

\

Theory 2

Figure 1 - Output Second Order Curnulant (or Variance)

0.15

Theoty3

Fokker-Planck

0.15

high level of

_{nonlinearity,}

where the linear

_theory (ails.

3- APPLICATION

3-1- Output

_Probability

Density Function

-Edgeworth developed an_{asymptotic series}

probability density function

_{for the sum of random}

variables

related to the

_{law of error.}

The Edgeworth _series

probability density'function may be understood _{as a} perturbation of _{the Gaussian distribution in}

terms of the curnulants8 _{(see e.g.}

Longuet-Higgins [141). _In

practice, the series is _{therefore truncated assuming}

slightly non_{Gaussian randOm}

variables. Thefirst _few terms of the

Edgeworth series distribution, in the_case of a zero

rnean random variable, are given by: p(u) _{Z(u) +} 1y2Z()(u)/4l1 + _{.y4Z(6)(u)/6I + y22Z(8)(u)t2!(41)21} (3.1) where: U -_Y/'/<2 -Z(u) -_(21Th2)'/2 exp(-u2/2) ZW(u)_{- H1(u) Z(u)}

fle p4.1. of a _{random variable}

Is uniquely

determined by its

eumulsals (or its moments)

as long as they_{are finite and}

convergence of the Taylor serIes (2.9) or_{(2.10)is ensured.}

where H1(u)_{denotes the}

usual Hermite _polynomial order i.

It is rather_{interesting to}

note that the output statisli and p.d.f. can be expressed

completely in _{terms of} unique

_{parameter a2,}

although the _{problem w} originally described by two_parameters.

The output _{p.d.(. obtained}

using several methods I shown in figure 4 in the case of the_{surging motion}

of tension leg

platform with nonlinear_{mooring force:}

with the _following

data: w0 - _0.1055

rad.s.-1; - 0.2057; r _{0.0057 m.2 2}

The _{equivalent white} noise excitation _{spectrum is S0}

- 3.5825 10-5 _rn.2 which, in _{this case,}

corresponds to_{a significant} wave height H5 _{9.15 m. 1151.} This _particular eampIe _{corresponds to} the adimensionaljzed _{linear output}

variance _{- 0.12,}

Which is already

a relatively high level of_{nonlinearity.} Theories 1. 2 & 3 correspond

respectively to the terms in (K2), (K2. _{) & (K2,}

K4, _{K6) in the}

Edgeworth

_pd!.

asymptotic _{expansion (3.1).}

As already _mentioned,

theory 1 yields exactly_{the p.d.f.}

obtained through _the equivalent _{linearisation}

method. _The

two-term Volterra

series (Volterra 4th) yields _{very poor} agreement with the exact _{p.d.t. derived}

from the Fokker-Planck equation; it _{even becomes}

negative for large values_{of y.}

It is

_{seen that}

the. _proposed

technique _yields

satisfactory results, for _{this example,}

with only _the

terms corresponding to K2 & K4

(theory 2). Theory 3

does, as might _{be expected,}

provide slightly _better accuracy over theory 2; the _{results are}

not, however, shown on_{the figure for}

better clarity.

Theory 2

\ç.Volterra4th

_- Linear

Surging Displacement,_{y meters}

Figure 4 - Output Probability

Linear Output Variance,

Figure 3 - Output Sixth Order_Cuinulant

g+ 2.0g+w02y+ry3-

_x(t)

3-2- Output Probability Density FUnctjo

_of

Maxima

-The probability density function of maxima of a

parro7b and orocess is given by:

J:pm(y)dy- Va/V0 _(3.3)

where Va and v1

are respectively the expected

frequencies of a-level and zero-level up-crossing (see

e.g. 161). given by:

va

.jy p(a,y)

dy _(3.4)

If we use the fact that y and

are statistically

independent, an exact property derived from the

corresponding Fokker-Planck equation [4], _the

probability of maxima can be evaluated through:

Pm(a) - - l/p(0) [dp(y)/dyl_{y-a (3.5)}

We _{now substitute eq. (3.5) into eq. (3.1) and get an} approximate expression for the output probability density function of maxima:

p(u)

(Z(1)(u) + y2Z(5)(u)/4'

+ Z(9)(u)/2!(41)2)/(K,'/2 p(0)) (3.6)

Figure 5 shows the surge motion p.d.f. of maxima for

the same numerical values

_{as already used in 3-1.}

Volterra 4th

Theory 3

Fokker- Plànck Linear

Surging Maxima (Amplitude), _meters

Figure 5 - Output Probability Density FunctiOn of Maxima, ay12 0.12

293

The agreement of theories 2 and 3

with the exact

solution, derived: from the Fokker-Planck equation, is

excellent, while the 2-term Volterra series (equation (2.14)) yields a rather crude approximation of the

surge motion pd!. of ma*ima.

CONCLUSION

The method described yields good _{agreement with the} exact output cumulants and p.d.(. obtained using _the Fokker-Pianck equation over an extensive _dynamical domain. Yet, such an approximate method would not prove to be worthwhile if it were only able to achieve results which can otherwise be obtained exactly. The only merit of the particular example _{treated here was}

to assess the validity of the proposed

_technique, keeping in mind all along, that it does provide _added versatility over the methods based on the Fokker-Planck equation (with relatively little more computational effort), and sensibly greater accuracy than the Volterra functional series model.

This technique is currently applied _{by the authors to}

obtain the response statistics and

_{p.d.f. of simple}

nonlinear systems (analytic systems) _{subject to general} stochastic excitations for which _{no exact solution is} known, although the question related to _{our ability to} infer the validity of the proposed technique from this particular example will have to be addressed.

The validity of the mathematical model is also being assessed through scale model experiments, If necessary, dynamic nonhinearities (e.g. viscous damping) and memory effects will be implemented in

the mathematical model, in order

_{to get a more}

accurate mathematical description of the physical phenomena involved.

ACKNOWLEDGEMENTS

This work, is the result of research sponsored in part by NOAA, National Sea Grant College Program,

Department of Commerce, under grant number

NA85AA-D-SG 140, project number RIOT-i 2, through the California Sea Grant College Program. The US. government is authorized to reproduce and distribute for governmental purposes.

"

'!

The authors wish to express

their gratitude to Professor Theodore Kokkinis for his inspiring guidance throughout this study.

REFERENCES

I BEDROSIAN, E.. & RICE, S,O.,

1971, 'The output

properties of Volterra systems (nonlinear systems with

memory) driven by harmonic and Gaussian inputs."

Proceedings, IEEE, vol.59, no. 12. pp. 1688-1707. 2 BISHOP, R.E.D., BURCHER, R.K., & PRICE. W.G;, 1973,

"The uses of functional analysis in ship dynamics."

Proceedings, Royal Society, London, Serie A, no. 332. pp. 23-35.

3 CAUGHEY, T.K., 1963, "Equivalent linearisation

techniques,' Journal of the Accoustical Society of

America,vol.35.no. 11,pp. 1706-1711.

4 AUGHEY, T.K., 1971, "Nonlinear theory of random vibrations," A dvances in Applied Mechanics, voL

11. pp. 209-253.

5 CRANDALL. S.H., 1963, "Pertubation techniques for random vibration of nonlinear systems," Journal of

the Accoustical Society of America, vol. 35, nO.

I1.pp. 1700-1705.

6 CWTh1INS. W.E., 1962, "The impulse response

function and ship motions,". Rep. 1661. David Taylor Model Basin.

7 DALZELL, J.F., 1973, "A note on the distribution of maxima of ship rolling." Journal of Ship Research. voL 17, no. 4, pp. 2 17-226.

8 DALZELL, J.F., 1976. "Estimation. of the spectrum of nonlinear ship rolling: the functional series approach," Rep. SIT-DL -76-18.91', Davidson Laboratory. Stevens Institute of Technology.

9 DALZELL. J.F.,

1982. "An investigation of the

applicability of the third degree funconal polynomial model to nonlinear ship motions problems," Rep. SIT

DL -82-2-2275, Davidson Laboratory. . Stevens

Institute of Technology.

10 FL0wER,J.O., 1976, "A perturbational approach to nonlinear rolling in a stochastic sea," International

Sbipbuilding Progress, vol. 23. no.263. pp. 209-212. 11 HADDARA, M.R., 1974, "A modified approach for

the application of Fokker-Planck equation to the

nonlinear ship motions in random waves,"

International Sbipb u/Id/ag Progress, vol. 2 I, no,

242, pp. 283-288.

12 HASSELMAN, K., 1966. "On nonlinear ship motions in irregular waves," Journal of Sb/p Research, vol

10. no. 1. pp.64-68.

13 KAPLAN, R.. 1966.. "Lecture notes on nonlinear

theory of ship roll motion in a random seaway,"

Proceedings, Eleventh International Towing Tank Conference. Tokyo, Japan, pp. 393-396.

14 LONGUET-HIGGINS, M.S., 1963, "The effect of non-linearities on statistical distributions in the theory of

sea waves," Journal of Fluid Mechanics, voL 17, no. 3, pp. 459-480.

15 0Cm, ML & MALAKAR, S.B., 1984, "Non-linear response of ocean platforms with single degree of

freedom in a seaway," Report, University of Florida UFL/COEL-TR/050.

16 PAPOULIS. A. 1984,

Probability,

Random

Variables, and Stochastic Processes, McGraw-Hill, New York.

17 ROBERTS, J.B., 1982, "A stochastic theory for nonlinear ship rolling in irr'egular seas," Journal of

Ship Research, vol. 26. no. 4, pp. 229-245.

18 ST. DENIS, M., & PIERSON, W.J., 1953. "On the

motions of ships in confused seas," Transziciions SNAME, vol. 61. pp.280-357.

19 TIcK, U. 1959. "Differential equations with frequencydependent coefficients," Journal of Ship

Research, vol. 3, no' 2, pp. 45-46.

20 VASSILOPOULOS, L.A., 1966, 'The application of statistical theory of nonlinear systems to ship motion performance in random seas,"

International

Shipbuilding Progress, vol. 14, no. 150, pp. 54-65. 21 V ASSILOPOULOS, L.A., 1971, "Ship rolling at zero speed in random beam seas with nonlinear damping and restoration." Journal of Ship Research, vol. 15. no. 4, pp. 289-294.

22 VOLTERRA. V., 1930, Theory of Functionals

and Integral and IntegrodiTferent/al Equations.

Blackie, London.

23 WIENER. 1958,

Nonlinear

Problems

in

Random Theory, The Technology Press of MIT and John Wiley & Sons, Inc., New-York.

24 YAMANOUcHI, Y., 1966, "On the effects of nonlinearity of response on calculation of the spectrum." Proceedings, Eleventh International Towing Tank Conference, Tokyo, Japan, pp. 387-390.