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i1LTFIS IS A PREPRIT --- SUBJECT TO CORRECTQI
Some Observations on the Techniques for
Predictnq
the Osci I tations of Freely-FIoating Hut
in a Seaway
By
Manley St. Dents, U. of Hawaii
©Copyright 1974
Offshore Technology Conference on behalf of the American Institute of Mining, Metallurgical, and
Petroleum Engineers, Inc. (Society of Mining Engineers, The Metallurgical Society.and Society of
Petroleum Engineers), American Association of Petroleum Geologists, A.rnerican Institute of Chemical
Engineers, American Society of Civil Engineers, American Society of Mechanical Engineers, Institute
of Electrical and Electronics Engineers, Marine Technology Society, Society of Exploration
Geophysicists, and Societyof Naval Architects and Marine Engineers.
This paper was prepared for presentation at the Sixth Annual Offshore Technology Conference
to be held in Houston, Tex., May 6-8, l974.
Permission to copy is restricted to an abstract of
not more than 300 words.
Illustrations may not be copied.
Such use of an abstract shod coitain
conspicuous acknowledgment of where and by whom the paper is presented.
PAPER
Iu::F.
OTC 2024
ABSTRACT
The paper is a sequel to an. inquiry
presented last year into the validity of
spectral techniques for describing the
seaway (St. Denis, 1973).
The problem now
addressed is stated as follows:
Given a
seaway defined by its variance spectrum and
a hull whose dynamical properties are
linear and frequency dependent, what
tech-niques are
Tilable for fing statistical
expressions for the oscillatory motions in
each of the six degrees of freedom.
The
nonlinearities appearing in the hydrodynarnic
coefficients are either the consequence of
changes in hull geometry resulting from the
oscillatory displacements from the
mean
position, or they are the manifestation of
n2nhinear
henomenon of viscous damping
It is brought out that, although numerl
solutions can be obtained in the time
dornain,no generally applicable
non-numerical technie for solving nonlinear
differentiqtations either deteiistic
or stochastic is in hand, and that one is
forced to have rcourse to approximative
techniques.
Some available such techniques
References arid illustrations at the end
of paper.
are discussed as to their iLisefulness in
solving the second order set of stochastic
differential equations expressing hull
response in a random seaway.
There appears
to.be no simple choice.
1.
INTRODUCTIONDuring the past two decades, it has
become common practice to describe the
seaway through the spectrum of the variance
of the surface profile.
Some comments on
the validity of such a representation have
been made in a previous paper (St. Denis,
1973), where it is brought out that the
spectral technique can be applied with
con-fidence when the seaway is of mild
inten-sity; that it provides only a first order
description of seaways of moderate
inten-sity, but is otherwise inadquate for
revealing certain imPortant features that
hese seaways exhibit (haur.ching, cusping,
whitecapping and breaking); and, finally,
that it is entirely insufficient to
des-cribe the heavy seas of great severity.
The purpose of the present paper is to
examine the validity of the spectral
tech-nique for describing the oscillations
to
which vehicles and platforms
are subject
ther cpeating at sea.
The scope of the
SOME OBSERVATIONS ON THE TECHNIQUES FOR PREDICTING THE OSCILLATIONS OF 824 FREELY-FLOATING HULLS IN A SEAWAY
inuiry is
limited, to those motionsthat are induced by light seaways which themselves ate properly representable by a variance spectrum. Seaways of moderate and high intensities are nonlinear and are not contained within this restriction. Otherwise stated, the attention is focused øn the kinetic behavior in light, i.e., linear, seas of those systems that are characterized by nonlinear transfer func-tions.
Prediction of the oscillations that a vehicle or platform system experiences in a seaway is a problem in dynamics (the determination of the forces imposed by the seaway on the system) and in kinetics
(the determination of the motions resulting from the forces imposed by the seaway). Only in linear systems is it feasible to separate the two aspects; in nonlinear systems, the hydrodynamic loading is affected by the oscillatory response and the two are not properly separable; never-theless, such a separation is made herein for ease of exposition; however, the mutual interaction of loading and respoñe will need be borne in mind.
The causal relationship between a seaway and the oscillations that a floating system undergoes therein is developed in two steps:
Seaway Dynamics - Kinetics In the first step (dynamics), derivation is made of the forces acting on the plat-form; these consist in:
a) The seaway-induced excitation Fi(t) corresponding to each degree of freedom i of the system.*
b) The systeri's reactions corresponding to the as yet unknown systemts motions x(t) in ach degree of freedom i. The
reactions are the terms in th equations of motion and are formed by the product of a hydrodynamic coefficient and of a kinematic variable of the motion
(acceleration, velocity, displacement). The second step (kinetics) is the solution of the equations of motion.
Motions can be defined either as sta-tistically steady, as evolutionary or as transient. Only the first are considered herein: they are characterized as
processes whose statistical parameters are time-invariant. Obviously, the oscil-lations of floating systems, like the seaway that excites them, are steady only over a short run.
The waves of the sea undergo continual change in height, length and direction of travel, and such a change manifests itself in a randomness, of greater or lesser degree, whose descriptive parameters tend themselves to vary with time, the interval over which they remain reasonably constant ranging from a few minutes to a few days or even weeks, depending on locality and meteorological circumstances. This random-ness, which is a basic characteristic of the seaway, can be properly described only by statistical models.
The random character of the seaway in turn reflects itself in the oscillatory response of the systeni, so that the latter also has a statistical expression.
2. DYNAMICS 2.1 INTRODUCTION
In a seaway, a hull is subjected to a cyclic pressure field generated by the continuous.interplay of the hydrodyriamic excitations of the waves and by the hydro-dynamic reaction induced by the motion of the hull. If the hull Is moored, there
is in addition an elastic reaction. Hydrodynarnic forces are separable according to the role played by viscosity:
if this not significant; the forces are derivable on the assumption that the fluid
is inviscid -and can described in terms of a velocity potential; if viscosity plays a non-negligible yet far from dominant role, the forces derived on the assumption of
inviscid fluid can sometimes be suitably
See discussion on this term in the previous paper (OTC 1819).
x,y,z are the coordinates of a right-handed orthogonal system of body axes with origin at the center of gravity of the hull and positively directed forward, to port and truckward. Statical rotations about x, y and z are referred to as heel, change of trim and change of heading. x(t), y(t) and z(t) denote motions of the body's center of gravity along its
prin-cipal axes, the time-dependent displace-ments being measured with reference to the mean (calm water) value of the xyz system; ther are respectively termed; surge, sway or sidle, and heave. The corresponding rotations about the x,y,z axes, called roll, pitch and yaw, are denoted by aCt), 8(t), y(t). The
substi-tutions xx1,
yX, ZX3,
ct=x4, 8X5, yx6 are made when convenient. The arnpli-tudes and phase lags of the oscillatory motions are respectively written i and 5..= -pg<
R..
13x.(t)
3
in which case, the coefficients Rjj are given as follows; R33
= -pg
-pg A+ x22Jds
= -pg I +x1,].ds
= -pg R35 = R53-pgO[x1Z3
+x321]ds
= -pg Mwhere A is the waterplane area, M is the first moment of the waterplane area about the transverse (or pitch) axis, while I and I are the second moments of waterplane area about the longitudinal (or roll) and the transverse (or pitch) axes respectively, and £1, 2.2, 2.3 are direction cosines.
R35=R530 only if there is fore-and-aft
asymmetry of waterplane
areaThe quasi-hydrostatic reactions in
roll and pitch are accompanied by a static
upsetting moment of weight equal to pgzV, where Zb is the depth of the center of buoyancy below the center of gravity (z0), and V is the volunie of displacement. The net reactions thus become:
_PgEI -
zb.V].a(t) E
Cc(t)
-pgtI
- z]D]8()
-. gI8(t)C55s8(t)
These are restorations. There being no adjustment to be made in heave and in coupled heave and pith, one can write for
conformity of symbols
R33 = C33,
R35=
C35and
R53 =C53.
The restoration coefficients C33 and C55 are usually written in terms of the transverse and longitudinal metacentric heights
C pgV0z
C55 = pgV.z
The expressions for the restoration coefficients Cj are reasonably valid only for small linear and angular displacements - adjusted by an amount which, at tle present
state of knowledge, can be given only enpi-rical formulation; if viscosity plays a dominant role, the hydrodynamic forces have purely empirical statements and the concept of a velocity potential ceases to have meaning, In what category a force falls
depends essentially on the geometry of the body, its orientation to the flow and its depth of submergence. If I have just stated something that has long been common knowledge, it is to point out that a
technique for adjusting the inviscid forces for the action of viscosity is not quite in hand; and while there is reason -to believe that this adjustment may be small except possibly in the case of the lateral motions of all hulls and in the case of all the
motions of platforms with blunt hulls
(e.g.,
SEDCO 135), there is little exper-imental evidence to upport or deny this belief.2.2
THE REACTIONSThe time-varying pressure acting on the hull as the result of any forced oscil-lation to which she may be subjected con-sists in three component terms:
A pressure difference corresponding to the change in position of the hull: this is the quasi-hydrostatic restoration.
A hydrodynaxnical pressure in phase with the oscillatory velocity of the hull, A hydrodynamical pressure in phase with the oscillatory acceleration of the hull. The speed of advance acts to modify the second component and the cross terms in the first component.
The foregoing pattern corresponds to a somewhat simplified model of the pressure field which is, nevertheless, sufficient for the theme of this paper. For a fuller exposition, the reader is invited to the surveys, by Ogilvie (l964), Newman (1970)
Wehausen (1971) and St. Denis (l97), 2.2.1 Quasi-Hydrostatic Reaction
The quasi-hydrostatic reactions obtains only in heave, roll and pitch and is written
where the functional relation depends on the geometry of the hull and on the ampli-tude of motion. It is usual to assume that the relation is linear and expressible as
such that the relevant characteristics of the underwater
hull
form do not change appreciably. For ships of conventional form, this means amplitudes of heave less than about one-fifth of the draft, ampli-.tudes of roll less than some 15 degrees and amplitudes of pitch less than about5 degrees. These are, of course, somewhat gross values which apply to an average hull of a geometry not rigorously defined. The ranges in motiOn displacements for which the expressions are valid for sea-going platforms can be stated as follows: the combined heave, ±'oll and pitch must neither cause the deck structure to submerge nor cause main submerged elements of buoyancy (hull, pods,
..) to broach the surface, or main surface elements of buoyancy to submerge. The speci.fic va].ues of the displacements depend critically on the operating draft: they can be very low.
An appreciation of how nonlinear the restoration can be is obtained from
Figure 2.2.l.a, which is a plot of righting arms in roll for the platform "Aquapolis" now being constructed for the Ok-inawan Exposition to be held in 1975. The depar-ture from the straight line, which yields a restoration proportional to the meta-centric height, is significant.
In the general case, the quasi-hydrostatic restorations can be written
formally as
C..(s.)x.(t)
lj ] J
where the restoration coefficients depends on the amplitude of Inoti9n
Sj.
2.2.2 Hydrodynamic Reactions
The hydrodynamic reactions can be written as
- iB..
1Jwhere the first terms is of phase opposite to the oscillatory displacement of the hull;
hence, the factor Aij is interpreted as adding inertia to the hull and is referred to as the hydrodynamic inertia. For a deeply submerged ul1, B1O and the only hydrodynarnic reaction is the inertial one. In the above expression, is the frequency ofwave encounter: it is related to the wave frequency u through the expression
cos(>)/g
vhere U is- the hull's speed of advance, X is the relative wave direction and g is the gra'itational acceleratiOn.
For i,=l,2,3, A1 is the hydrodynarnic mass; for 11,2,3, j4,5,6 or vice versa, Aij is the (first) thoment of the. hydrodynamic mass; for i,j=L,5,6, is the moment or
pro-product of inertia of the hydrodynamic mass, The hydrodynamic inertia can be expressed in terms of coefficients obtained as the ratios to the corresponding inertia of the basic hull:
K.. = A../m
= A../J.. i,j=4,5,6
where in is the mass of the hull, is the moment or product of inertia; ther is no coefficient of hyth'odynainic moment for i and j from different sets inasmuch as there is no reference hull moment with the system of coordinates adopted.
For a hull at or approaching the water surface, two changes manifest themselves.: the hydrodynamic inert Ia undergoes a modi-fication which can be quite large, and the component
Bi1
appears whiCh is in quadrature with the motion. The modification to the hydrodynamic inertia is related to the gén-eration of a standing wave whose 'amplitude peaks at the hull and dies off rapidly with distance; the component in quadrature gives rise to an outwardly radiating wave system which, inasmuch as it carries off energy, acts so as to damp the motion: it is the wave damping coefficient.In principle, there are 3.6 coefficients of hydrodyn-amic inertia and wave damping, but symmetry makes for a powerful reduct-ion
in number. All hulls have lateral symmetry (about the vertical centerplane) and for these only the parameters having either an even-even or an odd-odd combination if i,j
indices remain, indicating that the longi-tudinal motions do Oct couple with the lateral ones; the resulting matrix of para-meters, indicated only by the indices, is
rll
O 0 22 131 0 or {B. .} =I)
-O 42 51 O 62If, in addition, the hull has fore-and-aft synrnetr-y, as if often the case, the cross-diagonal parameters, A26A52A35A53=A46= A54=O and similarly for the wave damping
B3.
13 0 15 0 24 0 26 33 0 35 0 -0 44 0- 46053
0 55 0 0 64 0 66 SOME OBSERVATIONS ON THE TECHNIQUES FOR PREDICTING THE OSCILLATIONS OFOTC 2024
The coefficients of hydrodynarnic inertia and wave damping depend on several factors, including hull geometry, depth of water, depth of subme'gence, closeness to other hulls, speed of advance; but the fac-tors of chief concern herein are those of frequency and of amplitude of motion; thus, one writes the hydrodynamic reaction of inertia as
2
d x. A. .(a,s.)
dt and that of wave damping as
dx.
B..(CT,s.)'1
i.j j dt
The foregoing discussion is based on the assumption that the water is inviscid, and question naturally arises as to how much the values of the hydrodynamic inertia and damping coefficients are affected by viscosity. The answer is not readily forth-coming because of lack of parallelism, for all the tests are made in a real fluid. There are both happy coincidences as well as serious and sometimes puzzling deviations between theory and test. And whenever such deviations occur, it is viscosity that is usually invoked as the cause--and indeed
it appears that viscosity may have been introduced into physics simply to explain unexplainables.
It is a common observation that the streamlined flow pattern about an oscil-lating body can be greatly distuibed by the presence of the turbulent wake which the body itself generates. However, to intro-duce viscosity into a mathematical model of hydrodynamic inertia and wave damping so as to take into account the wake phenomenon would turn a formidable problem into a quite
inaccessible one, Thus it remains that at least for certain bodies of very full or re-entrant sections (as occurs, indeed, in many platforms), the assumption of irrotationality must be viewed with some suspicicion.
In some cases, the frictional aspect can be taken into account. separately and included: except possibly in roll and surge, it is negligible thid does not upset the general pattern of the flow. So can the viscous pressure drag provided the body has clearly defined surfaces disposed nor-mally to the flow (bilge keels, cut-off
sterns, etc.); but even gross, empirical methods are unavailable when a large part of the hull becomes cyclically engulfed in a turbulent wake.
Of all the motion predictions, that of roll suffers most in accuracy if the
contri-MANLEY ST. DENTS
bution of viscosity is neglected.
E:er.
ment data upon which to beean estire
the viscous effect can be fcund in ier-'-(1965); and it can be verified that c two components of the viscous reactic, frictional one is dominated by that of vie-cous pressure (or eddy-ma-king), An empir-ical technique for accounting for the latter is that of Tanaka (1960). It appearsthat the techniqueworksweu. for ship hulls; but corroborative evidence on platforms
is wanting.
Thus, viscosity enters as an essential influence in surge and roll and reflects itself in the appearance of additional damping terms
Idx.1 2
B..(a)'1-1'
LdtJ
where for i1, =l,3,5
14, j2,4,6
2.2.3 Elastic Reaction
When a hull is moored, there is added to the hydrostatic retoration an elastic restoration which results from the changes
in tension that occur in the mooring lines as the hull is displaced from her position of static equilibrium. The changes in ten-sion are conveniently separable into two components respectively termed quasi-static and dynamic, The former is the difference in tension that obtains when a mooring linets point of support at the hull moves so slowly that no inertial reaction or change in hydrodynamic drag is excited, the only parameter of motion entering into the calculation being the displacement of the point of support; the latter is the corres-ponding difference in tension that the line experiences when its point of support moves so rapidly that it becomes necessary to consider also the rates of motion of the latter, the inertial reaction and the incre-ment in hydrodynamic drag associated with the motion of the line.
Because of the usually (but not always) large mismatch between the modal periods of hull motion and the natural periods of transverse and longitudinal
oscillations of the mooring lines, it is customary to assume that the hull
motions are affected only by the quasi-static increment in line tension; these motions are then introduced as inputs into a determination of the oscillations of the lines; the dynamic reactions of the latter are not, however, fed back into the
platform,
The restoration is obtained in two steps: In the first, the position of
SOME OBSERVATIONS ON THE TECHNIQUES FOR PREDICTING TJ- OSCILLATIONS OF
828
FREELY-FLOATING HULLS IN A SEAWAYstatic equilibrium is found corresponding to the mean values of wind intensity, current strength, wave drift and mooring line tensions acting on the hull. In the second step, the hull is perturbed about the point of static equilibrium and the changes in mooring line tensions are
determined: these are then resolved in :the directions of interest and summed.
Even in the simplest case of a hull moored in calm water with uniform lines, the
restoration is a nonlinear function of her motions; furthermore, when a current is
present, the mooring lines follow space curves which have the shape of distorted catenaries (the sole exception occurring in the case of a single line mooring when only a uniform sea current is present).
The variation in current intensity with depth is, generally, quite complex, and if to this is added the nonlinear variation in tension with the displacement of the terminal of the mooring lines at the hull, it is apparent that an analytical solution cannot be expected. Numerical solutions for the
single cables are however, available (see, e.g., Casarella and Parsons (1970) for a survey).
The programs on mooring lines yield, either directly or through subsequent inter-polation between parametric solutions, the vector, of the mooring tension at the hull and the horizontal projection of the hanging length of the mooring line (should part of the line rest on the sea-bed); both outputs are functions of the input values of the physical characteristics of the line, (net weight, lift and drag per lineal foot), depth of water, height of hull terminal above (or below) the mean free surface, total scope of mooring line, tension at the anchor and current intensity vector, the last being stated as a function of depth and relative to the vertical plane passing through the anchor and hull terminal. The component of mooring line tension at the hull terminal n directed along the principal axis x is Tn'iin where Lin is the corrés-ponding direction cosine. The elastic
reaction from all the mooring lines is by summation
N T 9..
n in
n1
The condition of reference of the cal-culations is that in which all the mooring 1!.nes are adjusted so as to result in 2astic balance (zero net reaction in any
ectjon) when the environment is calm. :.? hull (hence, all the terminals) is
-- m parametric displacements xk in all six
s (k=l to 6). The elastic reaction in
the mode i corresponding to a further dis-placement in the mode j measured
from
the disturbed condition defined by the set [xkJ is given by 1 NtT (x /[x ])'
n i k(x./[x])
Ifl 1nl
where LTn(xi/[xkJ) is the difference in tension resulting from the displacement
Xj
given the disturbed condition [xk], and £in(xi/[xk]) is the corresponding new direc-tion cosine at the terminal n. When ij, one has the uncoupled restoration, when ij, the restoration is cross-coupled.The important point is that the elastic restoration parameters Eij and Ei are non-linear and there is considerable cross-coupling between motions. This poses no difficulty when a solution is sought in the time domain, but it does complicate the out-come when the intent is to seek appropriate functions for resolving the problem in the frequency or period domain.
In practice, Ej(x±) is often fitted by a polynomial
E..(x.) = k. + k. +
11 1 i3O a,2 1
or, more conveniently for linearization, by the power trend
a E..(x.)
= k..x.
11 1
111
When these expressions are extended to account for concurrent motions (xi/[xkj) in other nodes of oscillation, the coeffi-cients and power index undergo change, and this results in a computational maze.
But even for the uncoupled motions there are complications: the polynomial
fit
is easily found provided that th±'oughout the excursior ±Xj, either:A part (however short) of the mooring line remains always on the sea-bed, or
At
no tine does any part of the mooring line rest on the sea-bed.When neither condition is attained, it becomes necessary to fit two expressions to Eij(xj/Exk]) depending upon whether the displacement x is less than that which takes up the slack or more; but the critical dis-placement is itself a fiinctiob of all con-current displacements.
OTC 2024
As can be judged from the foregoing, there is a superabundance of computational complications to be found in a determina-tion of the elastic restoradetermina-tion suitable to an analysis of platform motions in the frequency domain. This has led to the use
of.a linearized elastic restoration based on the average amplitudes of all motions. The drastic simplification is made with
serenity, based in the belief that there are more serious approximations in an analysis of platform motions than are to be found in the variable springiness of a mooring line.
The elastic restoration is written in linearized form as
E.(x.) =
11 1 11 1 1
where kjj(si) is a linearized coefficient,
2.3 THE EXCITATION
The conversion of a seaway into an ex-citation is carried out on the assumption that the dynamical, actions of the plurality of wave components which constitute the seaway are superposable. Such an assumption is necessary if use is to be made of the spectral description of the seaway. The first step, therefore, is to formulate the excitation imparted by a single wave train of arbitrary amplitude and direction.
The wave excitation consists in three components which correspond directly to the three components of hydrodynamic reaction, namely,
+ iaB. + C.
1 3. 1
(the single indexing stthscript being employed). The third component is the so-called Froude-Krylov action: it is derived on the assumption that the exciting waves are unmodified by the presence of the hull. The first two components jointly account for the distortion to the train of incident waves that is induced by the hull:
together they constitute the wave diffrac-tive action.
The change in excitation arising from wave diffraction consists in turn of two components: the first is parallel to the hydrodynamic inertia and is the accelera-tive excitation, while the second, which
is parallel to the wave damping, is the
MANLEY ST. DENIS 829 velocity excitation. The parameter of
accelerative excitation Ai is in essence Abut modified to account for the atten-uation of wave action with depth and resolved for wave direction; the velocity excitation B5 is E5.j similarly modifed and resolved. It may be noted that the dependence on depth can be very strong; and it is this dependence that makes it feasible for platforms to respond gently to the action of heavy seas by being designed to attain deep draft by ballasting.
Whereas in the linear system the hydro-dynamic coefficients of the reactions, namely, Aij, Bij, Cij are constants, the corresponding coefficients of the excita-tion are funcexcita-tions of the wave frequency w and relative direction X and of the speed of advance U; the excitation compo-nents are written, consequently:
' A.ccelerative excitation (of opposite phase)
x U)
Velocity excitation (in quadrature)
icJB.(w,X,U)
Displacement or Froude-Krylov excitation (in phase)
C.(ü,X,U)
If the system is linear, the complex transfer function which converts the wave system into an excitation is
V.(w,X,U)
C..(W,x,U) -
cl2A.(w,X,U) + iclB.(w,X,U)Such a description implies that the viscous component of the velocity excitation is zero.
x=
omega chi
One should be careful to distinguish be-tween components of the hydrodynamic ex-citation (which are always wave-induced) and those of the hydrodynámic reaction
(which are always motion-induced). The qualifiers of inertia and damping are in-consistent when applied to excitation: the former denotes a physical property by which a body resists a change in motion; the latter, a property by which a motion is attenuated; both concepts are contrary to that of exciting a motion. The
literature is abundant in such inconsistency.
The excitation.arnplitude per unit wave amplitude results as
[[C(w,x,U)
-+ c2B?(w,X,U)3h/2
'and the dynamical phase lag is given by
c.(w,U,X) E arctan
'C (,u,x)_C2A(w,U,x)
'i
Thus, the excitation imposed on a hull traveling at the speed U in the direction x relative to that of a regular wave train of amplitude a and frequency w is, in the
linear case, F.(a,w,X,U;t) = aF.(w,x,U;t) =
aV.(w,X,U)exp(iat)
= a.[V.(w,X,u).V!(w,X,U)JZ/2 exp(icYt_c(w,X,U)) =a.F.(w,X,U)exP(it_E(W,X,U))
where Vj( ) is the complex conjugate of
V( ). It follows that the first transfer
function introduces a (dimensional) change in amplitude and a change in phase.
But for motions of amplitudes not aithogether small, the hydrodynamic coeffi-cients A(W,X,U),
B(w,x,U) and
C1(w,x,U) are not constants and in certain modes, especially roll, the viscous damping coef-ficient Bf(w,x,U) cannot be neglected since usually dominates the wave dampingcoefficient.
3. HULL KINETICS
3.1 OVERVIEW
The system of forces acting on the hull being known, derivation of the response is a straightforward process thich is
simple when the seaway is described by a variance spectrum and the transfer function which converts the excitation into a
motion is linear.
As for the hull dynamics, the kinetics are
also considered in
two steps transfer function and short-term spectral description.3.2
TRANSFER FUNCTIONS OF THE LINEAR SYSTEMThe transfer functions are derived from
the statement of dynamical equilibrium
existing between the excitation and the
restoring (if any), damping and inertial reactions of the system, When the system is linear, the hydrodynamic coefficients of inertia, wave damping and quasi-hydrostatic restoration are assumed con-stant; also, viscous damping is neglected, and the elastic restoration is not taken into account either because the hull is freely-floating or, if moored, the elastic constraint is deemed to be insignificant. The condition is defined by the equation
{L..(wxU)}{x}
{F.(ci,x,U)}where {L1(w,x,U)} is the square matrix of elements ij (i,j1 to6),{x} is the row matrix and {F(w,X,U)} is the column matrix. The generic form of the operator Ljj( ) is expanded to read L..(w,U,X)
[m.M
+ d2x.(t) + b. .B. .(w,X,U)dt2
'J
'-J dx (t) dt +with nj denoting the
radius(ij)
orcross-radius (ij)
of gyration; alsoa.. ,b.. E
1 if both
i and 5 are either even or odd0
otherwise
1 if ij3,4,5 or if ij3,5
M
mass of hull
For the purpose of ulterior discussion,
the known linear solution results as
follows: assume sinsoida1 motion and
write,
x..(a,w,x,U;t) s. .(a,w,x,U;t) 3. 31 exp(iat-C..((A),X ,U)) where m.1]
= 1 0 0 0 0 0 O 1 O o O O 0 0 1 o 0.0
0 0 0 0 0 0 0 0 0 r 0-r6
0r5O
2 2-r6O
r56
SOME OBSERVATIONS ON THE TECHNIQUES FOR PREDICTING THE OSCILLATIONS OF
where Xjj( ) is the oscillatory
displace-ment in mode j at instant t-induced by the excitation Fj(a,w,X,U;t) in the mode i at the same
instant;
-ji( ) is the amplitude of the oscillatory aisplacement and c-jj.( )is the corresponding kinetic phase lag. The linearized matrix becomes
M + a. .A. .(w,,U)J +
13 13 13
Cc. .C. .(w,X,U)] + i:yB. .(w,X,U)}
13 1313
= a'{F.(w,X,U)} i.e.,
{M. .(w,X,U) + iN. .(w,X,U)}'{s.}
13 13 3
= a{F.(w,X,U)}
1
Let D(w,X,U) represent the square determi-nant in M( ), N( ); and let further
Djj(W,X,U) represent the minor of D(w,X,t.J) in which row i and column j have been omitted; then, the complex displacement in the direction j resulting from an excita-tion in the direcexcita-tion i is
s..(w,x,U) = D(,X,U) P..( 13
F.(,x,U) = aR
( 1ij
= a[T..( )+iU..( )]'F.( ) 13 13 1 a'W..(,X,U)F.(w,X,U)which yields the amplitude of motion
= a[T(w,X,U) +
+
and the kinetic phase lag
= arctan
Thus, the second transfr function, which converts an excitation into a motion, also makes for a change in amplitude and in phase.
The amplitude of the total response in the direction 5 to the full set of excita-tions F(w,x,U) is
6
s.(w,X,U)
3
i13
3.3 RESPONSE SPECTRUM OF THE LINEAR SYSTEM Given the spectral description of the excitation in the mode 1,, Sfi(w,x,U) and the set of linear transfer functions of the response Wjj(c,X,U), the variance spectrum of the response in the direction j is
Sj(wxU)
from which the mean squared value of the response is
r2lT
s(U)
.2_
I I S (ci,x,U)dXd3 27rJ J
xi
00
4. VALIDITY OF THE SPECTRAL DESCRIPTION 4.1 RECAPITULATION
Up to this point the exposition has been in the nature of a prologue for the purpose of bringing out two points which are of fundamental import in a validation of the spectral description of system response in seaways which themselves are validly described by a variance spectrum. The two points are:
b) The nonlinearities of all the components of reaction and excitation.
The validity of the spectral represen-tation of the seaway, being assumed, the inquiry is on the transfer
functions
which convert the seaway into hull motions.4; 2 FREQUENCY-DEPENDENCE OF THE
HYDRODYNAMIC REACTIONS AND EXCITATION When discussing the subject
of
this section, one is likely to recall TickTs (1959) statement: "Differential equations with frequency-dependent coefficients are very odd objects." I should like to claim credit, not for the statement or for the exposition of the concept, but only for having suggested to Tick the nature of the problem that was then unclear to me,namely: does a ship in a confused sea respond to every wave component of differing frequency as if no other wave component were present? My
concern
was obvious, for the above wording is thestatement of what has been sometimes referred to as the St. Denis-Piersori hypothesis.
)+iQ.. (
13 )
Fc
a) The frequency-dependence of the hydro-dynamic components of reaction and excitation.
)+S..(
1J
)
832
SOME OBSERVATIONS ON THE TECHNIQUES FOR PREDICTING THE OSCILLATIONS OFFREELY-FLOATING HULLS IN A SEAWAYFor the reply, Pierson and I are indebted, in addition to Tick, also to Cummins (1962) and to Ogi].vie (1964) who developed with rigor the argument of validity and in doing so provided clear insight into the physical nature of the problem. But the validation is only for linear systems.
For the purpose of analysis and pre-diction, it is convenient to write the hydrodynamic coefficients so as to account for the influence of the frequency of osci.i-lation in separate factors thus, formally at least,
A. .(cl) A.. f. .(a)
iJ iJ
1]
B1(ci) E B..g..(a)
where B± are frequency independent while fi(°) gij(c) express explictly such dependency. As will be seen below, the hydrodynamic coefficients are of somewhat more complex expression and involve other dependencies; however, the preceding expressions are sufficient for the moment.
In the case of the hydrodynarnic inertia, Ajj represents the value corres-ponding to infinite submergence, while fij(G) is a correction for the influence of the free surface. In the case of the hydrodynamic damping, only the wave damping is frequency dependent and the factor gjj() is not a correction factor but rather a statement of the law at which the frequency function varies.
It should be mentioned that the
frequency factors f() and gij() have
been computed for a small number of (three-dimensional) bodies of which the ellipsoid is the most complex, and for an even smaller set of boundary condi-tions (floating, submerged close to the surface, finite depth of water); however, two-dimensional solutions for cylinders, of a great variety of forms and for a small selection of prisms are in hand
(see, e.g., St. Denis, 1974). So long as the hull is of slender proportions, use can be made of such two-dimensional solutions, the results for the actual hull being derivable through the proper integration along the axis of slender-ness of the distribution of the sectional values of hydrodynarnic inertia and
damping obtained for the two-dimensional forns.
There are but few experimental yen-f.ction of theoretical computations; ':.ese include Yu (1960) and Paulling and
Porter (1962), for the semi-immersed heaving circular cylinder; and, more recently, Ainey and Pomonik (1972), who in addition to reporting tests to determine the hydrodynamic inertia and wave danoing of circular cylinders, rectangular prisms and spheres, derived empirical formulae for estimating the influence thereon of frequency of oscillation and depth of submergence.
4.3 NONLINEAR HYDRODYNAMIC REACTIONS AND EXCITATION
The equations of motion are a set of stochastic differential equations whose coefficients are not only frequency depen-dent but also nonlinear. The present engineering practice has been that of ignoring the nonlineanities with a great deal of unconcern. To be sure, so long as the resulting motions do not exceed moderate amplitudes, the computational
task of accounting for the nonlinearities may appear somewhat more burdensome than warranted by the anticipated increase in the accuracy of the results. This is, of course, a subjective evaluation. At all events, engineers as a group hold to the mystic belief that a simple solution is a good solution, particularly if the outcome is tempered with judgment, a quality with which every engineer believes himself to be abundantly endowed. When reality departs from prediction, one invokes the statistical theory. of error and coins or modifies an empirical factor to allow for the divergence. Engineering analysis and prediction is a field domi-nated by empirical factors -- most of them havIng values rather intuitively derived.
The essential role of nonlinear. solutions is to reduce the guesping or arbitrariness that is the companion of linear solutions. Engineering practice can still be based on linear techniques provided fatcrs of adjustment 'are applied which have been priorly derived from a more general solution to the nonlinear equation(s) which define a specific problem in hand.
Nonlinear solutions can always be obtained by solving the equations in the time domain by numerical techniques; however, the process is exceedingly
lengthly if statistical motion parameters are sought in whose derived values one can repose an adequate measure of confidence. Such solutions are not considered
herein: they are not within the scope of the exposition,
b) Perturbation technique
Such a technique (of which.there are several variants) applies to systems tbat are only slightly nonlinear; however, one of the limitations of the previous tech-ni4ue no longer applies, i.e., the non-linearity is not constrained to reside in the restoration term. The transfer func-tion is again deterministic and for a rnonoperiodiá excitation, the solution is obtained as an expansion in ascending powers of a small parameter which defines
the magnitude of the nonlinearity; to derive the terms in the expansion, use is made of a recursive technique: the assumed series solution is introduced into the equation (or set of equations)
governing the motion: the coefficients of like powers of the parameter of non-linearity..are equated: this leads to a chain of linear differential equations whose solution yields the terms in the assumed expansion, each one of which can be considered as the linear response to an excitation which itself is a nonliner function of the previously determined next lowest term. If the expansion is limited to the first term, the result is the linear approximation; if limited to the second term, the outcome is the first order nonlinear approximation.
The perturbation technique can run. into the serious difficulty of
non-convergence either because of the appear-ance of secular terms (which results in a divergence of motion with the passage of time) or because the parameter of nonlinearity is not sufficiently small. A criterion (of Poincar) does exist for concluding what is meant by small, and in an analysis of hull motions this criterion is often exceeded when the sea state is low and almost invariably
exceeded when the sea state is high. The secular terms result from the truncation of the expansion: they can be eliminated but only at the price of considerable additional labor. Nevertheless even an expansion truncated after the first non-linear term is sufficient to reveal, at least qualitatively, the basic features introduced by the nonlinearity.
I-f. the excitation is deterministic, the order of the approximation is, in principle, unlimited, although the compu-tational work increases very rapidly with number of terms and in practice the expansion is not carried beyond the first nonlinear term; but if the excitation is stochastic, an additional difficulty is encountered in attempting to proceed beyona the first nonlinear term, a At the present, there is. no
gen-erally applicable non-numerical technique for solving nonlinear, differential equa-tions be they either dete±'ministic
or' stochastic.
The problem of present interest is stated as follows: Given a seaway defined by its variance spectrtmi and a hull whose dynamical parameters are nonlinear, find statistical expressions for the oscillatory motions in each of the six degrees of freedom.
There are several techniques by which to seek a solution to a set of nonlinear stochastic differential
equa-tions' of the second order; some remarks are presented on the application to the problem in hand of the following
techniques:
Fokker-Planck equation Perturbation technique Equivalent linear.ization Iteration
Except for the first, all the techniques are approximative and the accuracy of the. solution (when this can be established) depends on the relative magnitude of the nonhinearities and on the number of
approximations made. In general, the non-linearitjes are assumed to be weak and only the first approximation involving the nonlinear terms is develOped; it is somewhat impractical to proceed' further. a) Fokker-Planck equation
A technique based on this equation is listed only because someone is bound to raise the question as to its applica-bility to the problem of interest. The appealing aspeot of an approach based on the Fokker-Planck equation is that the derived solution is exact; unfortunately the scope of application is so restricted that it is not a useful tool fOr solving practical problems. As presently devel-oped (Caughey, l962a), the following conditions must be met if a solution is to be found: (a) The nonlinearity is only in the restoring tezm, (b) The damping is linear', (c') The excitation is Gaussian and characterized by' a uniform
(white) spectral distribution, and (d) The excitation, and damping matrices are correlated. This set of conditions is', clearly, of no direct relevance to the problem under discussion.
SOME OBSERVATIONS ON THE TECHNIQUES FOR PREDICTIN( THE OSCILLATIONS OF
834
FREELY-FLOATING HULLS IN A SEAWAYdifficulty that arises because this term has a probability distribution that is non-Gaussian and therefore requires third or higher order moments for its defini-tion. The conclusion to be reached is that the perturbation technique has not proved to be sufficiently useful in ship oscillations and does not hold great promise of further development for this application.
Equivalent linearization
The second approxirnative technique consists in replacing the nonlinear system of equations by an equivalent linear one so selected that the difference between the two systems is minimized, the measure of difference introduced being the mean squared error. The deterministic aspect of the technique was first presented by Krylov and Bogoliubov (19'49) and it carries their name; the stochastic exten-sion is primarily the work of Caughey (l962b).
In principle, the nonlinearity is not restricted to any term, but is required to be weak. It is also required that the systeinbe lightly damped, ãnd this tends to eliminate application of this technique to oceanic systems, inasmuch as the
damping of such systems can be relatively high, i.e., it can amount to or even exceed half the critical value. Addi-tional conditions to be satisfied are that the stochastic excitation be Gaussian and that the variance spectrum be smooth: in the present discussion, this last set of conditions is satisfied by restriction of the scope to hull oscillations caused by light seas. If modes of oscillation are coupled, the excitations in these modes must be uncorreJ.ated. This last condition does not obtain for hulls: the excitations in all the modes are all derivative from the same generating cause, namely, the seaway. Thus, the method of equivalent linearization has not yet .been developed to the point of usefulness in the analysis of hull oscillations.
A Heuristic Iteration technique Th'is technique combines both an averaging procedure as well as one of linearization. Whereas the previous techniques. are valid only for small deviations from linearity, the present nethod is, in principle at least,
appli-:Th1e
to considerably greater deviations; .rtunately., it is not ?SSibie tothe magnitude of the error,
In the application of this technique, the hydrodynamic parameters are linearized
on the basis of equivalence of work done
by each component during a cycle.
Consider first the restoration: if the amplitude of oscillation is sufficient to alter significantly the waterplane geometry, the resulting variation in the term needs be taken into account. To this end,
write
the nonlinear restorationC. .(x.)x.(t) C. .h. .(x.)x.(t)
1]
3 3 131J
3 Jwhere Cj corresponds to Cj
(xj) as xj#O
and h(x) is expressible as a polynomialin the displacement
c(i,j ;k)
h. .(x.)
3. +r(i,j;k)x.
k
where for each element ij the coefficients
r(i,j;k) and the powers c(i,j;k) are obtained by curve fitting. Equivalence
of work yields the linearized restoration
C.(x1)x(t)
= c..[l
r(i,j;k)C. 'h. .(s.)
3 3 13 13 3
x.(t)
C. .(s.)x.(t)
J
1]
3 3Note that
the linearized restoration is
a function of amplitude of the oscilla-tory displacement.The hydrodynainic damping is
sepa-rable into two components: a wave damping, which is usually assumed to be linear in
the velocity and a viscous damping, which is usually assumed to be quadratic in the
velocity; the resulting expression being
dx.(t)
rdx.(t)]2
i-B..
BW.
J v [j
i
dt
1]
dtThe common assumptions of linearity of wave damping and of quadratic behavior of viscous damping are not fully supported by experinent, and it is pertinent to call attention to a recent experimental study by Arney and Fomonik (1972) which reveals that not only is the wave damping a non-linear function of amplitude, but that for prisms it increases with the square root of the amplitude of oscillation, while for circular cylinders and spheres, it decreases with the inverse suare root of the amplitude. No explanation for this strange anonaly is suggested.
OTC 2O2L 1ANLEY ST. DENIS 835 It is, nevertheless, somewhat
dis-turbing that it is only recently that such strange behavior of the damping reaction has come to light: our strongly held belief in the linearity of wave damping and quadraticity* of viscous damping must now be examined.
To keep the discussion sufficiently general to encompass not only the hitherto usual assumptions of linearity of wave and quadraticity of viscous damping, but also other possibilities, the damping components are re-written as
B'I
]
+dt B
ii
L dtNote that the coefficient of wave damping is a function of the frequency of wave encounter; and should the hull oscillation
in the generic mode attain sufficient
amplitude that
the underwater portion ofa floating hull undergoes significant
variation in geometry, both the wave as
well as the viscous damping become
func-tions
of the amplitude of displacement.
Consequently, the damping reaction iswritten
dx.(t) inB.(a,s.) .[
]
+ B.(s.)
dx (t)n
[_t ]
-
B.g(a)
g(s1)
[dx.Ct)
1j
+B5'
dx.(t) n [ ] dtThe wave damping factor gi(sj) is related to the
form of the hull, whereas
the viscous damping factor g(s) is
related to changes in
wetted surface.In
The generic velocity law [dx(t)/dtJ is generalized through the equivalence relation
rdx(tyIm
rm-1
m-1 2
r(l+m/2)
],
Ldt
]
='S
r(l[m1J12)
dx(t) - dx(t)dt
- g(c,s,m)
dt
Here r( ) is the gamma function. (For m1, g(a,s,m)=l; for m2, g(a,s,m)=8cs/37r).
Possibly, this word does not exist; its meaning, nonetheless, should be clear.
,
The factor g.Y(sj, i
e>:nc: into
g..(x.)
E1 t
g(i,j;k).x.b(1j)
13
k
hence,
the linearized viscous dampingX'eaction is written
dx.(t) inB4gj(s.).[
t
]
=dx.(t)
where:g.(s.)
1 i-
V(ji.k).
2 k-
b'(i,j;k)+l
btCi,j;k)lS.
J g1v(a,s3 - fl1s.
n-l 2-
rcl+n/2 3 r(L+[+l]/2The factor gj'(xj) is similarly
ex-panded for each value of the frequency into
E
1 +
qw(ij;k).xbw(i,i;k)
k
and this leads to the linearized wave
damping reaction
rdx. Ct)
w w 13 3 Ldt
B.. g .(cl)'g. .(s.)
B.g.()'[l
1J +qW(i
jk)
1)
k
b (i,j;k)+l
s.bw(i,i;k)_l].[m_l.s m-l. 2j
dx. (t)r(lm/2)
1. i B r(1+[m+l]/2)J dt ijdx. (t)
W Wg. .(s.)g..
13 3 13(cY,s.,m) dtThe hydrodynainic inertia is also a
function of oscillatory frequency and of
the amplitude of motion.
To this end,
one writes
A. .(c,s.)
EA .'f. .(c)'f. .(s.)
13 3 13 13 13 3
where is the hydrodynamic inertia in
an infinite medium, fi(G) is a
factor which accounts for the variation in hydro-dynamic inertia with frequency, whilef(x-j) accounts for the parallel variation
witFi amplitude of motion.
The latter is
836
SOME OBSERVATIONS ON THE TECHNIQUES FOR PREDICTING 'THE OSCILLATIONS OFFREELY-FLOATING HULLS IN A SEAWAY 2 f. .(s.) 1 +P(ii;k)1a(ij;k)+1
k a(i,j ;k)-1S.
where, again, p(i,j;k) arid a(i,j;k) are obtained by curve fitting, Thus, the hydro-dynamic inertial reaction results as
d2x.(t) A..(a,s.) 13 dt2 d2x.(t) = A. .f. .(c)f. .(s.)' 13 13 13 dt2
The same logic is followed with the coefficients of the excitation, Ai, Bi, Cj with the result that excitation compo-nents are written:
accelerative excitation _c72.f±(a).f(a).A(a,X,U) velocity excitation W( W ia g. cY)g.!.a.)'g.(a,a.,m)' 1 1 1 1 B(c,X,U) +
V'
11
BI (w x U)displacement or Froude-Krylov excitation
The (single-indexed) factor f(a) is related to the corresponding
(double-indexed) factor entering into statements of the reactions through the relation fi(ai)Efii(si), where
Sj
has been made equal to ai; and similarly for gj(ai) and hi(aj).The consequence of this is that the first set of transfer functions (the dynamic transfer functions which convert the wave characteristics into excitations) ae no longer linear and contain the wave amplitude as a parareter. Thus, in lieu of V(wx,U) one writes Vi(w,X,U,a) and similarly for V( ) ad F( ).
The foregoing discussion has been purely deterministic;
it
now remains tontrodce the stochastic aspect. To this use is rre a suggestion by ilopou1os (1571) in a study on
non-roll in which he considers the a damping tern that is partly Partly quadratic; he also
OTC 20214
considers the case of a restoration of the Duffirig type, i.e., partly linear and partly cubic and derives for a uniform spectral density distribution of the excitation, the interesting result that the equivalent damping and restoration are
B B' wa B"
and
C C' t 3a2C" (cY=x2)
where the modal subscripts i, j have been omitted and where a2 is the variance of the response spectrum. Upon comparing these parameters with those corresponding to
regular, waves (B=B'+8wsB"/3ir; CC'+l/2s2C") it follows that the equivalent damping terms are equal provided that for the motion ampli-tude one substitutes [8/3iiJ/[/7].s/a=
0.849/1.596 s/a=0.534 s/a. This relation is based on the assumption that the response spectrum is narrow (a justifiable hypotheses inasmuch as the damping in roll is rela-tively small). Vassilopoulos does not develop the condition for equality in the equivalent restoration terms, which can be shown to reduce to
/i7s/c = 0.408 s/a
Vassilopoulos then employs a technique of equivalent linearization to obtain the nonlinear transfer function, hence the variance of the motion: the former is obtained through expansion in a Taylor Series of the coefficients of damping and restoration: this restricts the technique to weakly nonlinear systems.
It is suggested that a preferable technique is to combine the methods o.f equivalent linearization and of iteration. The procedure is outlined following an introductory observation.
To focus on the solution to the non-linear problem, the problem is simplified to that of a hull hove-to in a seaway and headed so as to line up with the wave direction; thus, U=0, X=0, a=w.
1. Introduce the variance spectrum of the seaway. For illustration, the Bretschneider formulation of the scalar spectrum is used (see St. Denis (1973)). where S (T,0) E S (T)f (8(T)) n n n
S CT)
S (t)
n
S nOTC 20214 MANLEY ST. DENIS and H is the significant (observed) wave
height, T5 is the significant (observed) wave period and S(t) is t'ne normalized wave spectrum.
S(t) E 4t3exp(-t)
where TET/T5. The range of this spectrum
iS
01T15/3T.
For simplicity of exposi-tion, let the wave spreading function f(8(T))1 in what follows.2. Determine the set of hydrodynamic cofficients: .(a,s.) 13 3 B.'(a,s.) 13 ] .(a,s..) 13 3 .f. .(a)f. .(s.) 13 13
1]
3B.g.(a)g.(s.)g."(a,s.,m)
131)
13 3 13 3 .'g. .(s.)g. .(a,s.,n) 13 13 3 13 3 C. .(s.) E C. .h. .(s.) 13 31]
1]
3(The frequency of encounter a has been retained to facilitate correlation with the preceding text). Let initially f(a)=
fi(s )gj(a)=gi(a,sj ,m)gj(sj )=
(cY,s ,njh (Sj )=i.
3. Derive the natural period of oscillation in mode i. Solve by interation
T. .(a. .
,s.) 2Tr/A. .(a. .,s.)/C. .(s.)J3 33 3 33 33 3 3] 3
where
a..
= 2 IT. .(a. .,s.)33 3] 33 3 Obtain ratio T..(a. .,s.) .3] 33 3 T S
4. Derive the critical damping coefficient at the natural period
(B. .(a. .,s.) =
IA. .(a. .,s.)C
33 3] j cr 3] j. .(a. .,s.) J] 3]
and the corresponding normalized damping coefficient
B W( .,s.)+B.Y(a..,s.)
K. .(a. . ,s.) = 3 33 3] 3
3] 3] B. .(a. .,s.)
3] 33
5. Calculate the dynamical transfer
functions for small amplitudes for the range of periods and wave amplitudes of
interest
-V.(J,X,U,a.) V.(a,a.) = C.(w,X,U). h.(a.) - o2A (w,X,U)f.(a)f (a
11
1 1
ii
+ I ).g(a,a. ,m) 1 2. 1 1 2. .i-+ B(w,x,tJ).gY(a.).gY(a,a.,thfl w v Initially, let(The functional relations (u.i,X,U) have been retained to facilitate correlation with the preceding text).
6. Determine the kinetic transfer functions
W..(W,X,U)
W. .(a,s.)1]
1]
3in the manner indicated in Section 3.2, but with the coefficients determined from step 2.
7. Derive the response spectrum S.(w,X,U,a) E
6
= S .(a,a.),IV.(a,a.)12.Iw..(a,s.)12
1 1
1]
3i=l
8. Obtain the mean squared values (i.e., the variances of the motions x.
3
-x. = I S .(a,a.,s.)'da
Ja X) 1 3
0
where a0 corresponds to the minimum wave period of interest (T=0) and in corresponds to the maximum such period (T5/3.(T5)).
9. Calculate the deterministic linear equivalent amplitude of motion
Corresponding to the hydrodynamic inertia and estoration
Sjl
= O.5314(x)'2Corresponding to the damping reaction
S.
32 = O.408(x?)'23 10. Return to step 2 with
Sjl
beingentered in the coefficients of hydrodynamic inertia and restoration and Sj2 in the coeffficients of damping.
11. Continue to convergence.
In the foregoing development, no statement was made as to the nonlinear char-acter of the hydrodynamic coefficients. Only in rofl is the nonlinearity symmetric; for the nonlinearity to be symmetric in heave and pitch, the hull should be floating
SOME OBSERVATIONS ON THE TECHNIQUES FOR PREDICTING THE OSCILLATIONS OF
838
FREELY-FLOATING HULLS IN A SEAWAYat a horizontal plane of symmetry (as does, e.g., a cylinder with axis in the waterplane). This is not a real condition to be commonly encountered.
The method can be extended to cope with asymmetric and antisymmetric non-linearities (as occurs, e.g, with. a V-shaped hull), and the least complicated
.iay is to solve separately for the posi-tive and negaposi-tive excurs ions, obtaining
21/2
21/2
in this manner (x..) and (xt) , i.e., the statistics of3the crest aid the
trough. But this is only a tempoDary solu-tion pending the development of techniques
based on higher order spectra.
5. SUMMARY AI'fD CONCLUSION
So long as the seaway is of modest intensity and the hull can be validly modeled as a linear system, spectral tech-niques are in hand that provide reasonably accurate engineering predictions of hull motions. These techniques fail when;
The intensity of the seaway ceases to be modest.
The hydrodynamic parameters which describe the hull are nonlinear and coupled.
When the seaway becomes severe, the spectrum of its variance provides an insufficient description: the waves exhibit high asymmetries of form and are no longerrepresentableby the random superposition of Airy wave trains. The case of heavy seas was excluded by the restricted scope of this paper, but it is well to recall this exclusion and to restate that it is the high sea states that are of greatest importance and con-cern to the designer and mariner: it is in such sea states that vehicles and platforms are likely to come to grief rather than when the surface of the sea is perturbed by gentle undulations.
Even when the seaway is itself of such modest proportions that it can be suitably described by a variance spectrum, spectral techniques of motion prediction available at present become inadequate when the nonlinearities are rio longer weak, or if weak, when the hull geometry
is such that her modes of oscillation are coupled. Nonlinearities of any kind, wiether in the seaway or in the system,
reduce the validity of a spectral repre-entation of hull notions in proportion
the agmitude of the departures from Learity measured relatively to the
OTC 2O2L magnitudes of the linear or linearized
parameters to which they correspond. Also, inasmuch as the same seaway is the gen-erating cause of all motions, the excita-tions in all modes of oscillation are correlated. Such correlation makes for an additional difficulty and at present no dependable solution appears to be in hand. I have made a modest attempt in this direction (St. Dens, 1967) by what I admit to be a somewhat intuitive ap-proach; but it appears that the success has been more modest than the attempt.
In concluding the theme of this paper, I would like to make a vigorous plea for concentration on studies, both theoretical and experimental, on the non-linear aspects of hull oscillations. I
do not wish to de-emphasize any work on the linear response of hulls: recent years have witnessed brilliant achieve-ments in this subject, and there is still so very much to be undertaken and brought. to successful conclusion before one can relax one's scientific preoccupation with hull systems that are linear. But the serious, the important, the challenging problems to be solved are all on the non-linear behavior. The measure of relative importance may be given by analogy: linear hull oscillations are related to the incip-ience of sea-sickness; it is only when hull oscillations become nonlinear that a hull may capsize with ensuing loss of life.
Nonlinear problems present greater difficulties of solution than their linear counterparts. In general, they are solved more readily in the time domain than in
the frequency domain, But solutions in the time domain of nonlinear equations with stochastic inputs become brute force
solu-tions if they are employed to derive statis-tical parameters useful in predictions.
When seeking the solution to any problem, one must strive for elegance. By this I mean that one must strive to seek that solution which will minimize the expended effert, For linear systems sub-ject to stochastic inputs, the spectral technique is indeed, the elegant one. The question that now seeks an answer is: Can the spectral technique be adapted to describe the behavior of nonlinear systems with stochastic inputs and still retain its place of preference? If not, what is to be done?
The answer will not come except as the outcome of a long sequence of arduous studies. The need for nonlinear solutions to the problem of hull oscillations in heavy seas is now. And we must now
dedicate ourselves to their search if we are ever going to gain insight into the
seaworthiness of hulls and replace with objectivity the rules-of-thumb by which even today we assess its measure. The task will be an onerous one, but the rewara will be more than commensurate with the effort.
6. REFERENCES
Amey, H.B. and Pomonik, G. (1972), "Added Mass and Damping of Submerged Bodies Oscillating Near the Surface" Offshore Technology Conference, Paper 1557.
Caughey, T.K. (l962a), "Derivation and Application of the Fokker-Plarick Equation to Discrete Nonlinear Systems Subjected to White Random Excitation" 64th Meeting of the Acoustical Socity of America, Nov. 9, 1962, pp. 1-35. Caughey, T.K. (1962b), "Equivalent Linearization Techniques" 64th Meeting of the Acoustical Society of America, Nov. 9, 1962, pp. 87-109.
Cuinmins, W.E. (1962), "The Impulse Response Function and Ship Motions" Schiffstechnik, Vol. 9, pp. 101-109, reprinted as David Taylor Model Basin Report 1661.
Kryloff, N. and Bogoluiboff, N. (1943), Introduction to Nonlinear Mechanics, Princeton University Press.
Ogilvie, F.T. (l94), "Recent Progress toward the Understand and Prediction of Ship Motions," Fifth Symposium on Naval Hydrodynamics, Wageningen, Publication ACR-fl2, Office of Naval Research, Washington, D.C.
Paulling, J.R. and Porter, W.R. (1962), "Analysis and 1easurernen of Pressure and Force on a Heaving Cylinder in a Free Surface," Proceedings, Fourth U.S. National Congress of Applied Mechanics, Berkeley.
St. Denis, M. (1967), "On a Problem in the Theory of Nonlinear Oscillations of Ships," Schiffstechnik, Vol. 114, Nc. 70, January 1967.
St. Denis, M. (1973), "Some Cautions on the Employment of the Spectral Technique to Describe the Waves of the Sea and the Response Thereto of Oceanic Systems" Offshore Technology Conference, paper no 1819.
St. Denis, M. (1974), "On the Motions of Oceanic Platforms" International Symposium on the Dynamics of Marine Vehicles and Structures in Waves, University College, London. Tick, L.J. (1959), "Differential Equations with Frequency-Dependent Coefficients," Journal of Ship Research, Vol. 3, No. 2,'pp. 45-46. Vassilopoulos, L. (1971), "Ship Rolling at Zero Speed in Random Beam Seas with Nonlinear Damping and
Restoration," Journal of Ship Research, Vol. 15, No. 4, December 1971,
pp. 289-294.
Wehausen, J. (1971), "The Motion of Floating Bodies," Annual Review of Fluid Mechanics, Vol. 3.
Yu, Y.S. (1960), "Surface Waves Gen-erated by an Oscillating Circular Cyl-inder in Shallow Water," Dissertation, Massachusetts Institute of Technology, 93 pp.
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-2 25 20 15 10 5 50 20 30ANGLE OF ROLL, OC (DEG.)
Fig. 2.2.1.a - Righting arms in roll of platform. AQUAPOLIS for- four loading conditions at normal draft of 16 m.
