THE

DEPARTMENT OF

RE AtiD

T. Francis Ogilvie

THE UNIVERSITY OF MICHIGAN

COLLEGE OF ENGINEERING

MINE ENGINEERING

ARCH1EF No. 021

May 1969

Lab. v. Scheepsbouwkunde

Technische Hogeschool

Delft

THE DEVELOPMENT OF

SHIP MOTION THEORY

DOCUMENTATiE : r2 1DATUM'

THE DEVELOPMENT OF SHIP-MOTION THEORY

T. Francis Ogilvie

s1TY e Department of Naval Architecture44,111rt+4 and Marine Engineering...me Col lege of Engineering=.0 4.. The University of Michigan

Ann Arbor, Michigan 481047811

rjsre HocinchooNo. 021

May 1969

Udeling Schee eepvaarthunde

PRFNACE

This report is written in the attempt to present an essentially

non-mathematical discussion of the analysis needed to predict the heave

and pitch motions of a ship in head seas. About half of the report is

devoted to a description of the basic assumptions used by previous inves-

tigators, explaining their logic and showing why their results had to be

abandoned by engineers. The remainder concerns the assumptions which

have recently been found to lead to quite reasonable predictions; these

hopefully may soon become available to the ship designer.

The actual mathematical analysis is very complicated, and we hardly

hint at it here. The details may be found in Ogilvie & Tuck (1969),which should be released very soon, and in Ogilvie (1969), which is in

preparation. Corresponding computer programs are under development.

May 1969 T. Francis OgilvieAnn Arbor, Michigan

INTRODUCTION

A heaving and pitching ship behaves qualitatively like a two-degree-

of-freedom spring-mass system with fairly heavy damping. This is quite

obvious if one gives a ship model a heave or pitch displacement from its

equilibrium configuration and then releases the model; it oscillates a

few times with rapidly decreasing amplitude and then comes to rest. A

frequency-response experiment shows the same qualitative behavior; for

example, a plot of heave amplitude/wave height indicates the presence of

a resonance condition, with typical spring-mass-system behavior above and

below the resonance frequency.

This analogy is only qualitative. Golovato (1959) showed many years

ago that the response in a transient pitch experiment could not be

matched by the simple exponential-decay curves which are typical for the

response of a spring-mass system, and the usual equations of motion for

sinusoidal excitations involve coefficients which are functions of fre-

quency. Nevertheless, the general features of a damped resonant system

are present, and a successful theory for predicting heave and pitch

motions must include such features.

This report is essentially an account of the search for such a

theory. The modified strip theory developed primarily by Korvin-Krou-

kovsky (1955) includes all of the necessary components, and his theory has

been quite successful in its predictions. Nevertheless, it is not that

kind of theory which is to be discussed, for Korvin-Kroukovsky's approach

was strictly an engineer's approach. He patched together all of the con-

cepts which he thought to be significant and produced formulas corres-

ponding to each of the pieces. What I shall discuss is the attempt to

derive a satisfactory perturbation theory for ship motions beginning with

a precisely stated boundary-value problem, that is, the most nearly exact

mathematical model of the system possible. Such a model is easily pro-

duced, but it is intractable, and so one must develop an appropriate

procedure for obtaining an approximate mathematical model with the essen-

tial properties of the physical system. This is done by introducing an

-1-

4-

- 2 -

appropriate small parameter and using it consistently to simplify the

exact problem. When the appropriate small parameter has been found,

along with its relationship to several variables and parameters in the

overall problem, one has only to analyze the logical consequences of

treating that parameter as being vanishingly small. Then the predictions

which result must be tested against experiment, for there is no other way

of being sure that the exact solution of the approximate problem will be

the same as the approximate solution of the exact problem.

A considerable portion of the report is devoted to discussing the

history of some unsuccessful attempts to establish the appropriate approxi-

mate mathematical model. This discussion provides a considerable amount of

insight into the nature of the model which is finally adopted and which

appears now to be leading to generally correct predictions. The funda-

mental modifications in assumptions from theory to theory are quite small,

but the discussion points up how slowly these necessary changes have been

recognized.

Following this history, there is a discussion of some consequences of

adopting the most recent model. Some of the results are as expected, but

others are somewhat surprising.

The required comparison with experiments is given short shrift for

two reasons. On the one hand, most of the final formulas are rather similar

to those of the modified strip theory, which has been compared with experi-

ment many times; the new analysis should not give grossly different answers.

On the other hand, the aspects in which the new results differ from strip

theory have not been subjected to thorough numerical analysis, and so it is

not yet clear just how much difference there really is.

The details of the most recent analysis are not presented herein; only

their rational basis is discussed. The complete analysis may be found in

Ogilvie & Tuck (1969) and in Ogilvie (1969).As a final introductory note, it should be mentioned, that we are

treating only the case of heave and pitch motions in sinusoidal head seas.

The extension to irregular seas is straightforward provided that the sea

state is not extreme and provided that the hull geometry does not accen-

tuatenonlinear phenomena. Under these circumstances, it has been amply

demonstrated that a simple superposition of sinusoidal waves and sinu-

soidal motions yields valid results. Other degrees of freedom and other

headings with respect to incident waves have not been satisfactorily

analyzed yet.

FORM OF THE EQUATIONS OF MOTION

The equations of motion for coupled heave and pitch are often

written as follows:

(m+a) + 1,1* cz. + d.E; + e9 + gf). = F( t.)

(A+I)a + Be' + ce + ft. + Ei Gz M(t) ;

t = z(t) = heave displacement;

9 = 9(t) = pitch rotation;

F(t) . heave force due to external causeb;

M(t) pitch moment dueto external causes;

m. mass of ship;

I .= moment Of inertia of ship about pitch axis.

The other coefficients correspond to added masses, damping coefficients,

and restoring-force coefficients. In. thisreport, we shall generally use

the following indicial, notation to reexpress the same equations:

_+a. .) t _ E . + 0.. Fi(t) ; = 3,5. (11.)la j 1J Jj

In this notation,, 't3(t)z(t)and5(t) e(t) ; clearly the meaning

'can be extended to more degrees of freedom.,, but we shall here interpret

the sum to be taken over j, only. 'If the products of inertia are

zero, as implied. in Equations (1), then; m._ =0 for 1 j . Note thatij

F5(t)denotes.. a Moment, and

m55denotes a moment of inertia.

The equations above are not really equations of motion, of course,

(1)

3

+

=

=

= 3,5

-4

since they are valid only if the excitation terms are sinusoidal in time.

Thecoefficientsaijandb.are all functions of frequency, (4) .ijHowever, we may interpret the equations as a description in the frequency

domain, and then they are valid for arbitrary motions, through the use of

Fourier-integral and generalized-harmonic-analysis techniques. See

Ogilvie (19O4). We shall include in thec.j j

terms only those forcesiwhich are dependent strictly on displacement; such forces do not depend on

frequency. All frequency-dependent forces are included ina.j j

andib. . Such a convention is required to make the coefficients unique.

It should be noted thata33

anda55

are of the same order of

magnitude asm33

andm55

, respectively. This may be taken as an

experimentally observed fact. Also, the coefficients must have values

such that free oscillations are highly damped.

From an engineering point of view, the equations above are quite

reasonable. They include the qualities generally evidenced by a heaving

and pitching ship, as already mentioned in the Introduction. But they do

no good unless we can calculate the individual terms. Therefore we now

consider some of the approaches which have been used in solving the hydro-

dynamic problem.

Thin-ship idealization (one parameter). Peters & Stoker (1954) took

advantage of the simplicity of the thin-ship idealization to develop a

complete, mathematically consistent description of heave and pitch motions

in head seas. They argued that the oscillation problem must be superposed

on the steady-motion problem, and approximations must be made in such a way

that both problems are simplified simultaneously. In view of the long

history of thin-ship theory for the prediction of wave resistance, it was

natural that they should try representing the ship in the Michell-Havelock

manner even for the unsteady problem. A special advantage of this approach

is that the body boundary condition is easily satisfied; there are no

integral equations or singular perturbation problems to be solved. Once

the boundary condition is written down for the ship hull, the solution of

the linearized problem follows almost trivially.

-5

Peters and Stoker assumed that there is a "thinness" paramater, say

, which may be thought of as the beam/length ratio or, more accurately,

as the maximum slope along any waterline. It is often convenient to think

of the ship length as being a quantity which is 0(1) as beam approaches

zero, in which case we may write:

beam = 0() (2a)

There are two other quantities which one would expect to be small in

a linear theory of ship motions, namely, the amplitude of ambient waves

and the amplitude of ship motions. Peters and Stoker assumed simply that:

motion amplitude . 0(6) ; (2b)

wave height = 0(8) . (2c)

Assumption (2a) allows us to recover the Michell-Havelock thin-ship

theory for wave resistance in the absence of oscillatory motions, and

assumptions (2b) and (2c) lead to the development of solvable, linear

boundary-value problems for the prediction of hydrodynamic disturbances

caused by the oscillating ship in waves. In fact, the above assumptions

allow one to formulate a sequence of problems from which even nonlinear

effects can be calculated, although one never encounters the need to solve

an actual nonlinear boundary-value problem.

But there is a rub when we consider the equations of motion which

result from these assumptions. The orders of magnitude of several of the

terms in (1') can be estimated easily:

F.(t) (wave height).(waterplane area) = Oki5 ) . (3a)

This part of the generalized exciting force requires only a buoyancy

calculation, from which the estimate follows directly. There will also be

higher-ordercontributionstoF.(t) caused, /or example, by the diffrac-ition of the incident waves, but we are considering now just the lowest-order'

parts. The restoring-force terms are similarly calculated from buoyancy

considerations:

-6

hydrodynamic force (water disturbance).(waterplane area)

. (3d)

Thus, the hydrodynamic-force terms are higher-order than those terms

estimated in (3a)-(3c), and so they do not belong in the lowest-order

equations of motion. The first approximation to the equations of motion

is accordingly:

[ra+ c.. = Fi(t) .

lj j 1J J

(Cf. (1').) There is no damping and there are no added masses; the water

provides only the restoring forces, -c.. . . Because of the lack ofj

(4)

c. (t) -a (motion amplitude)-(waterplane area) = 0(02) . (3b)

The terms involving the inertia of the ship are:

m. . j(t) (mass ot ship).(motion amplitude) ()1k2) (3c)ij

(Thelastrelaticrl10110wsfromthefactthatm..-"(length)-(depth).(beam)ij

. 0(5) . In thin-ship theory, depth or draft is not to be considered as a

small quantity.) Thus, the terms above are all of the same order of magni

tude.

It is different with the hydrodynamic-force terms, but it requires a

bit more argument to show wiw. The water disturbance which results from

the ship motions is proportional to (amplitude of motion)-(waterplane area) .

By "water disturbance" is meant fluid velocity components, pressure tluctu

ations, velocity potential, and so on. Similarly, the water disturbance due

to the presence of the snip in the ambient waves varies like (wave height

(waterplane area) . Thus, in both cases the disturbance is 002) . The

force on the ship from such disturbances must be tound by integrating the

pressure appropriately over the hull surface. Since only vertical compo-

nents of force will matter for heave and pitch, the pressure must be multi

plied by the direction cosine of the hull normal vector with respect to the

vertical reference axis. Then, in effect, the pressure is integrated over

an area which is comparable with the waterplane area, and it follows that:

.^0

=

7

added mass, Equations (4) predict resonances at highly erroneous frequen-

cies, and, because of the lack of damping, they predict arbitrarily large

amplitudes of response near the resonance frequencies.

Both of these results are unacceptable, of course, for engineering

purposes. There is also a difficulty in principle mathematically, for it

was assumed initially that the entire disturbance caused by the ship is

very small, and yet the disturbance can be infinite in amplitude. There is

no fault to be found in the way in which Peters and Stoker worked out the

consistent expansion. Rather, the original assumptions must be re-examined,

and several approaches are possible.

Inconsistent thin-ship model. Haskind (1940 had already formulated

the ship-motions problem for a thin ship some years before the work of

Peters &Stoker (1954). However, Haskind made no attempt to develop a

consistent perturbation analysis. Rather, he simply assumed that the

ship was thin enough that the appropriate boundary-value problem could

be formulated, he solved that problem, and he included the resulting

damping and inertial effects in his equations of motion. Mathematically,

his approach is unsatisfactory, since he includes some terms which are

0(183) but he makes no pretense of finding all terms of that order of

magnitude. From an engineering point of view, his results are more satis-

factory than those of Peters and Stoker. Thin-ship predictions of added

mass and damping are not too bad, compared with experiment, and it is cer-

tainly preferable to have some estimate of these effects rather than to

relegate them to a higher-order approximation. (In fact, it should be

observed that the anomalous results of Peters and Stoker cannot be corrected

by using a higher-order theory. By definition, the next order of approxima-

tion must involve perturbations which are small compared to previous-order

terms, and so they can never eliminate the infinities in the lowest-order

theory.)

Multi-parameter thin-ship idealization. Newman (1959) remedied the

difficulty in principle of the Peters-Stoker analysis by introducing three

small parameters: beam, motion amplitude, and incident wave height. Not

8 -

all three are independent; motion amplitudes must be small, and near

resonance this requirement produces a relation between the other two

parameters. Newman obtains in fact a different lowest-order problem near

resonance from the problem away from resonance. Unfortunately, it has not

been found feasible to use Newman's results for practical purposes.

Flat-ship idealization. The entire problem can be linearized in an

entirely different way by assuming that the ship is thin in its vertical

dimensions. This approach has the attraction that it more nearly corre-

sponds to the realities of shipbuilding practice than does the thin-ship

model. Also, it has been recognized for many years that it eliminates in

principle the difficulties found above in the thin-ship model (as least

for the modes of vertical motion). However, there is another difficulty

introduced here: It becomes necessary to solves. singular integral

equation in two dimensions. This fact alone held up the development of

flat-ship theory for many years and will probably continue to make it use-

less for practical ship-motions problems in the foreseeable future.

Slender-body theory. Starting with the work of Vossers (1960),

another approach was intensively investigated during the early 1960's,

namely, assuming the ship to be small in both beam and draft. The result

of applying such an assumption is very interesting in terms of the consis-

tent mathematical analysis developed by Peters and Stoker. There is one

principal change wrought in the lowest-order equations of motion: The

inertia terms disappear altogether. This happens because ship mass is

now 0(2) , and so the inertia terms are 0(33) , while the excitation and

static restoring forces still depena primarily on waterplane area and are

thus still O(j20 ) . The equations of motion become simply:

cij = Fi(t) . (5)

The mass of the ship re-enters the problem in the second-order theory, at

which point the hydrodynamic forces also appear.

It may appear that Equations (5) are too trivial and lack too many of

the required system properties to be even worth considering. But this is

-9

not necessarily true. What Equations (5) really say is that, to a first

approximation, the displacements in heave and pitch are determined by the

springconstants,c...A second-order theory would predict typicalij

resonance-type responses, since ship mass, water inertia, and wave damping

all appear at that stage of approximation. All that is required in principle

is that such resonance phenomena should involve disturbances which are small

compared to those implied by (5).

For a typical ship in a typical seaway (head seas only), these condi-

tions are in fact satisfied if the ship has zero forward speed. This was

shown by Newman 8: Tuck (1964), from whose paper Figures 1 to 4 are repro-

duced. The information presented concerns an unspecified aircraft carrier.

In Figures 1 and 2 are shown the heave and pitch response-amplitude opera-

tors, as found experimentally and as predicted by Equations (5). For fre-

quencies less than 0.6 cycles per second, the agreement between experiment

and theory is remarkably good.

Impressive as this agreement may be, it is somewhat of an accident.

It may be noted that there is a scatter of data at higher frequencies, but

nevertheless there is a clear suggestion of the existence of secondary

peaks in the response curves. These are not predicted by the lowest-order

theory. What is really happening physically is this: The pitch and heave

resonance frequencies lie in this high-frequency region. Incident waves

of such frequencies are rather short compared to ship length, and so they

produce little net excitation of either heave or pitch motion. Most of the

excitation occurs at low frequencies (longer waves), well below resonance,

for which the system is essentially spring-controlled. Thus, the derivative

terms in the equations of motion are small because: (a) at high frequencies,

F.(t) is very small, and, (b) at low frequencies, differentiation of

effectively multipliesi by co , the frequency, making successive

derivatives smaller and smaller.

The situation is altered when the ship has forward speed, as shown in

Figures 3 and 4. An incident wave of any given frequency of encounter now

has a longer wavelength, and so it may be expected to have greater effec-

tiveness in exciting heave and pitch. In particular, the high-frequency

disturbance which caused the secondary peaks in Figures 1 and 2 are now

12

- 10 -

.2 .4 .6 .8 10

Frequency in cyc les per second

Fig. 1 - Pitch response calculated from firstorder theory and compared with zero speedexperimental data

.2 .4 .6 .8 10 12

Fr equncy in cyclic per second

Fig. 2 - Heave response calculated from firstorder theory and compared with zero speedexperimental data

1,1

T14,

o

LI ULARREG

0 TRANSIENT

LEGEND

,

THEORY

WAVES

1

TEST

,

-

o

o

o 6ao

REGULAR

0 TRANSIENT

LEGEND

THEORY

WAVESTEST

----

o o pi

o

aDo

a o0o

o

.8

.6

.40

00

0-a

.2

.6

.4

.2

.8

-

.

-

--

1.8

1.6

2

.2 .4 .6 .8 1.0 12

FREQUENCY IN CYCLES PER SECOND

Fig. .3 - Pitch response calcu-lated from first order theory andcompared with experimental dataat 0.14 Froude number

.2 .4 .6 .8 1.0

Frequency in cycles per second

Fig. - Heave response calculated from firstorder theory and compared With experimentaldata at 0.14 Froude number

I

A REGULAR

- THEORY

'LEGEND

-!----

I

1

gr

WAVESTEST

-1-

°tO.,0c. ' o ,

-1

-o

o

IC

0

LEGEND

REGULARTRANSIENT

1 1

WAVESTEST

THEORY00 A0 ,t, 0 0CAA

A0-

00

4 o9)oo.,,

0,-0 '.. 0

ad

1.2

1.0

.8

.6

.4

.2

-

00

.4 -

o

At

- 12 -

much larger and they cause quite large motions in the band around resonance.

The theoretical curves in Figures 3 and 4 are calculated just as in Figures

1 and 2, the only correction being an allowance for the Doppler shift in

frequency caused by the relative motion of ship and waves. It is clear that

the error in the lowest-order theory is quite considerable, particularly at

the higher frequencies (near resonance).

Presumably, a second-order theory would correct some of this discre-

pancy. Newman and Tuck showed that at zero speed the second-order theory

made little change. Unfortunately, in order to obtain the second-order

results with forward speed included, one must solve the complete second-

order problem. That fact reduces the utility of this approach greatly,

even if the results were highly accurate. No one has yet succeeded in

making this into a practicable procedure.

High-frequency slender-body theory. We have seen that the first

attempt to use slender-body theory has not been successful for predicting

ship motions. It is also clear that the situation will not be improved

unless we retrieve the inertia terms, so that resonances are predicted,

but we must do this in a manner such that added-mass and damping terms

also appear in the lowest-order theory. We find that these objectives

can be reached if we make explicit use of the fact that we are seeking to

raise the relative importance of the derivative terms. We do this by

inking one more order-of-magnitude assumption:

00-1/2) (6)

Formally, it is readily shown that this does indeed bring the inertia

terms back into the lowest-order equations:

, .

mij j = 0(mass of ship)0( .)*Oku)2)J

= 002).00).00-1) = 002) .

Thus, the inertia terms are the same order of magnitude as the exciting

forces and the restoring forces.

Before considering the effect of this assumption on the hydrodynamic-

force terms, let us digress a moment to consider the meaning of the assump-

- 13 -

tion. The previous order-of-magnitude assumptions have all been made in

terms of quantities with length units. It is natural to ask how one may

relate a quantity measured in units of time to other quantities which have

length units. The answer lies in the relationship between the frequency of

a iree-suriace wave and its wavelength. It is easily found that, if a wave-2,

motion has irequency = 1/) it has a wavelength A = c(ii) .

Thus the assumption concerning implies that corresponding waves have

length comparable to ship beam.

For the ship with no forward speed, this implication may be taken in

the obvious way. Strictly, the theory pertains to situations in which the

incident waves are very short. In Figures 1 and 2, we would be concerned

with the extreme right-hand portions of the data.

For the ship with forward speed, the interpretation is not so obvious.

The frequency (..) is the frequency of encounter, that is, the actual fre-

quency of oscillation. If the ship is moving with speed U into head waves

which have a frequency (,30 in an earth-fixed reference frame, the frequency

of encounter will be:

= (JO ULJ12/01g(7)

In order to find the order of magnitude of the wavelength of the incident

waves, we must solve (7) for k.)0 in terms of The result is:

_ + A2U

gt g2

U4112

'(8)

For large (A) , the right-hand side is dominated by the term containing

Lc; and so we find that:

LAO = 0(t; -1/4) (9)

The usual relationship between frequency and wavelength then shows that

the incident waves are long compared with ship beam and short compared

with ship length. Note that we have tacitly assumed that the quantity

g/U can be treated as a quantity which is 0(1) .

Thus it appears that the new assumption, that w= Oc6-1/2)

restricts the results to short waves in any case and to extremely short

waves in the case of zero forward speed. Whether this is as serious as

,

,

- 14 -

might appear is not entirely clear. We shall return to the question when

we consider further implications of the new basic assumption.

Now we return to a consideration of the hydrodynamic-force terms., 2

We shall find that the leading-order terms are 00 ) , and so they must

be included in the lowest-order equations of motion.

We may expect the fluid velocity components and the velocity potential

to vary directly with the vertical velocity component of the ship and with

the waterplane area:

Fluid disturbance

-a (vertical velocity component of ship)(waterplane area)

= 0(64))O) = 003/2) . (io)

In linearized theory, the pressure is given by the product of fluid density

times the partial time derivative of the velocity potential. Thus:

pressure = 0(aPi9) = 0() .

As before, we obtain the hydrodynamic force by integrating the pressure

over an area which has the order of magnitude of the waterplane area:

hydrodynamic force ^, (preesure).(wateriolane area) = O(g2) . (12)

Thus, the hydrodynamic force has the same order of magnitude as the pre-

viously considered terms in the equations of motion, and the equations

must then be of the form of Equations (1) or (1').

This is the result we set out to find. We see that the high-frequency

( slender-bodyapproximation is the only approximation considered that leads

consistently to a set of equations of motion which has the basic form that

observations say it must. Whether the assumptions lead to correct values

of the coefficients in the equations is another matter altogether which

cannot be answered by these order-of-magnitude arguments.

- 15 -

CONSEENCES OF THE HIGH-FREQUENCY SLENDER-BODY ASSUMPTIONS

First we should recall for a moment the role of small parameters in

formulating approximate boundary-value problems. The small parameter is

just &convenient device which provides a means of deciding what terms are

most important and what terms may be neglected in a first approximation (or

in subsequent approximations). One never knows a priori how small the

parameter must be for the results to have any specified degree of validity.

Indeed, it may not have to be small at all. One may make an analogy to a

Maclaurin series: The fact that we represent a function by a power series

in, say, z does not mean that the series is necessarily valid only for

small iz1 . The series might converge for any

On the other hand, our series might not converge for any value of the

small parameter. We may have an asymptotic series on our hands. We simply

do not know, and there is not much likelihood of our finding out in any

rigorous way what properties our series may have. We just use the series

and observe whether the results agree with observation.

In accordance with usual practice, the small parameter never appears

in our final formulas. We assume that all quantities can be expressed in

asymptotic series in terms of the small parameter, and we use this assump-

tion to order all terms that appear in each equation. When we have

decided how many terms to retain in any particular expansion (usually not

more than one or two), we drop the rest and solve the remnant of the problem

as nearly exactly as we are able. The fact that each term is assumed to be

of a certain order of magnitude is now thoroughly camouflaged. The fact

certainly does not appear explicitly in the final formulas.

After this discussion, we should not be terribly surprised if the

analysis gives fairly accurate results even in some cases in which the

order-of-magnitude assumptions appear to be violated (although it would

be naive to start out hoping that this would be the case).

Such a case occurs in this problem. At zero speed, we have observed

that the theory can be expected to be valid only for extremely short-,waves, for which w= 001/2) and A. = 0(4) . However, it has been

found that the theory gives rather good predictions even for low frequen-

cies. We do not have to look far to find the reason. At low frequencies,

- 16 -

the terms in the equations of motion which involve time derivatives are

numerically small, simply because they involve factors containing c,) .

Thus, even though these terms were retained because of the high-frequency

assumption, they have little effect. We would have found the same equations

of motion if we had assumed that op = 0(1) and then inconsistently included

some higher-order terms. After all, there is no real harm in being incon-

sistent and keeping some higher-order terms if the higher-order terms are

really very small. For low frequencies and zero speed, all that matters

is that we correctly predict exciting and restoring forces.

It is tempting next to speculate that the same result may occur for

forward-speed problems. We have observed that the basic assumption requires

that incident wavelengths be large compared with ship beam and small com-

pared with ship length. The latter restriction is certainly objectionable,

because it is well known that maximum ship motions usually occur in waves

of length comparable to ship length (or somewhat longer than ship length).

Can we reach the same conclusion that the theory will still be valid even

for moderate frequencies, such that wavelength is comparable with ship

length? Only with some qualification. The character of the waves generated

by ship motions and by diffraction of incident waves changes grossly from

very low speed and/or frequency to high speed and/or frequency. At zero

speed, it is intuitively clear that waves are radiated in all directions.

However, at high speeds the motion-generated waves are included within a

wedge-like region behind the ship. The difference in the character of these

wave systems suggests that the problem solution will also show a correspond-

ing difference.

The boundary between these regimes is marked by the parameter 1.7 =c4U/g

-2.having a value of 1/4 . With our assumptions, 17 = 001/ ) , and so our

theory is a "high-7" theory. If speed is high enough, the theory will

possibly be good down to fairly low frequencies, but certainly the frequency

cannot approach arbitrarily close to zero. For = 1/4 , there will be

infinities in the higher-order solutions, at least, and so we must consider

the whole analysis void. It is not possible to say how closely we may

approach Z = 1/4 from above and still have any usable results. The

behavior of the system near this singularity is probably rather well-behaved

except in a very small neighborhood, and so we may hope to get down even to

F

F

.= 17 =

the range in which, say, 1: -= 0.3 But there is only one way to find

out by trying it.

At present it is not possible to reach, A Conclusion on this matter

even if we make a great many calculations, because the system behavior at

small values of t makes experiments very difficult. Near 7.7,= 1/4

the radiated waves move &Way in part nearly broadside from the ship, and

tests in ordinary towing tanks are subject to great wall-effect errors.

We badly need some definitive experiments at low speed and low frequency'

in abroad seakeeping basin.

The result of applying the high-frequency slender-body assumptions

is a modified strip theory. The arguments leading up to (12) pertain

only to the lowest-order hydrodynamic effects; the added-mass, and -

damping forces are those which would be obtained in the most ,naive strip,

analysis, and the same is true of the exciting forces. That is, they

contain no effects of forward speed*. Although they do represent the

qualitative properties of the heaving/pitchingship in waves, the effects

of forward speed are not negligible. Therefore it is desirable to inves-

tigate the next order of approximation, and the availability of the small

parameter for ordering all quantities makes this in 'principle a rather

reasonable task, As it turns out, even in practice it can be done, and

several consequences are found':

,A fluid disturbance is found .which may be interpreted as a

coupling between the steady flow around the hull and the oscillations Of

the hull, This part arises analytically in requiring that the hull

boundary condition be satisfied on the exact, instantaneous position of the

hull; since the steady-motion boundary condition is satisfied on the -mean

position Of the, hull, a correction to the steady-motion velocity field

arises. The resulting 'hydrodynamic force id found to be 0(S5/2)

This suggests that, notwithstanding the previous discussion, these termsmay yield a valid first approximation for all values of If . Nevertheless,this is doubtful, for the singularities at T 1/4 which appear in thesubsequent terms of the approximation cannot be ignored.

.

--

(1)

=

-18 -

A fluid disturbance is found which may be interpreted as a

coupling between the steady incident flow and the oscillations of the free

surface. It is quite analogous to the effect mentioned in the previous

paragraph, although it is more complicated, inasmuch as two boundary condi-

tions are involved. Again, the resulting hydrodynamic force is found to

be 0((55/2) .

The lowest-order excitation is entirely consistent with the

Froude-Krylov hypothesis, that is, it may be calculated from the pressure

field which would exist in the absence of the ship. The next-order excita-

, ,

tion is 001/2 ) higher in order of magnitude, and it includes the first

approximation to the diffraction problem. One remarkable result is that

this contribution does not depend on forward speed except for the Doppler

shift in frequency. Such a result is perhaps rather surprising, but it

appears to agree well with observations.

At this stage, the theory has some remarkable similarities to the

modified strip theory of Korvin-Kroukovsky, which has been adapted and

modified by many workers. But it also has some remarkable differences.

The interactions between hull and steady flow are similar, although not

identical. One fact of special importance is that the added-mass and

damping coefficients have the symmetry which they must have, in contrast

to previous results. The interactions between free-surface oscillations

and steady flow are new; their importance is not yet evaluated. The lack

of forward-speed effects in the excitation forces has not been found

previously, although, as remarked above, experimenters have already observed

that this conclusion is valid.

-19-

REFERENCES

Golovato, P. "A study of the transient pitching oscillations of a ship,"Journal of Ship Research, 2:4 (1959) 22-30.

Haskind, M. D. "Oscillation of a Ship on a Calm Sea," Bulletin of theAcademy of Sciences of the USSR, Technical Sciences Class, 1 (1946)23-34. Translation available in: T&R Bulletin No, 12, Society ofNaval Architects and Marine Engineers, New York.

Korvin-Kroukovsky, B. V. "Investigation of Ship Motions in Regular Waves,"Trans. Soc. of Nay. Arch. & Mar. E., 63 (1955) 386-435.

Newman, J. N. A linearized theory for the motion of a thin ship inregular waves, M.I.T. doctoral dissertation (1959). Reprinted:Journal of Ship Research, 541 (1961) 34-55.

Newman, J. N., Tuck, E. O. "Current Progress in the Slender-Body Theoryof Ship Motions," Proc. Fifth Symposium on Naval Hydrodynamics, Bergen,Norway, Sept. 1964. Office of Naval Research ACR/112, Washington.

Ogilvie, T. F. "Recent Progress Toward the Understanding and Prediction ofShip Motions," Proc. Fifth Symposium on Naval Hydrodynamics, Bergen,Norway, Sept. 1964. Office of Naval Research ACR/112, Washington.

Ogilvie, T. F., Tuck, E. O. A Rational Strip Theory of Ship Motions:Part I, Report No. 013, Dept. of Naval Architecture and Marine E/Agi/leering, University of Michigan, Ann Arbor (August 1968).

Ogilvie, T. F. A Rational Strip Theory of Ship Motions: Fart II, in prepa-ration.

Peters, A. S., Stoker, J. J. The Motion of a Ship, as a Floating RigidBody, in a Seaway, Report No. IMK-203, Inst. of Math. Sci., New YorkUniversity, New York (1954). Also: Comm. Pure &Apol. Math. 10(1957) 399-490.

Vossers, G. Some Applications of the Slender Body Theory in Ship Hydro-dynamics, Dissertation, Tech. University, Delft (196077 Also:Publication No. 214, Netherlands Ship Model Basin, Wageningen (1962).

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