The development of ship - motion theory

THE

DEPARTMENT

_OF

RE AtiD

T.

Francis

Ogilvie

THE UNIVERSITY OF MICHIGAN

COLLEGE OF ENGINEERING

MINE ENGINEERING

May

1969

Lab. v.

Scheepsbouwkunde

Technische Hogeschool

Delft

THE DEVELOPMENT OF

SHIP MOTION THEORY

DOCUMENTATiE :

r2 1

DATUM'

THE DEVELOPMENT OF SHIP-MOTION THEORY

T. Francis Ogilvie

s1TYe

_{Department of Naval Architecture}

44,111rt+

4

and Marine Engineering

...

me Col lege of Engineering

=

.0 4..

The University of Michigan

Ann Arbor, Michigan 48104

7811

rjsre Hocinchoo

No. 021

May 1969

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 Ogilvie

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

4-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

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

_J

j

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 that

ij

F5(t) denotes.. a Moment, and m55 denotes a moment of inertia.

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

(1)

3 + = =

= 3,5

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

Thecoefficientsaijandb.are all functions of frequency,

(4) .

ij

However, 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 the

c.j

i

j terms only those forces

which are dependent strictly on displacement; such forces do not depend on

frequency. All frequency-dependent forces are included in

a.j j and

i

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

It should be noted that

a33 and a55

are of the same order of

magnitude as

m33 and m55 , 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)

_i

caused, /or example, by the diffrac-tion of the incident waves, but we are considering now just the lowest-order' parts. The restoring-force terms are similarly calculated from buoyancy

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 of

j

(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 perturbaapproxima-tions 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

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 typical

ij

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 first

order theory and compared with zero

speed

experimental data

.2 .4 .6 .8 10 12

Fr equncy

in cyclic per second

Fig. 2 - Heave response calculated

from first

order theory and compared with zero

speed

experimental data

T

14, o LI REGULAR 0 TRANSIENT LEGEND , THEORY WAVES 1 TEST ,

-o o o 6 a o REGULAR

0

TRANSIENT LEGEND THEORY WAVES TEST ----o _o

pi

o aD o a o0 o o .8 .6 .4 0 0 0 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 and compared with experimental data at 0.14 Froude number

.2 .4 .6 .8 1.0

Frequency in

cycles per second

Fig.

- Heave response calculated from first

order theory and compared With experimental data at 0.14 Froude number

I A REGULAR - THEORY 'LEGEND -!----I 1

gr

WAVES TEST

-1-°tO.,0

c.

' o

, -1 -o o IC 0 LEGEND REGULAR TRANSIENT 1 1 WAVES TEST THEORY 00A0,t, 0 0 CAA A 0

-00

4

o 9) oo.,, 0,-0 '.. 0

a

d 1.2 1.0 .8 .6 .4 .2 -0 0 .4 -o At

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( J.)*Oku)2)

=

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 assump

-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:

_

+ A

2U gt g2 U 4112 ' (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

,

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-body

approximation 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

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 equaassump-tion. _{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}

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

.= 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 terms} may yield a valid first approximation for all values of If . Nevertheless,

this is doubtful, for the singularities at T _{1/4 which appear in the}

subsequent terms of the approximation cannot be ignored.

.

--(1)

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

-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 the}

Academy of Sciences of the USSR, Technical Sciences Class,

1 (1946)

23-34.

Translation available in: T&R Bulletin No, 12, Society of

Naval 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 in} regular 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 Theory of 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 of}

Ship 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 Rigid

Body, in a Seaway, _{Report No. IMK-203, Inst. of Math. Sci., New York}

University, 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:

001

"Computer-Aided Ship Design," Edited by

Robert S. Johnson,

Horst Nowacki and T. Francis Ogilvie,

Course Notes, May

1968--$18.

002

"Wave Resistance: The Low-Speed Limit," by T. Francis

Ogilvie, Aug. 1968--No Charge.

003

"The Economics of the Container Ship Subsystem," by Dave

S. Miller, Oct. 1968--$2.

004

"Analysis and Statistics of Large Tankers," by Virgil F.

Keith, Oct. 1968--$1.

005

"Reactions on Independent Cargo Tanks,"

by Finn C. Michelsen

and Ullmann Kilgore, Sept. 1968--$1.

006

"Experiments on the Resistance of a Family of Box-like

Hull Forms for Amphibious Vehicles," by Horst Nowacki,

J. L. Moss, E.

D. Synder, and B. J. Young,

Sept.

1968--No Charge.

007

"Measures of Merit in Ship Design," by Harry

Benford, Feb.

1968--$2.

008

"The Control of Yaw in Towed Barges," by Harry Benford,

Presented at the Gulf Section of SNAME,

1955--$1.

009

"Economic Criteria in Fish Boat Design," by Harry Benford,

Presented at the Conference on

Fishing Vessel Construction

Materials,

Montreal, Oct. 1968--$1.

*010

"On the Steady-State Ship Hull Response," by Horst Nowacki,

Presented to the Sociedade Brasileira de Engenharia Naval,

Rio de Janeiro, Aug. 1968--$1.

*011

OUT OF PRINT:

"Feasibility Study of a Subseatrain," by William

White and George Lamb, presented at the San

Diego Section of

SNAME, Feb. 1969, $2.

For further information, please write

Prof. Harry Benford, Dept. of Nay.

Arch. and Mar. Eng.

445 W. Eng., U of M., Ann Arbor,

Michigan 48104

012

"The Practical Application of Economics to Merchant Ship

Design," by Harry Benford, Reprinted from

Marine Technology

Vol. 4, No. 1, Jan 1967, (Also available in Spanish)

Feb. 1969--$1.

*013

"A Rational Strip Theory of Ship Motions: Part

I," by T.

- 2

--*014

"Ferro-Cement with particular reference to Marine Applications,"

by Charles Darwin Canby, presented to

the Pacific Northwest

Section of SNAME, March 1969--$2.50.

015

"The Cost Savings of Multiple Ship Production," by John C.

Couch, reprinted from AACE Bulletin,

Vol. 6, No. 2, 1964

pp. 50-58--$0.50.

*016

"The Engineer's Role in Managing Marine Transport Systems,"

by Hugh C. Downer, remarks to the

Quarterdeck Society,

March 28, 1969--No Charge.

017

"Notes on the Design and Operation of Automated Ships,"

by Harry Benford, Seminar on the

Labor Problems Resulting

from Automation and Technological

Changes on Shipboard,

Sept. 1965--1.

*018

"Class Notes on Computer-Aided Ship Design," Used for graduate

course NA 574, by Horst Nowacki, April 1969--$7.00

*019

"Shallow-Water Performance of a Planing Boat," by Andras

Istvan Toro, presented at the Southern Section of SNAME,

April 1969 (To be available July 1969)--$2.

020

"Hydrodynamic Aspects of Tracked Amphibians," by Ullmann

Kilgore, Presented at the First

International Conference

on Vehicle Mechanics at Wayne State University,

1968