Stress analysis of ship-suspended heavily loaded cables for deep underwater emplacements

REPORT NO. 1370863 a At. / 56

PROJECT TRIDENT
TECHNICAL REPORT

STRESS ANALYSIS OF SHIP-SUSPENDED
HEAVILY LOADED CABLES FOR DEEP
UNDERWATER EMPLACEMENTS

— ARTHUR D. LITTLE, INC.

— 3 CORN PARK _ CAMBRIDGE, MASSACHUSETTS

~ AUGUST 1963

O @205€00 TOEO oO

MAC A A

IOHM/18N

REPORT NO. 1370863

PROJECT TRIDENT
TECHNICAL REPORT

STRESS ANALYSIS OF SHIP-SUSPENDED
HEAVILY LOADED CABLES FOR DEEP
UNDERWATER EMPLACEMENTS

ARTHUR D. LITTLE, INC.

35 ACORN PARK CAMBRIDGE, MASSACHUSETTS

DEPARTMENT OF THE NAVY
BUREAU OF SHIPS

NObsr-81564 $-7001-0307

AUGUST 1963

PREFACE

This is one of a series of Technical Reports being issued by
Arthur D. Little, Inc., under contract NObsr-81564 with the Bureau of
Ships as part of the TRIDENT Project.

Arthur D Hittle Ine.

S-7001-0307

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TABLE OF CONTENTS

Page
LIST OF FIGURES AND TABLES iv
I. SUMMARY 1

A. PURPOSE AND SCOPE 1
B. CONCLUSIONS AND RECOMMENDATIONS 2

II. STATIC STRESSES DUE TO THE WEIGHT OF THE ARRAY AND
CABLE 5
Ill. THE EFFECT OF OCEAN CURRENTS 7
IV. THE EFFECT OF MOTIONS OF THE SEA SURFACE 10
A. FORMULATION OF THE PROBLEM 10

B. THE STEADY STATE SOLUTION FOR SINUSOIDAL

INPUTS 13

C. PARAMETRIC ANALYSIS OF THE MAGNITUDE OF THE
MAXIMUM STRESS FOR SINUSOIDAL INPUTS 17

V. THE DYNAMICS OF THE VESSEL AND SPECTRAL CHARACTER-

ISTICS OF A REALISTIC SEA 25

VI. BUCKLING 29

APPENDIX A - FRICTION ON THE CABLE 32

APPENDIX B - THE DRAG ON THE ARRAY 34

LIST OF SYMBOLS 36

PROJECT TRIDENT TECHNICAL REPORTS 39

DISTRIBUTION LIST 42

iii

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LIST OF FIGURES AND TABLES

Page
Definition of Space Coordinates 5
Bending of the Cable by a Horizontal Current 7
The Normalized Maximum Dynamic Stress Versus the
Normalized Angular Frequency for Various Values of the
_ 4apA|Uo| _/wL
Parameters 8 (- eqn and uU = =) 18
Response Amplitude Operators for Cuss I and Maximum
Safe Input Amplitude to the Cable as Functions of Frequency.
(Cable is a Special Flexible Hoisting Wire Rope of USS.) 24
Amplitude Characteristics of Fully Developed Sea 26
Frequency Characteristics of Fully Developed Sea 26
iv
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I. SUMMARY

A. PURPOSE AND SCOPE

This report relates to the operational aspect of lowering heavy array
structures to the bottom of the deep ocean. Because of the weight of the array
and the depth to which it will be lowered, the lowering cable can be expected to
be under a high stress. Ocean currents, surface waves, and motions of the
lowering vessel will also contribute to the stress placed on the lowering cable.
Therefore, a critical design problem may exist for the lowering cable.

The objective of this report is to develop a reasonable theory on
which to base the ultimate design of the lowering cable. The study is limited
to the case of a single cable lowering the array vertically. Each factor con-
tributing to the stress on the cable is considered separately, and the conditions
under which the analyses are valid and of practical significance are given.
From the results of this theory some general conclusions are drawn as to the
feasibility of the operation with regard to dynamically optimum array configu-
rations, stability of the lowering platform in a rough sea and degree of rough-
ness of the sea (sea state). The results of the study are presented in plot form
involving a non-dimensional maximum (dynamic) stress versus a non-dimensional
frequency for various mass and drag parameters.

Other methods of operation, which are not treated in this report, but
which will be considered in later investigations, are:

(1) Cancel out most of the dynamic inputs to the cable by an auxiliary
mechanical system on the vessel. There are two crucial problems here: the
horsepower of the driver of such a system and the rapidity with which the over-
all system can respond.

(2) Introduce considerable damping along the cable, which will have
the effect of flattening out resonant peaks. This can be effected by attaching
bluff bodies to the cable at certain intervals. The drag of the water on them
will give the desired friction. This idea seems quite attractive, since these
bodies can also provide buoyancy to reduce the rather high static stresses in
the cable, but its plausibility must be investigated quantitatively.

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(3) Use more than one cable. However, it should be borne in mind
that each cable must actually be designed to withstand much more than an equal
share of the total load because of some adverse conditions that can readily oc-
cur. Also, the cables can be tangled, a situation which can lead to serious
problems.

B. CONCLUSIONS AND RECOMMENDATIONS

The static stresses due to the weight of the array and cable can be
taken care of by practicable cables. For a maximum depth of 20, 000 feet and
50- and 100-ton steel arrays, the maximum static stress in a steel cable of
three square inches metallic area is 100,000 and 130,000 psi, respectively,
when no buoyancy is introduced anywhere. For the same depth and a 15-ton
aluminum array (which in water weighs about 10 tons), the maximum Static
stress in a steel cable of one square inch metallic area is 100,000 psi. The
ultimate tensile strength of such cables is about 220,000 psi; therefore, these
cables can handle these stresses with a factor of safety of about two. However,
the above static stresses must be reduced, because higher factors of safety
(about 3) are usually required for such an operation, and in addition, there are
other stresses (dynamic) in the cable.

Static stresses can be reduced by increasing the metallic area of the
cable or by making the array and cable more buoyant. Since a large portion
of the static loading is due to the weight of the cable (the maximum length of
cable which can hold itself without any factor of safety is only 62,000 ft), the
most effective method is to introduce buoyancy in the cable. An equivalent
way of effecting this is by using a tapered cable.

Nylon type ropes are desirable in this respect, because they are
much lighter than steel ropes while their ultimate strength is just about as high
as that of the strongest steel ropes. The velocity of sound of nylon ropes is
also much smaller than that of steel ropes. With respect to the dynamic stres-
ses that can be induced by a rough sea, these two factors (small weight and
velocity of sound) render nylon ropes undesirable. Of course, the small modu-
lus of elasticity, characteristic of nylon ropes, is a favorable factor and can
probably compensate for the undesirable effects of the above two factors. In
any case, in this operation there are some environmental hazards which make
steel ropes preferable to nylon ropes.

The effects (stress and array offset) of reasonable ocean currents
are negligible, provided that the vertical forces due to gravity minus buoyancy
are much larger than the horizontal drag forces due to the currents. This will
be the case in practice, because buoyancy must not be used to the extent of

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canceling the gravity forces completely. It is highly desirable to have the
cable in an appreciable static tension so that compressive dynamic stresses
will not buckle the cable. Buckling of the cable is not in itself dangerous, but
the formation of kinks, which can occur when the cable buckles appreciably,
is. Some consideration of this problem is given in Section VI. Perhaps, the
problem should be pursued further.

The motions of the vessel in a rough sea (heave, roll, etc.) will
induce dynamic stresses in the cable. Due to its great length, the cable can-
not be considered nonflexible for the kind of inputs that exist in a realisticsea.
In fact, for sinusoidal inputs, resonance can occur at frequencies as small as
1.5 rad/sec. These frequencies correspond to periods as large as four sec-
onds, which are included in the spectrum of a disturbed sea.

In order to obtain an accurate value of the maximum dynamic stress
induced in the cable by a regular (sinusoidal) surface wave, the dynamics of
the vessel, cable, and array must be analyzed simultaneously. The parame-
ters necessary for such an analysis are too many, and the problem can become
formidably long and difficult. However, it can be safely assumed that the dy-
namic loading of the vessel by the cable and array is negligible, except possi-
bly when the cable is attached to the vessel by a boom extended from the side.
Thus, the problem can be simplified by considering the dynamics of the vessel
and of the cable-array system separately.

A theory is developed which takes into account the propagation of
longitudinal elastic waves in the cable and the complete dynamics of the array
in water. It is shown that friction on the cable by the surrounding water is
small and can be neglected, even near resonance, since friction on the array
is much larger. The drag on the array must be taken as quadratic because of
the large Reynolds number involved. This is the only nonlinear term in the
theory. It is linearized to a quasi-quadratic form. Thus, a formula is de-
rived for the maximum dynamic stress due to a sinusoidal input. It gives this
stress in a parametric form as a function of the input frequency and amplitude,
characteristics of the cable, and of the weight and shape of the array. This
formula is plotted in a dimensionless form with two parameters.

It is shown that resonant frequencies as small as 1.5 rad/sec can
induce very large stresses for input amplitudes of one or two feet; furthermore,
that dynamically heavy arrays increase the dynamic stresses a lot, especially
near. resonance. Entraining or displacement of large amounts of water in-
creases the dynamic mass of the array. Therefore, the array should be a
trussed open structure. Also, the velocity of sound in the cable should be as
large as possible. This makes the resonant frequencies large so that they lie
on the cut-off edge of the spectrum of a given sea state.

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A 15-ton trussed aluminum linear array, whose length is about 600
feet and whose cross dimensions are a few feet, and a 1-5/8-inch diameter
6 x 37 Special Flexible Hoisting wire rope (USS) are selected as a numerical ex-
ample. A curve is computed giving, for each frequency, the maximum safe in-
put amplitude for any length of the cable up to 20,000 feet. This curve shows
that, as the frequency increases, the maximum safe input amplitude decreases
sharply. Ata frequency of 1 rad/sec (six-second period), the maximum safe
amplitude is 20 feet, while at a frequency of 2 rad/sec (three-second period),
itis only 1.7 feet. This shows that the vessel must be rather stable if this oper-
ation is to be carried out successfully in a moderately rough sea. Also, it indi-
cates that ways in which the cable can be attached to the vessel are limited.

Two obvious ways of attaching the cable to the vessel are: (1) from
the center of gravity, and (2) from the side through a boom. In the first, heav-
ing motions of the vessel in beam seas will be critical, while in the second, rol-
ling motions in beam seas will be the most critical. Amplitude response curves
for heaving and rolling of Cuss I (see footnote of page 27 for a description of this
vessel) in beam regular (sinusoidal) waves are plotted.

The problem now is that a realistic sea is highly irregular, with a
continuous probabilistic spectrum of amplitudes and frequencies. To overcome
this problem, we adopt the procedure, used by many other investigators, of rep-
resenting an irregular sea by a regular one. In this way, we can draw some con-
clusions as to the chances of the success of the operation, although much depends
on the definition of the regular sea representing an irregular one of a specified
State.

It is shown that, by attaching the above cable to the center of gravity
of Cuss I, the chances for the success of the operation are quite good in seas
excited by a wind velocity up to 25 knots (beginning of sea state 6). It is also
shown that, when the cable is attached to the side of Cuss I through a boom, the
chances of success can be acceptable only in very small wind velocities.

The above results lead one to the idea of using either an unconventional
vessel or a lowering platform detached from the main vessel, in order to achieve
higher stability. The use of a very stable platform for lowering arrays into deep
water will be investigated further and discussed in a later report.

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Il. STATIC STRESSES DUE TO THE WEIGHT OF THE
ARRAY AND CABLE

The distribution along the cable of the static tensile stress, o, due
to the weight of the array and cable is linear and is given by:

ae é (L - x) (wb) + W, -B, | (1)
where:
x = Space coordinate measured as indicated in Figure 1.
L = Length of the cable.
Ss = Metallic cross-sectional area of the cable.
Ww = Weight of the cable per unit length.
b = Buoyancy force on the cable per unit length.
We = Weight of the array.
Ba S Buoyancy on the array.
ae

FIGURE 1 DEFINITION OF SPACE COORDINATES

Arthur D Little Inc.

This stress is maximum at the top of the cable and minimum at the
array. For a steel cable with S=3 in? and L=20,000 ft, anda 50-ton array
(without any buoyancy), the maximum and minimum static stresses are 103,000
and 33,300 psi, respectively. For a 100-ton array these values become 130,000
and 66,700 psi. This cable could be a steel wire rope of the Galvanized Bridge
Strand type with a diameter of 2.25 inches. The ultimate tensile strength of
this rope is about 220,000 psi. Therefore, this rope can take the above stres-
ses safely. This type of rope has a ratio of ultimate tensile strength to weight
per unit length equal to about 62, 000 ft, which is equal to the maximum length
of rope that can hold itself.

On the other hand if buoyancy is utilized, the static stresses can be
decreased considerably. In this respect, nylon type ropes are preferable, since
they are almost weightless in water while their breaking strength is just about
as high as that of the strongest steel ropes.

Arthur D Hittle Inc.

S$-7001-03 07

Ill. THE EFFECT OF OCEAN CURRENTS

Due to the frictional forces on the cable, a current will bend the
cable as shown in Figure 2. (Such frictional forces will also act on the array
and vessel if they are within the current.) Of interest here are three quanti-
ties: the additional stresses in the cable, the offset (D) from the target, and
the horizontal force (F) which must be exerted by the vessel in order to remain
Stationary against the current.

This problem can be solved very accurately by balancing the compo-
nents of the various forces acting on an infinitesimal element of the cable. The
results for all pertinent variables can be expressed in terms of integrals which
have been tabulated (David Taylor Model Basin Report 687 and its Supplement).
However, if the net vertical force (gravity minus buoyancy) is much larger than
the horizontal frictional forces due to the current, bending of the cable will be
very small, and the following simple analysis can be applied.

ARRAY

FIGURE 2 BENDING OF THE CABLE BY A
HORIZONTAL CURRENT

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Consider a uniform current of velocity V and width X (see Figure 2).
The horizontal force, F, which is equal to the total drag of the current on the
cable, is given by:

I Oh ents) W~ Ox (2)
where:
a = Drag coefficient. It is a function of the Reynolds
2aV
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ity of water.
a = Radius of the cable.

0 = Density of water.

The stress at the top of the cable (which is the maximum stress) will
be altered only slightly by the current. The offset, D, is given approximately
by:

ChEL o) W/E

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a a

which is an overestimate of the actual D.

As a numerical example, let us consider a current with V = 1 ft/sec
and X = 5,000 feet.* The cable described in Section II will be used. The
drag of this current on the cable is 0.225 lb/ft, which shows that the above ap-
proximation is valid. The horizontal force, F, is equal to 1,100 pounds. For
the 50-ton array without any buoyancy in the cable and array and for L = 20,000,
10, 000 and 5,000 feet, D is equal to 18, 28, and 37 feet, respectively. These
offsets are, indeed, very small. Even for lighter cables and arrays the offsets
will be small as compared to the length of the cable.

*A current with this velocity and width is about the largest that will be found in
the Atlantic.

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When the array is in the current, an additional horizontal force,
due to the drag on the array, must be included. The magnitude of this force
will vary for arrays of different shapes. Assume a linear array composed of
three horizontal hollow cylinders each 600 feet long. Two of the cylinders are
assumed to have diameters of two feet, and the third a diameter of one foot.
For normal incidence, the drag of the current on the array is equal to 1.8 tons.
Since this force is small, compared to the weight of the array, its contribution
to the stress will be small. The offset and the horizontal force, F, will in-
crease accordingly.

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$-7001-0307

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IV. THE EFFECT OF MOTIONS OF THE SEA SURFACE

A. FORMULATION OF THE PROBLEM

A realistic wavy sea will cause the vessel to move in the vertical
direction, and, thus, generate waves in the cable which will interact with the
array. This Section deals with the analysis of the dynamic stresses induced
in the cable by such waves.

This problem can become as complicated as one wishes to make it.
Here we will aim at a reasonable linear analysis. Therefore, we must, first
of all, assume an elastic cable. Metallic cables remain elastic as long as the
total stress does not exceed their elastic limit. For the cable considered in
Section II, the elastic limit is about 1.5 x 10° psi. The maximum static
stress calculated for the cable is below this elastic limit, and therefore, this
cable is still elastic. Nylon type ropes are more plastic than elastic, unless
they have been prestressed sufficiently. On the other hand, the dynamic stress
wave propagating in the cable can be a compressive one. Then, if this com-
pression exceeds the static tension in the cable, the cable will buckle, and the
propagation of stress in the cable becomes a nonlinear problem. Buckling is
treated briefly in Section VI. Since the present linear analysis assumes that
the cable does not buckle, any results which indicate that the cable should have
buckled are not valid.

Consider the cable and the array in a vertical position. It has been
shown in Section III that the deflection of the cable by a reasonable horizontal
current is small under conditions stated there; therefore, even in the presence
of such a current, the cable can be considered to be vertical for our present
purpose. As the upper end of the cable moves vertically, waves will be propa-
gating in the cable. If u(x,t) is the dynamic displacement at time t of an ele-
ment of the cable Ax, which in the absence of waves is located at x (see Fig-
ure 1), the governing equation for the propagation of waves is:

o7u O72 u ou
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where:

e. = Density of the cable

S = Metallic cross-sectional area of cable

E = Modulus of elasticity of cable

K = Constant of friction on the cable by the surround-

ing water. It should be kept in mind that K may
not be actually a constant.

In an actual operation, these waves can appear while the array is
being lowered at some rate, i.e., L in general changes with time. However,
if we assume that the rate of lowering is small as compared to the velocity of
the dynamic displacement due to the waves, then L can be considered constant
in time and the following dimensionless variables and parameters can be de-
fined:

ce alt Wess GA dene,
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where c is the "velocity of sound” in the cable.

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o7u OM . OF m
See Tle ae = ss (7)

In order to solve Equation 7, one needs two boundary conditions
(most likely one at each end of the cable) and, if transient solutions are sought,
initial conditions as well. One of the boundary conditions, which we will apply
here, is the specification of u at x’ = 0 for all times. Practically speaking,
we should specify the motion of the ocean surface for all times. But, then the
dynamics of the vessel and the way in which the cable is attached to the vessel
must be considered simultaneously with the dynamics of the cable and array,
which is a very complex problem. A discussion of the dynamics of the vessel
is presented in Section V; here we will assume that we know u at x' = 0 for
all times.

1l

Arthur DHittle Inc.

S-7001-03 07

ane iy Lise: ai alti shang wae sanhate |
fh We TOWNE seri Ye Bis y hate | ‘ia a | a: an

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‘t
-. er) bantu bios od KA .! jit sboren ah © gah
- on Act. nay a7" ae etna hrm walls man / seed v7 si

=
¥

4B)

7 tou deat wt

iin

| a 4

a>

The other boundary condition is obtained at the array (x = L),
where we must have:

ou
= = 0 (8)

where:

M_ = Dynamic mass of the array. In general, it will
be composed of three parts: the actual mass of
the materials of the array, an apparent mass due
to the motion of the array in water, and the mass
of any water trapped in the array and having to
move with it.

a = Drag coefficient of the array. This is a function
of the Reynolds number, and for certain array
configurations such as cylindrical or spherical,
its value can be found in the literature.

A = Area of the array projected in the direction of
motion.

Notice that a quadratic form has been taken for the hydrodynamic
drag. (The two vertical bars in the first velocity factor mean "absolute value
of," and they are necessary since the drag must always oppose the motion).
This is absolutely necessary because the Reynolds numbers involved are high.

Let us define the following two parameters:
p SL
Cc

u = ae (9)
a a

Notice that u is the ratio of the mass of the cable to the mass of
the array. Equation 8 can then be put in the following form:

Te a
2 iF Xi at’

= 0 (10)
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Arthur D Little Inc.
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ltt 2c Te ee SAN SHA SE RN UCHR sala nope uid tin “all ’
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| en wh Pe

Minit 30 ney Ake ie Ie. ET of sil
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B. THE STEADY STATE SOLUTION FOR SINUSOIDAL INPUTS

We will take u at x’ = 0 as sinusoidal, of amplitude U., andof
angular frequency w and try to find the steady state solution of the problem.

In Appendix A it is shown that, under conditions which are valid
here, the friction on the cable is proportional to the velocity, and an expres-
sion for the friction constant (8.) is derived. In Appendix B it is shown how
the drag term in Equation 10 can be linearized and how good such an approxi-
mation is. Once this linearization is introduced, the entire problem is linear
and therefore, for sinusoidal inputs we will have sinusoidal waves.

It must be emphasized, once and for all, that the approximation of
the nonlinear quadratic drag on the array, developed in Appendix B, is not
linear but "quasi-quadratic."' It replaces the drag, which is actually propor -
tional to the square of the instantaneous velocity, not by a term proportional
to the instantaneous velocity but by a term which varies in time as the velocity
and whose amplitude is proportional to the square of the amplitude of the vel-
ocity.

In what follows a capital letter will be used to denote the complex
amplitude (magnitude and phase) of the respective instantaneous sinusoidal
variable. For instance, U(x') means the complex amplitude (magnitude and
phase) as a function of x' of the instantaneous sinusoidal dynamic displace-
ment u(x’, t’).

Let us define a normalized displacement amplitude U' as being
equal to U divided by Uo| . If U, is the value of U' at the array (x' = 1),
then the solution for U’ asa function of x’ is:

U = U, cos w'y' + C sin w'y’ (11)
where C is a complex constant and

x _ wl
Te » W ae c (12)

This solution satisfies Equation 7 if the friction of the water on the cable is
neglected. As is shown in Appendix A, this is valid for the frequencies of in-
terest here. Only at very sharp resonant peaks can the friction on the cable

be important, and even then its role can be insignificant if damping of a greater
order of magnitude exists somewhere else in the system (as, for example, in
the array).

13

Arthur D Little Inc.

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ca of) diirgn 4) seaplane eal(nice oo mirent (aed ein el ae |
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The two unknown constants U; and C will be determined by the
two boundary conditions. By defining the origin of the time (t') properly, we
can take U; as being real and positive. Then, according to the linearization
introduced in Appendix B, substituting Equation 11 in Equation 10 we obtain:

- (w')? U; - wuc + iB(w')? (,)2 =O) (13)
where:
8 4apA [Vol (14)
37M,

This parameter is the ratio of the drag to the inertial force of the
array. Therefore:

t

SC SEA aay at
Se a U, (leu) (15)

Then Equation 11 reduces to:

U' = U, sec @ cos (w'y'+¢) +i8 (U,)? tan @ sin w'y’ (16)
where:
ian 6 = —, 0<e<t (17)

Finally, the unknown real positive constant U; will be determined by requiring
that |U'| at y’ = 1 be equal to 1. This gives:

, 2 t Baame) wt ase. 2
(U1)? — Cosi (ON seo) (+ 8“sin* w' sin a8). 1 (18)
2 6? sin? ¢ sin? w' cos™ (w' +4)

If we denote the amplitude of the dynamic stress by © and define a
: : Lz
normalized stress amplitude, =" , as being equal to , then the dis-
Us E

tribution of »'' is given by:

mo os OP U, sec ¢ sin (w'y’ + #@) - iw8 (U,)? tan ¢ cos w'y’ (19)

14

Arthur D Little, Inc.

S-7001-0307

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; a Ay (* YW) PED. ce hy "aha i” eee | BR M
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9 | J: i
ba VA Sat tots rt of ou five waite hati
(R23) ‘vw aos 8 ae 8 ah 9 a ae
: : wr iY

Equations 16, 18, and 19 determine completely the amplitudes of the
normalized displacement and stress waves as functions of y', w' and the two
parameters wu and B.

1. Summary of Formulae for the Magnitudes of the
Amplitudes of the Array Displacement, Maximum
Stress, and Stress at the Two Ends of the Cable

Equation 19 shows that the distribution of =’ along the cable is sinus-
oidal. Therefore, depending on the frequency and length of the cable, &' can
have absolute maxima and minima. These extreme values of >", as well as
their locations, can be computed analytically. Denoting the magnitude of the

maximum of ~%’ as (sea the location(s) of this maximum as y',,,,, the mag-

nitudes of =" at x' = 0 and x' = 1 as |=}| and {=| , respectively, one can
show that:
2 eas
2 = (w')? (U,) E + tan ¢ (tan ¥ + sec ¥) | (20)
ad = ia = =
ZO Man as fd ve Mm © iy Sy D coc (21)
Z| = (HP (ye E + 2 tan @ cos? w' (tan ¥ + tan w)| (22)
a = (w')? Wye E +2 tan @ tan ¥ | (23)
where:
2
2 ’ Ve) ted an are)
wy? _ _G9s2 (MO) (+ sin* w' sin 28) 1 (24)
28 sin? @ sin? w' cos- (w' +o)
@ = arctan ron. (25)
¥ = arc tan 1 g2 (U')? tan @-cot2¢|, -~<¥<2 (26)
2 a =” =P
15
Arthur D.Hittle, Inc.

S-7001-03 07

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Ait
hi

y

: yea iS Dati we me ened h F nee
7 ia '¢ sor es # a bal Beri ;
ve i i Ls | ye ry < We eee poe

For convenience, we repeat here the following definitions:

La

wL U
Os peace ees | er x Ho
sa er aw. IP UME 20)
fe) © |
4 p. SL on 40 A|U,| ai
S Ne 37M
a a

If the values of y',,,,» aS given by Equation 21, are outside the
range 0 to 1 for all of the indicated n's, the maximum stress, as given by
Equation 20 does not occur in the cable. In that case, the maximum and mini-
mum stresses in the cable are the stresses at the top and the bottom of the
cable, respectively. In order to have 0 $s Ymax * 1, the condition that

2w'+V¥2 5 must be fulfilled.

2. Special Limiting Cases

Case 1
w' = mm, where m = 1, 2, 3..., andu< 1 so that @ = >
Then U, = 1 and:
1 m= Li
’ = |r = ' = 2\3
| acl : Ba ee) (29)
Case 2

L- 0 so that w’ and u- 0 but ¢ is a finite number other than zero.
Then everywhere in the cable U and © are constant, and they are given by
U= Ce and:

Pye Eee) 2 ys
m= = w UM, (1 +6?) (30)

16

Arthur D.Little, Inc.

S-7001-0307

Wo, eM halide

y

LP Ber Ast

C. PARAMETRIC ANALYSIS OF THE MAGNITUDE OF THE
MAXIMUM STRESS FOR SINUSOIDAL INPUTS

The magnitude of the normalized amplitude of the maximum stress,
as given by Equation 20, is obviously a function of only three variables,namely
w', u, and 8. Itis plotted in Figure 3 versus w' with u and 8 as parameters.*
As it is stated at the end of Section IVB1 the maximum stress as given by Equa-

: A T i yas
tion 20 occurs in the cable if 2 w' + ¥ 2 7° When this condition cannot be

fulfilled, which can happen only for small values of w', the stress at the top of
the cable (Equation 22) is used in Figure 3, because it is greater than the stress
anywhere else in the cable. Figure 3 then shows the dependence of the maximum
stress on the input characteristics (t) and U,), the characteristics of the cable
(c, E, andS), the length of the cable(L), and the weight and shape of the array.

Let us see whether Figure 3 is compatible with the well known results
of simpler systems. For 8 = 0, there is no damping in the system. In this
case the resonant frequencies of the system are the roots of the equation
tan w' = u/w'. The smallest of these roots lies between 0 and 1/2, the next
between 1 and 31/2, and so on. At these resonances, the reflections of the on-
coming stress waves by the hanging mass have the proper phase so that their
individual contributions result in an infinite total stress. As the hanging mass
is decreased indefinitely (u-~), the resonant frequencies approach the values
1/2, 31/2, etc., and we have the case of the free end spring. As the hanging
mass is increased indefinitely (u-0), the resonant frequencies approach the
values 1, 21, etc., and we have the case of the fixed end spring. On the other
hand, when 8 # 0, energy is dissipated by the hanging mass, and the amplitudes
of the reflected waves are diminished. Resonance can still occur but with finite
amplitude. The actual value of 8 (amount of damping) should have a small ef-
fect on the values of the resonant frequencies and a very profound effect on the
amplitudes at resonance. One could expect that the amplitude at resonance will
be decreased as the damping is increased.

Figure 3 shows quite clearly all these expected trends, except one;
that is, for the resonance occurring near w' = 7 (and 2n, 3m, etc., as well),
the amplitude increases when the damping is increased beyond a certain value.

*In computing these curves, the values of ae and | x have also been

recorded. These values can be of use in the design of the joints at the top and
bottom of the cable.

IL7/

Arthur 7 Little Inc.

S-7001-0307

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NosmalizediAng Ularieirequency. wis cn) FIGURE 3. THE NORMALIZED MAXIMUM DYNAMIC STRESS VERSUS THE NORMALIZED ANGULAR FREQUENCY
U 7_WL
FOR VARIOUS VALUES OF THE PARAMETERS 8 (= 1 Alt ») and w=)
7 M, \ M, A

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I. SUMMARY

A. PURPOSE AND SCOPE

This report relates to the operational aspect of lowering heavy array
structures to the bottom of the deep ocean. Because of the weight of the array
and the depth to which it will be lowered, the lowering cable can be expected to
be under a high stress. Ocean currents, surface waves, and motions of the
lowering vessel will also contribute to the stress placed on the lowering cable.
Therefore, a critical design problem may exist for the lowering cable.

The objective of this report is to develop a reasonable theory on
which to base the ultimate design of the lowering cable. The study is limited
to the case of a single cable lowering the array vertically. Each factor con-
tributing to the stress on the cable is considered separately, and the conditions
under which the analyses are valid and of practical significance are given.
From the results of this theory some general conclusions are drawn as to the
feasibility of the operation with regard to dynamically optimum array configu-
rations, stability of the lowering platform in a rough sea and degree of rough-
ness of the sea (sea state). The results of the study are presented in plot form
involving a non-dimensional maximum (dynamic) stress versus a non-dimensional
frequency for various mass and drag parameters.

Other methods of operation, which are not treated in this report, but
which will be considered in later investigations, are:

(1) Cancel out most of the dynamic inputs to the cable by an auxiliary
mechanical system on the vessel. There are two crucial problems here: the
horsepower of the driver of such a system and the rapidity with which the over -
all system can respond.

(2) Introduce considerable damping along the cable, which will have
the effect of flattening out resonant peaks. This can be effected by attaching
bluff bodies to the cable at certain intervals. The drag of the water on them
will give the desired friction. This idea seems quite attractive, since these
bodies can also provide buoyancy to reduce the rather high static stresses in
the cable, but its plausibility must be investigated quantitatively.

Arthur D.Uittle Inc.
$-7001-0307

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Ill. THE EFFECT OF OCEAN CURRENTS

Due to the frictional forces on the cable, a current will bend the
cable as shown in Figure 2. (Such frictional forces will also act on the array
and vessel if they are within the current.) Of interest here are three quanti-
ties: the additional stresses in the cable, the offset (D) from the target, and
the horizontal force (F) which must be exerted by the vessel in order to remain
stationary against the current.

This problem can be solved very accurately by balancing the compo-
nents of the various forces acting on an infinitesimal element of the cable. The
results for all pertinent variables can be expressed in terms of integrals which
have been tabulated (David Taylor Model Basin Report 687 and its Supplement).
However, if the net vertical force (gravity minus buoyancy) is much larger than
the horizontal frictional forces due to the current, bending of the cable will be
very small, and the following simple analysis can be applied.

SURFACE — O56

ARRAY

FIGURE 2 BENDING OF THE CABLE BY A
HORIZONTAL CURRENT

Arthur D Little Inc.

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BTA RIAA EM TE)
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This should be expected, because when the damping of the hanging mass be-
comes very large, the mass cannot move very much, and we have again the
fixed end spring with resonant frequencies at 1, 27, 3n, etc., and infinite am-
plitudes. To put it another way, if a very large amount of energy is dissipated
by the hanging mass, the energy propagating along the spring must be very
large, since the spring is the only link between source and load.

Two remarks are in order here. First, as Figure 3 shows, there
are resonances (some of them highly sharp depending on the values of u) at
w' =m or w=". For c = 10,000 ft/sec and L $ 20,000 ft, this corre-
sponds to frequencies as small as 1.5 rad/sec or periods as high as four sec-
onds. Since such periods can be found easily in a realistic sea, it is not cor-
rect to consider the cable nonflexible when an accurate estimation of the stres-
ses is desired, unless such frequencies are filtered out by the vessel very ef-
fectively. Also, for small values of 8, there are resonances at values of w'
much smaller than t, corresponding to periods much larger than four seconds --
which are quite frequent and prominent in a rough sea. Second, an idealized
system, such as the free end or fixed end flexible cable, will give portions of
the curves of Figure 3 rather accurately. However, this does not render the
present analysis, which includes the dynamics of the array, superfluous. Our
objective here is not to display a rough picture of the phenomena, which could
be done by the free end or fixed end cable, but rather to find as accurately as
possible the dependence of the maximum stress on the various quantities. In
terms of a realistic and sound design of the cable, which hinges on the magni-
tude of the maximum stress, simple models--such as the nonflexible and the
free end or fixed end (flexible) cables--are useless, and a complicated model,
like the present one, is indispensable.

In order to demonstrate how these curves can be used, let us con-
sider a cable and array as specified below:

L = 20,000 ft M, = 600,000 lb

ie 6: ..; re 2
E = 20x 10° psi A = 3,000 ft
S = 3in2 @% 8 1.2
w = 10.6 lb/ft
c = 13,600 ft/sec

19
Arthur D.Hittle, Inc.

S-7001-0307

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ident, F

The array described at the end of Section III has been taken here.
This array has a weight of 50 tons. Because it is hollow with rather thin walls,
approximately (600) (62)(2.25m) = 250,000 pounds of water are trapped in it
and have to move with it. Also, the apparent additional mass of the array in
water is about 250,000 pounds. Hence, the total dynamic mass of the array,
M,,» is about 600,000 pounds. The value of A is equal to (600) (5) = 3,000 ft2 .
The value of 1.2 for a has been obtained from the literature. Therefore,
we have:

p SL
Wale 7s
u = M mange e 0.353
a a
4apA|U |
B = 3M = 0.16 |U |
a
pe SS Say
Cc
je oe ft oe PS oon fue
max L o ma max
Now, if we take | Us = 1 ft and w = 1.26 rad/sec (which corresponds to a
period of 5 sec), then 8 = 0.16 and w' = 1.85. Then, from Figure 3,
[pz | = 1.9 and therefore, > | = 1,900 psi. Also, for the same w,
max max

and|U_| = 10ft, |= _| = 20,000 psi. Similarly, for |U_| = 19 ft and
O max fo)
w = 0.63 (which corresponds to a period of 10 sec), eee = 23, 700 psi.

The static stress distribution is linear, and for this cable and array without
any buoyancy, the maximum static stress (at the top of the cable) is 103,000
psi while its minimum value (at the bottom of the cable) is 33, 300 psi.
Stresses due to currents of the type considered in Section III are much smaller
than these static stresses. Even without determining the locations of the above
dynamic stresses, it is obvious that this cable will be able to withstand the
maximum total stress with a factor of safety of about 2. If a larger factor of
safety is desired, the metallic cross-sectional area of the cable can be in-
creased and/or buoyancy in the cable and array can be utilized in order to de-
crease the large static stresses. However, if the static stresses are de-
creased below the dynamic stresses, the cable will buckle (see Section VI).

20

Arthur D Little, Inc.

S-7001-0307

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od +f 7 i p ‘ ma ¥ M
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On the other hand, as Figure 3 shows, certain resonant frequencies
can induce very large dynamic stresses. The frequencies of these resonances
are close to w’ = 7, and their peaks are very high for small values of u.
For the above cable and U, = 1 ft, we have (at resonance) w’ = 3.2 and

a | = 90. Therefore, > = 90,000 psi and w = 2.2 rad/sec, corre-
max max

sponding to a period of about three seconds. If we take | Us = 3 ft, then

[Bell = 195,000 psi at w = 2.2 rad/sec. Clearly then, resonant frequencies

with rather small amplitudes can break the present cable easily. Depending on
the state of the sea, the above resonant frequency with the corresponding am-
plitudes can be found in a realistic sea. However, the input amplitude to the
cable at this frequency can be much less than the amplitude of the sea, depend-
ing on the stability of the vessel (or lowering platform) and the method used to
attach the cable to the vessel. These aspects will be treated in the following
Section, but the following two general conclusions must be emphasized here.

: : ee mC Shoe
First, since the resonant frequency is given by w = Te it is de-

sirable to have as high values of c as possible so that this frequency can fall
in the region of the input spectrum which is characterized by small amplitudes.

Second, it is highly desirable to have a large value of u. This means
that the dynamic mass of the array must be small. The present cylindrical
array is poorly designed with regard to this aspect, because of the large quan-
tities of displaced and entrained water.

A linear array made of aluminum trusses can serve the same pur-
pose as the 50-ton cylindrical array. The aluminum array does not entrain any
water, it displaces a very small amount of water, and it weighs only about 15
tons. It appears that a reasonable value for the dynamic mass of this array in
water is about 20 tons. (Compare this with the 300-ton dynamic mass of the
50-ton cylindrical array.) Let us consider with this array a 6 x 37 Special
Flexible Hoisting wire rope (USS) with a diameter of 1-5/8 inches, which has
the following characteristics:

E = 11x 10° psi S = 1.1 in2
c = 10,000 ft/sec w = 4.1 lb/ft
Notice that, for L = 20,000 ft, u is now equal to 2.
21
Arthur D.Uittle Inc.

S-7001-0307

ths yi
f , mm ra gd pri Oy hy. .
sak amino vit crits Gh pat til

vu % fiisn 7 ane

a eT oh

nr,

porate ry

er oo Na 2 |

‘ tis 4 AN ed ‘a V4

eee i hi
iF ) fan cl 4 i psn uh
»! ow

The only unknown factor in the system is the drag coefficient 8 for
the array. For the present array, we cannot obtain a value of 8 from the
literature. For the cylindrical array, however, we found that B = 0. 16|U,| j
Since the present array has the same dimensions as the cylindrical array,
values of 8 a few times smaller than that for the cylindrical array seem to be
reasonable.

If we plot the value of | U,| versus the value of w which will give
a specified maximum dynamic stress for various values of L, we will have
curves like those shown in the following diagram. The envelope (dashed line)
drawn through

lUol

S

L, = 20,000' ~

the first minimum of each of these curves has the following significance. In-
put characteristics | Wall and w corresponding to a point which lies on or below
this envelope will never produce a maximum dynamic stress greater than the
specified one for any value of L up to 20,000 feet. Then, if the specified
maximum dynamic stress is the design stress for dynamic loads of a particular
cable, this envelope is the boundary of safe (points lying below it) and unsafe
(points lying above it) input characteristics.

730)

Arthur D.Aittle, Ine.

th ine j

sivig ive ahr, im W ‘wa bay: a nye b bd
gt EW at te cathand SUEY ee, has spine pis sae
oe fins meta ‘at? eels ue? Left wo, nt 7

c ‘ OM. oti art ij ; iol aah hee lo Gh) Ta ie i. Bova lal te Hy",
wise Pn th a ‘ j att Nhe iad pais a yas ae ihe, | AP t Ve le a bie iit,

be Gert) rulbwty ir Hs ety ight) Wy patel, ae TW givenE High 8,
edits cutie dab 8 pil Oe) 2 se aa Al RAY tere a a
Toit Ate Day Ee eet ty ae 1 SD od eee eas, 1 ing eel
qrceral tri ti) shih 1 Pert BE a) vedio, (4 : at * eT 19 13% ty) ee lh

ote ita gee a 7 a

Such curves are plotted in Figure 4 (dashed line) for the aluminum
array and above specified Special Flexible Hoisting wire rope. The maximum
tolerable dynamic stress is 32,000 psi, about one-sixth of the ultimate strength
of this cable. The values of 8 assumed are stated on the curves. Notice that
more friction moves the curve to the right. Assuming that 8 =0.05 |U,] 5

we see that at w = 2 rad/sec the maximum dynamic stress in the cable will
not exceed 32,000 psi as long as | UL is equal to or less than 1.7 feet and so
on for other frequencies. If B = 0.1 | U, , then for the same frequency the

maximum allowed | U,! is 3 feet.

23

Arthur D.Little, Inc.

S-7001-0307

aes ee TR,
if wv :
ery aaah tpl ’
. nuance bat’ oo snr? yatiRGlt
mua

gone MOUs Het WY pat eae ony Derhkerany: " ‘in
LAG © # aii gtr AIH OEE

spied aes wht Al slewrse vata ine ovdsrotbagn aidan ik
|e Bike et A atl) wank a6 ct dandnit ot b ny wa geek aaa’
asl quinborpitey ratendie at dh pel oljats i

\ ee eh a hewathl: —

iy

a “a
- -
, i
:

\
\
\

IUol FOR 3=0.1 Up!

1Uol FOR @= 0.05 is
\

HEAVE ee

RESPONSE AMPLITUDE
OPERATOR FOR
ROLL(10 ‘rad /ft)
AND FOR HEAVE

MAXIMUM SAFE
75 INPUT AMPLITUDE
[Uo| (ft)

0.50 5.0

ROLL (CUSSI1)

(0)

() 05 1.0 1.5 2.0 25 3.0
FREQUENCY (rad/sec)
20 10 6 4 3.5 2.5
PERIOD (sec)
1500 500 200 100 60 40 30
WAVELENGTH (ft)

FIGURE 4 RESPONSE AMPLITUDE OPERATORS FOR CUSS I AND MAXIMUM
SAFE INPUT AMPLITUDE TO THE CABLE AS FUNCTIONS OF
FREQUENCY. (CABLE IS A SPECIAL FLEXIBLE HOISTING WIRE
ROPE OF USS.)

24

Arthur D.ULittle Inc.

wid ‘ paw ADM da
Ty he |
4 Lae

\
ee)

"acre i iA J Hd '} 5 yu ry 4 OT, , hia A ; aa iv : q Me ; ¥ ING san
MRT Mee 210AO SY OF. COUT TIGR a
“sayy Um Se ie LAO eee BR) ORR
. bik. ly 1) RAE Se my

i

V. THE DYNAMICS OF THE VESSEL AND SPECTRAL
CHARACTERISTICS OF A REALISTIC SEA

It has been already pointed out that, in calculating the dynamic stres-
ses induced in the cable by a disturbed sea, the dynamics of the vessel and the
way in which the cable is attached to it must be considered simultaneously with
the dynamics of the cable and array. Besides the elastic force exerted by the
cable, other forces acting on the vessel are inertia, dynamic buoyancy due to
the waves of the sea, viscous damping, and damping due to the "radiation" of
surface waves by the vessel. Then, in an analysis of the entire system, simi-
lar to the analysis carried out in Section IV for the system without the vessel,
at least three additional parameters must be introduced. This will be a very
complicated problem.

We will assume that the dynamic loading of the vessel by the cable
and array is negligible and, therefore, the motions of the vessel during the
operation will not be influenced by the cable and array. Because of the relative
dynamic masses of the vessel and the cable and array, this condition is well
satisfied in practice, except possibly when the array is lowered through a long
boom from the side of a vessel which is rather unstable in rolling.

In Section IV, we have shown that near resonant frequencies, which
can be as small as 1.5 rad/sec, the amplitude of the input to the cable should
be about 2 feet or less. This definitely shows that the vessel must be very
stable at such frequencies, and it limits the variety of ways in which the cable
can be attached to the vessel.

Let us first examine the spectral characteristics of a realistic sea
where such an operation will be carried out. A realistic sea has a continuous
probabilistic distribution of all possible amplitudes and frequencies. Depend-
ing on the weather conditions (mostly the speed of the wind) and the degree of
development of the sea (duration of the wind and extent of the fetch), amplitudes
above a certain value and frequencies outside a certain range are not very prob-
able, while amplitudes and frequencies within a certain range are most probable.
It appears that separate probability data for amplitude or frequency exist in
available literature, but data for the joint probability for a specified amplitude
and frequency do not exist; i.e., we do not really know what is the probability
for the occurrence of a certain amplitude with a certain frequency in a given sea
state. Tables I and II show some important characteristics of a fully arisen sea
(infinite fetch and wind duration) at various wind speeds. These data have been
obtained from "Observing and Forecasting Ocean Waves," H.O. Pub. No. 603,
U.S. Navy Hydrographic Office. The significant range of frequencies is defined
as the range outside which it is highly improbable to find a frequency at the indi-
cated wind speed.

29

Arthur D Little Ince.

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Ff

7 shes ac ak iooaion th $e ip sublha iy sah rahe fry wii oo vue
pasty cline way wid Toeuenieny aft pace ott Sita Shag ihiynnt et
biviteien 24 Re serene) ovat ty ban, qian mt yl be saalllne (At fou, {lyr nail
Howse: ‘aditibgas gin)’ Yuri hay, han watt Hide’ teawel pakd Ye intact ne
1 yal 6 ieilo’s a hevgeyney)! Ai KA hh mh radiw EE cane fyb “Ko, dapsone i bral

Bt {for ni otdnin is githaes Bh vigil sey i ey iain wh) Mapa Ed g

ssdalw wins Apo semitones? TW ject sein Bd felt ‘irk adel al =

Ediipite ehies eet od ujinh oft To petioles ay be Nba Baha Dian ep ms iat
: Yeas ad teed fenuey ot jee mwa. Yi tala hele al Me R aper'h Lanta aS
iy Ls sae i sy eee sails eld at) ban Lhd ere 1 tii ay iat ;
; San at 0 aieat alt at leper RS

i
tay

it ait i i ee Cee aah
ADORE 9 her cha MY has hs oo an ge 19 Sy. Sate
haga, , ea pau jy 3) Wee Seb BLL nh sh Hw 3 fe ts art wih EU rites a .
iY cla rl tare (oniw aH do ba vegan oar ae Petite), en QTD Mase ae HON iW atl
gebintiqne gay sds Do Varies bom haute dlr ko) Hobie) aoe 1% ne
“rere eRe 10% a Ouiley pleTied & ete aShtenps? bie OIE Minis ;
yekdalderety PADI Alle SUNY, AiATLOS A ale Gervawipngt rie gota atte IW. pie BAe
GR ata ictal Ata TT Lone FF

7.

HE PAS Ylmetipirat a Igri LOT mts i
ebiiqman by io? Moone a OL RRR MS jot FOL bit HAY uy YOM » con ehiieh Se mi
vi, treherd at wilt ee livier wood # Uae Moo ab 4 CS Lat, Mes " " CAH .
Loi ievig: Moat Yoweipos) ented on ate hamtalgeene Abie Ph TO OCG
a rate Viel ofp eset eas: hep peotyann OLN, ‘iL crit SPN Sie i
Ropd Grad wie wed? soronge bidiWy aiesbiay' se dejan Wate da MINOR § iia
B08 CH til) Ol ai dis O nt ease 2 bats ert epee) ern) (haat ie ;
famiiah di waconaiyper'to ages Dip anal ET WOUAEY O Duepeeyeayet Ital i ft
Hah Wa) 3m Youensapendt ae bail <0 plclaclennepssil: Clohytitvl 1 Atay pti eRiO mgm wit i ;
By ae Se aR ey OR AG “eh hao

ae oiswilaet a tt | apialaet ey silo ee dull

TABLE I

AMPLITUDE CHARACTERISTICS OF FULLY DEVELOPED SEA

Average of 30% Average of 10%

Most Frequent Average of Highest of Highest
Wind Speed Amplitude Amplitude Amplitudes Amplitudes
(knots) (ft) (ft) (ft) (ft)
20 2 7438) 4 5
30 5.4 6.8 iit 14
40 ll 14 22 28
TABLE II

FREQUENCY CHARACTERISTICS OF FULLY DEVELOPED SEA

Most Frequent Average
Wind Speed Frequency Frequency Significant Range of Frequencies
(knots) (rad/sec) (rad/sec) Lowest (rad/sec) Highest (rad/sec)
20 1.4 el .06 Po al
30 0.9 243 . 38 es
40 0.53 50) 529) oy
26

Arthur D. Little, Inc.

S-7001-0307

l ne ' it an . i

ne

oa id ve i it ;, i | | .
if et of | aly | . 7 PAS
Win a Buney |

are i ‘mors + ieee i

i a sy ts ane
a x ’ Mig) ia
aT iS hi ;

Tables I and II show that frequencies as small as 1.5 rad/sec (the
smallest resonant frequencies of the cable) are more likely to be found at
smaller wind speeds, but the amplitudes are much smaller at smaller wind
speeds. Obviously, as the wind speed decreases, the chances for the success
of the operation increase. The largest wind speed that is safe will depend on
the stability of the vessel and the manner of attaching the cable to it.

Two obvious ways of attaching the cable to the vessel are from the
center of gravity, and from the side through a boom. In the first, heaving
motions of the vessel in beam seas will be critical, while in the second, roll-
ing motions in beam seas will be the most critical.

In Figure 4, the heave and roll response amplitude operators for
Cuss I* in unidirectional sinusoidal (regular) deep ocean beam waves are
plotted versus the frequency w. Scales for the period (equal to 2 T/w) and the
wavelength (equal to 2ng/w*) are provided. For waves approaching the vessel
from other directions, heaving and rolling are less. These curves were ob-
tained from "The Motions of a Moored Construction-Type Barge in Irregular
Waves and Their Influence on Construction Operation,"’ Contract NBy~32206,
U.S. Naval Civil Engineering Laboratory.

In the case of attaching the cable to the center of gravity of Cuss I,
some meaningful conclusions as to the safe sea state can be drawn from Tables
I and Il and Figure 4. Assuming that we can represent an irregular sea by a
regular one with amplitude equal to the average amplitude of the irregular sea
and frequency equal to the most frequent frequency of the irregular sea, then,
when the wind velocity is 20 knots, the input to the cable is w = 1.4 rad/sec
and |U,| = (0.42) (2.5) = 1.05 ft. This input can be tolerated by the cable.
According to this representation of an irregular sea, a wind speed of 30 or
even 40 knots can be tolerated by this cable. Since the input amplitudes that
the cable can tolerate are small for high frequencies, perhaps a fairer repre-
sentation of an irregular sea by a regular one is by the average of the 10% of
highest amplitudes and the most frequent frequency or the highest frequency of

*Cuss I, originally a nonpropelled freight barge, is 260 feet long and has a beam
of about 50 feet and a draft of about 11 feet at 3,000 tons displacement. It has
been converted to a sea drilling vessel, and in March 1961, it was used suc-~
cessfully by Project Mohole to drill in almost 12,000 feet of water at a site 40
miles east of Guadalupe Island, Mexico. Cuwuss I is used for our preliminary
computations, because it is the only vessel, of the type that could be used in
the present operation, for which we can obtain the amplitude response curves
from existing literature.

27

Arthur D.ULittle Inc.

S$-7001-03 07

} nae. or
oe pi sipah ck Tee ws “sides ‘het sttiboed Prue

| i maa coat EL at prnsora attri i
ro Sule kasummieierdes aes nee

Wien Lams dbisMhanamnic Mahal

ee Pui ava a eohuhiajae ith bn cvind une ee er
‘yaa Rae EAM mated a Mpheate (anh ing ky Ate lentes lien
ial ha AVE Leh lunges bok zmg: ony nuk WeAbiaR Me leet .
Probar aidt Qillska nO LonpH secteN FO; ablVirnd hp, shag alta ath
“wl Le ehen ty pean | erst ea ctl ib Hi; 1% Pc Lahiri ii
po: Halrgeerey easy vr ual pinks Sonntag herd caey 4 he muah: eth" 9
pie Balas sul searseanen ia inlet sett iodine th ea Sa hbk ;
i i | a Lae te NS nes Rhona l diel ieid Mond inl:
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| Sapte? any ve ty wit teat mi nl ato ahi pk ae oa mer haiti phan yi
2 Pe ee afoursiga, AD 4! LaDy ATRIA eon tis UE aah
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ene oe das Ay eae faced nde eae OR ah ‘ygtaleahaa Rane an:
aca ait yioattiekat ink ainda WARE MT RO Leh AR RE a ‘Se na...
au key naeaea Wah bet ah. may 30: ra peice a dios)
ean ‘well duel w Vite, welty natok) tlelug: HageY Ye Load acedeligt st bis vp OY
fa eee lll) \odiatila bape ha FTN SERRE! 1) tie RR
a ne wh ie, YAR, #602 eA AMR 4 to) ps ane , ;
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herein

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7 ‘? oye sy Va Nai ey cer Hit aii Kogan Hint flee ee | eee pane seit ait “i
md: by hid Hh, Vieinegeterets berg ON) 8 ty fOet Hf idan hes ay le ‘Tene 1H? 9
ae Me isky pains ani by LOS) RRA eh Poel Near Shen HEF at Pe a

a pee, Oe tay bein wus Meee Le re ab Thess i “hint fae neetlial ae
wi mea eilaeiihicwy Bro at Cha wT wen ite on Haves til! sings ita ie Ade HO

if a er) ay fig Pe ey MPa at hh eee iE Ey) Wi, ‘f sagt iesatucd if
7 1 ae igh vou “er pal aid as pian) HAs, wr dasih aie .

the significant range of frequencies. Under these conditions, a 20-knot sea is
quite safe, while a 30-knot sea is marginally safe. Of course, in the final
analysis the reasonableness of our argument depends on the value of the fric-
tion coefficient 8 for the array. We only guessed its value here. Assuming
that we have not made a gross overestimate of 8, we can conclude that the
chances for the success of this operation are quite good for wind velocities up
to 25 knots.

If the cable is attached to the end of a boom extended from the side
of Cuss I, a rather quiet sea is required for a successful operation. Since
Cuss I has a beam of about 50 feet, a boom at least 30 feet long is required.
Heaving and rolling can occur in phase, and, as seen from Tables I and II and
Figure 4, only in a sea excited by a wind velocity of much less than 20 knots
are the chances for the success of the operation acceptable. Indeed, it will
take a vessel having much greater stability in rolling than Cuss I to carry this
operation successfully in a sea excited by a wind speed of 20 knots.

28

Arthur D Hittle, Inc.

S-7001-0307

ie: ATT ee ee
ip a

“Selamat Pas is
o Dien BARE Gel NE Thao. cee BN, “da |
Nes take T wala Cort mon my eo ynmeely at Zucce amp qaMelinn!

. 7 me jooedeaidcepuahahaennetbiriG ped todlaxs es
deli peta AAT ose py 1rOKN nicer tic} es Waerea NED OF
lose HED ett oily quritox a dite nit ghpr it My
nail DE Ne bison ht iy side cdanaei nit

aoe y De

VI. BUCKLING

We assume that the cable buckles as soon as the total stress at some
point becomes compressive. Since, without any dynamic input, the cable is
under tension due to gravity, not all kinds of inputs will make the cable buckle.
In order to find whether a given input will cause buckling, step by step we must
trace in the cable the propagation of the input, as well as its interaction with
and reflection by the array and vessel. Here we will investigate only one as-

pect of this complex problem, namely, buckling within the time T = = after
the onset of the input.

In the absence of any input, the static elongation of the cable, us:
is given by

DS E (w-b) + W, - B, | (31)

Now the simplest input dynamic displacement, ws , Which satisfies the condi-
: : : : ; a A

tion that at time t = 0 the vessel is stationary is Ness t®, where a is an
acceleration. Therefore, if buckling is to occur within the time T, uo must

become greater than Us in time equal to or less than T, i.e.:

aS 78 z (w-b) + W, - B, | (32)

where g is the acceleration of gravity. Thus, the acceleration a necessary
for buckling decreases with increasing L and with decreasing net weight of the
array. For L = 20,000 ft, w = 10 lb/ft and W, = 100 tons (and no buoyancy
anywhere), a> 3g. For We = 0, a>g regardless of L and w. Therefore,
it takes tremendous input accelerations to buckle the cable in this manner.

Suppose, though, that the cable has buckled. Then, depending on the
degree of buckling, the cable may or may not form kinks. When the vessel now
moves upwards and stretches the cable, these kinks will make the failure of the
cable easier. In order to determine the degree of buckling, let us assume that
the array falls freely in the sea after buckling.

29

Arthur D Little, Inc.

S-7001-0307

aa i a
6 x ie

Me mist lig vd id

j

ei ‘chomp rT ivy BA wt * ' vn Mb a

ey

nn ay is} tei rhe # ey al el Heh waitin ai Mi fe “ arent td ne

i

ne.

heya aye ¥ bs

‘ a4
al]

Shima MMB Rstindren

The forces acting on the array are inertia, gravity, buoyancy, and
friction. The downward displacement of the array, uy, (t), must obey the fol-
lowing differential equation:

2
sn)
= Li Peal
M me = WN, BOW B. Yapa | (33)

The second term in the right side of the above equation accounts for the gravity
on a volume V of water trapped in the array and moving with it. For the cyl-
indrical array discussed thus far, goV will be approximately equal to B, >

and the solution of this equation satisfying the conditions that at t = 0,

— Go a) §
Mis aaraee is:
uy = V_ ty, £n cosh 2 (34)
Oo t,
z
where Vi = (2W.,) /(%A)| is the ultimate velocity of the array and
ty = 1.41 M, (a oN) . For the 50-ton cylindrical array, ve = 5.3 ft/sec

and ty = 1 sec.

To determine the degree of buckling, uy and u, 7 Us must be

plotted as indicated below.

VELOCITY

e TIME

30

Arthur D Little Inc.

Wallace wingibity

ic

emains: Poat ys anle cme OR yet

hn Th, ou we on

The fact that the cable has buckled means that the curve ee must lie
above the curve uy for some time. Of course, after some time of downward
motion, the vessel will move upwards, as indicated by the dashed line. The
difference between Ue ewe and u is a measure of the degree of buckling,
and can be used in some way to determine whether the cable will form kinks.
At the time corresponding to point A (or at some previous time, if the cable
has formed kinks), the cable will be again under tension. The failure of the

cable by kinks should, perhaps, be investigated.

31

Arthur D.Little, Inc.

S-7001-0307

nd

Male Anns

APPENDIX A
FRICTION ON THE CABLE

In order to derive an expression for the friction on the cable by the
surrounding water, we will consider the mathematical model of an infinite
straight cable surrounded by a viscous fluid of infinite extent. Furthermore,
we will assume that the cable moves longitudinally like a rigid body with a spe-
cified velocity. This rigid-body approximation is valid as long as the distance
along the cable required for an appreciable change in the velocity is much
greater than the radius of the cable. This requirement is easily met in the
present case.

The governing equations of the motion of the fluid reduce to:

)
Lie (A-1)

where w is the velocity of the fluid along the cable, vy is the kinematic vis-
cosity of the fluid, and r is the radial coordinate.

If the velocity of the cable is sinusoidal of amplitude Wp, and fre-
quency ™, then the solution for the amplitude of the velocity of the fluid W is:

Bl (eae) en
vey SS. (A-2)

2 (agen 2)
O v
where a is the radius of the cable and HY) is the Hankel function giving out-
going waves.

Therefore, the amplitude of the force F exerted by the fluid on the
cable per unit length is given by:

HY) (a J-i =)

ane ee) Reh cna rf
F = 27 ap vw, fig tis (1) = (A-3)
H (a /-i — )
fo) WV)
where p is the density of the fluid.
32
Arthur DHittle Inc.

S-7001-0307

Jk ie. aoe sf
tS i Sean ‘4 Br Pies oa

a OM he pind tO
hi Pt nen 2 ny ca — aN a . 1 wip! Ny ;

A
i a ;
- Th i)
ae ai — ae
le 7
“és

: Fi cc tk ee One Prenre i f bil, Dae sedi
{ anes aus pry va ‘ ; be ; }

al $y

er yh aaa see tan

7 Ae “sbi x sono a une ih ie nang | ihc | i : ea
: m4 m4 i j r f fi : aH
f 7

Now, for water v = ‘lesa ee fees. Therefore, for a and w
even as small as 1 inch and 0.16 rad/sec, respectively, a | = >>1 and the
Hankel functions can be expanded asymptotically. Then

Hg (a i?)

—————— ar (A-4)
q) (a al 42 )
fo) ( —®
Therefore,
a UN
F = -27 ap Juv W, era (A-5)

This equation shows that the viscous force is out of phase with the velocity of the
cable.

Thus, it has been shown that the viscous damping on the cable is pro-
portional to the velocity of the cable. The coefficient B rs used in Section IV-A
is obviously given by:

Hy ie B/N
5 savaa/GM eS — sg (2 o (A-6)
c p,Sc P. \ca?

Now for a metallic cable p. = 500 lb/ft? and c = 12,000 ft/sec. Thus, with
the values of p and v for water and L and a equal to 20,000 feet and 1.125
inches, |8| = 0.013 w' #. For smaller L, 1s! is even smaller, and it is
hardly possible to have a of a smaller order of magnitude than 1.125 inches.
The inertial term in Equation 7 is of order w'2 , while the friction term is of

order w' | 8 | = 0.013 w'3/2, Therefore, for w' = 0.02 the friction term can

be neglected as compared to the inertial term (except at very sharp resonances).

33

Arthur D.Hittle, Inc.

S-7001-0307

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a} i: is hf i: marty: any er) cot) ‘te esti eluate arte
Fy NI aia "5, sage Bray palin nity alae y wht ™ eee Mga a} ie
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APPENDIX B
THE DRAG ON THE ARRAY

The problem formulated in Section IV is linear except for the term of
the drag on the array. This term makes the entire problem nonlinear, and
enormously more difficult than a linear problem. It is, therefore, expedient to
linearize this term.

For sinusoidal inputs, one can take the drag on the array, D,, as
being proportional to the velocity of the array (not the square of the velocity)
and then determine the constant of proportionality experimentally. For veloci-
ties and arrays of the type involved in this problem, this constant of proportion-
ality will be found to be a function not only of the frequency but also of the am-
plitude of the velocity of the array. However, such an experiment is not prac-
tical in the present case; no effective scaling, according to the principle of
similarity, of the variables is possible, because of the enormous dimensions of
the arrays and the appreciable velocities involved. We therefore propose the
following analytical linearization.

The drag D, on the array is given by (see Equation 8):

ou.
ot

ou

ID, 3 2G fo) AN vo (B-1)

For sinusoidal inputs, it is expected that the motion of the array will be periodic.
If the | du/dt| is replaced by a constant, then D, and the entire problem become
linear, and therefore the displacement of the array will be sinusoidal of ampli-

tude U, . We Set the constant which replaced |au/dt| equal to Ww, . This

selection results in the same amount of dissipation of energy by the array when
u is taken as sinusoidal in both factors in Equation B-1 and when only the sec-
ond factor is taken as sinusoidal. Thus, with U, a real positive number, the
amplitude of the drag D, will be given by:

ee Be 2 2 x
D, = an 0PM AU, (B-2)

34

Arthur D Little Inc.

S-7001-0307

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' 5 | a beni ¥, al. ie
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or iedld mnt} 4 LG FEN ai a TEAITAEYT mH ae Oo? we eed, ny Fisie ub Les ie Pent ait:

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(2°) x a eed
iy po
ay
a
‘a
my

mH ae Pas

To estimate the error involved in the above approximation, we consider the
array alone and a sinusoidal force acting on it (in the actual problem this force
is the force exerted by the cable). If we denote the velocity of the array by v,
then the dynamics of the array will be governed by the following differential
equation:

dv

Tage coe plviv (B-3)

where F and p areconstants. Then, according to the above approximation:
v= Vcos (wt -¢) (B-4)
where: 4

2
2 BNE
2 3m Ww 16pF
WS: Wl -14+]1+ B-5
I aa ees)

and @ can also be computed. (In obtaining the above result, \y| in Equation

8 ‘ : : : : d
B-3 was replaced by ae V.) This solution can be considered as a first iteration
toward the exact solution. Now we can compute a correction v’ through a sec-

ond iteration, by solving the equation:

a+ BY S jp) |) Vv (B-6)
where v and V areas given by Equations B-4 and B-5. By expanding the
term in the right hand side of Equation B-6 into a Fourier series, we find that
v' is given by a Fourier series, the frequencies of the components being 3w,
Sw, 7W,...(i.e., v' has no fundamental component!). This result is due to
the above choice of the constant replacing |v | in the first iteration. The am-
plitude B. of the leading component of v' of frequency 3w is given by:

w? \2
B, = 0.2V(1+12.5 — 5) (B-7)
> WV.

This shows that the correction v' is not more than 20% of v, and, therefore,
the error involved in the above linearization of the drag is at most of the same
order.

35

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S-7001-0307

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LIST OF SYMBOLS

Radius of cylindrical cable, except in Section VI where it stands for a
constant acceleration.

Area of the array projected in the direction of motion.
Buoyancy force on the cable per unit length.

Buoyancy force on the array.

Velocity of sound in cable.

Horizontal offset of the array from the target due to a horizontal cur-
rent.

Hydrodynamic drag on the array.

Young's modulus of elasticity.

In Section III total horizontal force of the current on the cable.
Acceleration of gravity.

Hankel functions of orders 0 and 1, respectively, giving outgoing
waves.

Length of the cable.

Dynamic mass of the array. (See page 12.)

Radial coordinate measured from the axis of the cable.

Metallic cross-sectional area of the cable.

Time coordinate.

Normalized time coordinate (=ct/L).

Characteristic time of the array in free fall in water (see Equation 34).

Time (=L/c).

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Dynamic displacement of an element of the cable due to dynamic
(time varying) inputs.

Dynamic displacement of the top of the cable.
Dynamic displacement of the array.

Static elongation of the cable due to its own weight and the weight
of the array.

Complex amplitude of u for sinusoidal variation with time.
Normalized U (= U/| U4 Ne

Value of U at the top of the cable.

Magnitude of Ue .

Value of U at the array.

Value of U' at the array.

Velocity of the array.

Correction term for v.

Terminal velocity of the array in free fall in water.

Weight of the cable per unit length, except in Appendix A where it
stands for the instantaneous velocity of the fluid along the cable.

Complex amplitude of the velocity w for sinusoidal variation with
time.

Value of W at the cable.

Weight of the array.

Space coordinate measured along the axis of the cable.
Normalized x (=x/L).

Width of current.

37

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$-7001-0307

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iy

Normalized space coordinate (= 1 - x’).

Value (s) of y' at which the amplitude of the dynamic stress in the
cable for sinusoidal variation with time becomes maximum.

Drag coefficient.

Dimensionless drag coefficient for the array, equal to the ratio of
the drag to the inertial force.

Dimensionless friction coefficient for the cable.

Parameter equal to the ratio of the total mass of the cable to the
dynamic mass of the array.

Kinematic viscosity.

Density of water.

Density of cable.

Tensile stress in cable due to static loading.

Complex amplitude of the dynamic stress in the cable for sinusoidal
variation with time.

Normalized = (=LE/E| U4!)
Value of &' at the top of the cable.
Value of &' at the array.
Maximum value of =".

w'
Angle (= arctan ane, j
Angle (see Equation 26).
Angular frequency (radians/sec).

Normalized w (=wL/c).

38

Arthur D.Little, Inc.

S-7001-0307

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Report No.

1011260

1021260

1031260

1041260

1051260

1061260

1071260

1080361

1090561

1100561

1110561

1120561

1130961

1141061

1150162

1160262

1170262

PROJECT TRIDENT TECHNICAL REPORTS

COLOSSUS I, December, 1960 (C)

THEORETICAL INVESTIGATION OF CROSS-FIX PROBLEMS
AND CORRELATION EFFECTS, December, 1960 (C)

THE SUBMARINE AS A SURVEILLANCE PLATFORM,
December, 1960 (5)

Title Classified, December, 1960 (S)
AIRBORNE JEZEBEL, December, 1960 (S)

SURFACE -SHIP SONARS IN OCEAN-AREA SURVEILLANCE,
December, 1960 (S)

Title Classified, December, 1960 (S)
LOW-POWER ENERGY SOURCES, March, 1961 (C)
SOLUS, May, 1961 (C)

NONACOUSTIC METHODS FOR SUBMARINE DETECTION,
May, 1961 (S)

ARTEMIS, May, 1961 (S)
COLOSSUS II, May, 1961 (S)

RELIABILITY OF UNATTENDED ELECTRONICS EQUIPMENT,
September, 1961 (U)

NUTMEG, October, 1961 (S)
DEEP JULIE, January, 1962 (S)

METHODS FOR ANALYZING THE PERFORMANCE OF DIS-
TRIBUTED FIELDS OF DETECTORS, February, 1962 (C)

MAGNETIC ANOMALY DETECTORS IN FIXED SHALLOW
WATER BARRIERS, February, 1962 (S)

39

Arthur D.HLittle Inc.
S-7001-0307

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}

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© ee a7 (| ee ® sae oe ae
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ih oa nn.

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ae 56 she dies How ST pcnlaress Avis: i ore
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ie, Masai Abe cada str zmcrrcirs yu aanegega OAADA: i
| i — ) | me bi) neo ; felt ‘Sara ui

am te Ry

KOH) et
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hi Mal ul ae Ji
Pe en ae

Report No.

1180262

1190462

1200562

1210562

1220562

1230662

1240762

1250862

1260862

1270862

1281262

1291262

1300363

1310363

1320363

ELECTRICAL CONDUCTIVITY, COMPRESSIBILITY, AND
VISCOSITY OF AQUEOUS ELECTROLYTIC SOLUTIONS,
February, 1962 (U)

A FEASIBILITY STUDY OF THE PASSIVE DETECTION OF
QUIET SUBMARINES, April, 1962 (S)

Title Classified, May, 1962 (S)

RADIATED NOISE CHARACTERISTICS OF DIESEL-ELEC-
TRIC SUBMARINES, May, 1962 (S)

DIRECTIVE RECEIVING ARRAYS, May, 1962 (C)

ANALYTICAL BACKGROUNDS OF COMPUTATIONAL METHODS
FOR UNDERWATER SOUND PROPAGATION, June, 1962 (U)

MARINE CORROSION AND FOULING, July, 1962 (U)
SURVEY ON AMBIENT SEA NOISE, August, 1962 (C)
DEEP SUBMERSIBLE WORK VEHICLES, August, 1962 (C)

THE EFFECT OF PRESSURE ON THE ELECTRICAL CON-
DUCTIVITY OF SEA WATER, August, 1962 (U)

ENGINEERING PROPERTIES OF MARINE SEDIMENTS,
December, 1962 (U)

AN INTRODUCTION TO MODULATION, CODING, INFORMATION
THEORY, AND DETECTION, December, 1962 (U)

SPHERICAL DIRECTIVE ARRAYS: A PRELIMINARY STUDY,
March, 1963 (U)

ESTIMATES OF SUBMARINE TARGET STRENGTH, March, 1963 (C)

PHYSICAL CHEMISTRY IN THE OCEAN DEPTHS: THE EFFECT
OF PRESSURE ON IONIC TRANSPORT PROCESSES AND EQUI-
LIBRIA, March, 1963 (U)

40

Arthur D.Little, Inc.
S~7001-0307

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ne
a) pee
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> ma Sa nan if

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hs ite : i

i) ma ei Tu ie Ki xt ih, wlatbonld sa vt

ie 1 _ i Al : ai Wi tel ' yi oH
1 ;

; ecbun’y.’ ; wend a sere ah

Report No.

1330563 SUBMARINE CABLES AND REPEATERS, May, 1963 (C)

1340663 APPLICATION OF ADAPTIVE SAMPLING STRATEGIES TO
THE PLANNING OF SURVEYS, June, 1963 (U)

1350663 CURRENT OPTICAL DATA PROCESSING TECHNIQUES -
ASW SYSTEMS, June, 1963 (S)

1360863 ACOUSTIC SCATTERING IN THE OCEAN, August, 1963 (U)

4]

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st Apnis ay. ‘sueado0 daap ayi Jo Wonog amy 0} Saimonzs Avrie
Aavay BulraMoy] Jo JOadse [euoneisado amy sassnosip i10der sTyL

(F9ST8-ISQON 1980p)

Z S21GeL ‘p SuONeSNI] PF + Ar -dd
‘ego isndny (eggOLET itoday [eoruysaL quaprazL 199f01g)

SLNAWAOV TdNd YALVMYSGNN
daad YO SAUTAVO CHANAdSNS-dIHS AO SISATVNV SSHULS

sasnyoesseP ‘Op aspiaquiep ‘uy ‘apy ‘dG anyimy

“ON uoTSsaDgY

fal
a

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-
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= 7

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