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711 O



NAVY DEPARTMENT

THE DAVID W. TAYLOR MODEL BASIN

Washington 7, D.C.



A METHOD OF DETERMINING OPTIMUM
LENGTHS OF TOWING CABLES



by
Leonard Pode





April 1950



Report 717
NS 880-019




i □
: rn





INITIAL DISTRIBUTION

Copies

, 10 Chief, BuShips, Project Records (Code 362), for distribution:

5 Project Records

1 Technical Assistant to Chief of the Bureau (Code 1 06)

1 Research (Code 330)

1 Technical Information Disclosure (Code 367) for:

National Research Establishment, Halifax, Canada,
Attn: Mr. Richard Parr Blake

1 Minesweeping (Code 620)

1 Magnetic Defense (Code 66OM)

2 Commanding Officer and Director, U.S. Navy Electronics Laboratory,
Point Loma, San Diego 52, Calif.

1 Commanding Officer and Director, U.S. Navy Underwater Sound Lab-
oratory, Port Trumbull, New London, Conn.

1 Commander, Puget Sound Naval Shipyard, Bremerton, Wash.

1 Development Contract Officer, Kellex Corporation, Silver Spring
Laboratory, 936 Wayne Ave., Silver Spring, Md.

1 Director, Woods Hole Oceanographic Institution, Woods Hole, Mass.

1 Chief, Bureau of Aeronautics

Attn: Mr. John S. Attinello (Code RS3)
Rm 2W87, Navy Department
Washington, D.C.



A METHOD OP DETERMINING OPTIMUM LENGTHS OP TOWING CABLES

t>y

Leonard Pode

INTRODUCTION

A problem that arises frequently in connection with the design of
towing arrangements is that of choosing the design variables so that the size
of equipment and the magnitude of the forces involved are kept to a minimum.
Usually the preliminary choice of the design variables has been merely a guess
and the improvement of the guess has depended upon the results of extensive
exploratory calculations. Tables and charts are presented here which, for the
most frequent design problems, will help to reduce the labor of such calcula-
tions and to enable the designer to determine optimum conditions in a straight-
forward manner.

STATEMENT OP PROBLEM

Suppose that it is desired to tow a body at a stated depth, y, using
a specified cable. Since the hydrodynamic behavior of the cable may be as-
sumed to be known, the length of cable needed to reach the required depth and
the tension at the upper end of the cable are determined when the direction
and magnitude of the force that the towed body applies to the lower end of the
cable are known. The question of interest to the designer is the manner in
which the tension at the upper end of the cable, which is the greatest tension
in the cable, will vary with the direction and magnitude of the force applied
by the towed body.

A force of given magnitude may best be used to attain depth by ori-
enting it as nearly as possible in the direction of gravity because the com-
ponent of force perpendicular to the direction of gravity increases the ten-
sion in the cable without contributing to the attainment of the required depth.
The angle, <t> , that the force applied by the towed body makes with the direc-
tion of stream— which is equal, for equilibrium, to the angle that the cable
makes with the stream at the point where the cable meets the body— is there-
fore made as close to rr/2 as possible. If the downward force of the body is
developed by means of lifting surfaces, the angle <t> is limited by the lift-
drag ratios which such surfaces can attain; if the downward force is derived
from the weight of the body, the angle <j> Q is limited by the relationship of
the weight of. the body to its drag. Since the weight of the body is constant,



whereas its drag increases with the square of the speed, the weight required
to obtain a given value of increases very rapidly with speed. Hence the
limitation on </> becomes more severe as the speed increases. The value of <j>
for a body employing lifting surfaces is not affected appreciably by speed or
by scaling its dimensions. However, the magnitude of the force obtained from
such a body varies as the square of the speed and as the square of the factor
to which its dimensions may be scaled.

Let it be assumed that the direction of the force applied by the
body is known. The question is then how should the magnitude of the force be
adjusted. It is clear that if the magnitude of the force is very small, the
length of cable that is required will be exceedingly long so that the hydro-
dynamic force acting along the cable will cause the tension at the upper end
of cable to become very large. On the other hand the length of the cable is
shortest only when the magnitude of the force applied by the towed body grows
exceedingly large and then the tension in cable is also very large. Between
these extremes there must lie an optimum.

ANALYSIS
The calculation of this optimum configuration depends upon the spe-
cific assumptions made regarding the forces acting on the cable. It can be
shown from very general considerations, however, that regardless of these spe-
cific assumptions the solution of the cable configuration can be expressed by
equations of the following parametric form.

[1]



[2]



T


T




T o~


T o




Rs


a _


\


T o~


T

c


)


Ry =


V -


''o


T o


T o




Rx =


t -


*o


T o _


T o





[3]



W



where T is the tension at the upper end of the cable,
s is the length of the cable ,
y is the depth of the body,
x is the distance of the body aft of the upper end of the cable,



T is the tension in the cable at the point where the cable meets the
body, i.e., the magnitude of the force applied by the towed body,

R is the drag of a unit length of the cable when the cable is normal
to the stream,

r, o , r] , and f , are certain functions of <j> which depend upon the specif-
ic assumptions that are made regarding the forces acting on the
cable, where

<j> is the angle between the cable and the direction of the motion at
the upper end of the cable, and

T ~> ff ~> v~> a nd f . are the values of these functions for <t> = s. See

OO'O'O

Figure 1 .




Figure 1 - Cable Configuration For a Towed Body



Since y, R, and & are fixed, Equation
or T as an explicit function of



A general expression for the optimum configuration can be obtained
through the use of these equations.
[3] gives <f> as an implicit function of T ,
<t>. In Equation [1 ] T can therefore be considered to be a function of either
T Q ,<6, or any function of <t> . The optimum configuration is obtained by mini-
mizing T. This can be done most easily from the differential forms of Equa-
tions [1 ] and [3] taken simultaneously, which can be written



4



Ry/



t6



Ry



dr



o V T„ /



[la]



r^o Ry-1



= dr,



[3a]



For minimum T, dT must vanish. Hence from Equations [la], [3a], and [3]



Ry r r



t6v

~a7



[5:



77 _T



dr



[5a:



This equation may be used to determine the value of 4> that obtains
at the optimum configuration. From Equation [3] the appropriate value of T Q
may then be found; thence T, s, and x can be calculated from [1 ] , [2], and
[4].

The designer is usually interested in the optimum conditions for
high-speed operation. If the speed of towing is sufficiently high so that the
weight of the cable can be neglected, the functions r, a , tj , and 4 may be tak-
en as those given in TMB Report C-122, Appendix I, p. 27, i.e.,



t = 1 + f csc0; f =
a = cot <j>



[6]
[7]
t) = In cot ■?■ [8]

f = CSC0- 1 [9]

where F is the drag per unit length of the cable when the cable is parallel
to the stream. Equation [5a] then becomes



4. <t> 1 + f CSC^ , 4. ^0

ln cot 2 " f cot 6 = ln cot —



[5b]



Thus far only the case of towed bodies has been considered so that
the maximum value of </> is 71-/2 — since negative drag cannot be realized in this
case. There are, however, some cable configurations in which the values of
<t> greater than n/2 are possible; i.e., configurations where the cable forms
a loop. An example of such a configuration is a cable joining two self-
powered bodies such as two airplanes or two submarines. When the speed of tow-
ing is high, so that the effect of weight is negligible, the plane in which the
cable lies need not include the direction of gravity; so that a line strung be-
tween two surface vessels may also present such a problem (see Figure 2). In
such cases, if the angle of the cable at one end may be considered fixed this
end may be referred to as the lower end, the other end of the cable may be
called the upper end and the fixed angle can still be designated as <j> . Then
the foregoing analysis still applies and Equation [5b] gives the condition
that the tension at the upper end of the cable is minimal.

It is found that when the angle <t> is increased beyond n/2 the ten-
sion at the upper end of the cable will continue to be reduced. Nevertheless
there is a limiting condition beyond which minimizing the tension at the upper
end of the cable is no longer a reasonable procedure: When further increase
in </> will cause the tension at the lower end to become greater than that at




Figure 2 - Cable Configuration for Two Powered Vessels



the upper end of the cable. From the symmetry of the functions t, a , r] , $
about <t> = t/2, it is clear that this condition will occur when the tensions at
the two ends of the cable are equal and when the configuration of the cable
is completely symmetric about a line parallel to the direction of motion.

Solutions of Equation [5b], and pertinent values obtained therefrom
are listed in Table 1 and are graphically presented in Figure 3-

The cable configurations that are found by solution of Equation [5b]
are optimum configurations with respect to tension in one end of the cable



32

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10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

O - degrees



Figure 3a - Variation of <t> with <t> and f .

Figure 3 - Values Obtained for Optimum Cable Configurations
for High-Speed Towing



when the angle at the
fied. Other types of
sired that the drag at
tension; also, instead
cable the drag of the
distance aft might be
tion to such problems,
a first approximation



other end of the cable and the depth of towing are speci-
optimum problems may occur. For example it may be de-
the upper end of the cable be a minimum instead of the
of the angle <t> being specified at the lower end of the
body might be specified or the ratio of the depth to the
fixed. The tables presented here do not give the solu -
However in many cases these tables may be used to get
to a solution for such problems.



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1

Online LibraryLeonard PodeA method of determining optimum lengths of towing cables → online text (page 1 of 2)