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Anchor-last DeploymentProcedure for Mooring

by

Robert W. Thresherand

John H. Nath

SCHOOL OF SCIENCE

OREGONI STATE UNIVERSITY

Office of Naval ResearchContract N00014-67-A-0369-0007

Project NA 063.102and

Sea Grant Program

Reproduction in whole or in part ispermitted for any purpose of the

United States Government

Reference 73.3,dune 1873

9

School of Oceanography,-

andDepartment of Mechanical Engineering

Oregon State University.Corvallis, Oregon 97331

ANCHOR-LAST DEPLOYMENT PROCEDUREFOR MOORING

A Progress Reportfor the

Ocean Science and Technology DivisionU.S. Office of Naval ResearchContract N00014-67-A-0369-0007

Project NR083-102

by

Robert W. ThresherAssistant Professor of Mechanical Engineering

and

John H. NathProfessor of Mechanical Engineering/Oceanography

John V. ByrneDean

Oceanography1Reference173-5School of Oceanography

June 1973

Approved For Public Release: Distribution Unlimited

ACKNOWLEDGMENTS

This research was supported by the United States Office of Naval Re-

search through contract N00014-67-A-0369-oo07 under project NR083-102 with

Oregon State University, and by the Sea Grant Program. Part of the compu-

ter work was supported by the Oregon State University Computer Center.

TABLE OF CONTENTS

List of FiguresAbstract

INTRODUCTION ................ ....................................... 1

NUMERICAL MODELS FOR EXTENSIBLE LINES ................................. 4

Method of Characteristics ........................................ 4

Lumped Mass Model ................................................ 6

Spring Force ................................................. 6

Hydrodynamic Drag Forces ..................................... 7

Added Mass Forces ................................ ......... 8

Governing Equations for the Line ............................ 8

Governing Equations for the Anchor .......................... 9

Numerical Integration ....................................... 10

A LUMPED MASS MODEL FOR INEXTENSIBLE LINES ............................ 15

RESULTS OF TEST RUNS .................................................. 16Nylon Line .......................................................Dacron Line ......................................... ...........Steel Wire Rope ................... ............................

16

18

19

CONCLUSIONS AND DISCUSSION ............................................ 21

REFERENCES ............................................................ 23

FIGURES ............................................................... 25

APPENDIXES

A. Listing for the Extensible Line ............................... 39B. Listing for the Inextensible Line ............................. 59C. Program Sequence .............................................. 87

D. Tabulation of Constants ....................................... 89

E. Computer Run Times ............................................ 91F. List of Symbols ............................................... 93

LIST OF FIGURES

Figures

1 The Lumped Mass Model

Free Body Diagram of a Typical Mass

Free Body Diagram of the Anchor

Computation Flow Chart

Standard Catenary Configuration

Goose Neck Catenary Configuration

Nylon Line 2.5" Dia. (NVRO5)

7 Tension Contours ' KIP 15° Catenary Nylon Line 2.5" Dia.(NVR05)

8 600 Catenary Nylon Line 2.5" Dia. (NVRO5)

Tension Contours % KIP 600 Catenary Nylon Line 2.5" Dia. (NVRO5)

10 Anchor Link Tension for Nylon Line

Components of Anchor Velocity Nylon Line 2.5" Dia. (NVRO5)

12 60° "Copse Neck" Catenary, Nylon Line, 2.5" Dia. (NVR06)

13 Anchor Link Tension for Nylon Lines Starting from 60° StandardCatenary and 60" Goose Neck Catenary.

14 15° CAT Dacron Line (NVRO8)

15 Tension Contours ti KIP 15° Catenary Dacron Line 1.0" Dia. (NVRO8)

16 60° Catenary Dacron Line 1.0" Dia (NVRO8)

17 Tension Contours " KIP 60° Catenary Dacron Line 1.0" Dia. (NVRO8)

18 Anchor Link Tension for Dacron Line

Vertical Component of Anchor Velocity Dacron Line 1.0" Dia. (NVR08)

20 15° Catenary Steel Line 2.5" Dia. (NVR07)

21 Tension Contours ti KIP 15° Catenary Steel Line 2.5" Dia. (NVR07)

5b

11

19

ABSTRACT

The anchor-last mooring procedure is investigated in order to determine

the transient forces in the mooring line and the velocities of the anchor. Tran-

sient forces were determined and the results showed that no severe snap loads

occurred for the cases investigated. In addition, it was found that the

vertical velocity of the anchor can be small as it approaches impact with the

floor of the ocean.

Both extensible (nylon and dacron) and inextensible (steel wire rope)

lines were investigated. Lumped mass numerical models were developed for both

cases. For the extensible line case the equations of motion were determined

for each mass from Newton's Second Law, and they were integrated using a

second order predictor-corrector integration technique. Hamiltonian techniques

were utilized to determine the equations of motion for the inextensible line.

The predictions from the numerical models show the line tensions and positions

as a function of time.

INTRODUCTION

Oceanographic research buoys moored in deep water are often deployed by

the anchor-last deployment procedure. The general sequence of events in this

procedures is: the buoy is deployed from the ship with the mooring line attached

and distributed in some manner on the ship; while the buoy drifts away from the

ship the line is paid out until only the anchor, which is secured to the lower

end of the mooring line, remains on board; finally the anchor is cast overboard.

At sometime during the descent of the anchor it is possible for large transient

forces to be exerted on the line, which can be destructive to the line or to

attached conductors and instruments. One purpose of this research is to examine

the possibilities of such large forces for line scopes greater than one. The line

scope is the unstressed length of the line divided by the water depth.

The Woods Hole Oceanographic Institution has conducted a series of field

measurements of the launching transient in mooring lines where the anchor-last

deployment procedure was utilized (1). However, the study was restricted to the

consideration of moorings where the scope was less than 1.0. It was found that

a fairly steady increase in line tension occurred after the anchor drop and

during the free-fall stage. Then a pendulum action followed where the final line

tension was equal to the submerged weight of the anchor. During the pendulum

action, the line (nylon) was stretching until the anchor reached bottom. After

the anchor reached bottom a reduction in line tension occurred.

A similar problem was investigated by Froidevaux and Scholten (4) with a

numerical model which considered the line to be a discrete number of lumped masses

connected by weightless line segments. However, when the elongation properties

of the line were included, a prohibitive amount of computer time resulted and the

2

investigation only calculated the first and last few seconds of fall. A short

system was considered and results were extrapolated to apply to the 6500 Oceanic

Telescope. One scope which was greater than 1.0 was investigated but the

distance from the buoy to the anchor at launch time was not determined, nor the

influence of scope and line type on line tensions. The conclusion from the

study was that there may be an overstress shortly after the drop and that a severe

transient may occur when the anchor made impact with the ocean floor. Therefore,

the anchor-first mooring procedure was recommended. During this investigation

it was found that large transient forces shortly after anchor deployment can be

an artifact of the numerical program.

The Electronics Division of General Dynamics has employed the anchor-last

deployment procedure for several moorings of the forty-feet diameter oceanographic

buoy for the Office of Naval Research. It has been noticed that some conductors

in the upper portion of the line have experienced large stresses during the

deployment of some moorings but not for others. Thus it was suspected that large

transient forces occurred during the anchor deployment either concentrated at the

buoy,or propagating up the line to the buoy. This investigation predicts that

large snap loads should not occur for such moorings.

Goeller (5) investigated snap loads in steel cables but the study was restricted

to the consideration of straight lines with a scope equal to 1.0 where the mooring

was already in place and the upper end was excited with a sinusoidal motion.

A review of the current techniques for the dynamic analysis of morring

line systems was presented by Casarella and Parsons (3).

The first technique to be employed during this study was like the one used

by Nath (7, 8) where the line is considered to be a continuum and the partial

3

differential equations of motion are solved by the method of characteristics.

Later the analysis technique was changed to a lumped mass approach that was

very similar to the analysis made by Wang (10) except that the equations

developed were in a different coordinate reference frame and the integration

technique was different.

NUMERICAL MODELS FOR EXTENSIBLE LINES

The numerical models presented here pertain specifically to the anchor-

last mooring procedure. The basic equations and procedures for the method of

characteristics solution were presented by Nath (7, 8) and they will not be

reproduced here. However, certain modifications to the earlier program will

be presented which shows the attempts to utilize the method of characteristics

for the anchor-last procedure. The lumped mass model for the extensible mooring

line will be presented in detail. It is similar to the approach taken by Froid-

evaux and Scholten (4). The lumped mass model for the inextensible case will

not be presented in detail because it is available in the work by Rupe (9).

Method of Characteristics

The computer program. presented in (7) was modified to accept the new boun-

dary conditions of the anchor-drop problem. The first runs produced results

which gave unrealistic mooring line shapes. Attempts to improve on the numerical

procedure met with only limited success. The problem was caused by the extremely

tight curvature of the line in the first few seconds of fall. The calculations

of line positions did not correspond to the line velocities.

The solution technique was to solve for the normal velocity, VN, the tangential

velocity VT and the local angle, 0, made by the tangent to the line and the horizon.

For simplicity and as a first approximation the buoy attachment point

was assumed to be fixed. At first it was assumed that the mooring line was

composed of a series of straight lines between calculation points. Then the

coordinates of points along the line could be calculated using:

XiXi-1 + AS cos 0i

= Z + AS sin 0i-1 1

where AS is the distance between calculation points. It is easy to visualize

why this technique would give unreasonable results if the mooring line were to

be given a high curvature or "kink", which is exactly what takes place during

the early stages of anchor drop.

In an attempt to eliminate this problem the following more accurate approx-

imation was used to calculate the mooring line position:

dx1

ds= cos 0

i

(2)

dzi= sin 0.

ds 1

Since the 0.'s are known at constant intervals along the line, then all that is1

required is to integrate the above equations along the line from the attachment

point to the anchor. For example, the trapazoidal integration formula gives

X. X. +AS

2(cos 0i + cos

0i+1

zi-1+

AS(sin 0i + sin 0 1+1)

=

Z.(1)

=

(3)

=

which is equivalent to assuming that the line shape is quadratic between

calculation points. Both the above formula and Simpson's rule were used, however,

the curvature of the line is so steep that even these higher order approximations

did not eliminate the problem. Therefore, the decision was made to proceed with

the lumped mass model of the following section.

Lumped Mass Model

A mooring line can be modeled as a group of discrete masses interconnected

by springs as illustrated in Fig. 1. The various distributed forces which act

on a mooring line are gravity, hydrodynamic viscous drag and acceleration, and

the line tension, which is influenced by the internal, or structural, damping

and the stress-strain relationship. In the lumped mass model these forces are

discretized and it is assumed that they act at each mass.. Figure 2 shows the

various forces acting on a typical mass. The magnitude of the various forces

were calculated in the following manner:

Spring Force. From closed solution studies of taut line mooring dynamics it

has been found that the external hydrodynamic forces are considerably larger

than the internal damping force. For this study in particular, where periodic

oscillations are not a factor, the internal damping force was negligible.

Therefore it was not included into the equations of motion of the masses.

Hooke's Law was taken in the general form

(4)

calculating the strain, using the straight line distance between masses, then

1/2OQ [(x.1 - + 2

i) (Zi+lZ) J - Qi

eiQ

Q (5)i 1.

= f

2

where Q1is the unstressed distance between masses and X and Z. are the coordin-

ates of the mass M1,. This gives the spring force SF as:i

SFi = Area

The net spring force acting on the mass M. is given by1

spring force

Lxdirection = SFX. = SF

1. cos(O

1.) - SF1.-1 cos(6 1.

1 -

spring force

z directionSFZ. = SF. sin(O.) SF, sin(O. )

1 1 1 1-1 1-l

The Hooke's Law relationship was assumed to be in the form

where Cl and C2 are constants. The values used for the various lines are given

in the appendix.

Hydrodynamic Drag Forces. Using the quadratic drag law, the tangential and

normal drag forces on the mass M.1

are given by

{Tangential brag} = DFTi CDTpAT (VTi)IVTiI2

{Normal Drag} DFN, = CDNpAN (VN.)1 2

(8)

where tangent. and normal are referrenced to the angle OAVi as illustrated in

Figure 2, and

7

CC

(7)

8

CDT = Tangential Drag Coefficient

CDN = Normal Drag Coefficient

VTi = Tangential Velocity of Mass Mi

VN = Normal Velocity of Mass M.i ip = Fluid Density

AT = Tangential Area = 7DQ_i

AN = Normal Area = D.i

Defining VCX and VCZ to be the local velocities due to currents and waves,

then the tangential and normal velocities of mass Miare given by

VTi = (k. - VCX) cos (OAVi) + (zi - VCZ) sin (OAVi)

(9)

VNi = - (xi - VCX) sin (@AVi) + (zi - VCZ) cos (OAVi)

Added Mass Forces. The added mass forces due to the acceleration of the

line masses through the still fluid was considered in the usual way. That is,

the excitation force accelerates the line and the fluid medium around the line.

The pressure distribution on the line from the accelerating fluid is characterized

by the displaced mass of fluid times a coefficient called the added mass coefficient

and the product is called the added mass. The sum of the line mass and the added

mass coefficient for the line in the direction normal to the line was taken equal

to that for a smooth cylinder (0.5) and in the tangential direction it was assumed

to be zero.

Governing Equations for the Line. Substituting the above forces into Newton's

second law and including the hydrodynamic added mass effect with the actual line

mass (the sum is the virtual mass)

9

A>Ftangential = (Mi + CITpVoli) ACTT

Fnormal = (M. + CINpVol.) ACN.

. 1 1 1

where

CIT = Added Mass Coefficient Tangential

CIN = Added Mass Coefficient Normal

p = Fluid density

Vol = Displaced Line Volume =7FD2 ki

4

ACN = Normal Acceleration of the Mass M.1

ACT = Tangential Acceleration of the Mass M

Then the accelerations are given by

ACT. _ (SFXi)cos(6AVi) +(SFZi - WTi) sin (6AVi) + DFTi

1 i + CITp ° i)

- (SFXi) sin (6AVi) + (SFZi - WTi) cos (6AVi) + DFNiACN _.

1 (Mi + CINpVoli)

In the x, z coordinate system the accelerations are given by

= ACTT cos (6AVi) - ACN.1

sin (OAV1.)

zi = ACTT sin (6AV.) + ACN cos (6AV1 1 1

(10)

(12)

Governing Equations for the Anchor. Figure 3 shows a free body diagram of the

anchor from which the following governing equations can he obtained

1

10

cos (B) - 1/2 C() A p x VX = (MA + CIpVol)

(13)

T sin (S) - W - 1/2 CDA'pzVZ = -(MA + CIpVol)

where CD = the anchor drag coefficient

T = line tension at the anchor

CI = the anchor added mass coefficient, constant in all directions

A = Area = TFD2/4a

For simplification, the water velocities from current and waves have been ignored.

They can be easily added if necessary. The anchor was assumed to be a sphere,

with a high drag coefficient to recognize the fact it is not actually a sphere.

Numerical Integration. Numerical integration of Eqs. (12) and (13) was accom-

plished with the following second order predictor-corrector set (6) .

Velocity Predictor:

Let

Xi (t) = f (xi (t) , zi (t) , Xi(t), zi (t) )

(14)

and z (t) = g(x1(t),

1z. (t), x. (t), z. (t) )i 1 1

where the functions f and g are equations (12) above. The velocity predictors

are then

Pxi (t) = xi (t-At) + 2At xi (t)

11

and zP (t) = zi (t-At) + 2At 'zi (t) , (15)

Displacement Predictor: Similarly the predictors for displacements are

xP (t) = x. (t) + of (x (t)+ XP (t+ot)

1 1 2 i 1

(16)

and zP(t) z. (t) + At (zi (t)i 1

Velocity Corrector: The corrector equations are for velocities

xP p(t+ t) = f (xi. (t+ht) , z (t+At), xP P(t+At), z (t+At) )i i(17)

..PZ1(t+At) = g(x

1.(t+ot), z

1(t+ot),

1P(t+ot), t

1.P (t+ot) )

. C (t+At) = x. (t) + at (X (t) + XP (t+At) )1 2 i 1

(18)

and zC (t+At) z (t)1

P

(t) + L. (t+At) )At

2

Displacement Corrector: Likewise, the corrector equations for displacements

are

xl(t+At) = x.(t) + (xi(t) + xi(t+At) )1 --7

and z. (t+At) = z. (t) At

i (t) + zi (t+At) ) (19)

The corrector values are then considered to he the actual position and

velocity. The appropriate error at each step of the integration for each mass M

= +2

= + ('z

+

12

can be calculated (6) using the following formula

e(xi) = (xi(t) - x. (t) )/5.

c (zi) (zi (t) - ip (t) )/S.

(20)

These errors were monitored at all time during the integration and when

errors exceeded a specified limit, the time step was cut in half and the integra-

tion continued; conversely, if they fell below some minimum limit the time step

was doubled.

From the form of equations (14) through (19) it is evident that both position

and velocity must be known for the time (t - Lit) and (t). However, when starting

a solution or just after the time step has been changed the position and velocity

will only be known for the last time step, and therefore some other equations

must be used to generate this required data. In this particular case the Runge-

Kutta third order single step integration formulas were used.

Runge-Kutta: Equations (14) can be integrated to give

xi (t+Ot) = ki (t) + (21 + 422 + 2. ) /6

z1 (t+At) = zi(t) + (kI + 4k2 + k3 )/6

(21)

and

13

where

zi (t+Atj = zi (t) + ml + 4m2 + m3) /6

k1 = At f (xi (t) , zi (t) , Ai (t), z . (t) )

(22)

k2 = At f (xi (t) + nl/2, Z. (t) + ml/2, 1ci (t) + R,1/2, z. (t) + k1 /2)

k3 = At f (xi (t) - n2 + 2n2, zi (t) -

(t) - k1 + 2k2) ,

k1 = At g(x1(t), z

1. (t), k

1. (t), z

1. (t) )

+ 2m2, x (t) - k + 2ki 1 2

(23)

k2 = At g (xi (t) + n1/2, zi (t) + m1 /2, ki (t) + k1 /2, ii (t) + kl/2)

k3 = At g (xi (t) - n2 + 2n2, zi (t) - ml + 2m2, ki (t) - k1 + 2k2,

zi (t) - k1 + 2k2)

n1 = At ki (t)

n2 = At (X (t) + 1/2 R. l)

(24)

14

n3 = At (x (t) - Rl + 22 )

and

ml = At zi (t)

m2 = At (zi (t) + 1/2 k1)

(25)

m3 = At (zi (t) - kI + 2k2) (26)

Using the predictor-corrector set and the single step Runge-Kutta equation

for the first time step the governing equation for the lumped mass mooring line

and the anchor can be numerically integrated. In this routine, the error can

be automatically controlled by increasing or decreasing the time step At through

Eqs. (20). The general outline of the calculation scheme is illustrated in

Fig. 4.

It should be noted that higher ordered predictor-corrector schemes exist

which increase the precision of the integration. However, the accuracy of the

technique used may be well within the accuracy of the entire computations. Until.

data from physical models becomes available to prove otherwise it is felt that

this technique gives adequate results.

15

A LUMPED MASS MODEL FOR INEXTENSIBLE LINES

A tentative model for the inextensible case has been developed jointly

with the Oregon State University Sea Grant Program (9). Hamiltonian techniques

were utilized to develop the equations of motion. The explicit detailing of

the numerical procedures will not be presented here as for the inextensible

line case because this has been accomplished in Reference (9).

The model assumed the line to be perfectly inextensible. Thus the motion

of each mass is more directly influenced by the motion of each other mass than

for the extensible line case. A listing of the program is presented in the

Appendix B.

16 RESULTS OF TEST RUNS

Prior to the production runs several preliminary tests were made. The

first of these showed large dynamic loading in the segment of the line near

the anchor at about 20 seconds after the anchor was released from the "standard

catenary configuration" of Fig. 5a. This result was reported in the 1972 report

to ONR together with results where the anchor was released from the "goose neck"

catenary configuration of Fig. 5b. The second configuration showed much lower

dynamic loading. Since that time the segment lengths were made considerably

smaller and the control on the time step was improved by using the integration

method described in this report. It is now evident that the high dynamic loading

reported earlier was an artifact of the first lumped mass numerical program.

The results presented in the following section show only a small amount of

transient loading during the early stages of anchor deployment.

After this experience, each test case to be run was divided into several

time segments and the computer results were examined at the end of each time

segment. In this manner, links in a critical area along the line could be

subdivided while links in areas where the line tensions were more constant could

be consolidated. After these adjustments, the computer continued the solution

from that time until the end of the next time segment where the output would

again be examined and adjustments made in the link lengths. In this way, accurate

results could be obtained without using an excessive number of links and computer

time could be conserved.

Nylon Line

Figures 6 through 13 present the results for the 2.5" diameter nylon mooring

line for both the 15° catenary and the 60° catenary initial condition. Figures

17

12 and 13 show results for the "goose neck" initial position. Figure 6 shows

the line shape for various times during the drop for the 15° catenary. Figure

7 shows the tension contours for the 15° catenary as a function of position

and time. The most notable feature of these results is that the line has

essentially constant tension during most of the drop, and there is no snap

load during the fall or after the anchor touches bottom. Figures 8 and 9

present the same plots for the 600 catenary. Although the line tension is not

constant the tensions vary slowly with time and position due to the high hydro-

dynamic resistance forces acting on the line.

Figure 10 compares the tension for both catenaries at the anchor link for

the first 140 seconds of drop. Figure 11 compares the vertical anchor velocities

during the drop. It is interesting to note that the terminal velocity for the

12,000 lb. anchor in free fall (unrestrained by a mooring line and where drag

acts only on the anchor) is about 50 ft/sec. When the anchor is attached to a

mooring line, the velocity approaches the terminal free fall velocity during the

first few seconds of fall and then reduces considerably due to the drag on the

mooring line. Figure 11 also shows the horizontal anchor velocities. The

maximum tension during the drop was approximately equal to the anchor weight.

In general, the long run times for these solutions (see Appendix E) were

a result of the high curvature in the line near the anchor. When the anchor is

released it drops straight down which causes a "kink" in the line just behind the

anchor. Early during the drop the line motion tends to be mostly tangential

especially near the anchor, which requires that the line on the buoy side of

the "kink" must move around the "kink" at relatively large velocities. For the

lumped mass model, this means that a lump on the buoy side of the "kink" is

18

accelerated toward the "kink" and obtains a relatively large horizontal velocity.

As this lump moves around the "kink" its velocity must very rapidly change from

nearly horizontal to vertical. Because of this large acceleration the time step

is drastically reduced as a mass moves around the "kink".

The problem is compounded if the links are relatively large. As a large

heavy link is accelerated toward the "kink" it obtains a considerable amount

of momentum and it has a tendency to over-shoot the "kink" which causes jumps

in the line tension. In one particular case the horizontal anchor velocity

changed directions as a relatively large line mass overshot the "kink". This

result is shown in Figure 11 where the horizontal anchor velocity becomes positive

for a short time during the early portion of the drop.

Comparison of the output for the "standard catenary configuration" and the

"goose neck catenary configuration" of Figure 5 showed about the same results

provided sufficiently small link lengths were used. Figures 12 and 13 show the

results for the "goose neck" catenary during the first 70 seconds of the drop.

The comparison of anchor link tensions for the "goose neck catenary" and the

"standard catenary".shown in Figure 13 indicates very similar results except

that at any given time the "standard catenary configuration" has about 1000 lb.

more tension. Since the"standard catenary configuration" seemed to be a more

realistic starting configuration, and because the results are very similar, no

further runs were made using the "goose neck" catenary.

Dacron Line

Figure 14 through 19 present the results for a 1.0 "diameter dacron mooring

line for both the 15° catenary and 60° catenary initial conditions. Figure 14

shows the line shape for various times during the drop for the 15° catenary.

Figure 15 shows the tension contours for the 15° catenary as a function of position

along the line and time. The computer runs for this case were terminated after

140 seconds of drop because the results in general were not greatly different from

the 2.5" diameter nylong line presented in Figs. 6 and 7. There was no snap

and the tensions seemed to be fairly constant and approximately equal to the

anchor weight. Figures 16 and 17 present the same plots for the 600 catenary

initial condition. For this case the computer runs were terminated after 100

seconds of drop because of the general similarity to the results obtained for

the nylon line.

Figure.18 compares the tension in the anchor link for both catenary initial

conditions during the first 140 seconds of drop. Figure 19 compares the anchor

velocities during the drop and here again these results are very similar to the

results for the nylon line.

Steel Wire Rope

At the time of this writing a computer program to model inextensible steel

or chain mooring lines has just been completed. A few preliminary runs have been

made for the anchor drop problem. The results of one of these runs is presented

in Figs. 20 and 21. Figure 20 shows the line shape for various times during the

drop for the 15° catenary initial configuration of a fictitious equivalent 2.5"

diameter steel cable with a density of 350 lb/ft3. This diameter was selected

initially to compare with the 2.5" diameter nylon line results. Subsequent runs

will use more realistic diameters.

Although the program was only run for about the first 85 seconds of fall,

the results are considerably different from the previous results as would be

20

expected due to the large line weight and the inextensibility conditions. The

results show that the line falls almost as fast as the anchor, which is expected.

Figure 21 shows the tension contours in the line and the results are markedly

different with the maximum tensions occurring at the attachment point. Although

these results are preliminary, in that the link lengths should be reduced and a

more realistic case should be run, they do indicate general tendencies. The drop

was run for only 85 seconds due a limited computer budget.

21

CONCLUSIONS AND DISCUSSION

1. The lumped mass model together with an appropriate integration scheme

such as the predictor-corrector method is a useful tool for the dynamic

analysis of extensible mooring lines. It is easily adapted to almost any

line configuration and can readily handle non-linearities.

2. The results obtained for the anchor drop problem show no dynamic line snap

for the configurations that were considered. It appears that the high drag

of the long mooring line quickly disipates the kinetic energy of the anchor,

which slows the anchor to a velocity well below its free fall terminal

velocity of 50 ft/sec.

3. The elastic properties of the line do not seem to play a great role in the

line motion during anchor drop. The nylon and the dacron line showed

similar position time histories.

4. Mooring line diameter did not seem to play an important role in the

mooring line motion. Both the 2.5" line and the 1.0" line showed about the

same position time histories.

5. For an anchor drop from the 15° catenary as modelled here, the line tension

is essentially constant with respect to line position.

6. For an initial condition of a 600 catenary the tension increases more

slowly than for the 15° catenary.

7. At no time, for any run, was the line tension greater than the anchor weight

for the extensible mooring lines.

8. Time-velocity histories of the anchor show that the anchor settles onto

the bottom at a vertical velocity of about 5 ft/sec for the 2.5" nylon

rope.

22

9. Preliminary runs made for an inextensible steel line show a much different

behavior. At least for the limited runs made, no snaps loads were observed.

Further runs must be made to verify this result.

10. It is postulated that the high loads during buoy mooring implantment that have

been experienced at sea for scopes greater than one were not caused by shock,

or snap, loads in the line. Possibly, during the time of anchor fall, the

line was relatively taut and the dynamic loads from waves acting on the buoy

created the high tension forces close to the buoy.

23

REFERENCES

1. Berteaux, H. 0. and Walden, R. G., "Analysis and Experimental Evaluation

of Single Point Moored Buoy Systems", W.H.O.I. Reference No. 69-36.

2. Bulirsch, R. and Stoer, J., "Numerical Treatment of Ordinary Differential

Equations by Extrapolation Methods", Numerische Mathematik 8, 1-13 (1966).

3. Casarella, M. J. and Parsons, M.,"A Survey of Investigations on the Motion

of Cable Systems under Hydrodynamic Loading", Marine Tech. Soc. J.

Vol. 24, No. 4, pp 27-44 (Jul-Aug 1970).

4. Froidevaux, M. R. and Scholten, R. A., "Calculation of the Gravity Fall

Motion of a Mooring System", M.I.T. Instrumentation Laboratory Report.

E-2319, August, 1968.

5. Goeller, J. E. "A Theoretical and Experimental Investigation of Snap Loads

in Stranded Steel Cables", U. S. Naval Ordinance Laboratory Report NOLTR

69-215, November 26, 1969.

6. Lapidus, L. and Seinfeld, J. H., "Numerical Solution of Ordinary Differential

Equations", Academic Press 1971.

7. Nath, J. H., "Dynamics of Single Point Ocean Moorings of a Buoy--A Numerical

Model for Solution by Computer", Oregon State University, Dept. of Oceano-

graphy, ONR Progress Report, Ref. 69-10. July, 1969.

8. Nath, J. H., "Analysis of Deep Water Single Point Moorings", Colorado

State University, Technical Report CER 70-71 JHN 4, August, 1970.

9. Rupe, R., "An Inextensible Lumped Mass Dynamical Mooring Line Model",

M.S. Thesis, Oregon State University. To be completed, June 1973.

24

10. Wang, 1'. 11., "A Two-degree-of-freedom Model for the Two-dimensional

Dynamic Motion of Suspended Extensible Cable Systems". Naval Ship

Research and Development Center, Bethesda MD. Report 3663, October 1971.

25

FIGURES

26

Figure 1. The Lumped Mass Model

27

eAVi = (6i-1 + 0. )/2

Figure 2.. Free Body Diagram of a Typical Mass

Line Tension Z Velocity, 2T

Drag Force Weight, W

Figure 3. Free Body Diagram of the Anchor

At=

28

At/2

RungeKutta

=t+ot

=t +At

ReadInitial

Conditions

YES

Equations(l6) to (20)

at =At/2

Equations(9) to (11)

Predict

XP(t+ot)Equations(12) and (13).

P(t+At)xi (t +At)

ZP (t+At)

Calculate

XP(t)

Equations(9),(10), (14)

ZP (t)

Corr

Xi (t

ect

+At)

Equations(15),(16)

Zi (t +at)

Xi (t + At)

Zi (t *At)

Equation (17)

Initial Line Shape andInitial Velocities

NO

t

Figure 4. Computation Flow Chart

29

Figure Sa. Standard Catenary Configuration

Anchor

O

Figure 5b. Goose Neck Catenary Configuration

30

Attach Point 6 /0

POS/T/ON (f/)2000 4000 6000

Figure 6. Nylon Line 2.5" Dia (NVRO5)

//

9

Anchor0 400 800

1-110

r//WE (See) Anchor on Bottom

Figure 7. Tension Contours ti KIP15° Catenary Nylon Line 2.5" Dia (NVROS)

20

40

60

80

7

POSIT/ON (f /)2000 4000

Attach Point

6000

Figure 8. 60° Catenary Nylon Line 2.5" Dia. (NVRO5)

2 3 5 7 9 /0 II

200

o\° TIME (sect

J400

31

Figure 9. Tension contourstiKIP60° Catenary Nylon Line 2.5" Dia (NVROS)

60

32

I0

/50 CAT

6

NYLON LINE (NVR05)2.5" Diameter

' I I 1 1

40 80 120TIME (sec)

Figure 10. Anchor Link Tension for Nylon Line

VERTICAL COMPONENT600 CAT (HO

/50 CAT ( VERTICAL COMPONENTHORIZONTAL COMPONENT

200 400 Bottom 600 800

TIME (sec)

RIZONTAL COMPONENT

Bottom

Figure 11. Components of Anchor VelocityNylon Line 2.5" Dia. (NVRO5)

8

0

60° CAT

POS/T/ON(fl)33

2000 4000

Figure 12. 60° "Goose Neck" Catenary, Nylon Line, 2.5" Dia. (NVR06)

20 40TIME (sec)

Figure 13. Anchor Link Tension for Nylon Lines Startingfrom 600 Standard Catenary and 600 "Goose Neck Catenary".

60

34 POST/ON (ft)2000 4000 6000

Figure 14. 150 Cat Dacron Line (VR08)1.0" Dia.

8 9 9.5 9.7/ 2

0 40 80

TIME (sec)

Figure 15. Tension Contours % KIP150 Catenary Dacron Line 1.0" Dia. (NVR08)

t = 20 sec

f=50

Figure 16. 60° Catenary Dacron Line 1.0" Dia (NVR08)

Attach PointL

20

40

60

hO

80

Anchor20

POSITION (fl)2000 4000 6000

40 60 80TIME(sec)

100

35

Figure 17. Tension Contours ti KIP60° Catenary Dacron Line 1.0" L)ia. (NVR08)

0.2 0.6 /

0

36

/5°CAT

Dacron Line (NVR08)/.0 Diameter

40 80TIME (sec)

120

Figure 18. Anchor Link Tension for Dacron Line

vq)

19 Oft

15° CAT Steel Line2.5" Diameter NVRO7

40 80TIME(sec)

120

j

Figure 19. Vertical Component of Anchor VelocityDacron Line 1.0" Dia. (NVR08)

4000

POSITION (ft)4000 6000

Figure 20. 15° Catenary Steel Line 2.5" Dia. (NVR07)

129 KIP at t =0

2000

37

Figure 21. Tension Contours KIP15° Catenary Steel Line 2.5" Dia. (NVR07)

Attach Point

20

40

60

80

Anchor0 40 60

TIME (sec)

39

Appendix A

Listing for Extensible Line

PPOGRAf', LUMPCJMM )It/T I'IE/DELTAT IT TMEIT I'1EMAXCOMM 1 110 TPT/PRIttTIM, PRINTIVCall'-l )N/"'"JCM/PNCHCOO, REAOCOOCC11MC'i/FLAG/IFLAG, N2 , JFLAGIP* LA'-,J3 LAGCALL INPUTTIM: = 1.P,INtIM=0 INTIV-.3LU1CALLCALL

j= LL

C4`ILEICOUTPJTLET

1':r IF (Tt''E. G TIME" AY ) GO TO 2 `I

CALLC" LL

NO 1

NUM I,T2TII,E=Ti°"-+JELTAT

( T ;1 CALL "'UT PUT

?RI 1zIIM=°mY'JTI"1+P:INTIVLv "JtaTIvWE

GJ T 3 id'1C CDNTINIJE

IF CR'JJ'ECO1.cQ.1) CALL PUNCHCRPALL LENGTkkSLOP;:,N, FP

SUPRJLTI N" INPUT+,J 1m 111/CC,JST/PI,GC;JMM ):dACCEL/S (1J I ) 9 ARE A,0,WATP0EN ,DE NLINE ,T{1+ 1),{7;30l- 9 /ST"',AS/SCON,SEXPC J" "J43/T I' .=/OELTAT,T IMIE, TIMEMAXCJ'1MJ/PAIL/FAILSCJ'1MJr1/OTPT/PRINTIfi1, PRINTII

0 C1?",`T/0P,'41ASS,N3,STJTAL,SFINAL(") 3'1aRP0'SE; MAx,EKRlIW

.:'#V i/FNCHCon, REAJ0C0CM°1°J S A,4GLC/ THE T'J MJd/FL')C/IFLAS,P'2,JELAG

PRINT 1U .

iC a ; CRMAT (1 ti1, 5CX, 32HGI SCRETE MASS MOORING LINE MODEL)RZAr) 132,STOTAL,J_tLtNE,J,C,NMASS

RE BJ 1,'3 ,OP, WA T1?CEt,°R_tj t? 1.C7,cAILS

l 7 F^-';,rAT {F).C. 0)Iv3 PJRr1AT(2F111 .C)

REAL) 431,THETR-AG 72, SCUN,SE XP

32 t:CRmjT(E1J.3,F1C.))=P,0 S.C3,TIMEMAX,J34TAT;;" 1C 103 PR MA X, E"RF*4IN

R" AD +#:1 , PRINT IVRvPJ 4O1,PNGHC00REb0 4+ 1,READCOC

4`,1 FCRMAT (F1-). 3)PI=3.1415g27GC=? ,17AR FA=°I/4.*(0/12. 2J=11/12 .P-IINT 11+,STOTAL,0tNLINE,0,0,NM,"!SS

1v4 FC ',":T(///!4H2LINE LE'4G{ i =F6.194H FT./

DENSITY =F7.3,1'TM L8./FT `*3/21 ri-LI d OIAM-"TER ==`5.3,+F1 FT. /3131W'' DAI1PPIG CONSTANT =LIU.3,.13H L 3 SEC/FT's*2/4 1E14 OF MASS} S = 132)

Ni=N44SS+1N2=N1.+1PRINT lCr),DPWA'TR0EN,TIMEMAXj0ELTATPRINT 4:i2,ERRAAX,ERRMINPRINT 4t 5,THET

43 5, F3Rl,' AT (2 3140 CATENARY ANGLc Al SURFACE =F5.2,5H DEG. )T .-th T=THE T*PI/13.

4 2 FCRMAT(2 ,`VELOCITY ERROR BGUNDS,F8.4,3H TO,F8.4)1:;5 FO+ ' AT t 14H: WATER C PTH =F9.2, 3H FT /

1 16H 1W1AT DENSITY =F6.2,9h Lt3/FT-CU /? 1.'P.; +AX TIME =c8.2,4k SEC /

12W"TIME STEP =F5.3,41 SEC )PRINT 136,P?INTIV

1.6 R')kMT(1?"4PRINT LVFRY F5.3,4H SEC)PR1J1 1tr 8, NILS

1J3 FJC'`l^,T (2-4GLIN FAIL 'JR= LOAD _ F11.1,4H LF3S)t'Pit NT -,j :,PNCHCE)D

4 ? : r'i3 MAT tl 31=. ^U^;,'.j CC1E =FS. 1, 5X,S6HPUNCH CODE = 1. PUNCHS ALL POS AAT TI4L= TIME,1X

PR IN i Y j '+, RcAOCOO+ FR`^uTtl'41 R;A`l CCOr =F5.1,5X,57H REAL' CODE = I. READS ALL FOS A

1 J3 V -7L FO", TIM = TI MEMAX )IN t 4, 's, SCON,SEXF+

4 F; aT tlf.?_X,34 iSTP-SS-STCAIt RELATION STPESS ,E13.3.,-3H*STRAIN

C

L' ,F,.l)C S=GM=NT LENGTHSr

SL=Si0TALP14ASSS(I)=SL/2eDO ?'j I= 2,N'4ASSSt I) =SLC O NT [ IJE

(Ne1 ASS+ 1) =SL/?.RE TU RNEli)

Si; ?' )UTI i NUMINT :%;i.i lP J-i/T Im:-/Dli-I AT,T IMri, Tl,'IE:MAX

CuM.,+J/d`J"iy/XL {il),Z_L(1:1),XO(1°l1),ZD(11), XOO{11l9ZOD (131) 91 XLN(1C1),7LN(1+:1),XOt (iil),ZU.N(1C1),LRRORX(1J1),ERRORZ(1O1)

CCMI'C"4/I'SPT/OP,tNMASS,N1,STGTAL,SFINALCu '!J,d/AGv L/S (iii) , A+2.A, O, tAT20c,t, DENL INE,T (i31) , DC ""+' ad/4 L/FAILS

GGr`"Jd/)T"'T/P?I 3Tlmg L(4TIV

J

r

CC t`A )J/=) L-)fXLJL, ZLOL3(iu1),XUOLO(Ibl),Z'OOLO(I 1),XO90LO(1l1)))J_O(1.1)

G 1+','WFL:,S/IF L, N. ,JFL At.DIME SIO^i XUi)N(1-1),7OON(13ii)DIM 4 I0"; AKXi(I:,1),AK7_i(ILI),AKXZ(Imi),AKZ2(101),AKX3(IGI),AKZ3(I

Ix), )Xi(a.'1),0Z1(111),OX2(131),OZ2(1O1),OX3(1!1),QZS(131)

Oh. T ',,-_, CITY AN) °3SITION 4T TI"" T+D[LTAT

I ^_3 TI'1JLIi (ILL=t 0.C) GO TOC, C, I I = , N2'r (I GO TO 77CALL ACC LL(I,XL(I+1),XL(I),XL(I-i),ZL(I+1),7L(I),ZL(I-1),XO(I+1),

I XD(I),XC(1-1),ZO(I}1),ZD(I),ZD(I-1),XOO(I),ZJD(I))CyJ TJ 74

7;3 CALL -N)?(XL('N?),)(L(tdi),ZL(N2),ZL(N1),XD(N2),ZO(N2),S(Ni),AREA,1 XJD(P42) ,ZDD(N?))

IF(I...'D.N:.AND.ZL(N2)LT.-DP)XDCiLD(I)=C.1; (I.L0.+v?.ArdD.ZL(t2LI07f)ft 1J(I)=

-74 CCNTINUL

X

7 7tti{I)='_DO LOI+DcL7AT2ZDD{I

XL«(I)=XL.(I)+DELTAT* (XO(I)+XON(I)) /2.ZLN(I)=ZL(I)+O L T A T 4 (Z0(I)+ZDN(I))/2.

.).

C

.O'((I)=-XON(I)ER P02Z(I)=-ZON(I)

iL` CANT LNUE

CO . 0TO'Z

X'i=X.'4(2) ;ZM=ZLN(1) iXOM=XDN(1) SZOM=ZCN(1)C) 2 J I2N21 F (I.rO. 2) GO TO 75CMLL ."C-L(T,XLN(I+1),XL'J(I),XM ,7LN(I+1),ZLN(I),ZM

XCJ"J(I+1),XON(I),XC"7 ,ZCN(:+i),ZQN(I),ZDM ,DDX,DDZ)G O T ) 7 F

75 C I L L NX1XLN(N2XLfi((N1) 9ZLN(N2),ZL N(t i),XDN(N2),ZDN(N2),S(N1),I ARE ,JDX,JDZ)

'6 C2N,TT;JU''r

f,

X'=Y'-4(I) i7M=1-7LN(I) t'XDM=XDN(I) 3ZCM=ZOON(I)X'P.'t T -)=X)(_T)+C.LTAT*(XJC) (I)+')JX)/2.7J1,.(I)=7 (I)+O=LTAT*(700(1)+;)DZ)/2

X_tztl)=XL(I)+J.LTAT'(x0(I)+XJN(I))!?.ZLt'(C)=L (T)+%?=LTAT`(ZJ(L)+ZJPa(I)1/2.

X(I(cam;O,X{I)+X'7rd(I))/(5.*XON 1)r x :t Z(T)=('RR;);;Z{I)+ZJE.I))/(5.*ZONI1 ;,J2 7L t,2 LT -OP) z

2 41*4JZL t,,2 LI -0P)ERROKZ(N2)=L.

IF{( 3S(YI)).LT..1) X=10.IF IS XDWI)) LT.1. }F,<RX=1.IF ( A 3S{ -tORX(I)).CT.-RX .U+2.A3S( iRURZ(I) ).GT,ERRZ )1+3,2Q01L DELI AT=FiCLTAT/?.IFLC '`s=l,°RI!T 9. 3,U2L7AT,TIM,I,c!(JRX(I),ERRORZ(I)

F0k"'AT (132X ) )

IF ( :LTAT.LT.1.7 PTT 9YAP FUR T (2

STOP15E 3'- TO 12,.. C')NTI )U:.

155, i5

GE LTAT LT uc:l

)) 3,., I-1,v42XLCLJ(1) =YL(I)ZLCL)(T.) _7L(I)X,OL7(I) =XD(I)Z') CL. )(I) =7O(I)X_ U)=XLN(I)7r (I)=ZUN(I)X "(I)=XfN(I)ZJ (1 70N, (I)I=(ti3` (= ;tJ (T))LTIN.AND. A3S(ERFOPZ(I))LT.E R',4 IN)NOUNTE)2

O: ' 'CNTINUEIf- (^j,,JUNT»\.;0.fda)

35: 3uLTAT TAT2,IFLA3=(?)') + 7 I 1,N2XOLJ(I) XLN(I)ZLOLJ(I) =ZLN(I)X:)0L%) (I) =XON(I)

437 Z)OL J(I) =Z'DN(I)PRINT 91Ca,UELTAT,TIME

P1ta FORMAT (21HLOtLT ^,T I #CRE.ASLO TO F7.4,3H AT,F8.3,4H SEC)3O C3NTI'T E

Gi TO 2'; -.

1 : C C C NIT INUE

U,

TO

R'UNGE-KIJT1A (THIRD ORDER)

JJ 1 L 1=202Iw (I. r. k') GO TO I

-A C,r' L(1,XL( LOJ(II1),XLOLO(I),XLCLD(I-i),ZLOLD(I+i),ZLOLD(I),ZL`a(I-1) ,X` OLJ (I+1), XC0 LJ(I),X00LO(I-i),ZDOLD(I+1),7DOLD (I),ZOOLD(

1I-S) ,XJ7;J(I) ,Z)CL'3(I) ?GJ Ti) 11J,,

.L1 i -,ALL O (XLOL')( J2),XLOLD(N1),ZLOLD(N2),ZLOLO(Ni),XOCLD(N2),ZOOLO(1N ), S(,1),4KLA ,X)OOLO('42),7DDOLD(N2))

11'C C++PtiTI'it1

'< X1 (T)= LT -"T *K 3,)C L I)A<71 (I)=JMLTAT *ZDDCLO(I)i)x1(I)=)LTAT*XDOLC(I)O?1(I)=rLT T4ZrOLC,(I)Y)t! I=X--3LI+AKXL(I)tZ =77))Li1(I)+AKZi (I) / 2.XL N:(T)=XLULD(I)+OXI I)/2.ZL1ZL.AL`l(1 +0 Z1I)12.

I .) O TO 12 5(I,XidT+2XLNIXLhI-17LN(i+i)ZLN IZLNI-1)XON

T+1IXr)"tI-1L0ivI+1),ZD'd(I)2DN I- 1)XJD4 (1)9 ZDD fi)G C

T'1 12'10C1LL L-CNO?(XL`)(42),XLN(Ni),ZLti(N2),Z,-N(N1),XON(N2),ZON(N2),S(Ni),

12 rb T'TI U

1N( L)

44"X2 (I)= )c'LTAT*XO+i) )(I)AKZr (T)=JLTAT+ZD)N 1)OX 1)=D LTAT X004' I

2I)=DLTAT*7 DNIy2i, 3 NT INU:-

)(11 N

ia I=2,N21)=X;JOL0 1AKXI(1)+2.*,KX2(I)

Z3'I=73JL I IAKZ1 (I+2.'AKZ2 (IXLt,(I)=XLOL0(I)-(X1(I)+2.+DX2(I)

(I)=7LOLD(I)-QZ1(I)+2.4QZ2(I)13C CNTIu ;-

G" 1jiJ u i=2, N2I (I ,.:D. 'vc) CO TO 1---.15-

A CC L(I,XLN(1+1),XLi`a(I),XLN(I-1),ZLN(I+1)ZLN(I) 71 N(I-1)X'JNi1(T+13.X n(I),XON(I-1),ZDJI+1ZDN(IZ0 t4(I-1)X0ON(I),Z00N(I)G,j Ta 13,!6

1'C-- CALL 7-ND2(XLN(CSC),x'L"((i,41),ZL14(v2),ZLN(N1),X33 (N2),ZDN(N2),S( I),"R?=A,XDOIJ(W) 9ZION(N2) )

13: CJNTINUEI)=.tLTAT* X331v(I)

AKA (I)= =1F'LTAT+ZOcD ( 1)Q (1) TAT* XO ON (I)0'3(I) =)ELTAT*ZON (I)

14u i C) NT IN UEDO 1 U I=2, N2XJ(I)=XDCL O(I)+(AKX1(I)+,,.*AKX2(I)+AKXv(I))/6.t)ZJ (T)=Z3GL!n(I)+(AKZ1 (I)+4.#AKZ2(1)+AKZZ(I))/6.s0XL (I)=X,L3L.D(I)+(DX1(I)+4o*QX2(I)+OX3(I))/6.(3ZL(I)=ZLOLD(I)+(QZ1(I)+QZ2(I)*4.+QZ3(I))/6.C

1-3 ') CO SIT INUE117 LAtJ- i

2Jj0 CONTINUE03 73j I =2, N1CALL TENSION(I,XL(I+1),XL(I),XL(I-i),ZL(I+1),ZL(I),ZL(I-1))

38tU C3NTTNUL=no 4,). I=1,NiIF (T(I).GT.FAILS)41394G,

41C- CALL OUTPUTJJ 1-1: Y=RU OT (1 . )

4 (, . (:C NT INUEkE, TURN:jr)

1i4L/,,':LTAT IT 1M1-)TII-.MAX3,)+ )^alIS ( .13, A At Q, ,WAT OE1 9OEN INE T 1)D

);sfr,}CTXJP*44acSrfJ STOTALISF1NA.)"1')w'If ( ?L( ul),XO(l i3,Zt1.1),XJ {11),70f t1 1?s)tw(l ),_Zon(i,-' RCPX(1,3I),ERRORZ(1

L(1T

(2

SJmr,J!1TINE ACCrL(i,XP,X,XM,ZP,Z,ZM,XDP,XD,XDM,ZDP,ZO,ZOM,XDD,ZOD

C!°`°9 j/AiEL/S (1..,I),A,,,D,ATR'3,ZNLI?`lE,Tt1J1),dCr",!'l1 ft /I PT/DP,NMAcS,^N2,STOTAL,SFINALCJ 3'J/CJi T/PI,t C

C31 a);II/TI'?E/fELTAT, TI ,TIMEMAXCJM'9ON/OTPT/QP.INTIh', Pr;INTIJT t`Tw=,T"42((ZP-L), (XP-X) )T1ETA^A=aTZ-7M), (X-XI))

T J'J r OZCE O"! MASS I

°F '- ;T((x-XM)"*2+(Z-ZM)''2)/S(I-1)-1.P=S`)T((XP-X)4' 2+(ZP-Z)**2)/S(I)-1.

cr O*: X= IRS A*( STRASS (EP) *COS (THETA) -STR ASS (E PM) *COS ( THETAM) )SF OR'Jt Z=AREA4 (STPASS (cry) *SIN (TH ETA)-STRASS (EPM)' SIN (THEY AM) )

WA TES ;JA MPI!VC FORCES ON MASS ITH AV= (TM ETA+TH.TAM) / 2.

C

C '1. OLAD V7 ARE LOCAL VELOCITIES

vV X =,+ 4

V A°_J=(X'1-''IX)*COS(TH1V)+(ZJ-VZ)*SIN(THAV)V'40R A=-( )(0-M *SII(THAV)+(ZD-VZ)*(30S(1±AV)

E'(I.4(:.2)SL=S(I-1)fS(I)12.(1.0. 1)SL=S(I-1)!c+S(I)

A <FT=P.1+ 34 3LA_ L

;:T=1. +`« JCN=14O AGT=-;, T*WAT DFN*t ,ET*V T ANG*A3S (VTAU(;) / (2.4 3C)1at,' d=-,^,'" AT ,Jhl`FKe,V'Iu "k (VNOR14)/(2. C)

V-LT `J" L IAS:; CO CT ION

S SSL4ID _NL1'{ 4 .A /V0A 1AJWh T FZO_N* aK:I .CL /Gc : -f= i(

.

;CT :G= ((SFO }`CJS(1dA )+1 ^!(T"iH }+r;y C.T)I(S'ASS+CIT AMASS)

A"CN O+M= (-- (SFO 2C(:X ) SIN 1AV}.

C0S(TIaAV)± JZ4 6 `3)11S'dACIkx NMA S

FORCE Z

(SFORC E Z

Y 0= ;CTA,I=*!:OS (T-±CJ)-4CCNO M*Sl t(THAV)Z,JU= ACC, T01G4Sirk(Tt-AV)+4CCNORM* COS(THAV)

E D

-WT)*

-WT) *

(

SJ UTIN' LNU2(X,X",7_,Z`,XO,ZO,S,A ,XOO,ZOO)/FL.;GfiFL4G,ft2,JFLAG

C!MJ4/INPT/OpR AL ;

I ; : GO TO 95RhrI: iT=12o;c.ViL='4.r" =. i

W4.R 4i 2.P 4 2 2

^`vt IG HT-WO ids VOLnl 7.11 532,E

= E :,: 1G.1 1 /G+CI* VOL'`R#GA=rn#ARE A*K3O/(2.* 1)JF LA'a=1

5 ,0 "ITI,AUUIP(Z.L.T*-)F)53,S

55:, X i=Z1=XOO=7OOU= J.n=TUBA

6 =S )T ((r-XM)'2+(Z-Z1'?) 15-2.T=AR*ST ASS(EP)T E T A T AN2 (Z- ZM, X-XM) +PIF{-T*COS (THETA)FZ=T*SIN(THETA)-W9V=S`) T(XO**2+Zt9**2)XOO=EX/M-A*XD'"VZOO?=F'Z/M-A*'ZO * V

E TUI - !

7N, r)

FUMCTIO4 S1RASS(X)cc 1" 3 / TEAS/ScON,ScXP

s1 N STRAIN X, FINO STRESSIF{X.LT. .?1 2i'ST RA SS=;.

?. TU RNCC t, TIt;U7ST+y 3S-)CCia*X *S;XP

J T y cT t`.i I N,_,T =( x/-C0H) 44 (, * /SIX )

-1 7

T Y ?U7Tc IN, T

r'''AT(1X9 21-+ HAS 9c:.N RUPTUR O)Tt

F" ?UTI"J_ ENOICtiMM'1,4/i'I: /DLLTAT,TIMk:,TIME-MAXCl''!'N/'L1'^; /XL {1L`1),ZL (1u1),XD(131),Z(?(1v1),XtD{it31) ,?7D {131),

r,b(,.I :./."JIXLJLD {1CT1) ,ZL3L J (iv1, Xi)DLD (1:.1), Z3DOL0 (iu1),XDJDL(3(111)OO2 J( :)

T 3) ' -" Y O'+:1ITiONS xT END i9 (1)_... T 7 Ljd(I)=1a iXON(1)=2 4 Z)N(1)=uI

CILI(!) =XJOL`) (1)-ZLDL') =Z)OLD(1)=vL0-11 N

L r JUTI $ LENGTHIM )'/NNj 4I/XL (1':1) , ZL (1J1) ,XD(131) ,ZC)(1Q1) , XCO(1o1) ,ZOD (131),

1 <L(1;: ), ,f (11XD'4(11),ZOU(11E',90RX131),ERROR Z(131)

i',=T ',Nm4Ss, tkl a STOTAL, St= INAL

'T PA IN-1 -)

n3 1J

GTH OF CABLE

S;=I`JIL =,S;IN AL#S{)RT((X,.(I)-XL(I-1))*42+(Z.(I)-ZL(1-1))**2)

'DINT 1%1,S I i

IJ1 FO AT{//1 ?3,2}X,1: L1N LENGTH =,F8.1):

R, I JRt4d" N D

X

_N,

=,F8.1)

SUFRJUTINE OUTPUTCQM%I )N/T IMF/DELTAT, T IMt, T IMEMAXu.:`1M )r/ti1I/XLt11), ZL(11),Xpt11),ZD41fl1), XDQti{l1} ,ZD t171),

1 (l'fl «),?Li(lrl),x3tth1),ZIlPtl31),tRORXti1,ERf20R7tii,.C4mj',r/AC,;,;:-L/S (11), AREA,Q,WATRDENIOEt UNFIT (1C1)t:,O 1=i )v/Ivi)T/DP gNMASS,NI, STOTAL,SFTNAL--RI1iT iZI ,TIMEFOR'& T (1M1,//,7HOTIME =F6.2)P? Ip, T ii i

101 M"CP""'AT (/ /// 1H j , 5X, FH LINK 96H MASS , 6X,13H COORDINATES , 1OX, 7HTENS1ICX4 1,"iiVz`L;;CITi=S,14X,6FRR()RS)RIr,T

,7X,3HNO.,3)(,w,iN:U.,9X,1HX,9X,1HZ,25X,1HX,1OX,1HZ,BX,1 4HX )OT, iX, 4HZDOT)

PRINT 1! 3, (I,X-(I),ZL(I),XO(I),Z')(I),ERRORX(I),ERRO Z(I) ,I,T(I),1 I=i,'a1)

I:J7 c-JR'41T(13X,I3,2X,2(1X,F).2),17Y',2(1X,F1J.+),2(2X,E1o.3)/7X,I3,31X,1 E11.4)

2 =141uj t"PI"T 1,-',r,2,XL(N2),7L(hd_'),X')(" 2),ZO(td2),ERRORX(N2),ERRORZ(N2)RE T'J i"3EN 0

SU9'JUTi, CA?L ICOJ11Jr /SA.*JGL2/TH T:;CM4CIWOTPT/P INTI*', P'INTIV

M HLO0,REAJCOOC,J J I L T I TIM ' I

('u r 4ST/PI GOAR- a, 0,WATPO:Nti,O NLINFT (101)0

CC t^ NP// OP, fym 4 SS, N! t SP OTAL t S Fl N AL"4 /lJ3I1XL{iul),7,L L1)7X30d1"1i),ZJO(1G1,

<." (.I'a),ZL,a(11i),XO(iu1),ZCN(1:i),EFRC-X(1:1),ERRCkZ(1J1)"Mi UOL-/XLCLr (iLi),ZLJLJ(iLi)XDCLO(!Ll),ZJOL0(1'.ji)X)DOLO(IJI)

' ',.)3.O(i;1)" 1+1

IF (<2AOCUO.EO.1.) co TO 5UCW' - = ( J-NL 1' -WTR=OEN) A A

TIN!! Iv (Tr T) /COS(THET)31.=wfT)TAL

L 1 , 3SL /(2.`T4NO)

FL=T/WL*SINMI (TANG)

c 5 J=i,N2I;: (J.NE. 1) SL=SL+S U-1)x-(J)=FL/Z.+TO/wL*:INHI(SL* L/TO-SINH(WL*FL/(2.*TO)Z L (J) =T0/V L* (COSH (WWL /TO* (XL (J)-FL/2.))-COSH(WL*FL/ (2.*TO)) )T2=TJ*CJS14(WL/TO* (XL (J)-FL/Z.)).°,-,2=ST ;'A IN (T2/ AREA )IF (J.;:0. i) GO TO acsT(J-1)=(Ti+T2)/Z.O--:LTAL= (-.Pl+EP2) *S (J-1) /2.AN (,= IT 4N 2 (Z:. (,J) -ZL (J-1) , XL (J) -XL (J-1) )'(L (J)=XL (J)+Oc_LTAL*COS(ANG)

=ZL (J)+DcLTAL*SIN(ANG)SL=EL+ )CLTLOO I i T I1' L1 :V

T1=T?

Z7OLI(J) _jOxOcIL J(J) = r.7) (J)=f:.XD(J)=C.)(LN(J)=(L(J) IEZLN(J)=ZL(J)XL.fL)(J) =XL (J)ZLt)L3(J) =ZL(J)X') (1)=k . 170 N(J)=CE, FFI'RY (J)=L. 9FRRCRZ(J).

237 C3PNT INUE2"2 CONTINUE

RE TIJRN59" CCNTINUF

RFAO ,:):J,JELTAT,TIt IFP-IrTIM=TI4+PRINTIOno 63) 1=1,N2"hn :: ,XL(I),ZL(I},XJ(I),ZO(I}T(T)='.XL1,(I)=XL(1) 'ZL;J(I)=ZL(I})ON(I)=XJ(I) I70N(I)=Z3(I)YLOL )( 1) =XL (I)7LCL)(I)=7(I)

= X : ) )

7 iL)(1}=0(I)

R .O TTM 'lAXFOPM T2: _,.-)

0 1 FOc.., AT (4FI C, 2E12.5)R TUEND

SI(I ,XP ,X,X ,ZP,Z,Z41)I1i_ Q, T D£ta9 C Ni INE,T{131

IF (I. "c , 2) ;U TC° 1,

4''S(E P') F. tL ru J { J 1 i .....

T ( I) S T-R I'SS ( J-) *ARE Ar "f { _) t

(-X))

x) + (-X) )

59

Appendix B

Listing for Inextensible Line

FR O(= ?AM CARL: 1OI""F SIC?"d T44ETA(3 ), TO(3J),THLTA2(u T0 (3G),TOO(3t;To02(3.i},TOI

C (?C) ,THETA I (:, )DIPZ vSION Tlr)I(3Z)OIME,SION Vr_kROP(3C) , P`4kOk(3C)CO M1111 )N/ANC ASS/A(JC} ORCt) )N/POS/X(3 ),Y(3C)/T-zN/TEN(30)/VELIVX(3C),t/Y(33)/B3S/Xf,YQ,XQ,Y

r z), XD ), YO fl, Cl`-NS ITY, YCCCMHCw/3')FAG2/oRAG(?C)/G/GRAV(3C)C) ""-";/T/TT E,OFLTAT,OELTATNX.=' ThXO-Y("=XOC=YOC= ._A=t r,TI'"LCUUT,TP2ItJT,CI,CN,c;T,KORRECT

5 FCCMAT (FE.2,1X,F5.2,iX,FS.t+,tX,F4.2,1X,F5.4, 1X,I2)R=i»n 6,`)ELTAT,tUP,FLOW

6 FCPMAT (FE.F+, lX,F6.4, IX,Fli.9)CALL INPUT('d,OIA,OfN SITY,XC,YL,TOTAL,ANCHOR)CALL ADDMASS(N,OIA,CI)CALL TN( , UTA, CN,CT, A iCHt e:)F?I',.T 1-', ;3.,OENSITY,TOTaL,N,gCH0R,X:,Y( ,N

L1 E 1)IA= ,F4.2,/,15H LINE DENSITY= ,F5.1,/,14H LINCE LE IGTH= F4.51/114H ANCHOR MASS= ,F7.'=9/97H SPAN= ,F5.2,/,14H WACT,:P :Th= ,FS. ,/,18H NO. OF SEGMENTS= ,12)

PRI"'T 1 ,I1M CU ,TF ,INT,CI,CN,CT,KGRRECTTIMECtJT= F6.29/,i6H PRINT INCFEMENT= ,F5.2,/,1SH ADDED

C "Ass C OER= ,F4.2,/, l'H ORAC COEF,/,5-X, R,HNCRMAL= ,F4.2,/,5X,122HTANCG;hTIAL= NC OF CC ECTIONS= ,12)

P IC=T 1 ,,rUP, E LOWto C r,T t1 # tPOR LI'1ITS,/,5Y,7 R PEr,= ,F7.4, /,5X,7HLOWEk ,.F12.9,/

C////)CALL INITIAL (N,X",THETAI,TO1)CALL ANGLE (N,THETii)CALL COORlt9,XQ,YQ,YO,YO,TDI)FP It T 20

F:;

,P"..1+,2;

2 FFT(32X,1.$HINTTIAL. CDc$DZTIUN5',II/,1CX,1GHLINK MASS,9X,1HX,IjX,Ci"Y, 1X 97HT6'NSICNy 7X, 5HTHt.I 93X1, 91-1THETA 00T,/I)

= 1 ./ 141 -9>> T

(I)=T-i-'T ? ITTJc(I)=T:I(I)* A

r 's,=,I.T` I),TyETA2(I),TD2{I).I,X(I),Y(I)I;AT(tiyiT;,?uXyL1.+,k,F7.2_,3X,F515X912,5X,F7.2,1X,F7.2)

4t GC,TIhUPRII'T 5,)cLTAT

5aa c'C ;w1T(/III/,3. DELTA T ,Fo.Y,ll1I1)TII ='i,H_C<=k .

1_T ITl3=j;- T AT

FLA =

!ri LP:=..KC. .K=IF ( K 'z LA3.E 0.3) CG TO 62T;- (Y (4). LT.vr rO TO 52

CC Ti 1' Eb3 CALL "20TTDH Ni, TO ,TIwE}

KFLAG 1NF LA

62 a 6C TD 7aC:L.. CALL T4;'.TA,TD)CALL CL VE (',I ,igKC,hz2'CTGALL 01C (,N )r,LTA T,TDI9Tf!,T,jT9, i=)0,TifTA2fTD )

T-ftT 1.(I)=T' TA(I)

A E( J F

TOi(I)=TC(i)TJC4 (I)=r"OO(I)

61 C tNTI\!ECALL -,ALL (N,TH:TA2,TC2)CALL S(lLVr(4,T002tK0RRECT)CALL CC'R .CT(N,OF-LTA T,THETA1,T01,TOO1,T002,THETA,TO)CC c5 I=1,NVE' OR(I)=(TO2(I)-1O(I))/(S. TO(I))rFO(I)=(T9E:TA2(I}'-THtTA(I)}l(TiETA{I)5.1

65 CC V I",tip,?? 12 I=1,td

(,lE2n1R(I) ). w.Eup) GC TO 112G E E 3 0 i GO TO 138

PRINT 1; 5,'ELTAT,TI"1E,IF0R'A AT (174 DELTA T t#FLO TO ,F9.7,1C:H AT TIME= ,F11.7,i7H BECAUSE 0

CF LIv:K , 12)Go T) 11:'

108 ra:LTAFN=O'_' LTAT/(': .TIM! =T .1m _-')F.LT aTTOOH,.' ;K=TC'i`CK-}LL TAt01 LT AT=0ELTATNP +I"1T 10,", OFLTAT, TIME

1 J FO;'-'ST (??H ':)ELTn T OTC EASc3 TO 1 3 1 H AT TIME= q F11.7)Krf.;. =K;r4 CK+'1IF (u `=^Ks c_. W) CALL EXITJF Lj i=tGO T 1 EC

112 C:)t,T Ir,0 ECO it I=1,NIF (t ?S (d rte= n (1)) .,-LGw? GO TO 1iWOE.LTATN=JILTA1 *2.PRINT 3O,DELT1TN,TIME

60 FOP1"IT (22H DELTA T INC:2 ASEO TO ,F9.7,1 H AT TIME= ,Fi1.7)

7=)

Kwt,FO'=KCHCK+IIF (K0Hv:CK.Gr.3) CILL EXITJFLA ,='G C T ' 11C., I',T I1 uU,n^0 h,T INN17uUCL L ' (h,'? LTT,TH-TLI,TC1,TOc1,Tl#ETA,TO,KQ RECT)JWt.A=1

I I F TI'°= = T I"E+CrLTATT_ ,+'.=TCH -'CK+`,LLTT

I (T : K.f TT''ItT) Gv ToGC T r 6t'

13* 'CH CK=TCMICK-TPRI(T17G 77'.r rI,140 FCR''IT(////,7r TIME= ,F11.7,////, V.X,1LH INK MASS ,6X,11HCOOROINAT

C ,1 )x, }/TM LCCITY,r7X,22'T1T TH,-TA,7X,7HTENSION,13X,5HEPROR,//,2C7X,1,iX,5X,:.HY, gX,1HX,7X,1hY,15X,?HOCT,2;^X,5HTHEETA,4X,9HTHETA OOT,/

CALL At GLascw, T4ETA)GALL COORO('1N,XC',Yf ,X ,YO,TO)CnLL TE NSIOPlf ,YOJ,YOO,T #ETA,TO,TOO,OEFNSITY,XC,YO,YO)A=1}i ./3.141Fc;

T'1 T 2, (I )=THETA (I) '*AI C (1) =T C (I) *A

F;'IJT lc;" =,I,TI+=T42(I)sT 2(I),TEN(I)PEQRGR(I),VERRQR(I),I,X(I),Y(IC), /X(I)t JY(I)

1`,1 - aTt11X,22,45X,F6.212XIFC.214X,E22..4t4XIF4.513XyF11.81/117X,I2t4x IF 7.21 2X IF7.2s4XIF6.2? 2X9 F6.29/)

161? CCV T I \Ut:P INT

155 F' . ^ T t!/!/1)TF t ' 'L4G.E1).1) GO TO 63IF (TIFF. .TIM7-CUT) GO TO 170G'O T') E0

171 CC".TI rUFC4 Lt EXI TF N F,

ctjT Il,::,I!T(N9DIA,JttiTIYsX:9y11,9TOTAL 9ANCH R)i

T / L ( 3 2 ) / SS/ - ) / J-Kr N )!TOTMA S lTM(3 )y NcH0 0141D hvS17YxTCTALX1'9Y

Fig w , i ( I L k X j i f q ! )C 4 i ` Y T i f 1, X q P c > o 6`9 1X i F 5' J * C g l x 9;:c,;* C-)

( i ? j .. (I) C (I )

30 CCTINL

t" 1=t-

5 CCt,TIfyUFM ( ) = `s L (N) +4t C#_ORTV M) ="1( \)CC 41 I= 2, N=:+ *I

T(K)=T'1(K+1)+ (K)4 1 CC NT I'vUE

tr-: TURlEND

2

SL9 -1,ITIN AODMASS( ,CIA,CI)-,EAL L

CCM')'i/L L''+GTH/Lt37)IAQM4S/AM(3C)CI'i4.`.1+15g t IA/24.) 'ZPC 11 I=i, NAM (1)=A*L(1)

lfl CC KTI'YU:RE TIJI RN

T T \ Y I' r 1 <'X 1 ' j 4{ C T j 4 tom, J ri C1 h)

Iy','.fysJ'D"! )/cC'-p/JT ( ..) ,D`df?

i .` 1 159I./.L.A = ]E7N*f TA/12.

ON (T)=A LCT)T {1)=34LtI)CC t,T Nil:-'

* 1 rC`'f` 12. 43u = a NCHC /fL

,'tit':+.,.?=..,3r`13tiC!{:';l t+.'3e,t43JC67j )''0' .s/3+ C"'+_'Q=wasa 15g2Y.5` `+f1iH{3+(**2

=. i 'J-` -d*NANCHO / 2.R IE T

7N 0

SUP lUTI Nc IE'«dTIAE(tt,X.,,THETA,TD)`JSION Tr)4;TA(30)1T)(33TX(3G)9TYL,M

E GT4/L(30MASS/M(3j/TEN/tL-N3r=

4/L.NCTO'

iMSMI-S',)'1

QC i : I=1, NiTMASS=TMaSS+M(I)TC(I)=.4

1J CJt;TINUETO W) =u.TY (I)=.5*Tr ASSTX (PH) =TY tN)CAM"' A=.2 *TMASS

TX'S=TX(;))TYS=TY(N)

22 TX (i) =TY (11)TY(i)=TMA -TY(1)V !(1)=S CRT(TX(1)**,'+TY(1)**2)A=TY(1)/TX(1)THET(1) =ATAN(A)cX L(1) *CCS (THETA(:.) )E Y =_(1) *SIN (THETA(1) )CC f) I=2,MTY (i)=Tv(I-1)-,A(I-i)Tx(1)=TX(I-1)T: -(I)=SrRT(T)<(I)**2+TY(I)**2)A=TY(I)/TX(I)TtIETA(I)=ATAN(A)EX +==X*L(I)*t'7SCttETA{I))EYO=EYO+L(I)*SIN(TFETA(I))CC NT IN U;7

4

Cr0 L T F P YC.) : TO *if7 Ate;,",

( COs GC) Tu o.E xt _E xy

c VC EYE;T)S=TX (N)TYS=-TY(*,)

TK(=TxS+G A4(X(-=xIS)/S v

T Y (#`j) =TyS+Got M Y A * EY(S1SAVET O

C NT T',,'tl

q - 1 ?6'

CC 7) T= 1,N(T Ta(IGT.3.14!5965) T TA(I)=T H ET4tI?-470 CONTINUE

K`TURNEN 0

5'n

A=?.?-15

I

S U P =J T I E CALL (`4, THETA, TD)CCMtICN/C,S/XO,YO,XD,YC,X3O,Y+;J, OEF4SITY,YCCI4SI,Th THETA(33),Tr)(3CALL -A PISLES (N, THETA)CALL C0OR9(t4,XC,YC),X`),YJ,ID)COLL. 1tSS(4,Y'?)CALL R8 si.(N)CALL XYMASSI(N,X301YOc,YJ)CALL z'hVITY(u,Y(',LE14SIT Y)CALL )RAGA (""THETA)CALL SET U (J,TO)R: TL Mfrrt,C3

, - )UT I ". . W, L' S (Pv, R T H E T J- )",JCIvrNS ION :ITH TA(3( )

CCm,'-"I/AN.,/ (?T),C (ZZ)/OIF FANZ/32(3j,3L),C2CL 1 ? I=:,"S1 (I)= ST N(:T&,E TTA(TI

1)i)C Cr ) = 1 J T W i r 1 I I1) 1 J J=i,11CIF17= T!j , = (I)-;JHETA(J)S "I (I J) TN0.;, )

(I,J)=CJ (CIFF)CC NTINU2T)1 1

SURE .1U.TINtr COOROtt',XO,Y),XO,Y'J,TO)CI-q"} NSIJK T)(3 :)QEAL L.,.MM ld/L^'dGTf/Lt31)/A°1G/S1(:v),G1(?v)/CQS/X(?J) tY(3£)IVEL/VX(3(I3,v

CY(3L)X(1)=L(1)*C1(i)+XV(j)=_(i)4S1(1)+Y^VX (i) =X9 -L (1)4T,,3(1)*S1(1)Vy (1)=Y+L (1) * TC(1)*C1 (1))L 1' I=,, t.X(I)=y(T-1)+L(I} C1(I)Y(I)=Y(I-1) +L(I)4S1(I)VYI)=VXII LITJI} S1 I}VY I1 =VY (T-1? L (I TC(IC1(I)CC NT I. UL

Ttt'4r

SUR 0'1TT HASS(N,v )'R: AL "' ,L 9 1 !SS1,'1ASS2

"NITCTMAS/T'M(3rA34 AS,>1(3 0 IFF4NG/S2(3C,3C2(30,3Vo),LENG/L( p'SIt3a)Ihl*ASIMASS 1(3C,3P),NASS2(3L,3C)v

0'.. r*3 J=I, NiK = I

1 `-1 -A = J,

Ir I. ;T. J) U T0 IC

"1(i,J)=T'1(f) 3, IGmA*AJC2(I,J)1 J)TM(K)2 (i,J)TCTALM:=..TOTAL i2= .Cr 2) LI=K,MTOTA,.'11=TCTALMI+A (LI C'' (19LI.)#02 (J,Ll

23 CC It T Iv UEI: (J.Z). GO TO 4C'{I=J+iI (I. T. J) <1=ICC ?" L1=KI

Pr'TOTu;.'12=TJTALM2+A1(L1)+C2(I,L1.)*S2(J,L,)CC NT Tn1JE

40 J ) M19,1 SCI,J)= a7 (I,J)+TOTALM2

SS(I,J)="ASS2(I,J) `L(J)CCS:71°4UERETURN° 7

"4)

N

Sum }UTINE fRAGI (N)OI 4S ION Vt(3w),VT(3i )CO ","IONIVELI VX(3CVY(7' )/ANG/S1(3 ) ,Ci(30/0IFFA4GIS2(IVp30),C2(3CC,'0 /COEF/DT (3,3)-,Dt(3,:)/CURRENT/ VC (3 O),DRAG2,DRAG(3J)VN (1)=.5*((VC(1)-VX(1))*SI(1)+VY(1)*Ci(1))VT ( 3) = C. JwDC 1 _ I= 2, NJ=I-tV",(I)=.5*(VC(II+dC(J)-VX(I)-VX(J))*Sl(I)+.5* (VY(i)+VY(J))*Cl(I)VT (I)=.5*(VC(I)+V;"(,J)-VX(I)-VX(J)) *CI(I)-.5* (VY(I)+VY(9))*S1(I)

la CC NT I NUESC 71 I= 1, Nrr(I)=5.K(;C 'r W v

CC "'l J= I, N(I)="IRCC( I-IN0*Vd(JA75(V4 ?)C2IA-DT(J)*VT(J)A8S(VT((J))'?(I,J)

2(' C^TIUECC NT I'.tJFRwTU

'N C

1r

S4 esJTIf E xX1-1ASSI ( 0 1 , X ' J ',YcL

CC .'4/1 C r AyIT ( 3 A AS4 ?°) L Z.NGTH/ , (3 J iASSI+ 3 XYIXYMASSC>S :),G1(3;)/JI=FahG/S2 ( i,3r),C2dti;,3)no 2 1 I= 1, tiXYMAIS(I),-

Y" I='`v-5-'(1 v(J) `c: (J)*G2(I,J)*)(CC,-Am (J)+CI(J)*C2(I,J)*YCCD

C NT 1 U

XY ya4(I)=X"S"ASS (I)+;,X D*S1(I)-Y ')*C1(I)TM(I)CC NT INUER7 TU-RN

,';

SU9 J()TINE (-;,RAVITY(N,Y:3,JEN4SITY)R:-' AL v03 i 4/1 ;SS/MC:.vm"N/LIN+;,S1(3 ),C1(?-)/TJTr"ASJT<1t) PGS/X (?0) ty(30)/1GRA/( )WCEN =o4.G= 32.24 (DE SITY-WCEN) /!DENSITYcc I ; I=1,N

#j G(:lA v([)=G T9(I)'Cj(T)10 Cl. t, T IaNUE

IF (M(N).LT.IUC'zi0O.) GO '0 63cC 2 1 1= 1, N;

20G AV(I)=G' La tI) _'4(P) *C1(1)CGr T I"lU`.

I=1,11IF(Y{I).4T.Y_) GO TC 5u,DC 4, J=1,I

4CaKIaV(J)=CCC NT I'4U,--

HV(J)-1(I)Cj(J)'G5 CC J,T IMJEbJ CC NT INUE

R:IUUR,EN t0

TI`" TA)CI " I0 ( , 3 - '

Ct , 1 :Cat V; . VX f 3 L VY 3=) / Cf RAG 3IF Vx tO P+-.i . ,r} GC T0 2,..PHI

H= VY 4) / A 3 (V ( N

P,i ;T AN (H)

FCPb°. =04*( \X(N)**?-+VY(f)''`*2)ADC 4, I - i,Np3(I) =F N(Tri(TA(I)

40 C0 1, TI41 ,U-7

R N,

SLFRCUTINE SETUP(N,TO)REAL MASS1OASS2CIt NSION TO(3'),TCSQ(3"CC"14 3^lG/CrRA1t(30)/IJUMMAS/MASSI(3j,3;4),mASS2(3 v3G)IXY/XXMASS(30)/0

CRAG?PlRAG(3°) /EQUAL/ECUALS(3)CC MM O . /A P4C / A ORA G (3 t )DC I I=1, NTCSG(I)=TO (I)L`*2

12 CC NT P UEC I=l,N

GC c? J=1,NE'ALALS(I)=;.r}UALS(I)+MASS2(I,J)*TOSQ(J)

2'J CC ,TINUEEAU;A1.S(I)=I-QUALS(I)+A3rt4t(I)ECUA-S(I)=E^UAL (I)+GRAV(I}+XYMASS (I)+GRAG (I)C3hT I'tiUERETURN'END

SUP;;",UTIN4 OLVE(N,TD),KOKRECT)CIENSIC C^LTA(3),TCC{3RnAL a1, SSSS2,LCC?,"'WIN '-'4" AS/` ASS. : ,3,;) MASS 23 /EQUAL/EQUALS (3u /R 'SIC/R 3

C ) /W/WORK(*;',31)/LE_NGTH/L(3L)nC i 1 I= 1, NWCPK(I,N+1)=EQUALS (I)Cc 1 J=1,N

K. I,J) :;S1 (I,J)CCtTIN (3=-C-1 LL r,IlU ; (N,TCC)I;: T.2(I.") GC T0 5CC ' J K=1,KCK ECTC: I=1,Nq(I) ECLALS(I)

(I)= (I)# S)1(I,J)*TCC(J)CCNTI'^IUECC 3) I=19NWCRK(I,N+1)=P(I)CC 31 J=1,tti'NCoK (I,J)=MASS1 (I,CC NT INUSCALL -7 AU SS (I CELTA)C C 4 :i I=1.TOO( I)=T0 (I)-JELTA(I)

40 CC NT INU59 CONTINUE

O C 61 1=19NT'1C(I)=TC') (T)/L.(I)R=TURNEN 0

13

;;_

7).,

SLP0,)UTINE GAIJSS(N,X)JIM 9SION X(Z`)CCMMY4/w/WCpl< (3L, 31)

CC 71 I ,

WCRKII=WO K(I,I)IF (WCKI I.EC. . uu) GO TO 8GO TJ 5

8 PRINTFORMAT (53'4 AF#NOR4AL HALT CAUSED 9Y DIVIDING -3Y ZERO IN GAUSS ROUTI

CNE)CALL EXIT

5 CC 1) J=1041"WCRK (I,J)=WO,RRK (I, J)/WCRKII

13 CC NT INUEJ=1,t`

WCt*KIJ=WCRK (J, I)CC ?I K=1,ailIF (J. 'O. I) GO TO 3LtNORK(J,K)=WORK(J,K)-WO+ K(I,K)*WCRKIJ

24 CO NTrLNUt+

3tE CCikTINUE00 w.,; I-1,t4Y (I) =WCR K (I, t-) +1)

40 CONTINUERETURNENO

SL C " ,7 T I ? { P J , )L TA ,THE T 1,T019 TOO, THETA,TC,KG R GT)?Iif'4 ION Tit_'TA!(3L), `I)i(3,J)TO'DI(3 ,THETA (3 TO (30) pT 03(3J) ,THE

-T?( ', ),T' 2(!: TDC2(?),AK (?),AK2(2i)A)C(3w),Q1(3,),G?(3 ),G3(CTS)CALL O LL(P.,TLTAI,T' 1)CALL CLV ("y, )'J 71, K3 RCCT)'I: 1 ) i= 1, NA<1(T)=C'ELT AT*T^D1(I)01 ( I. ) - ; J LTt" TC1{IT

)e J ? ( I ) " ' I fl ( ) + A < I (1) /LTHE1 %2 ( I ) = T ! J E I Al (I)+C1(I)/r-.

1 CC TINUEECALL -ALL (`), IliETA2,TG2)CALL uGL (N,TC02,K0P 2CT)C C ? 1 1 , NAK2(I)=) LTAT*T,jD2(I)C2(I)=0 LiAT TJ'(i)

THETA? (I ) =THzTi1(I)-C1 (I)+2.4C2(I)

23 CONTINUECALL CALL(N,THETA2,T02)CALL SGL V (t4, TCU2, KGRRECT)GG t) T=1, NAK (I)=DELTT*TCG2(I)0: (T)=0E'l"TAT+T 2(I)T (I)=T0i(I)+(A!C1(I)+4.*AK2(I)+AK3(I))/b.

30 CC KT INUCALL CAL.L(N,THETA,78)CALLRE T') '< ",

SLRRJUTI N FFH )ICT (N, CELTAT , TDI 9 THETA *TD,TOQ, THETA2, TOE)t)ISICt T u'"},THETA(3C),TU( v),T'l(-7C)ITHFTA2(30),TD2(33)CL I I= 1, NTC2(I)=TOO (I) *2.*OFLTAT*TCo(I)T4FT12(I)=Tr- TA(I)+) LTAT*(T3(I)+TO2(I) )/2.

ID N11 TIM7 T1J

J I " c' NSIOP TRh

',T (N,D:LTAT,THETAI,TJ1,TflD1, TO2,TH_TA,TD)J1( ),TG 1(3,)T G2 (3 TH -z T A(3Ll T0(3C

TO (TT).! (:t) *)uLTA74 (Ti -, ( I ) + T D 0 2 ( D /2THCTI)=T 71Tt1 +3-iTAT TO1(I+TD;I)12

16 CC"TINU--:*' 1 V

SL,``PTUTlt; T--NSION MlYDD9THE, TA,T I,T -G, DENS ITYIXr7,YD,Y;aR E t L L,MDIMl-',4SIG"a THETA( 3'J),T! (?LTCDf?u),ATG),VN(3u),VT (3r-)CCM . N/L'J(,Tit/L,(3j),t-'ASS/P (:J)/CJEF/')T c3L),DN(3c)1VFL,VX {33) IVY {30G)/At"3/S1 7 ),C1(3,,)/0IFFANG/52(5,x,?,,) C^(3^s 3,;)/TEN/T(3t )CChMJ'1/CHHRRc.rjT/VC(, 0)/PuS/X(3O),Y(:.)CCMMCN/ACOEF/DAAT(1)=-X[ *C1(2)-YCD*S1(!)+L(I)*TO(1)**2!}e 13 I=2,rti11=1-1AT (I)=-xDD*C1(I)-Y^D*S1(I)+.(I)*TO(I)4*2iQ 1 1 J=1.,11AT(I)=AT(I)+L(J)*(TOO(J)*S2(J,I)+C2(J,I)*T0(J)**2)CCt.TINUEV' (i)=.5*L (?)*TD(1)-XD*S1t11+YD*:1(i} +VC(1) *S1ti3VT (1)=-Xrj*C1 (1) -YD*S I(!) +VC (1) *C1 (1)DD 2 t I=?, NI1=I-iVt (I)=.5*L{I)*TD(I)-VX(Ii)*S1(I)+VY (Ii) *G1(I)+VC(I) *S1 (I)VT (I)=-VX(II)*C1(I)-VY(Ii)*S1(I)+VC(I)*C1{I)

2 CC tTINUtar; ' #_,4 -

G=,;2,2*('?FNSITY-wtIp(VX"'A ^1) /')LNSITY(O TO 3

PHI=;. 1415P2E5/2,GO TO 2

3 ^=VY (N)/ltS(VX (I)IF(VX(to) .LT.",PHI=IT AN (A)GC TO 2

GO TO I

A=AS3(A)PHI= 3.141592t,5-ATAI,,

2 C0 NT 1NUET (l.) _4 (N) *AT (ti) +4 (P,) *C*Si (N) +.5*DT (',) *VT (:) *ABS(VT(#)) +D A* (VX(#V) **

VY (") ?) C,)S( 4I-TrwT` (1., )(f).'z-.:.:, . ) TCO :T('v)-MT:d)4;'S1U0-'1 (N')*AT(N)

Kw =.+

Kt=K+1'( (K;*,.T(-)+T4Y' (K,Kl)+'( (K)+.5*(OT(K)*VT (<)*ARS(VT';(JT(K;,))-CN(KI SKKI VN(K1 AP_}}+,T(.i)zCLfKKK JT(K.)"a-

(y It <) a I, } _ . V : ) ! (K. ) = ( ) - I., ( "1, ) * k7 * 1 ( K )7!S Lr.. , tv! ::T

T(T)=T(I)40 1, f, 1 w!J}i

L 9 JTI UTT0Mtt , TCC, TTEL

,M0I-3 LCr T)C` )

/L -',TFi/ t?:)IC:IFFANG/52433g3:C2t? 133

A4 'Y _ ,,

CC x j I- 1, t.t,=LII)TCCI)

1 C ' I"r)cTIC; I1 +IiLC i} 9 2 C",,t1))

P IT TI'21 rot,," 'AT `T(CaLL°.J AT Ti lt= , 11.7)

1(h) C'd*I. _. r

T"CZ)=TM ( )+nb31 (.)HT TL J)Z

c1C

87

Appendix C

Program Sequence

NVRO1 - The standard method of characteristic program reference [5]

debugged for the Bonneville Power Administration Computer.

NVR02 - The first modification of NVRO1 to allow the boundary conditions

for the anchor-drop problem.

NVR03 - An improvement of NVR02 which used equations (3) of the text for

calculating line coordinates.

NVR04 - An improved version of NVR02 using a Simpson's integration formula

to integrate equations (2) of the text.

NVRO5 - The first lumped mass program using a self starting predictor-corrector

for the "Standard catenary configuration" of a Fig. G.

NVR06 The same program as NVRO5 except the initial condition is the

"Goose neck configuration" of Figure E.

NVR07 The inextensible line lumped mass model of reference [9].

NVR08 - The final version of the lumped mass model using the second order

predictor-corrector of Section 11-3.0.

-

-

Appendix

Tabulation of Constants

Mooring Line

Nylon - 2.5" Dia.

Density = 71 lb./ft3

CN = 1.4

CT = (.008) (1.4)

CI = 0.t

CI = .5N

Length = 5660 ft. unstressed

Stress = CI(strain)C2

C1 = 1.67 x 108 PSF

C2 = 4.37

Dacron - 1.0" Dia.

Density - 1.0" Dia.

CN = 1.4

CT = (.008) (1.4)

Cit = 0.

CI = .5N

Stress = C1(strain)C2

C1 = 1.2 x 109

C2 =3.7

89

length 5660 ft. unstressed

90

Steel - 2.5" Dia.

Density = 350 lb/ft3

C = 1.3N

CT = (.008) (1.3)

CI = 0.tCI = .5

N

Inextensible

Anchor

Weight = 12000 lb.

Vol.- 24.5 ft3

Frontal Area = 12.6 ft2

CD = .4

CI = 1.0

Length = 5660 ft.

Water

Density = 64 lb/ft3

Depth = 5000 ft.

Appendix E

Computer run times for computations made on the Bonneville Power

Administration CDC-6400.

Runs made for nylon lines

1) 15° catenary

a) Early portion of drop

b) Late portion of drop

t.14 computer sec

Real Sec Mass

t _ .1 Computer Sec

Real Sec Mass

600 catenary

a) Early portion of drop t .28 Computer Sec

Real Sec - Mass

b) Late portion of drop

Runs made for dacron lines

1) 15° catenary

Early portion of drop

60° Catenary

Early portion of drop

t .21 Computer Sec

Real Sec - Mass

.84 Computer Sec

Real Sec - Mass

1.4 Computer SecL Real Sec Mass

91

2)

-

-

=

-

93

Appendix F

List of Symbols

A Frontal Area of Anchor

ACN Normal Acceleration

ACT Tangential Acceleration

C Constantsn

CD Drag Coefficients

CDN Normal Drag Coefficient

CDT Tangential Drag Coefficient of the Line

Cl Added Mass Coefficient for Anchor

CIN Added Mass Coefficient Normal for Line

CIT Added Mass Coefficient Tangential for Line

D Line Diameter

Da Anchor Diameter

DFN Drag Force in the Normal Direction

DFT Drag Force in the Tangential Direction

M.r

The ith mass representing the Line

MA Anchor Mass

SF Spring Force

SFX Spring Force in the X direction

SFZ Spring Force in the Z direction

T Line Tension at Anchor

VCX Local Water Velocity in X Direction

VCY Local Water Velocity in Y Direction

94

VOL Anchor Displaced Volume

Line Velocity Normal

Line Velocity Tangential

Anchor Weight

.WT Weight of the ith Mass0

ki Line Segment Length

kI to k4 Runge Kutta Parameters

to k3 Runge Kutta Parameters

mi to m4 Runge Kutta Parameters

nI to n4 Runge Kutta Parameters

Horizontal Coordinate

Vertical Coordinate

Arc Distance Along the Line

AS Equal Distance Taken between Calculation Point

At Time Increment

Angle from the Horizontal to the Anchor Line Link

Line Strain at the ith link

Angle Between the Horizontal and the Line Tangent

0AV The Average Between Two Adjacent Links

P

6

Mass Density of the Line

Mass Density of the Anchor

Stress in the Line

c Error in Velocity Computation

UnclassifiedSecurity Classification

DOCUMENT CONTROL DATA- R & D(Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified

1. ORIGINATING ACTIVITY (Corporate author) 2s. REPORT SECURITY CLASSIFICATION

School of Oceanography UnclassifiedOregon State University 2b. GROUP

Corvallis, Oregon 973313. REPORT TITLE

ANCHOR-LAST DEPLOYMENT PROCEDURE FOR MOORING

4. DESCRIPTIVE NOTES (Type of report and inclusive dates)

Engineering Research Report (1 July 1971 to 30 June 19735. AU THOR (S) (First name, middle initial, last name)

Robert W. Thresher and John H. Nath

6. REPORT DATEJune 1973

7a. TOTAL. N92O. OF PAGES 7b. NO. OF REFS 10

8a. CONTRACT OR GRANT NO.N00014-67-A-0369-0007

9a. ORIGINATOR$ REPORT NUMBER(S)

b. PROJECT NO. Oceanography Reference 73-5

NR 083-102C. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned

this report)

d.

10. DISTRIBUTION STATEMENT

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Office of Naval ResearchOcean Science and Technology DivisionArlington, Virginia 22217

13. ABSTRACT

The anchor-last mooring procedure is investigated in order to determinethe transient forces in the mooring line and the velocities of the anchor. Transientforces were determined and the results showed that no severe snap loads occurred forthe cases investigated. In addition, it was found that the vertical velocity of theanchor can be small as it approaches impact with the floor of the ocean.

Both extensible (nylon and dacron) and inextensible (steel wire rope) lines wereinvestigated. Lumped mass numerical models were developed for both cases. For the

extensible line case the equations of motion were determined for each mass from New-ton's Second Law, and they were integrated using a second order predictor-correctorintegration technique. Hamiltonian techniques were utilized to determine the equa-tions of motion for the inextensible line. The predictions from the numerical modelsshow the line tensions and positions as a function of time.

DD FORM (PAGE 1)I N o v 6 5 1 4/ 3 l l...- 1,«: 4;- A

Ilnrlassif iedSecurity Classification

tA.KEY WORDS LINK A LINK B LINK C

ROLE WT ROLE WT ROLE WT

Numerical Models for LinesAnchor-Last Mooring ProcedureAnchoringLumped Mass Models

D U^ r

rwRi"asNov

33 (BACK)Unc lass i_f i ed