Development of a time domain simulation for ship capsizing in following waves

4. Title and Subtitle

Development of a Time Domain Simulation for Ship Capsizing in Following Waves

5. Report Date

October 1973

6. Performing Organization Code 8. Performing Organization Report Na. 7. Author(s)

David M. Bovet

9. Performing Organization Name and Address Commandant (C-DST-2/62)

U. S. Coast Guard Headquarters

Washington, D.C. 20590

10. Work Unit No. (TRAIS)

11. Contract or Grant No.

13. Type of Report and Period _Covered

Final Report

12. Sponsorin Agency Name and Address Commandant (G-DsT-2/62)

U.S. Coast Guard Headquarters

Washington, D.C. 20590 14. _{Sponsoring Agency Code}

15. Supplementary Notes

16. Abstract

This paper describes the development of a time domain computer

simulation for ship capsizing in following waves. A survey of the

recent literature in this field is presented. The formulation of the

present approach is discussed, along with computer program limitations

and assumptions. _{The program developed is used to study the phenomenon}

of low cycle roll resonance as demonstrated by a two-dimensional section

in forced heave motion and by a fast cargo liner in following _{waves, both}

regular and irregular. _{The effect on capsizing tendency of variations}

in ship characteristics, initial conditions, and wave conditions is

investigated. _{Qualitative comparison of results with open water experiments}

is presented. _{Finally, the potential application of this program to the}

determination of merchant vessel stability criteria is discussed, and recommendations are made for further work.

17. Key Words

Following Waves,

Ship Capsizing, Time Domain Simulation, Capsizing Literature Survey,

Computer Program

18. Distribution Statement

Document is available to the public

through the National Technical Informatic:

Service, Springfield, Va. 22151

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages 158

22. Price

The work reported herein was accomplished for the IJ. S. Coast Guard's Office of Research and Development, Marine Safety Technology Division, as part of its program in Vessel Safety Technology.

The contents of this report reflect the views of LTJG David M. BOVET, USCGR, who is responsible for the facts and the accuracy of the

data presented herein. The contents do not necessarily reflect the

official views or policy of the Coast Guard. This report does not

constitute a standard, specification, or regulation.

Reviewed by: . L. FOLSOM, LCDR, USCG

Project Officer

Submitted by:

/4%d '.

.ONES CDR USCG

h f, Marine Safety Projects Branch

(ThPg,

Released by:

C.L.'iAPT,

USCG

chief, Marine Safety Technology Division Office of Research and Development

IT. S. Coast Guard Headquarters

BY

LTJG DAVID M. BOVET, USCGR

United States Coast Guard Headquarters Marine Safety Technology Division Marine Safety Projects Branch

This paper describes the development of a time domain computer simulation for ship capsizing in following waves. A survey of the recent literature in this field is presented. The formulation of the present approach is discussed, along with computer program limitations and assumptions. The

program developed is used. to study the phenomenon of low cycle roll resonance as demonstrated by a two-dimensional section In forced heave motion and by a fast cargo liner in following waves, both regular and irregular. The effect

on capsizing tendency of variations In ship characteristics, Initial conditions, and wave conditions is investigated.

Qualitative comparison of results with open water experiments Is presented. Finally, the potential application of this program to the determination of merchant vessel stability criteria is discussed, and recommendations are made for

further work.

This effort was undertaken in order to further understand a marine safety phenomenon of particular interest, that of

ship capsizings in following seas. The method of approach

was suggested by Professor J. H. Paulling, Jr., of the University of California, through his work on ship survivability. The

concept for the present effort along similar but simplified lines was greatly encouraged by Commander E. L. Jones, Jr., U.S. Coast Guard Headquarters.

My formulation of the problem was greatly aided by Dr. Nile Salvesen of the Naval Ship Research and Development Center, Carderock, who served as my thesis adviser. Discussions with him provided much impetus all during this project.

Assistance in computer programming was generously provided by Lieutenants H. K. Jenner and Frank Mlttricker, U.S. Coast Guard Headquarters. The graphics throughout this report are due to the drafting skill of Mr. Nonty Wing. Finally,

I wish to thank my wife for her moral support during this lengthy task. The author gratefully acknowledges the help extended by all these friends.

ABSTRACT

ii

PREFACE

iii.

LIST OF FIGURES

v

LIST OF TABLES

vii.

NOMENCLATURE viii.

CHAPTER

I. INTRODUCTION 1

Description of Problem 1

Past Research 7

II.

METHOD OF APPROACH 20

General Considerations

20

Detailed Formulation 2k

III. DISCUSSION OF RESULTS

38

A. Two-Dimensional Section Results

39

8. Results for Cargo Ship in Following Waves 50

IV.

CONCLUSIONS AND RECOMMENDATIONS 76

REFERENCES CITED 78

APPENDICES

DERIVATION OF EQUATIONS .F OR HULL

IMMERSION RELATIVE TO WAVE PROFILE 80

DERIVATION OF COORDINATE

TRANSFORMATION EQUATIONS 85

ESTIMATION OF HYTEODYNAMIC TERMS

FOR THE AMERICAN CHALLENGER 89

C OMPUTER PROGRAM RMOMT:

HYIOSTATICS PROGRAM

93

COMPUTER PROGRAM SHIPMO:

SHIP CAPSIZING SIMULATION

105

Figure Page Wave

Encounter Period

and. Frequency as

Funotion of Wc.velengthnd Ship :oa.ing

for a Given Ship Speed 55

Wave Enc.ounter Period as

a Function of

Wavelength, in Following Waves, for a

Given Ship Speed

57

Time History of Vessel Roll Motion, Roll Restoring Moment, and Wave Height for a

Capsize Situation 59

Vessel Roll Notion for Three Different Cases 62

Unstable Roll Region for the

AMICAN

CHALLENG in Regular Following Waves 63

Time to Capsize as a Function of

Wavelength 65

Time to Capsize as a Function of

Wave Amplitude 66

21.. Linear Roll T).mping Effect on Time to

Capsize 68

Increased Stability Results for the

AMERICAN CHALLENGER in Regular

Following Waves 70

Time to Capsize as a Function of

V

Fiui

Page

'1

Vessel With W.Ve Crest Amidships

2

Vessel With Wave

Trough Amidships 2

Time History of StabIlity and Roll Angle 2

Stable and Unstable Regions for the

Mathieu Equatioi

10

Approximate Regions of Stability for the

Mathieu Equation With Small Values of6

10

Experimental Results Compared

With Theory;

Paulling and Rosenberg

₍₁₉₅₉₎

12

Coordinate System

25

Section Areas for the AMICAN CHALLENG,

Station 6.

at Various Heel

Angles and

Depths of Immersion

33

Transverse Righting Moments About Keel

for the AMICAN CliALLENG, Station 6, at

Various Heel Angles and Depths of

ImmersIon..

34

Flow Chart of Computation Procedure

₃₇

Unstable Roll Region for a Two-DImensional

Section ln Forced Vertical 0scillatio

43

Time to Capsize

as a Function of Vertical

Oscillation Frequency

46

Time to Capsize as a Function of Vertical

OscIllatIon

Amplitude

47

Time to Capsize as a Function of

Initial Roll VelocIty

48

TIme to Capsize as a Function of

Initial OscIllatIon Phase Angle

49

Table Page

1. Characteristics of Two-Dimensional Section LO 2, Stability Variation for Two-Dimensional

Section 41

Vessel Characteristics for the

AMERICAN CHMTENGER 53

. Irregular Wave Rims 73

A section area

Az heave added mass coefficient

sectional heave added mass coefficient

A0 pitch added mass moment of inertia coefficient A9 roll added mass moment of inertia coefficient

a9 sectional roll added inertia coefficient

B ship beam

Bz heave damping coefficient

bz sectional heave damping coefficient B0 pitch damping coefficient

BlO linear roll damping coefficient

b19 sectional linear roll damping coefficient B29 nonlinear roll damping coefficient

BN d.istance from center of buoyancy to metacenter

C(t)

time-varying heave restoring force coefficient

C0(t) time-varying pitch restoring moment coefficient

C0(t) time-varying roll restoring moment coefficient wave encounter frequency

f(t) time-varying vertical wave excitation force g gravitational acceleration

GM transverse metacentric height

I moment of inertia

pitch mass moment of inertia viii

roll mass moment of _inertia

k _{ship gyradius}

height of center of gravity above keel height of metacenter above keel

1 ship length ship mass

S station spacing

Te wave encounter period T0 _{ship natural roll period}

t _{time variable} u wave displacement

V ship speed

X axis of coordinate system

X(t) general time-varying displacement x _{displacement in X direction}

Y _{axis of coordinate system}

Z axis of coordinate system

z _{heave displacement}

wave amplitude

constant In Mathieu equation time step

constant in Nathieu equation

'1

wave elevation

e

roll displacement

NOMENCLATURE

'D.)

mass density of water

pitch displacement

heading angle

radial frequency

Description of Problem

The problem of severe ship roll motion in following seas has long been recognized but not adequately studied. It is a problem well-known to seafarers from experience, although its underlying causes are not generally understood. The fact that following and quartering seas pose a severe

safety hazard to many vessel types under certain extreme conditions is generally accepted. One of the paradoxes of this hazard is that its occurrence can be so sudden that a vessel may capsize before her crew is aware of the danger.

A brief description of the physical phenomenon involved can be presented with the aid of Figures 1 and 2. _These

figures show a ship proceeding in the same direction as a two-dimensional regular wave system, with the vessel center-line perpendicular to the wave crests. With this orientation there is no external roll exciting moment present. _{In the} first figure, a wave crest Is amidships, while in the second, a trough lies amidships. For a normal ship hull form with flared sections at both ends and wall-sided sections amIdships, It can be seen that the transverse inertia of the waterp].ane Is different In the two cases. With the wave crest amidships, inertia Is lost at the two ends without any corresponding

Figure 1. Vessel with Wave Crest Amidships.

Figure 2.

Vessel with

Wave

Trough

Amidships.

N

ROLL 4NGL'

N

/

_/

/

_/

/

_/

/

TIMe. O

'

QOLL

/

,/

I'

"WI",

I

/

/ N_{N /}

/

Ro

TA.tLtY AXI5

- TIME

at the ends, the value of the transverse inertia of the water-plane is increased. Thus the position of the transverse

metacenter changes, and hence the stability varies, with a period equal to the wave encounter period.

This periodic variation in the vessel's transverse stability may lead to serious consequences under certain conditions. Suppose that the stability variation, caused by the vessel's encounters with regular following waves, has a

frequency twice that of the natural roll frequency. _Starting with an initial roll angle and a wave trough amidships, _as shown In Figure

3,

_{the vessel has a high stability value and} returns rapidly to the upright position. Now as the vessel rolls to the other side, a decreased level of stability _In the wave crest allows a heavy roll to occur. _{At the peak of} this roll, Increased stability Is again caused by _{a wave trough} moving Into the amidships position. _{This causes the vessel}

to right herself quickly and roll heavily to the _{other side} against a decreased stability level caused by a wave crest amidships. _{In a regular wave situation, this process can} continue over several wave encounter cycles, _{increasing the} roll angle until capsizing ultimately occurs. _{In simplified}

form, this is the capsizing phenomenon that will be investi-gated In this paper.

A recent case which illustrates the practical dangers of following seas concerns two 79-foot herring selners that

were brand new and were

tions in moderate seas. The designs had apparently been approved for government subsidy as meeting the 1)1CC stability criterion for fishing vessels. The conditions of the losses were both in following seas, with significant wave heights of about four feet, and speeds of about nine knots. The vessels

cap-sized suddenly causing the deaths of most of the crewmen.

Model tests confirmed the possibility of capsizing under these conditions, and pointed to the stability loss in following

seas as the likely capsizing mechanism.

The loss of a large West Coast tuna clipper in the

1950's

was analyzed in depth by Paulling

(1960).

This vessel had

an overall length of ₉₉ feet and a loaded displacement of Lj75

tons, with an initial metacentric height of 2.06 feet. In

still water, the righting arm curve peaked at around 15 degrees, declining then to yield a range of 60 degrees. Paulling

calculated the righting arm curve for two following sea condi-tions, one with the wave trough amidships, another with the crest amidships. A hydrostatic pressure distribution was assumed for the wave, and the vessel was posed in static

equilibrium on the wave. Wave length was taken equal to ship length, and height equal to length of wave over twenty. This calculation showed the vessel to have vanishingly small or negative stability at all angles for the case with wave crest amidships. It was theorized that the vessel capsized

in a wave crest, thus causing her loss during the maiden

voyage.

Bole and Kastner (l96) reported on a thorough

investi-gation carried out to determine the cause of capsizing of the German motorship LOHENGRIN In January 1963. This vessel was a 1,500 dwt cargo ship of 66.70 meters overall length,

built in 1958 for coastal service. At the time of the casualty, the LOHENGRIN was standing in to the Ki.el Fjord with some ice accumulation aboard as well as a full load of wood pulp.

Following seas with a significant height of about 2.0 _meters were present when the vessel suddenly developed a k5 degree

list to starboard. _{This caused flooding of an internal} com-partment through a deck vent which eventually led to the

sinking of the vessel. _{A major question for study was the cause} of the Initial heavy list.

In a detailed study of this casualty, a 2.0 meter model of the vessel was built and tested under natural conditions on Pl5ner Lake. Wave conditions were obtained that

corres-ponded to actual conditions during the full-scale casualty. The model was self-propelled at various headings and with different stability levels. Capsizing of the model occurred In following seas with stability levels in the _{range of the} estImated 0.13 to 0.22 meter metacentric height present with the actual ship. _{These stability values took into account} the ice buildup, which alone was not sufficient to cause the

The tests demonstrated that it was through the effect of the following seas, chiefly related to the reduction of the righting arm in a wave crest, that the model reached an equilibrium

position at about a 5 degree heel.

The LOHENGRIN actually remained in this position for some two hours before down-flooding eventually caused the

vessel to sink. The existence of a second equilibrium position at a heel of LI5 degrees resulted from the particular hull form

and low initial stability of the vessel. However, this angle could only have been reached through the sudden large roll caused by reduced stability while the vessel was being over-taken by a wave crest. Thus, loss of stability in a following sea was a major cause of this casualty.

In fact, capsizing of small vessels in heavy stern or quartering seas is a significant safety problem. U.S. Coast Guard statistics (l9o8-71) indicate the loss of some 292 U.S.

fishing vessels and 25L1. tugboats due to foundering, capsizing, and flooding over a four-year period. How many of these casual-ties were directly related to stability loss in following

seas is not known. These small vessels are the most suscep-tible to such a hazard by the nature of their small size and low freeboard. A number of stability-related small ship

casualties were reported and analyzed by Rahola (1939) In his pioneering work which first advanced the concept of a

stabil-ity criterion based upon the area under the righting arm curve. Recently, Storch (1972) presented an excellent overall analysis

of stability-related problems of tugboats and a discussion of applicable safety criteria. However, the problem of deter-mining effective stability standards to withstand the hazards of following seas is not presently resolved.

This project has been undertaken with the purpose of developing a computer program for use as a research tool in the study of the phenomenon of ship capsizing in a following

sea. _{It is believed that a computer simulation will prove}

to be the most efficient means of testing a number of hull forms under various following sea conditions. The simulation developed here is necessarily simplified, but it is believed that it can be used effectively to evaluate stability situations In pure following seas. The knowledge gained by extensive

simulation tests can be used to set safety levels guarding against this capsizing phenomenon. Once such criteria have been determined, they can be incorporated with other _stability requirements f or wind heel, icing, towrope pull, etc. It is essential, though, that the safety hazard of following seas should be considered In the determination of overall vessel stability standards.

B. Past Research

Investigations into the problem of ship stability in waves extend back to Sir William Froude (1861). PaullIng

(1961)

briefly discusses the history of such research. It

study of ship rolling in following waves have been accomplished within the past twenty years. This body of work may be grouped under four headings: (1) work based on the study of the

Mathieu equation in the 1950's; (2) experimental and theore-tical work carried out up to the present time by Wendel,

Arndt, Roden, Kastner, and others in Germany;

(3)

experImental and theoretical work begun four years ago by a team under

the direction of Professor Paulling at the University of

California, Berkeley; and () other recent research. These

major research efforts will be briefly described here.

1. The Mathieu Eauation

An important realization in the study of ship roll motion in following seas was that of the applicability of the Nathieu equation to this problem. This equation describes the pheno-menon of parametric self-excitation, in which subharmonic oscillations of a system are caused by the periodic variation

of a time-dependent parameter of the system. This equation

governs a wide variety of problems in mathematical physics, and consists of a linear, single degree of freedom, system with a harmonically-varying restoring term. In the ship roll application the harmonic variation is caused by periodic

changes In the vessel's transverse roll restoring moment. These stability changes may occur due to changes in the

trans-verse moment of inertia of the waterplane or due to changes in the underwater volume, both of which may be caused by

relative vertical ship-water surface motion. In turn, this motion can be induced through forced heaving in still water, or through the slow variation of the relative wave profile in a following sea as illustrated in Figures 1 and 2,

The Mathieu equation can be written as:

+ (+eco3t)6 = o.

This equation does not consider damping, and assumes a pure harmonic variation in the restoring moment. _{However, the} study of this equation does yield significant insights Into the general physical phenomenon.

Despite its limitations, the Mathieu equation can be used to determine regions of stable and of unstable _solutions. Minoraky (1947) states that, in the linear form of the equation given above, a boundary between these regions can be found

based on the values of the constants,

_b

_{and e}

. In our

ship roll application, 6 is the square of the ratio of natural roll frequency to the frequency of stability _{variation; E}

is directly proportional to the amplitude _{of the stability} variation.

Mapping the

£-6

plane results in stability charts

showing unstable regions expanding up and outwards from points along the

b-axis as

follows:

3 .

)

(=

o,%)iZ,±3)....)

Figure l. Stable and. Unstable Regions for the Nathieu Equation.

Figure

5.

Approximate Regions of Stability for the Mathieu Equation with Small Values of E.

approximate representation of the main regions of interest, also due to Stoker, is shown in Figure

_5.

These charts Imply the existence of unstable solutions In regions where the period of stability fluctuation has a ratio to the natural roll period of 1/2, 1, 3/2, 2, 5/2, etc. Within the unstable region, roll motion must increase inf in-itely, while outside the region any initial roll angle will be suppressed. At the borders ofthe region, periodic rolling

to some arbitrary maximum angle Is possible.

Paulling and Rosenberg

₍₁₉₅₉₎

were able to accomplish the experimental verification of the Mathieu

4

diagram for the case of a )larIner ship model. The inclusion of nonlinear

second-order couplings between heave and roll motions was

shown to cause the possibility of instability in roll, through excitation from the heave mode, as predicted by the Mathieu equation. This predicted Instability was verified by exper-Iments with the ship model where unstable roll motion was induced through forced heaving at certain frequencies and amplitudes. The comparison between experiment and theory

Is shown in FIgure 6.

Qualitative agreement between these model tests and the theory was good, although quantitative agreement was only fair. Differences observed between experimental results and theory

were attributed to the exclusion of damping and of certain

second-order derivatives in the theory, and to experimental problems of wave reflection from tank walls. _{It was stated}

that inclusion of damping terms in the theory would tend to raise the amplitude required to produce instability at a given frequency. This can be understood as the absorption of excitation (input) energy by the damping (output) mechanism. Hence, values of the boundary line are critically dependent upon damping values, particularly in the vicinity of boun-dary minima along the 6-axis.

GO

LOCW.

0.70 M0LEL

GM: O.0215X eAM, ZE0

IZOLL MOTIOP'J

H'JPUCEP bY 1OCEP VERTICAL

0cILLzmOPJ, \/ITh IlJITIAL 5'

0LL A)4GLE

o.Ia 0.10

0.08

0.06

0.04

0.02

0

0.% 0.15 0.20 0.25 0.50 0.55

Figure 6. ExperImental Results Compared with Theory; Paulling and Rosenberg

(1959).

Kerwin

_(1955),

durIng his stay at the Delft University of Technology, was one of the first to show that the single degree of freedom differential equation for roll motion In following waves can be written in the form of a Mathleu

equa-tion. The main thrust of Kerwin's work was to calculate the change in stability due to following waves and to compute maximum roll angles for a given stability fluctuation. Cal-culations were based on the Mathieu equation using linear

and nonlinear damping, and some experiments were conducted with a fishing vessel model In following waves to confirm

the calculations. However, the stability of the model was such that maximum roll angles of only about 20 degrees were obtained, and took several hundred oscillations to build up. As Kerwin's computations were carried out by hand, the pro-cedure was rather laborious.

Following his work with Rosenberg, Paulling (1961)

carried out further Investigation of the transverse stability of ships In following waves, both theoretically and

experimen-tally. The appreciable decrease in stability (due to changed transverse moment of Inertia of the water plane) with the wave crest amidships and Increase with trough amidships, was

shown. Models were towed at fixed heel angles, free to pitch and heave, and righting moments measured for various following wave conditions. The variation of stability in waves was

shown to result from the altered characteristics of the im-mersed hull in waves. Further, the influence of model speed

stability variation was shown to be small also. Finally. Paulling pointed out that the effect of a following seaway on a vessel's ability to withstand capsizing should be con-sideréd when judging the vessel's ultimate performance at sea.

2. German Research

Since the mid-l950's, a team consisting of Wendel, Arndt, Roden, and Kastner has been studying ship stability in

follow-ing seas at the Universities of Hamburg and of Haimover. The body of work accomplished by this group represents the

only major concerted effort in the field outside of the Uni-versity of California. A major difficulty, however, is the

lack of English translations for most of the resulting

publi-cations. Brief discussion of two translated articles here

will have to suffice.

Kastner and Roden (1965) presented a paper to INCO describing the experimental approach to ship stability in following seas then underway in Germany. This consisted of model tests under different stability and sea conditions, conducted on a natural lake, to determine the minimum sta-bility required to withstand capsizing. As a result of these tests, required righting arms in a wave crest were determined

by statistical evaluation of the results. The authors sug-gest that vessel stability should be assessed by examining

the righting arm curve with a wave crest amidships, rather than by incorporating an arbitrary margin into the still water stability value.

Kastner

₍₁₉₆₅₎

carried out further analytical work based on the numerous ship capsizing model tests conducted. He utilized an analog computer to solve the equations of

motion, and represented the stability variation as a normalized righting arm spectrum for all heel angles. The treatment

considered an irregular pure following seaway In a probabil-istic fashion. Results were obtained in terms of the sta-tistical length of time until capsizing. _{Limitations included} accuracy problems of the analog computer, and the assumed spec-tra for righting arm curves. _{It is understood that Kastner} is continuing his research in Hannover.

3.

Research at the University of California

Beginning in 1969, an intensive investigation was begun

into ship stability in a seaway at the University of California, Berkeley. This work was directed by Professor J. Randolph

Paulling, Jr., under U. S. Coast Guard sponsorship. _Paulling approached the problem both theoretically and experimentally. As part of this effort, two frequency-based studies were

initially completed. Haddara (1971) studied the coupled,

three degrees of freedom, nonlinear motions of a ship in

regular oblique waves using the frequency domain. _By consi-dering just the single degree of freedom roll equation,

Haddara (1972) was able to extend this analysis to the con-sideration of irregular wave spectra. However, these frequen-cy-based analyses were limited in their ability to represent the roll restoring moment variation caused by following seas. A ma1or thrust was carried out in the experimental direction to obtain improved understanding of the phenomenon of ship capsizing in following seas. An 18-foot model of the fast cargo liner AMERICAN CHALLENGER was equipped with extensive motion recording devices and was tested under remote control in San Francisco Bay. During these tests, variations to ship heading, speed, stability, and freeboard were carried out.

Results of these tests were presented in basic form by Paulling,

et. a].. (1972a), and analyzed by Kastner (1973). A good

summary of the overall project results was presented by Paulling,

et. a].. (l972b).

The most important contribution of these extensive model tests was the identification of three primary modes of capsizing. These may be described briefly as follows.

Mode 1, Low Cycle Resonance: This refers to a roll buildup during a series of particularly large following or quartering waves. As the crest of a wave is amidships, the vessel's stability is greatly reduced, causing a heavy roll. Next a wave trough moves into the amidships position, causing

increased stability and rapidly returning the vessel to the upright position. Now it a crest appears amidships, the vessel will roll slowly down on the other side against a weak

restor-Ing moment. If a wave trough now appears, the vessel will snap upright again. This process continues until either the model capsizes or it moves out of the wave group and

the motion dies down. This is the process which was described earlier in the paper.

This mode of capsize was characterized by a very reg-ular rolling motion which, in a group of three or four waves, grew rapidly to a large amplitude. The rolling motion was observed to occur at a frequency corresponding to the first

1'lathleu resonant frequency. Rolling sometimes occurred at the same frequency as that of the stability variation, hence corresponding to the second Mathleu resonant frequency. In

either case, the roll motion built up rapidly and. capsizing occurred as the model rolled into a wave crest.

Node 2, Pure Loss of Stability: It was sometimes ob-served that, with the model operating at high speed in very high waves, sudden capsizing might occur with little prelim-inary rolling when a crest amidships caused dramatic stability loss and capsizing. A high model speed was necessary to

induce this capsize mode, to produce a relatively stationary situation with the vessel caught in the wave crest. _{A Froude} number of 0.4 is required for vessel speed to equal wave

speed, for a wavelength equal to ship length.

Node

_3,

Broaching: This is the most dynamic mode, caused when the model is struck from astern by several steep breaking seas in succession. The waves cause the model to

yaw off course to such an extent that the steering system cannot bring the model back on course between waves. The breaking seas striking the model combine with the dynamic heeling moment from the turn to capsize the model, again with the crest of a wave amidships.

The identification of these three modes of capsizing

and many other valuable insights were gained from the model tests conducted in San Francisco Bay. However, it was deter-mined that the number of test runs required to statistically

define the stability limit for safety from capsizing with adequate confidence was prohibitive. Therefore, Paulling's

current effort concerns the development of a six degree of

freedom numerical capsize simulation program, using as input fully directional wave spectra. Such a program would

vir-tually allow free-running model tests to be run on the computer. This represents a major digital computer programming effort, and results are not available at this writing.

M.. Other Recent Research

Several other investigators have recently attacked

certain aspects of the ship capsizing problem in following

waves. The Dutch government, concerned with fishing vessel safety problems, sponsored model tests in following waves

at the Delft University of Technology. The results, presented by Beukelman and Versluis (1971), show a fair comparison

waves for a series of four beam trawlers of varying prismatic coefficients. De Jong (1970), also at Delft University of Technology, carried out a probabilistic analysis of ship roll motion in beam seas and in following seas. His analy-tical treatment is based on the application of the Duff ing

equation to the beam sea problem, and on the use of the Mathieu equation for following waves. Recently in this country, a

group at the Virginia Polytechnic Institute and State Uni-versity has studied the phenomenon of nonlinear coupling between different modes of ship motion. Nayfeh, Mook, and Marshall (1973) reported analytical results indicating the existence of unstable roll motion Induced through nonlinear coupling with the pitch mode in the first and second Mathieu resonance regions.

Although there are undoubtedly other researchers working in this field, it Is believed that the efforts described

in this chapter represent the major advances to date. It

Is evident that a substantial body of excellent research has been carried out by investigators both in this country and abroad. However, we are still lacking the practical tools to permit detailed examination of ship capsizing situations to be made. It is in this direction that the present effort hopes to make some progress.

A. General Considerations

The objective of this effort is to develop a computer program for use as a research tool to study the phenomenon of ship capsizing in a following seaway. In particular, it is hoped to simulate the low cycle resonance mode of

cap-sizing. The method of approach chosen for this program

con-sists of a computerized time domain analysis of a ship's roll motion in pure following waves. This method utilizes the numerical integration of the equations of motion. The typical frequency-based equations, thoroughly discussed by Salvesen, Tuck, and Faltinsen (1970), are not adequate for this problem. Here the accurate time-varying representation

of the roll restoring moment is critical, and it can only be calculated by knowing the exact wave profile alongside the

vessel. This wave profile can easily be traced in a time domain solution.

The programming effort was carried out on an IBM 1130 computer until the computational procedure appeared to be completely functional. The program was then switched to a CDC 3300 computer using the remote job entry mode. The

pro-gram had to be slightly altered, but core capacity and run time were greatly improved. Fortran IV was used throughout

the effort.

The salient features and assumptions inherent in the approach used may be grouped as follows:

Three degrees of freedom.

Dominance of hydrostatic forces. Pure following waves.

Large roll amplitudes.

Numerical Integration techniques. These points will be briefly discussed here.

1. Three degrees of freedom

The analysis is limited to the consideration of three degrees of freedom. Heave and pitch are Included as they are considered essential In the determination of the Immersed

hull geometry, which Is critical to the transverse stability analysis. These modes are solved using linear,

frequency-Independent coefficients, except for the hydrostatic restoring force and moment, which are computed for the instantaneous hull-water surface position. Roll Is computed similarly, but with provision for a constant nonlinear damping coefficient. The instantaneous roll restoring moment is computed for each

ship section, thus incorporating the exact wave profile along

the ship's length, and Integrated lengthwise.

The vessel is assumed to advance at a steady speed on a constant course. Yaw and sway are Ignored, since the whole analysis Is limited to perfect following seas. Any

consider-ation of oblique quartering seas would require inclusion of these two modes, as well as inclusion of a wave exciting term in the roll equation. Finally, surge is ignored In order to simplify the computation of relative ship-wave

position. It Is hard to say what effect this assumption has on the stability analysis. Although it is believed that surge may affect the results, it is very difficult to obtain the necessary coefficients. In short, this assumption is made to allow a reasonably simple statement of the problem.

2, Dominance of hydrostatic forces

The treatment of static and dynamic forces adopted is intended to simulate specifically the condition of low encoun-ter frequency present In the following wave situation. Thus, the main effort is placed on a very accurate computation of the static forces acting at each instant of time. Conversely, the dynamic terms in the equations of motion are estimated using frequency-independent coefficients. It is anticipated that the dynamic terms are relatively unimportant compared to the dominant effect of the static forces under the low en-counter frequency condition.

3.

Pure following waves

The wave environment considered consists of waves whose orestlines are perpendicular to the vessel's heading, arid which are advancIng In the same direction as the vessel.

The term "foflowing" seas is used to mean waves proceeding in the same direction as the vessel, regardless of whether the wave speed is greater or less than the vessel speed. Regular waves are represented by single sine waves.

Irregu-lar wave conditions are generated by a combination of sine waves of varying wave lengths, amplitudes, and initial phase angles. Vessel motion response is based upon this overall

irregular wave surface input, rather than upon the individual sine wave components utilizing the superposition principle of linear, frequency-based analyses.

Large roll amplitude

No restrictions are placed on the amplitude of roll, as would be the case in a linear, frequency-based analysis. This is a major advantage of the time domain approach. A roll angle of 90 degrees from the vertical is defined arbi-trarily as a "capsize" situation In an attempt to standardize the occurrence of this event. Although no special restric-tions are placed on pitch and heave amplitudes or on wave steepness, certain second-order inaccuracies are Introduced due to geometrical calculation methods if vessel motions in the longitudinal vertical plane are too violent. See Appendix A for details of the calculation procedure involved.

Numerical integration techniaues

mathematical techniques employed in this time domain approach. The equations of motion used to represent the physical situa-tion must be integrated time-wise. Nany techniques are avail-able for accomplishing this, including Runge-Kutta and pre-dictor-corrector methods. Here a Runge-Kutta method of the

second, order has been employed, using a straight-forward extrapolation of the motions based on constant accelerations across each time step. This is accurate so long as the tl.me

step used is sufficiently small. Areas and volumes used in the hydrostatic calculations are approximated using the trapezoidal rule. Thus, certain inaccuracies are inherent

in the results due to the mathematical approximations employed.

B. Detailed Formulation,

The specific formulation of the approach is discussed in this section, beginning with the coordinate system and the equations of motion. The calculation procedure employed for the time domain solution is then discussed in some detail.

1. Coordinate System

The coordinate system adopted for this study is shown in Figure

7.

Let (X,Y,Z) be a right-handed coordinate system fixed with respect to the mean position of the ship with Z vertically upward through the center of gravity of the ship, X in the direction of forward motion, and the origin in the

Figure 7. Coordinate System.

plane of the undisturbed free surface. Let the translatory

displacement in the Z direction with respect to the origin

be z, the heave displacement. Furthermore, let the angular displacement of the rotational motion about the X and. Y axes

be

0

and

0,

respectively, so that

0

is the roll angle and

0

is the pitch angle. _{Free surface elevatIon Is}

p.

As shown in Figure 7, the crestlines of the waves are perpendicular to the vessel's centerplane and are advancing in the same direction as the vessel for the pure following wave situation. Since the ship is advancing with a steady forward speed V, the expression f or the wave profile in this

coordinate system must account for the difference between

ship speed and wave speed. A derivation of the necessary equations Is given in Appendix A. Since the Z axis is per-pendicular to the undisturbed free surface, points on the

ship hull surface in the transverse plane must be transformed from their symmetrical coordinate system to that shown here, accounting for the roll and heave displacements. The equations

for this transformation are derived in Appendix B.

2. Eauations of Motion

The equations of motion adopted for this study are as follows:

Heave

(M+A),4e

+ _;

£)z.

I f(x,t) ijx)

c&

Pitch

(I##A.

+e =

Roll

(t0+P18)Ô61é 466(81

+c (t)e = 0.

Alternatively, the accelerations may be expressed as:

=

-(e,,3. #c(e)

.

-J'F

,v1x)ax) / (M#A)

(6

C#(e,,

)czjj*f(Z,t)It(Z)x

_4.i

_)/(104-A0')

0

-

(è

_{te!éI +C9( t.,it;1)Q}

)/(t,+A9)

The basic approach here involves the estimation of linear,

time-independent coefficients for the added inertia and damping terms, and the computation of nonlinear, time-varying values for the restoring and wave exciting terms. Heave-pitch cou-pling in the added inertia and damping terms is ignored, in keeping with the simplified treatment of dynamic terms in

the low encounter frequency situation.

storing forces and moments are nonlinear and fully coupled, since they are computed at each step in time according to the actual elevation of the free surface, including the effects of heave, pitch, and roll along with the exact wave elevation,

For the wave exciting force in the heave and pitch equa-tions,. two assumptions are made. First, the wave diffraction effects are ignored, leaving only the free-wave profile (or Frou.de-Kriloff force) and discounting any effect of the vessel

on the wave structure. Second, the Froude-Kriloff force is

approximated by using the long wave approximation, thus ignoring the so-called Smith effect. This is equivalent to setting the term e in the Froude-Kriloff force equal to 1.0, which

leads to an error of less than twenty percent for a wavelength

equal to the ship1ength. This approximation for the exciting

force is necessary because of the difficulties involved in the time domain approach, where a specific wave profile is considered. possible improvement would be the use of con-volution integrals and Fourier transforms to compute the ex-citing force (and other dynamic terms as well) in the fre-quency domain, and then transform the results to the time domain. In practice, the hydrostatic restoring force and the exciting force, shown here as separate terms, are calculated as a single difference quantity by the hydrostatics subroutine. Finally, it should be pointed out that the inaccuracy in the

wave exciting force does not directly affect the roll motion

pure following seaway. Thus, the effect of this approximation on the capsizing results is not expected to be significant.

3.

Calculation Procedure

The calculation procedure adopted f or determining the vessel trajectory in terms of heave, pitch, and roll can be outlined as follows:

a. Calculation of new position in time.

I. Vessel velocities and displacements are found by

integrating accelerations over the time step.

Ii. Wave position is found by Integrating wave velocity

over the time step.

III.. Relative immersed hull-wave geometry is determined

from items (i) and (ii).

b. Calculation of forces acting at a given point in time.

Hydrostatic forces are computed in a subroutine

util-izing item (a.Il.i).

L*mpirig forces are a constant function of velocities computed in item (a.1).

Added inertia terms are incorporated as constants within the vessel inertia terms.

c. Calculation of accelerations present at a given point

in time.

1. Accelerations are found with Newton's second law,

using the forces from item (b).

d.. Advano ement of time.

i. Time is advanced by a given step.

ii.. Entire procedure beginning with item (a) is repeated.

This is the basic process used to trace the vessel's motion in the time domain. Several simplifications are introduced

to keep calculation time at a reasonable level. _{Since the} main emphasis of this simulation is on accurate representation

of the vessel's time-varying transverse stability, _{the largest} calculation effort is spent on computation of instantaneous hydrostatic properties. _{The four Individual sections of this} procedure will now be discussed In further detail.

a. Calculation of position

I. _{Vessel displacements and velocities}

To start the program, vessel displacements and velocities are specified as initial conditions, along with the initial wave profile. _{Vessel accelerations are initially set equal}

to zero. _{At the begInning of each time loop, vessel}

displace-ments and velocities are calculated using a simple Integration procedure. _{According to Dorn and McCracken (1972), this}

procedure can be considered a Runge-Kutta method of the second order, since it relies on evaluation of _{derivatives at only} one point in time, and since the method used agrees with the Taylor series through terms in the second order in time step. The previous acceleration and velocity values are used to determine current displacement and velocity. Displacements and velocities In each of the three degrees of freedom are obtained as follows:

Displacements

X(t

_X(t4

_#1

₊

if

or

x

(4:.)

+

(h-)

(M.'2

Velocities

(4k)

+

I

(q)

at

or

x(-L''

(

th-IS)

+

Here, X(+.1)represents the new heave, pitch, or roll displace-ment. The time step Is

At,

which Is equal to tn-tn_i.

ii. Wave position

The wave position at any point In time is found by

taking the initial wave position and adding to It the integral of the wave velocity up to that point In time. The first-order, small-amplitude linear wave theory is used, which

implies that the regular wave profile is sinusoidal. A possi-ble improvement would be the use of third-order Stokes wave

theory, as discussed by Salvesen (1969). Wave velocity Is computed according to the deep water assumptions. For irreg-ular waves represented by a sum of sine waves, the velocity and position of each sinusoidal wave component is computed and summed before the wave surface is applied to the vessel. This yields simple basic wave forms, but avoids any assumption

of linear superposition for the responses. Equations for the position of the wave crest in the X direction at a given point in time, with respect to the origin at initial time to,

are as follows:

u(t) = u(t0)

+f

(t) dt or

u(t) = u(t0) +

i t where i =

and u(t0) = initial position of wave crest.

iii. Relative hull-wave geometry.

Using the knowledge of the present positions of both the vessel and the water surface, their Intersect ion can be determined. An accurate estimate of this intersection Is needed In order to calculate accurately the hydrostatic restoring and exciting coefficients, c(t) and f(t), In the equations of motion. The detailed derivation of the equations

used to determine sectional immersion, or station "drafts,

is given in Appendix A.

b. Calculation of forces

I. Hydrostatic forces

Given the current relative position of the vessel in terms of roll angle and station drafts, it Is a simple matter to determine the required hydrostatic forces. _{The approach} used is to pre-compute an array of data for a given hull _form; this is accomplished by Program File RMONT, described in

Appendix D. The dimensions of the two arrays -- one for

section area, one for sectional transverse righting moment

- are heel angle, station draft, and station number. For

the cargo ship studied, ten heel angles, ten drafts, and eleven stations are used. Graphical representation of the arrays for section area and for righting moment are shown

in Figures 8 and 9, for one station of a cargo ship.

In order to find the hydrostatic values for the precise heel angle and draft present, a linear interpolation scheme

is employed. At this point in the computer program, the

values of heel angle and draft are fed to a subroutine (RSIM1) which carries out linear interpolation within the two arrays and returns sectional values to the mainline program.

A

trapezoidal integration is then performed along the vessel's

length to obtain the exact buoyancy (heave) force and the exact roll restoring moment. The pitch restoring moment is obtained from the longitudinal distribution of the buoyancy

forces.

ii. tamping forces

These are easily obtained at each point in time, since the coefficients are time-independent constants, and since the velocities are known for the given instant of time. The derivation of the damping coefficients for the cargo

ship studied is given in Appendix C. The only point of inter-est is the introduction of a nonlinear coefficient, B29

3500 3000 2500 w

2000

/500 /000 500

0

)JOTE I7o5ITVE PAFT !6

0FJ.9 A'

SI1oil&i

11EL

t'EGE

Pojró

6ttoir'J

d

PO

A&e'

OF

AVA1LAL TO

6Up MoTloj

_?OGL1Ak1,

?ornV 2AFT

Figure 8. Section Areas for the ANERICAN CHMTENGER, Station 6,

at Various Heel Angles and Depths of Immersion.

pgF% 58'

PZ4FTs Q' I

\

70,000

60,000

5000O

O)00O

30,000

20,000

10,000

1

PZAFT°

PAFT

QRArT:

II

38

28'

,;:;;1t:c

_v-

.-&

:

-U

-/0°

200

30

40°

50°

60°

_70°

80

HL ALE, P-3

POIP.iT6 Sr(O\/M S2E FOI't AE' O VILUE3

AVAJLASLE 10 5111? t4OTIOJf

PfQIM..

Figure 9. Transverse Righting Moments about the Keel for the

AMERICAN CHALLENGER, Station

6,

at Various Heel

to account for some of the viscous effects and other

nonlin-cantles introduced in roll damping at large amplitudes of

motion. A value for this coefficient would have to come from an analysis of an experimental roll decrement curve from a model or full-scale vessel. In the case of the cargo ship studied, the value of this coefficient has been provisionally set equal to zero.

iii. Added inertia terms

The added mass and added mass moment of inertia coeff

i-dents, A, A# and A&, are taken as time-independent _constants

for the hull form used. Thus the hydrodynamic inertia terms are incorporated with the vessel inertia terms. _{The derivation} of the constant added inertia terms used for the cargo ship example is presented in Appendix C.

Calculation of' accelerations

The initial accelerations are set equal to zero. Then

accelerations at each instant in time are easily obtained from the forces calculated above, using Newton's second law. Expressions given for the accelerations in the earlier section on equations of motion are used.

Advancement of time

The last step of the time loop used for the numerical integration process is simply to increase the value of time

by some constant amount. The size of this time step is crit-ical to the stability of the numercrit-ical solution. Although more sophisticated integration procedures would permit larger time steps to be used, the approach adopted here is simply to select a sufficiently small time step such that further change in the time step would not produce any appreciable

change in the results. Following the addition of an incre-mental time step, the entire procedure beginning with item

(a) above is repeated.

This is the overall procedure utilized to obtain the vessel motions in the time domain. Further details are con-tained in the appendices, particularly In the two sections describing the computer programs developed. These are Appen-dix D, describing the hydrostatics subroutine, and AppenAppen-dix E, which presents the ship capsizing program file. A simple flow chart summarizing the overall computation procedure

T1M

The computer simulation developed was applied to the study of two particular cases. For the first example, a case was selected which was computationally simple and which clearly demonstrated the low cycle resonance mode of roll instability. This example consisted of a two-dimensional

section of rectangular shape which was harmonically oscillated in still water In the heave mode. The section was given

an initial roll velocity, and was allowed freedom In the roll mode. The heaving motion caused a periodic variation of the roll restoring moment, since the transverse stability was affected by the change In immersed area of the section. With the proper vertical oscillation frequency and amplitude, unstable roll motion was excited In a fashion similar to that

of the model experiments of Paufling and Rosenberg

(1959)

which were described earlier.

The second example studied was the cargo ship AMERICAN CHALLENGER. This hull form was chosen because of the exten-sive open water tests conducted with a model of this vessel by Paulling In San Francisco Bay. Computer simulations were conducted for both regular and irregular following waves. The irregular wave case was selected to allow direct conipar-ison with Paulling's recent experimental results. The

investigations carried out with the A1ERICAN CHALLENGER were quite extensive, and represent a practical example of the type of vessel stability and capsizing analysis made possible by the time domain computer program.

Results obtained from the computer analyses of these two cases are presented in the following sections. The two-dimensional section results are described first, followed by the more extensive AMERICAN CHALLENGER results.

A. Two-Dimensional Section Results

1. General Observations

The rectangular section studied had a beam of twice the draft and other characteristics as presented in Table 1. This section was harmonically oscillated in the vertical direction, while being allowed freedom to roll. The poten-tial for roll instability lies in the periodic variation of transverse stability caused by the forced heaving motion. The transverse stability varies due to changes in the posi-tion of the center of buoyancy and of the metacenter for different degrees of section immersion. The change in meta-centric height (GM), which governs the roll restoring moment for small angles of heel, is computed for several depths

of immersion in Table 2. Inclusion of the section roll angle in this computation would yield the same general result,

Length 1.0 foot Beam 10.0 feet Depth 10.0 feet Draft 5.0 feet 4.07 feet 0.0967 feet

Roll gyradius in air 4.0 feet

Roll gyradius in water 4.58 feet

Pitch gyradius 100,000.0 feet

Heave damping term 0.56 x mass

Pitch damping term 0.00 x mass

Roll damping term 3.80 x mass

Heave added mass term 1000.00 x mass

Roll added mass moment of inertia term . . 5.00 x mass

B

A

C---10.0'

Two-Dimensional Section Section Immersed to Waterline A.

= 4.000 feet 3M = I/A = 1.041 feet KM = 5.041 feet = 4.000 feet

GM 1.041 feet,

Section Immersed to Waterline B.

= 2.500 feet 3M = I/A = 1.667 feet KM = 4.167 feet KG = 4.000 feet

= 0.167 feet.

Section Immersed to Waterline C.

= 1.000 feet 3M = I/A = 4.167 feet KM = 5.167 feet = 4.000 feet

= 1.167 feet.

where Height of center of buoyancy above keel

3M Distance from center of buoyancy to metacenter KG = Height of center of gravity above keel

GM = Transverse metaoentrio height

I = Transverse moment of inertia of waterline

same frequency as the vertical oscillation and an amplitude variation determined by the section shape and the vertical

oscillation amplitude.

This oscillation procedure simulated the case of an experiment conducted in calm water by force-heaving a body which' is free in the roll mode to respond according to the equation of motion for roll given earlier. In other words, the example studied was similar to the model experiments carried out with a MARINER hull form by Paulling and Rosen-berg (1959). The only difference lies in the simplified section shape used for the computer simulation. The object of these tests was to demonstrate the same type of roll in-stability that occurred in the experiments.

The basic results of the two-dimensional section runs, excluding preliminary tests, are given in tabular form in Appendix E.6 and are shown graphically in Figure 11. The figure clearly shows a pattern of results similar to that shown in Figure 6 for the experimental results of Paulling and Rosenberg (1959). Although the amplitude-frequency plane has not been rigorously divided into stable and unstable

regions, it is clear that the same upward and outward extend-tug unstable region is present here. The region shown cor-responds to the first roll resonance region, since the natural roll period was about six seconds; a stability variation

corres-0

T\f0- PMEU6I0KJAL

5.CT10PI, ZERO 6PP.

0.0967 FEET.

VA.I0U

IIJrflAL

_{IZOLL VLOCT5 AMP}

IJITIAL PHM. A)JGLE3.

Figure 11. Unstable Roll

_Region

for a Two-Dimensional

_Section

In Forced Vertical

Oscillation.

U&ITASLE poim POWJT

0 STABLE.

\

_\

o \

A

AA AA

AAc

,

c

0

I.bo 1.90 2.00 2.10 2.20

220

2.O

2. 0

2. Sensitivity Analyses

most runs were directed toward defining the first resonance region, capsizing also occurred in the second resonance region, with the period of stability variation equal to the natural roll period.

A typical computer print-out for the two-dimensional

section runs is given in AppendixE.5. In general, the results

fell into one of three categories: (1) the initial roll imposed on the section decayed with time; (2) the initial roll built up to some steady-state value which was maintained indefinitely; or

(3)

the initial roll built up over a period of from three to twelve cycles to the point where a capsize

(90 degree heel) occurred. The result obtained was dependent upon the oscillation amplitude and frequency present, causing

either a stable or an unstable roll condition. The

synchron-ism of roll with the heave oscillations was obvious, with the section rolling heavily when the stability was low and.

snapping back upright when the restoring moment was high. In some cases, the initial roll disturbance actually subsided for a time until the roll motion reached the proper phasing with the heave oscillation, at which time the roll built

up again to finally capsize.

by the time required until capsizing occurred, was analyzed as a function of oscillation frequency and amplitude

separ-ately. In Figure 12, for a constant oscillation amplitude of 3.0 feet, the section clearly has the greatest amount of instability in the vicinity of an ostillat ion frequency of 2.24 radians per second. For frequencies outside the range of 1.92 to 2.37 radians per second, capsizing did not occur with a

3.0

foot amplitude. Of course, as one moves to lower frequencies, the second resonance region will eventually be reached. But each resonance region itself is clearly defined.

Choosing a frequency of 2.24 radians per second, Figure

13

shows time to capsize as a function of the oscillation amplitude. As would be expected, a certain threshold ampli-tude is required to cause capsizing, above which the capsizing tendency increases with amplitude.

For a given oscillation amplitude and frequency, the sensitivity of capsizing to initial conditions was

investi-gated. First the initial roll velocity was varied, with the results shown in Figure 14. These results indicate that, with regular heave oscillations, capsizing will occur

even-tually with even the smallest initial roll velocity, provided that the proper amplitude and frequency are present. The same general result was obtained by varying the initial phase angle of the heave oscillation. The results, shown In Figure

100

90

80

0 V

Ii

*

60

50

'10

F

20 /0 0

VE11CAL 06CILL4T0PJ AM?UTU

3.0 FEET,

IMITIAL PHA6E

2

AQA16.

J I I I I I I I I I I I I I I I I I I I I I $ I I I I I I -I I I I I I I I I I I

4

"ii

_'4 I I

Ui

0

I4

I

-I I

0

12

I I I I I I I

I.

S I I I I I I I I I I I I I I I I I $ I I I I I I I

1.80 190 a.00 2.10 2.20

2.O

2.qo 2.50

OCILLAT0PJ

QU3UC',

t6/6CO1u0

- Figure 12. Time to Capsize as a Function of Vertical

50

10

30

20

Fio

0

CLLTI0

2.2'4 2AVA/6ECOkJ2

1MrnAL PH4&$ MM3LE

2 7 A/il

At'L&U6.

I

0

.0

2.0

3.0

40

5.0

VERTICAL 0CILLATI0U AMF'LITUPe, FEE.T

FIgure 13.

_{Time to Capsize as a Function}

_{of Vertical}

l00

90

80

7o

60

'Ii

c3O

20

I0

0

R0 6PW.

0.0967

FteT. O3CILtAT1Ot4

AMPurUP

3.0FEET,

2.25 RAPJ5/6ECOUP, IAJrrt&L. PIW6e AJGLE 90.

0

10 15

ILLITIAL

0LL vL.ocrr'i', s/.cow

Figure ll.

_{Time to Capsize}

_{as a Function of Initial}

Roll Velocity.

60

40

30

I0

0

TIO- PM&IUoIOUAL 5ECT0U

,

_{R0 PaEP.}

0.0967 FaT, RJtTIL 0LL V

Crr'= i6.7 PE62EE./6EcoWP.

VERTICAL 06CILL4TI0IJ FQUJ'JCY: 2.25

4MPL(TUP. = 3.0 FET

I I I I I I

90'

18d

Z70'

360

IPJITlL O5CILLATIOU PljME AMGL, V4U

Figure 15.

Time to Capsize as a

_{Function of' Initial}

initial conditions than with others, but that the final result is the same. The zero degree phase angle refers to an initial

immersed heave position; as the section moves upward from this position, stability decreases as indicated in Table 2. This critical initial phase angle causes the section to lose roll

restoring moment at the same time as it is heeling over from

the upright position due to the initial roll velocity.

The main purpose of the test runs with the two-dimensional

section was to demonstrate the capabilities of the computer

program. The results presented in this section indicate the successful simulation of unstable roll motion caused by forced vertical oscillation. The runs carried out represent the

type of experiment carried out by Paulling and Rosenberg

(1959),

and led to similar results. The buildup of roll

motion through periodic changes in the roll restoring moment can easily be understood by this simulation in still water. Thus the fundamentals of the phenomenon under study have

been successfully demonstrated using the time domain computer

program.

B. Results for Cargo Ship in Following Waves

An extensive Investigation of the AMERICAN CHALLENGER hull form was carried out in both regular and irregular following waves. Digitized hull offsets for the AMERICAN CHALLENGER, connected with straight lines as input to the computer program, are shown in Figure 16 for the eleven

FIgure 16.

stations used. The vessel characteristics and hydrodynamic coefficients for this 521-foot freighter are given in Table

3;

estimated values for the hydrodynamic terms are presented in Appendix C. Although not particularly critical from the stability viewpoint, the AMERICAN CHAT.LENGER was used since it was the subject of intensive model tests in San Francisco

Bay. It was hoped to relate computer results to this exper-imental data.

Altogether, well over 150 runs were made with the AMERICAN CHALLENGER, first on the IBM 1130 computer and then on the

CDC 3300 computer. Very close agreement was found between results using the two different computers, although slight differences were present due to truncation errors. A number

of capsizings were obtalned;for example, 58 out of the 128

cases investigated on the CDC 3300 computer capsized. The results were exactly repeatible using the same computer and

input data, and appeared entirely consistent.

1. Determination of Critical Roll Resonance Condition

For a vessel proceeding with a forward speed in following

waves, the mechanism of stability variation is somewhat more complex than for the forced vertical oscillation of a section in still water. For the case in waves, the frequency of

stability variation 'is equal to the wave encounter frequency since this is the frequency of alternation of wave crests and troughs along the vessel length. The encounter period

TABLE 3.

VESSEL CHARACTERISTICS FOR THE AMERICAN CHALLENGER,.

Length between perperndicuiars _{521.00 feet}

Maximum breadth, molded _{75 00 feet}

Design draft, molded _{29 25 feet}

Displacement at 29.25 foot draft _{18,871.21 tons S.W.}

at 29.25 foot draft

_{31 64}

_feet

Pitch gyradius _{143.00 feet}

Roll gyradius In air _{25 50 feet}

Roll gyradius in water

_{27 50}

_feet

Ship sieed

_{16 50}

_knots

Heave added mass term ...1.00

_{x mass}

Heave damping term ₀

₄₅

_{x mass}

Roll added mass moment of inertia term. . .

105.60

x mass

Roll linear damping term

_{44 40}

_{x mass}

and frequency for a ship in deep water may be expressed as functions of wavelength, ship heading, and ship speed by the following equations, derived from Lewis (1967):

L/(J

cosp)

v

je -

cos9)

where Te = wave encounter period, seconds = wave encounter frequency, Hz = wavelength, feet

V = ship speed, feet per second

and = ship relative heading angle; zero degrees for following seas, 180 degrees for head seas. Wave encounter periods and frequencies for a vessel at a given speed with various wavelengths and vessel headings, adapted from Kastner (1973), are shown in Figure 17 for the case of a vessel speed of 16.5 knots. This graph clearly

shows the condensation of the encounter spectrum for the quar-tering and following sea cases. It can be seen that waves

of almost any length will all encounter a vessel at the same frequency in a quartering sea situation. The seriousness

of this situation can be imagined, if this encounter frequency happens to bear a critical relationship to the vessel's

natural roll frequency.

Since we are only considering the pure following sea