Report No. CG-D-12-94

AD-A282 7161111l111111111111lll ll

Experimental Study Of An Automatic Pitch Control System

On A Swath Model

Edward M. Lewandowski

Davidson LaboratoryStevens Institute of Technology .. Ok

Castle Point StationHoboken, NJ 07030 EL

FINAL REPORT UJUNE 1994

This document is available to the U.S. public through theNational Technical Information Service, Springfield, Virginia 22161

&NC 94-23928Prepared for:

U.S. Coast GuardResearch and Development Center1082 Shennecossett RoadGroton, CT 06340-6096 QUALI INSPECTED 5

and

U.S. Department of Transportation , I.United States Coast GuardOffice of Engineering, Logistics, and DevelopmentWashington, DC 20593-0001

94 7 27 099

NOTICEThis document is disseminated under the sponsorship of theDepartment of M'wnsportation in the interest of informationexchange. The United States Government assumes no liabilityfor its contents or use thereof.

The United States Government does not endorse products ormanufacturers. Trade or manufacturers' names appear hereinsolely because they are considered essential to the object ofthis report.

The contents of this report reflect the views of Le Coast GuardResearch & Development Center. This report does not consti-tute a standard, specification, or regulation.

D. L.Motherway

S Technical Director, AUnited States Coast GuardResearch & Development Center1082 Shennecossett Road

'Groton, CT 06340-6096

ii

Technia I Reoort Documentation Paae1. ROprt No. 2. Government Accession No. 3. Recippleie Catalog No.

CG-D-12-94

4. Title and SubItle 5. Report DateJune 1994

Experimental Study of an Automatic Pitch Control System 6. Performing Organization Codeon a SWATH Model

8. Performing Organization Report No.7. Author(s) Edward M. Lewandowski R&DC 0591

9. Performing Organization Name and Address 10. Work Unit No. (TRAIS)

Davidson Laboratory 11. Contract or Grant No.Stevens InstituteCastle Point Station N00014-84-C-0644Hoboken, NJ 07030 13. Type of Report and Period Covered

12. Sponsoring Agency Name and Address Department of Transportation Final ReportU.S. Coast Guard U.S. Coast Guard 14. Sponsoring Agency CodeResearch and Development Center Office of Engineering, Logistics,1082 Shennecossett Road and DevelopmentGroton, Connecticut 06340-6096 Washington, D.C. 20593-0001

15. Supplementary Notes

16. Abstract

The report presents the results of a series of tests which was carded out in order to develop asmall physical model of an automatic pitch control system for a SWATH ship, and to study theinteraction of the control surface with the twin hulls In calm water and regular waves.

A four phase test program was followed to assess the control system: fixed trim, free to heave testsof the unappended hull, tests of Isolated canards, fixed trim tests in calm water and in waves withinstrumented canards, and free to trim and heave tests in regular waves with and without automaticcontrol. Drag, pitching moment, canard lift and drag, etc., were measured and analyzed. Themaximum lift on the canards in waves was found to be significantly larger than that measured instatic tests. Results are presented in graphic and tabular form. Tests in regular waves with pitchcontrol showed that a 50% reduction in pitch amplitude is possible in following seas.

17. Key Words 18. Distribution Statement

SWATHautomatic control Document is available to the U.S. public throughunsteady lift the National Technical Information Service,control surfaces Springfield, Virginia 22161seakeeping

19. Security Classaf. (of this report) 20. SECURITY CLASSIF. (of this page) 21. No. of Pages 22. Price

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TABLE OF CONTENTS

ABSTRACT ............................... o....... . . . . . . . i

LIST OF TABLES...................... ....... viii

LIST OF FIGURES. . ...... o... o...... o..... oo-..... . .. . . .. . . .... X

NOMENCLATURE and SIGN CONVENTION ............................ xii

ACKNOWLEDGEMENT ........................................ *.xqY.i

INTRODUCTION ........... .... ... ............. . ....... .. .... I

MODEL. ...... . . . . . . . . .. . . . . . . . . . ....... ._. .2

APPARATUSoo. . . . . . . . . . . .. . . . . . . . . .......o. o.. 3

TEST PROGRAMo ... o... S ... ... . ... .......... ............. 5

TEST PROCEDURE ..... Hev t............ ...... a ...... 6Calibration ......... ......... ................ .......... 6Fixed-Trim Tests in Calm Water .. .......... .... ... .. ... 7Tests in Regular Waves ................................. 7

DATA PROCESSING .................... .........................Calm Water ............. o...............o.................... 8Regular Waves ...................................oo.......... 9Expansion to Full Scale ..... ........ ................. 10

RESULTS*..*.. *....... ........ ... . . ..... ............ io.. 10Fixed Trim Tests in Calm Water .......... o.......o......... 10OFixed Trim and Heave Tests with Instrumented Canard ..... _10Free to Pitch and Heave Tests with Control System ........ 11

DISCUSSION..o... o.... ooooooosoooooooolTests in::Cal::Water.........::... :::::::::..... :::::::::2Tests Fixed in Regular Wvs...............1Tests with Automatic Pitch Control .............. ...... 16Large Motions in Following Seas. .... o................... 08

CONCLUS IONS- o o .... o o ......... . . .. . . .. . . .. . . .. . .... _ 19

RECOMMENDATIONS...................... .................. o20

REFERENCES ......... ... . . .. . ......... . .. .o.o..... ...... _ 22

TABLES 1-10 ..... o ... _ o..... o ......... .. . .. .. . .. .......... 24

FIGURES 1-17 ..... o... . . . .. ....... .. . ... .... . ........... 42

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TABLE OF CONTENTS (Concluded)

APPENDIX A INCLINING EXPERIMENT .............................. Al

APPENDIX B VIDEOTAPE SCENARIOS .............. .s. ... .... ... .B1

APPENDIX C FORCED OSCILLATION TESTS .......................... C1

APPENDIX D CALIBRATION OF FIN BALANCE ........................ D1

APPENDIX E TABULATION OF WATER TEMPERATURES .................. E1

APPENDIX F COMPUTATION OF ANGLE OF ATTACK OF CANARDSINDUCED BY REGULAR WAVES ..................... * ... F1

APPENDIX G EVALUATION OF AN EXPRESSION WHICH OCCURS IN THEANGLE OF ATTACK COMPUTATAION ..................... G1

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ABSTRACT

A series of tests was carried out to develop a model of anautomatic pitch control system for a SWATH ship, and to study theinteraction of control surface (canard) with the ship in calmwater and in regular waves. In one series of tests, a canard wasinstrumented to measure lift and drag; maximum lift on the canardin waves was found to be significantly larger than that measuredin static tests. Tests in regular waves with pitch controlshowed that a 50% reduction in pitch amplitude is possible infollowing seas.

KEYWORDS

SWATHAutomatic ControlControl SurfacesUnsteady LiftSeakeeping

A8oesston 1orN TIS GRA&Il ["

J I~ieaon

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LIST OF TABLES

TABLE A PITCH CONTROL OPTIMUM GAIN FACTORS .................. 12

TABLE B LOCATION OF APPARENT CENTER OF PRESSURE ............. 13

TABLE C CONDITIONS FOR MODEL SINKINGS IN FOLLOWING SEAS ..... 18

TABLE 1 SHIP PARTICULARS ........................ o ........... 24

TABLE 2 DIRECTORY OF DATA TABLES ............................ 25

TABLE 3 CALM WATER TESTS OF UNAPPENDED MODELFIXED IN TRIM ..................... %................. 26

TABLE 4 TESTS IN CALM WATER WITH INSTRUMENTED CANARD;FIXED TRIM AND HEAVE .................. . ............ 29

TABLE 5.1 TESTS WITH INSTRUMENTED ASPECT RATIO 1 CANARD INREGULAR WAVES. FIXED TRIM AND HEAVE. MODELSCALE UNITS ...... ..... .. .. ...... .... ......... 30

TABLE 5.2 TESTS WITH INSTRUMENTED ASPECT RATIO 1 CANARD INREGULAR WAVES. FULL SCALE EXPANSION OF MEANQUANTITIES ......................................... 31

TABLE 5.3 TESTS WITH INSTRUMENTED ASPECT RATIO 1 CANARD INREGULAR WAVES. CANARD LIFT AND DRAG COEFFICIENTS...32

TABLE 6.1 TESTS WITH INSTRUMENTED ASPECT RATIO 2 CANARD INREGULAR WAVES. FIXED TRIM AND HEAVE. MODELSCALE UNITS ........................................ 33

TABLE 6.2 TESTS WITH INSTRUMENTED ASPECT RATIO 2 CANARD INREGULAR WAVES. FULL SCALE EXPANSION OF MEANQUANTITIES ......................................... 34

TABLE 6.3 TESTS WITH INSTRUMENTED ASPECT RATIO 2 CANARD IN

REGULAR WAVES. CANARD LIFT AND DRAG COEFFICIENTS... 35

TABLE 7 TESTS IN REGULAR WAVES WITHOUT ACTIVE CONTROL ....... 36

TABLE 8 TESTS IN REGULAR WAVES TO OPTIMIZE CONTROL SYSTEM... 37

TABLE 9.1 TESTS IN REGULAR HEAD WAVES WITH AUTOMATIC PITCHCONTROL. CANARD ASPECT RATIO 1 ..................... 38

TABLE 9.2 TESTS IN REGULAR FOLLOWING WAVES WITH AUTOMATIC PITCHCONTROL. CANARD ASPECT RATIO 1 ..................... 39

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LIST OF TABLES (Concluded)

TABLE 10.1 TESTS IN REGULAR HEAD WAVES WITH AUTOMATIC PITCHCONTROL. CANARD ASPECT RATIO 2 ..................... 40

TABLE 10.2 TESTS IN REGULAR FOLLOWING WAVES WITH AUTOMATIC PITCHCONTROL. CANARD ASPECT RATIO 2 ..................... 41

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LIST OF FIGURES

SKETCH A PARTICLE VELOCITIES AND PITCH MOMENT IN HEAD ANDFOLLOWING SEAS ..........................**........... 17

FIGURE 1 SWATH CONFIGURATION ................. .............. 42

FIGURE 2 PLAN AND SECTION VIEW OF MODEL FINS ................. 43

FIGURE 3 SWATH MODEL RUNNING WITH FIXED TRIM IN CALM WATER... 44

FIGURE 4a LIFT AND DRAG BALANCE INSTALLATION IN MODEL ......... 45

FIGURE 4b FIN BALANCE INSTALLATION IN MODEL ................... 46

FIGURE 5 SCHEMATIC DIAGRAM OF PITCH CONTROL SYSTEM ........... 47

FIGURE 6a BEHAVIOR OF PITCH MOMENT WITH SPEED FOR UNAPPENDEDSHIP AT VARIOUS TRIM ANGLES ........................ 48

FIGURE 6b BEHAVIOR OF PITCH MOMENT WITH TRIM ANGLE FORUNAPPENDED SHIP AT VARIOUS SPEEDS .................. 49

FIGURE 7 PITCH STABILITY VS SPEED ............................ 49

FIGURE 8 RESULTS OF INSTRUMENTED FIN ON BODY TESTS INCALM WATER........................ ............... 50

FIGURE 9 PITCH MOMENT VS FIN LIFT AT TWO SPEEDS INCALM WATER ......................................... 51

FIGURE 10 BEHAVIOR OF FIRST HARMONIC OF FIN LIFT COEFFICIENTWITH WAVE LENGTH AND WAVE HEIGHT IN FIXED TRIM ANDHEAVE TESTS IN WAVES ........................... 52

FIGURE 11 BEHAVIOR OF FIRST HARMONIC OF FIN LIFT COEFFICIENTWITH WAVE LENGTH AND WAVE HEIGHT IN FIXED TRIM ANDHEAVE TESTS IN WAVES ............................... 53

FIGURE 12 EFFECT OF ACTIVE CONTROL ON PITCH MOTION,SPEED = 15 KNOTS ..... ................................. 54

FIGURE 13 EFFECT OF ACTIVE CONTROL ON PITCH MOTION,SPEED = 20 KNOTS ................................... 55

FIGURE 14 EFFECT OF ACTIVE CONTROL ON HEAVE MOTION,SPEED = 20 KNOTS .................... %.............. 56

FIGURE 15 EFFECT OF FIN ASPECT RATIO ON MOTION RESPONSES WITHCONTROL, SPEED = 15 KNOTS .......................... 57

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LIST OF FIGURES (Concluded)

FIGURE 16 EFFECT OF FIN ASPECT RATIO ON MOTION RESPONSES WITHCONT'ROL, SPEED - 20 KNOTS .. .. ... . .o . .. .. .. . *.... 58

FIGURE 17 BOW PLOW-ININ FOLLOWING SEAS................. o.....59

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NOMENCLATURE and SIGN CONVENTION

A A constant in a Fourier series

a Wave amplitude, ft; also acceleration ft/sec2

B a constant in a Fourier serids

CD1, CL, Amplitude of first harmonic of drag or lift coefficient

CD2, CL2 Amplitude of second harmonic of drag or lift coefficient

CTM Model total resistance coefficient

CTS Ship total resistance coefficient

c Fin chord, ft

D Drag, lb

d the differential operator

EHP Effective horsepower

g Acceleration of gravity, ft/sec2

9l Pitch displacement gain factor, deg/deg

92 Pitch angular velocity gain factor, deg/(deg/sec)

h Water depth, ft, or height of towpoint, ft

I moment of inertia, slug ft2

L Lift, lb

LBP Ship length between perpendiculars, ft

IShip length between perpendiculars, ft

M Pitching moment, ton ft (ship) or lb ft (model),positive bow up

Mp Pitching moment measured about an axis passing throughthe towpoint, ton ft (ship) or lb ft (model), positivebow up

m Mass, slugs

N The highest harmonic used in a finite harmonic series

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NOMENCLATURE and SIGN CONVENTION (continued)

n Frequency coefficient in a harmonic series

Phase Phase lag, degrees

R A calibration matrix

Re Reynolds number, Vc/v

Rij An element in a calibration matrix

rps Radians per second

RS Ship resistance (full scale), lb

S Projected area of fin, ft2

T Wave period, sec

Te Wave encounter period, sec

t Time, sec

u Wave particle horizontal velocity component, fps

V Velocity, knots (ship) or fps (model)

v Wave particle vertical velocity component, fps, orvoltage reading

W A weight, lb

w A tare weight, lb

y Vertical coordinate of canard relative to calm watersurface, ft (below water surface is negative)

z Heave, positive upward, ft

z1, z2 Amplitudes of first and second harmonics of heave, ft

a Angle of attack of canard relative to hull, deg,positive leading edge up

01 , 02 Amplitudes of first and second harmonics of canardangle, deg

A phase angle, radians

0 Pitch angle, deg, positive bow up

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NOMENCLATURE and SIGN CONVENTION (Continued)

0 1, 02 Amplitudes of first and second harmonic of pitch, deg

iPitch angular velocity, deg/sec

A Wavelength, ft

Kinematic viscosity, ft2/sec

p Density, slug/ft3

oFrequency ratio, "1o

Frequency, radians/sec

We Encounter frequency, radians/sec

Wo Natural frequency, radians/sec

Subscripts

1 First harmonic, also first channel

2 Second harmonic, also second channel

e Encounter

i An index, e.g. the ith channel or the ith harmonic

m Model

n Number associated with a harmonic in a harmonic series

o The lowest order harmonic, or mean, in a harmonicseries; also an amplitude

s Ship

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NOMENCLATURE AND SIGN CONVENTION (Concluded)

-------- --------

SIGN CONVENTION (Positive senses shown)

A t it 4.*

0 7rIKx

Sign convention for wave particle motions

(Positive senses shown)

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ACKNOWLEDGEMENT

The author is indebted to Dr. T. Glenn McKee, Chief of theMathematical Services Division at Davidson Laboratory, for thecalculation of the Fourier series expansion of the canard angleof attack in waves and for preparation of Appendix G.

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INTRODUCTION

The advantages of Small Waterplane Twin Hull (SWATH) shipsover conventional hulls in applications requiring minimalmotions and high sustained speeds in heavy seas are welldocumented. Because of its reduced waterplane area, the naturalperiods of the motions of the SWATH are longer than those ofconventional hulls, effectively detuning it from the modalperiods of the most commonly occurring seaways. Additionally,the wave forces on the submerged hulls of the SWATH are smallerthan those on a monohull ship of equivalent displacement,because thp5 wave-induced velocity field decreases exponentiallywith depth'5 [Superscripts refer to References on page 22].

SWATH hulls are typically fitted with fins to enhancestability at high speed in calm water. The fins also provideincreased damping of motions in waves. Further reductions inthe motions when the vessel is underway are poffiJe by activelycontrolling the fins13 Theoretical analyses PL and limitedfull-scale experience have indicated that an active controlsystem would significantly increase the already substantialoperating envelope of the SWATH vessel and would certainlycontribute to increased comfort of passengers and crew in allsea states. However, prior to the present work no model testshave been carried out to validate the theoretical work and tostudy the positive benefits and limitations of active motioncontrol.

As part of its extensive research program on SWATHhydromechanics being conducted at Davidson Laboratory, the U.S.Coast Guard has undertaken the development of a towing tankmodel of a possible SWATH pitch control system and a study ofthe associated modeling laws. This pioneering work is thesubject of the present report.

To study the hydrodynamics of a SWATH with automatic pitchcontrol, the following four phase test program was developed inwhich the major components of the system - hull and controlsurfaces - were first tested independently and then assembledfor the final evaluation of the control system:

1. Fixed trim, free-to-heave tests of the unappended hull.

2. Tests of isolated canards.

3. Fixed trim tests in calm water and in waves withinstrumented canards.

4. Free-to-trim and heave tests in regular waves with andwithout automatic control.

The unappended hull was tested fixed in trim, in calm water, inthe first phase. This was done to determine the relationshipbetween pitch moment and trim at various speeds. In the secondpart of the program, the canards were tested alone on a

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groundboard at various speeds and angles of attack. The thirdphase consisted of tests with an instrumented canard mounted onthe model. The tests were conducted in calm water and inregular waves with the model fixed in trim and heave. In thefinal phase of testing the model was run free to pitch and heavein regular waves with and without active control. Phase 2 isreported separately in Reference 1; phases 1, 3 and 4 arereported herein.

Tests were carried out in the High Speed Test Facility(Tank 3) of the Davidson Laboratory in January, March, May andAugust, 1987; some of the tests were observed by Mr. James A.White of the U.S. Coast Guard. Testing was funded underContract N00014-84-C-0644 (Task 7), Office of Naval Research.

MODEL

An existing 1/24-scale model of a U.S. Coast Guard SWATHdesign, designated as SWATH 10, was used in these tests. Withthe exception of the cylindrical midbodies of the lower hullsand the wet deck, the model was constructed from pine. The wetdeck was made from 1/2 inch marine plywood and the hullmidbodies were made from foam-packed ABS plastic tubing. Figure1 is a drawing of the configuration, which gives all majordimensions (ful1-scale). Particulars are listed in Table 1.

Canard and stabilizer fins, fitted for phases three andfour of the tests, were made from plexiglass. Based on theresults of Reference 1, NACA 0015 section canards, and existingNACA 0015 section stabilizers with an aspect ratio of 1.195 wereused for these tests. The canards were fixed at various anglesin phase 3 and active in phase 4. Figure 2 is a drawing of thecanards.

To help induce a turbulent boundary layer, Hama strips wereplaced on the lower hulls, struts and appendages of the model.The strips consist of a double layer of electrical tape cut withpinking shears to form a serrated leading edge. They wereplaced on the hulls and struts at five percent of the hulllength aft of the nose, and five percent of the strut length aftof the leading edge. On appendages, the strips were placed fivepercent of the local chord length aft of the leading edge. TheHama strips can be seen on Figure 3, which is a photograph ofthe model running in calm water.

The model was ballasted to the 14.5 foot waterline,corresponding to a full scale displacement of 591 long tons.For the fixed trim tests (phases 1 and 3), the center of momentswas located 51.25 feet aft of the strut leading edge (thenominal LCG of the SWATH 10 configuration, as reported inReference 16) and 27.25 feet above the baseline (a convenientlocation for the deck-mounted instrumentation, also consistentwith the tests reported in Reference 16). Measured pitchmoments were transferred to a point on the full-scale thrustaxis, which was taken to be on the centerline of the lowerhulls, 5 feet above the baseline. This was done to remove theeffects of thrust from the moment results.

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For the free to pitch and heave tests (phase 4), the modelwas towed from a point 27.75 feet above the baseline (the "pivotpoint"). The construction of the existing SWATH 10 model didnot permit towing from the thrust line, which would have been abetter arrangement. The towpoint was located at the apparentlongitudinal center of flotation, 63.25 feet aft of the strutnose. This was done to minimize pitch-heave coupling at thepivot point (the location of the pitch sensor, which producedthe input signal to the control system). Because the contractStatement of Work limited the present study to pitch control, itwas desirable to eliminate coupling with heave.

An inclining experiment was performed to determine thelongitudinal GM, which was 24.45 feet. The natural pitch periodat zero speed in the water was 11.76 seconds. Prior to thedynamic tests, the pitch gyradius of the fully equipped modelwas determined to be 38.4 ft; this is close to the value of 38.2ft reported in Reference 2 for this model. Other particularsappear in Table 1. Details of the LCF determination and theinclining test are given in Appendix A.

APPARATUS

Tests were conducted in the Davidson Laboratory High SpeedTest Facility (Tank 3). Figure 3 shows the apparatus used inthe fixed trim tests. A pivot box was mounted in the model atthe second deck level. Two screws on the pivot box permit theadjustment of trim to any desired angle. Trim was measured bymeans of an inclinometer, also located on the deck. Above thepivot box was a moment balance, used to measure pitch moment,and a stainless steel drag balance. These were attached to thecrosspiece of a free to heave apparatus. Heave, the verticalmotion of the towpoint relative to the static floating location,was also measured.

For the third phase of the tests, the model wasinstrumented to measure lift and drag on the starboard canard,as well as the total drag and pitch moment on the model. Thefin balance was located inside the lower hull as shown on Figure4a. Figure 4b is a photograph of the installation in the model;the photograph is taken looking aft from a position forward ofthe model (the hull nose has been removed). The angle of attackof the canard was adjustable by means of a clamp on the shaft.These tests were conducted at zero nominal trim; however, trimwas monitored using the inclinometer on deck. The phase threetests were conducted with the model fixed in heave at the 14.5foot waterline level.

The fourth phase of the tests employed an automatic pitchcontrol system. The system is shown schematically on Figure 5.For these tests the model was free to pitch and heave but fixedin surge, sway, roll and yaw. The rotary differentialtransformer in the pivot box produced a voltage signalproportional to pitch angle. This signal was passed to thecontrol box, mounted on the towing carriage. The output of thecontrol box to the servo motors was a signal linearly

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proportional to pitch angle and rate of change of pitch angle;thus the fin angle was related to pitch angle as follows:

a - -gl1 + 92;

where a is the canard angle of attack relative to the hull, 0and 0 are the pitch angle and pitch rate, and g, and g2 are thegains (degrees per degree and degrees per (degrees per second),respectively). It should be noted that a positive pitch angle(bow up) gives rise to negative canard angle (leading edgedown). The gains were adjustable using potentiometers on thecontrol box, in the following ranges (model scale):

0< g, < 7.07 degrees/degree

0 1 g2 ; 1.97 degrees/(degrees/sec)

The response of the control system to the input signal wasessentially instantaneous, due to a tight feedback loop of finposition and rate to the servo motors. The output signal to theservo motors was used to monitor the canard angle of attack.Mean drag was measured during these tests, as well as pitch,heave and canard angle.

A wave strut attached to the towing carriage, forward ofthe model near the tank edge, was used in the phase three andfour tests to measure encounter period and to establish thephase of the model forces and motions relative to the waves.

Voltage signals from the balances, proportional to forcesor moments, and from heave and pitch or trim transducers,proportional to linear or angular displacement, were amplifiedby signal conditioners on the carriage and transmitted throughoverhead cables to a shore-based analog to digital converter andMASSCOMP computer for processing and storage. Processed data(for example, mean forces and moments in engineering units) wereprinted out at tankside. Time histories of transducer signalswere monitored at tankside on an oscillograph recorder. Alldata has subsequently been backed up from the computer disk toone quarter inch magnetic tape.

Regular waves were generated using the dual-flaphydraulically-driven wave machine3 located at the end of thetank. The wavemaker was controlled by a PDP-11 minicomputer;desired wave lengths and wave heights were entered on a tanksideconsole to start a run.

All good runs were recorded on VHS videotape. Videotapescenarios are given in Appendix B. In addition, still colorphotographs were taken of selected runs.

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TEST PROGRAM

As explained above, the test program had four parts. Theprogram for each phase of the test is described below:

Phase 1: Calm water fixed trim tests of the unappendedmodel. Model free to heave but otherwise fixed.

Speeds: 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,20 knots

Trim angles: -2, -1, 0, 1, 2 degreesMeasure: Drag, pitch moment, heave

Phase 2: Tests of canard fins on a groundboard. Thesetests are reported in Reference 1.

Phase 3: Calm water and regular wave tests of model withinstrumented canard. Model fixed in all degrees of freedom,including trim and heave.

a) Calm water tests

Speeds: 10, 15, 20 knotsCanard angles: 0, 5, 10, 15, 20 degreesCanard aspect ratios: 1, 2Measure: Drag, pitch moment, fin lift and drag

b) Regular wave tests: Head seas

Speeds: 10, 20 knotsWave length/Ship length (LWL): 1, 1.5, 2, 2.5, 3, 3.5Wave heights: 4, 8 ft (8, 12 ft for two longest waves)Canard aspect ratios: 1, 2Canard angle: 0 degreesMeasure: Mean drag and pitch moment; mean and

oscillatory canard lift and drag

c) Regular wave tests: Following seas

Speeds: 10, 20 knotsWave length/Ship length (LWL): 1, 2 (10 knots)

1, 2.5, 3.5 (20 knots)Wave heights: 4, 8 ft (8, 12 ft for two longest waves)Canard aspect ratios: 1, 2Canard angle: 0 degreesMeasure: Mean drag and pitch moment; mean and

oscillatory canard lift and drag

The number of conditions in following seas was reducedbecause of the impossibility of encountering a sufficient numberof waves for proper analysis in a tank of finite length.

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Phase 4: Regular wave tests with automatic pitch controlsystem. Model free to pitch and heave but restrained in surge,sway, roll and yaw.

a) Tests with inactive fins (baseline)

Speeds: 15, 20 knotsHeadings: Head seas, following seasWave length/Ship length (LWL): 1.5, 2.5, 3.5Wave heights: 4, 8 ft (8, 12 ft for longest wave)Canard aspect ratio: 1Measure: Mean drag; heave, pitch

b) Tests with active fins

Speeds: 15, 20 knotsHeadings: Head seas, following seasWave length/Ship length (LWL): 1.5, 2.5, 3.5, 4.0Wave heights: 4, 8 ft (8, 12 ft for two longest waves)Canard aspect ratios: 1, 2Measure: Mean drag; heave, pitch, canard angle

Part b) of the phase four tests was conducted with the controlsystem optimized, that is, with the gains set to minimize theamplitude of the pitch motion. To find the optimum settings, 10runs were made in head seas and 15 runs were made in followingseas with various gain settings (see Results).

It was not possible to run all of the conditions in thetable above in following seas, in some cases because asufficient number of. waves was not encountered for properanalysis. In other cases a dangerous condition arose in whichthe model attained a pronounced bow down attitude, eventuallyplunging into the water. These results will be described inmore detail later.

Wave heights listed above are nominal values. Actual waveheights (given in the data tables) were determined from the wavemachine calibration.

In addition to this test matrix, a series of runs was madein calm water at two speeds with the canards oscillating, over arange of frequencies, to study the response of the system tosinusoidal excitation. These tests are described and theresults are presented in Appendix C.

TEST PROCEDURE

Calibration

All transducers were calibrated immediately prior to thetest. The drag and pitch moment balances were calibrated byapplying known forces and moments, taking readings, and fittinga straight line to the results, the slope of which is thecalibration rate. The heave and pitch transducers and theinclinometer used for steady trim were calibrated by setting

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known heave or trim displacements and taking readings. Allcalibrations were well represented by straight line fits.

The two component fin balance was calibrated by applyingknown weights in the direction of lift, drag, and combinationsof the two, taking voltage readings on both channels, and usinga multivariate least squares fit to express the digitizedvoltage readings as linear functions of both the lift and thedrag. The resulting matrix of coefficients was inverted toobtain the calibration rates. In addition, because the balancewas to be used to make dynamic force measurements, a dynamiccalibration was carried out, which showed the response of thebalance to be flat in the range of frequencies to be encounteredin the tests. The calibration procedure is explained in detailin Appendix D, which includes calibration results, plots, and aphotograph of the dynamic calibration rig.

The wave machine was calibrated by running waves of thenominal length and height past a stationery wave wire located130 ft from the wave machine; wave heights and periods weremeasured.

Fixed-Trim Tests in Calm Water

Zero readings were taken on all transducers, with theexception of the trim inclinometer, with the model floating atzero trim. The inclinometer reference (zero voltage) washorizontal. The model trim was next set to the desired valueand a run was made. Data were collected in a 50 foot run lengthafter the model had reached steady speed, and the resultsaveraged for the time of the run. The resulting readings, minusthe zero readings, were multiplied by the calibration rates toobtain measured quantities in engineering units.

Tests in Regular Waves

Zero readings were again taken on all channels with themodel floating at level trim, when the water was sufficientlycalm (generally in the morning before the first run and afterlunch break). To start a run, the desired wave length andheight were entered into the PDP-l1 computer and the wavemachine was activated. For the head seas tests, the model wasstarted just before the waves reached the beach (located at theopposite end of the tank from the wave machine). Data werecollected in a 110 foot run length after the model had reachedsteady speed. For the following seas tests, the model wasstarted after a sufficient length of the wave train had gone byso that the model would not overtake the waves before the end ofthe run. Data were collected in an 80 ft run length; theoscillograph traces were then carefully examined for the effectsof wave reflections from the beach which often occurred near theend of a run in the longer waves. If necessary the timehistories to be processed were truncated to eliminate theapparently corrupted portion.

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Preliminary tests in calm water with the model free topitch and heave showed that the model had a pronounced bow-downtendency. This was due in part to the towing arrangement: Themodel was towed from a point above the deck, introducing a bow-down moment not present on the ship, where the thrust is appliedalong the centerline of the lower hulls. In addition, the flowover the hulls introduces a bow-down pitching moment; the highpressure at the nose is indicated by the bow wave. The particlevelocities of the bow wave induce a downward force on thecanards. To achieve zero mean trim on the ship it was assumedthat the SWATH operator would adjust the stabilizers (the aftfins), allowing the canards (controlled automatically) tocounteract the oscillatory pitch induced by waves. For themodel tests the stabilizers were set to -150 (the downward forceon the stern stabilizers brings the bow up). Final adjustmentto near zero mean trim was achieved by shifting a small amountof weight on deck:

Speed Weight Distance

15 kt (5.17 fps) 1.50 lb 4 ft20 kt (6.89 fps) 0.50 lb 4 ft

The weight shifts were symmetrical about the CG so that thepitch moment of inertia was unaffected.

Because the weight of the model and apparatus exceeded itsdisplacement at the 14.5 ft waterline, it was necessary tounload. This was accomplished by attaching weights to an armwhich applies an upward force, equal to twice the appliedweight, at the pivot point. During dynamic tests, this meansthat the mass being accelerated by the waves was about 7 percentgreater than the mass of the model plus hydrodynamic added mass.Since the effectiveness of the pitch control system was judgedby comparison of results for this same configuration with andwithout control, the extra mass was not perceived to be ofcritical importance in interpretation of the results.

Water temperature was monitored daily during all tests. Atabulation of water temperatures is included as Appendix E.

DATA PROCESSING

Calm Water

For the calm water tests, the data, sampled at a rate of100 Hz, were averaged over the duration of the run. Theresults, minus the zero readings, were multiplied by thecalibration rates to obtain forces, moments and displacements inengineering units.

he effect on the pitch moment of the height of the towpointabove the full-scale thrust line was accounted for by

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mathematically transferring the thrust to the centerline of thelower hulls:

M = Mp + D h

where M is the reported pitch moment, Mn is the moment measuredabout the axis passing through the towpdint, and h is the heightof the towpoint above the centerline of the lower hulls, 1.8542ft model scale. This is equivalent to transferring the momentreference to the thrust line. All reported pitch moment datacontain this correction.

Regular Waves

In the regular wave tests, time histories of voltagesignals on all channels were recorded on the computer disk. Thefundamental period of the excitation, the wave encounter period,was determined by counting the number of zero crossings in thetime history of the wave strut signal, as described in Reference4. The signals from each channel were then fit to an expressionof the form

N Nv i = aio + E ain cos(2*nt/Te) + E bin sin(2*nt/Te)

n=nO n=nO

by the method of least squares, where Te is the encounterperiod, N and no are the orders of the highest and lowestharmonics used in the fit, and the coefficients ain,. bin are theamplitudes determined by the fit. The expression is thenwritten in the form

Nv i = Cio + E Cin cos(2,nt/Te - in)

n=nO

where CiQ is the mean, Cmn is the amplitude of the nth harmonic,and 0i is its phase relative to a specified reference channel,which is the pitch channel in this report unless otherwisenoted.

For certain channels only mean quantities are of interest;thus, for drag and pitching moment, only the means Cio(multiplied by appropriate calibration rates) are reported. Forother channels it was found that only the amplitudes of thefirst and second harmonics, Cil and C., were significant, sothat even though a four-term series ?= 4) was used in thefits, only the amplitudes of the first and second harmonics arereported in addition to the means.

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Expansion to Full Scale

Mean drag and pitch moment, heave, wave height, and waveperiod and frequency have all been converted to full scale asnoted in the Results section. Resistance expansion has beencarried out according to the method described in Appendix A ofReference 2. Pitch moment was scaled up according to thefollowing formula:

Ms = (ps/Pm) Mm (LWLs/LWL) 4

where p_ is the density of salt water at 590 F and pm is thedensity of the tank water. Full scale heave and wave heightwere obtained by multiplying model quantities by the lengthratio, and period is scaled by multiplication by the square rootof the length ratio.

RESULTS

Results of this investigation are presented in Tables 3through 10. Table 2 is a brief directory of the data tables.

Fixed Trim Tests in Calm Water

Table 3 contains the results of the calm water fixed trimtests of the unappended model (phase 1). The table contains runnumber; model speed, drag and pitch moment; measured trim; full-scale heave; model and ship resistance coefficients; and shipspeed, resistance, EHP and pitch moment. Figure 6a is a plot ofthe behavior of pitch moment with speed; on Figure 6b the pitchmoment is plotted against trim at several speeds. The slopes ofthe pitch moment vs. trim contours are plotted against speed onFigure 7.

Fixed Trim and Heave Tests with Instrumented Canard

Results of the tests with the instrumented canard in calmwater are given in Table 4. The table lists run number; canardangle of attack relative to the hull; model speed, drag andpitch moment; measured trim (nominal trim was zero degrees);model scale lift and drag on the canard; model and shipresistance coefficients; and ship speed, resistance, EHP andpitch moment. The fin lift and drag coefficients are plottedagainst angle of attack on Figure 8, which also shows theresults of the tests of the fin on a groundboard from Reference1. Pitch moment is plotted against fin lift for two speeds andtwo canard aspect ratios on Figure 9.

Tables 5 and 6 contain the data from the model fixed inregular waves tests, with the aspect ratio 1 and 2 canards,respectively. The tables have three parts. Tables 5.1 and 6.1

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contain all measured data in model scale. The tables list runnumber; model speed; wave height, length and period; encounterperiod and frequency; mean drag and pitch moment on the model;mean, first and second harmonics of fin lift and drag; and thephase lag of the oscillatory forces with respect to the arrivalof the wave crest at the fin shaft. Thus a phase of zerodegrees for Ll, say, would indicate that the first harmonic offin lift coincides with the wave crest at the shaft (actualvalues are near 2700, as would be expected; see Appendix F, pageF2).

Tables 5.2 and 6.2 contain mean quantities from Table 5.1and 6.1, expanded to full scale as described above. The tableslist run number; ship speed; wave length to ship length ratio;wave height and period; encounter period and frequency; totalresistance coefficient; resistance; EHP; and pitch moment.

Tables 5.3 and 6.3 include quantities pertaining to theinstrumented canard, presented in coefficient form or in full-scale units. Listed are run number; ship speed; canard Reynoldsnumber based on mean chord length; wave length/ship length; waveheight; encounter period and frequency; and mean, first andsecond harmonics of canard lift and drag coefficients and theirphases with respect to the wave crest at the canard shaft.

Behavior of the first harmonic of canard lift (which wasthe quantity of primary interest in this phase of the tests)with incident wave length and height are shcwn on Figures 10 and11 for the aspect ratio 1 and 2 fins, respectively. Acomparison is made with the results of a simple theory which isdescribed in the Discussion.

Free to Pitch and Heave Tests with Control System

Results of the fourth phase of the tests, which include theautomatic pitch control system, are presented in Tables 7through 10. The first seven columns of each of these tablescontain (in full scale units where applicable): Run number;ship speed; wave length/ship length; wave period; encounterperiod and frequency; and wave height. Negative encounterperiods in following seas indicate overtaking waves. All phaseangles in these tables represent phase lags relative to thepitch signal; thus the phase of the first harmonic of pitch isalways zero and is not listed.

Table 7 lists results for baseline tests with the controlsystem inactive; canards were fixed at zero degrees relative tothe hull. In addition to quantities described above, the tablegives the phase of the wave (arrival of the wave crest at thepivot point); mean drag (model); mean trim and heave; mean shipresistance and EHP; and first and second harmonics of pitch andheave, with phases.

Table 8 contains the results of tests of various controlsystem gains to minimize pitch motion. In addition toquantities described above, this table contains the firstharmonic of the fin angle and its phase, and the gains g,(degrees per degree) and g2 (degrees per (degrees per second),

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full scale). These tests showed that the optimum gains(settings resulting in the lowest pitching motion amplitudes)were as tabulated below:

TABLE A. PITCH CONTROL OPTIMUM GAIN FACTORS.

I gisilacemen&I (rate. full-scale)

Head seas 0 9.65 deg/(deg/sec)Following seas 6.36 deg/deg 6.76 deg/(deg/sec)

Results of tests in waves with active control are given inTables 9 and 10 for the aspect ratio 1 and 2 canards,respectively, with the gain settings in Table A. Tables 9.1 and10.1 are for the head seas condition; Tables 9.2 and 10.2 arefor following seas. The upper portion of the tables list meanquantities: model drag; trim; mean heave; mean canard angle;and mean ship resistance and EHP. The lower portion of thetables gives oscillatory quantities: first and second harmonicsof pitch, heave and canard angle, and phases. Phase angles areagain with respect to pitch motion.

Effectiveness of the pitch control system is shown onFigures 12 and 13, which show nondimensional pitch amplitude(normalized by maximum wave slope wH/A), against wave length,for speeds of 15 and 20 knots, respectively. The effect of thecontrol system on heave motion (which no attempt was made tocontrol) is shown for 20 knots on Figure 14. Figures 15 and 16compare motion results for the two canard aspect ratios used inthe tests.

DISCUSSION

Tests in Calm Water

Figure 6 shows that the pitch moment is an oscillatoryfunction of speed; this is evidently due to the influence of theship wave system. It can be seen that moment minima occur nearspeeds of 10 and 15 knots; maxima are observed at 12 and 20knots. Thus an increase in speed from 15 to 20 knots results ina large change in pitching moment, from bow-down to bow-up. TheSWATH operator must be alert to this behavior and adjust thecanards and/or stabilizers accordingly.

Figure 6b shows that the pitch moment on the unappendedSWATH is linear with trim in the range -20 to 20 and that theship is statically stable in pitch at least up to 20 knots.Static stability is indicated by the slopes of the lines on theplot, which are negative, indicating that the moment tends tocounteract any change in trim. The trend of static stability,as indicated by pitch moment per degree of trim, with speed isshown on Figure 7; extrapolation would indicate that theunappended vessel could become statically unstable between 25and 30 knots.

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Figure 8 contains the most important results of theinstrumented canard tests in calm water. The figure indicatesthat the lift curve slope of the fin on the body is virtuallyidentical to that for the fin on a groundboard; thus thecombined effects of the free surface and the curvature of thehull on the lift rate are apparently small. Due to the bow wavethere is a downward flow at the canards, inducing a speeddependent negative angle of attack. The downward force on thecanards produces a bow-down moment which would add to thegenerally bow down moment on the hull at zero trim indicated onFigures 6a and 6b. Thus the canards, if not at least passivelycontrolled, will tend always to pull the bow down. This wouldbe destabilizing if the ship develops a negative trim (which isthe tendency of the unappended hull below 20 knots).

If the lift force on a canard is known, it might be assumedthat the pitch moment induced by this lift force is simply theforce multiplied by the distance from the pitch axis to an"effective center of pressure" on the canard. This approachdoes not account for fin-body interaction. To investigate thevalidity of such an approach, a plot of pitch moment against finlift was prepared (Figure 9). It can be seen that the momentincreases linearly with the fin lift, and straight lines havebeen fitted as indicated. The slopes of the lines are equal totwice the distance from the moment reference point (on the hullcenterline, 51.25 ft aft of the strut leading edge) to theeffective center of pressure, since the lift is that on a singlecanard and the moment contains the effects of both canards.Results are summarized below:

TABLE B. LOCATION OF APPARENT CENTER OF PRESSURE

Speed Aspect CPknots ratio location (ft)

15 1 65.7715 2 50.0820 1 76.5020 2 59.23

The distance from the moment axis to the canard shafts is 45 ft;thus the CP locations given above are forward of the canards.It can be concluded that fin-body interaction cannot beneglected when estimating the pitch moment due to the canards.The fact that the apparent lever arm is larger than thegeometric one indicates that some of the low pressure inducedabove the canard when at an angle of attack is "spilling over"onto the hull, so that the upward force induced by the fin islarger than the force on the fin itself (this is the essence offin-body interaction). Additionally, the downwash induced bythe canards may interact with the stabilizers to induce anincreased bow-up moment.

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Tests Fixed in Regular Waves

An interesting aspect of the instrumented fin results inregular waves was that the maximum oscillatory lift coefficientsmeasured were substantially higher than the stall values foundin the steady state tests. The data of Reference 1 show themaximum lift coefficients of the NACA 0015 fins tested on agroundboard to be between 0.7 and 0.8. Figures 10 and 11 showthat oscillatory lift coefficients approached 1.0 in the 12 footwaves at 10 knots. In Run 111, for example, Table 6.3 showsthat the mean and first harmonic of lift coefficient were -0.316and 0.968, respectively, so that during this run CL oscillatedbetween -1.284 and 0.652. No indication of stall was evident inthe oscillograph records, and for this particular run theamplitude of the first harmonic of lift (0.49 from Table 6.1)was practically identical to the RMS of the signal multiplied byJ2:

0.3549 x 1.414 = 0.50

indicating that almost all of the energy of the signal wascontained in the first harmonic. Thus the sinusoidal fit is agood representation of the signal, which would not be the caseif stalling was taking place. It would therefore seem that theoscillatory flow somehow delays the onset of stall.

Though data for airfoils or hydrofoils operating inoscillating flows is relatively scarce (particularly for casesin which flow angles are near the stall angle of the foils),there is a fair amount of data for foils oscillating in pitchand/or heave in a steady stream. The situations are different,but it is expected that general trends with frequency ofoscillation, for example, will be similar, particularly at lowfrequencies where the effects of unsteadiness should be sall.

In the case of pitching oscillation, Halfman et al pointout that "several investigators... have noted that... the stallmay occur at an angle of attack considerably above the staticstalling angle". More recently Wickens , in an investigation ofan NACA 0018 airfoil oscillating in pitch, found that "dynamicstall... occurred about 5 degrees later than for the equivalentsteady flow case. This phenomenon resulted in an increase innormal force of about 20%...when the wing was pitching to 39degrees". This phenomenon is attributed by Ericsson and Redingto the "accelerating flow on the leeward side of (the] pitchingairfoil (which] causes a decrease in the adversity of thepressure gradient, resulting in a large overshoot of the staticstall." They express the dynamic stall overshoot Aa s as8

a= (c&/V) A

Examples of oscillograph records of lift on an oscillatingairfoil can be found in Reference 5: Records taken when stalloccurred were definitely non-sinusoidal and in one case not evenperiodic.

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where c is the chord length, a is the rate of change of angle ofattack, V is the free stream velocity, and af is a dimensionlesstime lag due to the oscillation. The quantity in parentheses isequal to the product of the pitching amplitude and the "reducedfrequency", cw/V, where w is the frequency of oscillation.tEricsson and Reding's analysis of the data of Reference 5indicates that the value of Af is 2. The increase in maximumlift coefficient would then be

(dCL/da) Aa s - (dCL/da)(c&/V)A,.

Referring again to the example of Run 111, the lift curve slopeof the aspect ratio 2 fin is given in Reference 1 as0.060/degree. The measured C amplitude would thus correspondto an angle of attack amplitude of 16.130, and a maximum rate a- 79.52 degrees/second. The formula above then gives the result

ACLmax - 0.41

for this run, so that the stall would be delayed up to a CL of1.2. This is roughly the magnitude observed in Run 111.

In Reference 1 it was concluded that the only importantReynolds number related difference between ship and between shipand scale model appendage lift was a reduction of maximum liftcoefficient at model scale. The present results indicate thatthis reduction is counteracted by the effects of oscillatoryflow. The data from the tests in waves would thus appear to befree of any scale effects on lift, particularly in head seaswhere the encounter frequencies were high. In following seas,the encounter frequencies were low but Tables 5.3 and 6.3 showthat the lift coefficients were generally well below the staticstall values.

The theory of two dimensional airfoils in non-unifo mmotion has been applied to sinusoidal oscillations by Sears ,who treats both the case of a foil undergoing pitching and/orheaving oscillations and the case of a foil penetrating asinusoidal gust. The problem of a fin moving in waves issimilar to the latter case but not identical, since in the gustproblem only the vertical velocity of the fluid is assumed tovary, whereas in waves both velocity components vary. In thetests described here the particle velocity was as much as 28% ofthe model velocity, which cannot be considered negligible.Hence Sears' results are not directly applicable to the case ofa foil in waves.

A three-dimensional theory for predicting hydrodynamicforces and moments on a hydrofoj moving in waves has beenpresented by Tsakonas and Henry'u; the theory leads to anintegral equation which must be solved numerically. However,they note that their results agree quite well with a "quasi-

TThe reduced frequency parameter occurs frequently in unsteadyairfoil theory, and is a dimensionless measure of the angularexcursion per chord length of travel at speed V.

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steady" prediction in which a time-dependent angle of attack iscomputed based on particle velocities calculated from linearwave theory, and simply multiplied by the steady-flow lift curveslope.lt Their quasi-steady theory is strictly only applicableto the gust problem, since the above-mentioned effect ofparticle velocity on the horizontal fluid velocity component(more precisely, the horizontal component of the total fluidvelocity relative to the foil) does not seem to have beenconsidered. A unique, simplified quasi-steady prediction isdeveloped in Appendix F for the case in which the ratio ofparticle velocity to ship speed is small (high speeds or lowfrequencies) but which does not completely neglect thehorizontal particle velocities. It is emphasized that "quasi-steady" refers to the computation of angle of attack and not tothe maximum lift coefficient, which is affected by unsteadinessas discussed above.

The results of the "quasi-steady" prediction of canard liftin the high speed (20 knot) tests are shown on Figures 10 and 11as broken lines. It can be seen that the theory generallyunderestimates the lift, by an amount that increases with waveheight. The simple theory thus yields a conservative estimateof canard lift at 20 knots. A more sophisticated theory, notsubject to the "small particle velocity" restriction, isrequired for accurate predictions at lower speeds.

In following seas, the waves overtook the ship in all runsexcept for the shortest waves and the highest model speed. Theovertaking waves become distorted they pass the ship, so it isreasonable to expect that particle velocities associated withthe waves would be reduced near the bow (at the location of thecanards). Videotapes of the phase three tests show thatconsiderable distortion of the waves does occur at 10 knots; thewaves appear to break just aft of the strut nose. At 20 knots,the waves pass by without much distortion. This would accountfor the lower canard lift coefficient in following seas at 10knots and the near agreement of the lift coefficients in headand following seas at 20 knots.

It should be reemphasized that these tests were carried outwith the model fixed in heave. When free to heave, the modelundergoes considerable sinkage (in excess of 5 feet under someconditions) so that the measured fin forces are not necessarilywhat would be expected were the model free to pitch and heave.

Tests with Automatic Pitch Control

The effectiveness of the automatic pitch control is shownon Figures 12 and 13, for speeds of 15 and 20 knots,respectively. The figures show a moderate reduction in pitchmotion in head seas, and a large reduction in follow'grseas.This is in accord with previous theoretical predictions' 1 and

T T It should be noted that Tsakonas and Henry were surprised bythis agreement, and suggest that it might be somewhatfortuitous.

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full scale trial data13 , but is somewhat surprising in light ofthe force data (Figures 10 and 11) which indicate lower canardlift in following seas. The increased effectiveness of thecanards in following seas may be due in part to the phase of theparticle velocities at the canards relative to the pitchingmotion. For waves longer than about twice the ship length, thewave-induced pitch moment on the hull lags the wave byapproximately 900. In head seas the flow over the canard actsto reinforce this pitch moment, pulling the bow up on the faceof the wave and pushing it down on the back side. In followingseas, the situation is reversed as shown on Sketch A below.

Wave inducea moment

Head seas

t4- D d, 4t

--- ow Wave direction

Following seas

SKETCH A. PARTICLE VELOCITIES AND PITCH MOMENT INHEAD AND FOLLOWING SEAS.

It might also be noticed that these same observations apply tothe stabilizers; this would tend to worsen the pitch motion infollowing seas. More will be said about this later.

The magnitude of the encounter frequency relative to thenatural frequency of the vessel in pitch also has an importanteffect on the magnitude of the motions. For an exciting momentof a given amplitude, such as that induced by the movingcanards, the amplitude of the response will be greater atfrequencies below the natural frequency than at frequenciesabove the natural frequency (in fact, the motion will approachzero at high frequency). Reference to Tables 9 and 10 showsthat the encounter frequencies in head seas were in the range of1 to 2 rps, whereas those in following seas were in the range of0.2 to 0.3 rps. The zero speed natural pitch period of thevessel is 11.76 seconds (Table 1), corresponding to a frequencyof 0.53 rps. The qittural pitch period of a SWATH typicallyincreases with speed-l; thus the encounter frequencies in headseas were well above the natural frequency of pitch motion.Smaller motions would then be expected in head seas than in

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following seas, where encounter frequencies were near or belowthe natural frequency of pitch.

Figure 14 shows that although no attempt was made to reduceheaving motion, heave was also reduced by the action of thecontrol system, again to a greater extent in following seas.Further reductions would be possible by using heave and rate ofchange of heave as additional inputs to the control system.

The effect of aspect ratio of the canards on theeffectiveness of the control system is shown on Figures 15 and16 to be small. Differences are insignificant at 15 knots; at20 knots the aspect ratio 2 fins reduce heaving to a slightlygreater extent.

Large Motions in Following Seas

During several runs in following seas the model attained apronounced bow-down attitude, taking on water over the main deckat the bow. This occurred only at the high speed (20 knots) inthe 8 foot waves, when the wave speed was close to the modelspeed. When the model speed slightly exceeded the wave speed,the model seemed to ride up over the first wave crestencountered in a run, and plow into the back of the next wave.Figure 17 shows one such occurrence. In longer waves, justovertaking the model, the model survived the first encounter butnear the end of the run the stern seemed to be picked up by thesecond overtaking wave, plunging the bow into the water.Conditions under which these phenomena were observed aretabulated below:

TABLE C. CONDITIONS FOR MODEL SINKINGS IN FOLLOWING SEAS

Run A/LWL Ship Wave Wave Automaticspeed, kt speed, kt height, ft control

85 1.0 20 15.1 8.24 no88 1.5 20 18.3 8.26 no97 1.0 20 15.1 8.24 no

101 2.5 20 23.5 8.66 not opt.216 1.0 20 15.1 8.24 yes217 1.5 20 18.3 8.26 yes219 1.5 20 18.3 8.26 yes220 2.5 20 23.5 8.66 yes

After run 101 a large coaming was placed on the deck to keep thedeck mounted electronic equipment dry during the swampings. Thecoaming is visible in Figure 17 and in the videotapes.

A similar phenomenon was observed by Fein et al duringself-propelled model tests of the SWATH SSP Kaimalino:

"Largest motions were found in following seas whenship speed was close to the wave speed. In that casea large bow-down static trim occurred due to the

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action of the wave on the full span aft foil. Thiscondition, which could lead to the upper structure bowbeing buried in the wave and propellq broaching, waslater observed in full scale trials."

The following passage pertains to the full scale trials:

"The only case where deck wetness occurred was infollowing seas when the wave speed approached the shipspeed and a large amplitude but gentle bow 'plow-in'occurred. Propeller broachinY3 occurred in similarconditions in quartering seas."

The severity of the "plowing in" in the present tests mayhave been exaggerated by the towing method. As explained inTest Procedure, the mean hydrodynamic moment on the hull and thecouple due to the height of the towpoint above the thrustlinewere compensated for by setting an angle on the stabilizers andshifting a small amount of weight on deck. When the bow beginsto plunge, the drag increases, giving rise to an increase in thebow-down couple which is not compensated for. However, thetendency for "plowing in" is in accord with the self-propelledmodel and full scale observations quoted above.

In the discussion under Automatic Pitch Control above, itwas pointed out that in waves longer than about twice the shiplength, the flow over the stabilizers gives rise to a momentacting to reinforce the wave-induced pitch moment on the hull.This is the mechanism alluded to by Fein et al in the quoteabove as a cause of the plow-in phenomenon. However, no plow-inoccurred in a 12 foot wave at a wave length to ship length ratioof 3.5, in which case wave-induced stabilizer angles of attackare slightly larger than for the shorter waves in which theproblem occurred. Clearly, further study of this potentiallydangerous phenomenon is warranted.

CONCLUSIONS

This hydrodynamic study of a SWATH vessel has provided muchunique data and has demonstrated the effectiveness of activatingthe canards in reducing the pitching motion in regular waves.Several other important conclusions can be drawn from the dataand discussions above:

1. The active control system employed in this study is mosteffective in following seas, where reductions in pitch motion ofmore than 50% were realized. This is in accord with previoustheoretical predictions and full-scale trial data for otherSWATH configurations.

2. When the vessel operates in following seas, at speeds nearlyequal to the wave speed, large amplitude pitching motions candevelop, possibly leading to the bow plowing into the waves.

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This tendency1Aas also been observed during full-scale trials ofSSP Kaimalino .

3. Measurements of unsteady canard lift in waves indicate thata higher maximum lift coefficient is reached than during statictests. No evidence of stall was detected in the data from thesetests; the lift produced by the fins in oscillatory flow is thusexpected to be fully representative of full scale canard lift.

4. The lift curve slopes of the canards on the hull are thesame as thfse found previously for the canards tested on agroundboard . Pitch moment data indicates that fin-bodyinteraction, and possibly canard-stabilizer interaction, cannotbe neglected in predictions of the moment due to canarddeflection, however.

5. Measured canard lift in regular waves is larger than thatpredicted by a simple quasi-steady approach in which the angleof attack is computed using particle velocities from linear wavetheory. The prediction improves with increasing speed anddecreasing wave height (that is, decreasing angle of attack).

6. The aspect ratio of the canards has little effect on theperformance of the SWATH in waves.

RECOMMENDATIONS

The results of this study have indicated that further workis necessary in order to gain a better understanding of thehydrodynamics of a SWATH in waves:

1. A potentially dangerous phenomenon has been observed infollowing seas. In a situation where the ship just overtakesthe waves, bow plow-in took place quite quickly, occurring justafter the first encountered wave. A SWATH operator thus wouldhave a limited amount of time in which to take corrective actionif such a wave component were present in a seaway (the automaticcontrol system was not adequate to prevent this phenomenon). Itis recommended that a careful study of the behavior of a SWATHin following seas be carried out, in regular and irregularwaves, to establish the conditions under which bow plow-inoccurs, and to explore ways to alleviate the problem. For thesetests, the model should be towed from a poin4 on the propellershaft line, as recommended by the 18th ITTC 4, and should befree to surge. This will minimize any influence of the towingsystem on the behavior of the model.

2. Measurement of the stabilizer forces in calm water and inwaves is recommended, in light of their possible role in causingbow plow-in in following seas. Tests with and without canardsshould be conducted to assess the possible effect of the canardtrailing vortex system on the stabilizer forces. Flow

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visualization tests could also be conducted to study thetrajectories and strength of these vortices.

3. It is possible that further reductions in pitching andheaving motions could be achieved by activating the stabilizers.This possibility also warrants further study.

4. The control system should be extended to include heavenotion control. This would involve the use of heave amplitudeand rate as inputs to the control system in addition to pitchamplitude and rate.

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REFERENCES

1. Lewandowski, E.M., "The Effects of Aspect Ratio, SectionShape, and Reynolds Number on the Lift and Drag of aSeries of Model Control Surfaces", Davidson LaboratoryTechnical Report 2598, December 1987.

2. Klosinski, W.E., and Numata, E., "Powering and SeakeepingModel Tests of a U.S. Coast Guard SWATH Cutter Design",Davidson Laboratory Report 2515, October 1984.

3. Dalzell, J.F., "design, Construction and Installation of aDual-Flap Wave Generator", Davidson Laboratory TechnicalReport 2367, August 1983.

4. Yu, H., "DAP5 Handbook", Davidson Laboratory Technical Note970, January 1988.

5. Halfman, R.L., Johnson, H.C., and Haley, S.M., "Evaluationof High Angle of Attack Aerodynamic Derivative Data andStall-Flutter Prediction Techniques", NACA TN-2533,November 1951.

6. Wickens, R.H., "Wind Tunnel Investigation of Dynamic Stallof an NACA 0018 Airfoil Oscillating in Pitch", NationalResearch Council Canada Aeronautical Note NAE-AN-27,February 1985.

7. Ericsson, L.E. and Reding, J.P., "Dynamic Stall SimulationProblems", Journal of Aircraft, Vol. 8, No. 7, July-1971, pp. 579-583.

8. Ericsson, L.E. and Reding, J.P., "Unsteady Airfoil Stall,Review and Extension", Journal of Aircraft, Vol. 8, No.8, August 1971, pp. 609-616.

9. Sears, W.R., "Some Aspects of Non-Stationary Airfoil Theoryand its Practical Application", Journal of theAeronautical Sciences, Vol. 8, No. 3, March 1941, pp.104-108.

10. Tsakonas, S., and Henry, C.J., "Finite Aspect RatioHydrofoil Configurations in a Free Surface Wave System",Davidson Laboratory Report 1118, June 1966.

11. Lamb, G.R., "Influence of Seakeeping Requirements on SWATHShip Geometry", paper presented at the June meeting ofthe Chesapeake Section of the Society of NavalArchitects and Marine Engineers, June 1987.

22

TR-2601

12. Zarnick, E.E., "Vertical Plane and Roll Motion Stabilizationof SWATH Ships", David W. Taylor Naval Ship Research andDevelopment Center Report SPD 1199-01, September 1986.

13. Fein, J.A., Ochi, M.D., and McCreight, K.K., "The SeakeepingCharacteristics of a Small Waterplane, Twin Hull (SWATH)Ship", Thirteenth Symposium on Naval Hydrodynamics,Tokyo, October 1980.

14. Report of the High Speed Marine Vehicles Committee, 18thInternational Towing Tank Conference, Kobe, Japan,October 1987.

15. McCreight, K.K., "Assessing the Seaworthiness of SWATHShips", Trans. SNAME, vol. 95, 1987.

16. Burtness, M.N., and Numata, E., "Trim Stability Tests of aU.S. Coast Guard SWATH Cutter Design", DavidsonLaboratory Report 2535, October 1985.

23

TR-2601

TABLE 1

Ship Particulars

Strut length (LWL), ft 125Hull length, ft 124Hull diameter, ft 10Maximum beam, ft 59Displacement (14.5 ft WL), LT 591Cross structure clearance, ft 10

(to 14.5 ft WL)

Strut wetted area, sq ft 2305Hull wetted area, sq ft 6710

VCG, ft above baseline 18.98LCG, ft aft of strut nose 51.72GM, ft 24.45Pich period (zero speed), sec 11.76Pitch gyradius, ft 38.4

24

TR-2601

TABLE 2

Directory of Data Tables

Table Description

3 Unappended, fixed-trim tests in calm water

4 With instrumented canard in calm water

Tests Fixed in Trim and Heave in Regular Waves:

5.1 All data in model scale; canard aspect ratio 1

5.2 Mean quantities expanded to full scale; canard aspect ratio 1

5.3 Canard lift and drag expressed in coefficient form; canardaspect ratio 1

6.1 All data in model scale, canard aspect ratio 2

6.2 Mean quan' expanded to full scale; canard aspect ratio 2

6.3 Canard lift Irag expressed in coefficient form; canardaspect rat.,..

Tests Free to Pitch and Heave in Regular Waves:

7 Baseline tests without active control; canard aspect ratio 1

8 Varying control system gains to minimize pitching motion;canard aspect ratio 1

9.1 Head seas with optimal control gains; canard aspect ratio 1

9.2 Following seas with optimal control gains; canard aspectratio 1

10.1 Head seas with optimal control gains; canard aspect ratio 2

10.2 Following seas with optimal control gains; canard aspectratio 2

25

IR-26o1

TABLE 3

CALM WATER ESM OF UNAPPID) IO.D FIXED IN TRIM

Run Vel Drag M Trim Heave CTMI CTS Vel RS EHP M

no. f'ps lb ft-lb dog ft (xlO00) (xlO00) knots lb ton-ft

Nwirul trIm - -2 deg-es

83 0.00 0.00 3.66 -1.94 -0.3684 1.72 0.23 3.22 -1.95 -0.46 5.091 2.891 5.00 1846. 25.3 490.485 2.41 0.49 2.96 -1.96 -0.46 5.509 3.549 7.00 14445. 95.6 450.886 3.10 1.02 2.59 -1.97 -0.60 6.883 5.084 9.01 1050. 291.5 394.587 3.45 1.13 1.73 -1.98 -0.62 6.152 4.416 10.02 11324. 348.4 263.588 3.79 2.02 1.44 -2.01 -0.68 9.179 7.497 11.00 23202. 784.0 219.389 4.14 2.38 2.58 -2.00 -1.06 9.077 7.444 12.02 27474. 1013.9 393.090 4.48 2.08 1.95 -2.00 -1.04 6.756 5.166 13.00 22327. 891.6 297.091 4.83 2.15 0.50 -2.02 -0.96 5.987 4.437 14.03 22306. 960.8 76.292 5.17 2.77 0.18 -2.04 -1.20 6.735 5.223 15.01 30066. 1386.1 27.493 5.52 3.84 0.68 -2.06 -1.58 8.214 6.732 16.03 44187. 2174.6 103.694 5.86 4.84 1.20 -2.02 -1.82 9.205 7.752 17.02 57382. 2998.9 182.895 6.22 5.64 1.72 -2.02 -1.86 9.535 8.111 18.04 67481. 3739.0 262.096 6.22 5.70 1.78 -2.04 -1.90 9.624 8.202 18.05 68309. 3786.7 271.197 6.55 6.40 1.93 -1.99 -2.12 9.746 8.3:49 19.01 77151. 4505.2 294.098 6.91 7.00 1.95 -2.00 -2.24 9.589 8.216 2.05 84375. 5194.0 297.0

Nocirnl trim - -1 degree

35 0.00 0.00 1.93 -1.04 -0.1236 1.72 0.24 1.64 -1.06 -0.22 5.202 3.000 5.00 1914. 29.4 249.837 2.41 0.50 1.23 -0.97 -0.26 5.544 3.583 7.00 1448B. 96.5 -187.350 2.41 0.49 1.19 -0.93 -0.22 5.456 3.496 7.00 4379. 94.2 181.238 3.10 1.00 1.12 -0.98 -0.36 6.750 4.949 9.00 10247. 283.2 170.639 3.45 1.12 0.28 -1.00 -0.30 6.117 4.380 10.02 11232. 345.5 42.640 3.79 2.01 0.53 -1.01 -0.46 9.143 7.460 11.00 23)87. 780.2 80.741 4.14 2.33 1.64 -1.01 -0.74 8.855 7.223 12.03 26696. 985.9 249.842 4.48 2.02 0.49 -1.01 -0.68 6.550 4.960 13.00 21427. 855.5 74.643 4.83 2.09 -0.85 -1.03 -0.66 5.801 4.251 14.03 21391. 921.7 -129.544 5.17 2.70 -1.16 -1.04 -0.86 6.548 5.033 15.02 29012. 1338.0 -176.745 5.52 3.68 -0.53 -1.06 -1.18 7.873 6.391 16.02 41915. 262.0 -80.746 5.86 4.69 0.22 -1.02 -1.52 8.908 7.456 17.02 55187. 2884.2 33.547 6.21 5.48 0.67 -1.03 -1.84 9.267 7.843 18.03 65190. 3610.3 102.048 6.55 6.11 1.00 -1.05 -1.96 9.294 7.895 19.01 72958. 4260.3 152.349 6.91 6.66 1.16 -1.06 -2.06 9.120 7.745 20.05 79541. 4896.4 176.7

26

TR-6

TABLE 3(Continud)

CAL WAT TESTS C NAP IED !O)EM FIXED IN TRIM

Run Vel Drag M Trim Heave CfM cis Vel RS P m

nM. ts lb ft-lb deg ft (x00) (xlO0) knots lb tan-ft

Naninal trim - 0 degrees

16 1.72 0.24 -0.04 -0.06 0.00 5.171 2.970 5.00 1896. 29.1 -6.199 1.72 0.24 0.02 -0.04 0.04 5.221 3.024 5.00 1929. 29.6 3.010D 2.06 0.35 -0.01 -0.04 0.02 5.337 3.271 5.98 2992. 55.0 -1.5

17 2.41 0.49 -0.09 -0.06 -0.02 5.407 3.446 7.00 4320. 92.9 -13.7101 2.41 0.50 -0.05 -0.05 0.00 5.547 3.588 6.99 4484. 96.3 -7.6102 2.75 0.66 -0.26 0.02 0.00 5.687 3.815 7.98 6217. 152.4 -39.6

18 3.10 1.00 -0.11 -0.07 -0.06 6.748 4.947 9.01 10264. 284.0 -16.8103 3.10 0.99 -0.14 0.01 -0.06 6.721 4.924 9.01 10208. 282.4 -21.3

19 3.45 1.08 -0.89 -0.07 -0.02 5.896 4.159 10.03 10685. 329.0 -135.620 3.80 1.89 -0.68 -0.09 -0.22 8.583 6.900 11.02 21399. 723.9 -103.621 4.14 2.32 0.35 -0.09 -0.50 8.816 7.183 12.03 26547. 980.4 53.322 4.49 2.02 -0.55 -0.10 -0.42 6.539 4.949 13.02 21436. 857.0 -83.823 4.83 2.01 -1.85 -0.11 -0.36 5.596 4.045 14.02 20319. 874.8 -281.8

24 5.17 2.54 -2.23 -0.12 -0.54 6.172 4.656 15.02 26840. 1237.9 -339.625 5.53 3.48 -1.74 -0.13 -0.84 7.425 5.943 16.04 39077. 1924.8 -265.0

26 5.86 4.44 -0.88 -0.14 -1.26 8.432 6.978 17.02 51654. 2699.5 -134.027 6.22 5.24 -0.19 -0.15 -1.52 8.847 7.422 18.05 61813. 3426.6 -28.928 6.55 5.89 0.33 -0.16 -1.68 8.947 7.547 19.03 69831. 4080.2 50.329 6.91 6.44 0.56 -0.16 -1.88 8.813 7.437 20.05 76379. 4701.8 85.3

Nominal trim = 1 degree

51 0.00 0.00 -2.02 1.03 0.2652 1.72 0.24 -1.90 1.03 0.24 5.212 3.012 5.00 1921. 29.5 -289.453 2.41 0.53 -1 .75 1.03 0.20 5.862 3.902 7.00 4888. 105.1 -266.567 2.75 0.70 -1.64 1.07 0.18 5.986 4.111 7.98 6699. 164.3 -249.854 3.10 1.04 -1.64 1.03 0.14 7.021 5.221 9.01 10825. 299.4 -249.855 3.45 1.15 -2.38 1.01 0.18 6.258 4.522 10.03 11617. 357.7 -362.556 3.79 1.89 -2.22 1.00 0.04 8.580 6.898 11.01 21358. 721.9 -338.157 4.14 2.36 -1.20 0.98 -0.26 8.988 7.355 12.03 27186. 1004.0 -182.858 4.48 2.08 -2.04 0.97 -0.16 6.735 5.145 13.00 22236. 888.0 -310.759 4.83 2.04 -3.29 1.00 -0.10 5.674 4.123 14.03 20731. 892.9 -501.160 5.17 2.51 -3.47 1.02 -0.24 6.099 4.584 15.01 26405. 1217.3 -528.561 5.52 3.42 -2.90 1.01 -0.58 7.314 5.831 16.02 38247. 1881.6 -441.7

62 5.86 4.32 -2.06 1.00 -0.92 8.208 6.755 17.01 49948. 2609.0 -313.863 6.22 5.05 -1.36 0.98 -1.22 8.532 7.108 18.04 59137. 3276.7 -207.165 6.55 5.67 -0.69 0.96 -1.42 8.618 7.218 19.01 66706. 3895.3 -105.166 6.91 6.22 -0.39 0.98 -1.52 8.502 7.127 20.06 73281. 4513.7 -59.4

27

TR-2601

TABLE 3(Cc, bzded)

CAL WATM IESS CF U NPPED MODEL FfIMEDI TRIM

Run Vel Drag M Trim Heave CIM CIs Vel RS E M

no. fps lb ft-lb deg ft (x1000) (x1000) knots lb tak-ft

Nominal trim - 2 degrees

32 0.00 0.00 -3.58 1.94 0.4068 0.00 0.00 -3.82 2.09 0.3833 1.72 0.24 -3.24 1.93 0.38 5.269 3.068 5.00 1957. 30.0 -493.569 1.72 0.25 -3.42 2.07 0.42 5.371 3.171 5.00 aD23. 31.0 -520.934 2.41 0.57 -3.18 2.03 0.40 6.312 4.351 7.01 5460. 117.5 -484.370 2.41 0.57 -3.20 2.07 0.44 6.314 4.354 7.00 5459. 117.4 -487.471 3.10 1.06 -3.15 2.06 0.40 7.181 5.381 9.00 11142. 3)8.0 -1179.872 3.45 1.23 -3.66 2.06 0.44 6.701 4.971 10.02 12754. 392.5 -557.473 3.79 1.88 -3.63 2.05 0.34 8.542 6.860 11.00 21230. 717.4 -552.974 4.14 2.44 -2.55 2.04 0.00 9.291 7.659 12.03 28334. 1046.9 -388.475 4.48 2.17 -3.25 2.04 0.12 7.048 5.458 13.01 23610. 943.2 -495.076 4.83 2.05 -4.52 2.03 0.20 5.708 4.158 14.03 20907. 900.5 -688.477 5.17 2.50 -4.72 2.02 0.06 6.075 4.560 15.02 26289. 1212.4 -718.978 5.53 3.38 -4.05 2.01 -0.241 7.213 5.731 16.04 37684. 1856.2 -616.879 5.87 4.23 -3.15 2.01 -0.52 8.020 6.567 17.03 48661. 2544.4 -t79.880 6.22 5.00 -2.25 2.00 -0.84 8.442 7.018 18.04 58387. 3235.1 -342.781 6.55 5.62 -1.64 1.99 -1.08 8.537 7.138 19.03 66041. 3858.8 -249.882 6.90 6.11 -1.26 1.98 -1.22 8.365 6.990 20.03 71704. 1411.4 -191.9

28

TR-260 1

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TR-2601

TABLE 5.2

TESTS WITH INSTRUMENTED ASPECT RATIO 1 CANARD IN REGULAR WAVES.FULL-SCALE EXPANSION OF MEAN QUANTITIES.

Run Vel A/L H T Te We CTS RS EHP M

no. knots ft sec sec rps lb ton-ft

Head seas

117 10. 1.0 4.20 4.95 2.98 2.11 0.00610 15638. 481. -289.4119 10. 1.0 8.04 5.00 3.00 2.09 0.00887 22745. 699. -359.4127 10. 1.5 4.16 6.07 3.92 1.60 0.00588 15069. 463. -322.9129 10. 1.5 8.16 6.07 3.93 1.60 0.00721 18481. 568. -361.0131 10. 2.0 4.52 7.01 4.75 1.32 0.00582 14927. 459. -324.4155 10. 2.0 8.84 7.01 4.76 1.32 0.00632 16206. 498. -380.8135 10. 2.5 4.24 7.89 5.52 1.14 0.00599 15353. 472. -318.3154 10. 2.5 8.56 7.89 5.52 1.14 0.00616 15780. 485. -385.3152 10. 3.0 8.44 8.67 6.23 1.01 0.00621 15922. 489. -391.4153 10. 3.0 12.64 8.67 6.24 1.01 0.00671 17201. 529. -507.2150 10. 3.5 8.52 9.50 6.94 0.91 0.00649 16633. 511. -393.0151 10. 3.5 12.80 9.50 6.94 0.91 0.00671 17201. 529. -511.8118 20. 1.0 4.20 4.95 2.13 2.95 0.00774 79081. 4853. -406.7120 20. 1.0 8.04 5.05 2.18 2.88 0.00816 83694. 5144. -409.7128 20. 1.5 4.16 6.03 2.88 2.18 0.00768 78512. 4818. -406.7130 20. 1.5 8.16 6.07 2.92 2.15 0.00793 80865. 4955. -400.6132 20. 2.0 4.52 7.01 3.60 1.75 0.00760 77866. 4786. -402.1134 20. 2.0 8.84 7.01 3.61 1.74 0.00779 79649. 4888. -357.9136 20. 2.5 4.24 7.84 4.24 1.48 0.00760 77866. 4786. -383.8138 20. 2.5 8.44 7.89 4.26 1.48 0.00776 79365. 4871. -350.3140 20. 2.7 8.44 8.28 4.56 1.38 0.00772 79145. 4864. -341.2142 20. 3.0 12.64 8.67 4.86 1.29 0.00770 79209. 4875. -359.4144 20. 3.5 8.52 9.50 5.46 1.15 0.00758 77723. 4777. -262.0146 20. 3.5 12.80 9.50 5.46 1.15 0.00758 77723. 4777. -325.9

Following seas

169 10. 1.0 4.20 4.95 -14.62 -0.43 0.00566 14501. 446. -316.8170 10. 1.0 8.04 4.95 -15.22 -0.41 0.00560 14358. 441. -325.9171 10. 2.0 4.52 7.01 -13.30 -0.47 0.00582 14927. 459. -321.4172 10. 2.0 8.84 7.01 -13.24 -0.47 0.00638 16348. 502. -412.8173 20. 1.0 4.20 4.95 15.22 0.41 0.00761 78008. 4794. -385.3174 20. 1.0 8.04 4.95 16.25 0.39 0.00747 76174. 4668. -245.2175 20. 2.5 4.24 7.89 -52.37 -0.12 0.00757 77581. 4768. -406.7176 20. 2.5 8.56 7.89 -52.37 -0.12 0.00765 78434. 4820. -333.6177 20. 3.5 8.52 9.50 -35.38 -0.18 0.00783 80076. 4914. -303.1178 20. 3.5 12.80 9.50 -35.16 -0.18 0.00772 79145. 4864. -9.1

31

TR-.2601

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TR-2601

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a 33

TR-2601

TABLE 6.2

TESTS WITH INSTRUMENTED ASPECT RATIO 2 CANARD IN REGULAR WAVES.FULL-SCALE EXPANSION OF MEAN QUANTITIES.

Run Vel X/1 H T Te We CTS RS EHP M

no. knots ft sec sec rps lb ton-ft

Head seas

93 10. 1.0 4.20 4.95 2.97 2.11 0.00605 15496. 476. -327.595 10. 1.0 8.04 5.00 3.00 2.10 0.00854 21892. 673. -388.497 10. 1.5 4.16 6.07 3.91 1.60 0.00571 14643. 450. -339.699 10. 1.5 8.16 6.07 3.93 1.60 0.00671 17201. 529. -357.9

101 10. 2.0 4.52 7.01 4.75 1.32 0.00588 15069. 463. -345.7103 10. 2.0 8.84 7.01 4.75 1.32 0.00655 16775. 515. -385.3105 10. 2.5 8.56 7.89 5.52 1.14 0.00621 15922. 489. -386.9107 10. 2.5 12.64 7.89 5.53 1.14 0.00788 20186. 620. -511.8109 10. 3.0 8.44 8.67 6.23 1.01 0.00616 15780. 485. -380.8111 10. 3.0 12.64 8.67 6.24 1.01 0.00738 18907. 581. -505.7113 10. 3.5 8.52 9.50 6.93 0.91 0.00638 16348. 502. -351.8115 10. 3.5 12.80 9.50 6.94 0.90 0.00704 18054. 555. -453.994 20. 1.0 4.20 4.95 2.13 2.96 0.00774 79081. 4853. -435.696 20. 1.0 8.04 5.00 2.16 2.91 0.00808 82841. 5091. -420.498 20. 1.5 4.16 6.07 2.90 2.17 0.00773 79287. 4873. -389.9100 20. 1.5 8.16 6.07 2.91 2.16 0.00784 80424. 4943. -356.4102 20. 2.0 4.52 7.01 3.60 1.75 0.00769 78654. 4827. -397.5104 20. 2.0 8.84 7.01 3.61 1.74 0.00779 79856. 4908. -370.1108 20. 2.5 4.24 7.84 4.24 1.48 0.00760 77659. 4766. -382.3106 20. 2.5 8.56 7.89 4.25 1.48 0.00772 79145. 4864. -345.7110 20. 3.0 8.44 8.67 4.86 1.29 0.00768 78719. 4838. -321.4112 20. 3.0 12.64 8.72 4.87 1.29 0.00733 75165. 4620. -394.5114 20. 3.5 8.52 9.50 5.48 1.15 0.00774 79081. 4853. -240.6116 20. 3.5 12.80 9.50 5.47 1.15 0.00757 77581. 4768. -278.7

Following seas

179 10. 1.0 4.20 4.95 -14.41 -0.44 0.00549 14074. 432. -310.7180 10. 1.0 8.04 4.95 -13.98 -0.45 0.00605 15496. 476. -315.3181 10. 2.0 4.52 7.01 -13.24 -0.48 0.00549 14074. 432. -312.2182 10. 2.0 8.84 7.01 -13.15 -0.48 0.00632 16206. 498. -383.8183 20. 1.0 4.20 4.95 15.97 0.39 0.00771 79003. 4855. -403.6184 20. 1.0 8.04 4.95 16.92 0.37 0.00750 76664. 4705. -155.4185 20. 2.5 4.24 7.89 -49.87 -0.13 0.00771 78797. 4836. -428.0186 20. 2.5 8.56 7.89 -51.61 -0.12 0.00739 75527. 4635. -260.4187 20. 3.5 8.52 9.50 -34.71 -0.18 0.00768 78512. 4818. -379.2188 20. 3.5 12.80 9.50 -34.89 -0.18 0.00687 70474. 4331. -144.7

34

TR-260 1

cc %D'~ MN M O N "A cN W 0 t N in %n~ UO. f Cu t0 -Y -y M LnU M~

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TR-2801

TABLE 8

RUNS IN R B3JAR WAVES TO OFrIKffE 03TM L SYSI E1. CANAR ) ASPET RATIO 1.

Run Vel )/2. T Te w H eve 01 z1 we t ns1 e g, g2

no. knots se ae r ft frtmase de f de

Head seas

135 15.00 3.50 9.46 6.119 1.03 8.68 39 2.60 2.08 126 10.42 154 3.54 1.93137 15.00 3.50 9.46 6.114 1.03 8.68 46 2.42 2.02 135 14.36 131 3.54 4.83138 15.00 3.50 9.46 6.119 1.03 8.68 49 2.31 2.00 139 17.36 123 3.54 6.76139 15.00 3.50 9.46 6.119 1.03 8.68 52 2.38 2.00 142 19.84 131 4.95 6.76140 15.00 3.50 9.46 6.114 1.03 8.68 52 2.17 1.96 143 19.97 118 3.54 8.69141 15.00 3.49 9.46 6.109 1.03 8.68 54 2.13 1.96 145 21.29 116 3.54 9.65142 15.00 3.51 9.50 6.124 1.03 8.68 54 2.25 1.94 146 22.16 125 4.95 8.69143 15.00 3.51 9.50 6.124 1.03 8.68 47 2.01 1.94 140 18.23 96 0.00 9.65145 20.00 3.50 9.46 5.472 1.15 8.68 35 2.89 2.28 113 22.29 121 3.54 6.76147 20.00 3.50 9.46 5.472 1.15 8.68 30 2.64 2.30 112 18.71 96 0.00 6.76

FoUowing seas

98 20.00 1.50 6.07 73.485 0.09 4.08 217 1.99 0.90 51 0.04 181 2.12 0.00100 20.00 1.50 6.07 68.586 0.09 4.08 222 1.56 0.74 67 5.35 179 3.54 1.45111 20.00 1.50 6.07 61.727 0.10 4.08 184 2.99 1.92 25 1.36 87 0.00 4.83113 20.00 1.50 6.07 63.687 0.10 4.08 220 1.3 0.64 76 6.65 174 4.95 4.83103 20.00 2.50 7.84 -52.968 -0.12 4.28 123 1.72 0.02 316 5.86 181 3.54 0.00104 20.00 2.50 7.84 -54.678 -0.11 4.28 112 1.75 0.02 328 6.00 175 3.54 2.41105 20.00 2.50 7.84 -52.718 -0.12 4.28 115 1.93 0.02 331 6.70 171 3.54 4.83116 20.00 2.50 7.84 -57.823 -0.11 4.28 52 1.10 1.08 300 5.36 172 4.95 6.76117 20.00 2.50 7.84 -56.559 -0.11 4.28 69 0.97 1.20 314 5.33 173 5.66 6.76119 20.00 2.50 7.84 -58.175 -0.11 4.28 72 0.96 1.24 318 6.60 175 7.07 6.76120 20.00 2.50 7.84 -54.031 -0.12 4.28 53 0.99 1.28 298 6.18 173 6.36 7.72107 20.00 3.50 9.46 -36.978 -0.17 8.68 33 1.81 3.34 306 6.2 174 3.54 2.41108 20.00 3.50 9.46 -35.870 -0.18 8.68 28 1.55 3.36 297 5.53 166 3.54 4.83109 20.00 3.50 9.46 -35.714 -0.18 8.68 29 1.31 3.20 299 6.42 169 4.95 4.83110 20.00 3.50 9.46 -34.092 -0.18 8.68 31 1.38 2.92 299 9.26 173 7.07 4.83

37

TR-2601

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540

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04

TR-260 1

0 0

-W C

o w o

r4C1 C.4w. .

-- 4

00

42-

6.92

5.03 7.12

apect ratio 1 1.-2.22.4

Aspect ratlo 2

Senla. vuwRACA 0015

6.94

t ~12.67 -

-_b£1.00

5.7.5 -

STA5LZm

FIGURE 2 PLAN AND SECTION VIEW OF ODEL FINS.ALL DIDUSIONS IN FEET.

113

-amp -*W1

J#4

..... ..... ..... .....

to-

J44

TR-2601

Forward

S Drag balance

\-Lift balance

Fin (canard)

PLAN VIEW

FIGURE 4a LIFT AND DRAG BALANCE INSTALLATION IN MODEL.

45

TR-2601

displacement rate

Gain settings

Fin RatePitch sensor and

Position

Pitch motion

-Servo motor

Active fin

FIGURE 5 SCHEMATIC DIAGRAM OF PITCH CONTROL SYSTEM

47

TR-2601

600

400

200A

00ea

* 0 ---- --- lE0

.200

-400R6

-600

-500

0 5 10 I5 20

Speed, knots

FIGURE Ga BEHAVIOR OF PITCH MOMENT WITH SPEED FOR UNAPPENDED SHIP

AT VARIOUS TRIM ANGLES. FIXED TRIM. FREE-TO-HEAVE.

48

TR-2601

600

400

200

g4

0 Speed 20 kt0

r.

-200 5 ft

-40010k

-600

-800 1-2 -1 0 1 2

Trim, degrees

FIGURE 6b BEHAVIOR OF PITCH MOMENT WITH TRIM ANGLE FOR UNAPPENDED SHIPAT VARIOUS SPEEDS (FIXED TRIM, FREE-TO-HEAVE).

E

-r -I0

o

0 I I

-1 00

0.

C.

0

030

0 5 10 15 20Speed, knots

FIGURE 7 PITCH STABILITY VS. SPEED.

49

TR-2601

.8

.8 II

Aspect ratio = 1

.6

On ground boardl

.4

.2 20 knots

C L 15 knots

0 10 knots

-.2

-.40 5 10 15 20

Fin angle, degrees

Aspect ratio = 2

On ground boardl

.4

20 knots

-15 knots

.2 -10 knots

FIGURE 8 RESULTS OF INSTRUMENTED FIN ON BODY TESTS IN CALM WATER.

50

, , , a I I-I2

TR-2 601

aoo00-0_

2500 -

12000

A 2, V 1801 20 knots

2100A - --- ------------

100

EE 500

a U * 109.96 4.100.161.

0

-500 A - 1, V 15 knots

.2 -1 0 1 2 3 4 5 6 7 a 9

Fin lilt, LT (one fin)

FIGURE 9 PITCH MOMENT VS FIN LIFT AT TWO SPEEDS IN CALM WATER(MODEL FIXED IN TRIM AND HEAVE)

51

TR-2601

Speed - 10 knots - i - 12 ft

Sfin aspect ratio 12 f

B-8 ft

.4 ® 0

ra B ft

.2

0 2 34

0 Wavelength/Ship length

Wave Head Folloving

Height Seas Seas

4 a a

8 0

12 b a

Speed - 20 knotsFin aspect ratio - 1

.4i 12 ft

CLI ' H-8 ft

.2 u ft

.2~M - --- - -- - - heory, HW4 ft

0 1 2340 Wavelength/ShiP length

FIGURE 10 BEHAVIOR OF FIRST HARMONIC OF FIN LIFT COEFFICIENT

WITH

WAVELENGTH AND WAVE HEIGHT

MODEL FIXED IN TRIM AND HEAVE

52

TR-2601

Speed - 10 knots

1.0 Fin aspect ratio - 2

H 12 ft

.8

.61

H =8 ft.6

.4 ft

.2

0 I I0 1 2 3 4

Wavelength/Ship length

Wave Read FollowingHeight Seas Seas

4 0 U

8 0 0

Speed 20 knots 12 A

.6 Fin aspect ratio - 2

.4 --- .H H12 ft%1 _ __- -- _ - - - _ ory,H-12 ft

-H - ft

-- H-4ft

Theory, H = 4 ft

0 I Io 1 2 3

Wavelength/Ship length

FIGURE 11 BEHAVIOR OF FIRST HARMONIC OF FIN LIFT COEFFICIENT WITHWAVELENGTH AND WAVE HEIGHTMODEL FIXED IN TRIM AND HEAVE

53

TR-2601

1.4 Head Seas

1.2U

1.0

4' .8 .8 No control

eq .6

S .4 wi-rlth control

.2

1.5 2.0 2.5 3.0 3.5 4.0

Wavelength/Ship length

Wave No WithHeight Control Control

4 ft 0

8 ft12 ft

1.4 Following Seas

1.2

1.0 0

00

No control4 .6

.4 - With control

-4 .20.

.

1.5 2.0 2.5 3.0 3.5 4.0Wavelength/Ship length

FIGURE 12 EFFECT OF ACTIVE CONTROL ON PITCHING MOTION, SPEED - 15 KNOTS.FIN ASPECT RATIO - 1.

54

TR-2601

1.4 -Read Seas

c.1.21.

~ .0

~a .6

0 Yf1.-.4.02 53 0 . .

Waeegh.hplnt

Wavlenth/hi lengthh

Height Control Control

4 ft 0 a-a ft '-12 ft

1.4 Following Seas

0' 1.2 No control

4' 1.040'

.80

S .6-40.

4' .4

.2

1.0 1.5 2.0 2.5 3.0 3.5 4.0Wavelength/Ship length

FIGURE 13 EFFECT OF ACTIVE CONTROL ON PITCHING MOTION, SPEED =20 KNOTS.FIN ASPECT RATIO - 1.

55

TR-2601

s1.24

m. 1.0 Riead Seas

.8

-~.6

00

Wav .2it

Height Control Control

4 ft 0 0

8 ft aD

12 ft A

A1.2 Following Seas9.4

S1.0 No control

> .8

V .6

.2

0

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Wavelength/Ship* length

FIGURE 14 EFFECT OF ACTIVE CONTROL ON HEAVE MOTION, SPEED - 20 KNOTS.

FIN ASPECT RATIO = 1.

56

TR-2601

o 1.0 Pitch response

.4

S.2

S-

0.

Wav

40t

.6

.4

1.0 1.5 2.0 2.5 3.0 3.5Wavelength/Ship length

FIGUR 15 EFECT F FINASPEC RATI ON MTIOavESPNE IHCNRL

Height A- 15KOTS

57t 0

TR-2601

. 1.0- Pitch response0

o .8

- .6

-4 .4-4

S .2X

0 L011.0 1.5 2.0 2.5 3.0 3.5Wavelength/Ship length

WaveHeight A-i A=2

4 ft 0 0

8 ft D12 ft

1.0 -- Heave response f A =2

, .6-2

0 1.5 2.0 2.5 3.0 3.5

Wavelengthj'Ship length

FIGURE 16 EFFECT OF FIN ASPECT RATIO ON MOTION RESPONSES WITH CONTROL,

SPEED - 20 KNOTS.

58

, 0

~... ~ -

V

-

*I

0I0

* S

A'

AS.

I* 59/60

'I~

TR-2601

APPENDIX A

Inclining Experiment

An inclining test was done just prior to the tests with the automatic

pitch control system, to determine the longitudinal GM. A five pound weight

was shifted on the deck to produce a moment about the pivot point. The trim

was then read using the inclinometer mounted on the deck. Results are

tabulated below.

Weight Moment Trim

shift, ft ft-lb deg

0 0 .076

0.25 -1.25 -.609

0.50 -2.50 -1.415

0.75 -3.75 -2.170

1.00 -5.00 -2.952

1.25 -6.25 -4.373

-0.25 1.25 .853

-0.50 2.50 1.602

-0.75 3.75 2.345

-1.00 5.00 3.101

-1.25 6.25 3.908

-1.50 7.50 4.580

A straight line was fit to the data in the range -5 - M S 7.5, with the

following result:

e - 0.093 + 0.6033M

Using the theoretical relationship between trim and moment,

A.GML*- M

for small angles where A is the vessel displacement and E is in radians, oneobtains in the present case that

Al

TR-2601

APPENDIX A

(Continued)

(1/A-GML)(57.296 deg/rad) - 0.6033

and using the model displacement A - 93.22 lb,

GML - 1.019 ft (model scale)

- 24.45 ft (full scale)

Determination of LCF

Before the free to pitch tests were conducted, a determination of the

longitudinal center of flotation (LCF) was made as follows: with the pivot

point at its initial location, 2.135 ft aft of the strut nose, a 5 lb weight

was placed at various stations on the deck and the resulting trim change

noted:

Location Trim Change (deg)

Pivot -1.545

5 in aft -0.246

6 in aft 0.000

9 in aft 0.813

12 in aft 1.597

The center of flotation is the point at which the addition of a weight

causes no trim change, which in this case is 6 in aft of the pivot. To

avoid coupling between heave and the measured pitch, which was used as input

to the control system, the pivot point was moved to the LCF for the free to

pitch tests, to 63.25 ft aft of the strut nose.

A2

TR-2601

APPENDIX B

Videotape ScenariosVideotape No. 1

Calm Water, Unappended, Fixed Trim

Run Velocity Trim Videoknots deg start

16 5 0 1:0017 7 2:0518 9 3:1719 10 4:0920 11 4:5521 12 5:3922 13 6:2823 14 7:0424 15 7:4025 16 8:1426 17 8:4727 18 9:1828 19 9:5129 20 10:1633 5 2 10:4234 7 2 12:0836 5 -1 13:2037 7 14:5338 9 16:0039 10 16:5140 11 17:3941 12 18:2142 13 19:0143 14 19:3744 15 20:1145 16 20:4546 17 21:1747 18 21:4748 19 22:1049 20 22:3751 5 1 23:0253 7 24:3254 9 1 25:3355 10 26:2556 11 27:0757 12 27:4758 13 28:2459 14 28:5660 15 29:3761 16 30:1162 17 30:4063 18 31:1064 19 31:3666 20 32:05

B1

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APPENDIX B(Continued)

Videotape ScenariosVideotape No. 1

Calm Water, Unappended, Fixed Trim

Run Velocity Trim Videoknots deg start

67 8 32:3069 5 2 33:2870 7 324:5671 9 36:0072 10 36: 14973 11 37:3674 12 38:1675 13 38:5576 14 39:3077 15 40:0378 16 40:3279 17 41:0080 18 41:2881 19 41:5282 20 42:1884 5 -2 42:4185 7 44:0786 9 45:0687 •10 45:5488 11 46:3589 12 147:1790 13 47:5691 14 -2 48:3092 15 49:0293 16 49:3294 17 50:0095 18 50:2897 19 50:5098 20 51:20

B2

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APPENDIX B(Continued)

Videotape ScenariosVideotape No. 2

Tests with Instrumented Canard; Zero Trim and Heave

Tests in Calm Water Varying Canard Angle

Run Velocity a Aspect Video Commentsknots deg ratio start

12 20 0 1 2:0013 10 5 2:2314 15 5 2:52

15 20 5 3:1516 10 10 3:5417 15 10 4:3318 20 10 5:0019 10 15 5:2721 15 15 6:1022 20 15 6:3623 10 20 7:0124 15 20 7:4125 20 20 8:1334 10 0 2 8:3235 15 0 9:1236 20 0 9:4337 10 5 10:0938 15 5 10:5139 20 5 11:1942 10 15 11:4443 15 15 12:1544 20 15 12:4545 10 15 13:0946 10 10 14:0347 15 10 14:4348 20 10 15:1249 10 20 15:3051 20 20 16:4652 20 20 17:06 Closeup of fin53 15 20 17:30 Closeup of fin54 10 20 18:00 Closeup of fin

B3

TR-2601

APPENDIX B(Continued)

Videotape ScenariosVideotape No. 2

Tests In Regular Waves with Instrumented Canard. Zero trim and heave; a-00.

Run Velocity Aspect A/1 H Video Commentsknots ratio ft start

Head Seas66-74 23:00 Preliminary runs

93 10 2 1 4.20 26:1494 20 1 4.20 27:0595 10 1 8.04 27:3096 20 1 8.40 28:2097 10 1.5 4.16 28:4498 20 1.5 4.16 29:3499 10 1.5 8.16 30:00100 20 1.5 8.16 30:47101 10 2.0 4.52 31:13102 20 2.0 4.52 31:59103 10 2.0 8.84 32:27104 20 2.0 8.84 33:14105 10 2.5 8.56 33:40106 20 2.5 12.60 34:11107 10 2.5 12.60 34:51108 20 2.5 4.24 35:38109 10 3.0 8.44 36:08110 20 3.0 8.44 36:55111 10 3.0 12.64 37:19112 20 3.0 12.64 38:07113 10 3.5 8.52 38:34114 20 3.5 8.52 39:21115 10 3.5 12.80 39:46116 20 3.5 12.80 40:34117 10 1 1.0 4.20 41:03118 20 1.0 4.20 41:38119 10 1.0 8.04 42:05120 20 1.0 8.04 42:32127 10 1.5 4.16 43:21128 20 1.5 4.16 44:07129 10 1 1.5 8.16 44:30130 20 1.5 8.16 45:16131 10 2.0 4.52 45:44132 20 2.0 4.52 46:29133 10 2.0 8.84 46:54134 20 2.0 8.84 47:46135 10 2.5 4.24 48:10136 20 2.5 4.24 48:58137 10 2.5 8.56 49:23138 20 2.5 8.56 50:15139 10 3.0 8.44 50:40

B4

TR-2601

APPENDIX B(Continued)

Videotape ScenariosVideotape No. 2

Tests in Regular Waves with Instrumented Canard. Zero trim and heave;a=00.

Run Velocity Aspect A/1 H Video Commentsknots ratio ft start

140 20 3.0 8.44 51:27141 10 3.0 12.64 51:32142 20 3.0 12.64 52:40143 10 3.5 8.52 53:08144 20 3.5 8.52 53:57145 10 3.5 12.80 54:03146 20 3.5 12.80 54:52150 10 3.5 8.52 57:39151 10 3.5 12.80 58:31152 10 3.0 8.44 59:18153 10 3.0 12.64 1:00:08154 10 2.5 8.56 1:00:55155 10 2.0 8.84 1:01:46

Following Seas

163 10 1 1.0 4.20 1:03:02164 10 1.0 8.04 1:03:57165 20 1.5 8.16 1:04:54166 20 2.5 8.56 1:05:26167 20 3.5 12.80 1:05:58168 10 1.0 4.20 1:06:32170 10 1.0 8.04 1:07:26171 10 2,0 4.52 1:08:12172 10 2.0 8.84 1:09:22173 20 1.0 4.20 1:10:14174 20 1 1.0 8.04 1:11:05175 20 2.5 4.24 1:11:31176 20 2.5 8.56 1:12:00177 20 3.5 8.52 1:12:35178 20 3.5 12.80 1:13:02179 10 2 1.0 4.20 1:13:29180 10 1.0 8.04 1:14:14181 10 2.0 4.52 1:15:02182 10 2.0 8.84 1:15:49183 20 1.0 4.20 1:16:37184 20 1.0 8.04 1:17:02185 20 2.5 4.24 1:17:27186 20 2.5 8.56 1:17:52187 20 3.5 8.52 1:18:16188 20 3.5 12.80 1:18:44

B5

TR-2601

APPENDIX B(Continued)

Videotape ScenariosVideotape No. 3

Tests with Automatic Pitch Control

Run Velocity Aspect A/£ H Video Comments

knots ratio ft start

26-45 0:46 Preliminary runs

Baseline Tests Without Automatic Control. Head Seas.

46 15 1 3.5 8.66 4:1747 3.5 8.66 5:0248 3.5 8.66 5:4849 1.5 4.10 6:38

59 3.5 12.92 7:1860 2.5 4.28 8:02

61 2.5 8.66 8:37

62 1.5 8.26 9:15

63 1.5 4.08 9:5264 20 3.5 8.68 10:3765 3.5 8.68 11:0866 3.5 12.92 11:44

67 3.5 12.92 12.19

68 2.5 4.28 12:56

69 2.5 8.66 13:3370 1.5 8.26 14:0871 1.5 4.08 14:44

Following Seas

72 15 1 0.58 4.08 15:20 No oscillations

74 1.5 4.22 16:34

75 2.0 4.08 17:1376 2.0 8.26 17:56

77 2.5 4.28 18:4778 2.5 8.66 19:3779 3.5 8.68 20:26

80 3.5 12.92 21:15 Need more heave travel

81 3.5 12.9- 21:58 Need more heave travel

82 3.5 12.9 22:41 Need more heave travel

83 20 0.89 4.26 23:28

84 1.0 4.22 24:00

85 1.0 8.21 24:34 Model nosedives

87 2.5 4.08 25:00

88 1.5 8.26 25:30 Model nosedives

89 15 2.0 4.25 25:45 4 in coaming added

90 15 2.0 8.50 26:31

91 20 2.0 4.25 27:20 Zero encounters

92 2.5 4.28 28:00

93 3.5 8.68 28:3394 3.5 12.92 29:07 No data

B6

TR-2601

APPENDIX B(Continued)

Videotape Scenarios

Videotape No. 3

Following Seas

Run Velocity Aspect x/I H Video Commentsknots ratio ft start

95 3.5 12.92 29:4696 2.5 8.66 30:2997 1.0 8.24 3:02 Model nosedives

Tests To Find Optimum Gain Settings; Following Seas

Run Velocity Aspect X/L H Video g1 92knots ratio ft start

98 20 1 1.5 4.08 32:00 2.12 099 1.5 4.08 32:45 no data100 1.5 4.08 33:33 3.54 1.45101 2.5 8.66 34:21 model nosedives103 2.5 4.28 34:56 3.54 0 added PVC

windshield104 2.5 4.28 35:35 3.54 2.41105 2.5 4.28 36:06 3.54 4.83106 3.5 8.68 36:51107 3.5 8.68 37:40 3.54 2.41108 3.5 8.68 38:08 3.54 4.83109 3.5 8.68 4.95 4.83110 3.5 8.68 7.07 4.83111 3.5 8.68 0 4.83112 1.5 4.08 zero encounters113 1.5 4.08 4.95 4.83116 2.5 4.28 4.95 6.76117 2.5 4.28 5.66 6.76118 2.5 4.28 6.36 6.76119 2.5 4.28 7.07 6.76120 2.5 4.28 6.36 7.72

B7

TR-2601

APPENDIX B(Continued)

Videotape Scenarios

Videotape No. 3

Tests in Following Seas with Optimal Control Gains

Run Velocity Aspect A/1 H Video Commentsknots ratio ft start

121 20 1 2.5 4.28 44:17122 20 1.5 4.08 45:15123 20 1.34 4.20 46:04124 20 1.34 4.20 46:48126 15 2.0 8.50 47:30127 15 2.0 8.50 48:20128 15 2.5 8.68 49:00129 15 2.5 12.92 49:38130 15 2.5 4.28 50:22131 15 1.5 4.08 51:07132 15 1.5 8.26 52:07133 15 1.5 8.26 53:04 Servos off134 15 1.5 8.26 54:09

Tests in Head Seas to Find Optimum Gain Settings

Run Velocity Aspect X/£ H Video g1 92knots ratio ft start

135 15 1 3.5 8.68 55:03 3.54 1.93137 15 3.5 8.68 56:20 3.54 4.83138 15 3.5 8.68 57:02 3.54 6.76139 15 3.5 8.68 57:44 4.95 6.76140 15 3.5 8.68 58:26 3.54 8.69141 15 3.5 8.68 59:08 3.54 9.65142 15 3.5 8.68 59:40 4.95 8.69143 15 3.5 8.68 1:00:33 0 9.65144 20 3.5 8.68 1:01:15 0 9.65145 20 3.5 8.68 1:01:48 3.54 6.76146 20 3.5 8.68 1:02:22 Servos off147 20 3.5 8.68 1:02:55 0 6.76

Tests in Head Seas With Optimum Control Gains

148 20 1 3.5 12.92 1:03:27149 20 4.0 12.50 1:04:02150 20 4.0 12.50 1:04:33151 20 2.5 8.66 1:05:05152 20 1 2.5 4.28 1:05:40153 20 1.5 8.26 1:06:15154 20 1.5 4.08 1:06:48155 15 4.0 12.50 1:07:20 Servos off156 15 4.0 12.50 1:08:03 Servos off157 15 4.0 12.50 1:08:47

B8

TR-2601

APPE'.iiIX B(Continued)

Videotape ScenariosVideotape No. 3

Tests in Head Seas With Optimum Control Gains

Run Velocity Aspect A/L H Video Commentsknots ratio ft start

158 15 3.5 12.92 1:09:32159 15 2.5 8.66 1:10:15160 15 2.5 4.28 1:10:57161 20 2 4.0 12.50 1:12:26162 20 4.0 12.50 1:13:02163 20 3.5 12.92 1:13:37164 20 3.5 12.92 1:14:09165 20 3.5 12.92 1:14:42166 20 2.5 4.28 1:15:16167 20 2.5 8.66 1:15:53168 20 2.5 8.66 1:16:26 Controls not active169 20 1.5 8.15 1:16:57170 20 1.5 4.72 1:17:30171 20 3.5 12.92 1:18:05172 15 1.5 8.26 1:18:38173 15 2.5 8.66 1:19:17174 15 2.5 4.28 1:20:00175 15 3.5 8.68 1:20:42176 15 3.5 12.92 1:21:26177 15 4.0 12.50 1:22:06

Tests in Following Seas With Optimal Control Gain

178 15 2 1.5 8.26 1:22:49179 15 1.5 4.08 1:23:31180 15 2.0 4.25 1:24:04181 15 2.0 8.50 1:24:40182 15 2.5 8.66 1:25:20183 15 2.5 4.28 1:26:00184 15 3.5 8.68 1:26:37185 15 3.5 12.92 1:27:17186 20 1.34 4.20 1:28:57187 20 1.5 4.08 1:28:30188 20 1.5 4.08 1:29:25189 20 2.5 4.28 1:30:14190 20 2.5 8.66 1:30:42191 20 3.5 8.68 1:31:14191 20 3.5 12.92 1:31:45

B9

TR- 260 1

APPENDIX B(Continued)

Videotape ScenariosVideotape No. 3

Tests In Calm Water With Aspect Ratio 2 Canards Oscillating

Run Velocity Frequency Videoknots Hz (model scale) start

193 20 2 1:32:21194 2 1:33:56195 1 1:34:25196 .5 1:34:53197 .75 1:35:20198 .625 1:35:47199 .5 1:36:15200 .3 1:36:41201 .4 1:37:11202 .22 1:37:39203 .15 1:38:07204 15 1 1:38:35205 .75 1:39:08206 .625 1:39:43207 .5 1:40:19208 .4 1:40:54209 .3 1:41:30210 .25 1:42:05211 .2 1:42:37212 .15 1:43:12213 .1 1:43:48214 .45 1:44:24215 .55 1:45:00

Final Runs in Following Seas With Automatic Control

Run Velocity Aspect X/i H Video Commentsknots ratio ft start

216 20 2 1.0 8.21 1:45:52217 20 1.5 8.26 1:46:48 Nosedive219 20 1.5 8.26 1:47:25 Nosedive220 20 2.5 8.66 1:48:35 Nosedive

B10

TR-2601

APPENDIX C

FORCED OSCILLATION TESTS

A brief series of tests was conducted in calm water toobserve the response of the SWATH model to a sinusoidal pitchingmoment while underway. A signal generator was used to drive thecanards at various frequencies, resulting in constant amplitudepitching motion for the duration of the run. The model was runfree to heave and pitch. A range of oscillation periods from2.47 to 48.63 seconds (full scale) was investigated, at speedsof 15 and 20 knots. Time histories of heave, pitch, and canardangle were measured. A harmonic analysis of these signals wascarried out as described in the report under Data Processing.Results are given in Table Cl below. The phase angles in thetable are with respect to the canard angle.

Pitch and heave amplitudes per unit canard amplitude andtheir phases are plotted on Figures Cl and C2 for 15 knots andon Figures C3 and C4 for 20 knots, respectively. Figures Cl andC2 show an apparent heave and pitch resonance peak at a canardperiod between 12 and 16 seconds. At 20 knots there is a peakin the pitch motion at a canard period of just under 8 secondsand a peak in the heave motion at a period of about 10 seconds.Phase angles are near 900 at the resonance peaks, indicatingthat the exciting moment (induced in part by the canards butalso influenced by the hull) is nearly in phase with the canardangle.

Cl

TR-2601

=0 - A 0-rA =-t- 0tnU -0 M NinMfn en NNN N m C'.Jli CUN Ne--

0 0 0 0 0 0 YCn0%Nt-00 ON o 0 O'0N'.N r.

CU '.0'.. 0 M' - 0 t=L U,%no - t--o UN'O cno'.ar.M t -=. OCO WtL n c'c %0MM N N m N - Mo U% -t- rt- --. D l%M

0. - - --- - - rC N - - -- N

o o = M3 co '0 N Go '.C 0 '0 -- %D0 = 'o.0 0 '.D 0 cm =~ c 0 0 =N N- n % NN-N-.e 0 NO -rn M0 0 0 0C7%moounN

N I-

4)e X: C c zc z4c %;U

to ( %0 0%i t~. % UN0% - M N %0 r O a'0-O.-m-0%m nI-- ~ M v = MsM0t- . -co .C NNNNNNNN VNNNN-0'.N'No'..N N 'NoM.N

CMi ie ii ia gg -~ 0 MU 0Mgn0Ng 0MM

I-3

C tn .n .nN-U DM0 -

L) a e

0

< Es) o o* C ;C

0Us.

Cw enen M0MN LM wr c % o % 0%%Dor= N 0 N N N MM -M

0%~n M f U%0 N\ Ma en%0 t- a %0 %D 00%0t-N N CM LnO'0

- -- -- -- -- - -- - - - --

MMvoM -=%.OO0-0000 NMO -.-- N-NO-0L

a enM M mN00 - Fn '. ' 0 N t--. t- ' 00' t-- 0m 0 a- eo u L

N) M - -) - 0%r

-- U %D00t- -X000 N MU' -0OM0 00 000e

n 0 a -- .- N en C>- - - -Y %0 ' - N en0M NC YNN-- UMC

C2

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0.30 10

e 0.25 1500

- 0.20 120. Amplitude

* S

*. U

€ 0.16 90

" 0.10 00" C

ES

0.0-- I IUII I 1

10 0.10 4060

0.05E C PIC30PNEI FRE S LTONTSV1 NT

C3

0.120 IS0

0.110

0.100IS

* 0.090___ ___

a.0.050 120E

0.0700

* 0.060 90

U* 0.00 Go

E I0

0.030

M .2 0 .

* 0.010

0.000 00 10 20 30 40 s0

Period, *so

FIGURE C2 HEAVE RESPONSE IN FORCED OSCILLATION TEST, V-15 KNOTS

C4

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0.20 Igo

V

0.15

Amliud 120ES

c 0.10 90

0.0

30

0.05

0 10 20 30 40

Period, see

FIGURE C3 PITCH RESPONSE IN FORCED OSCILLATION TEST, V-20 KNOTS

C5

TR-2601

0.10 ISO

0.03

ISO

m * 0.03

* 0.07

Amplitude 120

E 00

0.06 90• CI

* 0.04 h0s*

U 00 0.03

mm

0 10 20 30 40

Period, seeo

FIGURE C4 HEAVE RESPONSE IN FORCED OSCILLATION TESTS, V-20 KNOTS

C6

0.04 m I

TR-2601

APPENDIX D

CALIBRATION OF FIN BALANCE

To measure lift and drag on a canard, a two-componentbalance was designed which would fit inside of a lower hullof a SWATH model (Figure 4). The balance consisted of astrain-gaged lift spring used in the previous tests(Reference 1) and a specially modified drag balance.

The balance was calibrated on a tankside calibrationstand by application of known weights at the approximatecenter of pressure location of the fins. The apparatus wasrotated so that lift, drag, and combinations thereof could beapplied. The digitized voltage readings were expressed aslinear functions of both lift and drag:

"2} = [C] [L

V D

The coefficients in the matrix [C] were determined by meansof a multivariate least squares fit. Inversion of thismatrix gives the calibration rates Rij:

ti = [R] (V:} [R] = MC]1where off-diagonal elements R1 2, R21 account for cross-coupling.

Results of the calibration are summarized below:

Lift Lift Difference Drag Drag Differenceapplied computed applied computed

0.000 0.003 0.003 0.063 0.064 0.0010.000 0.002 0.002 0.125 0.126 0.0010.000 -0.005 -0.005 0.250 0.252 0.0020.000 -0.003 -0.003 0.500 0.497 -0.0030.000 0.000 0.000 0.750 0.748 -0.0020.000 0.001 0.001 1.000 0.998 -0.0020.000 0.004 0.004 1.250 1.249 -0.0010.985 0.989 0.004 0.174 0.176 0.0021.970 1.972 0.002 0.347 0.352 0.0050.940 0.933 -0.007 0.342 0.348 0.006

-0.940 -0.948 -0.008 0.342 0.347 0.0050.500 0.507 0.007 0.000 -0.001 -0.0011.000 1.000 0.000 0.000 -0.003 -0.0033.000 2.997 -0.003 0.000 -0.001 -0.0012.000 1.998 -0.002 0.000 -0.003 -0.003

D1

TR-2601

The calibration rates are:

Lift - -0.0070998 V1 - 0.0000739 V2

Drag - 0.0000141 V1 - 0.0016039 V2

where V and V2 are the digitized voltage readings from thelift anA drag channels, respectively. The calibrations areplotted on Figure D1.

Dynamic Calibration

Because the balance was to be used to measure unsteadylift and drag in waves, a dynamic calibration was performedin addition to the static calibration described above. Thiswas accomplished by mounting the balance on a scotch yokewhich could oscillate sinusoidally in a horizontal plane withadjustable frequency and amplitude. Weights of variousdenominations were fastened to the balance at the approximatecenter of pressure location of the fins, and time historiesof lift and drag were recorded for several frequencies ofoscillation in the range of encounter frequencies expected inthe tests. Readings were also taken with no weights fastenedto the balance to evaluate the "tare mass" of the balanceacting on the springs. Time histories of displacement andacceleration were also measured. A photograph of theapparatus is included as Figure D2.

Time histories of the signals were fit to a harmonicseries as described in Data Processing. The amplitude of thedynamic force on the balance is given by

F = (W + w) a/g

where W is the applied weight, w is the tare weight, and a isthe acceleration of the scotch yoke. Results are tabulatedbelow.

D2

TR-2601

Lift Calibration

Added Radian Applied MeasuredWeight Frequency Accel Tare Force Force nlb rps ft/sec lb lb lb rps

0 11.38 14.35 0.050 13.87 21.32 0.080 15.46 26.49 - 0.10

0.56 8.20 7.45 0.03 0.16 0.16 196.2 .0420.56 11.26 14.05 0.05 0.30 0.30 196.2 .0570.56 13.78 21.05 0.08 0.45 0.46 196.2 .0700.56 15.19 25.57 0.09 0.54 0.56 196.2 .0771.06 5.91 3.87 0.01 0.14 0.15 132.9 .0451.06 7.84 6.81 0.02 0.25 0.26 132.9 .0591.06 11.04 13.51 0.05 0.49 0.51 132.9 .0831.06 13.80 21.11 0.08 0.78 0.80 132.9 .1041.06 18.12 36.39 0.13 1.33 1.40 132.9 .1362.07 8.53 8.06 0.03 0.54 0.56 88.4 .0972.07 11.62 14.97 0.05 1.01 1.06 88.4 .1322.07 13.90 21.41 0.08 1.46 1.53 88.4 .1572.07 15.25 25.78 0.09 1.75 1.85 88.4 .173

Drag Calibration

Added Radian Applied MeasuredWeight Frequency Accel Tare Force Forcelb rps ft/sec2 lb lb lb

0 7.57 6.24 - 0.160 10.31 11.62 - 0.300 13.22 19.08 - 0.500 14.76 23.80 - 0.620 16.31 29.06 - 0.750 16.87 31.14 - 0.800 18.18 36.32 - 0.940 20.40 43.33 - 1.18

0.24 9.19 9.21 0.24 0.31 0.310.24 11.39 14.14 0.37 0.47 0.470.24 14.09 21.68 0.56 0.72 0.720.54 8.29 7.50 0.20 0.32 0.320.54 11.37 14.10 0.37 0.60 0.600.54 14.47 22.87 0.60 0.98 0.971.04 7.68 6.44 0.17 0.38 0.371.04 11.65 14.80 0.39 0.86 0.851.04 14.35 22.46 0.58 1.31 1.29

The natural frequency, wn, of the balance varies withthe applied mass. Natural frequencies of the lift springwere determined from oscillograph records taken while"ringing" the balance. The results are included in the tableabove, along with the ratio a = / wn"

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The frequency response of the lift balance is shown onFigure D3, where gain (ratio of measured to applied force) isplotted against the frequency ratio. The natural frequencyof the balance with the fin as used in the tests wasestimated using Figure D4, which is a plot of naturalfrequency against weight on the springs. Using the knownweight of the fin (with shaft) and an estimated added mass,the effective weight on the springs during the tests wasfound to be about 0.40 lb. According to Figure D4 thecorresponding natural frequency is 280 rps, or 44.6 Hz. Themaximum encounter frequency expected in the tests was about 2Hz, so that the maximum frequency ratio would be 0.045.Figure D3 shows that the response of the lift balance isessentially flat in this range.

Results for the drag balance show a flat response forthe entire range of weights and frequencies used in thecalibration.

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I.n

~ __. _ .. . .z ..... .

0 -- M

- _ )

ua

Ma F C0

s3Tq 'Su~pra-g

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tow~

16!)

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- 1.5 7 L,___

S 0.81 --- 7- ......

S0.41 H7__ __ _ _ - _____ T t

.01 .02 .03 .05 .10 .15 .20 .30

Frequency ratio fl - /w

FIGURE D3 FREQUENCY RESPONSE OF LIFT BALANCE

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L .. : ... ~

IJI

SAYj u v

rU FF ijp P.m

4- .1 7

.......... . i ...................

-4.b =7 ** :N jqga 2 1,h7.:

L ft..74 T ::~~ ~±J ..

.2 3.4.510 5-

Appie I egfl

FIGURE14 Ell!~LFEQEC O IT AAC

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APPENDIX E

Tabulation of Water Temperatures

DATE RUNS TEMPERATURE(1987) (OF)

Phase 1: Tests of Unappended Model in Calm Water

1/29 12-36 73.01/30 37-95 73.02/2 96-103 73.2

Phase 3: Tests with Instrumented Canard

4/28 10-25 72.34/29 34-60 72.34/30 65-83 72.35/1 88-120 72.15/4 127-155 72.35/5 163-188 72.5

Phase 4: Tests with Pitch Control System

8/18 45-71 74.98/19 72-88 75.08/20 89-113 75.38/21 116-145 75.28/24 146-177 74.98/25 178-206 74.68/26 207-220 74.3

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APPENDIX F

COMPUTATION OF ANGLE OF ATTACK OF CANARDS INDUCED BYREGULAR WAVES

The angle of attack of a canard, under the assumption thatthe canard and model do not disturb the fluid, can be expressedas

a = arctan (v/(u+V))

in head seas, where u and v are the wave-induced particlevelocities in the horizontal and vertical directions,respectively. If the wave profile is written as

q = a cos (kx - wt)

where x is increasing in the direction of wave advance and k isthe wavenumber, it may be shown (see Reference Fl) that theparticle velocities are

u - uo cos Wt v -v o sin wt

at x=O (for example), where

u0 = aw cosh k(y+h) / sinh khvo = aw sinh k(y+h) sinh kh

and a is the wave amplitude, y is the vertical coordinate of thepoint of interest (the canard centerline) relative to the calmwater surface, and h is the water depth. Thus

a = arctan [-v 0 sin wt/(V + uO cos wt)]= arctan [-(vo/V) sin wt/(l + (uo/V) cos wt)]

Let

sin wt/[l + (uo/V)cos wt] = F

be represented by a Fourier series

F = A0 + A1 cos wt + B1 sin wt + A2 cos 2wt + B2 sin 2wt +

As shown in Appendix G, the coefficients in the series have thefollowing form:

Ai = 0

Bi = -2(11 _ !7 - l)i/ 0i+l

where = Uo/V. For small D, the coefficients have thefollowing asymptotic form (see Appendix G):

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Bi - (-0/2)i-i

so that

B1 - 1

B2 -0/2

B 3 " B2/4

Thus if the particle velocity is sufficiently small relative tothe ship speed to justify neglecting B2 and further terms, oneobtains

a - arctan [-B1 (vo/V) sin wt]

or

tan a = (B1 Vo/V) cos(wt + w/2)

so that the angle (and thus the lift force in a "quasi-steady"theory) leads the wave crest by 900 (or, equivalently, lags thewave crest by 2700) which is in agreement with the test results.In addition, in the tests the second and higher harmonics of themeasured fin lift were insignificant, which would justifyneglecting B2 and higher terms in the Fourier series in thepresent case.

In following seas the ship has velocity -V in the directionof wave advance. Thus -V is to be substituted for V in theformulas above for following seas. The only effect of thischange is to reverse the sign of the angle of attack. Thus infollowing seas if the ship is overtaking the waves, the angle ofattack lags the wave crest by 900 rather than by 2700. If thewaves are overtaking the ship, a negative phase angle isinterpreted as a phase lead; so in overtaking waves the finangle of attack leads the wave crest by 900 (or lags the wavecrest by 2700) as in head seas.

The canard lift coefficient is predicted by multiplying theangle of attack from the expressions above by the steady-flowlift curve slope of the fin (including the "groundboard effect"of the hull as discussed in the main text). This is referred toas a "quasi-steady" theory, because effects of unsteadiness onthe angle of attack ("memory effects") are neglected.

Figures 10 and 11 show that this theory consistentlyunderpredicts the amplitude of the canard lift coefficient byabout 15% at a ship speed of 20 knots; the ratio of particlevelocity to ship speed was between .05 and .15 in these tests.Predicted phase angles are in accord with the observations,indicating that there is no appreciable time lag in thedevelopment of lift in the range of observed conditions.

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REFERENCE

Fl. Newman, J.N., Karin* Xydrodynamics, MIT Press, 1977.

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&PPENDIX G

EVALUATION OF AN EXPRESSION WHICH OCCURS IN THE ANGLE OF ATTACKCOMPUTATION

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1 Problem StatementConsider the periodic function

F(t) = sinwt/[1 + 8coswt] (1)

and find its Fourier series of the form

F(t) = Ao + A, coswt + B sinwt + A 2 cos2Wt + B 2 sin2wt +... (2)

2 DerivationThe coefficients of the Fourier expansion of F(t) are given by [1]

Ao = F(t)dt (3)

A. = . F(t) cos(iwt)di (4)

B, = - [rI" F(t) sin(iwt)dt (5)

for i = 1,...,ooNow, by definition of the function F(t), it follows immediately that

F(-t) = -F(t) (6)

and thus that F(t) is an odd function. Since the integration interval issymmetric, it follows immediately that all the integrals involving the termcos(iwt) must vanish because cos(iwt) is an even function, and therefore

A, = 0, (7)

for i = O, 1,2,..., oo.To find a form for the remaining Fourier coefficients, use the product

formula for sines [2]

2sin(z1 ) sin(z2) = cos(zI - z 2) - cos(zI + zI) (8)

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to rewrite the expression for Bi into the equivalent form

Bi = C_- - C,+1 (9)

where the quantity C is defined by the integral

Ci- cos(wt)/(1 + cos(wt))dt (10)

Using the fact that the integrand is even and the integration interval issymmetric, then it can be rewritten as

C,=d - ' r"cos(iwt)/(1 + 06 cos(wt))dt (11)

and then a change of variables to z = wt yields

1 cos(ix)/(1 + P cos(z))dx (12)

which is an integral in standard form and can be looked up in an integraltable.

From [3] 3.613.1, provided that fl2 < 1, then

C1 = v,, P (( /' - 2 - 1)/6)' (13)

for = 0, 1,... , oo. As examples, note that

Co = 1/r - #2

C 1 = (V/ "- #2 - 1)/(,8V'- #)

C2 = (2-#2-2 ')/(#2V -_6)

C3 = (-4 + 3# 2 + (4 - #2)Vkl - #))&63VF- j

C4 = (7- 8#2 + 4-4(2-_ #)V f1#2)&64V1-#2 ) (14)

Using these results in the expression above for Bi yields explicit valuesfor the non-zero coefficients in the Fourier expansion, namely

B, = C.-I - C,+j

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= f P - i)/81ii/V/j? -

X[#2 - (1 - #2 + 1 - 2%f1 - )]/p2

-- 2(V=I -02 - 1)'/P#'+ (15)

This finishes the calculation of the Fourier expansion.As a final issue, consider the behavior of the coefficients as P goes to zero,

then

Bi =- 2 - V/042

-2(_#21/2)'/# '+'

(-P/2)'- ' (16)

which gives the asymptotic behavior of tne coefficients in the physically in-teresting case when 8 becomes small.

References

[1] M. R. Spiegel Theory and Problems of Advanced Calculus, Scham'sOutline Series, McGraw-Hill Book Company, New York, 1963,Chapter 14.

[2] M. Abramowitz and I. A. Stegun Handbook of Mathematical Func-tions, Dover Publications, Inc., New York, 1972 4.3.31

[3] I. S. Gradshteyn and I. M. Ryzhik Table of Integrals, Series, andProducts Academic Press, New York, 1980

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