I: TECHNICAL'<REPORT

NCSL TB. 327-78

f~FF.BRUAIYY178

PREDICTION OfA ACELERATOHYDRODYNAMIC coE F'If-N TSORUNDERWATER YEiJCLE$

*~.~,GEOMETRIC PAýRAM'ETlft.$

-D. E,.,HUM-PHREYSLA-'

-tIA.VAL CO ISTAL SYMMSf~ cAsoRAT00.

A3 --4''

APPROVED ~oR PJUB= =W5AL--.

~F~B~fON UNL

~'-~ '2

NAVAL COASTAL SYSTEMS LABORATORYPANAMA CITY, FLORIDA 32407

CAPT JAMES V. JOLLIFF, USN GERALD 0. GOULDCommanding Officer Techniaii Dirvqclr

ADff N1STRAf1W 1NFOP44A~TION

Th~is wotk was perfotued under& the joint eponsorship o~f the proj ectsVehicle Hydrodynaxde*, vnak Arm& Number ZF 61-11.2-00l, Work Unit Number00112-5O-1,A Analytical 1nve tig&Ltiovs-ýf Co0untei.~aure Launcherx. TaskArea Nuzmber- 'Sr 34-371-208o work Unit Nuimber JfA~.&. and ths Advaxncel

Submarin# 25 i 1~brjc u~~ 027, Work Unit 20259.

R~.alav-d by 11Jndar Autharity of14, J. W4yna Head .G(.oiCoaot;1 Technology Departa~ent. Tecbaica1 D~irector:Februaty 1978

Mk.

NA-TI0NAL. 3$1C0RTY IN4FORMATIONUWAU'rHOtRIZE 01$Ckv . ABJWWC TO CRWINAL SAOCTIc*4S

UNCLASSIFIED _____

SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered)

REPOT DCUMNTATON AGEREAD INSTRUCTIONSREPOT DCUMNTATON AGEBEFORE COMPLETING FORM

NCSLýTR-32- 782. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

T-T I TLE (uind S'b t ItlIe) L" YPE OF REPORTA& PERIOD VQE;ED

.- a-a Technical pet~

7. SUHR.Lý . CONTRACT OR GRANT NUMBER(*)

I() KE.l umphreysK..W'Watkinson

S. ERORMNGORANIATONNAME AND ADDRESS 10 I. PROGRAM kELEMENT. PROJECT, TASK

Naval Coastal Systems Laborator'y-dAE OKUI UURPanama rity, Florida 32407

7I 7c Ot TRtOLLING OFFICE NAME AND ADDRESS 12REPORT qZJ_ /

108

S 14. MONITORING AGr6NrY.KA.MEJ& 1;A j!.RSS(ti dtig.eut from Goo~tUaAi~4 IS. SECURITY CLASS. (of this report)

. UNCLASSIFIED

,/~~~10 FD~7~~.( IaECLASSI FICATION/ DOWNGRADING

IS. DISTRIBUTION STATEMENT (of this Report)

Approvcd for Public Release: Distribution Unlimited.

Prediction of Acceleration Hydrodynamic?Coefficients for Underwater Vehicles(from Geometric Parameters different from Report)

16. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse aide it necessary and Identify by block number)

Hydrodynamic coefficients Simulation Identifiers:Underwater vehicles Hydrodynamics AccelerationDynamic response Equations of .,lotion hydrodynamic coefficientAnalytic functions Fluid mechanicsCoefficients

20. ABSTRACT (Continue on reverse aide It necessary ad identify by block number)

A compilation of methods for analytically calculating the accelerationhydrodynamic coeff zients 'from the geometric and mass distribution charac-teristics of an underwater vehicle is presented. Each of the coefficientsin the longitudinal and lateral planes are treated in the order introd'ic-tory description; relative importance in characterizing the vehicle dynamicresponse; causal relationship; and analytical expression for calculating -fi

SECURITY CLASSIFICATIbN OF THIS PAGE (When Dotm Rred)

. .. . k .ffrnq~3. .- ...... =- -

JN_-LASSIFPTED -

SECURITY CLASSIFICATIOM OF THIS PAGE (flth DAN Z1A66e

10.Task Area No. ZF 61-112-001, Work Unit No. 00112-50-lA iTask Area No. SF 34-371-208, Work Unit Number 16436Subproject No. S0207, Work Unit 20259

20.

the value of the coefficient. Relationships for cross-coupled and non-linear hydrodynamic coefficients arising from potential flow theory arederived. All of the techniques have been implemented as part of the com-puter program GEORGE, a program for computing underwater vehicle dynamicstability characteristics in use at NCSL since 1972. Comparison of experi-mental and computed results is presented, along with a physical descriptionof the generalized underwater vehicle used as the basis for the calcula-tions and graphic representations contained in the report.

I

S/N 0102- LF.014-6601

SECURITY CLASSIFICATION OF THIS PAGt('Vhen Date Rnterod)

NCSL TR-327-7t

GLOSSA.--Y OF TERMS

All symbols, subscripts, and nondimensionalized hydrodynamic coef-

ficient expressions appearing in the body of this report are defined

below. Any dimensions involved will be consistent with the foot-pound-

second system of units. All angles are in degrees.

SYMBOLS

Symbol Definition

2AR Aspect ratio of surface = b Is.

B Buoyancy force which is positive upwards, B pV (except

where used in fluid kinetic energy equations).

b Exposed root-to-tip span of surface.

c Chord.

c Mean chord of surface s/b b/AR.

c Root chord of surface.r

c Tip chofd of surface.t

I Added moment of inertia.a

Moment of inertia of an underwater vehicle about thex

x-axis.

I Added moment of inertia about the x-axis.x

a

I Moment of inertia of an underwater vehicle about they y-axis.

I Moment of inertia about the y-axis of the fluid

Ydf displaced by the body.

i Moment of inertia of an underwater vehicle about the

z-axis.

K Moment about the x-axis exerted on the body by the

dynamic pressure of the surrounding fluid.

k Lamb's inertial coefficient for a prolate ellipsoid1

in axial flow.

NCSL TR-327-78

Symbol (Continued) Definition

k2 Lamb's inertial coefficient for a prolate ellipsoidii cross flow.

k' Coefficient for added moment of inertia for a flat plate.pk •' Lamb's coefficient for added moment of inertia for a

prolate ellipsoid body.

Overall length of the vehicle.

M Moment about the y-axis exerted on the body by thedynamic pressure of the surrounding fluid.

m Mass of the vehicle including the water in the free-flooding spaces.

m Additional mass.a

mdf Mass of fluid displaced by hull envelope.

N Moment about the z-axis exerted on the body by thedynamic pressure of the surrounding fluid.

p Component of angular velocity about th- bodyfixed x-axis.

q Component of angular velocity about the bodyfixed y-axis.

r Component of angular velocity about the body

fixed z-axis.

s Exposed planform area of surface.

sb Maximum cross-sectional area of the body.

T Total kinetic energy of the body-fluid system.

Tf Kinetic energy of the fluid representing the system cCimpulsive pressures exerted by the surface of the bodyon the fluid during acceleration.

T Kinetic energy of the vehicle representing linear andTveh angular momentum of the body.

t Time.

U Linear velocity of origin of body axes relative to ajfluid-fixed axis system.

ii

NCSL TR-327-78

Symbol (Continued) Definition'

u Component of U along the body x-axis.

I Displaced volume of the hull envelcpe.

v Component of U along the body y-axis.

w Component of U along the body z-axis.

X Force along the x-axis exerted on the body by thedynamic pressure of the surrounding fluid.

X x-distance from nose to cb.

X x-distance from nose to cg.ncg

Xt x-distance from nose to centroid of the horizontaltail fins.

Longitudinal axis of the body fixed coord.~iate axissystem.

Y Force along the y-axis exerted on the body by thedynamic pressure of the surrounding fluid.

y Transverse axis of the body fixed coordinate axissystem.

Z Force along the z-axis exerted on the body by thedynamic pressure of the surrounding fluid.

z Vertical axis of the body fixed coordinate axissystem,

Zcb The z-distance to the center of buoyancy (cb) from thecenter of gravity (cg), positive down.

Angle of attack.

a Angle of side-slip.

6 Deflection angle of bowplane (or sailplane).bp

6 Deflection angle of rudder.r

6 Deflection angle of sternplane.s

NCSL TR-327-78

Symbol (Continued) Definition

X Planform taper ratio of surface - c /ct r

w Angular velocity.

Yaw angle.

*, Roll angle.

P Mass density of seawater.

e Pitch angle.

The component in the plane of a surface of the perpen-dicular distance between the axis of rotation and thecentroid of the area of the surface.

A prime over any symbol signifies nondimensionalization.

A dot over any symbol signifies differentiation withrespect to time.

SUBSCRIPTS

Subscript Definition

a Added or additional inertia.,

b Body.

bp Bow plane.

cb Distance from cg to cb.

cg Distance to cg.

df Displaced fluid.

"f Fluid.

ht Horizontal tail.

xvt Lower vertical tail section.

iv

NCSL TR-327-78

Subscript (Continued) De fin=tion

n( ) Referenced from nose to some point, (), e.g., ncbimplies nose-to-cb.

p Plate.

r Rudder.

s Sternplane.

sh Shroud.

t Tip.

uvt Upper vertical tail section.

veh Vehicle.

x Pertaining to the x-axis.

y Pertaining to the y-axis.

z Pertaining to the z-axis.

v

NCSL TR-327-78

NONDIMENSIONALIZED HYDRODYNAMIC COEFFICIENT EXPRESSIONS

X. X. x.w U - # - V

a NWp9 (U =1 Np

Y. Y. Y.

ul (½pR 'l v (½9 0 -

K.Z.Z.ZuI u )i a -v - T 0 z -

x x

YY Y.

-l yl . - r

k.p q Np, h

rV' I. uA 4 r

K.K. K.K . K'w 'v K I v K'

M. M. M.

u Mt.pt v mtpt w ½)

N. N. N.N' I ½)~ N'. -N'.-q- Tv (½p'w (½) P t

K'. K'. K'. Kp ()pR 'q (½)ptr (½pt

- ' ' Mtcv (i)p :' 44 mt)t

N. N. N.NV N'.-

Note that X'Ci a x'/aai, X'f-lp/' etc.

vi

NCSL TR-327-78

TABLE OF CONTENTS

P Ste No.

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

Background . . . . . ....... ........... 1

Acceleration Coefficients .............. . . . . . . 3

ACCELERATION COEFFICIENTS IN POTENTIAL FLOW THEORY. . . . . .. 4

METHODS FOR COMPUTING ACCELERATION HYDRODYNAMIC COEFFICIENTS. 9

Introduction ....... ............... . . . . . . . . 9

Longi idinal Coefficients. . . . . ............. 9

XV . . . . . . . . . . . . . . . . . . * . . . . . . . 9Uu........................................................ *. . . . . . . 14

MO . . . . . . . . . . . . . . . . . . . . . . . . . . 16u

V' . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Z' 17

Mo.' . . . . . . . . . . . . . . . . . . . . . . .. . . 26w

X1 . . . . . . . . . . . . . 29

M' .................... .......................... 29q

Lateral Coefficients .......................... 38

y1 . . . . . . . . . . . . . . . . . . . . . . . . . . 38

p

vii

NCSL TR-327-78

TABLE OF CONTENTS (Cont'd)

Page No.

Y 0 44

K. . ......... . . ........... . . . . . . 44

......... ** * * * * *50t

K .. . . . . . . . . . . . . . . . . . .. . . . . 56V

N', 56

N' . ........................ 56

N' . . . . . . . . . . . . . 59

SUMMARY . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 59

APPENDIX A - DERIVATION OF HYDRODYNAMIC COEFFICIENTSRELATIONSHIPS FROM ENERGY EQUATIONS ............ A-I

APPENDIX B - COMPARISON OF ANALYTICAL PREDICTIONS WITHEXPERIMENTAL DATA. ........ ......... B-i

APPENDIX C - DESCRIPTION OF TYPICAL HYDRODYNAMIC VEHICLEUSED FOR SENSITIVITY ANALYSIS.. . . . ....... C-I

APPENDIX D - DESCRIPTION OF MANEUVERS USED INTRAJECTORY SIMULATION. . . . ........... D-1

viii

NCSL TR-327-78

LIST OF ILLUSTRATIONS

Figure No. Page No.

1 Positive Directions of Axes, Angles, Velocities,Forces, and Moments 10

2 Trajectory Simulation Plot of X'' Contribution tou

X-Force During a 6 - ±300 Dive/Climb Maneuver 12s

3 Percent Contribution of X'. to X-Force During au

6 - ±300 Dive/Climb Maneuver 12s

Lamb's Inertia Coefficients k1 , k2 and k'

Plotted as a Function of Vehicle £/d 15

5 Trajectory Simulation Plot of Z'' Contribution tow

Z-Force During a 6 - 300 Steady Turnr

(Automatic Depth-Keeping) 18

6 Percent Contribution of Z'. to Z-Force During aw

6 - 30* Steady Turn (Automatic Depth-Keeping) 18r

7 Trajectory Simulation Plot of Z'. Contributionw

to Z-Force During a 6 = ±300 Dive/Climb Maneuver 19B

8 Percent Contribution of Z'. to Z-Force During aw

6 ±30° Dive/Climb Maneuver 195

9 Trajectory Simulation Plot of Z'' Contribution tow

Z-Force During a 6 = ±30'" Turn Reversal Maneuverr

(Automatic Depth-Keeping) 20

10 Percent Contribution of Z'. to Z-Force During aw

6 - ±30* Turn Reversal Maneuverr

(Automatic Depth-Keeping)

11 Root Locus Plot of u/6 for a Generalized5

Underwater Vehicle, Varying Z' 21w

ix

I

NCSL TR-327-78

LIST OF ILLUSTRATIONS (Cont'd)

Figure No. Page No.

12 Root Locus Plot of w/6 for a Generalized

Underwater Vehicle, Varying Z' 21

13 Root Locus Plot of 6/6 for a Generalized

Underwater Vehicle, Varying Z' 21w

14 Coefficients of Additional Mass for RectangularPlates 25

15 Top View of Body and Horizontal Tail FinConfiguration with Descriptive Dimensions 26

16 Trajectory Simulation Plot of M'. Contribution toq

Pitch Moment During a 6 = 30* Steady Turnr

(Automatic Depth-Keeping 31

17 Percent Contribution of M'. to Pitch Moment Duringq

a 6 - 30° Steady Turn (Automatic Depth-Keeping) 31r

18 Trajectory Simulation Plot of M'. Contribution toq

Pitch Moment During a 6 - ±300 Dive/ClimbManeuver 32

19 Percent Contribution of M'. to Pitch Momentq

During a 6 ±30* Dive/Climb Maneuver 32

20 Trajectory Simulation Plot of M'- Contribution toq

Pitch Moment During a 6 - ±30* Turn Reversalr

Maneuver (Automatic Depth-Keeping) 33

21 Percent Contribution of M'- to Pitch Moment Duringqa 6 = ±30° Turn Reversal Maneuver (Automatic

rDepth-Keeping) 33

22 Root Locus Plot of u/6 for a Generalized

Underwater Vehicle, Varying M'. 34q

x

INCSL TR-327-78

LIST OF ILLUSTRATIONS (Cont'd)

Figure No. Page No.

23 Root Locus Plot of w/6s for a GeneralizedS

Underwater Vehicle, Varying M'- 34

24 Root Locus Plot of e/6 for a Generalized

Underwater Vehicle, Varying M'- 34q

25 Coefficient of Additional Moment of Inertia forK Rectangular Plates 37

26 Trajectory Simulation Plot of Y'. Contribution tovY-Force During a 6 = 300 Steady Turn 39

r

27 Percent Contribution of yV. to Y-Force Duringa 6 - 300 Steady Turn V 39

r

28 Trajectory Simulation Plot of Y'. Contribution tov

Y-Force During a 6 = ±300 Turn Reversal Maneuver 40r

29 Percent Contribution of Y'- to Y-Force During av

6 - ±30° Turn Reversal Maneuver 40r

30 Root Locus Plot of v/6 for a Generalizedr

Underwater Vehicle, Varying Y'- 41v

31 Root Locus Plot of 1,16 for a Generalizedr

Underwater Vehicle, Varying Y'. 41v

32 Root Locus Plot of 4/6 for a Generalizedr

Underwater Vehicle, Varying Y'. 41

33 Trajectory Simulation Plot of K'. Contributionv

to Roll Moment During a 6 - ±300 Turn Reversal"Maneuver r 47

34 Percent Contribution of K's to Roll Moment Duringv

a 6 r = ±30* Turn Reversal Maneuver 47 I

xi

NCSL TR-327-78

LIST OF ILLUSTRATIONS (Cont'd)

Figure No. Page No.

35 Root Locus Plot of */6 for a GeneralizedUnderwater Vehicle, Varying K'. 48

V

36 Dependence of the Additional Moment ofInertia on Taper Ratio 53

37 Variation of the Additional Moments of Inertia

of a Single Plate With Dihedral Angle 54

38 Trajectory Simulation Plot of N'i Contribution tor

Yaw Moment During a 6 - ±30* lurn Reversal Maneuver 60r

39 Percent Contribution of N'- to Yaw Moment During ar6 = ±30* Turn Reversal Maneuver 60r

40 Root Locus Plot of v/6 for a Generalizedr

Underwater Vehicle, Varying N'. 61

r

42 Root Locus Plot of 4/,6 for a Generalized

rVehicle, Varying N'. 61

xii

NCSL TR-327-78

LIST OF TABLES

Table No. Page No.

IA Longitudinal Sensitivity Analysis of CharacteristicEquation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying X'. ±100% 13

U

IB Longitudinal Sensitivity Analysis of U NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying X' 6100% 13

IC Longitudinal Sensitivity Analysis of W NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying X'. ±100% 13

u

ID Longitudinal Sensitivity Analysis of 0 NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying X'. ± 100% 13A

2A Longitudinal Sensitivity Analy3is of CharacteristicEquation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying Z'. ±100% 22

w2B Longitudinal Sensitivity Analysis of U Numerator

Non-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Z'. ±100% 22

2C Longitudinal Sensitivity Analysis of W NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Z'. ±100% 22

w

2D Longitudinal Sensitivity Analysis of 8 NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Z'. ±100% 22

3A Longitudinal Sensitivity Analysis of CharacteristicEquation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying M'. ±100% 28V

3B Longitudinal Sensitivity Analysis of U NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying M'. ±100% 28w

3C Longitudinal Sensitivity Analysis of W NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying M' ±100% 28

xiii

NCSL TR-327-78

LIST OF TABLES (Cont'd)

Table No. Page No.

3D Longitudinal Sensitivity Analysis of 0 NumeratorNon-Dimensional Roots for a Generalized UnderwaterIVehicle, Varying M' * ±100% 28

4A' Longitudinal Sensitivity Analysis of CharacteristicI: Equation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying Z' ±100% 30j

4B Longitudinal Sensitivity Analysis of U NumeratorI Non-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Z'. ±100% 3

[4D Longitudinal Sensitivity Analysis of W NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Z'. ±100% 30

5A Longitudinal Sensitivity Analysis of 0CNumractorisiEqainNon-Dimensional Roots for a Generalized UdraeUnewtrVehicle, Varying M. ± 100% 30

5B Longitudinal Sensitivity Analysis of ChaumracterisiEqainNon-Dimensional Roots for a Generalized UdraeUnewtrVehicle, Varying M'. ±100% 35

5C Longitudinal Sensitivity Analysis of U NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying M'. ±100% 35

5D Longitudinal Sensitivity Analysis of W NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying M'. ±100% 35

6D Longtudal Sensitivity Analysis of 0ha NumerastorEqainNon-Dimensional Roots for a Generalized UdraeUnewtrVehicle, Varying Y'. ±100% 35

6B Lateral Sensitivity Analysis of ChaumracterisiEqainNon-Dimensional Roots for a Generalized UdraeUnewtrVehicle, Varying Y' .±100% 42

xV

NCSL TR-327- 78

LIST OF TABLES (Cont'd)

Table No. Page No.

6C Lateral Sensitivity Analysis of *p NumeratorNon-Dimensional Roots f or a Generalized UnderwaterVehicle, Varying Y'. +100% 42

6D Lateral Sensitivity Analysis of 4) NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Y'. ±100% ~42

7A Lateral Sensitivity Analysis of CharacteristicEquation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying Y'. ±100% 45

7B Lateral Sensitivity Analysis of V NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Y'. ±100% 4

7C Lateral Sensitivity Analysis of *~ NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Y'. ±100% 45

7D Lateral Sensitivity Analysis of 4) NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Y'. ±100% 45

8A Lateral Sensitivity Analysis of CharacteristicEquation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying Y'. ±100% 46

8B Lateral Sensitivity Analysis of V NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Y'.f ±100% 46

8C Lateral Sensitivity Analysis of 4p NumeratorIF Non-Dimensional Roots for a Generalized Underwater

Vehicle, Varying Y'. ±100% 46

8D Lateral Sensitivity Analysis of 4)NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying Y'.i ±100% 46

9A Lateral Sensitivity Analysis of CharacteristicEquation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying K'. ±100% 49

vv

NCSL TR-327- 78

LIST OF TABLES (Cont'd)

Table No. Page No.

9B Lateral Sensitivity Analysis of V NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying K'. ±100% 49

9C Lateral Sensitivity Analysis of * NumeratorNon-Dimensional Roots for a Generalized UnderVehicle, Varying K'. ±100% 49V

9D Lateral Sensitivity Analysis of 4 NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying K'. ±100% 49

IOA Lateral Sensitivity Analysis of CharacteristicEquation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying K'. ±100% 52

p

10B Lateral Sensitivity Analysis of V NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying K'. ±100% 52

P

lOC Lateral Sensitivity Analysis of i NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying K' ±100% 52

10D Lateral Sensitivity Analysis of 4 NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying K'V. ±100% 52

p11A Lateral Sensitivity Analysis of Characteristic

Equation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying K'r ±00% 55

lIB Lateral Sensitivity Analysis of V NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying K'. ±100% 55

lIC Lateral Sensitivity Analysis of i NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying K'. ±100% 55

r

l1D Lateral Sensitivity Analysis of 4 NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying K' ±100% 55

xvi

INCSL TR-327-78

LIST OF TABLES (Cont'd)

Table No. Page No.

12A Lateral Sensitivity Analysis of CharacteristicEquation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying N'. ±+00% 57

v

-12B Lateral Sensitivity Analysis of V NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying N'. ±100% 57

12C Lateral Sensitivity Analysis of iP NuimeratorNon-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying N'. ±I00% 57

12D Lateral Sensitivity Analysis of 0 NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying N'. ±100% 57

v

13A Lateral Sensitivity Analysis of CharacteristicEquation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying N'. ±100% 58

p13B Lateral Sensitivity Analysis of V Numerator

Non-Dimensional Roots for a Generalized UnderwaterVehicle, Varying N'. ±100% 58p

13C Lateral Sensitivity Analysis of i NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying N'. ±100% 58

p

13D Lateral Sensitivity Analysis of ý NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying N'. ±100% 58

p

14A Lateral Sensitivity Analysis of CharacteristicEquation Non-Dimensional Roots for a GeneralizedUnderwater Vehicle, Varying N'. ±100% 62

r

14B Lateral Sensitivity Analysis of V NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying N'. ±100% 62

r

Sxvii

NCSL TR-327-78

Table No. PaeNo.

14C Late a&l Sensitivity Analysis of i4NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying N'. ±100% 6

r 6

14D Lateral Sensitivity Analysis of 0 NumeratorNon-Dimensional Roots for a Generalized UnderwaterVehicle, Varying N'. ±100% 62

r

15 Root Sensitivity Analyses for Derivative Variationof ±100% 64

xviii

NCSL TR-327-78

INTRODUCTION

BACKGROUND

F This report is primarily a compilation of methods for analyticallypredicting the acceleration hydrodynamic coefficients from the geometricand mass distribution characteristics of an underwater vehicle. It rep-resents the current results of an effort to fin' the most generally ap-plicable and accurate techniques for calculating these coefficients.

Teneed for such analytical. techniques grew from the many uniqueunderwater vehicle requirements inherent to the Naval Coastal SystemsLaboratory mission areas. Typically these requirements include design,modification, or simulation of towed, -nthered, or free vehicles forwhich analytical calculation of the hv :odynamic. coefficients is neces-sary. This necessity arises from the need for some combination of thefollowing:

1. Flexibility. The user's project is in a design phasewhere vehicle requirements imposed by hardware sub-systems are subject to change and it is desirable toassess the effect of many more vehicle configurations

E than would be practical for experimental determination.

2. Responsiveness. The project is of a research and develop-ment nature such that there is a requirement for rapid andfrequent updating of the vehicle's predicted performancein response to changes in performance goals and/or subsys-tem hardware.

3. Economy. Cost constraints on the project will not permituse (or extensive use) of model basin testing, but theproject still requires some degree of assurance that thereare no potential catastrophic failures due to problemssuch as dynamic instability.

To more effectively meet each of these require~ments, all of themethods presented herein, as well as methods for computing all of the

remaining hydrodynamic coefficients, have been computer-implemented inthe complete stability and control analysis program GEORGE. The GEORGEcomputer program uses vehicle geometric and mass distribution informiation

k to compute vehicle dynamic characteristics.

NCSL TR-327-78

It should be pointed out that the procedure yielding the highestdegree of confidence f or an underwater vehicle design involves the useof both model basin testing and analytical techniques. An example ofa desirable sequence of design steps would be the implementation of acomputerized hydrodynamic math model of the proposed vehicle 'using con-ceptual information available in the very early design phases. 11singthis model and incorporating updated information as it becomes avail-able, the effect of various sizing, distributing, and shaping ideas onthe vehicle dynamics can be assessed.

As soon as a reasonably firm design configuration is laid out,model basin tests can be performed to validate the computer modeling.These tests can be much less extensive than those necessary to producethe parametric information required as a basis for new desigv tech-

f niques-echniquea sufficiently general to maintain predictive accuracyfor the redesigns or modifications frequently necessary in vehicle de-sign. This reduction in model testing requirements is primarily theresult of having a computer model to use in defining what needs to beverified experimentally, and in providing a sophisticated tool fordescribing complicated trends, thus reducing data requirements.

The validated computer simulation model can then be used in manyways to support the project. Trajectory simulations can be made forgiven operational conditions so that motion information can be genera-ted for performance evaluation or definition of hardware specifications.Examples of work performed using simulations in this manner are trajec-tory analyses of submarine-launched acoustic countermeasure devices andmotion prediction of a swimmer delivery vehicle to assess the motion in-fluence on the design of a high-resolution, side-looking sonar.*

Another use has been to provide the hydrodynamic coefficients foranalog or hybrid simulation models to study man-machine interface prob-lems or to provide training. An example of recent work in thisa area isthe implementation of a hybrid model of the Mine Neutralization Vehicle(MNV) as a performance evaluation and training device. The bflNV simula-tion solves the nonlinear equations of motion for a tethered vehicle(including four propulsors, bowplanes, and cable tension) and producesa motion display for either body or inertial reference frame, all inreal time. Of course, the computer simulation model is most often usedto assess the influence of design variations of the vehicle hydrodynamicand control system parameters on dynamic stability and controlcharacteristics.

*Trajectory simulations have usually been performed using suitable modi-fied versions of the standard six-degrees-of-freedom submarine simulationcomputer program, ZZMN, developed by DTNSRDC.

2

IMc.;L 71R-327- 78

Having briefly stated the background for the prediction and simula-

tion capability regarding the stability and control of underwatervehicles at NCSL, attention will now be focused on the analytical tech-niques for assigning a value to acceleration, nunlinear, and cross-coupled hydrodynamic coefficients.* These techniques have been appliedto a wide variety of body types, including torpedoes, submarines, swim-mer delivery vehicles (SDV's), and many other special purpose vehiclessuch as the Deep Submergence Rescue Vehicle (DSRV), the Shipboard Mine-hunting System (SMS) towed sonar body, and submarine-launched torpedocountermeasures devices.

Generality in applicability has been sought; that is, use of amethod that yields accurate results for only a specific class ofvehicles, such as torpedoes, bas been avoided. A more generally appli-nable and, in the experience of the authors, more accurate, formulationbased on theoretical derivation and experimental verification is pre-sented. The relative importance of each hydrodynamic coefficient as itcontributes to the responsive motion of the vehicle is noted. In addi-tion, important relationships between acceleration, cross-coupled, andnonlinear hydrodynamic coefficients are derived from potential flowtheory.

ACCELERATION COEFFICIENTS

The forces represented by acceleration hydrodynamic coefficientsare attributed to the acceleration of fluid particles surrounding a sub-merged body. Some of the forces appear in the equations of motion asadditions to terms representing the inertia of the body. They are knownas adted mass or mass accession terms and, when combined with the in-ertial parameters of a body, have been called apparent, virtual, andhydrodynamic mass.

Analytical methods for estimating the acceleration coefficients ofthe bare hull differ from the methods used for the contributions of ap-pendages in that they are not estimated using semi-empirical formulae.Rather, the body is assumed to be moving through an infinite, inviscid,circulation-free fixed fluid. For these conditions, Lamb(1) derives thecomplete expression for the kinetic energy of the body-fluid systemwhich, coupled with inertia coefficients for prolate ellipsoids(),yields estimates of body bare-hull acceleration derivatives.

*Although techniques are also available for the prediction of staticand dynamic velocity terms with viscous effects included, the scopeof this report is limited to the accaleration coefficients.

(1)Lamb, Sir Horace, Hydrodynamics, Dover Publications, New York,

Sixth Edition, 1932. pps 172 and 154.

3

NCSL TR-327-78

The basis for the added masses and moments due to body appendages,such as fins and bowplanes, is largely semi-empirical. Theoretical ex-pressions for thin, flat plates moving in a perfect fluid have beenderived from hydrodynamic theory with experimentally determined correc-tion factors for non-ideal conditions and shapes, A great deal of thebyMn(2)(3),primary work done in this area was accomplished by M withlatar significant experimental work by Gracey(4), and Malvestuto andGale(s).

ACCELERATION COEFFICIENTS IN POTENTIAL FLOW THEORY

The forces and moments represented by the acceleration hydrodynamiccoefficients can, to a very great extent, be modeled as potential flowphenomena. Neglecting the details of the boundary layer in modelingacceleration-dependent forces and moments acting on a submerged bodyyields quite satisfactory results for most stability and control simula-tion. Coefficient contributions due to appendages are most noticeablyinfluenced by boundary layer effects. Acceleration coefficient valuespredicted using potential flow theory are probably most suspect in theirapplication to fine that have a sigrnficant portion of their planformarea in separated flow. There are a few considerations concerning thesesources of error that should be mentioned before launching into furtherdevelopment.

Acceleration-dependent forces can arise from changes in the bound-ary layer, resulting, in turn, from changes in angle-of-attack or driftangle. There are several interesting facts in this regard. One is thatthe speed/power relationship for underwater vehicles generally encouragesa designer to fair his vehicle so that the boundary layer has a signifi-cant thickness over as little of the vehicle as possible. The result

(2)National Advisory Committee for Aeronautics T.N. No. 197, Some Tables

of the Factor of Appazent Additional Mass, by Max M. Munk, 1924.

(3)Munk, Max M., Fluid Mechanics, Part II, Aerodynamic Theory, Vol. 1,edited by W. F. Durand, Dover Publications, Inc., New York, 1934.

(4)National Advisory Committee for Aeronautics T.N. No. 707, The Addi-tional-Mass Effect of Plates as Determined by Experiments, byWilliam Gracey, 1941.

( 5 )National Advisory Committee for Aeronautics T.N. No. 1187, Formulasfor Additional-Mass Corrections to the Moments of Inertia ofAirplanes, by Frank S. Malvestuto, Jr. and Lawrence J. Gale, 1946,

4

NCSL TR-327-78

is that the pressure distribution over 80 percent of the surface areaof a typical underwater vehicle can be described very accurately bypotential flow theory. Consequently, the acceleration coefficientsmost important to the description of underwater vehicle dynamics canusually be accurately computed as potential flow phenomena. Some lessimportant acceleration coefficients, especially those resulting fromthe imbalance of added mass effects distributed fore and aft of the cg,can be significantly erroneous if the boundary layer is neglected. Un-fortunately, those coefficients influenced by boundary layer effectssuffer from similar problems in experimental model testing; e.g., im-proper Reynolds number scaling leads to nondynamic similarity in themodeled boundary layer.

Describing acceleration-dependent effects on fins is simplifiedsomewhat by the vehicle designer's avoidance of the sizing and place-ment of fins so that they lose effectiveness in regions of separatedflow. Consequently, more often than not, treatment of the added masseffects of fins as if they are in unspoiled flow provides good results.Tube-launched devices are often exceptions to this rule. They are gen-erally constrained by tail size and after-body taper so that large por-tions of the fins are in separated flow.

Decisions on "effective" fin aspect ratios based on past experi-ence with similar vehicles may be necessary to achieve atcuracy inthose coefficients with significant fin contributions. Since the finson a torpedo are generally small, their contributions to the major coef-ficients of importance are less significant than is the case with othertypes of underwater vehicles. Roll coefficients are 4 important ex-ception since they are due almost completely to fin ca; 1txibutions.

The acceleration, cross-.coupled, and nonlinear coefficient equali-ties derived in Appendix A must be used with great caution when simu-lating vehicle trajectories. The NCSL approach has been to assess theimportance of coefficient accuracy using two computer-implemented tools.One computer program performs root locus sensitivity studies for thelinear coefficients on the vehicle of interest. The required accuracyis determined on a coefficient-by-coefficient basis, taking advantageof the insight that can be gained by a control system analysis approachto the vehicle system.

A more thorough evaluation of the math model simulating the vehicle

fluid system, including the nonlinear and cross-coupled terms, must bedone on a maneuver-by-maneuver basis. To assess the relative importanceof each contributing force and moment, the authors have modified thestandard six-degrees-of-freedom submarine simulation computer program(r)

(')Naval Ship Research and Development Center Test and Evaluation ReportP-433-H-01, User's Guide NSRDC Digital Program for Simulating Subma-rine Motion ZZMN-Revision 1.0, by Ronald W. Richards, June 1971.

5

NCSL TR-327-78

to plot the following on any maneuver: (1) time histories of the force(or moment) of each contributing coefficient, propulsor, buoyancy/gravity, and control surface term together with the total force (ormoment); and (2) time histories of the motions of the vehicle with thecontribution of a single term removed, and repeated for each term. Know-

ing the degree of accuracy expected in the computed value for each c If-ficient, one can analyze the plots on the maneuvers of interest and assign

a level of confidence. This procedure will give an indication of whichcoefficients are most critical in achieving accurate simulation on agiven maneuver.

Using these techniques and a multitude of comparisons with planarmotion mechanism and rotating arm results, the methods presented in thisreport are, on the basis of the authors' experience, considered to bequite accurate on any hydrodynamically-faired vehicle for all but ex-treme maneuvers. This is especially true for linear models, as will beshown by examples in Appendix B.

The modeling accuracy achieved for acceleration terms under theseassumptions is quite contrary to the extremely poor results obtainedwhen the same assumptions are applied to velocity-dependent terms. Theassumptions are: (1) the body is submerged so that there are no near-surface or near-bottom effects; (2) the fluid is inviscid; and (3) thereis zero circulation.

Under these conditions the momentum equations for the body-fluidsystem may be written in the six Lagrange equations:

d 3T 3T DTdt --u r q- X,

d3T 3T 3Td v =p - r - -Y,(i3w 3u

d aT DT DT Z;dt 3w 3 -u- p - ; i

d 9T 3T 3T 3T 3Td- -p= W-•- v-•w+ t -- - q-•- K, (2)

d DT DT DT aT 3Tdt 7-U- - W-5u + p - r M, (2)

d DT 3T aT 9T 3Tu- p -N, (2)

where X, Y, Z, K, M, and N represent the forces and moments exerted onthe body by the pressure of the surrounding fluid, and the total kineticenergy T may be written

T = Tveh + TfV

6

NCSL TR-327-78

with Tveh representing the linear and angular momentum of the vehicleitself, and Tf representing the system of impulsive pressures exerted,by the surface of the solid on the fluid in the supposed instantaneousgeneration of motion from rest.

If we rewrite Equations (1) and (2), isolating the terms due to Tf,we obtain expressions for the forces exerted on the body by the dynamicpressure of the surrounding fluid, e.g.:

adTf BTf DTfX dt Du + (3)

and

DT DT aTf T2f f 1Tf 4j 3TfSK dt --f + w •-ý- v •-w+ r --- q•- (3)

dtaDp a w raq aqr

etc.

The fluid kinetic energy, Tf, can be expressed as a quadratic formof the body axis velocities u, v, w, p, q, and r with no explicit time Jdependence.* The quadratic expression for Tf may be written( 1 ):

2 2 2 2 22Tf - Au + Bv + Cw + 2A'vw + 2B'wu + 2C'vu + Pp + Qq

+ Rr4 + 2P'qr + 2Q'rp + 2R'pq + 2Lup + 2Mvq + 2Nwr

+ 2F(vr + wq) + 2G(wp + ur) + 2H(uq + vp) + 2F'(vr - wq)

+ 2G'(wp - ur) + 2H'(uq - vp). (4)

The body axes are generally chosen so that the xz-plane is a planeof symmetry (port-starboard symmetry). This eliminates nine of thecoefficients, A', C', R', L, M, N, G, and G', since by symmetry argu-ments, sign changes in v, p, or r should not change Tf. Letting

F + F' = F1 , F - F' - F 2 , H + H' - Hi, and H - H' H2 , (5)

(±)ibid.*Recent investigations indicate that there can be significant time vari-"ant tcrms arising from vortices shed from the forward appendages andhull. The mathematical modeling of this phenomenon is accomplishedusing force and moment terms in the equations of motion that are convo-luted in time. These terms are outside the set of acceleration, non-linear, and cross-coupled hydrodynamic coefficients under considerationhere.

7

NCSL TR-327- 78

the remaining 12 coefficients are included in the equation

2Tf = Au2 + Bv2 + Cw2 + 2B'wu + Pp2 + Qq2 + Rr2

+ 2Q'rp + 2FIvr + 2F 2wq + 2Hluq + 2H 2vp . (6) jThe complete expressions, assuming no planes of symmetry, are pre-

sented in Appendix A.

Proceeding as indicated in Equation (3), the full six-degrees-of-freedom momentum equations can be written in terms of the 12 remainingcoefficients') . The coefficients are written in hydrodynamic coeffi-cient notation as follows.

Added inertia terms:

A = -X., P f-K., B -YO, (7)u p

Q = -M., C =-Z., and R -N. (7)q w r

Inertial intermodal coupling terms:

F1 -Y -N., F Z. -1, and H2 -Y. -K.. (8)q p v

The remaining three are

B f-X. = -Z., Q' = -N. =-K., and H, -X. -M. . (9)w u p r1 q u

If, as shown in the complete expressions in Appendix A, there isalso fore-aft symmetry (symmetry about the yz-plane), then, in additionto the aforementioned zeroed terms

B' = Q' =F 1 =2 f= 0

Vehicles having symmetry about both the xz-plane and the xy-plane havethe additional zeroed terms

B' 1 Q H H = 01 2

( 7 )Massachusetts Institute of Technology, Instrumentation LaboratoryReport R-570-A, Deep Submergence Rescue Vehicle Simulation and ShipControl Analysis, by Charles Broxmeyer, Pierre P. Dogan, et al,1967, p. 31.

8

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

NCSL TR-327-78

1 14,These relations will be used extensively in the following coeffi-cient formulations. It should be noted that the coefficient identities,such as Equations (8) and (9), result from derivation under no assump-tions of symmetry. The existence of symmetry results in some of "'heterms being equal to zero. The complete set of identities of accelera-tion, nonlinear and cross-coupled coefficients is presented in Appendix A.

METHODS FOR COMPUTING ACCELERATION HYDRODYNAMIC COEFFICIEN.

INTRODUCTION

Methods for estimating the acceleration-dependent hydrodynamic co-efficients are given in this section. Each coefficient is described

using the following format:

1. Introductory description.

2. Relative importance - major or minor, and under whatcircumstances.

3. Causal relationship - important assumptions.

4. Analytical expression - including normal range of coefficients

in the expression and necessary references.

Appendix C provides a physical description of the underwater vehicleused as the basis for calculations illustrated graphically in this report,and Appendix D contains time history plots of the four maneuvers fromwhich the trajectory simulation and percent contribution plots were ob-tained. Reference frame conventions are as shown in Figure 1.

A word of caution is appropriate here. When, comparing computed ac-celeration coefficient values with experimentally determined values,attention must be given to the location of the reference frame origin.If the origin during the experimental tests is not the vehicle cg, thenerroneous comparisons will result for those coefficients where distancesfrom the cg are involved.

LONGITUDINAL COEFFICIENTS

1. X1. is an acceleration coefficient resulting from the resistingforce of thue fluid on a vehicle accelerating in the x-direction. This

(Text Continued on Page 11)

9

NCSL TR-327-78

r-4

WUQ 4- 4/

P4-)

LA-

100

"NCSL TR-327- 78

force is required to change the flow field about the vehicle, whereasthe static coefficients represent the forces necessary to maintain aflow field configuration at a constant vehicle velocity. More simply,X'. may be thought of as a mass of fluid that must be accelerated alongwith the vehicle, thus giving the vehicle an "added" or "apparent" masswhich is manifest during vehicle acceleration.

2. Insight to the relative significance of V. can be gained bynoting that X' appears in the linear equations of motion in the massterm

(ml - X1O

Thus, in cases where the value of X'. represents a significant contribu-tion to this term, accuracy in prediuting X'. is important. The X'.u u

contribution to vehicular motl .. during a control surface step inputis generally minor, and dissipaLes rapidly as the vehicle attains asteady state. In control surface reversal maneuvers, as illustratedin trajectory simulation and x-force percent contribution plots inFigures 2 and 3, V. becomes slightly more significant and longer-acting,but still remains a Urelatively minor influence. For nearly neutrallybuoyant faired vehicles having an L/d ratio higher than 8.0, X'. typi-cally contributes less than 3 percent to the mass term. X'. has in-creasing relative significance for lower t/d ratios. Root Yocus studiesfor typical submarine and torpedo-like bodies has corraborated the fore-going, indicating very minor sensitivity to large percentage errors inpredicting X'..u

When performing a two-degree-of-freedom linear analysis in the Ion-gitudinal plane (M and Z equations), three roots of the characteristicequation are obtained. The investigator may find a fourth root arisingfrom the x-degree of freedom equation by assuming that the only signifi-cant terms are V' and V'. (this is often acceptable for an axisymmetricu uvehicle). The fourth root is found by the relation

X1ua I 'X'. (10)

u

Tables 1A through 1D present the results of a sensitivity analysis ofnondimensional roots of the characteristic equation (transfer functiondenominator) and u, w, and e (numerators) for a generalized underwatervehicle, varying V' from -100 percent to +100 percent.

3. Mathematically, given the linearizing truncation of the Taylorseries expansion to the first term, X'V. is the force additional to theinertial force occurring with a vehicular x-acceleration. In the com-putation of K' the body is approximated by a prolate ellipsoid having

11

NCSL TR-327-78

182.7

x .-0

108.2

S33.?7

Toa

-40.8'. xt• -C-,

-115.2

-189.7 TII I1 I

0 I 7U14 21 28 35 42 49

Time In Seconds

r FIGURE 2. TRAJECTORY SIMULATION PLOT OF X'ý CONTRIIBUTION TO X-FORCE

DURING A as t 30" DIVE/CLIM INNEUVER

100

80-

60-

S40-

20

00 7 14 21 28 35 42 49

Time In Seconds

FIGURE 3. PERCENT CONTRIBUTION OF X'. TO X-FORCE DURING Au

Ls - ±30° DIVE/CLIMB MANEUVER

12

NCSL TR-327-78I

I E

1-13

lieNCSL TR-327-78

like length and maximum diameter, and neglecting appendages to the bodydue to their comparatively small froutal area. Using the results ofLamb"1 ), derived assuming potential flow, we have:

km4X. kl 1mdf4. b6b

wherea

k 1 20- a0 (12)

with

Ll J20 2 49n I--e- e (13)0e3 L1 - eJ

e being the eccentricity of the rotated ellipse expressed in commonterms as

e - •(14) A

The inertia coefficient for axial flow, kI, ranges from 0.5 to 0for fineness ratios of 1.0 to -. kI is graphically presented in Figure 4.

Z'.

1. The acceleration coefficient Z'. represents a z-force resultingfrom an acceleration in the x-direction. U

2. Root locus sensitivity studies on typical submarine and torpedo-like vehicles show extreme insensitivity to large magnitude excursionsand sign changes of Z'. . Thus the coefficient is generally assumed

Unegligible and set to zero.

3. Vehicles exhibiting symmetry about the xy-plane produce balancedz-forces during acceleration along x, indicating V. - 0. In potentialtheory, symmetry about the xy-plane or yz-plane cauges Z. - 0 uai shownin Appendix A. U

4, In the vast majority of cases, the combination of near symmetryabout the xy-plane and minor influence on roots of the characteristic

equation leads to setting Z'. 0.

(")ibid. (Text Continued on Page 16)

14

NCSL TR-327-78

-I -W'lid -h -19J "

-4b 0I:~Le -A -I -'

Lo-

6-4I~ - -

L6-t en C%4 -

- - - - - - - - 0j

NCSL TR-327- 78

MI.U

1. The term M'. represents the pitching moment resulting from anacceleration paralleY to the x-axis.

2. Root locus sensitivity stlidies on typical submarine and torpedo-like bodies show extreme insensitivity to large magnitude changes in M'.

U

3. As was the case with Z'., vehicles having symmetry or near symme-try about the xy-plane will creade a negligible M-moment due to x-acceleration. Potential theory predicts i. - 0 for yz-plane symmetry(see Appendix A). Extreme vertical separation of the cb and cg, coupledwith a large XI. force (arising from low R/d ratio), could conceivablycause a significantly large M'.. For free-flooded vehicles, launcher de-vices, and torpedo-like bodiesu this is a highly unlikely occurrence.

4. The combination of near syimnetry about the xy-plane and minorinfluence on the roots of the characteristic equation usually leads tosetting M'. - 0. For a vehicle with an extremely large vertical cg - cbuseparation and/or an V. value that is significant when compared to m',H'. is calculated as u

u

M. X' Zcbu

where z - z-distance from the cg to the cb, positive down.

X1.w

1. The acceleration derivative X'. represents the x-force resultingwfrom an acceleration in the z-direction.

2. As with Z'. and M'., the roots of the characteristic equationshow negligible change withuX'. set equal to zero, then varied'positivelyand negatively over a range much greater than would be likely in reality.

3. Near balanced z-force during z-acceleration due to near fore-aft symmetry yields resultants of a largc z-force component and somepitching moment, but very little x-force. It is significant to notethat Z'. -'. according to potential theory (Equation (9)). If the cgand cb are significantly separated along both the x- and z-axes then,since those forces arising from potential flow theory are in a referenceframe with a cb origin, the pitching moment term M'. will yield an X'. termw wwhen transferred to the cg. As with Z., though, even variation throughextreme ranges showed that setting X'V.- 0 was justified for applicationto the vast majority of underwater velicles.

4. X'. =0.

16

NCSL TR-327-78

ZI

1. The resisting hydrodynamic force resulting from a z-acceleration is represented in the equation of motion by Z'.. Theterm (m' - Z'V) is the nondimensional "apparent mass" of the vehiclewhen one is considering heavn accr leration.

2. Z' is one of the most important acceleration derivatives.Trajectory simulation and z-force percent contribution plots in Fig-ures 5 through 10 illustrate its significance to vehicular motionduring various maneuvers. The root locus plots in Figures 11, 12, and13 illustrate the wide variation in the locations of the poles due tovariation in Z'.. Tables 2A through 2D present the results of a sensi-tivity analysisWo. nondimensional roots of the characteristic equation(transfer function denominator) and u, w, and 8 (numerators) for a gen-eralized underwater vehicle, varying Z'. from -100 percent to +100 per-cent. As in X'V. and M'W., insight into Yhe influence of Z'V. can be gainedby noting the sYgnificance of Z'. in the hydrodynamic masswterm (m' -

wwZ'•) Vehicles with Z/d ratios greater than 8.5 will usually have

nearly a 50 percent V. contribution to the apparent mass. That is, thew

added mass will be roughly equal to the mass. Horizontal fins will con-tribute to Z'. and "requently cause it to be larger than m'.

3. As with 4 body is approximated by a prolate ellipsoidhaving like lengta ai maximum diameter. Lamb's"1) inertial coefficientfor such a body in crsswise flow is used as follows:

kmf k(15)

3 3

where

k - 0 (16)

with2

1 e- e 1 +_e (17)" 2 3 1 (1ee 2e

e being the eccentricity of the rotated ellipse as noted in Equation (14)."k is graphically presented in Figure 14. Fortunately, this very impor-

tant body coefficient has proved to be accurate to ±5 percent in compari-son with experimental data on the majority of underwater vehicles

( 1 )ibid.(Text Continued on Page 23)

17

NCSL TR-327-78

' I

I I . I S '- ... 44

fIGURE 5. TRAJECTORY SIMULATION PLOT OF l'l CONTRIBUTION TO Z-FORCE DURNG A

ar 30* STEADY TURN (AUTOMATIC DEPTH-KEEPING)

ii,"

e~me

FIGURE 6. PERCENT LUNTRIBUIION OF ,* TO Z-FORCE DURING A

6r * 30* STEADY TURN (AUTOMATIC DEPTH-KEEPING)

18

NCSL TR-327-78

42.0))

,-15. 5

-44.1

Time In Se~ondq

FIGURE 7. TRAJECTORY SIMULATION PLOT OF Z.ý CONTRIBUTION TO Z-FORCE

DURING A as ±30* DIVE/CLIMB MANEUVER

1001

80

60)

40

20

v Timie In Seconds

FIGURE 8. PERCENT CONTRIBUTION OF Z,. TO Z-FORCE DURING A

as ±30' DIV[/CLIMB MANEUVER

19

NCSL TR-327-78

-Te 1 1 2I 2 4Tim In5od

too-,

S I-41.

____

7 14 212 8 35 42 49Time In 3econds

FIGURE 9. TRAJECTORY SIMULATION PLOT OF Z* CONTRIBUTION TO Z-FORCE DURING A 6= ±30 TURN REVERSAL MANEUVER (AUTOMATIC DEPTH-KEEPING)

200

0 0 7 14 2 1 28 B 3 15 42 49

"Time In Seconds

FI..URE 10. PERCEIKT CONTRI BUTION OF Z'. TO Z-FORCE DURlING A 6r - 30 ° TURNREVERSAL MANEUVER (AUTOMATIC DEPTH-KEEPING)

20

NCSL TR-327-78

VZl variation using the formula

1100+Con tant)

"where Coostant ± ±100.

44

W .5

".5

FIGURE 11. ROOT LOCUS PLOT OF u/6s FOR A GENERALIZED UNDERWATER VEHICLE, VARYING Z',

Z'" variation using the formula

.100+2±Nstant) x (Z'4 ),

where Constant ± -100.

N -.5

444

~i

FIGURE 12. ROOT LOCUS PLOT OF w/6, FOR A GENERALIZED UNDERWATER VEHICLE, VARYING Z'

Z'w variation using the formula

I(lO0+Con tan t) x~

where Constant - ±100. -t

I I I~J

-I~I

FIGURE 13. ROOT LOCUS PLOT OF 8/6s FOR A GENERALIZED UNDERWATER VEHICLE, VARYING Z',

21

SNCSL TR-327-78 I•

I .N N

La eN~OL~o NNN 1

N N N.N

'N NH

- -

NCSL TR-327-78

compared. In applicatioio to a wide variety of underwater vehicles, ithas been our experience that if a body is nearly axisymmetric thenLamb's inertial coefficient is a reliable formulation for Z'. . Thisis especially true for large X/d ratios. wb

This is the first hydrodynamic coefficient with significant con-tribution from appendages to the body. The discussion and formulationsthat follow can be applied to any surface projecting from the body thatcan be approximated as a flat plate in accelerated fluid flow normal toits surface. The equations given will be for horizontal tail fins, butthey are applicable to any horizontal surfaces such as bowplanes,' sail-planes, fairwater planes, depressors, wings, faired struts, sails, etc.Tail contributions to other longitudinal hydrodynamic coefficients areformulated using the expressions given below. Theoretical values of theadditional mass of a number of types of bodies (bodies of infinite length,bodies of revolution, ellipsoids or elliptic plates of finite dimensions)have been derived by Lamb(") and Munk(2)13). Values for bodies of finitedimensions not covered by the theory; for example, rectangular plates andwings, have been the object of many experimental research programs on thephenomenon.

Extensive test programs on additional-mass and additional-momenteffects were conducted by the National Advisory Committee for Aero-nautics (NACA) from 1933 to 1940. The coefficients given here are basedon the results of these tests as reported by Gracey(4), and Malvestutoand Gale(5). A very similar formulation is given by Pastor andAbkowitz(e).

For a thin flat plate of infinite span moving in a perfect fluid atconstant velocity, V, along the normal to its surface, the momentum im-parted to the fluid per unit span is given by hydrodynamic theory as

2 *Trpc V4

For plates of finite span, this expression must be corrected by theintroduction of coefficients whose values depend on the dimensions of

( 1 )ibid.(2)ibid.

(O)ibid.

()ibid.(r)ibid.

(e)Naval Underwater Ordnance Station T.N. No. 120, Hydrodynamic Stabilityand Control Derivatives, by D. L. Pastor and M. A. Abkowitz, 1957.

23

NCSL TR-327- 7 8

the plate. The additional mass, m for translation of a plate of spanab, V is thus determined from the equation of linear momentum

2m V k pc bV

-• a 4

so that•0 k~pc b

.m 9 (18)Sa 4

where k is the coefficient of additional mass shown in Figure 14. Thisformulation is for one fin. So, with the proper nondimensionalizingterms added, the expression for two identical horizontal fin surfacesin a tail configuration is written

2 _ 21 kfpc b krc bZ'h " (19)½p3 23

Note: Pastor and Abkowitz substituted a curve fitted expression for k:

k=

(1 + 1/AR)t

An effective chord, c, is usually used since most surfaces are not simplerectangular shapes. If a sternplane or any movable horizontal tail sur-face is present and is an extension of the fixed horizontal tail surface,then it should be included in the area for calculating c and s. If it isnot a part of the fin, then its contribution to Z'. can be calculatedseparately, as can the bowplane contribution, by using Equation 19 toyield Z' and V . For completeness and clarification, Figure 15 and

s Wbpthe following relationships are given.

2AR = aspect ratio of surface = b /s

b = root-to-tip span of surface

s = planform area of surface

c = root chord of surfacer

ct = tip chord of surface

c = mean chord of surface s/b = b/AR

S= planform taper ratio of surface = ct/cr.

(Text Continued on Page 26)

24

NCSL TR-327-78

1.1 - _ _ __ __ _ _ _ _

1.0

.8q-

4- .

: 0

4- .4

.3Curve is Average of

NACA 1944 & 1940 Tests.2I

NATIONAL ADVISORYCO MITTEE FOR AERONAUTICS

0-i1 2 3 4 5 6 7 8 9I

b2Aspect ratio of plate, A S

FIGURE 14. COEFFICIENTS OF ADDITIONAL MASS FOR RECTANGULAR PLATES

25

~C~tb4

the ~ ~ i to ha

f~~.atot rohc S fda

treatuiOT ',

ttea thet t atva~of 9;e-to

it as~ at tarUtree s r the %Y p ai

VWe e s ý rh thet

ZT e q a t aVj 'b eV

'he the ti0aT ett z

vhei ton i

tio4 q at Z I s ab ?u o

be b

aatirat~r eatl5 n

NCSL TR-327-78

Tables 3A through 3D present the results of a sensitivity analysis ofnondimensional roots of the characteristic equation (transfer functiondenominator) and u, w, and e (numerators) for a generalized underwatervehicle, varying M'. from -100 percent to +100 percent.

w

3. The cg and cb of a submerged vehicle are generally in verticalalignment to i chieve level trim at zero speed. Since the body addi-tional mass effects are lumped as acting through the body cb (cautionmust be exercised when the cb is not coincident with the centroid of

* the volume displaced by the outer hull surface--as with "wet" vehicles)and the vehicle equations of motion are typically written about a. cgorigin, the resisting force to a z-acceleration is a pure V'. * term.

wbThat is, there is no x-moment arm to cause the generation of a momentM'. *. The added mass and moment effects of the horizontal surfacesgenerally are thi primary contributors to M'. . Just as any force on

wa moment arm is calculated to produce a moment, so with '. we havew

(X -X )4. M'. - v. (21)

wb Wb

(X -x )(nht - ncg),

M'. = . (22)t Wht £

where

X = x-distance from nose to cbncb

X = x-distance from nose to cgncg

X = x-distance from nose to centroidof the horizontal tail fins .

Note: The distances are all positive numbers.

Substituting the x-distance to bowplanes or shroud into Equation (22)yields MI. and M' . The cunplete expression is the sum of the com-

ponent con4ibutions; e.g.,

M'. =M'. + M'. + M'. + W'.w wb wht Wbp ws

(Text Continued on Page 29)

27

NCSL TR-327-78

IT

-is

rd

E-2-

28

/I

NCSL TR-327-78

XV.

1. Forces along the x-direction caused by angular acceleration inpitch are represented by X'.

q

2. & 3. X'. is usually an insignificant term as is its near-equal, Aq

M' , (Equation (9)).u

4. X'. = 0q

1. z-forces arising from pitch angle accelerations are termed Z'..

q

2. Z'. , as its counterpart M'. , is a minor derivative, yet itcan have saRe influence over the characteristic equation of a vehicle.It should therefore be calculated. Tables 4A through 4D present theresults of a sensitivity analysis of nondimensional roots of the char-acteristic equation (transfer function denominator) and u, w, and 0(numerators) for.a generalized underwater vehicle, varying Z'. from-100 percent to +100 percent. q

3. & 4. VZ' = M'q "

MI.

1. When a submerged body is accelerated in pitch, it must changethe flow pattern in the surrounding fluid. The added moment needed toeffect this change in flow pattern is represented in the equations ofmotion as an apparent moment of inertia, (V' - Mt.)

yq2. M'. is a major derivative in influence on the characteristic

qequation. As with Z'. and X'., M'- enters the equations of motion asan addition to the velicle inertiaqterm, in this case the moment ofinertia about the y-axis. Trajectory simulation and pitch momentpercent contribution plots in Figures 16 through 21 illustrate its sig-nificance to vehicular motion during various maneuvers. The root locusplots in Figures 22, 23, and 24 illustrate the wide variation in thelocations of the poles due to variations in M'. . iaules 5A through 5D

qpresent the resulto of a sensitivity analysis of nondimensional rootsof the characteristic equation (transfer function denominator) and u,w, and 0 (numerators) for a generalized underwater vehicle, varying M'.from -100 percent to +100 percent. q

(Text Continued on Page 36)

29

NCSL TR-327-78

m10110 0 U

0 0 , .

0 � �

0 0

- 0 0 0 0

00 , I -�� � 0 N 0 0 0 -

0 0 0 0 01I - Ci 010 010 010 tO �

N N N N N - �

I l�

01 0 0 01 IC -

N N - 0 t 0

0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 -�

01 0 0 0 - 01 01 - � -

tO 010 010 tO

� I

1 �I 4

0 0 0

0 0 0 0 N

010 0 0 0 01 N

0 0 191 0 Ia 0

�o �0 �o �0 -

I-a I'll *� '� '� *T 0 ITi 0 01 N

0.l '-'-a 0 0 0 0

=. ZI 01 0

to -if. 0 00 00 tO %.0 00

0101 N * 0 0 N

0 00 0 00

-� � 10 �0 I00 OOjjj 00 00 00 00

� 10 � 0

F � 001 �0 �

I-

30

i�- %�iC01001C4001.�.O�. -� I-

NCSL TR-327-78

169, 5

"- 02-

-94.0-

TotalS o7 14 1 18 A, 42 49

Time. In Seconds

FIGURE 16. TRAJECTORY SIMULATION PLOT OF M'- CONTRIBUTION TO PITCH MOMENT DURING Aar -30" STEADY TURN (AUTOMATIC DEPTH-KEEPING)

1001

80-

60-

• 40-

20

0+0 7 14 21 28 35 42 49"Time In Seconds

FIGURE 17. PERCENT CONTRIBUTION OF W4 TO PITCH MOMENT DURING Aor 30" STEADY TURN (AUTOMATIC DEPTH-KEEPING)

31

NCSL TR-327-78

t~.0

-40.7

0 7 14 21 is 4 4 9Ttmv In gec,,nds

FIGURE 18. TRAERCETOR SIULTONTRPLOTI OF WCNTRBTO TO PITCH MOME NT DRN

DUIN as t30* IVE/CLIA MANEUVER

232

NCSL TR-327--78

234.3 -

148.9o

63.5 Total '0

-21.9-

-107.3-

-192.7 I I I |

0 7 114 21 28 35 42 49

Time In SecondsFIGURE 20. TRAJECTORY SIMULATION PLOT OF M'- CONTRIBUTION TO PITCH MOMENT DURING A

ar -30* TURN REVERSAL MANEUVER (AUTOMATIC DEPTH-KEEPING)

[00-

80

2

~ 60

• 0.

40-

20-

II

7 14 21 28 35 42 49Time In Seconds

FIGURE 21. PERCENT CONTRI3UTION OF M;4 TO PITCH MOMENT DURING A

a +30" TUIN REVERSAL MAKCUVER (AUTOMTIC DEPTH-KEEPING)

33

NCSL TR-327-78

1 variation using the formula

(100+Constant) x

where Constant - ±100. Ju

x .-. uo 2.'0 1.

-- o

FIGURE 22. ROOT LOCUS PLOT OF u/6, FOR A GENERALIZED UNDERWATER VEHICLE, VARYING M'I

Mi4 variajion using the formula

(I00+Cons tant)

where Constant -t1-0O.

oo z

-- 0

ox w we x

mu>; -2.0

-2.0

FIGURE 23. ROOT LOCUS PLOT OF w/I6 FOR A GENERALIZED UNDERWATER VEHICLE, VARYING M'

M'. variation using the formulaq

(iOO+Constant)

where Constant - ±100. J

0i.

2.0

FIGURE 24. ROOT LOCUS PLOT OF 0/6 FOR A GENERALIZED UNDERWATER VEHICLE, VARYING M' A

34

a.

NCSL TR-327-78

Sie,

w-a NN CaN N:LNa a, '---N

- 0 til ',

' • i- -

tggi

CJ - . .. .

>• CL " C

'_,

I

--- N LN z N ~

~4 N N a,-z I5

NCSL TR-327-r78

3. & 4. The body contribution to M'. is calculated assuming theqbody is a prolate ellipsoid. Thus, we use Lamb's coefficient for

4 •added moment of inertia and the parallel axis theorem

-k' bIydf (XIncb _ Xnc) 2

.M,. q + Z'.b: o.

where *4e- ('0-aO)

0 0vb7 2 2 2

(2- e ) {2e - (2- e )( - a 0)}

a0 and 8 are defined in Equations (13) and (17). These equations arerepeated here for convenience:

a0 =2(1 - N e (½in[I +ee) (13)

0 2 3 3 - ee 2e

I = moment of inertia about the y-axis of theYdf mass of the fluid displaced by the body.

The tail contribution to M'. is in the same form as the body terms. Thatqjis, there is an added moment if inertia term and a term representing anadded mass acting on a moment arm from the cg. The added moment of in-ertia of a flat plate resulting from rotation about an axis in the planeof the plate and parallel to the span is given by

k' pb 2-3

I Pa 48

where k' is the coefficient of added moment of inertia for a flat platepplotted on Figure 25. When c/b is sufficiently small (AR sufficiently

large), this term becomes negligible compared with the second term inM'. given as

Z . (X nht - x ncg)2

Wht £

(Text Continued on Page 38)

36

NCSL TR-327-78

4- (A

S- aC.

at <

0~U-0

I*-

0

*1~U OZLA.£i~w

GJC~ CD

- -~ a~i w cm< <

C..) CD

- s-i- - - - z tL 4

OW 4-h LiU-w (5 C0

C..))

U;

.L37

NCSL TR-327-78

The horizontal tail surface contribution is then

1 (k' wpb 2 3 ) (Xnt- Xn 2p nht ncg

qpht qZ5 48 Wht £2

Similar expressions may be written for bowplanes and a shroud so thatSomplete M'. formula would be

q

M'. = M'. + MI. + M'. +M'.q q ht qbp qsh

LATERAL COEFFICIENTS

yV.V

1. Forces along the y-axis proportional to accelerations along

the same axis are characterized by the added mass term Y'. . This termis the lateral counterpart of V. in the longitudinal equations.

w

2. Just as Z'. was seen to be a major stability derivative in thelongitudinal plane,Wy'. is a very important coefficient in the lateralcharacteristic equation. Trajectory simulation and y-force percent con-tribution plots in Figures 26 through 29 illustrate its significance tovehicular motion during various maneuvers. The root locus plots in Fig-ures 30, 31, and 32 illustrate the wide variation in the locations ofthe poles due to variation in Y'. . Tables 6A through 6D present the

Vresults of a sensitivity analysis of nondimensional roots of the char-acteristic equation (transfer function denominator) and v, *, and 4(numerators) for a generalized underwater vehicle, varying Y'. from

V-100 percent to +100 percent.

3. & 4. For axisymmetric bodies

Y. = Z'.vb Wb

Most underwater vehicles have nearly axisymmetric hull forms. Of thosewhose shapes are not a body of revolution, quite often good approxima-tions can be made by treating the body as everywhere circular in cross-section with a cross-sectional area equal to the actual body. Thisapproximation is often used in the analysis of underwater vehicles.

Appendages that contribute to Y'. are those with significant pro-V

jections onto the xz-plane. Usually these consist of vertical tail fins

(Text Continued on Page 43)

38

NCSL TR-327-78

13.6

-16.8

S-47.2, To ta I•

77.

-107.9

-138.2I 07 1'4 211 218 35 42 49 •

Time In Seconds

FIG•JRE 26. TRAJECTORY SIMULATION PLOT OF Y'i CONTRIBUTION TO Y-FORCEDURING A sr- 30 STEADY TURN

100-

80"

60"

40-

20-

0-0 7 14 21 28 35 42 49"Time In Seconds

FIGURE 27. PERCENT CONTRIBUTION OF Y'i TO Y-FORCE DURING A ar 30" STEADY TURN

39

I'

I. INCSL TR-327-78

127.51

73.7

C 20.0 -

v

'-33. 8

-87.5

-141.30 7 14 21 28 35 42 49

Time In Seconds

FIGURE 28. TRAJECTORY SIMULATION PLOT OF Y'ý CONTRIBUTION TO Y-FORCE

DURING A 6•. "30" TURN REVERSAL MNAEUVER

100-

80-

60

S40-

20

74 21 28 35 42 49

"Time In Seconds

FIGURE 29. PERCENT CONTRIBUTION OF Y'4 TO Y-FORCE DURING A

6r ±30* TURN REVERSAL MANEUVER

40

NCSL TR-32'-789i.

Y" variation using the formula L O{ .1 2O +u ,n s t a n t ) , ( . , / "

where Constant , . oo. 8 /

"8 -40

FIGURE 30. ROOT LOCUS PLOT Ov/ FUR A GENERALIZED UNDERWATER VEHICLE, VARYING Y'9

o+ jwY'v variation using the formula • •

( 100+con tant) K

whore Constant P flO0,o.

SF T SI 1I

"1 .-. o -4,0 -. -. -T -.D - -p -I - -

0

FIGURE 31. ROOT LOCUS PLOT OF v/r FOR A GENERALIZED UNDERWATER VEHICLE, VARYING Y,

SY'ý variation using the formula .

(I Qoicon, tan (y.

8where Constant t 100. 4

008

-eo L

-5,0~ -O&2V3.0

-3-30

FIGURE 32. ROOT LOCUS PLOT OF €/6r FbR A GENERALIZED UNDERWATER VEHICLE, VARYING Y'

41

- - - -� - -- - - �-

NCSL TR-327-78

I 41

No . - 410 4Q N .0

040 NO NO - .00 NO

� N - 0, -

. U 2

-

-1-n

= >4. at I�Il 4 4

NO N NO *O�IN NO 400

N N N 0 N N N N

*� N � ;t;LJ �. 2.� :z.NI

4 4 N�N � - ON NO NO �

S*0 - 00

+

40.

0

WO NN 0 N N

ON NO .00 N N 0 01-4

0 � N N N N N N N

00 N .. 4� N N *

�j N N N N N N 0

N N N N NN 0 0 N 04 0*4 NO NO 400 N N NO ON 0 0

2 . . . . . - �,N N 0 0 N NMN 4 4 I I

04

[TiN 4� N N N I.-.

Wa NN 0 N N N �b4 �NO 00

ON N

N

-0

4-4I.-4 �N 0 N O� NN N N N N N� L1i� - �4* ON 00 NO

0, 0 0*1 a� 0400.0 N N N

30 ON N', .0.0 N * NO �UN N N 4

�N � N ON 0 0 N N '4

4. '.14 II II 41 N N N

Na � N N .0400 N N N - N N 0 N

N NO 0 40 N '41 -- N N N N

NO 010 N N '00 4t4� N N N

�N � .. 0. '0 40

I*4I� N44 II II 44 I II

44.�'N 0 N .0 00 F �

ONO 0 40 =04 - 4.� .0 N N N �*O 00 -. 4 -

NO 40 N 0 4. 4

I.4� 44 II44��4�444 -

4 4 I

BES1.-AVAI�B�E COP'(42 I-

NCSL TR-327- 78

(or stabilizers), rudders, canards, and shrouds. The contributions ofthese and any other surfaces that can be treated as flat plates inaccelerated fluid flow normal to its surface can be calculated using

Equation (18). It is repeated here for convenience.

2m knpcb (18)a 4

Since it is common to have unequal upper and lower vertical finsin the tail, the two will not be combined as were the two horizontaltail fins. The expressions for the upper and L:wer vertical tail finsare written as

kirc bY'.v 3uvt(tvt) 2Z

If the rudder is an extension of tht vertical tail fins, then itshould be included in the area for calculation of c. If the rudder isseparate, then it can be treated separately using the same expression.

Since the bowplanes have no significant vertical surface area,they do not contribute to Y'. . The shroud has the same projection invthe vertical plane as in the horizontal plane, 30

v s W sh

The complete expression of Y'. for a typical underwater vehicle iswritten

v

y'. =y'. + YV. + y'. + V'.v vb vvb Vuvt VXvt sh

yV.

1. y-forces on a submerged vehicle that are due to angular accel-eration in roll are represented in the equations of motion as Y'.

p

2. This term can be significant on vehicles with large Z cb/,Z At, and Z /t ratios, whereuct Zct

Zcb - z-distance from body cg to cb

Z - z-distance from the x-axis to the centroiduct of the upper vertical tail, and

ZEct z-distance from the x-axis to the centroidof the lower vertical tail.

43

NCSL TR-327-78

Tables 7A through 7D present the results of a sensitivity analysis ofnondimensional roots of the characteristic equation (transfer functiondenominator) and v, p, and 0 (numerators) for a generalized underwatervehicle, varying Y'. from -100 percent to +100 percent.

3. Due to intermodal coupling, Y'. - K'. as shown in Equation (8).Since the conceptual description of thepphysical phenomenon causing K'.moments is more easily understood than that for Y'. , only the equalityrelationship is given here; that is

4. Y'. - K'. (See below).p V

Y'.r

1. When an angular acceleration in yaw results in a force in they-direction, the y-force is represented in the equations of motion bythe term Y'. . The longitudinal counterpart of Y' is Z'.

r r q2. Tables 8A through 8D present the results of a sensitivity

analysis of nondimensional roots of the characteristic equation (trans-fer function denominator) and v, p, and 0 (numerators) for a generalizedunderwater vehicle, varying Y'. from -100 percent to +100 percent.

r

3. Intermodal coupling yields the relationships in Equation (8.).That is,

4. Y'V N'.r v

K' .v

1. Rolling moment due to acceleration along the y-axis is termedk'. v

2. KV. couples the roll and yaw motions of a vehicle. As a

vehicle enters a turn, the changes in the drift angle, 0, will appear asa change in the sway velocity, v; i.e., an acceleration ii. The rollingmoment produced by v in this situation can be significant on a vehiclewith unsymmetrical vertical fin arrangements, like a submarine's sail.Trajectory simulation and roll moment percent contribution plots inFigures 33 and 34 illustrate its significance to vehicular motion duringa turn reversal maneuver. The root locus plot in Figure 35 illustratesthe variation in the locations of the poles due to variation in K'.Tables 9A through 9D present the results of a sensitivity analysis ofnondimensional roots of the characteristic equation (transfer functiondenominator) and v, ', and ý (numerators) for a generalized underwatervehicle, varying K'. from -100 percent to +100 percent.

(Text Continued on Page 50)

44

NCSL "TR-327-78

0 ;10 1 I, ON 'O I N S,1S SO ,

,-,-., ,-,-,NN N N S O S NO

•il -

LU$

' 2~~~~. -EIAALBECP

' 45

S........ . ..... .... ..

NCSL TR-327-78

K..,.

mII"I

IA � �. .. � 4r Lii �

II - �

I I I I

mI I* � �w

III,�j �h �: �*

IA I I IS.� �

2

CC ha

-c U ISIS ISIS ISIS IS IS

IASS. IS'�

1515 IA + * *

BESTIAVAItABLE COPY 46

NCSL TR-327-78

109.2

71.

.,• -K'( =01

--.--- To.

K .

-41. 3-

-78.4-tII

) 7 14 21 28 15 4 "' ±1

I"|me Inl S-me',d•.

FIGURE 33. TRAJECTORY SIMULATION PLOT o0 K' CONTRIBUTION TO ROLL MOMENT

DURING A A = ±30* TUR, RKVERSAL MANEUVERr

6O0 -

"4f)_

14 2 1. 1 4 2 4

Trim I n se--, ((

FIGURE 34. PERCENT CONTRIBUTION OF K'. TO ROLL MOMWNT DURING

'A 6

r -30* TURN REVERSAL MANEUVER

47

NCSL TR-327-78

0.o )ca +1 -L

W c

4J 41C uO C) c Lu

0 41 s41 L) 0n~3:9'(U + C i

-LJ

3 o3

00 LT.3SVG-:-5 U-

-J1-C I

V)

OO-J C..1

48

NCSL TR-327-78

.-

-':,i•' •- -

to " ' N •' a

;N NV. NV -IN ON .'tO N.

l~ !..........N N .aN - ._

NN

•.+ . ..•, N-N

494

NCSL TR-327-78

3. & 4. This coefficient is evaluated in a similar fashion tothat of M'I. It is the representation of added mass effects actingwIon moment arms from the cg.

K'. = -Y ' Z bvb Vb i

As noted before, it is not uncommon for the uppet and lower verti-cal tail fins to be of different size and shape. Also' since the cg isgenerally below the body axis of symmetry, there are unequal momentarms to the centroids of the upper and lower fins even if the fins aresymmetric about the center line. The vertical tail surfaces' contri-bution to K'. is written

Kt. -- {Y' Z + '. Z }v t Vt uvt uct V vt ct d

K'. + K'. + V.

v vb

Vuvt Vkvt

Z distance from the x-axis to the centroid i

AUct of the upper vertical tail

n z distance from the x-axis to the centroidact of the lower vertical tail.

Note: The vertical tail fins are assumed to lte in the xz-plane.

The complete expression, then, is written br i

V. V.K + V.' b vt

Any other vertical surfaces that can be approximated as flat plateslying in the xz-plane can be treated similarly to the upper and lowervertical tail surfaces.

KV.

pi1 . Moments additional to the 1x moment of inertia resisting angu-laracelraio i rllare included in the term V . . This added

Smoment of inertia is one that has no Lamb coefficien?(") for the bodyJS~contribution. This is because a homogeneous body of revolution can

rotate about its axis of symmetry and not experience any forces from an

inviscid fluid.

(5)ibid.

so

NCSL TR-327-78

2. Tables 10A through 10D present the results of a sensitivityanalysis of nondimensional roots of the characteristic equation (trans-fer function denominator) and v, *, and ý (numerators) for a generalizedunderwater vehicle, varying K'. from -100 percent to +100 percent.p

3. Since P, angular acceleration in roll, is defined about thex-axis passing through the cg, there will be some added moment effectof the body due to an added mass effect acting at the cb. It is writtenas

Z2

K'. . cb

Pb Vb Z22

Any surface projecting radially from the vehicle (such as tail fins,bowplanes, etc.) will ccitribute to the added moment effect representedby K'. . Each surface contributes in two ways. One is as an added mass

pacting on a moment arm from the x-axis. The general form for this isthe same as the body contribution above. Considering any "flat plate"surface, such as the upper vertical tail fin, this term would be written

Z 2K'. ffi yuc____t

Puvt uvt £2

The other form of contribution to K'. is by the individual added momentof inertia of the flat plate rotating about an axis parallel to thex-axis but passing through the centroid of the flat plate. Again, weturn to Malvestuto and Gale(r) for experimental data on flat plates withvarying dihedral angle and taper ratio. For rotation about an axis inthe plane of the plate, parallel to the chord, and passing through thecentroid, the equation is

I =-- D D k'c b3x 48 • Drp

a

where k' is the coefficient of added moment of inertia for a flat plateplotted in Figure 25, DA is the correction factor for taper ratio plottedin Figure 36, and D is the correction factor for dihedral angle plottedrin Figure 37. Thus, for any "flat plate" surface such as the upper verti-cal tail fin

2K'. i y, uctPýuvt ffi 5 X Ia v

")~ibid. (Text Continued on Page 54)

51

V

NCSL TR-327-78

~TII

_l-I

;z z

-AVAILABLE COP

5

NCSL TR-327-78

1.2

1.1

1.0

.9

"U . -8

€T 0

.3

.1 - - - NATIONAL ADVISORYCOMMITTEE FOR AERONAUTICS

1 2 ICr

A,, Taper Ratio, C

FIGURE 36. DEPENDENCE OF THE ADDITIONAL MOMENT OFINERTIA ON TAPER RATIO

53

.3 - --

NCSL TR-327-78

1.2

1.0

-3I 01xl½

- A

.6

L -

V NATIONAL ADVISORYCOMMITTEE FOR AERONAUTICS

0 1 2 3 5 6 7 8 9

r, Dihedral Angle, Deg

FIGURE 37. VARIATION OF THE ADDITIONAL MOMENTS OF INERTIAOF A SINGLE PLATE WITH DIHEDRAL ANGLE

The first term is often negligible compared to the second. The completeexpression for a common vehicle would be

KW. 'K'. + K'. + K'.P Pb Puvt P~vt

K'.r

1. Rolling moments caused by angular accelerations in yaw are rep-resented by K'.

2. Tables 11A through liD present the results of a sensitivity analy-sis of nondimensional roots of the characteristic equation (transfer func-tion denominator) and v, $, and * (numerators) for a generalized underwatervehicle, varying K' from -100 percent to +100 percent.

3. K'. arises in the same manner as the K'.i term except that th.

linear acceleration normal to the vertical surfaces is the result of anangular acceleration in yaw acting about the z-axis.

(Text Continued on Page 56)

54

NCSL TR-327-78

7 V Ir-

-N

-

C N

NWN ' - N

BEST AVAIIBLRE CmY

55

" "IbisN N N N

NCSL TR-327- 78

W.K' - K' (Xnuvt -Xng) (n+vt - )r vt + K'

where

X - distance from nose to upper verticaltail surface centroid, positive aft

Xn distance from nose to lower verticaltail surface centroid, positive aft.

N'.

1. Linear accelerations in the y-direction causing yawing momentsgive rise to the stability derivative N'. to describe these resultingmoments. v

2. Tables 12a through 12D present the results of a sensitivityanalysis of nondimensional roots of the characteristic equation (trans-fer function denominator) and v, *, and 4 (numerators) for a generalizedunderwater vehicle, varying N'. from -100 percent to +100 percent.V

3. Primarily due to the vertical tail surfaces, a vehicle changingdrift angle will experience a restoring force proportional to 4 (result-ing from ý). N'. is the term representing this force.

v(Xnc Xn)

4. N' =-Y'. ncg

vb 'b

(Xnuvt -X ncg) (X Xn)N' -Yvt ncg nM.vt neg

t uvt Vvt 4

N'.

1. Yawing moments caused by angular accelerations in roll are rep-resented in the vehicular equation of motion by N'.

p2. Tables 13A through 13D present the results of a sensitivity

analysis of nondimeasional roots of the characteristic equaticn (trans-fer function denominator) and v, *, and 4 (numerators) for a generalizedundetwater vehicle, varying N'. from -100 percent to +100 percent.

3. & 4. Equation (9) shows

N'. =K'.p r

(Text Continued on Page 59 )

56

NCSL TR-327-78

S9- %I I I,,

Nm ON

!9 ' 1 -1 01.0

Ow

NCSL TR--327-.78

m~omr

-,I

o< 3

~ w ~ -

BEST AAILABL COPY

NCSL TR-327-78

N'.r

1. Yawing moments additional to I are denoted N'. • This is theadded moment of inertia term whose longftudinal counterpart is 111.q

2. Trajectory simulation and yaw moment percent contribution plotsin Figures 38 and 39 illustrate its significance to vehicular motionduring various maneuvers. The root locus plots in Figures 40, 41 and42 illustrate the wide variation in the locations of the poles due tovariation in N'.. Tables 14A through 14D present the results of a sen-sitivity analyses of nondimensional roots of the characteristic equation

(transfer function denominator) and v, *, and 0 (numerators) for a gen-eralized underwater vehicle varying N'. from -100 percent to +100percent.

3. & 4. Like M', there are two parts to the N'. bodycontribution q r

- ½(z + Y2- zdf (ncb _ ncg)

b vb

for an axisymmetric vehicle body I = I Again, similar to M'.Zdf Ydf t

the upper vertical tail fin contribution is

2-3 2-- 1 (K•rpb3) (X Xncg)N*. + Yt.

r" = - ý- 48 v Y' XuT-£

uvt uvt

The lower vertical tail fin is ±dentica. in form. The complete expres-sion, then, is written N'. = N'. + 1'. + N'.r rb r r

b uvt Xvt

SUMMARY

A compilation of methods for analytically predicting the accelera-tion hydrodynamic coefficients from the geometric and mass distributioncharacteristics of underwater vehicles has been presented. A theoreti-cal development from potential flow theory is given to show the equiva-lences among acceleration and coupled terms. The detailed discussion ofeach coefficient presented includes descriptions intended to provide rhereader with an understanding of the physical relationships involved.

(Text Continued on page 63)

59

NCSL TR-327-78

81 2-

47. 5- •

-53.5

0• '7 '14 h21 28' 4' 2 49•Time In Seconds

FIGURE 38. TRAJECTORY SIMULATION PLOT OF N'ý CONTRIBUT!ON TO YAW MOMENT

DURING A ar "30* TURN REVERSAL MANEUVER

100.

80

60

S40

20N't

00 7 14 21 28 35 42 49

Time In Seconds

FIGURE 39, PERCENT CONTRIBUTION OF N'ý TO YAW MOMENT

DURING A 6r ±30" TURN REVERSAL MANEUVER

60

"NCSL TR-327-78

jWN virlation using ýhe w'.,aa 3.0

IOC+Const ant) /

where Const4nt -:100.

(5 0

o14 01. -7. -6. 0% 0V AD .0

FIGURE 40. ROOT LOCUS PLOT OF v/•,r FOR A GENERALIZED UNDFRWATER VEHICLE, VARYING N'.

N'r variation using the formula 1

0

S~(IO0+Constant)" wheie Constant =±-100.

I . IoI

S-z. c

l -10.0 -%0 -o7 -.0 -4 .-5 .0 -M .1V -.0 -ljj 1.0

ISI

S•L -U

I

FIGURE 41. ROOT LOCUS PLOT OF v/fr FOR A GENERALIZE! UNDERWATER VEHI VARYING N'.

S~r

jWF

N,', variation using the formula 3300

(IO0+Constant) xN')

00 r /.0

where Constant o ±100.

-2

1-.0

x

FIGURE 42. ROOT LOCUS PLOT OF T/r FO; A GENERALIZED UNDERWATER VEHICLE. ARYING N'

61

NCSL TR-327-78

-~ nC

IMIII.

BS- AVIAL CO

NCSL TR-327-'78

Results of _ensitivity analyses are presented to show the relative con-tributions of each coefficient to the equations of motion. A discussionof the important assumptions involved in mathematically computing eachcoefficient is given, along with the analytical expression used to com-pute each coefficient. An example case (Appendix B) is given to show acomparison between computed coefficients and experimental data.

Table 15 presents a summary of the sensitivity analyses conductedfor each acceleration coefficient. These results are for t100 percentcoefficient variation. It can be seen in Table 15 that five coefficientsmake major contributions to the vehicle dynamics. These are Z'., M'.,w

Y'., K'., and I . . Variations of ±i00 percent change both the numneatorr

anX denominator roots by better than 50 percent, except for K'. . ChangesV

in K'. significantly effect only the roll numerator, N6 %. It should bev r

noted that the ý terms, such as K'. and V'., do not affect the v-numerator

because no v-term appears in the v-numerator. This is easily seen whenone applies Cramer's rule to form the transfer functionsWs). This sameobservation can be made about each of the other numerators; i.e. now-term in the w-numerator.

( 9 )Naval Coastal Systems Laboratory Technical keport 287-76, Developmentof the Equations of Motion and Transfer Functions for UnderwaterVehicles, by D. E. Humphreys, July 1976.

63

S - Xa " - • . .. .... . . '. . .•.... . d , ~ ha l ,l • • . . .. . . .• .. . .. . . • , . ,. .. ..

NCSL TR-327-78

TABLE 15

ROOT SENSITIVITY ANALYSES FOR DERIVATIVE VARIATION OF -100%(Sheet I of 2

LONGITUDINAL DERIVATIVE ROOT VARIATION PERCENTAGES

f CHAR. E NUAT U NUMERATOR W NUMERATOR 0 NUMERATORDerivative Variation Derivative Variation Derivativey Varlaton erivative Variation

-lOOt +100% -lOOt +100 -o00t +100 -loot +loot

V 0 0 0 0 0 0 3.302 -3.10%3.30% -3.10% 0 0 3.30% -3.10% 0 0O 0 0 0 0 00 0

I n. o8; 0072 079

-0.122 0,12% -0,21% 0.23% 0 0 0 00 0 88.30% -32.30% 0 0 97.00% -33.00%

91.20% -32.50% 4.00% -0.98% 0 02.40% -0.582

Maj.,

M -0.01% 0.01% -0.02% 0.02% 0 0 0 0

0 0 0.43% -0.42% 0 0 -0.79% O.79%0.27% -0.27% -o.44% 0.43t 0 0

-0.29% 0.29%

neg. I

LONGITUDINAL DERIVATIVE ROOT VARIATION PERCENTAGEST CHAR. EQUATIOR U NUMERATOR V NUMERATOR 0 NUMERATOR

Derivative Variation Deri e Varation Derivative Variation Derivative VariatWo-1002 +100% -100t +lO0t -100 +00ot -lotO% +100%

Z'4 -0.007% 0.007% -0.008% 0.01% 0.02% -0.03% 0 0O 0 0.21% -0.212 0 0 0 0

0.102 -0.18% -0.232 0.23% -4.002 4.30%-0.21' 0.21%man.1

M4' -052% 0.53% -o.68% 3.71% -0.302 0.30% 0 00 0 1.20% -1.50% 0 0 0 0

0,68% _-0.85% 88.40% -31.50% 97.10% -33.10%

89.10% -31.80 J

LATERAL DERIVATIVE ROOT VARIATION PERCENTAGES

SCHAR. EQUATION V NUMERATOR V NUMERATOR N NUMERATOR

Derivative Variation Derivative Variation Derivative Variation Derivative Variatlon-1002 +100% -1002 +1002 -lOOt +.002 -1002 +100%

Y', 1 -3.90% 1.10% 0 0 -1.50% 0.07% 24.00% -11.30%t -3.90% 1.10% 0 0 -1.50% 0.07% 640.40% -40.402

2.102 -0,51% 0 0 96.90% -33.00% 0 091.ot -3?,50 % o 0

0 0

K'. tt 4.40% -4.30% 0 0 ** 1.70% -1.702 44.30% -36.40ott 4.40o -4.30% 0 0 * 1.70% -1.702 -62,002 789.00%

-O.052 0.05% 0 0 0.06% -0.06% 0 06% -0.16 0 0

0 0

N'K +t÷-t.02% 0,02% 0 0 *** -0.02% 0.022 -0.95% 0.96%ttt -0.02% 0.02% 0 0 *** -0.02% 0.02% 0.84% -0.84%

-0.27% 0.27% 0 0 -0.79% 0.802 0 00.25% -0.25% 0 0

0 0ne~g.I

IMAGINARY ROOT VARIATIONS: f -O.ld to 0.10; * -0.20 to 0.09; tI 0.15 to -0.15;** 0.19 to -0.19; t+t -0.003 to 0.003; *** 0

64

NCSL TR-327-78

TABLE 15

(Sheet 2 of 2)

LATERAL DERIVATIVE ROOT VARIATION PERCENTAGES

[ CHAR. EQUATION V NUMERATOR Y NUMERATOR * NUMERATORSDerivative Verlitlon Derivative Variation Derivative Variation Derivative Vriatlor-1 100 + 200t -100% +100o -100% +100o -100o +100o

.00-1. 1.00% * 3.40% -3.302 # -0.91% 0.90% 0 0-1.00% 1,002 * 3.40% -3.30% # -0.91% 0.90% 0 00.03% -0.03% 0.71% -0.70% -0.07% 0.07% 0 0

-0.06% 0.06% 0 0

neg. 0 0

K'ý 11.10. -1.00% tt 1.002 -1.00 tt 1.002 -1.00% 0 0++ 1.102 -1.002 It 1.00% -1.00% It 1.00% -1.002 0 0

0.002% -0.002% -0.0062 0.007% 0 0 0 00 -0.001% 0 00 0

ne g.

N' ittt 0.05% -0.05% * 0.04% -. 04o N### -0.03% 0.03% 0 0ttt 0.05% -0.05% 0.04% -0.04% ### -0.032 0.03% 0 0

0.012 -0.01% 0.01% -0.012 0 0 0 0

re. 0 -0.001% 0 0-0 0d

IMAGINARY ROOT VARIATIONS: t 0.24 to -0.24; * 0.13 to -0.13; # 0.26 to -0.26; tt 0.52 to"-0.51; ttt-0.003 to 0.007; *** -0.007 to 0.003; ### -O.00W to 0.003

LATERAL DERIVATIVE ROOT VARIATION PERCENTAGES

fI_ CHAR, EQUATION V NUMERATOR T NUMERATOR 4 NUMERATOR

Derivative Variation Derivative Variation Derivative Variation Derivative Variatlo;-1002 +I10 -1002 +100% -100% +÷100 -1002 +100%

Y'I t -0.052 0.052 * 0.06% -0.072 0 0 -0.27% 0.26%t -0.05% 0.05* 0 o.o62 -0.072 0 0 3.70% -3.50%

-0,19i 0.19% -3.90% 4.20% 0 0 0 000.15 -0,152 0 00 5 0

rin. IIK'* It -0.062 0.06% *f 0.14% -0,15% 0 0 0.78% -0.75%

t- -o.06. 0,062 ** 0.14% -0.15% 0 0 6.60% -5.70%

-0.01% 0.01% 0.03% -0.03% 0 0 0 0m0.001 0 0 0

in.

N ý t- 1.10% 0.95% *** 1.30% -0.65% 0 0 15.50% -10.30%ttt-1.102 0.95% *** 1.30% -0.652 0 0 50.20% -21.70%

88.20% -31.60% 96.20% -32.80% 0 0 0 00.53% -0.67% 0 0o0 0

IMAGINARY ROOT VARIATIONS: t 0.003 to 0; * -0.003 to 0.003; ++ 0.007 to -0.007;'1" 0.003 to -0.003; tII 0.05 to -0.007; ft** 0.05 to -0.07

65

NCSL TR-327-78

APPENDIX A

DERIVATION OF HYDRODYNAMIC COEFFICIENTS RELATIONSHIPSFROM ENERGY EQUATIONS

Beginning with the momentum equations given in Equations (1) and(2) in the body of this report, a derivation is given here for the com-plete expressions, showing the equalities and relationships Detween allof the acceleration and nonlinear hydrodynamic coefficients possiblefor a vehicle moving in an infinite, inviscid, fixed fluid with zerocirculation.

A summary of the identities and relationships derived is presentedon Page A-6 with simplifications arising from vehicle symmetry on PageA-8.

Note: The hydrodynamic coefficients in parenthesis with a subscriptare added together to get the value of the parenthesized hydro-dynamic coefficient, e.g.,

X (x + +(-X)rq rq) rq 2

X =(M)+ (-N)rq

X =M-N.rq

X-FORCE, SURGE

d6T 6T 6d f f -Tf-X =d u r--+ qwdt 6u 6V 6

d •Tf

•Tf-r�v + = -Brv - A'rw - C'ru - Mrq - Frr - F'rr - Hrp + H'rp

6 Tf

q -1-= Cqw + A'qv + B'qu + Nqr + Fqq - F'qq + Gqp + G'qp

letting F = F + F, F2 =F- F', GI = G + G•, G2 = G - G', H1 = H + H',

H H- H'

A-1

NCSL TR-327-78

X - -A6 - B - C'Cv - G- 2- H1 + Brv + A'rw + C'ru +

Mrq + F rr H 2 - Cqw - Aqv B'qu - Nqr - F2qq - 1lqP1 -X L -X. F1 X F2 -G-X

u rw qv p rr 2 qq

B X B' -X. =-X M = (Xq) G = -x G -- X.rv w qu.r 1 qp 2 r

C=-X C,=-X.X N -- (Xq) H -X H 2qw v ru qr 2 12 rp

Other resulting relations: Xrq M N

Y-FORCE, SWAYd 6Tf 6Tf 1Tf

at 6v 6P• u

d 6Tf

d 6v- Bv + A'* + C'I + M4 + Yk + FVi +'+ H'IPdt Sv

6Tf-p • = -Cpw - A'pv - B'pu - Npr - Fpq + F'pq- Gpp - G'pp

6 Tfr = Aru + B'rw + C'rv + Lrp - Grr - G'rr + Hrq + H'rq

Y =-B - A'* - C'IA - M - FI - H2 + Cpw + A'pv + B'pw + Npr +

F2 pq + Glpp - Aru - B'rw - C'rv - Lrp - G2 rr Hlrg

A=-Y A' =-Y. Y M =-Y. F -Y F 2 =Yru w pv q r pq

B -Y, B' =-Y = Y N ) G -Y G =-Yrw pu pr pp 2 rr

C=Y C' - -Y. = -Y L = (- " -YrHq YC = pw u rv rp 2 rq 2 p

Other resulting relations: Y N - L.rp

A-2

NCSL TR-327-78

Z-FORCE, HEAVE

d f f + 6f-z 6U - ---

d6T~ ..- !C* + A'iT + B'&ý + Ni + F'4 + Gb + G'j

6STf -Aqu -B'qw - C'qv -Lqp - Gqr + G'qr -Hqq -,H'qq

p6i-- Bpv + A'pw + C'pu. + Mpq + Fpr + F'pr + Hpp - H'pp

Z - C - A'4ý - B'Zi - Ni~ - F2 - Gh+ Aqu + B'qw + C'qv

+ Lqp + G qr + Hlqq - Bpv - A'pw - C'pu - Mpq -Flpr -H~p

A= Z A'= -Z. -Z L (Z) F -- Z F= -Z.qu v pw qp 1 pr 2 q

B =-Z B' =Z =-Z. Hm (-Z ) G =-Z G = zpv qw u pq 2 12 qr

C -Z. C, Z =-Z N =-Z. H, -Z H -Z

wqv Pu r 1 qq 2 pp

pq

K-MOM~ENT, ROLL

d T 6TTfd ff 6Tff

dt 6p 6v 6q 6r

d f -P Q'i R'4 - Lt- -j

dt 6p2

6TfwE Bwv + A'ww + C'wu +Mwq + Flwr + H2wp

6T f

6Tfr f Qrq + P'rr + R'rp + Mrv + F rw + H ru

6q 2 1

-q - Rqr -P'qq -Q'qp -Nqw -Flqv -G~q

A- 3

NCSL TR-327-78

A' K -- K L--K. F1 - (K) - (-Kqvvv vv1 1qv

B (K'v) B' -- K H (K) (K G K -K.wv 1 vu wq 1 vp w

C (-Kvw )2 C' K u N (-Kvr)2 (-Kqw)2 HI -Kru

"F2 (-K v) - (K) P -- K. P' -K -- K2 q 2 2 p rr qq

G =-K -. -2 qu Q (Krq) Q r qp

H =-K.• =K R -(-Kr) R' -K.- Kwp 2qr q rp

Other resulting relations: K - B - C

K -MH Nvr

K -M-N"qw

Krw - F1 +F 2 - 2F ,

K -- F -F -- 2Fvq 1 2

qr -Q-R

M-MOMENT, PITCH

d 6Tf 6T f 6T f STf 6T f

M j dt- q -'- w u 6r •'p

'STSd Tfdt Q 4 - - P'i - R'f - M-t - F2 , -HIdt 6Sq1

6Tu Cuw + A'uv + B'uu + Nur + F2 uq + G Up

6TfS-Awu B'ww - C'wv - Lwp - Gwr -,q

A-4

NCSL TR-327-78

6T fp f - Rpr + P'pq t Q'pp + Npw + Flpv + G2 Pu

56Tf

-r-p - Prp - Q'rr - R'rq - Lru - Glrw -H rv

A (-Hwu) A' -Muv L (-MWP ) = (-M ru) F- flpv

B' -M -M -M-. (M- )ww uu v UP 1

-(-M)rwi

c (M) C' -- M N- ) - (Mu) H1 -- M.uw wv pw2 u

2 2.

-Mwq

F2 •H-M.= M P - (-MrP) P' - -M. - Mpq M

2 wr)2 Pu)2 qQ'- - p

H -M- R- (M) R' --.--M2 rv pr 2 p rq

Other resulting relations: M - C - Awv

M -N-L

M -N-Lru

Mpu G 1 + G2 "2G

M -G- -G -- 2Grw 1 2

M •R-P•rp

N-MOMENT, YAW

d 6Tf 6T fTf 6T dTfdtdr + u -v + q

A-5

NCSL TR-327-78

16Td fAdt 6r -Ri - P'- Q'- N- F - G2

6TfvTf - Avu + B'vw + C'vv + Lvp + G2vr + Hlvq

6T

f- -Buv- A'uw - C'uu - Muq - Flur - H2 up

6Tf

q =p Pqp + Q'qr + P'qq + Lqu + Glqw + H2 qv

6T f-p =• - Qpq - P'pr - R'pp - Mpv - F2 Pw - Hlpu

A =(N ) A' N L =(N) -(N) F =-N.1u uwVP qu 1 1 v

" -N ur

B -(-N) B' -N M -(-N) -(-N) G -Nu 2 vw uq 2 pv qw

C' N -- N N -- N. HI = (Nvq

vv uuwvq

S (-Np )pu 1

F2 -- N P (N P' -N.

2 pw qp q pr

G - -N. = N Q -(-Nq) Q' -- N. -N2 ut yr Q pq p qr

H -(-N (N qv) R=-N. R' -Nqq -- Npp

2 P u 2 q 2 rqq p

Other Resulting relations: N - A - B

N -L- 1

N -L-Mqu

A-6

NCSL TR-327- 78

N - H + H 214vq 1 221

N -H- H 2

N iP Qpq

A uX ~ru zqu v(H) (u

B -Y.-X -- Z -(K) (-N)v rv pv 1v UV 2

Cn -K. -- K -M -N

w w w vw Wv vvu

A' --Z.--Y.X -X Y -Z -K --K M -N

w u qu rw Pu qv vu wu uu vw

C, - -X. --Y. X -- Y Z -- Z -K --M -N -- Nv u ru rv qv pu wu wv vv uu

L--X.--K.-(Z ) (-Y (-M ) (-N )i -(N ) (N)p u qp 1 rp 1 WP ru 1 P1 qu 1

M -Y. --H. -(X) (-Z (K) -(K) -(-N) -(-N)q v rq 1 pq 2 wq 1 rv 1 uq p

N--Z.--N. (-X ) (Y ) (-K) -(-K) -(M) (M)r w qr 2 rp 2 yr 2 qw 2 pw 2 ur 2

F Y. -- N. X -- Z -(K) -(K) M -N--1 r v rr pr yr 1 qv 1 pv ur'

G -Z. -- K.--X Y -K -(M) -(-M) -N1 V qp pp VP UP 1 rw 1 qw

H -X. -- M.--Y Z -K -- -(N ) (-N)1 q u rq qq ru wq vq 1 pu 1

F --Z. --M.-X Y -(K) -(K) M --N2 q V qq pq vq 2 rw 2 uq pw

G--X. -N.--Y ~-Z -- K -(-M ) (H ) N2 r u rr qr qu yr 2 p yr

H--Y -K.-X -- Z -K --H -(-N) (N)2 v rp pp WP rv UP2 qv)2

A-7

NCSL TR-327-78

P -KS " (-Hr) - (N )rp1 qp 1

Q--M.- (Kr) - (-N pq)i ~2

R--N. -(-K) (Mr)r qr 2 pr 2

P' -1. -N K -K M -- Nr q rr qq pq pr

""-K -N-K - M Nr p qp rr pp qr

R' -- K. -M. -K -M -N -- Nq p rp rq qq pp

When xz-plane symmetry exists, then the following coefficients are zero;A', C' P' , R' , L, M, N, G, G'

Table Al gives a summary of force and moment coefficients for a bodywith xz-plane of symmetry. When xy-plane symmetry exists, then thefollowing coefficients are zero; A' B' P') Q'p L, M, N, H, H'.

When the yz-plane symmetry exists, then the following coefficients arezero; B". C', Q', R', L, M, N, F, F'.

When two planes of symmetry exist, all of the terms that are zero inthe single plane cases are zero.

A-8

NCSL TR-327-78

TABLE Al(Sheet 1 of 2)

POTENTIAL FLOW FORCE AND MOMENT HYDRODYNAMICCOEFFICIENTS FOR BODY WITH XZ PLANE SYMMETRY

X Y Z K M N

x. 0 0 x. 0u w q÷ ¢* . €Y. Y.v p r

x. z. 0 z.w w q

0 Y. 0 K. 0 K.p p r

x. 0 z 0 M. 0q q q

0q Y. 0 K. 0 N.r r r

2 -- * -X. 0

uv X X. 0Y. - X.w v4 u

uw .- 0 0. 0u wup -K. 0 - X. + Y.w q p

uq x - -X.0 Y -z

yr Y. Y

" u - - . 0 - -.. 0 Z .u q r

2

V .p.q

vw - -Z. Y. -x.w V w

vp 0Y. . -Y. 0v r+ z.0 + -(x .+ Y .

II

A-9

, Jiv r• .. _, 0. ....

p.

S • • • , • • ." -' - --- - --, - -,-. . - - . ,-. ., . . - . • ,- • .,• • •, . .. . . ..- °-•

NCSL TR-327-78

TABLE Al

(Sheet 2 of 2)

SX Y Z K M N

2 ••2 0 Y. -- K.

p r 0I, pq 0 -Z. 0 K. 0 M. -K.q r q' p

pr -Y. 0 Y. 0 K. - N. 0p r p r

2q Z. - -x. 0 - 0q q

qr 0 X. N N. M. -K. iq r q r

2r -Y. 0 0 K.r r

*Ccefficient zero because of xz symmetry

**Coefficient not present for arbitrary body with no planesof symmetry

A-10

- .I.- - ,

rJ

NCSL TR-327-78

APPENDIX B

COM0PARISON OF ANALYTICAL PREDICTIONS WITH EXPERIMENTAL DATA

In this appendix hydrodynamic coefficients of an operational U.S.Navy submersible, obtained at the David Taylor Naval Ship Research andDevelopment Center are compared to corresponding data calculated by themethods set forth in the body of this report. Table Bl lists those 15acceleration coefficients generally considered to be important, theirexperimental and calculated values, and the percent difference betweenthe two. Percent difference in this table was obtained as follows:

Experimental - CalculatedExperimental

As can be seen in Table Bl, percent difference is significant inonly four of the 15 cases, that is, those above 12 percent.

Three of those six coefficients (M., Z., and K.) have been shown inw q p

sensitivity analyses illustratpd in the body nf this report to be of rela-tively minor significance.

Some questions exist concerning the validity of using the PMM ex-perimental technique to obtain roll derivatives, and specifically, inthis case, the fourth problem coefficient listed in table, K..

V

It should be noted that not all of the coefficients obtained byDTNSRDC were measured. Some by necessity were calculated. Those markedwith an asterisk in Table Bl were calculated by some DTNSRDC method andnot by NCSL methods.

i

B-1

.........

NCSL TR-327-78

TABLE BI

COMPARISON OF CALCULATED VERSUS EXPERIMENTALLY-DETERMINEDCOEFFICIENTS OF AN OPERATIONAL SUBMERSIBLE

Coef. Experiment Calculation % Error

1- 16800E-03 - .60300E-04 64.0

Z -. O1llOE-O1 -. 10100E-01 0.1

M - .499C !:, '-03 -. 54880E-03 10.0

Z4 -. 171OOE-03 - .6300E-04 65.0

K.* -. 33000E-05 -. 21500E-06 93.0p

% -. 10880E-01 -. 10001E-01 8.0

uX. -. 20900E-03 -. 18500E-03 11.5

SK -. 51000E-o4 -. 28500E-04 44.o

N. -. 48400E-03 -. 53570E-03 10.7

Xwq -. 10110E-01 -. 10100E-01 0.1

Xvr o10880E-01 .10001E-01 8.0

y* .10110E-01 .1010OE-01 0.1wp

N -. 49600E-03 -. 54900E-03 10.7pq

Zvp -,10880E-01 -. 10001E-01 8.1

Mrp .48070E-03 .53550E-03 11.4

B-2

NCSL TR-327-78

APPENDIX C

DESCRIPTION OF TYPICAL HYDRODYNAMIC VEHICLEUSED FOR SENSITIVITY ANALYSIS

This appendix contains a physical description of the underwatervehicle used as the subject of zhis report. Table Cl is a listing ofpertinent geometric facts describing the vehicle, and Tables C2 throughC6 list calculated derivativfs, moments and symmetry terms. Figure 2-1is a set of orthographic projections illustrating the vebi,,le.

C-1

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

NCSL TR-327-78

TABLE C1

SUBJECT UNDERWATER VEHICLE GEOMETRY

BARE HULL

Vehicle Total Length, Ft ........... ......... . 23534E 02Hull Maximum Cross-Sectional Area, Ft 2 . .. . 78824E 01Vehicle Total Weight, LBF . . . . . .... ........... . 79482E C4Vehicle IXX Moment of Inertia, Slug-Ft2 . . . . . .. .. 82148E o4Vehicle IYY Moment of Inertia, Slug-Ft 2 . . . . . . .. 21616E 06Vehicle IZZ Moment of Inertia, Slug-Ft 2 . .. . 21616E 06Vehicle Displacement Weight, LBF ....... ..... .. 79482E 04Displaced Fluid IYY Moment of Inertia, Slug-Ft 2 . 2 16 1 6 E 0 6

X-Distance from Nose to C.G., Ft. ...... .. 10391E 02X-Distance from Nose to C.B., Ft ...... ...... .. .. 1 0391E 02Z-Distance from C.G. to C.B., Ft ..... ...... .... 10000E 00Bare Hull Wetted Area, Ft 2 . . . . . .. . . . . . . .. . . . . 18421E 03Bare Hull Lateral Area, Ft 2 . . . . .. .. . . . . . .. . . . . 58070E 02Bare Hull Volume, Ft 3 .. .. .. .. .. . . . . . . . . .. . . . . . 12411E 03

TAIL STRUCTURE

Overall

X-Distance from Nose to A.C. of Tail, Ft.. ...... 21298E 02Z-Location of A.C. of Tail WRT C.G ..... ........ ... OOOOE 00Sweep Angle, Deg ...... ................. . .OOOOOE 00Taper Ratio ........ ....... .................... 1OOOE 01Effective Aspect Ratio . ........ ........ 24020E 01Exposed Span, Root to Tip, of Fins (each of 4, Ft .13200E 01

Horizontal Tail

Total Exposed Planform Area (2 fins), Ft 2 . . . . . .. 29010E 01Dihedral Angle, Deg ............ ................. OOOQE 00Incidence Angle, Deg ......... ................. OOOOOE 00

Vertical Tail

Total Exposed Planform Area (each of 2 fins), Ft 2 , .14510E 01

C-2

NCSL TR-327-78

TABLE C2

S N A M E NON-DIMENSIONAL LONGITUDINALSTABILITY DERIVATIVES

XU a -. 21408E-02 ZU - .0 MU - .0

XW - .0 ZW - -. 32464E-01 MW - .25379E-02

XQ - -. 15275E-03 ZQ - -. 15275E-01 MQ - -. 73062E-02

XTH -' .0 ZTH - .0 MTH - -. 29769E-03

XUD - -. 62385E-03 ZUD - .0 MUD - .0

XWD - .0 ZWD - -. 18182E-01 MWD - -. 14340E-03

XQD - .0 ZQD - -. 14340E-03 MQD - -. 83506E-03

XX - .0 ZX - .0 MX - .0

XZ - .0 ZZ - .0 MZ - .0

XDELT - -. 57867E-03 ZDELT - -. 57867E-02 MDELT - -. 28238E-02

M * .19043E-01 IY * .93512E-03

C-3

NCSL TR-327-78

TABLE C3

NONLINEAR X-Z SYMMETRY TERMS

XUU - -. 10704E-02 ZUU - .0 MUU - -.0

oXW - -. 11127E 00 ZWW - .0 MWW - .0

XWQ. - -. 18182E-01

XQQ - -. 14340E-03

XVV - -. 11127E 00 ZVV - .0 MVV - .0

ZVP - -. 18182E-01 MVP - -. 14350E-03

XVR = .18182E-01 ZVR - .0 MVR - .77258E-0!l

XRP - 77258E-O4 ZRP .14350E-03 MRP .83473E-03

XRR - -. 14350E-03 ZRR - .0 MRR - -. 60977E-06

I

TABLE C4

S N A M E NON-DIMENSIONAL LATERALSTABILITY DERIVATIVES

YV - -. 33046F-01 KV - -. 15537E-03 NV - -. 22682E-02 J

YP - -. 17294E-03 KP - -. 13896E-04 NP - .80153E-04

YR = .15280E-01 KR .53435E-04 NR = -. 73083E-02

YPHI - .0 KPHI - -. 29769E-03 NPHI - .0

YPSI - .0 KPSI - .0 NPSI - .0

YVD - -. 18182E-01 KVD - -. 77258E-04 NVD - 14350E-03

YPD = .77258E-04 KPD - -. 37211E-06 NPD - .60977E-06

YRD = .14350E-03 KRD = -. 60977E-06 NRD - -. 83511E-03

YY - 0 KY .0, NY -. 0

YDELT =.13285E-01 KDELT -. 56451E-04 NDELT --. 64829E-02

C "4

NCSL TR-327-78 I

TABLE C5

MASS MOMENTS OF INERTIA

M - .19043E-01 IXZ n .0

IX - .35537E-04 IZ , .93512E-03

TABLE C6

NONLINEAR X-Z SYMMETRY TERMS

YUU - .0 KUU - .0 NUU - .0

YWP - .18182E-01 KWP - -. 77258E-04 NWP - -. 14340E-03

KQR - -. 45882E-07

KVW - .21335E-06

YPQ - .14340E-03 NPQ - -. 83469E-03

I

I,

I1

I

C- 5

[ NCSL TR-327-78

I -L

0-4

wiLo

I-

20-

C-6,

NCSL TR-327- 78APPENDIX D

DESCRIPTION OF MANEUVERS USED IN TRAJECTORY SIMULATION

Figures Dl through D4 are time history plots of the four basicmaneuvers used with the subject underwater vehicle to generate tra-Jectory simulation and percent contribution plots of the various hydro-dynamic coefficients discussed in the body of this report. Thesemaneuvers were performed at a simulated vehicle speed of 14.35 feet persecond.

The following definitions are provided to assist the reader in theinterpretation of Figures D1 through D4.

DELS - Stern plane deflection, in degrees.

PITCH - Pitch angle, in degrees.

Z - Depth, in feet.A

ALPHA - Angle of attack, in degrees.

DELR - Rudder deflection, in degrees.

BETA - Sideslip angle, in degrees.

R - Yaw rate, in degrees per second.

PHI - Roll angle, in degrees.

Y - Off-track distance, in feet.

X - Along-track distance, in feet.

D-1

2NCSL TR-327-78

.4 .4

4l U

I I

4'

KF

SI. a.

In-4

z

-4

2 8 wdill Jill ASS 305- lOll- 0 �

lOll- OS, Q19 SIt lOt- Ill- finS'

HOLId VHd�W W

-. 5

I u�- -4

0U � 0

N+1

* 2

L)hi

a -

wF -2 U..

* p6

5'

___________________________ I __________________________________________

01 SI S 9- 61- 06- *�S SO6l 06111 10611 lAOS M�

6,30 z

D-2

I

NCS TR-327-7

IL

U Ul

w0 cc

D-3-

N'NSL TR-327- Mi

rA

I-JIW

amaEZI

00,0 9611- 1,11 Wat Zel- 4.,90

A0

D-41

NCSL TR-327-78

oo

0

ce

r I-;

.10

". G

- V )

+ I

LjJ

V.1.39 hi0

D-5

.r1 .O .O .ll .L L .I .O .~ .3 .i .I ... ..-

NCSL TR-327-75 ý

C~I

t0*0 *OS 2I*I- CCU0- gait- go*

A

D-6I

NCSL TR-327-78

o C)

00 I

vi 0

D-7-

(nadtov.&l Son Syate'stmsna2SS Cj, MV- !ýoJ4, Mr. lco)(u~s <2

(sZA84) (oy~(SPA 05, xr. Pet'e 0Cp 4

Peirce) (Cop 7MSA 91 )(4avjy Y)

(FM8 l~t Copy 6)

(Ebvtkry) C Cy.2)1

Cblef of vixvai. C-ast" copy 12)I

(Copy 1.3)

Cn aa ir, 0 1 taylo~r Sanl Mhir )Sssarch Ot-relopncnt

C 'ktr #athgsdctte, ytt~

(t~brary~(Copy- IS)(coi 144,Kr Pdi~- (opy 10)

(Cala53$'s(Cvpy 21)

(Celle4 5*30Ž! (copy 22)ýext h1Ary)ICopy 24ý)

Dncc~tor, MW va tfSaeJ-ch tAba ;4tory -

(lAg4vmy) .(Cop 25)Cajjihx(an-ery5k (Copy Ž6)

No n~ aval. SorfactA wcnpanae Ceuorr, L'ah1.jiwl(Copy 2))

iiorabs~np Nyol I~co~a, Araipol (Lbta-y)(Copy 28)

<Copy 79)Sopa~ntndut. aval, -o~tgraduate lic~,ol oot, *P rey

)X~bray I(Copy 30)(ClNIR 211~ PT Hr Sn) (Copy 31)

(MLf 438 -z Mr op Co r) (Copy 32)

(ORR 211 M )r. ýWt~tflaA) (ACo-py 33)I

Hoboken, WK) 076k;Sen. 0nttleo hzn)~,Ta~~n)aopoy

Applied %Uaoar'ch LRtl:to-y eaaS-e UIvnlveroity(I-. Nleearcr. Ta:4t) (n' eryc)TXl

(Hr. V.R.NWall) (Ccpy 371

O'-Srary . Copy 38)

Applied .'hynlsi~tottoy 94bi 7MktI, tuaaW

Mlibvary) . %y q9

RCo# 41.1VIrgIate M)olyttcs-- tnr-Lrtetlusd-a, VA 2 40-5

ctanv arat Twr"itarte of "-c-i'p~~ Ga 021,39l

(Mv. Akostr) \Cpvy 84)LQrvof C4tmron.m Waahxngton, 1Y' j

*(Scienvrce * T.ech'nolosy Wlbravy) (Copy 45)Direr-i-or, Vobdr flo~t ctafilograph),c !xatl rute .5±O ýDirector, ScrtppeQ Tn&atlii-ua of' 0crnog'vy0p 47)Camanding Officer, Nival wear's Rewoarct) P :vý pAcitivfty, bay St. l4ustst, 4S 39520 (Copy 4A)3ocsýe-Y of KVAl Arctebltwfv. & Arine Enpionra (SIXAME

74! Tr.;mssy M1., New lock, XY 10in3 '(Copy 44")Director of Nrlttaseý i(Žoinch ft Sgieorl V. waah-11nton xfvnq SO)Dtirec~tox aIllv Rpqsorcd V,-jrtt Agqmy- ahtýý (ny3

The nolrtca~ Sn-aes or-p. 1;14i Jacob) WarY. Reading.KM4A 31807

Sysemn C-ntroi Lx,, itute, 13401 hacillS Rd., Pain U~to.

t~rO ~tvetjN4 v. LAI-ri-Lme. W 660-4A 0.py -4)

Niolsgen tinginaiv ;xi YmRerc.i... oWmktkuiln Vija- CA ciCC"t3, )tdier. Up'a

7TNe n1%R,Ž,. Stork Drapo- a- Utbox-mrory, Vtit-iIdgý', 1*1 0:1114V 4 -'. K41V 4