Simulation of maneuvering characteristics of a destroyer study ship using a modified non-linear model

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Approvi for public release; distribution uiiriited

PROPLJLS!O D AL'XILAY 5YTE. DEPARTMENT

Annapolis

RESEARCH ANL) DE\'ELOPMENT TEPOFT

ii

August 197k

Report

27-745

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41 ) _{.!F fl '}

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;:uL %.jL (J!I

-Bthcd

Md. 20034

--SIMULATION OF NEUiTRIN C1iAPJCT:RISTIOS OF A DESTROYER STUDY SHIP USING

A NODlFI.E) IcONL:CNEAR MODEL

by

Samuel F1 Brown and Reidr Aivesta

v.

Technische Hogeschool

3 JULI ?75

bL

D_eIf

(

REPORT ORiGNATOR OFFICE 7-IN-CHARGE CARDE ROCK Os SYSTEMS DE V EL C Pu EN I DEPARTMENT 11 SHIP PERFORMANCE DEPARTMENT SHiP ACOUSTICS DEPARTMENT MAlE RIALS DEPARTMENT

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TECH4ICAL DIRECT9 28

MAJOR NSRDC ORGANI2ATtOAL COMPONENTS

AVIAT0N AND SURFACE EFFECTS DEPARTMENT 16 COMPUTATION AND MATHEMATIC, DEPARTMENT 13 PROPULSION AND AUXILIARY SYSTEM DEPARTMENT 21 CENTRAL I H ST RUM EN TA TI ON DEPARTMENT 29

The Naval Ship Reaearch and Devrlopment Center ia a U. S. Navy center (or laboratory

frt directed at chevin- ioprovd p.r. and air te.icle'j. It vi. torn-ed ri Msrch 1957 by rpin th David Taylor i.ic,det fair. at Card'rock, Maryland with tie Marine Engineering

Lahcrtcry at Ain*poli; Maryl and.

Ship Reae-ch and Developtrient Center

Betheai, Md. 20034 IN-ARCE ANNAPOLIS 04 STRUCTURES DEPARTMENT 17

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SECURITY CLASSIFICATION OF THIS PAGE (WRen Oat. Entered)

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I I JI I 1JJUI1LI II ILIII I READ INSTRUCTIONS BEFORE COMPLETING FORM I. REPORT NUMBER

27-745

2. GOVT ACCESSION NO. 3. RECIPIENTS CATALOG NUMBER

4. TITLE (and SubtItle)

SIMULATION OF MANEUVERING CHARACTERISTICS OF A DESTROYER STUDY SHIP USING A MODIFIEI' NONLINEAR MODEL

5. TYPE OF REPORT & PERIOD COVERED

R&D, July thru Nov 1973

6. PERFORMING ORG. REPORT NUMBER 1. AUTHOR(e)

Samuel H. Brown Reidar Alvestad

8. CON1RACTORGRANTNUMBER(a)

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Naval Ship Research and Development Cente Annapolis, Maryland 21402

-10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS

Element _635N, Task Area

s4659,

Work Unit 2731-120

It. CONTROLLING OFFICE NAME AND ADDRESS

NAVSHIPS (SHIPS

03414)

12. REPORT DATE

August ₁₉₇₄

13. NUMBER OF PAGES

69

14. MONITORING AGENCY NAME & ADDRESS(iI different from Controlling Office) 15. SECURITY CLASS. (of fbi. report)

UNCLASSIFIED

ISa. DECLASSFICATION/DOWNGRAD,NG

SCH EDULE

16, DISTRIBUTION STATEMENT (offRi.Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of 11,. ab.trcl enl.red in Block 20, if different froot Report)

IS. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on ,.ver.e .id. if nec....ry d identify by block n.tmb.r)

destroyer maneuvering, simulation, mathematical model, ship propulsion, thrust, torque, gas turbine, fuel flow

20. ABSTRACT (Continue on r.ver.. aid. if necessary end Identify by block ntmb.r)

This report describes an analog computer maneuvering simulation of a destroyer study ship. The mathematical model which is used includes the ship propulsion machinery dynamics and the ship equations of motion. The model couples the ship propulsion dynamics equations with nonlinear maneuvering equations. The power plant representation consists of a simplified mathematical

UNCLASSIFIED

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.LC..UR1TY CLASSFICATIO? OF THIS PAGEO.r Data FnC.r.d)

20. Abstract (Cont)

model of a General Electric LM2500 gas turbine engine _{and is} primarily an engine mapping of engine torque versus engine speed using fuel flow rate as a parameter. _{The simulation is} used to accurately predict slow transients in ship speed during maneuvers resulting from slow increases in the fuel flow rate to the gas turbine. _{All dynamics including the engine mapping} are simulated on EAI

₆₈₀

_{and TR-48 analog computers. Simulation} results are obtained for trajectories in standard maneuvers and also for performance of the propulsion plant. _{The advantage of} the modified model presented in this report over those not

including propulsion dynamics is that it permits simulations of the effects of maneuvering on the propulsion plant.

ft

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AB STR7 CT

This report describes an analog computer maneuvering simulation of a destroyer study ship. The mathematical model which is used includes the ship propulsion machinery dynamics and the ship equations of motion. The model couples the ship propulsion dynamics equations with non-linear maneuvering equations. The power plant representation consists of a simplified mathe-matical model of a General Electric LM200 gas turbine engine and is primarily an engine

mapping of engine torque versus engine speed using fuel flow rate as a parameter. The

simu-lation is used to accurately predict slow tran-sients in ship speed during maneuvers resulting from slow increases in the fuel flow rate to the gas turbine.

All dynamics including the engine mapping are simulated on EAI

₆₈₀

and

_TR-48

analog

computers.

Simulation results are obtained for trajec-tories in standard maneuvers and also for per-formance of the propulsion plant.

The advantage of the modified model presented in this report over those not including propulsion dynamics is that it permits simulations of the effects of maneuvering on the propulsion plant.

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ADMINISTRATIVE INFORMATION

This report satisfies milestone 1 of fiscal year 19711., Work Unit Summary Prototype Surface Ship Bridge Control System Development, of 1 July 1974. This work was funded by NAVSHIPS

under Element 63585N, Task Area S4659, Work Unit 2731-120. The work presented here was started in July 1973, and was successfully completed in November 1973.

ACKNOWLEDGMENTS

The authors would like to express their appreciation to J. Hadler, G. Hagen, Dr. M. Martin, and J. Strom-Tejsen

(formerly head of Powering System Branch) of the Center for many helpful technical discussions while this work was in progress.

Also, the authors would like to express their appreciation to C. J. Rubis, head of Propulsion Dynamics, Inc., for intro-ducing them to the subject of ship propulsion dynamics, and to Dr. H. Eda, Stevens Institute of Technology for contributing the destroyer study ship hydrodynamic coefficients used in this study.

Further acknowledgment should be made to W. J. Blurnberg (head of Control and Simulation Branch) of the Center for orig-inally proposing this effort and giving continual advice and support while this work was in progress.

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TABLE OF CONTENTS Page ABSTRACT iii ADMINISTRATIVE INFORMATION iv ACKNOWLEDGMENTS iv

DEFINITIONS AND TERMINOLOGY vii

INTRODUCTION 1

GENERAL DISCUSSION OF THE SIMULATION 1

GENERAL CHARACTERISTICS OF DESTROYER STUDY SHIP 2

MODIFIED NONLINEAR MANEUVERING MODEL FOR A DESTROYER

STUDY SHIP

4

Modified Nonlinear Maneuvering Equations 14

Destroyer SLudy Ship Propulsion Equations 11

Engine Torque Map

Advantages of Modified Nonlinear Maneuvering

Model 14

Destroyer Hydrodynamic Coefficients 15

ANALOG COMPUTER MANEUVERING SIMULATION RESULTS 15

Spiral Maneuver 15

Zig-Zag Maneuver ₁₇

Turning Maneuver 17

Acceleration Maneuver to Steady State 19

DISCUSSION AND CONCLUSIONS 19

TECHNICAL REFERENCES

₂₃

APPENDIXES

Appendix A - Additional Destroyer Study Ship Characteristics

(6

pages)

Appendix B - The Analog Computer Simulation

(29

pages) INITIAL DISTRIBUTION

c

DEFINITIONS AND TERMINOLOGY Propeller torque coefficient

Propeller thrust coefficient Propeller diameter

Propeller expanded area ratio

= Propeller pitch (distance the propeller advances while making 1 revolution without slip)

.4

.

Ho/D = Propeller pitch ratio

I _{= Polar moment of inertia of drive train (referred}

to

propeller speed)

Ship moment of inertia about the z-axis Ship length between perpendiculars (LBP) Propeller angular speed (r/min)

Ship yaw moment

Engine turbine speed (r/min) Propeller torque

= Engine torque developed on propeller shaft = Engine turbine torque

T _{= Propeller open water thrust}

U _{= Velocity of the origin of the body axes relative} to

the fluid

T(l-t) = Propeller net thrust W = Ship weight = L =

N

= N = NE = Q = Co = = D = EAR =

Qf = Reduction gear and bearing friction torque

t

R

= Total ship resistance in calm water when ship is

[

moving on a steady course (

Wf = Engine fuel flow rate (lb/h)

X,Y,Z _{= Hydrodynamic force components in ship body axis} (longitudinal, lateral, and normal components, respectively)

g = Acceleration due to gravity kg = Reduction gear ratio

In = Ship mass

n _{= Propeller angular speed (defined as positive for}

steady-state forward ship motion, r/sec)

r = Ship yaw angle velocity (1J=r)

t _{= Thrust deduction fraction or time}

u,v,w _{= Velocity components of the origin of the body axes} relative to the fluid (Longitudinal, _transverse, and normal components, respectively)

u,v,w up

-I

w

x,y,z

xo,

yo, zo

27-7)45

= Acceleration components of the origin of the body axes relative to the fluid (longitudinal, trans-verse, and normal components, respectively)

= Propeller speed of advance (defined as positive for steady-state forward ship motion)

= Wake fraction

= Coordinate axes fixed in ship. Origin of axes system need not be at the center of gravity of the ship (positive direction forward, starboard, and downward, respectively)

= Coordinate system fixed with respect to the surface of the earth

XOG,YOG,ZOG = Coordinates of the center of mass of the ship relative to the coordinate system fixed with respect to the surface of the earth

a _{= Propeller second modified advance coefficient}

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xG,yG,zG = Coordinates of center of mass relative to the ship body axis

= Angular displacement of the ship rudder = Ship yaw angle

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4FL ( 1 ( -a INTRODUCTION

One of the major limitations of current ship control systems is the dependence upon manual operation which results in slow reaction times, errors at human interfaces, and large manning requirements. Commercial fleets during the past several years have recognized and used the advantages of semiautomatic and automatic systems for performing such bridge functions as

collision avoidance, navigation, and steering. The technology that has been developed for commercial ships is available for adaptation to naval ship use.

The objective of this work is to develop a maneuvering analog computer simulation of a destroyer study ship which will be used for the design and evaluation of ship bridge console displays. The simulation developed uses a modified mathematical model in which the ship propulsion dynamics are combined with nonlinear maneuvering equations. Simulated destroyer study ship maneuvering trajectories and propulsion plant responses are also presented for different standard maneuvers.

The destroyer study ship was designed theoretically from ship data in the public domain and does not simulate the power-ing and maneuverpower-ing characteristics of a specific U.S. Navy

ship.

GENER1JJ DISCUSSION OF THE SIMULIATION

An analog computer simulation for destroyer maneuvering has been developed. The nonlinear equations for the ship powering,' which includes the effect of varying the advance coefficient on

the propulsion dynamics, were coupled with the destroyer non-linear maneuvering equations for motion in calm water in the horizontal plane. These equations were programmed on the EAI

680

analog computer.

The basic destroyer gas turbine power plant1'2 simulated

includes an LM2500 gas turbine engine, reduction gear, propeller shaft, and fixed pitch propeller. The gas turbine engine was simulated on the analog computer by an engine mapping of torque versus engine shaft revolutions per minute for various fuel flow rates. Maneuvering is achieved through rudder commands and

changes in fuel flow rate.

Superscripts refer to similarly numbered entries in the Technical References at the end of the text.

The advantage of this nonlinear maneuvering model over those which do not include propulsion dynamics is that the gas turbine propulsion plant characteristics can be simulated and displayed on a console for bridge design or powering studies.

Several limitations and assumptions exist in the present mathematical model:

The simulation does not include the effects of cross flow into the propeller during a turning maneuver.

The maneuvering simulation presented here cannot predict accurately transients in ship speed and power plant dynamics resulting from rapidly increasing the fuel flow rate to the gas turbine. - However, for slow increases in fuel flow rate and associated slow transients in ship motion, the steady-state thrust deduction factor, wake _{fraction, propeller thrust,} and torque curves are assumed to give a good representation of actual conditions.

The propeller side force and associated moment

induced by propeller-hull interactions resulting from transients in advance coefficient which tend to produce sway and yaw are neglected.

The effects of roll, trim, and sinkage _during turning maneuvers are neglected.

Future work is being planned in another program to study the above limitations and to remove them insofar as possible from the simulation in order to more accurately simulate the effects of maneuvering on the gas turbine engine.

The first section of this report describes the general characteristics of the destroyer study ship, the second, the development of the modified destroyer nonlinear maneuvering equations, and the third, the maneuvering and powering record-ings generated on the analog _computer.

GENERAL CHARACTERISTICS OF DESTROYER STUDY SHIP

The destroyer characteristics used in _{this work are taken} from the basic destroyer study ship designed by Rubis.''2 The powering and speed characteristics _{of the study ship provide a} Consistent set of data over the speed range of interest. The ship characteristics are presented in table 1. _{Additional ship} characteristics are presented in appendix _A.

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TABLE 1

CHARACTERISTICS OF THE DESTROYER STUDY SHIP1'2 Ship

Mass, slugs

2.8 x 1O5

Length, overall, ft* 14O

Maximum beam, ft 146

Depth, amidships, ft ₃₀

Full-load displacement, long tons 4,000

Propulsion plant Single gas turbine,

single screw with fixed pitch propeller

Base turbine rating, hp _25,000

Reduction gear ratio

Polar moment of inertia of drive train, lb-ft-s2

Number of blades

2 77)45

Propeller

3

*Abbreviations used in this text are from the GPO Style Manual,

1973,

unless otherwise noted.

-Diameter, ft _15.0

Propeller pitch ratio

_1.05

Propeller expanded area ratio

_0.8

9.8 x

3 ( 1'

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4

MODIFIED NONLINEAR MANEUVERING MODEL FOR A DESTROYER STUDY SHIP

MODIFIED NONLINEAR MANEUVERING EQUATIONS

As shown in figure 1, a set of coordinate axes (x,y) with the origin fixed at the center of gravity of the study ship is

used to describe the motions of the ship in the _horizontal plane

'OG

Note: All velocity components, angles, and dis-placements are positive as shown

Figure 1

Orientation of the Space and Ship Coordinate Systems

*Equation (24) does not include propeller thrust, which will be treated later.

27-7245 5

The equations of motion in this ship coordinate system3

(

can be expressed as:

t

_m(ürv)

_x

m(i'+ru)

= Y

= N.

Note that XG, YG and ZG have been assumed to be zero. Also note that vertical motions and rolling are neglected as they are

considered to have little influence on steering and maneuvering. X, Y, and N are the components of the total hydrodynamic forces and moment due to the overall ship motions in the hori-. zontal plane, as well as those due to the rudder, propeller, and waves. In the calm water case considered in this study, the forces and moments are considered to be functions of hull motions and rudder only.* Thus, X, Y, and N are given by:

x

-Y = f(u,v,r,ii,,,5), - (24)

N

where the effects of the parameters 8 and 8 are assumed negli-gible and therefore are not included in

the

above equation.

The functions describing the forces and moments3 can be

'fl developed by use of a Taylor series expansion of a function of

several variables. By considering a sufficient number of terms, the forces and moment can be expressed to any degree of accuracy.

The Taylor expansion for the X force including terms to the third order is:

x

x

± [xu + X\Tv + Xrr

+ X

+

Xjr + Xo5]

+

[Xu +

Xv2

+

+

+ 2.XurAur

+ +

2-x5

_o]

1

+

3.x,Lu2v + ).Xu2r +

+ 6Xuvr1UTr

+ 6.xvuuv +

with similar expressions for Y and N (see tables 2 thru 4 for

hydrodynamic coefficient values). TABLE 2

SUMMARY OF COEFFICIENTS IN X-EQUATION

Taylor Expansion Variable And Dynamic

Response Terms -u (m-X) 1/2 LBP NondimenSional Factor Non- dimen-sioa 1 Coeffi-cient .x105

-

-0.08m Definitions 2 yr vrLu V Lu r t Lu 1/2 X,,, (1/2 X+mxG 1/2 x 1/2 1/2 Xrru

1/2 x1

(Xvr+m) xv X_r X,_ru XV U r U x 1/2 LBP2U 1/2 LBP2 1/2 LBP2/u 1/2 LBP2 1/2 LBP4 1/2 LBP2u2 1/2 LBP2/u 1/2 LBP4/u 1/2 LBP2u 1/2 p LB?3 1/2 LBP2u 1/2 p LBP3u 1/2 p LBP3/u 1/2 p LBP2 1/2 p LBP3 1/2 p LBP2u2 -400 -95 876

Nonlinear term, 2nd order partial derivative of longitudinal force with respect to v

Nonlinear term,2nd order partial derivative of longitudinal force with respect to

Nonlinear term, 2nd order partial derivative of longitudinal force with respect to v andr(-)

(1)The estimated value of the nonclimensional coefficient (Xvr+m) used in this study appears to be too large. This has the effect of increasing the speed loss in

turns. However, for the purpose of this report, this is not deemed important.

r2

v2LU

11

TABLE ₃

SUMNARY OF COEFFICIENTS IN 7EQUATION

Variable Taylor cpansiofl And Dynamic Response Terms Nondimensional Factor Non- dimen-sional Coeffi-dent x105 Definitions r (m-Y)

(G-)

1/2 p LBP3 1/2 LBP4 531 0

Mass minus partial derivative of lateral force with respect

to v v3 yr2 v82 viu

vu2

1/6 1/2 _rr 1/2

_v8

1/2 1/2 p LBP2u 1/2 LBP2/u 1/2 p LBP4/u 1/2 p LBP2u 1/2 p LBP2 1/2 p LBP2/u -1049 6712

746

Derivative of lateral force with respect to V

Nonlinear term, 3rd order partial derivative with respect to v Nonlinear term, 3rd order partial derivative with respect to v and r

rv2 rtu

ru2

(Yr-mu)

1/6 'rrr

1/2 Y 1/2 '1r8 ru 1/2 _ruu 1/2 p LBP3u 1/2 LBP5/U 1/2 p LBP3/u 1/2 p LBP3U 1/2 p LBP3 1/2 p LBP3/u -52 6126

Derivative of lateral force with respect to r minus mu

Nonlinear term, 3rd order partial derivative with respect to r and v

6 o3 6u2

1/6 y6

1/2 1/2 _ôrr 1/2 _6uu 1/2 p L8P2u2 1/2 p LBP'u2 1/2 p LBP2 1/2 LBP4 1/2 p L3P2u 1/2 p LBP2 209 -247

Derivative of lateral force with respect to 8

Nonlinear term, 3rd order partial derivative with respect to 6

vró _'1vrâ 1/2 p LBP3 -Y*u 1/2 p LBP2u2 1/2 p LBP2u 1/2 p LBP2 7 - 27714.5

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-TABLE 4

SUMMARY OF COEFFICIENTS IN N-EQUATION

r

variable

Taylor Expansion And Dynamic Response Nondirnensional Factor Non- dimen-sional

Coeffi-dent

Definition

Terms x105

(G-N)

1/2 p LBP4

(I5-N)

1/2 p LEP5 31.8

Ship moment of inertia about z_axis

minus derivative of yawing moment

wth respect to t

V N 1/2 p LBPu _{-330 Derivative of yawing moment with}

respect to v

v3 2 yr 2 1/6 1/2 Nvrr -1/2 p LBP3/u 1/2 p LBP5/u 1023 -23.3

Nonlinear term, 3rd order partial

derivative with respect to V

Nonlinear term, 3rd order partial

derivative with respect to v and r

vtu

1/2 Nw5 1/2 P LBP3U

1/2 p LBP3 1/2 1/2 p LBP3/u

r

_{(NrnlXGu) 1/2 p LBPu} -240 _{Derivative of yawing moment with}

respect to r minus mnXGu r3

_{1/6 Nrrr}

_1/2 LBP6/U

rv2 1/2 Nryv

1/2 p LEP/u

-1127

Nonlinear term, 3rd partial

deriva-tive with respect to r and v

r2

1/2 Nr 1/2 p LBP4u

rtu

Nru 1/2 p LBP4

ru2

1/2 Nruu 1/2 p LEP4/u

1/6 N5

1/2 p LBP3u2

1/2 p LBP-1U2

-89.3

112.1

Derivative of yawing moment with

respect to

ó

Nonlinear term, 3rd order partial

derivative with respect to ô

1/2 1/2 p LBP3 r2 tu

1/2 Nrr

1/2 p L3P4u 1/2 p LBP3U 1/2 Nouu 1/2 p LEP3 yr5 Nyr 1/2 p LBP4

-

N* 1/2 p L3P3u2

Nu

1/2 p LBP3U Lu2 _{N*uu 1/2 Q LBP3}

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If unimportant terms are dropped, the hydrodynamic force and moment expressions reduce to polynomials of reasonable

length. These dropped terms are identified on the basis of the following assumptions:

The ship asymmetry with respect to the center line plane is minor. This permits neglecting certain odd-ordered terms for the X expansion and certain even-ordered terms for the y and N expansions.

As a further consequence of the ship body symmetry,

Yu' uu' uuu' Y, Nu, Nuu, Nuuu, and N are equal to zero.

0 All terms higher than the third order can be neglected.

The second or higher order acceleration terms are negligible.

Equating the nonlinear Taylor expansions for X, Y, and N with the appropriate dynamic response terms (see equations

(l)-(3)) and taking the above considerations into account results in the nonlinear maneuvering equations which are presented by Strom-Tejsen9 A detailed discussion of the derivation of these equations, the assumptions made, and the definitions of terms appears in an earlier NSRDC report.9

The nonlinear maneuvering equations are modified by replacing

the terms X* + Xu + 1/2 Xu2 +

1/6

by the expression T(1-t) - R, which is coupled to the power plant torque equation:

dn

= df0

(6)

Using the notation of Stom-Tejsen,9 the modified functions are:

X-equation: (m_X)ii = f1(u,v,r,ó)

Y-equation: (mY)i + (1G-Y)r

= f2(u,v,r,ô)

N-equation: (mG-N)

+ (I-N.)i = f3(u,v,r,5), where:

-f1(u

, v, r,

6) = T( 1 _t) -R + 1/2 X,v2 + (1/2Xrr+mxG )r2 + l/2x 6662

+ 1/2Xvvuv2AU

+ 1/2Xrrur2Au + l/2xoouo2u

+ (Xvr+1fl)Vr + Xv6 + Xrôrô + XvruVrAu

+ XvVôtU + Xr6uró.0

(10)

f2(u,v,r,) = y* + y*u + Y*u2 + yv + 1/6Yv3

+ 1/2yrrrvr2 + l/2Yô,vô2 +

+ 1/2YuuvAu2

+ (Yrmu)r +

l/6yrrrr3 ± 1/2Yrvvrv2 + 1/2Yróãrô2

+ yruru ±

1/2Yruuru2 + Yô + l/6y00003

+ 1/2Yovov2 + 1/2Yórrôr2 + Y6u + 1/2YooLu2

+ Yvrâvrô

(11)

f3(u,/,r,6) = N* +

+

N*U2 +

Nvv +

1/6Nvvv3

+

1/2Nvrrvr2 + l/2Nvôôvô2 +

Nuvu + 1/2Nuuvu2

+ (NrflXGu)r + 1/6Nrrrr3

+

_1/2Nrvvrv2

+

1/2Nrôôró2 +

Nruru +

l/2Nruuru2 + N66

+

l/6N66663

+ 1/2Na6v2 +

1/2Nôrrôr2

+ N66U

+ 1/2Nouu5u2 + NvrôVr6

(12)

The method of simulating the power plant by using a torque

map to represent the gas turbine LM2500 engine of the destroyer

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DESTROYER STUDY SHIP PROPULSION EQUATIONS

The ship propulsion equations' (equations

(6), (7)

and (10)) are solved on the computer according to the flow diagram in

figure 2.* Programming details of the analog computer simulation are presented in appendix B.

cr:nD4 2 2 2

/Up

n D

L7CQ9

2 2 22 T:CTpD (Up n D) Q: CQPD (up +nD2) J0: _(1-t)T

0

60K9

IT

REDUCTION GEAR 'I, NOTE:

0di0

:0.98KgQQ

m-X)

)] dt

NE Figure 2

Flow Diagram for Ship Propulsion Equations

*Note that the equation for u at the bottom of figure 2 does not include a number of terms shown in equation (10) since, accord-ing to table 2, their coefficients are taken to be zero.

27-7l5

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N

Q

The total ship resistance versus ship velocity u (figure 1-A of appendix A) is stored on a digitally controlled function

generator (DCFG).

For a fixed pitch propeller of given configuration, the propeller thrust is a function of the propeller angular speed n

and the propeller speed of advance _{Up. The speed of advance is} generated on the analog computer from equation (A_2) of appendix A, where the wake fraction w' is stored as a function of ship velocity (figure 2-A of appendix A), on a DCFG.

It should be noted that the thrust and power plant torque equations are coupled.

The direction of propeller and free turbine rotation (as viewed from the shaft end) is clockwise. The propeller angular speed is defined as positive for forward ship motion. All

torques are defined as positive when acting in the direction of positive n.

The torque Qd produced by the gas turbine engine on the propeller shaft is:

= kg _E (13)

where k9 is the reduction gear ratio. The open-water propeller torque is calculated from the equation:

Q = CQpD3(Up' + n2D2) (114)

using a stored computer function of torque coefficient CQ versus

0,

the second modified advance coefficient (see figure 3-A of appendix A). The coefficient is defined as:

0 = nD/

j2

+ n2D2

and is calculated during the simulation from the propeller speed of advance u, and the propeller speed n.

The propeller speed n, measured in revolutions per second, is related to the engine speed NE measured in revolutions per minute, by the reduction gear ratio:

(15)

The frictional torque of the drive train is assumed to be 2% of the engine torque.

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NE = 6Okgn

(16)

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n

The net thrust of the propeller T(1-t) is calculated during the simulation by multiplying the propeller open-water thrust T by the thrust deduction factor (l-t). The thrust deduction factor

(l_t) versus ship velocity u (see figure 2-A of appendix A) is stored as a computer function.

The open-water propeller thrust T is calculated from the equation:

The torque

map''2

presented in figure ₃ and used in this simulation, shows the steady-state engine torque as a function of engine speed at a specified fuel flow rate.

'2 0 Li 0 Li 2 2 Li 70 60 50 ; 40 Li 0 0 30 I.-20 2

22

T = CTPD2(UP + n D

using the stored function thrust CT versus G (figure 4-A of appendix A).

ENGINE TORQUE NAP

-(17) r ENGINE FUEL FLOW RATEWf 10.000

S..

-s 9.000 8,000 -S 'S 7.000 LB/HR 6.000 _____________ .000

-.000

:

-S.000 4,000 500 bOO 500 2000 2500 3000 3500 4000 4500

ENGINE SPEED NErpm

This figure is from the paper "Acceleration and Steady-Stare Propulsion Dynamics of a Gas Turbine Ship with Control/able-Pitch Propeller" by C. J. Rub/s. published in 1972 TRANSACTIONS of The Society of Naval Architects and Marine Engineers; included herein bv permission of the above-mentioned

Society.

Figure

3 -

Engine Torque Versus Engine Speed

Although this engine torque representation is only correct for steady-state conditions, it represents with sufficient accuracy the torque during slow speed transients which occur in the

maneuvering studies presented in this report.

The General Electric Company has provided the Navy with a

simulation of the LM2500 gas turbine which realistically repre-sents non-steady-state engine torque versus speed during large fuel flow rate increases. However, this gas turbine simulation was not used in this analysis because of its large size and complexity. For instance, during large fuel flow increases, where the gas turbine accelerates quickly from idle to full

power, the true turbine torque is much 1ess1' than that predicted from steady-state characteristics. Therefore, under large fuel increases to the gas turbine, resulting in fast ship speed changes in the simulation, the gas turbine powering model used in this study would not predict the proper ship speeds; thus, the propel--ler thrust and torque would also be in error. _{Also, during fast}

transients in speed, sway forces and yaw moment are not properly represented. Thus, the maneuvering model presented here may not accurately predict maneuvers during rapid increases in gas turbine fuel flow rate. However, the engine model could be extended to. include rapid accelerations by including a time-varying engine torque coefficient, as developed by Rubis.12 Rubis' method is only applicable for predetermined fuel flow schedules which would greatly limit a maneuvering study.

ADVANTAGES OF MODIFIED NONLINEAR MANEUVERING MODEL

When the ship is moving straight ahead at constant velocity, the propeller thrust multiplied by the thrust deduction fraction equals the resistance of the ship for that speed (x = T(l-t)-R =

0). This equilibrium condition is disturbed as soon as a maneuver is initiated. At a specified speed during a maneuver,

Strom-Tejsen9 calculated the ship resistance from the effective horse-power data; the corresponding propeller thrust values were computed by using certain assumptions. _{These assumptions depend upon the}

type of engine and the engine setting maintained during the maneuver.

Examples of these assumptions are constant propeller revolu-tions and constant turbine power output (engine torque varies inversely to propeller revolutions).

These simplifying approximations, which essentially remove the power plant torque equation, do not allow the true propeller speed n to be determined or the power plant parameters to be simulated during the maneuver.

t.

I

I

k

27-745

15

These simplifying assumptions are not used in this report.

In the gas turbine power plant model for the study ship presented here, the entire basic power plant is simulated. _{This means that}

the modified nonlinear maneuvering equations can be used to

simulate responses of the power plant during maneuvering. _The power plant parameters may be displayed on a control panel or on a ship's bridge console as is presently being done in the

labora-tory at the Annapolis site for bridge design purposes. DESTROYER HYDRODYNAMIC COEFFICIENTS

The nondimensional 'hydrodynamic coefficients used in the modi-fied nonlinear destroyer maneuvering equations are presented in tables 2 thru

4.

The values of most of these coefficients were de-termined experimentally (at a Froudé number Fn of 0.20) and were supplied by Eda of the Stevens Institute of Technology. _However, the coefficients l/2Xvv, l/2Xóã, and Xvr were estimated for the study ship.

ANALOG COMPUTER MANEUVERING SIMULATION RESULTS

The dimensional modified nonlinear maneuvering equations were scaled for programming on the analog computer. _{The details of the} analog computer programming are presented in appendix B.

The following results are not presented as a definitive study of destroyer maneuvering but rather to demonstrate the validity and versatility of the modified maneuvering model. Such

simula-tions are intended for use in the design and evaluation of surface ship bridge console displays or for power plant powering simula-tion studies during ship maneuvers.

The primary advantage of the modified model over maneuvering models which do not include propulsion plant dynamics is that the effects of maneuvering on various propulsion plant variables can

be studied, as shown in the following simulation recordings. SPIFAL MANEUVER

The spiral maneuver7 indicates whether a ship is stable, marginally stable, or unstable. This maneuver begins with the

ship on a steady-state straight-line course. The steady-state rate of change in heading angle for discrete rudder angles, as the rudder varies from starboard to port and then back to star-board (or port to starstar-board and back), are recorded. It is

important in the spiral maneuver that for each rudder angle the heading rate of change be allowed to reach a steady-state

condition before the data are recorded and a new rudder command is ordered. If this precaution is not observed, the recorded results will be misleading and may indicate instability (in the form of a

false uhysteresisl? loop) when the ship is actually stable.

The results of a representative spiral maneuver for the study ship carried out at a constant fuel flow rate are shown in figure

)4 _{This plot indicates the stability characteristics of the ship}

under the conditions shown. The single curve (for starboard rudder to port and back) with negative slope indicates that the destroyer has controls-fixed, straight-line stability.

r DEGREES/SECOND

-20 -15 -0 -5 5 10 15 20

STARBOARD -0.1 _PORT

Figure 24.

The Spiral Maneuver

r

L

L

ZIG-ZAG MANEUVER

One of the most important and frequently used maneuvers for testing naval ships is the zig-zag maneuver. The ability of the rudder to control the ship is indicated by results of this

maneuver. The results of this test depend on the stability characteristics of the ship, as well as the effectiveness of the rudder.

A representative zig-zag maneuver for the destroyer study ship at a constant fuel flow rate is shown in figure

_5.

Before the actual maneuver is executed, the ship is brought to a

steady-state speed of say, 15 knots in a straight line. A maneuver

begins when the rudder is deflected +20 degrees. When the change in heading angle reaches -20 degrees, the rudder is deflected -20 degrees and the procedure repeated. By the conventions

defined here, a positive rudder angle produces a negative change in heading angle and vice-versa. The switching of the rudder command takes place automatically during the analog computer simulation.

The effects of this maneuver on some of the propulsion plant variables are also shown in figure

_5.

The engine speed NE drops off approximately

50

r/min (steady_state value l,642 r/min), while the engine torque QE increases by approximately 1,2400 lb-ft

(steady-state value = 12,960 lb-ft). The propeller thrust T and torque Q increase approximately by l0 pounds (steady-state value

0.633 x io5

pounds) and 0.2 x

io5

lb-ft (steady-state value =

1.77 x io5

lb_ft), respectively. The ship speed drops

approxi-rnately 2 ft/s (or 1.2 knots) from the steady-state value.

One of the main characteristics of the zig-zag maneuver is the overshoot, defined as the difference between a maximum

heading angle and the heading angle that existed at the instant of the reversal of the rudder command. For the study ship used in this simulation, the overshoot is approximately 24 degrees. A rudder rate of

3.5

degrees/second was used in the simulation.

TURNING MANEUVER

The turning maneuver7 indicates the effectiveness of the rudder in turning the ship. A simulation of a representative turning maneuver was performed in which the ship was brought to a steady-state speed of

₂₅

knots in a straight line by using a constant fuel flow rate to the gas turbine. The rudder is then deflected 35 degrees right at a rate of

_3.5

degrees/second and held at this value during the maneuver.

27-7245 ₁₇ 1 t L I 1' (c ft

Figure ₅

-

Zig-Zag Maneuver with Initial Steady-State Speed of _{15 Knots}

3. .# FTISEC

35.4

I) 25.3 17.2 FTISEC

-FT/S

+10

o -10 FT/S

- +35

- 35

20,160 LB-fT of 12960 -- 5,760 R/MN - 1,950 1,642

- t334

1.71x105 LB I Q.633x105 -45x105 LB-FT .4x1O5 0 177x105 - -0.9x105 LB-FT MINUTES 0 2 3 4 5

Figure ₆ shows various ship arid propulsion plant variables during the test. The ship speed falls off approximately 10 ft/s

(steady-state value

= 45.6

ft/s), and the engine speed decreases approximately 300 r/min (steady-statevalue = 3080 r/min). The propeller thrust and torque increase approximately 0.4 x

io5

pounds (steady-state value = 2.0 x i05 pounds) arid

0.05 x i5

lb-ft (steady-state value =

0.5 x

i5 lb-ft), respectively, above their steady-state values.

Figure ₇ is a plot of the computed turning path that occurred during the simulated turning test. This figure gives a good indi-cation of the actual turning circle, even though a small drift in the computer was encountered in the turning simulation. _This

drift is peculiar to the analog computing system and is manifested in the coordinate transformation (coordinate system in ship axes to the stationary reference axis) portion of the simulation. Repeated multiplications generate a small error which feeds into the integrator that produces lateral displacement y0. The effect, although small, can be seen in figure ₇ where the curve is

slightly elliptical rather than a true circle. ACCELERATION MANEUVER TO STEADY STATE

In the simulation of this maneuver, the ship was accelerated from 0 to 26 knots by changing the fuel flow rate to the gas tur-bine from 0 to 10,000 lb/h in 10 seconds as shown in figure

8.

This maneuver is similar to that used in the propulsion dynamics

studies by Rubis.1' The steady-state values recorded in figure

8 compare favorably with the results obtained by Rubis under similar operating conditions.

DISCUSSION AND CONCLUSIONS

The object of this work was to develop a nonlinear mathe-matical model, including propulsion plant dynamics, for use in performing maneuvering simulations of a destroyer study ship. The maneuvering simulations will be used in the design and

evaluation of shipst bridge console displays.

The mathematical model was derived by coupling the nonlinear ship propulsion dynamics equations of Rubis' with nonlinear maneuvering equations of motion for motions in the horizontal plane.3 This mathematical model was programmed to permit simulation on EAI

680

and TR-248 analog computers.

F

27-745

19

r

Figure

6 -

Turning Test with an Initial Steady-State Speed of

₂₅

Knots

2500 FT _{NONLINEAR MANEUVERING} -RADIUS TURNING 2500 FT +3500 FT OG 3500 FT 45.6 FT/S SURGE 45.6 FT/S 5FT/S SWAY________ 5 FT/S 450

-T

HEADING ANGLE 450 3080 R/MIN NE - ENGINE SPEED

--

4.3xIQ5LB3080 R/MIN

-- T PROPELLER THRUST 43x105 LB L04x105 La-FT PROPELLER TORQUE --1.04 x105 LB-FT MINUTES 0 2 3 4 5

F I. V..,'

L

iI

L

:1 L,..

[T

IT

C

27-745

FEET -1400 -700 700 1400 2100 2800 3500 3500 FEET OG Figure ₇

Turning Test Maneuver

21

X

Figure

8 -

Straight-Line Acceleration Maneuver

I

45.6 FT/S SURGE 6FT/s NONLINEAR MANEUVERING lOFT/S STEADY STATE r0 SwAt - lOFT/S 72,0OL 3-FT I NGtNE TORQUE LB-FT 72,000 3080 R/MIN N ENGINE SPEED 3080 R/MIN TOX6TB PROPELLER THRUST 1.06 x 10 LB 26x106 LB-FT a / .PROPELLER TORQUE 2.6x106 LB FT 10,000 LB/H FUEL FLOW

a

---1.0

-

---2ND MODIFIED ADVANCE COEFFIENT I I MINUTES 0 1 2 _.3

Because of the intended application of this computer simu-lation, certain refinements which could have been included in the simulation of propulsion plants have been omitted. Among these is the engine torque coefficient, developed by Rubis''

and used in his work. This time-varying coefficient is used to augment the engine torque during rapid accelerations. However, maneuvers requiring rapid accelerations, such as crash ahead and crash reverse, will not be required of the present simulation.

In verifying the suitability of the simulation for use in ship control console development, simulated performance data were taken in two steps. First, the standard maneuvers (spiral, zig-zag, and turns) were simulated at constant fuel flow rates. The results of all these simulations compare favorably with data published in the open literature. Second, the ship was acceler-ated to full speed with the rudder angle set to zero. The

steady-state propulsion plant variables thus obtained were com-pared with those reported by Rubis'' in his propulsion studies. The two sets of data showed good correlation.

The advantage of the modified maneuvering model over models previously used for this purpose is that it includes the gas

turbine propulsion plant characteristics during maneuvering. The

simulation using this model can thus be used to study the effects of maneuvering onthe propulsion plant. In particular, the

maneuvering simulation presented here can be used to accurately predict slow transients in ship speed during maneuvers caused by slow increases of the fuel flow rate to the gas turbines but not

for accurate prediction of fast transients in ship speed.

TECHNICAL REFERENCES

1 - Rubis, C. J., "Acceleration and Steady-State Propulsion Dynamics of a Gas Turbine Ship with Controllable-Pitch Propeller," presented at the Annual Meeting of SNAME, New York, N.Y. (16 & 17 Nov 1972)

2 -

Rubis, C. J., "Braking and Reversing Ship Dynamics," Naval Engineers Journal (Feb 1970)

3 -

Eda, H., and C. L. Crane, Jr., "Steering Characteristics of Ships in Calm Water and Waves," presented at the Annual Meeting of SNAME, New York, N.Y. (11 & 12 Nov 1965)

14 - Eda, H., "Maneuvering performance of High Speed Ships and

Catamarans," Davidson Lab. Rept SIT-DL-72-1626 (Dec 1972)

L..

5 -

Eda, H., "Directional Stability and Control of Ships in

Restricted Channels," presented at the Annual Meeting of SNAME, New York, N.Y. (11 & 12 Nov

1971)

27-745

-

23

L

ft

6 -

Eda, H., "Directional Stability and Control of Ships in Waves," Journal of Ship Research (Sep

₁₉₇₂₎

7 -

Comstock, John, P. (ed.), Principles of Naval Architecture," SNAME

(1967)

8 -

Abkowitz, M. A., Stability and Motion Control of Ocean Vehicles," MIT Press, 2nd printing

₍₁₉₇₂₎

9 -

Strom-Tejsen, J., "A Digital Computer Technique for Predic-. tion of Standard Maneuvers of Surface Ships," NSRDC Rept 2130 (Dec

₁₉₆₅₎

I

9I

-APPENDIX A

ADDITIONAL DESTROYER STUDY SHIP CHARACTERISTICS REFERENCES

Rubis, C. J., "Acceleration and Steady-State Propulsion Dynamics of a Gas Turbine Ship with Controllable-Pitch Propeller,t' presented at the Annual Meeting of SNAME New York, N.Y. (Nov

16 & 17 1972)

Rubis, C. J., "Braking and Reversing Ship Dynamics,t' Naval Engineers Journal (Feb

1970)

Miniovich, I. Ya., "Investigation of Hydrodynamic

Characteristics of Screw Propellers Under Conditions of Reversing and Calculation Methods for Backing of Ships,' BUSHIPS Translation

_697,

Translated by Royer and Roger, Inc., International Div., Washington, D.C.

(1960)

Baker, D. W., and C. L. Patterson, "Representation of Propeller Thrust and Torque Characteristics for

Simula-tions;" Appendix C, "Data for 18 Propellers," MEL Rept

_202/67

(Mar

₁₉₇₀₎

This appendix contains the following study ship charac-teristics presented by Rubis (references (a) and (b)):

o Ship resistance. Wake fraction.

Thrust-deduction factor.

o .Propeller thrust and torque coefficients. SHIP RESISTANCE

A curve of the assumed total ship resistance versus ship speed (references

(a)

and (b)) in

knots is shown in

figure 1-A. The curve shows the clean-hull resistance at a full load dis-placement of 4000 tons. The velocity exponent in the expansion for resistance (R = bus) varies, from 2 to .4 in the speed range of 10 to 0 knots.

350

300

100

50

This figure is from the paper "A cceleration and Steady-State Propulsion Dynamics of a Gas Turbine Ship with Controllable-Pitch Propeller" by C. J. Rubis, published in 1972 TRANSACTIONS of The Society of Naval Architects and Marine Engineers; included herein by permission of the above-mentioned Society.

Figure 1-A

Total Ship Resistance (RT) Versus Ship Speed

25 30 35

0 5 10 IS 20

SHIP SPEED, knots

'-S

L

WAKE FRACTION

The wake speed is defined as the difference between the ship speed u and the propeller speed of advance Up. The wake speed u-uD divided by u is called the wake fraction (references

(a) and (T)):

(u-u)/u

(A-l)

/ Up

= u(l_W') (A_2)

The steady-state wake fraction (figure 2_A) used in the analysis is a function of ship speed. The assumed data are typical for ships of this type and were synthesized by Rubis (references (a) and (b)) from a number of published results of

test data.

-5 10 15 20 25 3C

SH(P SPEED, knots

W = W

in the notation used in this report

This figure is from the paper "Acceleration and Steady-State Propulsion Dynamics of a Gas Turbine Ship with Controllable-Fitch Propeller" by C. J. Rubis, published in 1972 TRAASACTIQVS of The Society of Naval Architects and Marine Engineen; included herein by permission of the abote-rncntioned

Society.

Figure 2-A

Wake and Thrust-Deduction Fractions Versus Ship Speed 0.12 L. 0.10 c 0.04 0.02

t

Vt

THRUST -DEDUCTION FACTOR

The propeller provides a thrust T when operating in _open water (free stream) but when the propeller _{operates in the wake} of a ship, a thrust of only T(l-t) is available _{to overcome the} ship resistance. _{The expression (l-t)}

is called the thrust-deduction factor (references _{(a) and (b)).}

The thrust-deduction factor as a function of ship speed for the destroyer _{study ship} is shown in figure 2-A.

PROPELLER THRUST AND TORQUE COEFFICIENTS

The propeller thrust and torque characteristics used by Rubis were based on work by Miniovich (reference (c)), _as modified by Patterson and Baker (reference (d)).

The torque coefficient CQ and the thrust coefficient _{CT so} obtained were plotted as functions of the second _{modified advance} coefficient c (see figures

_3-A

_and

_{4_A).}

1 C

I

C, F I C.

I'

-1.0

VP = u in the notation used in this report Courtesy of Naval ngineers Journal.

Figure 3-A

Torque Coefficient Versus

Second Modified Advance Coefficient

3 BtADES (AR:j

1kb:

).0

I

27-745

_A-5

Note: Vp = Up in the notation used in this report

Figure 4-A

Thrust Coefficient Versus

Second Modified Advance Coefficient

4

0 -9

-8 -6 -5 -4 -3 -2 CT 4 0 2 -1 1 2 4 5

Vp<OiJcO

Vp 'O,O

6 8 9 10

vp>o

-2

0 3 BLADES EAR:8 Ho/D:1.05

APPENDIX B

THE ANALOG COMPUTER SIMULATION

The simulation of the mathematical model was implemented on one EAI

680

and one EAI TR-48 analog computer. The complete simulation requires 30 multipliers. Of these, 214 are available on the

680,

while the remaining 6 multipliers are taken from

the _TR-248. The simulation consists of the equations of motion (surge, sway, and yaw), an engine mapping, a steering control system, and a coordinate transformation. The symbols used in the patching diagrams are defined in table 1-B.

On all patching diagrams, numbers 500 through 700 refer to components on the

680,

while components numbered in the 200's refer to the TR-148 computer.

THE STEERING CONTROL SYSTEM

The steering control system (see figure 1-B) is a simplified model of the steering control dynamics employed on destroyer class ships. For a rudder rate coefficient (RK) equal to 0.5, the actual rudder rate is 3.5 degrees/second. This rate is variable from 0 to 7 degrees/second.

GAS TURBINE ENGINE MAPPING

The propulsion plant used in this simulation is a simplified model of the General Electric LM2500 gas turbine (base engine only). Figure 2-B shows how the steady-state engine torque

varies with engine speed at specified fuel flow rates. Figure 3

of the text is repeated here as figure 2-B for convenience.

The lines of constant fuel flow in figure 2-B can be approx-imated by first order equations of the form:

= ANE + _E1

where A is the slope and QE1 is the ordinate intercept of a given curve. By tabulating data points for A and QE1 versus Wf and applying a least-squares curve fit to these two sets of data points, it is found that:

A = -0.0213 - O.72E14Wf + _O.227Wf2 (B-2)

L

27-7145 B-i

I

S rnbol

TABLE 1-B

SYMBOLS USED IN PATCHING DIAGRAMS

x Y x _y

_"

I

-

-I -V x

[+

I A +UL

-LI

6

'1 A LOGIC SIGNAL

Device

Integrator Su.rnme r Potentiometer Digital coefficient

Attenuator (DcA)

Reference

Test reference (on in ST mode only) Multiplier Track/s tore Comparator Y

= -[

xdt +

_x0]

A =

Lx + y + zJ

Y=kx

Y =-kx

Z -XY Expression A -- 1 x+y+z>O ) 0

x+y+zO

Variable

limiter

Y = UL, x, LL,

xUL

LL<x<UL

xLL

V

xO::

Y

-x

if A=1 -

store

if A=O

'C-TABLE 1-B (Cont) AND gate Trunk lines Divider Digital coefficient function generator Electronic switch Square root Zero limiter B

_-3

A =

ab.c

x=Y

x

Y = f(x) - x , A=1 0 , A=O

Y= -(sIGN x)..J[j

-x

, xo

0 , x>o

Substituting equations (B-2) and. (s-3) into equation _(B-i) yields:

=

(0.22f2-0.726wf-o.2l3)NE

+ l.24Wf-0.07 . (B-4)

Figure 3-B shows how this equation is implemented _{on the} analog computer. It should be noted that the engine map (figure 2-B) was normalized before the data points used _{in the}

curve-fitting procedure were taken. _{This greatly reduces the amount of} scaling required in figure 3-B. _{In this figure, amplifier 622} provides an automatic fuel flow ramp from 1,000 to 10,000 lb/h

for use during debugging procedures. _{For normal simulated}

maneuvers, the fuel flow rate is varied manually through the _use of potentiometer 527.

The simulation of the engine torque limit at 72,000 lb-ft is also shown in figure 3-B. _{This torque limit must be considered} whenever the fuel flow rate is greater than 8,300 lb/h and _the engine speed is less than 750 r/min. _{In this operating region,} it becomes necessary to determine the proper break point (BP) at which QE becomes less than the limit and is again governed by

equa-tion (B_i). The break point can be expressed mathematically as: BP = 1.59 (Wf-0.83) for _{wf>8,300 lb/h}

and NE<75O r/min . (B-5)

The implementation of the BP is also shown in figure _3-B. EQUATIONS OF MOTION

In the computer program (see figures 4-B and 5-B), the modified nonlinear maneuvering equations (see _{equations (6)}

through (12) in the text of the report) were put in the following

form: u = T-R/Dl + (c2/Djjvr - (C3/Dl)v2 _{- (C4/Dl)} 62U2 (B-6) . 1 n = 2111 [Qd - Qf - Q] QE1 = l.24Wf - 0.07

_{(B -3)}

IL

L

L

L. TA EL E OF COEFFICIENTS IN A2 = (l/2pL3)N' A3 = (1/2pL4)N'r A4 = (1/2pL3)N'5 A5 l/2(1/2PL4)N'rvv A6 l/2(l/2PL5)N'vrr A7 = 1/6(l/2pL3)N'vv A8 = 1/6(1/2pL3)N' A9 - (l/2pL5)N'. C2 = 1/2pL3X'vr+m C3

_{1/2(l/2pL2xv)}

= l/2(1/2pL2)X'óô 2-B - DEFINITIONS EQUATIONS (B-6) THROUGH (B-9) B2 = (l/2pL2)Y'v B3 = (l/2pL3)Y'r B = (l/2pL2)Y' B5 = l/2(l/2pL3"iY'_{/ rvv}

4,

LY

B6 1/2(l/2p _{/ vrr} B7 = 1/6(1/2pL2)Y'vv B8 = 1/6(l/2pL2)Y'555 B9 = m - (1/2pL3)Y' = 1.08m D2 B9

= A9

V = -(B2/D2)Uv +(B3/D2)ur + (B4/D2)5U2 + (B5AD2)v2r/U -(B6/D2)vr2/U + (B7/D2)v3./U -(B8A2)63U2

- mur/D2 (B

-8)

r = -(A2/D3)Uv -(A3/D3)ur -(A4/D3)U26 -(A5/D3)v2r/U

-(A63)vr2/U + (A7/D3)v)/U + A83U2o3

(B

-9)

where U = (u2+v2)1 has not been approximated by u, as is usually done. The coefficients C1, Bj, Aj7 and Dj are defined in table 2-B in relationship to the notation used by Strom-Tejsen at the Center.

In order to generate XOG and YoG' the sine and cosine of the yaw angle are required. The sine and cosine generators _on the ari

₆₈₀

_{analog computer are limited to angles less than or} equal to ±200 degrees. _{To circumvent this limitation, the} available continuous resolver was used which provides _{the sine} and cosine of any angle.

The continuous resolver is shown in figure 6-B, while the rest of the coordinate transformation is shown in figure _7-B. CHECKING AND DEBUGGING PROCEDURES

A DEC PDP-15 digital computer is used in the setting up

and static testing of the analog computer simulation. _{This saves} a great deal of time and also provides excellent documentation of

the simulation. An EAI 693 system is used to interface the digital and analog computers.

Figure

8-B

is a listing of the potentiometer setting

pro-gram. _{This program calculates all potentiometer settings from}

the data provided by the programmer and generates an output listing of these values (figure 9-B) as well as listings of the problem parameters (figure 10-B), the maximum values and _initial

conditions (figure 11-B), and the dimensional hydrodynamic

coefficients (figure 12-B). _{All potentiometers are then set to} the appropriate values by the digital machine.

Figure 13-B is a listing of the static check _program,

SCNLM SRC. This program is written in FORTRAN IV and is executed on the PDP-15 digital computer. _{It calculates the expected} out-puts of all potentiometers, amplifiers, and DCFG's from arbitrary initial conditions, and compares these values with those _{that it} reads from the analog computer. _{If the difference exceeds a} specified tolerance, then the particular component involved is

I I ( E

L

[I

1!i Figure 1-B

Steering Control System

t 8rri :2O = 7' K: 0.5 RUDDER COMMAND :20'

L

27-745

B-7

70 60 50 40 30 20 I0

j

-r

IO00

I

.5 -.

5

-ENGINE FUEL FLOW RATEWf 10.000 9.000 8.000

imiIi.

-S...-

5

-7.000 L8/H 6.000 z0O0 .5,---.5 5-30005-_S 5.000 4.000 500 l('O 500 2000 2500 3000 3500 4000 4500

ENGINE SPEED N ErPm

This figure is from the paper "A cceleration and Steady-State Propulsion Dynamics of a Gas Turbine Ship with Controllable-Pitch Propeller" by C. J. Rubis, published in 1972 _{TRANSACTIONS of The} Society of Na-va! Architects and Marine Engineers; included herein by permission of the aboe-mcnrioned

Society.

Figure 2B

I I

L

FR624 B 10 SEC FUEL FLOW RAMP FUEL FLOW COMMAND w Wf .07 .7264 .0213 .2217 715 -A [NE +LNEm 71 3 NE NE .83 685

1L.

.159 4 _(TRACK) LiM 685 lWt L 'm 684 i-68 .83 .1 714

Figure 3-B

Gas Turbine Engine Mapping

4

0

fur 658\ LUmnrr 64C TO P(638)

I

[0 Ja 00 1ttw1 LwJ WO + L)jJ J COt 69 rWi + L IJ 6 r, w1

[1+.

L Wf JA_J 1w L. 899 wJ I OL rw tdr I

L (a)

69 999 CL9

Figure 5-B - Surge Equation

.t.

- j+ rWn W 1+

\ L'J

LC9) WcJ + C99 [7A E99 _{L WJ}

tt' N) LV'm 63

I

r

['km L450 ó30

Figure 6-3

rj 69 F k COS 692 - cos'?' 664)

-'-OR

-

Continuous

LOGICI A

Resolver

C FQCALj5 VALP1 01.01 S ZQD-1 01.02 c 01,995 ZcC-1;S ZQP:o;S ZQI0 02.01 I (ZQC)2.02,1.02,1,02 02.02 C

g5,(9 I _{(ZQD)9,011I (-ZQD)9,55;OATA OUT POTS;G 9,55}

09.01 P ON;S ZQP:Z(W+1

09,10 1 ",,, NONLIN.AR MANEUVERING SIMULATION

09.33 I X3,"PAGE",ZQP, 0g.soTc!...11/1/73_1uu, 09.55 7 %10.05 10.01 C 10.10 S RHO1,905S L40;S \:O,5*RHO;S L2:L*L;S L3:L2L 10,11 S L'L2*L2;S U1:25,32.'S PI3.j4159;5 G32.2,'S t3:1,0 10,12 S D15;S KG:1;S MAS:3E5;S IS:2,3757E9;S IDT:9,E4 10.13 S NF1:(*L2;S NF2<*L3;5 NF3c*L;S NF':NF3*L.

20.03 S USa5,;5 VSM:10S CUMUSM2+VSM2

20.04 S NPM:218;S NPS:NPM/60;5 YAM;540;S ZZ:2L1S XIC:200S YIC:300 20,05 S PSI:O;5 RM:5;S RCM1;S DRM:35;S CCMFSQ1(CUM)

20,06 S DNM:218;S WKM:,6,s TM:.1;S wFM:100;5 HT:.200E6 20.07 S NEM3o8Ø;s QEM;,72E;S LJDM:,2*USH;$ VOMC,5*vSM;s CQM,08 20,08 S CTM,5:S UPML13;S V3UVSM3/CCM;S V?U;VSM2*(RM/57,3)/CCM 20,09 S VR2VSNi*(RM/57,3)2/CCM;S SV2UPM*UPM+(NPSW)2

20,10 S GM:RHO*CQM*D3*SV2;S NDM:1;5 TMRHO*D2CTM*SV2

20.11 S SyFSQ7(Sv2);s NU1;S D1,Ø8*MAS;s D253U-5NF2*VDM 20,12 S O3:31,8E5*NF4*RDM/57,3;S DEM7;S RK0.5

20,13 S A1D:4,92E5*NF2;S AO:_33OE_5*NF2;S A3D.2Z4E5*NF3 20.14 S ALD:..89,3E..5*NF2;S A5D-1127E5*NF3S A6D-.23.3E-5tNF4 20.15 $ A8Ou2E.5*NF2;S A7O1023E-5*NF27S A1J:31,8E5eNF4 20.16 S B1D:3.5E5*NFj;S B2D:..1049E.5*NF1;s 3300E-5*NF2

20,17 S Q239E-5*NFj;S b5D6j26E-5*NF2;S 6O-74bE-5*NF3

20,18 S B7U:6712E-5*NFI;S BbD:2L7E5*NFj;s B9531L-5*NF2 20,19 S C2D;550E-5*NF2,'S C3D-aø0,8E5*NF1$ C4D-.94.8E-5*NFj 20.23 S xDMUSM;S VDMVSM;5 XM25o0;S YM:3500

29,99 I (ZOO)33,01,50,01,50.01

30.01 7 UpRQBM PARAMETERS",1

30.02 T PROGRAM",

30,03 7 NAME VALUE _{VARIAt3LE",L1}

30,10 T H _{RHO ",RHO} ," WATER OENSITY", 30,20 1 " _{L ",L} , SHIP LENGTH 30,30 7 H _{K ,K ," ",} 30,40 7 L2 ",L2 ," 30.50 1 L3 '1,L3 ," FT3", 30,60 T u _111L4 , FT'4",l

30,70 1 U ",U1 ," QUIL. VL. FT/SEC",

30,80 1 P1 ",PI ,

30,90 T " G ",G ," GRAVITY',l

31,01 7 It _{B ",8} ," _8ETA",

31.11 7 0 ",D ," PROPELLER DIAM,",L

31.21 1 u _{KG ",KG}

," REDUCTION GEAR RATIO",I

31.31 1 MAS ",MAS ," _{SHIP MASS*1,08}

SLUGS",l

31.41 7 H IS ",I5 ," MOMENT OF INERTIA SLUGS.F12",l

31,51 7 _{101 ",IDT ," DRIVE TRAIN INERTIA LB-FT-SEC,1}

31.61 1 NFl ",NFI

,"

,1

Page 1, Figure _8-B

Page 2, Figure 8-B

27 -75

B-15 31.71 31.81 31.91 40.01 7 1 1 0 " " 9;T NF2 NF3 JF4 "MAXIMUM ',NF2 ," ", ",NF3 ,ft 'I, ",NF4 ," _n,j

VALUES AND INITIAL _{CONUITIONS",U}

40.02 D 3e,ea;D 30,03

40.10 1 "

USM "gUSH ," _{SHIP X VEL,}

FT/SEC",l

40.15 7 " YSM U,VSM _{," SHIP V VtL.,}

FT,'SEC",l

40.20 1 u

PSI ",PSI ," _YAW

ANGLE DEGREES", 40.25 1 " RM ",RM _{," YAw RATE} DEG/SEC",Z 40.30 40,35 40,36 1 1 7 " " ROM ORM OEM ',DiM ",DEM ," ," ,"

YAW ACCEL, _D[GfsEC2",

RUDDER COMMAND DEGREES", RUDDER ERHOR DEGREES",L

" DNM ",DNM ," RPM CCMMANIJ",Z 40,45 40,50 7 1 WKM TOM ",WKM ",TDM , ," WE FRACTION", THRUST DEDuCTION",l

40,55 1 " _WFM ",WFM _{," FUEL FLU HATE L/HR",j}

40.6 T " RTM ",RTM ," TOTAL SHIP RES, LBS",j

40,65 40,70 7 1 " NEM QEM ",NEM ",CEM ," ," ENGINE SPEED _RPM", ENGINE TURLUE L0-FT",

40,75 T " 0DM ",UL)Nl ," SHIP X ACCEL, FT/SEC2",

40,60 1 " VDM ",VDM _{," SHIP V ACCEL.U,}

40,85 1 " CQM ",CQM _{," PROP, TORQUE COEF.",L}

40,90 ¶ CTM ",CTM g' PROPr THf'UST COEF,"g

40,95 7 UPM ",UPM ," PROP, ADVANCE _F7/SEC",L

41,01 1 " V3U ",V3U , _{MAX, V3/Uu,}

41.06 T " VRU ",VRU _{," MAX. V2*/U",}

41,11 1 " VR2 _",VR2 ," MAX, V*R2/U",l 41,16 41.21 41,26 7 1 1 t' " NDM Tl1 OH ",NDM ",TM ",QH ," ,U SU

MAX N DOT _RPS/SEC",

u, 41.31 41,32 41,36 41,41 T T T 1 " " NPM NPS. Sy2 Sv ",NPM ,NPS ",8V2 ",Sv ,' ," , ," PROP, RPM", PROP, RPS', "d ic, 41.42 1 " XDM _{",XDM ," X DOT MAX ",} 41.43 41,44 41,45 1 ¶ 1 " " YOM XM YM ",YOM ",XM IT,YM ," ," ,n y DOT MAX ", X MAX FT I', Y MAX FT ", 41.46 7 " NU ",NU ," ,1,1 41.47 41.50 7 T " YAM RK ",YAM ",RK , ,"

TURN, RAD,MAX YAJ-DEG

RUDDER RATE CONSTANT",l

41,51 1 " Dl H,Dl _{,° SURGE DENOM",L}

41,56 7 " 02 ",02 ," _{SWAY DENOM",L}

41,57 7 CUM ",CUM ,"

U2*V2

",L

41,58 1 ' CCM ",CCM _," CAP U MAX FT/SEC 41.61 7 " D3

",03

,"

_{YAW DENOM",L} 41.62 D 911 "HYDRODYNAMIC COEFFICIENTs", 41.66 7 " AID ",AID ," ",L 41.71 1 " A2D ,A2D ," 41.76 1 A3D ,A3D _{," 11}

41,81 1 " A4D ",A4D _,"

_"Iz

41,86 1 " A5D _{",ASD ," ",}

41,91 1 " A6D ",A6D _,"

",L

41.96 1 " A8D ",A8D ," 't,

49.990 9;T "P01 SETTINGS',

50.1 C POT SETTING CLCULATI1JNS

60.2 S Io;s SP=DM/(1ø*)E1);D 9 60.03 S I,0i;S SP:1 M/(1)*uEVi);D 9j 60.05 S I:63;S SR*cDEM/(DRM*);D 9 60.C6 S Ibk14;S SP:USM/(Yt)M*10,)L) 90 60.07 S I=61'5;S SP:LJSM/XDM;1) 90 60.08 S I:6c'6;S SPySM/yDM*10,fl) 90 60,09 S I:677$ SP:MAS/D2;D 90 60,34 S I:632;S SPuSM2/CUM7() 90 60.39 S I637;S SP:VSM2/CUM;D 90 0,10 S I:68;S SP:VSM/xDM;O 90 60.12 S I6j0;S SP;YDM*1e,/yM;D 90 60.13 S I:611;S SPhXDM/XM;D 90 60,19 S I:6171S $P:(84D*CCM2*DRM/57,3),D2,D 90 60,20 S Ib18;S SP-B8D*(DRM/57,3)3*CCM2/D2;D 90 60.21 S t619;S SP:7D*v3u/L)2;O 90 60,27 5 1:625;S 5P0.999*vSM3/(CCM*V3u);D 90 60.28 S I:626;S $P0.5;D 90 60,32 S I=6301S SP0,5;D 90 60,33 S I63i;S SPrRM/YAM;D 9 60,35 S I:633;S SP:0,5;D 90 60,37 S I:635;S 5PZZ/DRM;D 90 60.38 S 1:63675 SPC_A140*CCM2e(ORM/57.3)/(D3*S.0)y0 90 60.40 S I:638;S SPC4D*(ORM/57,3)2*uSfr12/(usM*Dfl;D 90 60.41 5 I:639;S SP:A8D*(DRM/57,3)3*CC12/D3,D 90 60.4? S I:640;S SPA2D*CCM*VSM,(D3*iO);D 90 60,43 S I6131;S SPo.5;D 90 60.45 $ Z:643;S SPRDM/RM;D 90 60,47 S _{1.613575 SP:-82D*CCfr1*VSM/D2.D 90} 60,54 S I6S27S SPB3D*CCM*(RM/57,3),D2;D 9 60,49 S 1z647;S SP:ø,5;D 90

60,50 S I:648;S

SPyDMfV5M7D 90 60,51 S I:649;s SP.A70*v3u/03;D 90 60,52 S I650;S SPzA5D*VRU/D37D 90 60,53 $ 116511$ SPB5D*VRU/(B1e*UDM)70 90 60.57$ 1z655;S SP-A31*CCM*(RM,57,3)f(D3*1O);D 90 60,60 5 1c6581S SP:O,999*VSM2*(RM/57,3)/cC1*vR);D 90 60.62 S I6607S SPzPSI/YAM;0 90 60,63 S I661;$ SPO,5D 90 60,65 S 1s6637S SP0,57D 90 60,66 S I664;S SP'UPM2/SV270 90 Page 3,

_{Figure 8-B}

A

42,06 1 " A9D ,A9D _{," ",1} 42.11 1 " _{BiD ",BID} ," ", 42.16 42,21 1 1 " 2D 83D T,B2D ",831) ," ,'I

",

42.26 42.31 1 7 " " 850 ",84D 'I,85D ," ," ", ", 42.36 7 It ₈₆₀ ",86D ,'I fl1 42,41 42,46 42,51 7 1 1 " 870 880 810 ",87D tI,8E) ",BlO ," ," ," ", i', P1 42,56 42.bl 1' I " " C20 C3L ",CD °,C3[) , ,'I

",

4,b6 I "

CD

",C'40 ," ",

89.10 I c-FABSczrD))89.15;I _{ZtL89.12,89i5,89,15}

89,12 DATA CLDSE;T P0TS>1******NCT 5AvE0',flQ

69.15 I (ZCD)89.2;DATA C;T t'POTS SAVED',flQ

89,201 (ZQL)89,5,89,5,89,6

89,50 S ZQDj;P OFF;P _{OFF;DATA OUT POTS;G 1.02}

89.60 P OFF;A _{POT>1 SAVE POTS?",ZQ(7I (FA8S(ZQQ0YES))89,5,89,;U}

90,101 (-ZQD)90,5,90,5 90.1ST 3,'P(",I,") ',%B.04,SP;S 1:1+1 90.20 I (sP-1)90.21,90,21;T " _{****PQT>1*****";S ZQL-1} 90.21 I (,001_SP)90,2r5;S ZC:1;T _'I 90,25 S ZQIZQI+17I (46.Z01)90,271T iR 90.271 US ZOI0;D 9.99;R 90.501 %5,02,I,,Y.8,04,SP,flS II+1 90.55 I (FABS(ZQD))S0.6,90.6;I (1-SP)89,12,8q,12 90.60 R

Page 11, Figure

8-B 27-745 B-17 V ( 60.67 S 60,68 $ 60,70 S 60.71 S 60,72 S 60,75 S 60,82 S 60,83 S 60.85 S 60,86 S 60,87 S 60,88 S 60.90 S 60,91 S 60,92 S 60.93 S 60,96 S 60,97 S 61o2 S 61.04 S 61.05 S 61,09 S 61.18 S 61,19 S 61,20$ I:665;S 1r666;S I6b8;S I669,S

I6i;S

I673;S I:68;S

I68jS

I:683;S I684;S I:685;S I686;S I&88;S Ir6897S I693;S I691;S

I697S

I695;S

I698S

I700;S I7O1;S I705;S 1r714;S I715;S 117161S SP:NPS*L?/SV2;D 90 SP:_C3L1*VSM2/(USM*D1);D a SPi.q;C r SPrNPS*L)/SV;D 90 SPTN/(MAS*uDM);D 90 S'UDM/USM;D 0 SP-A6D*VR2/cD3;jJ 90 SP:0999*VSM*(RM/57.3)2,(CCM*vR2),r SP:0.0700;D 90 SP56D*VR2/D2;D 90 SP;0.8300;D 90 SF':0,j590;D 90 SP0,7264;D 90 SPe1j24e;D o SP:O,5;D 90 SPM/(2*PInIDT*NpS*10);D 90 SP(C2D*VSM*RM/57.3)/(UsM*D1);D 90 SP0.5;D 90 $PO,98*KG*CEM/(2*PI*IDT*Np$),D 90 SPRM/(2,c*PSI);D 90 SPR1M/(MAS*UuM);D 90 SPTDN*T./(MAS*uDM);D 90 SPEO,8300;D 90 SPcO.0213;D 90 SP:c.2270;D 90 90

Figure 9-B - Potentiometer Settings NONLINEAR MANEUVERING SIMULATION

...1 1 / 1/73-1 POT SETTINGS PC 600) 0.5000 PC 601)

0.500

PC t>'3) 0.1Ui0 P( 60(4) N PC 605) PC 60b) 0.10u0 PC 607) 0.1333 PC 608) 0,2193 PC 610) 0,c286 PC 611) 0,0183 PC 617) 0,2382 P( 618) 0,1050 PC 619) 0,1231 PC 625) 0,9990 PC 626) 0.5000 PC 630) a

Ø5Ø(Ø

PC 631) 0.o093 PC 632) PC 633) PC 635) 0,571 PC 535) 0.1107 PC 537) 0.0(459 PC 538) 0,0096 PC 639)

Ø5.77

PC b40) _0,j434 PC 5(41)

0500

PC 643) u PC 6L5) 0.4192 PC 6(47) 0.5000 PC 6(48) 0,5000 PC 6(49) U _0.20(40 PC 650) 0.8627 PC 651) : _0.2365 LI P( 652) 0.4603 PC 655) PC 656) 0,9990 PC 660) ; _0,37o4 PC 651) 0.5000 PC 663) 0.5000 PC 66'4) : _0.3837 PC 665) 0,6163 PC 666) 0,0052 F PC 668) 0,9000 PC 669) 0,7851 PC 670) PC 673) 0.2000 PC 680) 0,0685

I

ii:

Figure 9-3 (Cont)

L

L

L

27-745

B-19

NONLINEAR MANEUVERING SIMULATION ,..11/1/73'1 POT SETTINGS PC 681) 0.9990 PC 683) U

0.0700

PC

b8')

U _0,2017 PC 685) 0,8300 PC 6&6) 0.1590 PC 68) U _0.7264 PC

689)

U

0,1240

PC 690)

U

0,5000

PC 691)

U _0.1158 PC 694) U

0,0275

PC 695) 0,5000 PC 698) U _0.4416 PC 700) U

0.125

PC 701)

U _0.0731 PC 705) U _0,0395 PC 71L) U

0,6300

PC 715)

U _0,0213 PC 716) U _0,2270

NONLINEAR MANEUVERING SIMULATION

. 1 1/ii73i

PROBLEM PARANETRS

PP U IIR AM

NAME _{VALUE VARIABLE}

RHO 1,99050 WATER DENSITY

L L440.0000Ø SHIP LENGTH FT K - 0,99525 L2 19360.0000 FT L3 65184ø01,kø FT3 L4

_{0,37480961E+fl}

_FT4 25,32000

EQUL

VEL, FT/S.0 P1 3,114159 G

_32,2000

_GRAVITY B _{1,000O0 BETA}

U _{15,0i0OO PROPELLER DIAM,}

KG _{14,OV000 RL.DUCTION GEAR RATIO}

MAS _{3o0000,00ØO SHIP MASS*l,0 SLUGS}

IS 2375700000 MOMENT OF INERTIA _SLLJGS-FT2

IDI 98000.00000 DRIVE TRAIN INERTI.A

LB-FTSEC

NFl, 19268014000 NF2 84779376,00 NF3 _{0.3730292550E.11} NFI4 _{0,1641328720E+14} Figure 10-3 Problem Parameters