DAVID W. TAYLOR NAVAL SHIPRESEARCH AND DEVELOPMENT CENTER

u Bethesda, Maryland 20084

PERFORMANCE PREDICTIONS FORPLANING CRAFT IN A SEAWAY

I-.0

BY

E. NADINE RUBBLE

*1 * APPROVED FOR PUBLtC RELEASE: DISTRIBUTION UNLIMITED

SHIP PERFORMANCE DEPARTMENT REPORT

I ~ SEPTEMBER_1980 DTNSRDC/SPD-0840-02

'- - -. ~.-- - ~ --. .c.ý

o 9:

MAJOR DTNSRDC ORGANIZATIONAL COMPONE TS

DTNSRDC

COMMANDER 0TECHNICAL DIRECTOR

01

OFFICER-IN-CHARGE OFIEINCAGCARDEROCK I ANNAPOLIS05 0

SYSTEMSDEV ELOPMENTDEPARTMENT 1

SHIP PERFORMANCE ____ ___[ AVIATION AND

DEPARTMENT 1j SURFACE EFFECTSis DEPARTMENT

____ ___ ___ ____ ___ .116

STRUCTURES COMPUTATION, a

DEPARTMENT MATHEMATICS AND_____ ______ 17LOGISTICS DEPARTMENT4 __ __ _17_ _ 18

PROPULSION ANDSHIP ACOUSTICSAUIARSYTM19ATMN DEPARTMENT19______________ 27

*SHIP MATERIALS CENTRALENGINEERING - - -INSTRUMENTATIONDEPARTMENT DPRMN

28I 29

, A4w"

qECU,]Yyr C1LAS5IFICA~tIQN OIF THIS PAGE~ (ghmn nalm EIntcredI

REPOT DCUMNTATON AGEREAD INSTRUCTIONSREPOT DCUMNTATON AGEBEFORE COMPLETING FORM

)~)- ~2. GOVT ACCESSION NO. 3. RECIPIE1NT'S CATALOG NUMBER

7PORMANCE ~ePNFOR PLANING CRAFT IN ATL p 'SEAWAY ~,b~ J ~IA 1

9. CONTRACT OR GRANT NIUM11ERII1()

E. NDNE/ijUBBLE q///zMESS10. PROG11RA'M EKMET PrO7 Ed, TA K

David WH`.mTayy31o~r fNaval Ship R&D Center AE OKUI UBR

Bethesda, MD 20084 ...... Work Unit 1-1500-104Work Unit 1-1524-712

Ii. CONTROLLING OFFICE NAME AND ADDRESS / i IORT DATIE

Naval Sea Systesa Command (PMS 300) _______

Washington, DC 20362 13. NUMBER OF AE

Is. MONITORING AGENCY NAME 6 AODRESS(Ifi different from Controlling offi ce) is. SECURITY CLASS. (of thtis report)

Naval Sea Systems Command Detachment Norfolk UnclassifiedU.S. Naval Station Ila. DCAS I A9ATIowN*DOOAO-N*Norfolk, VA 23511 SHDL

I6. DISTRIBUTION STATEMENT (of this0 Report)

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

* ~17. DISTRIBUTION STATFMKNT (of the abstract entered In Block 20, It different troom Report)

ISI. SUPPLEMENTARY NOTES

IS. KEY WUPDS (Continuo an roverme aide It necessary Oldiu iUntiy by block 111,1106r)[ 1' Planing Crafts Resistance, Propulsion, Propeller Cavitation, Accelerations

IC0 AUSTRPAC T (Continue an a..,.. side it necessary and Identify by block number)Ar~rocedure for the prediction of powering requirements and vertical accerationsat the preliminary design stage is presented for planing hulls with propellerson inclined shafts. Envelope, of operatl.ng speed versus wave height are devel-oped based on (1) maximum speed in seaway due to limits of the prime moversand/or propellers and (2) human endurance limits due to vertical accelerations.Complete documentation of the computer program is provided in the appendix.Sample computations and plots are presentrni for a typical planing craft.

DD ', 1473 EDITION 01 NOV65 IS OBSOLETE Ucasfe

SECURITY CLASSItFICATION OF THIS FRAaI (%In Date kntffed)

IJA

TABLE OF CONTENTSPage

LIST OF FIGURES ii

ABSTRACT

ADMINISTRATIVE INFORMATION 1

INTRODUCTION 1

PROCEDURE A6 o 2

THRUST REQUIREMENTS 2

PROPELLER CHARACTERISTICS B2

POWER REQUIREMENTS D 3

ENGINE TORQUE-RPM LIMITS3

MAXIMUM SPEED I st a 4

HABITABILITY LIMITS 4

PROPELLER SELECTION 5

COMMENTS 6

REFERENCES 10

APPENDIX A - DOCUMENTATION OF SUBPROGRAMS 11

PROGRAM PHPRILM - EXECUTIVE ROUTINE 12

SUBROUTINE PRCHAR - PROPELLER CHARACTERISTICS 28

SUBROUTINE OWKTQ - OPEN-WATER CHARACTERISTICS 30

SUBROUTINE CAVKTQ - CAVITATION CHARACTERISTICS 31

FUNCTION TQMAX - CAVITATION KT, KQ 35

SUBROUTINE PRINTP - PROPELLER PERFORMANCE INTERPOLATION 36

SUBROUTINE TPLOT - THRUST VS SPEED PLOTS 39

SUBROUTINE HPLOT - WAVE HEIGHT VS SPEED PLOTS 41

SUBROUTINE PRCOEF - PROPULSION COEFFICIENTS 43

SUBROUTINE PHRES - BARE HULL RESISTANCE, SERIES 62-65 44

SUBROUTINE SAVIT - RESISTANCE & TRIM, SAVITSKY 47

•ii i_

ft• ,l•.• :••.••,•••.•, ,:.•._ -- ,,i-.,.,•' "••i, b•,''~l i,•i'O li•~li•, .,j,•,. '

TABLE OF CONTENTS (continued)

Page

FUNCTION CIDSF - SCHOENHERR FRICTIONAL RESISTANCE 51

FUNCTION SIMUN - NUMERICAL I' EGRATION 52

SUBROUTINE DISCOT - DOUBLE INTERPOLATION 53

FUNCTION MINP 54-

SINGLE INTERPOLATION ROUTINESFUNCTION YINTE 55

FUNCTION YINTX I 56

APPENDIX B - SAMPLE INPUT AND OUTPUT 57

LIST OF FIGURES

1 - Propeller Characteristics in Open-Water and CavitatingEnvironment 7

2 - Thrust Curves for Planing Craft with Twin Propallers andDiesel Engines 8

3 - Operating Limits for Planing Craft in Seaway 9

ii

F------------------------------------- - .i

I i*-*..- kU*

NOTATIONa Average 1/10 highest vertical acceleration at 90Z LOAaBOW forward of transom

a CG Average 1/10 highest vertical acceleration at center ofgrayvity

E PX Maximum breadth over chines

D Propeller diameter

EAR Propeller expanded area ratio

F nV Speed-displacement coefficient

g Acceleration of gravity

KI/ 3 Significant wave height

J Propeller advance coefficient

KT Propeller thrust coefficient

KQ Propeller torque coefficientQ

LOA Overall length of hull

Lp Projected chine length of hull

L /VSlenderness ratio

m Gear ratio9

n Rate of revolution per second; rp.

"N Rate of revolution per minute; rpm

:.4P Brake power

PD Power delivered at propeller

" PS Shaft power

Q Torque

,wii

¼',• -.. ... .. "

NOT AT ION(continued)

Qc Propeller torque load coefficient

Ra Resistance of appendaged hull in calm water

Raw Added resistance in rough water

Rb Resistance of bare hull in calm water

R Total resistancet

T Thrust

V Ship speed

VA Speed of advance of propeller

Z Number of propeller blades

1-t Thrust deduction factor

I-w Thrust wake factor

7 Displaced volume

A Displacement

n a Appendage drag factor

n D Propulsive efficiency

no Propeller open-water efficiency

r1R Relative rotative efficiency

P Water density

a Cavitation number based on advance velocity

a0.7R Cavitation number based on resultant water velocity at0.7 radius of propeller

T Trim angle

T c Propeller thrust load coefficient

iv

w

ABSTRACT

A procedure for the prediction of poweringrequirements and vertical accelerations at thepreliminary design stage is presented for

planing hulls with propellers on inclined shafts.Envelopes of operating speed versus wave heightare developed based on (1) maximum speed inseaway due to limits of the prime movers and/orpropellers and (2) human endurance limits due tovertical accelerations. Complete documentationof the computer program is provided in theappendix. Sample computations and plots arepresented for a typical planing craft.

ADMINISTRATIVE INMORMATION

This project was authorized and partially funded by the Naval Sea

Systems Command Detachment Norfolk (NAVSEADET Norfolk) Work Request

N64281 80 WR 0 0062 under Work Unit 1-1524-712. Development of theprogram was also partially funded by the David W. Taylor Naval Ship R&D

Center (DTNSRDC) 8hip Performance and Hydromechanics Program undev.

Work Unit 1-1500-104.

INTRODUCTION

NAVSEADET Norfolk requested DTNSRDC to develop a computer routine

to predict the operational limits of planing craft in a seaway based on

state-of-the-art technology.

The computer program predicts the resistance, thrust requirements,

and vertical acceleration of a planing hull for a matrix of speeds and

significant wave heights. It also estimates the maximum thrust, as afunction of speed, which can be developed with pre-selected prime movers,reduction gears, and propellers on inclined shafts. Speed, wave-heightenvelopes are then established based on the power limits of the propulsion

system and the endurance limits of the crew due to accelerations. This

program labeled PHPRLM is completely documented in Appendix A, with

sample input and output in Appendix B. A similar program for a waterjetpropulsion system is documented in Reference 1. Both programs augment

2the planing hull feasibility model program PHFMOPT and should eventually

be incorporated into the master program.

'References are listed on page 10.

.... ..f ' '

I,/

PROCEDURETHRUST REQUIREMENTS

The calm-water, bare-hull resistance Rb is generally derived fromthe synthesized Series 62-65 resistance curves presented in Figure 9 of

Reference 2. However, if more precise data are available, e.g., modelexperiments of the exact hull or a similar form, this resistance data

can be input directly for the matrix of speeds considered. Resistance

of the appendaged hull Ra is approximated using appendage drag factors na

either from Reference 3 or direct input if other data is available;Ra - Rb/na. Added resistance in rough water Raw is predicted from anempirical equation recently developed by a regression analysis of planing

hull rough-water experimental data. 4

0.5 (/1/3)-2.5R /A - 1.3 (Hl,/EJ)B FnV(~V~3 2Raw/ 1 , H/3/ PX) FnPV V

Where Fv* V/(g Vl/)

The total resistance for the hull then is Rt - R + Raw Thrust deductiont a awfactors l-t from either Reference 3 or direct input are used to calculate

the thrust requirement T - Rt/(l-t).

PROPELLER CHARACTERISTICS

Propeller open-water characteristics are derived as a function ofpitch ratio P/D, expanded area ratio EAR, and number of blades Z with

coefficients developed from the Wageningen B-Screw Series. 5 Thrust and

torque coefficients KT and K. for flat face, segmental section propellersQ6such as the Gawn-Burrill Series tend to be slightly higher than the

B-Screw airfoil section propellers. This difference can essentially

be taken into account by varying the EAR and/or P/D used in the open-

water equations. If actual open-water data are available for theselected propeller, these data can be input to the program.

The propeller characteristics in a cavitating environment areVderived from the maximum thrust and torque load coefficients Tc and Qc

developed as a function of cavitation number at 0.7 radius O0.7R for

several propeller series in Reference 7. Options are available in the

program for tabulating ai4/or plotting (see Figure 1) the propeller 1T

2

tl-J X " '

and K. as a function of advance coefficient J for the open-water conditionas well as the transition and fully cavitating regions at several cavitation

numbers a . These "transition" curves do not exactly match the shape of

experimental cavitation data since they represent only 80 percent of the

maximum thrust and torque lines developed in Reference 7 and are not faired

into the point of KT or KQ breakdown. The 80 percent criteria is based on

full-scale trial data which indicates actual thrust and torque in the

transition region to be less than the maximums derived from the propeller

series data -- see Figures 5 and 6 of Reference 7. For Gawn-Burrill type

propellers, the 10 percent back cavitation line shown in Figure 23 ofReference 6 is used as a design criteria for adequate blade area. Thetabulated output is marked with the letter C to represent thrust which

exceed the 10 percent cavitation criteria or with a * when 80 percentof maximum thrust is attained. The thrust at which 10 percent back cavitationoccurs is represented by a dash line on the thrust-speed curves shown inFigure 2.

POWER REQUIREMENTS

After the thrust requirements are computed for the matrix of speedsand wave heights desired, J, KT, and then K are interpolated from thepropeiler characteristic curves as functions of thrust loading /2 and

Corresponding propeller rpm N, torque Q, delivored horsepower PD'

propulsive efficiency • propeller effic-iency r, etc. are then calculated.

Thrust wake factor 1-w from either Reference 3 or input are used, with therelative rotative efficiency r assumed to be one, i.e., torque wake equal

to thrust wake (torque required in open water is equal to that requiredbehind the ship).

ENGINE TORQUE-RPM LIMITSEngine characteristics are input as an array of engine rpm versus

brake power PB values covering the operational range of the engines.Gear loss and shaft loss constants, generally about 2 percent each, areused to estimate the corresponding shaft power P. and the power delivered

at the propeller PD' For a given gear ratio m the corresponding propeller

3

.............................,...,... •... . .*** .. ••.,....

rpm and torque limits are then established. If the gear ratio is not input,

the program will compute an optimum for maximum speed in a specified sea state.

m engine rpm at max4 .mum power of enginesopt propeller rpm required in specified sea, at max, engine power

The available thrust at the torque limits of the prime movers is derived

after interpolation of the propeller data to obtain J, KX, then VIT as

functions of torque loading K /J2 and a . Similarly the thrust at maximum

rpm is obtained after interpolation for KT as functions of J and a

A sample illustration of thrust requirements and thrust limits due to

maximum rpm and torque of the prime movers is shown in Figure 2 for a

typical planing hull with diesel engines. Diesel engines generally

operate with nearly constant torque whereas gas turbines have nearly

constant rpm. The upper bound of the rpm limit curve corresponds

to the "transition" region of the propeller characteristic curves and

represents the maximum thrust which can be developed by the propellers

unless operating in the supercavitating regime, at low JVs.

* IMAXIMUM SPEED

The maximum speed attainable in a given sea state is represented by

the intersection of the thrust requirement curve and the thrust limit

curve due to either engine torque or rpm restriction, whichever occurs

first. With an optimum gear ratio, maximum speed for a specified sea state

is attained at maximum power of the prime movers, i.e., intersection of

:* torque limit and rpm limit curves. The maximum speed is derived for each

wave height. Figure 3 shows a sample graph of wave height vs maximum speed.

The computations and plots can be made for I to 4 different gear ratios for

each case, one of which can be the optimum mg derived in the program.

HlABITABILITY LIMITS

Average 1/10 highest vertical accelerations at the center of gravity

aCG and at the bow aBOW (90% of the overall length forward of the

transom) are estimated for the matrix of speeds and wave heights with8empirical equations recently derived from experimental data.

4

I',

I I-I I.I.I I*I i I*I I i i i

aCG- 7.0 (H1/3/BPX) (1 + T/2)0.25 -1.25 F

0.500.75 0.7510.5 (H /Bx (1 + r/2)050 -075 F

aBOW - 1/3 PX' (/PX) nV

The accelerations at any location between the CG and the bow, such as the

helmsman's station, can then be approximated by linear interpolation.

These data are interpolated to obtain speed, wave-height envelopes for

average 1/10 highest vertical accelerations of 1.0 g, representing the

endurance limit of the crew for 4 to 8 hours, and 1.5 g, representing

the endurance limit for 1 to 2 hours. Sample speed, wave-height envelopes

are shown in Figure 3.

PROPELLER SELECTION

The program can be run with any number of propeller sets to aide

in the selection of optimum parameters. P/D, EAR, and Z must be input

for each set. Propeller diameter D may either be input or selected by

the following method. Design speed and wave height are input and the

corresponding thrust requirement is computed. A design power, not

exceeding the maximum power of the prime movers, is also input. A minimum2

diameter Dmin is computed from the thrust loading KT/J corresponding to

the 10 percent back cavitation criteria from Reference 6. A maximumdiameter D is computed from KT/j2 at peak open-water efficiency. The

maxTpower and rpm requirements at design speed are also computed for several

D's in 5-inch increments from D to D . D is selected as the optimummin max, min

diameter if the power required with Dmin does not exceed the design power.

Optimum diameter is interpolated from the 5-inch increment array of D'r

if design power falls between the power requirements for Dmin and D ax.

When design power exceeds the requirements for both Di and Dmax, Dma

is selected as the diameter except for the following case. When Din is

actually greater than Da, indicating more than 10 percent cavitation

at peak efficiency, Dmin is the diameter selected. The optimum diametar

is always rounded up to the next full inch.

q5

Jd.. . ... s

No selection process for P/D or EAR is built into this program

since many different factors may influence a particular design. However,

numerous zets of parameters can be run for each case at minimal cost

for use in the development of design charts. In the diameter selection

routine, diameters corj'espondin• to even increments of 100 rpm are

interpolated from the array of 5-inch increment ditmeters. These values

can be used for plctting contours of rpm on the propeller design charts

to aide in the selection ot a propeller to match specific engine and/or

gear ratio requirements.

COMMENTSThis programn. together with the planing hull feasibility model

program PHFMOPT, is quita useful for making timely preliminary design

studies for planing craft. It can also be used for displacement shipssince the program has options for input of the resistance and propulsion

coefficients, but thia progr&m does not take into account any difference

in thrust and torque wake.

Some means of predicting roll at the preliminary design stage isalso desirable for enhancement of th,'.s program. A low-speed roll

criteria needs to be established for completely defining the speed,

wave-height envelope.

1k

II

1*

6I

.V .t. •,•. .

/D 1 .00 EAR 0.-70 IPROF I.

6.3

25.00

00.0

IN.bK 0

L' 0 0.0.40 0.O 0'.0 eo0 I'm.'0 l'.A0 1.60

Figur I rplejhrceitc noe ae n aiaigIvrrmn

07

.. ....

C3 Q3

H1-F

ENGINE LORaT 07MI 3ROELE

13 1.

(z C3liT

LO 1O t .4 LR AI

-tP(Um

UU

413 00 6.10.0 1

ENIN 1ORQ00 000T 242b00GV KNOTS)a

0'0 ' 0 4.0 ' o W00 I a 120 0 " GV /0C

Fiue2 Trs1uvsfrPaigCat3ihTi rplesadDee nie

138

CURVE 1 -- 1.0 G- 45,0 9"T FWDO TRANSOM

CURVE 2 -- lb G -- 45,0 FT FWO TRANSOM

CURVE 3..- POWER flItT -- 2.46 GEAR RATIO

CURVE 4..- POWER LIMIT -- 2.00 OGEA RATIO4, 3.

0

9.

i! oi

:I V (KNOTS )

V M./SEC

,'Figure 3 -Opertitng Limits for Planing Craft in BSe•wy

JL .' " .. . . . ..........."..:

REFERENCES1. Hubble, E. N., "Performance Predictions for Four Proposed Designs

of a 28-ft Survey Launch," DTNSRDC Report SPD-0902-O (Jul 1979).

2. Hubble, E. N., "Program PHFMOPT, Planing Hull Feasibility Model,

User's Manual," DTNSRDC Report SPD-0840-OI (Revised Aug 1979).

3. Blount, D. L. and D. L. Fox, "Small Craft Power Predictions,"Marine Technology, Vol. 13, No. 1 (Jan 1976).

4. Hoggard, M. M., "Examining Added Drag of Planing Craft Operatingin a Seaway," Paper presented to Hampton Road Section, SNAME (Nov 1979).

5. Oosterveld, M.W.C. and P. Van Oossanen, "Further ComputerAnmalyzed Data of the Wageningen B-Screw Series," International ShipbuildingProgress Vol. 22 (Jul 1975).

6. Gawn, R.W.L. and L. C. Burrill, "Effect of Cavitation on the

Performance of a Series of 16 Inch Model Propellers, Trans. INA, Vol. 99(1I9•7).

7. Blount, D. L. and D. L. Fox, " Design Considerations for PropellersI•in a Cavitating Environment," Marine Technology (Apr 1978).

8. Hoggard, M. M. and M. P. Jones, "Examining Pitch, Heave, andAccelerations of Planing Craft Operating in a Seaway," Paper presentedat High-Speed Surface Craft Exhibition and Conference, Brighton (Jun 1980).

iii

IIq

.. 1

;• 10 //

- ,..

APPEND IX A

*DOCUMENTATION OF SUBPROGRAMS FOR PHPRILt

I...

I.I

!i -•

• . .. _ ,. .... . . .. •] . I

-'.,• . '"• "'• :•'11•I;" : -- - • .• ,,,..¢!•:;:,••, :•!:.I...,,: ',•:•. ,•,• [ •" -• •'•• , ,"" •'• '••'-" :••""#"'• • '" • " • '• :.t.• .:-; • •-

NAME: PROGRAM PHPRLM

PURPOSE: (1) Predict the resistance, thrust requirements, andvertical accelerations of a planing hull for a matrix ofspeeds and significant wave heights.(2) Determine maximum speed obtainable at each waveheight with pre-seleoted prime mover(s) and propeller(s).(3) Determine wave heights versus speed correspondingto human acceleration endurance limits of 1.0 g and 1.5 g

SUBPROGRAMS: PRCOEF, PHRES, SAVIT, PRQLAR, PRINTP, HPLOT,

TPLOT, MINP, YINTE, YINTZ, CALCOMP Routines PLOT & PLOT$

j INPUT: Via Punched Cards Card Columns

NCASES Number of cases--repeat following 1 1-4set of cards for each case

TITLE Identification for hull design 2 1-80

XLOA Overall length of hull (LOA) in ft 3 1-8

PL Projected chine length of hull (Lp) 3 9-16in ft

BPX Maximum beam over chines (B ) in ft 3 17-24PxHT Draft at transom (TA) in ft 3 25-32

DLBS Displacement at rest ( A) in lb 3 33-40

BETAM Deadrise angle at midships (0m) in deg 3 41-48

XLCG Distance of center of gravity forward 3 49-56of transom (0n) in ft

VCOG Distance of center of gravity above 3 57-64

baseline (KG) in ft

XACC Distance. forward of transom in ft 3 65-72at which accelerations are to becomputed, in addition to CG andbow locations

FTRIM Fixed trim angle ( T ) in deg 3 73-80If not input, program will useSavitsky prediction of trim ateach speed

PEMAX Maximum brake horsepower of each 4 1-8

prime mover (P )max

12

,~~~~~~~~~..a .. -.• ........ •-:•...... i' .u.. H•,+,,•: _•. ••' • "...%. . .,,t,••....,•

PROGRAM PHPRLM

Card Columns

REMAX Maximum revolutions per minute (rpm) 4 9-16

of ;rime mover No.,), corresponding

to max

GL Gear loss ratio (K) I brake 4 17-24horsepower/shaf t hrsepover

SL Shaft losa ratio (K ) - shaft 4 25-32horsepower/hp2 developed at prop

RHO Water density (p) in lb x sec2/ft 4 5 1-8

VIS5 Kinematic viscosity of water (V) 5 9-16in ft2/sac x 105

GA Acceleration of gravity (g) in ft/sec 2 5 17-24

DC? Correlation allowance resistance 5 25-32predictions, generally zero

SD? Standard deviation factor for 5 33-40

resistance data. Use zero for mean

Series 62-65 curves

NV Number of speeds -- maximum of 20 6 1-4

NWH Number of wave heights, including zero 6 5-8-- maximum of 5

NPR Number of prime movers 6 9-12number of propellers (n pr)

NE Number of points input from engine 6 13-16characteristics curve -- maximum of 10

IPROP Control for type of propellers 8 17-201 for Gawn-Burrill type

"(flat face, segmental sections)2 for Newton-Rader type3 for Wageningen B - Screw type

(airfoil sections)

IPM Control for type of prime movers 6 21-241 for diesel engines2 for gas turbines

* 13

PROGRAM PHPRLM

Card Columns

IPLOT Control for graphical output, i.e., 6 25-28CALCOMP plots0 for no plots1 for plots of thrust vs speed and

speed-wave height envelopes2 for plots of propeller characteristics

in addition to plots above

IPC Control for propulsion coefficient, 6 29-321 if thrust deduction factor (l-t),

thrust wake factor (l-w), ardappendage drag factor (na) areestimated by Subroutine PRCOEF

2 if l-t, l-w, and va are input

IRES Control for bare hull resistance data 6 33-361 if resistance is estimated by

Subroutine PHRES based onSeries 62-65 data

2 if resistance is input

IOW Control for propeller open-water 6 37-40characteristics1 if computed by Subroutine OWKTQ

based on Wageningen B-screwcoefficients

2 if points from open-water curvesare input

NJ Number of points input from each open- 6 41-44water curve, if IOW = 2. Maximum of 60.

IPRT Control for amount of printed output 6 45-48'1 0 for complete output (pages 1-10)

1 to omit engine characteristics andpropeller characteristics (pages 2,4,5)

VKT Array of ship speeds (VK) in knots, 7 1-8in ascending order. Maximum of 20. 9-16Do not include zero speed.10 speeds per card. Use 2 cards if NV>10

14

_ t (

PROGRAM PHPRLM

Card Columns

H13 Array of significant wave heights (H1 / 3 ) 8 1-8

in ft, in ascending order. Maximum of 5. 9.16First wave height must be zero.

PE Array of operating horsepower values for 9 1-8

each prime mover (Pe), in asoending 9-16order. Maximum of 10.

RE Array of rpm values for prime mover 10 1-8

(Ne), corresponding to Pe values on 9-16card 9

TDF Array of 1-t values corresponding to 11 1-8

speeds on Card(s) 7. Use 2 cards if 9-16NVA>10. Omit Card(s) 1r if IPs c 1

TWF Array of l -w values at each speed 12 A e8

resOmit Card(s) 12 in IPC i 1 9-16

ADF Array of la values at each speed 13 1-8

Omit Card(s) 13 if IPC = E 9-16

RBH Array of bare hull resistance (Rb) 14 1-8

in lb at each speed. Appxndaged 9-16resistance may be input if ca is setto 1.0. Omit Card(s) 14 if IRES a 1

JT Array of propeller advance coefficients 15 1-8

(J) in ascending order. Maximum of 60. 971610 points per oard.

Omit Card(s) 15 if fOW z 1

KTO Array of propeller thrusts coefficients 16 1-8

(KT) in open-water corresponding to 9716input J's. Omit Card(s) 16 if IOW = 1

SKQO Array of propellor torque coefficient 17 1-8

"A (KQ) in open-water corresponding to 9-16

. input J's. Omit Card(s) 17 if 1OW a 1

NGR, Number of gear ratios considered 18 1-4-- maximum of 4

NPROPS Number of sets of propellers considered 18 5-8

GR Array of gear ratios (m_) 19 1-8SIf m = 0.0 is input, pfogram 9-16

willgcompute optimum value of r.

15

OR t43

PROGRAM PHPRLM

Card ColumnsDIN Propeller diameter (D) in inches 20 1-8

PD Propeller pitch/diameter ratio (P/D) 20 9-16

EAR Propeller expanded area ratio (EAR) 20 17-24

Z Number of blades per propeller 20 25-32

PEDES Brake power of each prime mover 20 33-40used for sizing propeller

X1 Index of ship speed, from Card 7, 20 41-48for sizing propeller

XJ Index of wave height, from Card 8, 20 49-56for sizing propeller and forcalculating optimum m

If DIN > 0, do not input PEDES and XX

If DIN a 0, progvam will calculate adiameter based on above speed, waveheight, and power

Values of P/D, EAR, and Z must be input,

Note; Repeat Card 20 for each set of propellers considered,Then repeat Cards 2 through 20 for each case,

I1

16 ;

.................. ....... .......... .�t.. ...... A . .,,..

PROGRAM PHPRLM1

OUTPUT:

Page 1 -- Echo of Input Data

Page 2 -- Characteristics of Prime Movers (Not Printed if IPRT > 0)

MAX.BHP Pemax = Maximum brake horsepower of each primemover, input from Card 4

MAX.RPM N1emax 2 Maximum rpm of prime mover, from Card 4

GEAR RATIO m - rpm of prime mover/propeller rpmfrom Card 4

BHP/SHP KI = gear loss ratio, input from Card 4

SHP/DHP K2 shaft loss ratio, input from Card 4

npr number of prime movers a number ofpropellers input from Card 4

BHP PER Pr'ENGINE points from engine charaoteristic curve,

IEIEP N0 input from Cards 9 and 10

, ENGINE RPM Net

TOTAL DHP PD' total delivered horsepower at propellers

Pe npr /K1 /K2

PROP. RPM N' propeller rpm a Ne/ mg

Q-FT.LB Q' Maximum developed torque in ft-lb33000 PD' / (2 it N')

Page 3 -- Propulsion Coefficients and Resistance

LOA-FT LOA X overall ship length in ft, input from Card 3

LP-FT Lp = projected chine length in ft, from Card 3

BPX-FT Bpx 2 maximum beam over chines in ft, from Card 3

HT-FT TA = draft at transom in ft, from Card 3

DISPL-LB A = displacement at rest in lb, from Card 3

BETA-DEQ 8 m a deadrise at midships in deg, from Card 3

17

__-• , __________________ f

PROGRAM PHPRLM

LCG-FT AO CCO from transom in ft, from Card 3

VCG-FT KG CO from baseline in ft, from Card 3

XACC-FT Xaca = distance from transom in ft, at whichaccelerations are computed, from Card 3

C-LOAD CA = beam loading coefficient2 /A/( P & Bp 3 /

1/3 PXPXLP/V13 L P/V Xn slenderness ratio

LP/BPX L/B C length-beam ratio, based on projected chinelength and maximum chine beam

D-IN Din propeller diameter in inches, from Card 4

P/D P/D propeller pitch ratio, from Card 4

EAR EAR = propeller expanded area ratio, from Card 4

NPR npr a number of propellers, from Card 4

V-KT VK ship speed in knots, input from Card 7

V-LOA VK/r =speed-length ratio based on overall length

FNV FnV speed-dis laoIo nt coefficientV ,(g ,93

SIVMA a = cavitation number

1-T 1-t a thrust deduction factor

1-W 1-w = thrust wake factor = torque wake factor

"EA na = appendage drag factor

1-t, 1-W, na generated from Subroutine PRCOEF if IPC =11-t, 1-w, na input from Cards 11, 12, 13 if IPC a 2

K RB/W Rb/W a resistance-weight ratio for bare hull instill water

Rb generated from Subroutine PHRES if IRES a 1Rb input from Card 14 if IRES = 2W weight of craft = displacement at rest A

RA/W Ra/W a risistance-weight ratio for appendaged hull"in still water(Rb/W) / na

18

', P,' I" a

:, "p.qilll ll ''•-: ,

PROGRAM PHPRLM

EHPB EHPb = effective horsepower of bare hullRb V / 550,

EHPA EHPa M effective horsepower of appendaged hull"- Ra V / 550

Page 4 -- Propeller Open-Water Characteristics (not printed if IPRT >0)

IPROP Indicator for propeller type, from Card 6

1 for Gawn-Burrill type2 for Newton-Rader type

3 for Wageningen B-screw type

D-IN Din - Propeller diameter in inches, from Card 20

D-FT D M Propeller diameter in ft - Din/12

P/D P/D - Propeller pitch ratio, from Card 20

EAR E.A.R = Propeller expanded area ratio, from Card 20

BLADEC Z Number of blades per propeller, from Card 20

DEPTH-FT ho depth of center of propeller belowwaterline in ft = TA + 0.75 D

SIGMA/VSQ O/VA2 - constant for cavitation number

S (PA + pH-

PA atmospheric pressure = 2116 lb/ft 2

PV = vapor pressure = 36 lb/ft 2

PH = static pressure = p g ho

AP-SQFT Ap Projected area of propeller in sq. ft('TD 2 /4) (EAR) (1.067 - 0.229 P/D)

VA speed of advance = (l-w) VV ship speed in ft/seo

* .J J - propeller advance coefficient VA/(nD)

n = propeller revolutions per second (rps)

KT KT s propeller thrust coefficient = T/(p n2 D)

T= thrust per iropeller in 2

19

A ii4 ______________________

* -' • T '-1 l "¶¶Vi+ +

PROGRAM PHPRLM

KQ KQ propeller torque coefficient = Q/(p n2 D )

Q torque per propeller ii ft-lb

Tables of open-water KT and KQ values generated from

Subroutine OWKTQ if IOW = 1 or input from Cards 15-17 ifIOW = 2.

EP -0propeller open-water efficiency0 (J/2w) (KT/KQ)

KT/J2 K /J 2 propeller thrust loading - T/(P D2 VA2TA

KQ/J 2 KQ/J 2 W propeller torque loading - Q/(p D3 VA2 )

3 2 3KQ/J3 KQ/J3 propeller power loading - Qn/(p D VA )

TC - thrust load coefficient - T/(Ap V0 .7 p/2)

" Y Kr/[. (Ap/D 2 ) (J2 + 4.84)]

.1 2QC Q - torque load coefficient - Q/(Ap VO.7R p/ 2 )

.K0/[ (Ai/D) (20 + 4.84)]

cavitation number based on velocity at 0.7R0.7R cavitation number based on advance velocity

2 2S 2/(J + 4.84)

EPMAX n0 maximum propeller efficiencymax

Page 5 - Propeller Cavitation Characteristics (Not Printed if IPRTXO)

IPROP, D-IN, etc. - Same as Page 4

Tl, T2, Ql, etc. - Constants from Subroutine CAVKTQ

SIGMA a cavitation number 2

S (PA + pH - / (1/2 pVA)

J J - advance coefficient - VA/(nD)

KT KT thrust coefficient - T/4) . 2 D4 )

KQ KQ a torque coefficient Q/(p n2 D5 )

Tables of and K at various r's generated* from Subroutine CAVKTQ

20

' Id ,. . L . *' .;• •I'• ... - .

..... W..,,e S "VI]n N Wa

PROGRAM PHPRLMPage 6 -Propeller Sizing (Not Printed if Diameter is Input)

VA (FPS) V A speed of advance In ft/secA a V (1-w) where V is design ship speed

P/D P/D = propeller pitch ratio

EAR EAR U propeller expanded area ratio

NPR n pr number of propellers - number of prime movers

BLADES Z = number of blades per propeller

DIN D in a propeller diameter in inches

MIN. DIAM D a minimum diameter based on 10% back cavitationtnin criteria from Gawan-Burrill propeller series

MAX. DIAM D ma Lmaximum diametev based on maximum propellermax open-water efficiency

OPT. DIAM D op - optimum diameter selected

DFT D - propeller diameter in ft

SIGMA aY - cavitation number based on V A

22

JT - advance coefficient at KT/J2

KCT KT = thrust coefficient at JT

ICQ KQ W torque coef ficient atJ

EP no propeller efficiency at J T

*after n0 indicates operation at J above peak efficiency

PC TiD W propulsive coefficient

T-LB T 0 total thrust requirement in lb, at design speedand wave height

C after T iridicates more than 10% back cavitation

*after T indicates max~mum thrust due to cavitation

Q-FT.LB Q 0 total torque in ft-lb

RPM N 0 propeller rpm

DHPI P D = total power developed at propellers

21

PROGRAM PHPRLMPage 7 - Powering Requirements and Accelerations

WOA-FT, LP-FT, etc - Same as Page 3

V-KT VK : Ship speed in knots, input from Card 7

H13-FT H1 /3 signifioant wave height in ft, from Card 8

RW/W Rt/W total resistance-weight ratio in seaway

(Ra/W) * (Raw/W)

1. Raw/W - added resistance-weight ratio in waves

'1.3 (HI3 /Bpx)0'5 (LpVll/3)"2.5 FnV (Reference 4)

T total thrust requirements in lbR t/(1.t)

KT/JSQ KT/J 2 thrust loading per propellerT/(npr P VA2 D2 )

"JT JT propeller advance coefficient corresponding"to KT/J

2

KT KT propeller- thrust coefficient at JT

KQ KQ - propeller torque coefficient at JT

EP - propeller efficiency at J

0 after n indicates operation beyond peak efficiencyi.e.,high JJT, KT, KQ, n0 generated fromSubroutine PRINTP as function of KT/J 2

and a

n = propeller rps = V / (JT D)

_-LB T a total thrust in lb KTpn 2 D4 nproheck; T z RT / (-t)

4 C after T indicates more than 10 percent back cavitation

* after T indicates that thrust required exceedsmaximum thrust limit of propeller due tocavitation -- unless the propeller canoperate in the fully cavitating range, ie.,high rpm, low JT.

Q-FT.LB Q total torque in ft-lb = KQ Pn2 D5 npr

' after Q indicates that torque required exceedstorque limit of the prime movers at the

required rpm

22

J"- -"" ' : : =• "•' '••: 't ' ' ' '11 ' L: L t l • ' ''• • ''-- ''" J'" { • ' . ld " " " ± -' • . • -•tpd '• • . . ,:r , • •• j

..................... i..i', -'

PROGRAM PHPRLM

RPM N upropel~ler rpm 60 n

*after N indiuates that. rpm required exceeds rpmlimit of the prime movers

DHP PD2total horsepower developed at propellersa2rT Q n / 550

FE total effective horsepowerRT V / 550

PC T1D propulsive coef'fioient EPcheck: n D 2 71 H n R

TIH 2hull ef'ficienoy x (1-t) /(1-w)

N = relative rotative efficiency a 1.0 sincetorque wake assumed equal to thrust wake

TR~IM T trim angle in deg 'Vfix~ed trim angle, if' input from Card 3otherwise trim generated by Subroutine SAVIT

CG ACC a CG 2 averatge 1/10 highest vertical accelerationsat the center of gravity in g '3

-7.0 (H A B~) (1 + T/2) 0 2 (LE/7 1/3)-1.25 F

BOW ACC a =O average 1/10 highest vertical accelerationBOWat 90% of LO forward of transom in g's

0 10.5 (H 1/3/B pX) (1 + T/2) 0 (Lp/BpX) 07 5 F 0 7

X ACC a X N average 1/10 highest vertical accelerationsat location X acc in g's

a (+B(a aC) (X -LCG)/(0.9 L -LCG)C BO CG acc OA

Equations for aC and a from Reference 8.

232

PROGRAM PHPRLM

Page 8 - 10 Percent Back Cavitation (Gawn-Burrill Propellers)

V-KT V K ship speed in knots, from Card 7

SIGMA a cavitation number

T-LB T - total thrust in lb at which 10% backcavitation occurs

*r 0, 494 4r 0.88c O,7R

Q-FT.LB, RPM, DHP a corresponding values of torque, rpm~deliveredpower

J, KT, KQ, EP a corresponding propeller characteristics

Page 8 - Maximum Speed with Optimum Gear Ratio (Input m. - 0)

V-KT VK ship speed in knots attainable in specifiedmax sea with maximum power

H13-FT H 3 a significant wave height in ft for computingoptimum gear ratio

DHP PD a maximum power delivered at propellersmax

RPM-P N M propeller rpm required at VK and H1 /3max

OPT.GEAR m - optimum gear ratioRATIO gopt W engine rpm at maximum power

propeller rpm required at VKma and H1 / 3,max

I.

24

'A.....,.. -

PROGRAM PHPRLM

Page 9 -- Thrust at Maximum RPM of Prime Movers

GEARRATIO m - gear ratio, input from Card 19 or optimumg m computed by program

RPM Nmax u maximum rpm of propellers= Nmax /

nmax = Maximum rps of propellers * Nmax/ 6 0

V-KT VK = Ship speed in knots, input from Card 7

SIGMA a cavitation number

Jmax - propeller advance ooeffioient at max. rpm= VA / (nmax D)

KT KT x propeller thrust coefficient at Jmax

KQ KQ a propeller torque coefficient at Jmax

EP n0 2 propeller efficiency at Jmax

KT, K, no generated from SubroutinePRINTP as function of Jmax and a

T-LB Tmax = maximum thrust in lb available at Nmax= KT Pnfax D4 npr

C after T indicates more than 10% back cavitationmax

*after T indicates limit due to cavitationI. max

Q-P'T.LB =ma torque in ft-lb at Nma

*aftor indicates that torque limit of' prime movers is exceeded

DHP P power delivered at propellers - 2n Qmax nmax/55C

25

,!

PROGRAM PHPRLM

Page 9 -- Thrust at Torque Limits of Prime Movers

DHP PD' 2developed horsepower calculated f'rom inputpoints f'or engine

RPM N' = propeller rpm characteristics-- Seeoutput Page 2

Q-FT.LB Q' = developed torque Jn'f = propeller rps z N'/60

V-KT VK' ship speed in knots at whichK9 z Q'/( Pn'D5 npr) matches3 = V '/(n'D) from propeller curvesobtained by iteration and interpolation

SIGMA a' = Cavitation number at VK'

S' = propeller advance coefficient at n'= VA,/(n'D)

KT KT = propeller thrust coefficient at J'

KQ KQ 2 propeller torque coefficient at J'

EP no = propeller efficiency at J'

KT, KQ, no generated from Subroutine PRINTPas function V' and cf

T-LB T' = thrust in lb available= KTP(n')2 D4 nprQ-FT.LB Q' = torque in ft-lb check = K Q n) 2 D5 np,

DHP PD' = developed horsepower check = 27TQ' n'/550

' Page 9 -- Speed-Power Limits

H13=FT H1/ 3 = significant wave height in ft, input fromCard 8

V-KT V ship speed in knots at which thrust requiredKVmax at H1/3 (output page 6) reaches the thrust

limit of the prime movers (output page 7)due to either rpm or torque restriction,whichever is lower

26

Ai -. I. . i

PROGRAM PHPRLM

T-LB T total thrust in lb at VKmax

C after T indicates more than 10% back cavitation

* after T indicates thrust limit due to cavitation

Q-FT.LB Q - total torque in ft-lb at V

Safter Q indicates torque limit of prime movers

RPM N % propeller rpm at VKmaxI after N indicates rpm limit of prime movers

DHP ID total developed horsepower at VKmax

Page .O -- Habitability Limits

V-KT VK a ship speed in knots, Input from Card 7

IOG L.Og = average 1/10 higiiest vertical accelerationsrepresenting the endurance limit of the crewfor a maximum of 4 to 8 hours

1.5G l.5g a average 1/10 highest vertical aooelo.rationsrepresenting the endurance limit of the crewfor I to 2 hours

H13-FT HI/ 3 significant wave height in ft. correspondingto habitability limit of 1.0 g (or l.5g) --

interpolated from accelerations at locationXaoc on output Page 6

Note: acceleration predicators areconsidered not valid for H1/3 > 0.75 BpX

:1

27

IMMUNE

NAME: SUBROUTINE PRCHAR

PURPOSE: Generate propeller open-water and cavitationcharacteristics. Select suitable propellerdiameter, if not Input

CALLING SEQUENCE: CALL PRCHAR

SUBPROGRAMS CALLED: OWKTQ, CAVKTQ, PRINTP, YINT2

INPUT:

DIN Din - propeller diameter in inches, from Card 20If not input, a suitable diameter will beselected by this routine.

PD P/D propeller pitch ratio, from Card 20

EAR EAR 0 expanded area ratio, from Card 20

Z Z N number of blades, from Card 20

VADES VA design speed of advance in ft/sec derivedfrom design ship speed, indexed from Card 20

TDES T * totsl thrust in lb required at design speedand wave height, indexed from Card 20

PDES P - total design power at propellers derived fromdesign brake power input on Card 20

PRN nr - number of propellers, from Card 6

PROPELLER OPEN-WATER CHARACTERISTICS: See Subroutina OWKTQ

PROPELLER CAVITATION CHARACTERISTICS: Sea Subroutine CAVKTQ

' i"

*|.

i!I

. . .. " w.. .

SELECTION OF PROPELLER DIAMETER: (When D is not input)

DMIN Dmin minimum diameter in inches based on 10Z backcavitation criteria for adequate blade areaof Gawn-Burrill type propellers

12 T / Cn 1p VA0 (V /,J2)]

2 .94f 0,88

where KT/J2 is derived from ¶ 0.494 0,88STC 0,7Rrepresenting the Gawn-Burrill 10X cavitationline

DMAX D m maximum diameter in inches based on maximummax propeller efficiency

same equation as above, with K%/J2 at pointof maximum •0

DIAM D i array of diameters in increments of 5 inchesfrom D mn to Dmax

DHP P - array of power requirements for above diameters

RPM N array of propeller rpm requirements forabove diameters

DIV D optimum diameter selected, rounded up toopt next full inch

Dopt - Dmin if power required < design power

D is interpolated from D array if design power isopt between requirements f~r Dmin and Dmax1max

SD opt Dmax or DMin, whichever is larger, if both

power requirements exceed design power

I29

•. 29

.r g

!i-

NAE SUBROUTINE OWKTQ

PURPOSE: Calculate propeller open-water characteristics asfunction of pitch ratio, expanded area ratio, and numberof blades from coefficients derived from WageningenB-Scraw Series.

REFERENCE: Oosterveld and Van Oossanan, "Recent Development inMarine Propeller Hydrodynamics," Proceedings of theNetherlands Ship Model Basin 40th Anniversary (1972),and "Further Computer Analyzed Data of the WageningenB-Screw Series", International Shipbuilding Progress,Vol. 22 (July 1975).

CALLING SEQUENCE: CALL OWKTQ

INPUT:

PD P/D - propeller pitch/diameter ratio

EAR EAR - propeller expanded area ratio

Z Z = number of propeller blades

OUTPUT:

N nj - number of J values generated -- max of 60

JT J - array of propeller advance coefficients inascending order from (J-O.) to (J at K--O.)in increments of 0.025 if P/D_<1.2in increments of 0.05 if P/D>t.2

KT KT - array of open-water thrust coefficients"- f (P/D, EAR, Z, J)

KQ K - array of open-water torque coefficients- f (P/D, EAR, Z, J)

K- and K developed from equations in above references.For GawngBurrill type propellers (IPROP-l) the equationsare modified to produce slightly higher % aiL Kthan the Wageningen B-Screw Series. T

30

ow

NAME, SUBROUTINE CAVKTQ

PURPOSE: Calculate propeller characteristics in cavitationregime as function of pitch ratio, expanded arearatio and cavitation number. Generate CALCOMP plotsof KT and KQ versus J.

REFERENCE: Blount and Fox, "Design Considerations forPropellers in a Cavitating Environment," MarineTechnology (Apr 1978)

CALLING SEQUENCE: CALL CAVKTQ

SUBPROGRAMS CALLED:TQMAX, CALCOMP Routines

INPUT:

IPROP Control for type of propellers= 1 for Gawn-Burrill type

(flat face, segmental sections)

= 2 for Newton-Rader typesS3 for Wageningen B-Screw (airfoil sections)

PD P/D a propeller pitch/diameter ratio

EAR EAR a propeller expanded area ratio

NJ nj a number of J values input from open-watercurves -- max. of 60

JT 3 = array of propeller advance coefficients

KTO KTo a corresponding array of propelleropen-water thrust coefficients

KQO KQo = corresponding array of propelleropen-water torque coefficients

NS ns number of cavitation numbers -- max. of 8,4: -- at which propeller characteristics are

to be computed and printed from thisroutine (if ns = 0 only the constantsare computed)

SIGMA 0 array of cavitation numbers

IPLOT Control of CALCOMP plots:2 for plots of KT and KQ vs J at each a

(no plots done if IPLOT < 2)

31

".... . ..- " . . . . ., ,,

SUBROUTINE CAVKTQ

GENERAL NOTATION FOR PROPELLERS:

VA = propeller speed of advance

n = rate of revolution

D = propeller diameter

T a thrust

Q torque

P x water density

Po a pressure at center of propeller • pA÷PH-PV

J a advance coefficient VA/ (n D)

KT = thrust coefficient T / (p n2 D4 )

KQ v torque ooefficient a Q / (p n2 D5 )

KT/J 2 = thrust loading = T / (pD 2VA2 )

KQ/J 2 = torque loading = Q/(p D3 VA2 )

KQ/J 3 = power loading v Q n/ ( pD2 VA3)

M cavitation number based on advance velocity" Po / (1/2 PVA 2)

VO.7R 2 velocity at 0.7 radius of propeller=VA2 + (0.7r nD)2 = VA 2(J2*4. 8J),/,j2

00.7R cavitation number based on VO.7RPo/(lt2PVO.7R2 ) =aJ 2 /(J 2 +4.84)

Ap projected area of propeller

(rrD 2 /4) EAR (1.067-0.229 P/D)

TO thrust load coefficient

* T / (1/2 p Ap VO.7R 2)

KT / 1l/2 (Ap/D 2 ) (j2+4.84)j

Q0 torque load coefficient

= Q ,'(1/2 P Ap VO.72R 2)

KQ / [1/2 (Ap/D 2 ) (J2+..84)]

32

."I

SUBROUTINE CAVKTQ

MAXIMUM THRUST AND TORQUE LOADS:

Blount and Fox (see reference) give equations formaximum thrust and torque load ooeffioients in acavitating environment based on regression ofexperimental data for the three propeller series usedherein.

TOm z maximum thrust load coefficient= a ao.7 b (transition region)= Tox(fUlly oavitating region)

QOm = maximum torque load coeffiiientM 0 O. 7 Rd (transition region)a Qox (fully cavitating region)

OUTPUT: IPROPTi a = 1.2 1

a = 0.703 + 0.25 P/D 2a = 1.27 3

T2 b = 1.0 1b = 0.65 + 0.1 P/D 2b = 1.0 3

Q1 0 0.200 P/D 1

0 O.240 P/D - 0.12 2o 0.247 P/D - 0.0167 3

Q2 d = 0.70 + 0.31 EAR0 ' 9 1d a 0.50 + 0.165 P/D Pd 1.04 3

TCX TOx 0.0725 P/D - 0.0340 EAR 1TOx= 0,0833 P/D - 0.0142 EAR 2Tax = O. 0 3

rQCX Qox = [0.0185 (P/D)2 - 0.0166 P/D + O.005941S~/EARII3

x 0.0335 P/D - 0.094 EAR 1/ 2 2Qcx a 0.0 3

SRMAX k =0.8Sinus full-scale trial data (see Figures 5 and 6 of

reference) indicates actual thrust and torque in thetransition region less than the maximums derived fromthe propeller series data, the factor k is applied to

TO and O,, in the transition region. The factork is not applied to To, and Q.

33

k% I'

• wq-

SUBROUTINE CAVKTQ

APD2 Ap/D 2/2 = Constant for calculation of TO and Q.

j J advance coefficient from input array

OPEN WATER K KTo= input values of open-waterKT KQ - 0Qo thrust and torque coefficients

SIGMA a cavitation number from input array

KT KT thrust coefficient as f (J, a)KTo or KTm, whichever is smaller

KTm TOm (1/2 Ap/D 2 ) (j 2 +4.8'1)

"Tom - (k a CO. 7 Rb) or (Tox),whinhever is greater

"LC 1. rtharacter Identifler for propelleccavitationC indicates more than 10% back cavitationfor Gawn props: >0.494 • O.7 RO8 8

* indicates thrust limit due to cavitationKT " KT

JT T

KQ KQ = torque coefficient as f (J, a )= KQo or KQm, whichever is smaller

= QQm (1/2 Ap/D 2 ) (j 2 + 4.84)

i. Qcm =(k o cO. 7 Rd) or (QOx),whichever is greater

KTm and KQ, generated by Function TQMAX

•,.. , •CALCOMP PLOTS: If IPLOT=2, open-water KT and KQ as well as KT andKQ representing the transition region and fully

"CO O cavitating region at ca:ho are plotted as a function

of J.

. 34

NAME: FUNCTION TQMAX

PURPOSE: Calculate maximum thrust or torque ooefficient in acavitating environment as function of cavitationnumber and advance coefficient

CALLING SEQUENCE: X TQMAX (SIGMA, JT, ITQ)

INPUT:

SIGMA U = cavitation number

JT J = advance coefficient

ITQ i = 1 if maximum thrust coefficient requiredi = 2 if maximum torque oooffioient required

Variables: a, b, c, d, T•c, QCx, k, 1/2 Ap/D 2

generated by Subroutine CAVKTQ

OUTPUT:

TQMAX K or K depo.nding on value of iTm QmTcn = maximum thrust load ooefficient

Sk a a b. or TC if greater

KT = cm (1/2 Ap/D 2 ) (J 2 +4.84)I ~MQCM = maximum tyrque load coefficient

zfi k c 0 .7R , or Qc if greater

It KQcm (1/2 Ap/D 2 ) (JP,4.84)

QM

i,

35

• 'W7*l4 *W

NAME: SUBROUTINE PRINTP

PURPOSE: Interpolate for propeller performance at specifiedvalue of (1) advance coefficient J, (2) thrustloading KT/J 2 , (3) tor uk loading, KQ/J 2, or(4) power loading KQ/J5.

CALLING SEQUENCE: CALL PRINTP (IP, PCOEF, SIGMA)

SUBPROGRAMS: TOMAX, YINTE

INPUT:IP Option 1, 2, 3, or 4

PCOEF input propeller coefficient,dependent on value of IP

JT = advance coefficient, input if IPuIKT/J2 a thrust loading, input if IP=2KQ/J 2 V torque loading, input if IPz3KQ/j 3 = power loading, input if IP:4

SIGMA a cavitation number

NJ nj number of J values defining propellercharacteristics

JT J array of advance ooeffioient, inascending order

KT KTo array of open-water thurstcoefficients

KQ KQo array of open-water torque0coefficients

PERFORMANCE AT SPECIFIC J:

JTP JT input advance coefficient

KTP KT thrust coefficient at JT= opeh-wator thrust coefficient,

interpolated from input array ofKT versus J, or maximum thrustooifficient in cavitating regimeKT calculated by FunctionPrqAX, whichever is smaller.

KQP KQ torque coefficient at JTopen-water value interpolated fromKQ vs J, or maximum cavitationva~ue KQ calculated from TQMAX,whicheve" is smaller

36

I' "11

V ...... ... ....

I.'.

, SUBROUTINE PRINTPIIPERFORMANCE AT SPECIFIC LOADING:

PLOD ln(KT/j 2 ) if IP=2 natural log ofln(KQ/j 2 ) if IP=3 input loadingIn(Kq/j3) if IP=4 coefficient

XLOG ln(KT /J 2 ) if IP=2 array of natural logs

ln(K /Jo2 if IP=3 of open-water leading3n(N/J it IP=4 ooeffiocent at J value

from input array

JTP JT X open-water advance coefficient interpolatedfrom array of open-water loadingcoefficients versus J at the specificloading required (logs are used because ofthe rapid change of loading coefficient atlow J's)

If JTo is in non-oavitating region (KTo< KTm)

KTP KT thrust and torque coefficients at JTo

KQP KQI interplated from arrays of KTo and KQo vs J

If JTo is in cavitating region (KTo > KTm)

XLOG ln(KT /j 2 ) if IPm2 array of natural logsln(KQm /j2) if IP=3 of loading coefficientsln(KQm /J3) if IP=4 based on KT or

Kom as funoTion J

JTP JT = advance coefficient interpolated from arraym of' cavitation loading coefficients vs J at

the specific loading required

KTP KT) maximum cavitation thrust and torqueKQP KQ coefficients at JTm calculated from TQMAX

OUTPUT:

JTP JT final advance coefficient

KTP KT final thrust coefficient at propellerperformance

KQP KQ final torque coefficient pointspecified by

EP n0 = propeller efficiencý PCOEF and2 JT KT/(2 Tr KQ) SIGMA

37

I I I I

SUBROUTINE PRIMTP

TALIC T thrust load coefficient

M KT (½A./D2 ) (J +44 84)]

S107 a 0.R - cavitation number based on velocityat 0.7 radius of propeller

- ~ ( ~Z +4.84) 4.84-(0.77T) 2

XSIG7 4 94 c 0.88 - term representing 10% back cavitation line for

M.R Gawn-Burrill propeller series

LT - 1 character identifier for propeller cavitation

*indicates thrust limit due to cavitation:KT -KT

m

C indicates more than 10% back cavitation forGawn-Burrill propellers, but less thanthrust limit cavitation

> 0494 0.88T c > o 494 aO 7R

38

Lu.IA-W- = w M

NAME: SUBROUTINE TPLOT

PURPOSE: Generate CALCOMP plots of thrust versus speed for(1) thrust required in various sea states(2) thrust limit at maximum rpm of prime movers(3) thrust at torque limit of prime movers

with one or more gear ratios(4) thrust at 10% back cavitation, if IPROP 1

CALLING SEQUENCEs CALL TPLOT

SUBPROGRAMS CALIOMP Routines, FACTOR, PLOT, AXIS) SYMBOL,NUMBER, FLINE, DASHL

INPUT:i

TITLE - identification to be printed at top ofgraph 80 characters maximum

VKT VK - array of ship speeds in knots

E13 H 1/3 a array of significant wave heights in ft.

NV n - number of speeds -- maximum of 20

SNWH n. - number of wave heights -- maximum of 5

NE ne - number of points defining torque limits of[ prime movers -- maximum of 10

NGOR ng - number of Sear ratios -- maximum of 4T T = matrix of values for thrust required as

* function of VK and H1 / 3

TC Tc - array of thrust values representing 10%

back cavitaiton, if IPROP a 1

TN TN w array of thrust values at maximum rpm ofN •prime movers

VQ V U array of speeds in knots at which torqueQ limits are defined

TQ T - array of thrust values corresponding toQ torque limits of prime movers

GR m - array of gear ratios

I,t

39

miTi

OUTPUT: SUBROUTINE TPLOT

X-axes Ship speed V in knotsShip speed V In m/sec

Y-axes Thrust in lbThrust in N

Curves (1) Thrust required at each wave height H1/3

(2) Thrust at engine rpm limit, for each gear ratio

(3) Thrust at engine torque limit, for each gear ratio

(4) Thrust representing 10% back cavitation (dash line)

Intersection of Curve (1) and the lower of curves (2)and (3) represents the maximum speed obtainable at eachwave height and gear ratio.

't.

40

min,

NAME SUBROUTINE HPLOT

PURPOSE: Generate CALCO0P plots of significant wave height versuaspeed corresponding to(1) endurance limit of crew for 4 to 8 hours operation(2) endurance limit of crew for 1 to 2 hours operation(3) power limit of propulsion system for 1 or more

gear ratios

CALLING SEQUENCE: CALL HPLOT

SUBPROGRAMS: CALCOMP Rout ins FACTOR, PLOT, AXIS, SYMBOL, NUMBER, FLINE

INPUT:

TITLE a identification to be printed at top of graphmaximum of 80 characters

VKT VK = array of ship speeds in knots

H13 HI/ 3 a array of significant wave heights in ft

NV nV - number of speeds -- maximum of 20

NWH nf - number of wave heights -- maximum of 5

NGR nu - number of gear ratios -- maximum of 4

HIOG H - array of significant wave heights as functionof V corresponding to average 1/10 highestvertical accelerations of 1.Og

H150 Hi~b - array of signifoant wave heights as functionl.g of Vx c orrsodlnsto verticalacceleration of 1.5g

S/ VKTW V- array of maximum ship speeds in knots asmax function of H derived from intersection

of required t ust at each wave height withthrust curve at engine rpm or torque limit,whichever is lower

XACC Xace distance forward of transom in ft at whichaccelerations have been computed

GR m a array of gear ratios

441

7141

/

* p * '

IOUTPUT: SUBROUTINE HPLOT

X-axes Ship speed V in knotsShip speed V in r/sec

Y-axes Significant wave height H 1 3 in ftSignificant wave height H i m "

Curves Wave height envelopes representing average I1/10 heighest vertical accelerations of

(1) 1.0 g -- habitability limit for 4 to 8 hours(2) 1.5 g "" habitability limit for 1 to 2 hours

(3) ) Envelopes of maxi.mum speed versus wave height(4) due to power limit of the propulsion system,etc.) for 1 to 4 gear ratios

34.1

:I

42

:Xi~.... ......... g::,,

NAME: SUBROUTINE PRCOEF

PURPOSE: Estimate propulsion coefficients for planing hullwith propellers on inclined shafts

REFERENCE: Blount, D.L. and D.L. Fox, "Small Craft PowerPredictions," Western Gulf Section of the Society ofNaval Arohitects and Marine Engineers (Feb 1975)

CALLING SEQUENCE: CALL PRCOEF (FNV, TDF, ADF, TWF)

SUBPROGRAMS: MINP, YINTE

INPUT:

FNV FnV z speed-disilacement ocefficient,V(g V 1 )1/2

OUTPUT:

TDF 1-t a thrust deduotion factor

""total horizontal resistanoe (RTJi total shaft-line thrust (T)

ADF a appenIdage drag factor

a resistance of bare hull (Rb)

resistance of appendaged hull (R.)

TWF 1-w u thrust wake factor = torque wake factor

PROCEDURE: 1-t, 1-w, and na interpolated from following table ofvalues at input value of Fnv . The tabulated datarepresent mean values from a bandwidth of data colleotedfor numerous twin-screw planing craft and reported inabove reference.

FnV 1-t 1-w ta

0.5 0.92 1.05 0.9511,0 0.92 106 0.9481.5 0.92 1.04 0.9422.0 0.92 0.99 0.9342.5 0.92 0.97 0.9253.0 0.92 0.975 0.9133.5 0.92 0.98 0.9004.0 0.92 0.975 0.885

43

I I,4*1 III.,~.d~flW., ,.Ae. . ~' .W WII, I

NAME: SUBROUTINE PHRES

PURPOSE: Estimate the bare-hull, smooth-water resistance of ahard-chine planing hull from synthesis of Series 62and 65 experimental data

REFERENCE: Hubble, E.N., "Resistanoe of Hard-Chine, SteplessPlaning Craft with Systematic Variation of Hull Form,Longitudinal Center of Gravity, and Loading," DTNSRDCReport 4307 (Apr 1974).Figures 9 and 10 of DTNSRDC Report SPD-0840-O0 (Dec1978, Revised Aug 1979)

CALLING SEQUENCE: CALL PHRES (DLBS, FNV, SLR, DCF, SDF, RH02, VIS, RLBS)

SUBPROGRAMS: DISCOT, YINTX, C1DSF

INPUT:

DLBS A z ship displacement in lb

FNV FnV a speed-displaoement coefficient (V/(g/VI/3)I/2

SLR Lp/ V1/3 a slenderness ratio

DCF 01oorrelation allowance; may be 0.0

SDF SDF a 0.0 corresponds to mean resistance-woight R/Wcurves derived from Series 62 and 65 data1.645 corresponds to minimum R/W curves

can be varied to approximate R/W for apartioular hull form from model experiments

RH02 0/2 .05 x water density in lb x seo2/ft 4

VIS V a water viscosity in ft 2 /seo

OUTPUT:RiU S R; a bare-hull, smooth-water resistanuo in lb

a (mean R/W - 3DF x c) x AH ci a standard deviation of Series 62-65 data from

mean R/W

PROCEDURE:XFNV array Tabulated values of Fn, from 0.0 to 4.0

ZSLR array Tabulated values of Lp/Vl/ 3 from 4.o

44

, f•', • ,. . .. ,• 0V~.h j

SUBROUTINE PHRES

YRWM matrix Tabulated values of mean R/W as f(FnV , Lp/VI/ 3 )for 100,000-lb planing craft derived from Series 62 and65 experimental data. See Table 1.

YWSR matrix Tabulated values of mean wetted area coefficients SiVfrom Series 62 and 65 hulls. See Table 2.

SD array Tabulated values of standard deviationa as f(FnV)See Table 1.

RWM R/W for 100,000-lb planing craft interpolated from YRWMmatrix of mean R/W values at input FnV and Lp/V' 1 / 3

WSR S/V 2/3 interpolated from YWSR matrix at input FnVand Lp/V 1 /3

Subroutine DISCOT used for the double interpolation

SDM a interpolated from SD array at input Fn7

Function YINTX used for single interpolation

RWM (RIW)m corrected R/W for 100,000-lb planing craft(mean R/W interpolated) - (SDF x a)

DLBM Am C displacement of 100,000-lb planing craft

XL - linear ratio of actual ship to 100,000-lb2 (A/Am)1/ 3

VFPSM Vm = speed of 100,000-lb craft in ft/seo

X (input Fnv) x 19.32

VFPSS Vs speed of actual ship in ft/seo Vm X1/2

PLM Lm = length of 100,000-lb qraft in ft= 11.6014 (input Lp/ VI/3)

PLS Ls - length of actual ship in ft n Lm

REM Rnm = Reynolds number of 100,000-lb craft= Vm Lm/ Vm

RES Hns Reynolds number of actual ship = V- Le/vs

CFM CFm * Sohoenherr frictional resistance coefficientfor 100,000-lb craft

45

a-. wl

SUBROUTINE PHE~S

CFS CF3 Sohoenherr frictional resistance coefficientfor actual shipFunction C1DSF used to obtain Sahoenherrfrictional resistance coefficients

SM sin wetted area of 100,000-lb craft in ft2

-134.5925 S1 V2 /3

SS Ss wetted area of actual ship in ft2 mS, X2

RM Rm a resistance of 100,000-lb craft in lb- (R/W)M am

CM CT M total resistance coefficient of 100,000-lbCT, craft

CR CR m / (Vm 2Sm pm/2)

CH CR aresidual resistance coefficientCT,- Cp.,

CTS CTs a total resistance coefficient of actual ship= Cys +' CR + ACF

VIS Vs =kinematic viscosity for actual ship

VISM V, a kinemiatic viscosity for tabulated data1.2817 x 10-5

RHH2 1052 =1/2 water density for actual ship

RH02M em/2 U 1/2 water density for tabulated dataa 1.9905/2

RLBS Rb Cresistance of actual ship in lb

2CT5 V3 2 S. Ps/2

46

NAME: SUBROUTINE SAVIT

PURPOSE: Estimate the bare-hull, smooth-water resistance andtrim for a hard-chine planing hull using, Savitsky'sequations for prismatic planing surfaces

CALLING SEQUENCE: CALL SAVIT (DISPL, LCG, yOG, VFPS, BEAM, BETA, TANB,COSB, SINB, HW, WDCST, RHO, VISO AG,DELCF, R, TD, NT, CLM, 0DB)

SUBPROGRAM: CIDSF

INPUT:

DISPL A=ship displacement in lb

LCOGr distance of center of gravity CG forwardof transom in ft

VCG KG =distance of CG above baseline in ft

VFPS V zspeed' in ft/sec

BEAM B x beam inft= maximum chine beam BpX

BETA 8 z deadrise angle in dogree3=de'adriss at midships $m

TANB tan B

COSB 00s8

SINB 33.n B

HW MW =hoight of center of wind drag abovebaseline in ft

WDCST CDWI horizontal wind force in lb /V2

RHO P =water density in lb x seo2/ft4

UIS v= kinematic viscosity of' water in ft2/seo

AG g acceleration of gravity in rt/3ec2

DELCF ACF ucorrelation allowance; may be 0

OUTPU7:R Rb bare hull, smooth-water resistance in lb

F 47

SUBROUTINE SAVIT

TD T = trim angle in degrees

NT Number of iterations to obtain trim angle

CLM A - mean wetted langth-beam ratio Lm/B

GDB - longitudinal center of pressure, distancetforwa,'d of transom, in ft

PROCEDURE:TD trim angle of planing surface from

horizontal in degfirst approximation of: LI It deg

CV CV Z speed coefficient = V/(gB) 1 /2

CLM = mean wetted length-beam ratio= Lm/B = (LK + LC)/2B

CLO CLo = lift coefficient for flat 3urfaee(0.012 X + 0.0055 XS/ 2 /Cv2)

CLB CL= lift coefficient for deadrise surfaceL 6 A/[V2 B2 p/2]= CL.-0.00 6 5 CL0 0.6

CL, and obtained by Newton-Raphson iteration

first approximations: CLo = 0.085; X= 1.5

XK 11 wetted keel length in ft[ 8(X+ tan $/2 IT tant

XC LC wetted chine length in ft : 2BX -LK

LK- LC = (Btans )/ ( TtanT

GDB AP longitudinal center of pressure forward oftransom in ft

-[] B ?[0.75 - 11(5.21 CV2 / ý2S+ 2.39)]

CLD CLd dynamic compnoent of lift coefficient- 0.012 X1/ 2 '1i.1

VM Vm mean velocity over planning surface in ft/sec

- V [1-(CLd - 0.0065 ý CLd 6)/(X cosTJl1/2

BE R : Reynolds number for planing surface- VmB A /v

48

.- .

4: , '- .

SUBROUTINE SAVIT

CF C + AC Schoenherr frictional rusiicgnoe coeffoientas f(Rn) plus correction allowarce

DFX DF viscous force due to wetted surface,parallel to the planing surface, in lb

- (CF+ACF) (0/2 (Vm 2) (B2 Xl/oso )

CCK 1.5708 (1 - 0.1788 tan2 0oos0 -0.09646tanp sin28)

CKi CKI CK tan T/sin$

Al al " [in 2 T(i-2CK) + CK-ts2t(i/sln28"ein2T)Jl/2AN al IS +(aliCK1 )/(aii I

cos T + Cy tan T sin T

TAN tan (4I÷CKI)/(1-al CK1)

THETA B angle between outer sp!,:,y edge and keel inradians

- arotan(tano oos$ )

DLM A,- effective increase in length-beam ratio dueto spray[ (tan$ (TrtanT)j- 1/( 2 tane ) /(2 oose )

Rn Reynolds number of sprayS - VB /(3cosB sill 6 ) /V

CF CF- Sohoenherr frictional resistance coefficientfor spray drag

DSX D= viscous force due to spray drag, parallel tothe planing surface, in lb

-CS (012) V2) (B2AX / cosB )

DWX Dw component of wind drag parallel to planingsurface in lb4 - CDI,' V2/oosT

DTX DT total drag force pa"allel to planing surfacein lb

= DF + DS + DW

PDBX PT 2 total pressure force perpendicular tosurface in 3b

= A/costý + DT tanT

49

SUBROUTINE SAVIT

EDB ep • moment arm from oenter of pressure to CO in ft

FF fF moment arm from center of viscous force toCa in ft ,

S- (B tan 0/4)

FW fw moment: arm from center of wind drag to COin ft

K - HW

RMT EtM a sum of momenta about CO In ft-lb

P PT OP + (DF+DS) fF + DWfW

Iterate with small ohanges in T until EM O.0016

NT Number of Iterations required to obtain equilibriumtrim; maximum of 15 itrrations

R R 2 net horizontal resistance force in lb

a DT 003T + PT sin*

50

0r.

I:

NAME: FUNCTION C1DSF

PURPOSE: Calculate Sohoenherr friotional resistance ocefficient

CALLING SEQUENCE: CF CIDSF (XN1RE)

INPUT:

XN1RE Rn Reynolds number = V L /v

* I OUTPUT:

I C1DSF CF Sohoenherr frictional resistance coeffioient

PROCEDURE: Iteration with Newton-Raphson methodSohoenherr formula: 0.242 /- ig1091 Rn CF

51

.0. S-..,~~~. . . . . . . . . .~~~. .... :. - -t.'... --- m nM.i ~ E~~4.',. m U i .

,,1• i• ,.,,:.

NAME: FUNCTION SIMPUN

PURPOSE: Numerical integration of area under ourve defined by --

set of (x,y) points at either equal or unequalintervals

CALLING SEQUENCE: AREA SIMPUN (X', Y, N)

INPUT:

X array Table of x values-- independent variable -

x values must be in ascending order

Y array Table of y values--dependent variable1

N Number of (x,y) %alnes

OUTPUT:

SIMPUN Area under nurve f fy e'x

52

NAME: SUBROUTINE DISCOT

PURPOSE: Single or double interpolation for continuous ordiscontinuous function using Lagrange's formula

CALLING SEQUENCE: CALL DISCOT (XA, ZA, TABX, TABY, TABZ, NC, NY, NZ,ANS)

SUBPROGRAMS CALLED: UNS, DISSER, LAGRANThese subroutines are concerned with the interpo-lation, and are not documented separately

INPUT:

XA x value (first independent variable) for interpolatedpoint

ZA z value (second independent variable) for interpolatedpointSome as x value for single-line function interpolation

TABX array Table of x values--first independent variableTABY array Table of y values--dependent variableTABZ array Table of z values--second independent variable

NC Three digit control integer with + signUse + sign if NX - NY/NZ w points in X array

Use - sign if NX - NYUse 1 In hundreds position for no extrapolation

above maximum ZUse 0 In hundreds position for extrapolation

above maximum ZUse 1-7 in tens position for degree of interpolation

desired in X direction

Use 1-7 in units position for degree of interpolationdesired in Z direction

NY Number of points in y array

NZ Number of points in z array

OUTPUT:

ANS y value (dependent variable) interpolated at x, zDISCOT is a "standard" routine used at DTNSRDC.Consult User Services Branch of the Computation,Mathematics and Logistics Department for additionalinformation.

II5

•! 53

NAME: FUNCTION MINP

PURPOSE: Select index of minimum x value to be used forLagrange interpolation, from an array of x valuesgreater than required

CALLING SEQUENCE: I -MINP (M, N, XA, X)

INPUT:

M m - number of points required for interpolation ofdegree m-l

N n - total number of points in x array > m

XA x value to be used for interpolation

X array Table of x values, must be in ascending order, but

need not be equally spaced

OUTPUT:

MINP Index of minimum x value from the array to be used byFUNCTION YINTE for Lagrange interpolation ofdegree m-I

SAMPLE PROGRAM USING FUNCTIONS MINP AND YINTE:

DIMENSION X(1O), Y(10)

N -10M -4

READ (5, 10) (X(J), J-1, N), (Y(J), Jul, N), XA

"I w MINP (M, N, XA, X)

fA - YINTE (XA, X(I), Y(I), M)

ALTERNATE PROGRAM USING FUNCTION YINTX:

DIMENSION X(1O), Y(1O)

N -10

M -4

READ (5, 10) (X(J), Ju1, N), (Y(J), J-1, N), XA

YA - YINTX (XA, X, Y, M, N)

The result from either program is the same. In either case, only the

M points closest to XA are considered in the interpolation formula. The

first combination should be used whenever several dependent variables are

to be interpolated at some value of the independent variable, since MINP

need only be called once. FUNCTION YINTE may be used alone whenever

N=M.

54.1

,'

.,a.• . ,, . .. .. . .- .,i''",- -. .;

NAME: FUNCTION YINTE

PURPOSE: Single interpolation of degree n-i for functionrepresented by n (x,y) points using Lagrange'sformula

CALLING SEQUENCE: YA - YINTE (XA, X, Y, N)

INPUT:

XA x value (independent variable) for interpolated point

X array Table of x values--independent variable

x values can be in either ascending or descendingorder and do not need to be equally spaced

Y array Table of y values--dependent variable

N n * number of (x,y) values defining the function

OUTPUT:

YINTE Interpolated y value (dependent variable) derivedfrom Lagrange formula of degree n-I

For example, when n - 4, cubic interpolation isperformed

Lagrange's Interpolation Formula

(x-x 1 ) (x-x 2 ) . . . (x-xn)Y" (x _Xo-1) (xo-X 2 (x 0, (X-n) O

(x-x0 (x-x) , ., (x-x)d

+ (-XO ), (X-Xn) Y2l

(x-xO (x-xI) (x-x ) . . . (x-xn

+ (x 2 -x 0 ) (x 2 -xI) (x2 -x 3 ) , . . (x 2 -xn) +2

(x-xO) (x-x 1 ) (x-x 2 ) . . (x-xn

S 0 nl) 1 n2 n-l

K (

55

i! sv/ !t

NAME: FUNCTION YINTX

PURPOSE: Single interpolatiou of degree m-l for functionrepresented by n (x,y) points using Lagrange'sformula. If n > m, only the m closest points areconsidered in the interpolation formula

CALLING SEQUENCE; YA a YINTX (XA, X, Y, M, N)

INPUT:

XA x value (independent variable) for interpolated point

X array Table of x values--independent variable x values mustbe in ascending order, but need not be equally spaced

Y array Table of y values--dependent variable

M m w number of (x,y) values considered for theinterpolation process of degree m-3

N n w total number of (x,y) values > m

OUTPUT:

YINTX Interpolated y value (dependent variable) derivedfrom Lagrange formula of degree m-1FUNCTION YINTX may be used instead of FUNCTION MINPand FUNCTION YINTE together

See Sample Programs using these three functions

56

,7 7,. iI.I I.

.I. .I

I. ."

APPENDIX B

SAMPLE INPUT AN~D OUTPUT FOR PHPRIM

57

W, nn

to.

dc z

L; 0 , 0 6 0' 0' 0 A

z I

ul e N 9' 0* *ccC OC

14 (z 1S *LAn 0 4 ; 00 Co C eea

- -j

N orIfl 4NC.- CCC S 0 0 -00P:

LA % 00 x e o"IC 0. 6 *

0 ~ ~ ~ ~ .j OM c k zJ- N N mi

4h X CLUCLAJU 0 -a. ii 4ca.9 0 JJ 1 1 A .LiC

*tj- L2 g 0 f 0~~- 0 ~~0..kou .0 - e'000 50 0

- w' m -4 4a*

z a a C 0 CL CL 30 30 1. 00 v. P-4.~0.. t~

t 58

0V 0 a

N $ 0N 0 0

o 0 N0 0 0

0~ a;NA Nn (P.

N i

og 0

ma

z r4

o 0 0

o (q 0

ILA

wN

o -0

0 0 0

N 0 N 0

0 In1- o a 3F0 0 0 0 0 0

u O to to 0Ato 0 N -n

to in-a - -

N z~c

- 0 0 0 0 0 0 59

..C4*p 4

UI a4 01 Ci* 01

* *, -

w ACL i

*0 114

0 - I , V

V9 m 4 L - mI* . n L

*U #

m U!

14 0 nI0 0

* 0IL 0

he* -41M 0 0 0

w.uii

a 5 o i n p Q k

Ix4

IA ID CL 0 0 0 0

zn cc I -T M N w 0 (4m IA flm-.LrI4lC4lhf 4flw4

(LI U

I Ix

w* 4I 0 0 C. 0 0 0 0 0 0 0

N 0.

(q it 0OT UN In tonz - N-a (0 I. I IA

* r -60

.. .L .. . .. .

w 40

CL C;06 (q C S

* o 0

3 0' N m N -o c t- N to 0 (14 t-t a. w 0- fS N ; CD WN v N CP to m'

-C -39 Cl CD t- to to 0 - O 0 0) 0 too t, " 0 t - t ) m0 W Ch

ft

m*0 0 t- M M t 0 t- 0V V 0 W' ot

* ~~ ~ .. . . .o . . . C. .- .o .o . .

C4 0. 0 N Cl 0 ty V) mD I'- 0 0 0 M)

> - - - - -.

3lt

LL 3L 0 0 0l 0 0 0 0) 0 a* N a0 Cl 0 -Nf ri) 4 C4 U) 0 N 0) 0) 0 N CN CN N C0) Im M 07 m 0 0 a) 0 al 0 0l

uJ . . . .U . . .

Wf 1. 0

0. c0

Z. In~ 0 0 N It to- 01 OF 0 tNN N C4* C, )

L 00

41 t- 0 N' q1- U') Nl0 C q -- m 0LA C.. 4 ) U 4 4 4 q 4 Cl C4 C4

wf UU' W 0)a )a a) a o) o) o) o) o o o o

0. 44 N C4 N"1

OfO

a.0

C) to I L) U o U) U 7 N 0 a 0 C. C. C.C- 0 0 ) 0 ) 0 ) 0

C 61

r.I0000000O c.)0OOOOOOOOO0 OO0 OOOO0oO.0 -------

og ~000M0R0 IT0 N0 000000 0`1 t- 000 00 in-t0MMMM00 1 0 00 000 000 1 L f 00 MM00

aL U 0 O0 M OM 00 t-t~- E n'r m ,) N N ----- - 0l' iI ,t n 1MNL 1i 04 -- 0 01C' M tO~ - ED U) C(1) & O-O alM - to

--- --- -- 00000.0 . . . . . . . . . . . . . .

to 0 M - --0l W V C4 4 M-

100 04 Clr'-0 h fl- Mr V

into0

c. .

W0 M -

0 .

CL -W ~ - w - - N m- i o o O O t

WO W

o EL 0 00 000000-00--0NN00"M0 MA m;,q00 irti0inin n G00 o D00 E E000toU:

UICL(A u.

W 0 00 0 00 0 00 00 0 00 0 00 00 0 0 -0000

cJ w

-j zO 3

a. I z Za0 00 111

62

is 0 " Da n Ga aa Ta00 ~~ )0000 ---------------- W-- - W 0 0 CU (0---*•oooo ooo oo o o oo00 oLoooo rNlo -

00•3000000000000000000000000000000000"•000

•oU U

0 m ) v v v v 0 v 0 0n 0 0n 0n 0n 0n in 0n 0 0 0 0- 0 0- 0- 0 0W 0 0n 0 0D 0 0M 0 ( 0 0 r 0 0o

""00ooc0ooooooooooooo000000000000OC000CA o00

I- 0L

.0 XOOO0 OO 000OaC .PO)(TaOO a 0000 ----- 0-0000NC)LA 0i- CY0"N00C ---

00 00000000000000000000000000000000 000.)0000

I0

S•0 .•000000000000000 0• •0000--0

in .4 . . . . . . . . *. .O L

vim 00000000000000000000000000000000--0-0-000

0mO

LA W000000000000000 •N UC U • --s

I0 0 0410000ton CI0 00m00 If00 0 000000 . 1 C if 4U.L...O0...

410 • • 0 0 0 0 0 0 -------------------------- ----N ( --

000OOOOOOOOOOOOOOOOOo0OOOOOOOO --- -- 000

..° . , , 0 . . . ., 0 . . . ... ...0 .... 0 , , . . 0 °. . . ° 0. 0.

0, 00

L O. 00 00 0 0000000000000

) 0000000000000000000000000000000000000000000

. .0 . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

W - O O0 0 0 0 0 0O O O OO0O 0 0 0 - - - - - - -- -: :-

-j0

.~000000000000000------------------000003

W IrC NOOOOOOOOOOOOOOOOOOOOO-----OOOOOcN --------- 00 0

0 .0I

.0 CY* (DUo o o o o oo o o o o o o oo o o o o o

-j 0

in Ln 000000000000000000000000000000000000000000

ty II

c 0 I . - 4 In . a . . . . . . . . ... . . . . .. .~ . . . . .. .. . .. . . .0 ..40 00 0

CC- 'a rr'' L 'I Dý "r M UI n I- 0 (DT lL L L AAA AA A A L UL A A = O 0 m - i0 () Il InM N0 c sW Ln I - 0 M D0 x 00000000000000000000------------N000-----------------000

CL I -

0. ~ ~ 0 I'm m m m-- - - 0S . a a. . . . . . . ..a . ..a.a.a.a. .a. ..a.a. . ..0 0. .

.1 - 0A -A -, IN NI A LA 1A 0A LA LIn L -LI C4 In t- 0 N In r- n' 14 Lin t- LI L 4 in 1-) 0 N)( in r-0 U,4 in N 0 c-4 In r,0~0000000000- -- - N0------------NC)NCMC)V1 Z In I n(N-----------V-r- ---------- 00 D0)

L63

00

*~ OD co in 14 v r'ý m 0 01 0 - 1C, 3 1Wn In U) w IN at to t- I n 01 0

* ~U M W IN M - - - 0' Ny 1--- 0 w* N*jIII I D : ) C - t n U D t D (z

0; 0; In CA m ('4 ON - ( r 1 m 0; -;

I * -. -1 -In - - - - r- 0

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