NAVAL POSTGRADUATE SCHOOLMonterey, California

' I !

THESISGENERALIZED HELICOPTER ROTOR

PERFORMAfNCE PREDICTIONS

by

James William Loiselle

September 1977A_U.O Thesis Advisor: L.V. Schmidt

.1 Approved for public release; distribution unlimited.

SECURITy CLASSIPICATION OF TMIS PAGE ("oi Daha Ca•med)

REPORT DOCUMENTATION PAGE 8FORS COMPLETU S "F"ORM"" MEOMT NUI ..... GOVT ACCSSION 3- MClPINT'S CATALOG NUMS ...

4. TITLE (mad Su&Ettf.) 00 COVERrOS... ..... . . .. ............. as l-e'r'psg Ki es i'snGeneralized Helicopter Rotor /esiL

Performance Predictions5 ' =ePtemmer' 3,//Si--6 PERPORNING ORG, ImEPOR•T NUMEER

7. AUTNOR•,- 1 1. CONTACT OR G-MANT NUMUSEC()

Y 4-James WilliamlLoiselle, ý /

m, --9. PCRMORMING OR&MNIZATION NAME AND AGODRSS 10. PROGRAMA ELEMENT. PROJECT TASK

CARA V IORK UNIT NUNSERSNaval Postgraduate SchoolMonterey, California 93940

11. CONTROLLING OFFICE NAME AND ADDRESS

Naval Postgraduate School Sepobr 177Monterey, California 93940 .138 NME Of' .....138

14. MONITORING A4OENCY NAMI2 & ADD t trom ComtIfplind Office) IS. SECURITY CLASS. (&( Chad ,jomtf)

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Its-. OR'CLGI I'- ICAO-N/ DOWN GM*OI NG

I~ ~ ~ ~~6 DG ISTRIOUT•ION STATIMENT"(of this Replort)i II .I

for public release; distribution unlimited.

17. OISTRINUTION STATEMENT (of the .aha~acl enterd On Wook 20, It 4111twont fires, R6114w IJ

13, SUPPLEMENTARY NOTES

IS. KEY wORDS ('Contine on tvwo* @id. II necessar and Ideminly byobk nbcmbo')

HelicopterHelicopter PerformanceHelicopter Computer Program

30, AISTRACT (Confinrm an reverse old° ii fteoseop mid Idendillo &Y 6101 0m4b6W)

The Generalized Rotor Performance (GRP) program is a computer

program designed for calculating forward fliit performance of a

helicopter rotor system at a specific flight condition. It can be

used to evaluate either ati articulated oT .3 hingele!,s single rotor

!YysLei inu LdiLwajd flIght or in a wind-tunniel test. -he pjo6mrdu was

DD , 'oA 2 1473 EOITION P, 1 NOV 65 IS OUSOLET,t Ig JAN I/ 10-1-60

ge1) SECURITY CLASSIFICATION OF TMIS PAGE (WhomIEn tered

5I

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

SCLA A ) T n

(20. ABSTRACT Continued)

originally designed by the Sikorswy Aircraft Company and purchased

by the United States Navy.

The goals of this thesis were (0) to reinvestigate the

theory and logic used in the program, (2) to add selected

desirable features to the program, (3) to produce a much needed•.} Users' Manual, and (4) to run an analysis comparing the program's•

Vcalculated results against manufacturer's data. These goalswere accomplished and the results of the analysis indicated that

the program produces highly accurate results within the normal

cruise range of a modern helicopter.4_.-

DIN

iiy• /

................. ......... . .. •.................................

DD Form 14731Jan 7301 Ja SguAIY CLhS4IVICATlON Of THIS PA49rWh" DO(& EnE.P.4)

s,'~ o'b2-04-66o

Approved for public release; distribution unlimited.

Generalized Helicopter Rotor

Performance Predictions

by

James William LoiselleLieutenant, United States Navy

B.S., United States Naval Academy, 1971

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOLSeptember, 1977

Authore-1sJso

Approved by: ---- tCL

"I~s igiMME1p

4WU

3

ABSTEACT

The -eneralized Rotor Performance (GRP) program is a

computer program designed for calculating forward flight

performance of a helicopter rotor system at a specific

flight condition. It can be used to evaluate either an

articulated or a hingeless single rotor system in forward

flight or in a wind-tunnel test. The program was originally

designed by the Sikorsky Aircraft Company and purchased by

the United States Navy.

The goals of this thesis were (1) to reinvestigate the

theory and logic used in the program, (2) to add selected

desirable features to the program, (3) to produce a much

needed Users' Manual, and (4) to run an analysis comparing

the program's calculated results against manufacturer's

data. These goals were accomplished and the results of the

analysis indicated that the program produces highly accurate

results within the normal cruise range of a modern

helicopter.

4

7" , , •i -• ' •" -• " "-/. "' ,",•....-'X •~-• ,• ,'•" t "A A ,.;

TABLE OF CONTENTS

I. INTRODUCTION - 9

II. METHOD OF ANALYSIS 15

A. GENERAL COMMENTS ----------------------------- 15

B. ANGLE OF ATTACK CALCULATIONS ----------------- 18

C. METHOD OF SOLUTION FOR THE FLAPPING EQUATION - 24

D. FORCES, MOMENTS AND RADIAL INTEGRATION 27

E. AZIMUTHAL INTEGRATION METHOD ----------------- 27

F. MAJOR ITERATION ------------------------------ 28

G. FLOW CHART ----------------------------------- 31

III. GRP USERS' MANUAL -------------------------------- 33

A. MAIN ITERATION OPTIONS ------------------------ 33

B. GRP DATA DECK ORDE- -------------------------- 36

C. AIRFOIL DATA DECK ORDER ---------------------- 36

D. SPAR DATA DECK REQUIREMENTS ------------------- 41

E. SAMPLE DATA INPUT FORM ----------------------- 42

F. CASE INPUT LISTINGS -------------------------- 48

G. CASE INPUT DEFAULT VALUES --------------------- 51

H. CASE INPUT DATA ------------------------------ 52

I. CASE INPUT FORMAT ---------------------------- 74

J. CASE OPTIONAL OUTPUT INDICATORS -------------- 75

K. IBM 360 EXECUTION CONTROL CARDS -------------- 77

L. SAMPLE PROGRAM OUTPUT 79

IV. GRP SAMPLE ANALYSIS - -------------------------------10

V. CONCLUSIONS -------------------------------------- 113

GRP COMPUTER PROGRAl- ----------------------------------- 114

LIST OF REFERENCES -------------------------------------- 137

INITIAL DISTRIBUTION LIST ------------------------------ 138

5. 1

LIST OF TABLES

TABLE I ROTOR HORSEPOWER WEIGHT = 16359 LBS ----------- 109

TABLE II ROTOR HORSEPOWER WEIGHT = 17321 LBS ----------- 110

TABLE III ROTOR HORSEPOWER WEIGHT = 19246 LBS 110

TABLE IV ROTOR HORSEPOWER WEIGHT = 19658 LBS ----------- 111

TABLE V ROTOR HORSEPOWER WEIGHT = 20829 LBS ----------- 111

TABLE VI ROTOR HORSEPOWER RATIO COMPARISION ----------- 112

dI

I

L 6.

LIST OF FIGURES

FIGURE 1 BLADE ELEMENT DIAGRAM ------------------------ 16

FIGURE 2 ROTOR BLADE MOMENT DIAGRAM ------------------- 16

FIGURE 3 SPANWISE FLOW DIAGRAM ------------------------ 19

FIGURE 4 UT DIAGRAM ----------------------------------- 21

FIGURE 5 UP AND UR JIAGRAM ---------------------------- 22

FIGURE 6 RESULTANT GRP FORCE CALCULATION DIAGRAM ------- 30

FIGURE 7 REQUIRED GRP FORCE DIAGRAM - ------------------ 30

FIGURE 8 GRP P'LOW CHART ------------------------------- 32

FIGURE 9 RESOLUTION OF GRAVITATIONAL FORCE ------------- 60

FIGURE 10 SHAFT AND CONTROL AXIS DIAGRAM --------------- 60

FIGURE 11 TIP SWEEP CALCULATION DIAGRAM ---------------- 73

7.I

I•

T681c OF 6YMNILPS

CI) Spctional Copffi~rmnnt o4~ f'r-7

Nl ý;rtjora1 rapffir~.1prt of I11,t

NI4 S>ncticnal Co,-ff Ir~nt of V4oment

U ppsultant o4 UP, UT an~d Un velocities

UP Volocity pprrnendle-.ular t(- tI' plane of rotnthlrn

UR Vplnilty ?r' rpcdi,- direpctior in r~lare m-4 rn1'tion

UTU' npSultant or UT Pr( U' vmlocitips

V Forv,,Prdi -FIIkýht v I of~- ry, F-f-t P- r smcor~ds

a L14t 'Purvn Slon~e, r.-

IAFAPPA., Inflcvi Catio 1- s'rft ixir.e system

XfL Plsdre r'tAticnal vp1cci ty, rT.'4ins per s-r-or'd

G.-Ig pitch Anmle at 0,r 75 p,-'r~nrt r,-flus nTnit I ~- S'

P-a .p ( zi u ýP

I. I-1 OODUCT ION

The Generalized Rotor Performance (GRP) program is acomputer program designed for calculating forward flightperformance of a helicopter rotor system at a specific

flight condition. It can be used to evaluate either anarticulated or a hingeless single rotor system in forwardflight or in a wind-tunnel test. The GRP series of programs

were originally designed by the Sikorsky Aircraft

Corporation with the cooperation of the United Aircraft

Research Laboratories. Work began on this series of

programs in 1955 and has continued to date. The UnitedStates Navy purchased this program in 1964 and has used it

in the Naval Air Systems Command as a helicopter performance

computer program.

The goals of this thesis were (1) to reinvestigate thetheory and logic used in the program, (2) to ensure that the

Navy's version of this program performed the calculations

according to the design theory, (3) to produce a much neededUsers' Manual, (4) to add certain desirable features to and

correct errors found in the program, (5) to run an analysiscomparing the program's calculated results against

manufacturer's data and (6) to document areas that may need

further attention. The work was done with the help and

cooperation of both faculty at the Naval Postgraduate School

and personnel in the Naval Air System Command.

The output available from the program includes rotorshaft horsepower, rotor profile horsepower, main rotortorque, lift, drag, thrust, H force, rolling and pitching

moments and other forces and moments calculated in the

shaft, control and relative wind reference axis systems.Also available are the azimuthal histories of the blade'sflapping angles, rates and accelerations, plub azimuthal and

9.

radial histories of angles of attack, CL, CD, Mach number,

sweep angles and air load distribution. in addition, the

program will calculate a Fourier coefficient series for

flapping angle, radial station air lcads and azimuthal Z

force distribution.

The program can be divided figuratively into three main

routines. The first routine determines a steady state

flapping solution. The second routiae determines the forces

and moments generated by this flapping solution. The third

routine compares the calculated results of lift, drag and

other options to those desired by the user. If the

calculated valves are not within a predetermined tolerance,

the program will, by use of a first order Taylor series,

generate new values to reenter into the blpia3 flapping

routine.

The program uses a combined blade element/momentum strip

theory. The blade's radius is divided into a maximum of 15

segments. The blade is advanced around the azimuth in a

prescribed number of degrees. At each particular azimuthal

position, velocities are computed in the perpendicular (UP),

tangential (UT) and radial (UR) directions. From these

velocities and a known pitch angle distribution, the local

angles of attack are calculated. From these angles of

attack, the forces at each radial blade segment are

determined. Once the forces are known, the local moments

can be found. From the summation of these moments at a

particular azimuthal position, the blade flapping

acceleration is calculated. This acceleration, combined

with the calculated flapping angle and rate, is used to

advance the blade to the next azimuthal position. A steady

state flapping condition is assumed tc exist when the blade

flapping angle and rate at the PSI equal zero and 360 degree

azimuthal position are within a prescribted tolerance of each

other.

111.

------

This method of calculation accounts for retreating blade

stall, the reverse flow region and compressibi-lity effects.

The program can accommodate a full range of geometric and

design variables including flapping hinge offset, elastic

flapping hinge restraint, first and second harmonic cyclic

inputs and spanwise variations in blade twist, local mass

densities, chord and tip sweep. The program uses no small

angle assumptions and has no restrictions on tiF speed,

forward velocity or advance ratio. The rotor system can be

oriented in any direction in space and can be given any

rotor shaft or aircraft roll, pitch or yaw angular

velocities. The program will perform calculaticns in

straight and level flight or a uniform induced velocity may

be added to simulate climbs and descents. The GRP will

accept up to five different airfoil data decks plus one

blade spar characteristics data deck. Lift and drag

information is entered into the program in the form of CL

and CD versus Angle of Attack Tables for up to 15 different

Mach numbers. The user has a choice of six methods of

solution depending upon the desired restrictions. The user

has available nine printout options, two of which are

program debugging options. There are also error printout

messages that will assist the user having difficulty with

the program.

In the method described above, the flapping motion

determined is about a flapping hinge offset, and with theelimination of all assumptions that the flapping angles are

small, the rotor control axis (axis of no feathering) andthe tip path plane axis (axis of no first harmonic flapping)

are no longer considered convenient for reference in the

analysis of rotor blade motion. Therefore, the axis

selected for use in the analysis is the rotor shaft axis

system. All forces, moments and angles are referred to theshaft axis system with the exception of some of the final

output forces which are referred to the relative wind and

control axis systems.

II.

With the use of the computer, the GRP program eliminated

many of the simpliflying assumptions of the earlier

classical theory originated by Wheatley and Bailey in Ref. I

and 2. The assumptions of the classical theory did not

impose serious limitations in low speed flight, but as

helicopter speeds increase, the inaccuracies inherent in the

theory seriously limit the usefulness of this method. The

assumptions of the Wheatley-Bailey theory which are not

Fresent in the GRP program include the following.

1. The flapping and inflow angles are assumed

small.

2. The lift and drag coefficients are approximated

by a linear and a quadratic variation,

respectively, with angle of attack.

3. The effects of Mach number on CL and CD are not

considered.4. Sectional characteristics in the reverse flow

region are the same as those in the

conventional flow.

5. The sectional characteristics of blade twist,

tip sweep, flapping hinge offset and root cut

out are ignored.

While the GRP program represents a refined approach to

the classical theory, there are still some basic assumptions

which make the GRP subject to error. These arc:

1. Steady state two-dimensional airfoil data are

used.

2. Quasi-static blade analysis is used.

3. The rotational speed about the shaft axis is

constant. There is no lead or lag motion in

the program.

4. The rotor blade is assumed rigid in bending and

torsion.

12.

5. The rotor • inflow is assumed uniform unless

varied by the user.

6. spanwise flow is incorporated into the

calculation of blade angle of attack; however,

it is assumed that the flow at one segment does

not affect the flow at any othar segment.

It is felt that 'the errors induced by the above

assumptions are relatively small in the normal cruise speed

region of a modern helicopter. This is the region from just

below the minimum power airspeed to the malimum allowable

cruise speed. The program can not calculate rotor hoverpower. This is because one of the factors in the

denominator in the main routine which estimates new flapping

routine reentry parameters is the advance ratio. Since the

advance ratio is zero in a hover, the computer would attempt

to divide here by the number zero. Highly accurate results

are not available in the airspeed region from hover to just

below the minimum power airspeed while using the normal

uniform inflow assumption. This is a region of highly

non-uniform flow. Variable inflow nay be inputed by the

user. However, at this time, no information is available on

how successful this technique is in accurately estimating

the required rotor system horsepower. while the program's

required technique for inducing harmonic variable flow is

cumbersome, Bramwell in Chapter ?cur of Ref. 6 describes

several methods which could be incorporated into the

prcgram.

The quasi-static blade analysis assumption is valid

except in the very high speed region where a largepercentage of the retreating blade is in a stalled

condition. This is an area of aerodynamic hysteresis that

is influenced by unsteady aerodynamics. While encountering

lift hysteresis, the amount of lift change above the steady

state CL is about the same as below. However, the lift

distribution on the rotor disc will change. Stall is

delayed and occurs at b lightly lat-e azimuthal station

18.

then predicted by quasi-static analysis. While difficult,

complicated and computer-time-consuming procedures can be

taken to reduce these assumptions there is no guarantee that

the accuracy of the solution will improve. It is felt that

the present program produces highly accurate results in the

flight range of interest in a modern helicopter.

14.

11. M1ETHODOZANALYSýIS

A. GENERAL COMMENTS

The GRP uses a combined blade element/momentum striptheory in its calcultions. The blade radius is divided into

a maximum of 15 segments or strips. Figure I illustrates a

typical blade element. In crder to obtain a steady state

flapping solution the program must calculate the time

history of the flapping motion. This necessitates a

complete knowledge of the flapping angles, rates and

accelerations. The time history of this motion must be

found by solving the highly non-linear equation for the

summation of moments about the flapping hinge. Figure 2

illustrates the forces which produce moments on the rotor

system. The blade has aercdynamic forces acting upwards,

blade weight acting downwards, centrifugal force acting

outwards, an optional elastic hinge restraint acting to

return the blade to a zero flapping angle and an inertia

force resisting any change in blade flapping. The moment

summation equation about the flapping hinge may be written

as

(1) M^WA + K + M, + Mcr + ÷. "' = 0.

The moment due to inertia force can be expressed as

where I1 is the blade moment of inertia about the flapping

hinge. Substituting this into equation 1 and solving for

the following governing equation is obtained.

(2) MA = M•O, + MW + •l + e

If the angle of attack and local lach number at each

blade segment were known, the CL and CD could be obtained

15l.l l ll l l

UpT

Figure 1 Blade Element Diagram

e~~p. C.IN'-akV4AhAL F0444 9

Figure 2 Rotor Blade Mcment Diagram

16.

from the inputed airfoil infcrmation. The local aerodynamic

forces and moments can be calculated and the flapring

acceleration determined. Equation 2 can be rewritten as

(3) = dMK, dM +÷L'.5' IbI

The flapping angle and rate at the (N + 1) azimuthal

position could be obtained from the following expressions.

(4) & 416I

However, since the flapping is periodic in nature and

has a direct relationship to the azimuthal angle, PSI, the

values for flapping are solved with respects to PSI, vice

time. The values of-.a, y and time are related by the

equation &IW =.M•t. Therefore

(7) -SL

The governing equation for the flapping motion now

becomes

(8)O + = +A + Mr-+ M

The flapping solution is based on the assumption that

the angle of attack is known. However, it is not and the

program must proceed through an iterative process in order

to determine the inflow ratio, collective pitch and cyclic

input angles required to generate the desired forces.

17.

r-

B. ANGLE OF ATTACK CALCULATIONS

A very important and basic part of this program is the

procedure by which the local angles of attack are

calculated. While the program will calculate angle of

attack with any angular velocity applied either to the rotor

system or the helicopter, the development here will describe

level flight only. The classical approach ignores radial

flow, UR, and the angle of attack would te obtained as shown

in Figure 1. However, as the blade rotates about the shaft,

it will encounter a large variation in radial flow. The

program attempts to compensate for this radial flow in the

following manner. Instead of the inflow angle PHI equalling

the arctangent of UP/UT, it is set equal to UP/UTUR where

UTUR is the resultant velocity in the tangential and radial

direction. This is illustrated in Figure 3. The pitch

angle is also reduced by the cosine of the sweep angle. The

angle of attack is now calculated in the sweep plane. This

three-dimensional angle of attack is lower than the

classical two-dimensional angle.

The program enters the CL, CD tables with this sweep

plane angle of attack and the sweep plane resultant Mach

number. The program computes the forces using the velocites

in the sweep plane, UP and UTUR, and the blade chord

geometry in the normal plane. Once the forces are computed

in the sweep plane they are resolved into their respective

directional forces.

This three-dimensional angle of attack, due to sweep,

will delay stall on the rotor by reducing the angle of

attack. This describes what actually occurs on the blade.

However, it is felt by previous personnel using the program

that there is a point where the sweep becomes so large that

it tends to wash out the lift being produced. The program

has been modified to reduce CL b.y one half for sweep angles

between 60 to 72.5 degrees and reduce CL to zero for sweep

angles between 72.5 and 90 degrees. These high sweep angles

occur normally only on the inboard Ilade segments of the

ltd.

Figure 3 Spanwise Flow Diagram

19.

retreating blade in and near the reverse flow region.

UP, UT, UR and pitch angle for level flight are shown inFigures 4 and 5. They are calculated in the followingmanner. The radial velocity, UR, is calculated as

(9) UR = (Vcos4)cos'I cosý

The tangential velocity, UT, has two components. Thefirst is the local rotational velocity,-O-r, the second is a

sinusodial component of the forward flight velocity. The

general expression for UT is

(10) UT =fIr + (Vcosckx)sinY

This expression contains a small angle assumption in the

termu ~-for blade flapping angle. The program acccunts for

the fact that the flapping angle does reduce the true radius

slightly by using the following formula

(11) UT = (E/R + (r - E/R) cos ý )A + (Vcos,) sin'#

The perpendicular velocity, UP, consists of three terms,

the inflow ratio, a flapping velocity and a small component iof forward velocity. The inflow ratio is defined as

(12) A - R

The second component is a verical flapping velocity

which is a function of flapping rate and radius. This is

computed as

(13) UP(2) = (r - E/R)•

The thire- component is due to the fact that there is asmall component of the radial flow which acts in the UP

direction due to blade flap angle. This is equal to

20.

4I

•• (Cos 4 S) css

UT" (E/R + (r- E/R)Oos•)-C.+ (VcosajlsinY

Figure 4 UT Diagram

21.

Lip \JlRcosý + (r ./) (VcosA3)cos',sinq

UR (Vcoso~i)cosjCOsQ

tr e (~ot'j CosSimSb

Figure 5 ifl and UT? Diagram

22.

(14) UP(3) - (Vcos4)cos' siný

The total formula for UP is

(15) UP2 =1..CRcosý + (r - E/R)~ + (VCOS4)COs'Tsiný

The pitch angle (9) is expressed in equation 16. e.75

is the pitch angle at the 75 percent radius position, 0' is

the twist depending on the relationship of the location of

the blade segment to the r = .75R location. The next four

terms are the first and second harmonics of cyclic pitch and

(tan V )( is the coupled effect of the flapping angle on

the pitch angle.

(16) 0 = (O.r + ' - AIScosV- BlSsinlf - A2Scos2W

- B2Ssin2'I - tan(s,,) cos-A-sots

The local Mach number is calculated as U/a where a is

the speed of sound and U is given in equation 17.

(17) U = (UP' + UT' + UR1)%

The angle of attack can now be calculated. Figure 1

illustrates that c = ;D- . Initially, the program requires

estimated values for the inflow ratio, collective pitch at

the PSI equal zero position, harmonic cyclic inputs, blade

twist and initial flapping angle and rate. The user can

either input these values or accept the program's automatic

default valaes of -. 02, 5, -1.2, 7.53, 0, 0 and 0

respectively. Using these assummed values, the initial

angle of attack can be determined as fcllows.up = 'ýARcosý + (r - E/R) + (Vcos4)cosysin

reduces to

UP = '--

23.

UT = (E/R + (r -E/R)cosý)JL + (Vcosc(j)sin'

reduces to

UT = (E/R + (r -E/R)) -CL

UR = (Vcos4)cos'coso

reduces to

UR = Vcos o,5 jIn the program Vcos.e• is replaced by the term At.OR whereA

is the advance ratio in the shaft axis system. Since the

local pitch distribution and inflow ratio are estimated, the

local angles of attack can be determined. The next section

describes the method used for calculating flapping angle and

rates at the N + 1 azimuthal position.

C. METHOD OF SOIUTION FOR THE FLAPPING EQUATION

In the preceding section, equation 8 was developed in

order to calculate the time history of the flapping moticn.

(8)1 C = W+ + MP + M0I• •.I

The relation involves a complicated second order

differential equation for establishing the flapping angle as

a function of PSI. The numberical solution is accomplished

by use of a finite difference equaticn and a step-by-step

procedure. An important characteristic of the solution is

that it is periodic in nature. T W! fun-tion whichrepresents a steady state flapping sol-it4.on has the property

S(Yi) = ý (Y-ti*2) . Using this fact, a Fourier harmonic series

can be written to describe the blade flapý.•ný mction.

(18) = AO - A1cosW - BlsinW - A2cos24 - E2sin2W-

A3cos3V - B3sin3 .

By assuming that the first harmonic flapping is much larger

2-1,

than the other higher harmonics, the series can be reduced

to

(19) hO - k1cos'V - Blsint

This equation can be differentiated with respect to PSI

to obtain

(20) A =lsinW - alcosW

(21) = AlcosW - Blsin4 rk

kssuming that the values of • , ( and ý are known at

some azimuthal position, the following equations must hold

(22) A = hO - Alcos'e - Blsinw

(23) • = AlsinV - Blcosw

(24) A•' AcosV + 3lsin'-

These equations can be solved for the N + 1 azimuthal

position by substituting 1t,+ V ='.* vW into the above

formulas. The flapping angle equation becomes

(25) %.% = AO - A1cos(L + WV) - EIsin(Wr ÷&14)

By using the following two identities equation t25) can

be rewritten as equation (28).

(26) cos4('W + h,) = cos'A cosh - finlsinjL

(27) sin(4 +A4) = sin0&cos5A + cos'Ysinh.4

(28) \ AO - A1(cosW%.cos& -si4 ~ sin&)

- B1 (sin'4WcoS& - cos%ý sinA%)

The terms can be rearranged into equation 29. The same

procedure can be used to develop equation 30 for

(29) = AO - cos4Q(A1cosW" + Blsin4f%)

+ sin~w(A1sinWn - B1cosT',)

(30) , = cos '(A1sinif% - B1cosu)

+ sin4W(A1COSW' + B1sin*W)

Substitution into equations 22, 23 and 2U reduces these two

25.-

expressions to the flapping equations used in the program.

The user can either enter the value for & or accept the

programs automatic default value of 15 degrees.

(31) co &, 'M%* -

(32) * -

While this integration scheme is not one of the standard

methods used, it is very useful in obtaining periodic

solutions for differential equations similar to the one used

here. Notice that for small values of 4.1 the trignometric

expression can be reduced to an ordinary Taylor series. By

assuming sin &W equals Wf and cos &W equals one equation 29

reduces to

4 41(33) *

By aqsuming sin A equals &W and ccsh.f equals the first

twc terms of the cosine series 1 - • equation 32 can be

reduced to

(34) 44

D. FORCES, MOMENTS AND RADIAL INTEGRATION

The forces acting on a rotor blade may be found by the

summation of the elementry forces along the span at any

azimuthal position. The forces considered by the program

are the resulting aerodynamic forces only, and are initially

summed in the shaft reference axis system. The program

computes forces in three axis systems. They are the (1)

shaft, (2) control and (3) relative wind axis systems.

The program radially integrates differently for lift and

drag calculations. The drag is calculated from the hinge

offset, E/R, to the rotor tip. The lift iL calculated from

26.

* -- - *-*--- ~- -;,~-*.-, -n- -

the hinge offset to the next to last rotor blade segment.

The last segment is considered a blade "Tip Loss Factor"'

segment. It is assumed that the tip trailing edge vorticescause no lift to be produced in this segment. The normal

procedure is to define this segment as the last threepercent of the rotor blade radius.

The first of the maximum 15 blade segments is

considered the spar or cut out segment. This is defined asthe area between the hinge cffset and the point where the

airfoil actually begins. If no spar data are entered, it is

asssumed that this first segment produces no lift and drag

is obtained. by using the CD verse Alpha Tables for the firstblade airfoil data deck entered. If spar data are entered,

the first segment and all other segments designated spar

segments will have the lift and drag characteristics of the

entered spar data.

E. AZIMUTHAL INTEGRATION METHOD

Onea the integration of the rotor forces and moments

along the blade is complete, an integration around the

azimuth must be performed in order to obtain the averageforces and moments. Since the solution of the flapping

equation is obtained by a step- by- step method, theintegrands of the integrals over Y are known only at a

certain number of equally spaced points around the azimuth.For equilibrium flapping the integral is a periodic function

of M , and for this case integration can be shown to beequivalent to an averaging process. This result makes the

azimuthal integration simple. The following method is usedwhere N is the number of azimuthal positions used in the

calculations and b is the number of tlades.

?4 K

' T -I -

F. MAJOR 1TERATION

Once the program has calculated a steady state flapping

solution and has determined the resulting forces and

moments, a question remains to be answered. Is this the

desired solution and, if not, what must be done to obtain

this solution? In order to solve the flapping equations

certain known or assumed values were used, including inflow

ratio (* A, collective pitch (0.75) and the cyclic pitch

(AiS, BIS, A2S and B2S). One method, first described in

Ref. 3, is to iterate on the required lift and drag thrcugh

modification of A and 4.75, where e.75 is the pitch angle

at the 75 percent radius station at the PSI equal zero

azimuthal position. The modified values are then reentered

into the flapping routine and the calculation is repeated

until it converges to within a specified tolerance on the

required lift and drag. Drag is useA here in the sense of

negative rotor propulsive force. The program procedure is

outlined below.

the required rotor lift and propulsive force are

expressed in terms of the magnitude and direction of the

resultant force in the longitudinal plane. These are shown

in Figure 6 where it can be seen that

(36) a" = c(s+ a'

) . T i

(38) L = Rcos (a") D = R~sin(a,,)

The required lift and propulsive force and theirresultant are shown in Figure 7. In a similar way,

(39) R (L' *D li

(40) a" = arctan (Da/Lg)

28.m

The differences between the required and the computed RA

and a" values are defined as

(41) t = RA - R,

(42) 6 a" = al - a"

In order to correct i and 0.75 to compensate for the

difference between( 4), and ( I,2.,o!) , the required

values are expanded in a Taylor series with ' and 9.75 as

variables. The first order equations are:

(44) C. . * 0 .S

Solving the equations for the iteration on 1 and 8.75

yields

•o'_ .0•- • t•o."

(4 5)

4-h•__ ,._. ~' _ ;A •_o.."

Now that RA, Rj, and a"I and a" are known, the corrected

values of ) and 0.75 can be approximated by:

(4-7)

In order to solve equations 45 and 46, the values of the

partial derivatives in these equations must be found. The

2-.9

C,.

F- U- -R

/ 7

- I

. _ -Fx$

4Figure 6 Resultant GRP Force Calculation Diagram

I \I

Figure 7 Required GRP Force Diagram

30.

procedure used in based on the Wheatley-Bailey method and

the formulas can be found on pages 186 and 207 of Ref. 4.

Reference 5 outlines the derivation and a complete

derivation was performed and verified in conjunction with

this thesis.

G. FLOW CHART

The flbw chart in Figure 8 is a block diagram showing

the relationship between the various parts of the GRP

program.

31.

GRP PLOW CHART

1. Inputs

I 2. Assumed Values3. Starting Values

I Part I

Flapping iteration(One Revolution)

1 I ~check Converge-nce_]

_ _ _ _ _ _..._.f

ChckTole'ance IoLift and Drag

LYes- ] NO'

IPart11 FreadMmn

I °Ii,,.

Print Options

'Figure 8

32.

4

III. GRP USERS' MANUAL

The GRP currently has the capability to enter over 200

individual case variables and option selectors. Presently,170 are available to the users. This manual explains the

input and output format of this GRP program. Examples of

both input and output are given in order to make the use of

this program easier. This manual is divided into the

following areas of discussion.

A. Main Iteration OptionsB. GRP Data Deck OrderC. Airfoil Data Deck Requirements

D. Spar Data Deck RequiresentsE. Sample Data Input Format

F. Case Input Listings

G. Case Input Default Values

H. Case Input DataI. Case Input FormatJ. Case Optional Output IndicatorsK. IBM 360 Execution Control CardsL. Sample Program Output

It is recommended that the user examine the GRP Case

Input listing carefully, since a few options require certainvariables to be inputed which are not necessarily located inthe same general area of the input listings. An attempt hasteen made to list all input variabl.s concerned with the

different options in the discussion of each option and input

variable.

A. MAIN ITERATION OPTIONS

The GRP offers the user six different options fordetermining a solution. They are as follows.

33.

1. Normal Routine

In the normal solution to the problem the computer

will vary the inflow ratio, LAMBDA, and the pitch angle,

0.75, in order to produce the required lift and drag. The

cyclic inputs, AlS and BIS, are considered constants.

Uniform inflow is assumed unless the user induces a

non-uniform inflow by the use of variables 117 and 118, LAML

and UVL. The program will calculate the required shaft axis

angle and position of the control axis from the fixed value

of BIS. In all of the options, the user has a choice of

using either a program calculated first estimate of flapping

angle, velocity and acceleration or an initial value of zero

for all of the above. Either method usually requires the

same approximate amount of computer time. The variable

PCNV, item 97, determines the method to be used.

2. Desired Flappin qAa~2e

This option allows the user tc specify the desired

longitundinal and lateral flapping angles with respect to

the rotor shaft axis. The program at each intermedient

force iteration will vary AlS, BIS, inflow ratio and pitch

angle. Variables 100 and 112, 113 and 114 control the use

of this option.

3. Short Iteration Scheme

This option follows the normal routine with one

major exception. The program will make only one pass

through the flapping routine on each major iteration. The

program may or may not arrive at a steady state solution for

flapping in the first few iterations. It is assumed that

the first few iterations in the normal routine are only

rough estimates of the way the variables should be changed

and that an exact flapping solution is not actually requiredif the user is only interested in transient flapping

34.

-- ~-.-----. -"- V : ~ ~ t V ~ ' . -- - - - - - --

behavior. In order to use this routine, the user must input

a negative number for the variable XITLIM, item 73. The

absolute value of XITLIM will still determine the maximum

allowable number of times the program will enter the major

iteration routine.

4. Trimmed Moments

This option will follow the Normal method described

in paragraph one. The Normal method will only converge on

lift and drag and will not consider the moments produced.

If the variable TRIM, item 160, were assigned a non-zero

value, the program would attempt to trim out the pitching

and rolling moments about the rotor shaft. It is suggested

that this be done in a two case run. The first case should

be a normal run, with the desired printout. Then, for the

second case, input the variable TRIM. This will do two

things. First, it will provide a ccnverged solution without

consideration of moments. Secondly, it will allow the full

number of iterations to be used to reduce the moments. The

program does this by a short routine varying AlS and BIS.

During this process, the whole flapping and force iterations

must be repeated, but it will at least start by using a

converged solution. The variable PCNV, item 97, should be

non-zero to allow the flapping solution from the previous

case to be used as a first estimate in this case. Setting

the variable SKIPIN, item 91, equal to zero will allow a

force and moment summation to be, outputed for each major

iteration. This will allow the user to see exactly how the

program is proceeding.

5. rQP Option

In this option, the program iterates upon the

required lift but ignores any values inputed for required

drag. This option can be used to simulate a wind-tunnel

test. The uscr munt input the shaft angle, iten 111, and

C5.

the pitch angle, item 87. These two inputs will be held

constant. The program will iterate on the inflow ratio,

LAMBDA, in order to obtain a solution. The variable TOP,

item 96, controls the use of this opticn.

6. ALO __ooption

This option, like TOP, is a wind tunnel option. It

also iterates on required lift only. However, here the

shaft angle, item 111, and the inflow ratio, item 88, are

required inputs and are held as constants. The program williterate on the pitch angle, item 87, in order to determine asolution. The variable ALOPT, item 110, controls the use of

this option.

B. GRP DATA DECK ORDER

Data is entered into this program in the followingorder.

1. Airfoil Lift Coefficient Table

2. Airfoil Drag Coefficient Table

(Repeat steps one and two as necessary.)

3. Case Irput Data

4. Harmonics ofj the Inflow Ratio **

5. Spar Lift Coefficient Table *

6. Spar Drag Coefficient Table *

** Optional Data

C. AIRFOIL BLADE DECK REQUIREMENTS

The GRP program requires that all CL and CD informationbe entered into the program in tabular form. Tables for upto 15 different Mach numbers and five different airfoils can

be entered. The program currently does not use or requirevalues for the Coefficient of Moment, CM. Since certain

segments of th? retroating blado arA in the reverse flow iregion, angle of attack tables are required to include

S~Iii

valued from -180 degrees to +180 degrees. If they are not

included, an error message will be printed when the program

can not locate a value for CL and CD at these large positive

and negative angles of attack. If complete angle of attack

information is not available for a particular airfoil, the

user can use the values provided in Section E. It is

realized that available data on airfoil behavior at large

angles of attack are very limited, but so is the region on

the rotor disc where the blade operates at these high

angles. Since this occurs only immediately around and

within the reverse flow region where dynamic pressure is

low, little precision is lost in performance calculation by

using one common representation for most airfoil behavior.

As a minimum, two values for CL and CD Ft two different

angles of attack and Mach numbers must be supplied. As a

maximum, 15 different values of Mach number may be entered.

Each Mach number may contain up to 48 different values of CL

and 48 different values of CD and theii associated angles of

attack.

The first Mach number must be equal to zero. This table

can be an exact duplicate of the lowest Mach number table

the user has available. The highest Mach number table

should be high enough in order to prevent the program from

stopping because of a local Mach number higher than that in

the table. A quick check can be obtained by adding together

the rotor tip velocity and the forward flight speed. This

combined velocity, divided by the local speed of sound, must

be less than the maximum Mach number entered into the

program. The program linearly interpolates between Mach

numbers and angles of attack in order to determine the value

of CL and CD. The subroutine BLIN4 does the interpolation.

Several options are available to help reduce the number

of data points that must be entered. If the airfoil is

symmetrical, the user only needs to enter values for

positive angles of attack. The program will assign the

0 7.

appropiate sign to the value of CL and CD according to the

sign of the angle of attack. This is accomplished by case

input variable 107, SYM. This option can also be used for a

cambered airfoil where values of CL and CD at negative

angles of attack are unknown.

Values for large angles of attack need not be entered

for each Mach number by making use of the program's input

variables 156 and 157, or their automatic default values.

Values for large positive and negative angles of attack need

only be entered for the two lowest Mach number tables. The

lowest Mach number table must be at a dach number equal to

zero. If an angle of attack is greater than variable 156

(HIALFA) or lower than variable 157 (LOALF) ., or the

programs default values of plus and minus 30 degrees, the

Mach number is set equal to zero. This ensures that only

the first two Mach numbers have to carry the whole range of

angles of attack from -180 to +180 degrees for a cambered

airfoil or from zero to +180 degrees for a symmetrical

airfoil.

The format of the table input will now be described.

The first data card contains the variable WELADE. This

controls whether or not the user receives an echo printout

of the Blade Airfoil Data being entered. If the value of

WBLADE is equal to zero, the user will not receive an echo

printout of the Blade Data. If WBLADE is a non-zero number,

the user will receive the echo printcut. The read format

for WBLADE is F10.0. WBLADE is the only item on the first

data card.

The next card also contains only one piece of data,

NBLADE. NBLADE is the number of different airfoil data sets

to be used and appears on this card in an 12 format. This

program will accept up to five different blade airfoil data

sets. It will also accept one blade spar data set. If the

rotor blade being analyzed were composed of thr'ze different

types of airfoils, NBLADE would equal three. However, if

38.

the blade consisted of three sections, of which the first

and third section were the same, NBLADE would equal two. It

is explained later how the blade segments are assigned their

respective airfoil type. This arrangement provides the user

with the ability to vary the make-up of the blade while only

having to enter into the program once a particular set of:4 airfoil data.

The above two variables, WBLADE and NBLADE, are only

entered once for each complete computer run which uses thesame set of airfoil data. The following information will be

entered twice for each type of airfoil used. It will beentered first for CL and secondly for CD for each type

airfoil. The overall format is summarized below.

card 1 WBLADE

Card 2 NBLADE

Card 3- CL's for airfoil number one

CD's for airfoil number one

CL's for a.irfcil number two

CD's for airfcil number two

Repeat as necessary.

Card number three contains the variable NZ, which is the

number of Mach numbers for which CL's will be entered for

the first airfoil. This number must be right ji.stified in

12 format. The maximum number of Mach numbers for each CL

and CD for one airfoil is 15. This is the only number

entered on this card.

Card number four begins the actual Mach number, CL

versus angle of attack tables. The format in this paragraph

must be repeated for each Mach number. This first card isdivided into 11 fields (12, 1OF7.0). The first field is a

two-digit, right-justified integer in 12 format. It is

equal to twice the number of data pairs for this Mach number

plus two. This tells the computer bow many numbers are

required to be entered for this particular Mach number. A

data pair consist of one angle of attack and its associated

LI

CL or CD. The remaining ten fields are each seven columns

long in floating point or F7.0 format. These fields begin

in columns 3, 10, 17, 24, 31, 38, 45, 52, 59, and 66. OA

this first card, the field that starts in column three

contains the number of data pairs at this Mach number. The

field that begins in columns 10 is the actual Mach number.

The remaining eight fields on this card are for the first

four data pairs starting with the lowest angle of attack and

increasing towards the highest angle of attack. If a

symmetrical airfoil option is used, the lowest angle of

attack is zero. If a non-symmetrical airfoil is used, thri

lowest value for the first two Mach numbers should be -180

degrees and for all the remaining Mach numbers the ialue of

LOMACH entered or -30 degrees.

All the remaining cards for this particular Mach number

will contain the data pairs. These cards contain ten fields

each, the first field consisiting of nine columns and the

remaining nine fields consist of seven columns beginning in

column ten and following the same format as the first card.

The format is (F9.0, 9F7.0). This card is repeated as often

as needed. Columns 73-80 are not read and may be used for

comments.

This procedure is repeated until all the CL's are

entered. Once completed the program is ready to enter the

values for CD. The whole procedure is repeated again,

starting with the value for NZ representing the number of

Mach numbers for which CD's will be entered. If more than

one airfoil data deck is to be used, the above procedure

will start over again by reading in the value of NZ for the

number of Mach numbers to be entered for values of CL for

the second airfoil. The number of data pairs for each Mach

number must be between 2 and 48. The number of "twice the

data pairs plus two" must be between 6 and 98.

Prior to eatering the airfoil data, tho user should

te view the following input variables.

40.

1. SYM (107) - Symmetrical and nonsymmetrical

airfoil data input ccntrol.

2. HIALFA (156) - The highest angle of attack for

which values of CL and CD will be found at all

Mach numbers.

3. LOAFLA (157) - The lowest angle of attack for

which values of CL and CD will be found at all

Mach numbers.

4. SPAR (103) - Number of segments using spar

airfoil data.

5. TIPSWP (158) - Amount of tip sweep in degrees.

6. TPSWST (159) - Blade segment number at which

the tip sweep begins.

7. BSPL (120) - Input control variable for spar

data.

8. RB(I) (161-175) - Controls the blade segment

airfoý`i data assignment.

D. SPAR DATA DECK REQUIREMENTS

The format for spar data are similar to that of the

airfoil data with the exception that only one set of spar

data can be entered into the program. Before the prcgram

will read spar data, input variable number 120, BSPL, must

have a non-zero value. In addition, variable 103, SPAR,

must indicate the number of blade segwents which are usingthe spar data. The program automatically assumes that the

first segment is a spar segment. This is further explained

in the Case Input section. The spar data are the last to be

entered into the program. This is an optional input and is

riot required. If spar data are not inputed, the programwill assume that the one automatic spar section creates no

lift and has the drag characteristics of the first airfoil

section entered.

41.

.~ ,. .~.-.~--.-----~*-~- . . ---. -

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

The format for inputing spar data are as fol ows. The

first card contains the variable WESPAR in Fi0 0 format.

A non-zero value of WRSPAR causes an echo printout of spar

data to occur. The remaining spar data are handled the

exact same way as the airfoil section data, starting with

the variable NZ. There is no input similar to that of the

airfoil section stating how many different spar data decks

are being entered since only one is allowed. As before, aminimum of two lach numbers are required to be entered. The

blade segment printout indicates spar segments by the use of

a "0' for that segment.Input variables associated with SPAR data are as

follows.

1. SPAR (103) - The number of blade segments using

spar data.

2. BSPL (120) - Input variable which controls theinput and use of spar data.

E. SAMPLE BLADE INPUT

The next several pages illustrates the sample fcrmat for

a blade which has the following characteristics; (1) anecho printout is not required, (2) there are two airfoil

decks to be read in and (3) the first airfoil deck has ninevalues of Mach numbers for which CL's are to be entered.

42.

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m ~ t 4% -3% 1 l

fis 4 mi0 91-o-- 0% U, ~ 0.1 * qNrg-. 9o-0

C4 .40C4 *. IN I *Nr . 0 *N "- 0 Ouco m -* Q11 ...

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*0 1 * s41 s o 01' - 0 0 a 0 c q q 0' C 4 t

46.

q-Onm *Irlow CNZmw LANm mm-. WN NWU)0O w1AiEm N-row %ONON '%omknL NoI*-ONOON NOON M~OON M-- 49- ýN-

* 004 g04f 4*44 0*06 4000 I0**

9 * 0 6; a 0 a

NqWC0,4* vC O .C0-LO o0*4vC %ONN* 00' *NN-0 oCN** -

CD cor'~ L~t ~ 9( l~

O~q ODO V"OOO V-OCC(0-*40 r-1-4CorAT *coo1 v-m c4 I 9- N 1 1AL -uc4 I1 09- N I I v-,r 1- N

I IJ' I0C~' oO N- I I I co -I 0~ 4 I ~o Ir Co.0IIc-00 cg

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NLA0-L'0v4*~rw ~r~ION ~~~O47, NC

F. INPUT CASE LISTING

GRP CASE INPUTS

ITEM PROGRA•iNO. DESCRIPTION VARIABLE DIMENSION REQUIRED

1 Tip speed OMEGAR FPS YES

2 Radius R FT YES

3 Speed of Sound SPSD FPS YES

4 Air Density RHO SLG/CJFT YES

5 No. of Blades XNB - YES

6 Forward Speed VEL KTS 89, 90

7 Offset Ratio of E- YESFlap Hinge (e/R)

8-22 Delta X DX(15) - YES

23-37 Local Twist TW(15) DEG 92

38-52 Local Mass Density XMASS(15) SLG/FT 78,79

53-67 Local Chord C FT YES

68 Delta PSI DPSI DEG *

69 Flap Iteration Limit FTRL - *.

70 Initial Beta BIN RAD *

71 Initial Beta * BPIN RAD/SEC *

72 Initial Beta * BPPIN RAD/SEC**2**

73 Lift and Drag XITLIM - *Iteration Limit

74 Required Lift RL LB YES

75 Required Drag RD LB 95

76 Lift Tolerance XLTOL LB *

77 Drag Tolerance XDTOL LB **

78 First Moment about FO1l SLG-FT 38-52Flap Hinge (M)B

79 Second Moment about SMOM SLG-SQFT 38-52

80-82 Shaft Orientation AGBGL,CG DEG *

83 Pitch-FltCopling TD3L DEGAngle (Delta 3•

84 Drag Increment DELD -

85 Lat. Cyclic Pitch AlS DEG **

48.

-tv-.- -- 'r.,v'___

86 Long. Cyclic Pitch BiS DEG *

87 Collective pitch T75 DEG *

88 Inflow Ratio LAMBDA 6 90

89 Advance Ratio KUL - 6, 90

90 V'EL Control UIN - 6, 89

91 Iteration Output SKIPIN - *-

92 Linear Twist TWIST DEG 23-37

93 No. of Blade XNSEG ---$

Segments

94 Climb Rate RCFPM FPM -

95 Flat Plate Area FPAREA FT**2 75

96 Thrust option TOP -

97 Flapping Re- Use PCNV *_

Indicator

100 HU Iteration ABIT -

Tolerance

101 Beta Tolerance BTOL -

102 Beta* Tolerance BPTOL -

103 Spar Segments SPAR -

104-105 Second Harmonic A2S,B2S DEG

Control Inputs

106 solidity RSL -

107 Symmetric SYMAirfoil

108 Spring Constant SFH FT-LB/RAD -

109 Damping Constant FDMP FT-LB/ -.RAD/SEC

110 Lift Only option ALOPT -

111 Shaft Angle ALL DEG -

112-113 Desired Al and RAlS DEG -

BI Flapping RBlS

114, Al, BI Tolerance TOLAB EG DE

115 Tangent Delta 3 TD3B -

116 Phase Angle for PHD3B DEG -Delta 3-

117 Induced Velocities LAML -

118 Induced Velocities UVL -

119 Not Used

120 Input Spar Data BSPL

49.

121 Azimuthal Printout PPSI DEG **Indicator

122 Minimum Lift ATEST .Curve Slope 71

123 iteration Gain IGC - **Factor

124-125 Not Used I126 Pre-Coning Angle PCR RAD

127-137 Not Used

138 Hub Moment Inplane INPL -Aero Forces

139-141 Aircraft Yaw, Roll PSIS, PHIS RAD/SEC -

and Pitch Angular THFSVelocities

142 CG Station ?SCG INCHES -

143 Rotor Center Station FSMR INCHES -

144 CG Waterline WLCG INCHES -

145 Rotor Waterline WLMR INCHES -

146 Aircraft Lateral VELY KT3velocity

147 Spar Symmetry SYMSPR V I156 Airfoil Tables HIALFA DEG

157 Airfoil Tables LOALFA DEG **

158 Tip Sweep TIPSWP DEG **

159 Sweep Station TPSWST - **

160 Rotor Moments TRIM -

161-175 Rotor Blade RD - *Airfoil DataAssignments•* Program has automatic default value.

50.: I

G. GRP INPUT DEFAULT VALUES

ITEMNO PROGRAM VARIABLE DEPAULT VALUE

68 DPSI 15.0

69 FTRL 15.0

73 XITLIM 15.0

76 XLTOL 100.

77 XDTOL 50.0

81 BGL 90.0

85 AlS -1.2 *

86 BIS 7.53 *

87 T75 5.0

88 LAMBDA - 32

91 SKIPIN 1.0

93 XNSEG 15.0

97 PCNV 1.0

101 BTOL .001

102 BPTOL .001

114 TOLAB 0.25

121 PPSI DPSI

122 ATEST 5.0 **

123 IGC 1.0

156 HIlLFA 30.

157 LOALFA -30.

159 TPSWST 16.

161-175 RB 1.0

*0. if ABIT, item 100, is non-zero

* -50 is TOP, item 96, is non-zero

51.

H. CASE INPUT DATA

Many of the case inputs are self-explanatcry by their

name listing alone, however, many are not. This section

will explain the input variables and case options available

to the user.

The program enters all the variables into a 200 element

array called V(I). Prior to entering case data, the program

will automatically do two things. It will initialize the

array V(I) to a value of zero. It will assign the default

values listed in the previous section to those particular

variables.

The V(I) array is associated with the variable names by

equivalent statements. The user needs only to enter values

for the variables that are different from the default or

initialized values. In a multiple case computer run, the

user need only enter variables that are different from the

preceeding case. If no new value is enter for a variable,

the value from the previous case is carried over.

1. I~tem_6 - Velocit~y _.-EL

The computer program will accept forward velocity in

one of two ways. The user will input either VEL, item 6, in

knots or advance ratio MUY, item 89. The value of UIN, item

90, determines which variable will be used. If UIN is zero,

VEL will be used. If UIN is non-zero, MUL will be used. If

VEL is used, the program calculates the advance ratio by the

following expression.

If MUL is used, the program will calculate the flight

velocity by the following expression.

"'A-A

2. Item_7=_ZiefU t -E

The offset ratio of the flap hinge, E/R, is the

distance from the center of the rotor shaft to the vertical

flapping hinge, normalized by the rotor radius, item 2.

3. Items 8-22 - Delta x- DX

Delta X is the non-dimensionalized width of eachindividual blade segment starting with segment number one.There may be up to 15 segments entered. The number of

widths entered here must equal the value of Item 93, XNSEG.

XNSEG is the number of segments intc which the blale is

divided. This can range from two to fifteen. It is

recommended that a value of ten or more be used for XNSEG.

If XNSEG were equal to 12, values of Delta X would beentered for items 8 to 19 and no values would be entered for

items 20 to 22. The sum of ER plus the summation of the

Delta X's must equal one. ER is the non-dimensional width

between the rotor shaft and the flapring hinge cffset. Item

number 8, which is the first segment width, represents thewidth between the flapping hinge offset and the point where

the actual rotor blade airfoil begins. This area is known

as the spar or cut o4t segment if no spar data were entered.

If item 103, SPAR, is zero, the program will assume that

this first section creates no lift and has the drag

characteristics of the first inputed airfoil data section.

Since this area experiences relatively low dynamic pressure,

the calculations in this segment do nct have an appreciable

effect on the outcome of the program. The area between theshaft and the hinge offset, ER, produces neithqr lift nor

drag in the program's calculations. The :Last segment is

53.

I

considered a tip loss factor segment. The lift is assumed

to be zero, but drag is calculated in the normal manner. In

previous runs, the width of this section has been set equal

to three percent of the rotor radius, or Delta X equal to

.03.

4 Ies2_3-37 -. Twist - TW

The program has two options fcr entering geometric

twist into the calculations. If the blade has linear twist,

item 92, TWIST, can be used. If the twist is non-linear,

the twist can be entered in items 23-31, TW. If a number is

entered for item 92, the program will assume linear twist

and ignore all values entered for TW. The local twist at

the center of each segment can be entered starting with item

23. If the blade contains ten segments, items 23-32 would),e entered and no values for items 33-37 would be entered.

A word of caution is necessary regarding the linear

twist option, item 92. The twist is considered to be zero

at the 75 percent chord point. The twist is calculated

assuming that the twist starts at the rotor shaft and varieslinearly out to the rotor tip. If the actual rotor bladeairfoil started at the 25 percent radius point with a twist

value of -9 degrees, a value of linear twist egual to -12

degrees would have to be entered in order for the correct

twist distribution to be calculated by the program.

5. Items 38-52 - XMASS

The program provides two methods for entering the

local mass density or the moment of inertia information.

The individual mass density for each section can be entered

in items 38-52, or the First and Second Moment, FMOM and

SMOM, about the Flapping Hinge, c ,, be entered in `.tems 78

and 79. If data are entered for item 79, the Second Moment

about the Flapping Hinge, the program will use the

information provided by items 78 and 79. If variable 79 is

54.

equal to zero, the program will calculate the First and

Second Moments fot items 38-52 and ignore any value entered

for items 78 and 79.

6. Items_53-57 - Chord

The program has two methods for entering local chord

data. The local chord at each segment can be entered

starting with segment number one. If the chord is a

constant chord, the amount .of data to be entered can be

reduced to only items 53 and 54. If items 53 and 54 are

equal, the program assumes that the chord is constant

throughout the radius and will ignore any information

entered in items 55-67. Therefore, for constant chord, only

enter values for items 53 and 54. If the rotor solidity is

not inputed in item 106, the program will compute solidity

as follows.

7. Item 68 - DPSI

DPSI, or Delta PSI, is the incremental azimuthal

value by which the program advances in its blade flapping

and force summation routine. This number must divide evenly

into 360 dcgrees. A minimum value of five degrees is

permitted. It has been found that for most cases decresing

the value of DPSI below 15 degrees does not improve the

accuracy but does .increase the computational time. As an

example, the rotor horsepower required for one particular

run was 1096 RHP for DPSI equal to 15 degrees, 1095 RHP for

DPSI equal to 10 degrees and 1097 RHE for DPSI equal to 5

degrees. DPSI has a default value of 15 degrees.

Sk .

8. Item 69 - FTRL

FTRL is the maximum limit on the number of times

that the program will enter the flapping iteration routine

in. search of a steady state flapping solution. The program

has an automatic default value of 15 iterations. The

program usually arrives at a steady state flapping solution

within three to four iterations. The only method by which

the user can actually determine the number of flapping

iterations required would be to use one of the debug output

options. However, these options will create a huge amcunt

of output and the user is cautioned abcut their use.

9. Items 70-72 BIN-BPPIN

The program has three options on how to assign the

values for the inital flapping angle, velocity, and

acceleration at the PSI equal zero position. The user may

(1) input the values, (2) have the program itself calculate

initial values, or (3) accept the default values of zero.

Option three is the most used option. It has been

discovered that there is little or no difference in solution

time between option two and three. Variable number 97,

PCNV, controls which option is to be used. If PCNV is

non-zero, the program will use either option one or three.

The program initially assigns values of zero to BIA, BPIN,

and BPPIN before the initial case data are entered. PCNV is

assigned the default value of one. Therefore, with no

values entered by the user for items 70-72 and 97, the

p rogram will start with an initial value for flapping angle,

velocity and acceleration at the PSI equal zero position of

zero.

If the u~ser sets PCNV equal to zero, the following jformulas are used for determining the inital values. These

formulas are derived .rom page 194 of Ref. 4.

$6.i

The term • is referred to as the Lock number where

In a run where more than one case is executed at a

time, a non-zero value for PCNV allcws the previous case

values for flapping angle, velocity and acceleration tc be

used as the initial estimate for these variables in the next

case. If PCNV is equal to zero, the inital values will be

calculated by the above fcrmlilas. M oLt caf-R9 ara run by

accepting the default value of one for PCNV.

After the program calculates a steady state flappinq

solution for its estimated values of inflow ratLo, pitch

angle and shaft tilt anq'e, the program determinomm tho

forces and momentn g )nerated by thin solution. If the

forces and, optionally, moments do not meet the ruuire d

amount3 entered by the user, the program will calculate ndw

values to re-ezter the flapping routino. XITLIM dtterm0ines

the maximum numiber of times the program will cswavo thu

calculated values to the desired valuea. The program has a

default va lu.e of 15 for XITLIM. A majocity of the

soluti.cns reguir:e approximately five iterations to cowVerge.

Once the XITLIM limit is exce',ded, thi progjram will utop ai,.

printout an "ExceaO.ed Limit" stat4mont. The pIogrAm

autorma-icrall y ,ori ¶.t A S Q 1 tJ.,'I t tIw fl19ut1 C f MRJQL

Itter&4 iuns It. us k, I iti ,a lcuiu t I rij t h' j I~ 1 A(ý•

F79

The sign of XITLIM controls the Short Iteration Option.

If XITLIM is a negative number, the program uses this

option. A complete describtion can be found in the section

entitled Main Iteration Opticns.

The required lift, RL, should equal the actual

weight of the helicopter that the rotor system is

supporting.

The required drag can be entered in one of two ways.

It can be entared directly in item 75 or it can be

calculated by the program by entering the aircraft's flat

plat* area in item 95, FPAREA. The drag is equal to the

flight path force that the rotor syatem must overcome to

sustain level flight at a certain velocity. Items 75 and 92

can be entard eithcr az positive or negative numbers and

the program will provide the correct fcrward flight solution

by assigning the proper sign value internally. If a value

is entered for PPAREA, the program will ignore any val.e

assigned to ED. The drag for PPARBA is calculated as

follows.

If item 92 is not entered. the user must supply the

value for the reguired drag by using the above fcrmula.

There dre certain options for which thQ program igncres or

does not iterate on drag. An example of this would be when

the program is used to estimate wind-tunnel test results.

If these options are desired, see variables 96, TO"V and

110, ALoPT.

r8.

VI1I

13. tejs 76-77 - Tolerances

XLTOL and XDTOL are the lift and drag tolerance..The program will iterate until both the lift and drag are

program has automatic default values of plus and minus 100

pounds for lift and 50 pounds for drag.

14. tems 78-79- Moments

Information regarding the use of FMOM and SMON can

be located in paragraph five on XMASS.

15. Items 80-82 - Shaft Axis

The gravitational or weight vector must be orientedto the shaft axis of the rotor system. Figure 9 shows thepositicns of these angles. The shaft can be oriented in any

desired direction. The program will automatically assignthe proper values for normal helicopter flight. The defaultvalues for AG, BGL, and CG are 0, 90, and 0 degrees,

respectively. This orients the shaft in the verticaldirection for normal flight. If any other orientation isdesired, the user must enter the appropiate values for items

80 to 82.

16. Item 83 - Delta_3

Delta three inputs are controlled by variables 83,115, and 116. These are TD3L, TD3B, and PHD3D,

respectively. If the flapping hinge is connected in such amanner as to cause the blade to change pitch due to

flapping, this is referred to as a Delta Three hinge. Item83 is the pitch-flap coupling angle; 115 is the tangent ofthe Delta Three Bar; and 116 is the phase angle for theDelta Three Bar. The program reduces the pitch angle at aparticular azimuth and segment by the quantity TD3 where

59.

:?1

IIlI,

Figure 9 Resolution of Gravlational Force

60. -- ;

Figure 10 Shaft and Control Axis Diagr~m

60.

lii• •_,] .- ;.Tj';••' r-r--E.---------• .- ,-. •*-r•llE.. . •z--.i• - -..- - . .. . . . .

TD3 TD3L + TD3B*sin(PSC + PHD3D)

Normally, variables 83, 115, and 116 are zero.

17- taj _DrU a__Illg__ me

The drag iucrement is a value of delta CD that is

added to the value of CD obtained from the airfoil input

data decks. This is added as a roughness factor that

naturally occurs on blades that are used on production

aircraft. A value of .002 is normally used. The value is

not added to the drag calculations for spar data.

18. Items 85-86 -- Cyclic Pitch

The lateral and longitudinal cyclic pitch is

controlled by four variables. The program allows the user

to enter both first and second harmonics of cyclic input.

The variables AlS and BIS, 85 and 86, control the first

harmonic inputs. The variables A2S and B2S, 104 and 105,

control the second harmonic inputs. The A's correspond to

the lateral inputs and the B's correspond to the

longitudinal inputs. The program has automatic default

vales for AlS, BIS, A2S and B2S of -1.2, 7.53, 0 and 0

degrees, respectively. The user may enter different values

if desired. Unlers other options are indicated, the program

will keep these cyclic values constant throughout the run

and will vary inflow ratio and pitch angle to obtain a final

solution. The program will automatically calculate the

postion of the shaft axis by the momentum theory and will

use the value of BiS to determine the position of the

control axis. Figure 10 demostrates this fact.

There is a different option available which allows

the program to seek specific values of longitudinal and

lateral flapping. This requires the use of variables 100

and 112 - 114. if this option is taken, the values of AlS

and 31S are se÷t gqual to zero iriiti~A1ly and will be changed

by the computer in its iteration of the required flapFing

....

-. -,r

angles. The program also has an cption to remove hubrolling and pitching moments. Once a solution is obtainedthat meets the lift and drag tolerances, the values of hiS

and BiS are varied to reduce the moments. This option iscontrolled by variable 160. Normally, runs are made with

the user not inputing values for iteiis 85, 86, 104, 105,

100, 112, 113, 114 and 160.

19. Ite 87 - Pitch Angle

Variable 87 is the initial value of the collective

p.tch at the 75 percent radius station at PSI equal zeroazimuthal station. The program has a default value of fivedegrees. This val.ue is used only to initiate thc program.The program, under the normal run option, will vary thisvalue in the process of iterating for a convergent solution.

There is an cption where the pitch angle remains fixed as ina wind-tunnel t-st. This is the TOP option, variable 96.

20. Item 88. -Inflow Ratio

The variable LAMBDA controls the inital estimate of

an uniform inflow. Since the equations of the program aredone in a gyrocopter mode, inflow is negative when air flowsdown through the rotor. This is the normal fcrward flight

mode. The program has a default value of -. 02 for LAMBDA.The program will iterate on LAMBDA in its normal iteration

rountine. The ALOPT, variable 110, o~tion will hold LAMBDA

constant.

21. Items 89-90 - MUL

Information regarding the use of KUL and UIN can befound in paragraph one on Flight Velocity.

22. ltem 91 - SKIINN

Informttion regarding the use of SKIPIN can be found

in the section of Case Optional Output Indicators.

62.

23. Item 92 -Linear Twist

Information regarding twaist can be found in

paragraph four on local twist.

Information regarding XNSEG can be found in

paragraph three on Delta X.

25. Item 94 - Rte of Climb

The program can be made to calculate a complete

solution for any given rate of climb or descent. Climb or

descent rate must be entered in units of feet per minute,

with positive values for climbs and negative values for

descents. The program assumes a ui 4iform down or up flow

across the entire rotor surface equal to the rate of climbor descent. This value is added as an incremental

correction into the calculations of UP and will effect PHI

and angle of attack.

26. -_Zrn9PPA

Information regarding the use of ?PAREA, flat plate

area, can be found in paragraph twelve on Required Drag.

27. Ltt.26_: TOP

The TOP option is one of the wind-tunnel optiocs.If TOP is a non-zero number, the program iterates to obtain

the required lift of variable 74, but will iqnore the

required drag of variable 75. Item 87, the collective pitch

at the 75 percent radius at PSI equal zero, will Le heldconstant. Item 88, the inflow ratio, will be varied in the

major iteration routine. A non-zero value of TOP will

result in a value of -50 for item 122, ATEST. ATEST is the

minimum accepabl.e value for the lift curve slope whin option

62.

96 or 110 is executed. If non-zero values for both TOP and

ALOPT are entered, the program will do the TOP option.

Shaft angle, item 111, must te inputed by the user.

28. Lej_97 - IPC•_Information, concerning the use of PCNV, the Flapping

Solution Re-Use Indicator, can be found in paragraph nine on

Initial Flapping Conditions.

29. Item 98 - PRINT

Information concerning the use of PRINT, the

program's main output indicator, can be found in the section

entitled Case Optional Output Indicators.

30. Item 99- XEND

Item 99 is the End of Case signal card. It is the

last data variable that will be entered for each case. If

XEND is a negative number, the program will stop after it

determines a solution for that particular case. However,

since an infinite number of cases can be entered for each

computer run, XEND also tells the program if there are more

cases to go. If XEND is equal to 2.0, the program will

assume that the next case will begin by reading in new

airfoil data. If XEND is any other positive real number,the program assumes that the next case will use the present

airfoil data and will enter only case input data and any of

the options which normally follow thp case input data. Each

time that the variable XEND is entered, be especially

careful to follow the format for NNUM for this card. NNUM

is the Dumber of inputs per data card. NNUM must be a

negative number for this XEND card. It is this negative

sign on NNUM which actually keys the computer to stop

reading data cards for a particular case.

64.

31. Item 100 - ABIT¶

Information regarding the use of ABIT can be found

in paragraph 40.

32. Items 101-102 - BTOL

In the blade flapping routine, the program searchesfor a steady state flapping solution for the given

conditions of inflow ratio, pitch angle, and cyclic input.

The program compares values of flapping angle and velocityat the PSI equal zero azimuthal positicn on each revolution.

If at this position, the difference between the n-th and the

(n - 1)th revolution values for flapping angle and velocity

is less than BTOL and BPTOL, respectively, the program

ac vumes that it has determined a steady state solution.

BTOL and BPTOL have default values of 0.000001 radians andradians per second, respectively. If other values aredesired, the user may enter those values for items 101 and

102.

33. Item 103 - SPAR

The number of blade segments using spar data can be

inputed in item 103. If no spar data are available, no

value for SPAR should be entered. For this case, the

program assumes that the first segment is the area betweenthe flapping hinge and the point where the actual airfoil

begins on the rotor blade. This area is also referred to as

the "cut out" segment. In this case, the program assumesthat this cut out area produces zero lift and uses CD

information from the first airfoil section for drag

calculations. If a non-zero number is entered for SPAR,spar airfoil data must be available. Case input variable120, BSPL, controls the spar input option. If spar data are

to be inputed, BSPL should be assigned a non-zero value. In

a multiple case run, where spar data are initially entered,

65.!

the variable BSPL is set equal to zero as soon as the spar

data are entered. If the program did not do this, the user

would have to enter a zero for BSPL for the next case if no

new spar data are to be entered. If in a multiple case run,

the user decides to enter new spar data, the variable BSPL

must be assigned a non-zero value for that particular case.

34. Items 104-i05 - A2S B2S

Information regarding the use of the second harmonic

control inputs can be found in paragraph 18 on Lateral and

Longitudinal Cyclic Inputs.

35. Item 106 - Solidi ty

Information regarding the use of RSL, rotor

solidity, can be found in paragraph six on local chord.

36. Item 107 - SY-

SYM is the non-symmetrical airfoil input control.

If the user assigns a rnon-zero value for SYM, the program

will assume that all blade airfoil data are non-symmetrical.

The user must enter values for CL and CD for the complete

range of angles of attack from -180 to +180 degrees. If the

value of SYM is zero, only tabular values from zero to +180

degrees need to be entered. The above holds true also for

variable 147, SYMSPR. SYMSPR applies to the spar data

exactly in the same manner as SYM applies to the airfoil

data.

37. Items 108-109 - SFH FDMP

Values for the spring constant, SFH, and damping

constant, FDMP, about the flapping hinge can be entered if

known. These variables can be entered to simulate a

hingeless rotor system or a system with flapping springs.

36.

• ;•• • • :' • • • . .. • '' . ': ''r. .•i `•.. . '• .- i• /i- * " • : .Ho ", . . . 2. 1, ... •. -. . . ... .. . . . .. .; .. . .,4 -

38. Item 110 - ALOPT

This is one of the wind-tunnel options. If ALOPT is

a non-zero input, the program will iterate to obtain the

lift required of variable 74 but will ignore ,the required

drag, variable 75. In this option, variable 87, collective

pitch, will be varied, but not variable 88, inflow ratio, in

the program calculations. This is the opposite of the TOP,

varialbe 96, option.

If a lift curve slope less than ATEST, item 122, is

calculated while using this option, the program will stop

and produce the following message. "Stall Criterion has

been violated -- will go to next case, if any." Item 111,

the shaft angle, must be inputed. If non-zero values are

entered for both TnP and ALOPT, the program will do the TOP

Option.

39. Item 111 - Shaft Angle

Item 111, the shaft angle, must be inputed whenever

the TOP or ALOPT options are used.

40. Items 112-114 - RAlS

If variable 100, ABIT, is non-zero, the program will

iterate the blade flapping solution in an attempt to obtain

the desired lateral and longitudinal flapping angles

indicated by variables RAlS and RB1S, respectively. The

program will iterate to the accuracy indicated by TOLAB,

item 114. Item 112 is RAIS and item 113 is RBiS.

41. Items 115-116 - Delta 3

Items 115 and 116 are the tangent and phase angle of

a Delta 3 Bar. Information regarding the use of TD3B and

PED3B can be found in paragraph 16 on the Pitch-Flap

Coupling Angle.

67.

42. Items 117-118 - LAML

The program allows the user tc induce a velocity of

any form onto the rotor system. This is done in a harmonic

series of the form (AO + A1*cos(PSI) + B1*sin(PSI) +

A2*cos(2*PSI) + B2*sin(2*PSI) + .... ) for each segment that

the blade is divided into. 112 variable 117, LAML, is a

non-zero number, the program will enter the harmonics of the

induced velocities. Dnring the Read routine, LA9IL, will be

assigned a value of zero. Therefore, for each case where

different values for the harmonics are desired, LAML will

have to be set to a non-zero number. Variable 118, UVL,

controls the use of the induced velocities. If UVL, is

zero, the induced velocities will all be set equal to zero.

A short example will now be given on the use of the

control variables in a multiple case run. Assume that no

harmonic induced velocites are desired for the first case.

The user would make no inputs for LAML and UVL. For the

second case, assume that harmonic induced velocities are

desired. LAML and UVL would be set equal to a non-zero

number. The harmonic variables would follow the case input

data for this particular case. For the third case, it is

desired that the same induced velocites be used. The user

woul'4 not enter any values for LAML and UVL since (1) LAML

has been automatically set to zero and hence nc new hdrmonic

data will be entered and (2) U VL is still equal to a

non-zeru numb-,r. It is desired for case four to use no

induced velocities. The user now would enter the value of

zero for UVL and program will zero out all the induced

velocity variables.

This paragraph will descLibe how to use the variable

induced velocity option. The inputs jýl, B1, A2, B2,are numbers in units of feet per second. If the direction

of the velocity is down, a negative sign is associated with

the Ms and B's. A necAtive sign will decrease the angle of

II-~ -~I..' -, -

attack for an element originally at a positive angle of

attack. It will increase the amount of negative angle of

attack for a blade element originally at a negative angle of

attack. The values of A's and B's do not effect the

uniform inflow ratio, LAMBDA, that the program iterates

upon. The format for using the variable inflow velocity is

as follows.

1. A value for NHARM is entered in 13 format.

NHARM is the maximum degree of the harmonics

that is to be used. If the maximum degree used

is A3*cos(3*PSI), then NHARM is three.

The next three paragraphs are repeated for each

blade segment.

2. The value for AO is entered in E15.6 format.

The IBM 360-67 at the NPS will accept F15.0

format.

3. The values for Al, A2, A3, ... , up to NHARM are

entered on the next data cards in format

5E14.6.

4. The values for B1, B2, B3, ... , up to NHARM are

entered on the next data cards in format

5E14.6.

5. Three cards are required for each blade segment

whether or not the values are equal to zero.

If the user has 15 blade segmen' , a minimum of

45 data cards are required.

43. Item 120 - B3FL

Information -oncerning the use of BSPL can be foundin paragraph 33 entitled Number of Spar Airfoil Data

Segments.

69.

44. Item 121 - PPSI

PPSI represents a delta PSI for printout purposes.

PPSI has a default value equal to DPSI, variable 68. PPSI

shall never be a smaller increment than the incremental DPSI

used to calculate the soluticn.

45. Item 122 - ATEST

ATEST is the minimum acceptable value of the lift

curve slope when option TOP or ALOPT is used. If no value

is assigned, ATEST has a default value of 5 (1/rad). When

the TOP option is used it has a automatic value of -50

(1/rad). The lift carve refers to the increase in rotor

lift with the tilting back of the tip path plane. A check

on it for a minimum is for convergence purpose only.

46. Item 123- IGC

IGC is the Iteration Gain Control factor. If the

major iteration fails to converge, choosing a fractional

value for IGC can greatly speed convergence. This may be

especially helpful when parts of the rotor are in stall.

The amounts by which the convergence algorithm changes theindependent variables is multiplied ty IGC. Setting item

91, SKIPIN, equal to zero may help thQ user decide if this

IGC option might be useful.

47. item 126 - PCR

The pre-coning angle in radians may be entered her.

48. Item 138_= IN.L

Input 1.0 for INPL in order to remove hiuh momentinplane aerodynamic forces from the calculatioct of

aerodynamic pitch and rull moments about the shaft axis.

70.

-~'.

49, ItemS 139-1~41

Variables PSTS, PHIS atid THYS tripro-sint ai[cLal

angualar vetoci teu. The si rcraft cat, be givnn anry ara9l,[aL

vo31Oity iii yaw ,~S pi-tch (PHTS) ,4ni I~u1 ('CUFS) in

radians per: second by the~ uue of thazo varaLl1es. Threy

eftect the calcui81tiotim of UP, UT1 ani U1;. I O1L 9raill 50

COntairifi a141.tio11B1 irfofQFWati~rI.

50.

It tile angular Vý,ochitite, ii 01'j ¶39 - I'll aro

*ntotea, thoy arvI u~e4 Iin the caicu latic~nt cif U[, UT aril UP.

it no valuea toc items 1142 - 145 atq *atitrn4, th 1'LOQLCW

aumi5i2O0 tha.t thes i1.tor kiY br to's'o 4oaz 2 due. tu t~h" an9'p1az

yelocitie'O &Lout 4thq cantat 01 th'- 9114!t. It vuljizu; ALq

antored fOE 1i0,w 142 - 145, thui callCuatiw'.~h 4111 a&hum-'

01&1,t tijO Cnt ii. 'rtot xh/Aft li i Lta 'i ij ai~mOul tbe'.~3l. ut

9! avi ty. Vatiab1ez 142 - iLUJ daro P rG , PSj, W; dPiraA~

r *c a im thi, 9 ignitudi4nal c; pozitton m i,*j f,;2I mvi X tllg

I v n g ri 6 1 Y ýiA ,Q A.1 1.C; . Ot thý- Maii 1, ?o~ctO . iVL(.4~ is ti-- (.G

01 VOLk tILj 1ý0410ti-a All if thIatia v~iugn Wt1;a;t L". uia¶VI ad

ill units of jinchotti

51, LO.-ak:ý3

If tht' tipar dat~a ear. xy.Invori[iIi, 5(j ne.,t hii t iv a

non- 8yh1.otLical~, qnftar any hIIQ--p1L1[) Y11)114 tol, NYMt1d'I' Tio,

pi o0'r*n Will. LaI'J01. VA114 fal!)M trm 1110 I U 4 160 11:ýl 119

53. Vu_>t>.ZAL

Irn or,1eL for the program to pr3perly calculate the

oejion in and around the reverse flow region, the valui s of

CL and CD are required to be known at high and low angles of

attacks appproaching Th0 degrees. However, in order to save

the user. from having to enter a whole range of angles of

attack for all Mach numbers, the program has an option where

for angles above HIALFA, 156, and below LOALFA, 157, the

Mach number is set equal to zero for table lookup purposes.

The U3SEr is only required to enter large angles for the

firnt two Mach number tables. HIALFA and LOALFA have

dufatilt values of +30 and -30 degress, respectively.

54.Itm½ :19Tnwe

The GRP pr-qaram will properly calculate the airflow

Swru, anglo, U'T, UR, arid pitch angle of a swept tip airfoil.

TIL1SWP i.s the amount of swuep of the tip measured in

Thgr•ee. Tf 'AST is tht hiAt. sergment nurnhcr at which 1he

swaee begi-I.. The program assumes that the remaining

outbonia ugmt.ntU Startihg with TPSWST are swept the number

of lugrue, indlcatod by TIPZWP. UT and U[ are modifjoed for

ti1Lh,. scgmontr. is shown is Figule 11.

55. aAQ.:IL

If TVIM i s a non-zero valuEý, the program will

attempt to adJu.n;t Al', arid DiS in order to reduce the rolling

an ,. ithtcnh',j iwuents to l,,.ess than plus or minus 100 pounds.

It w.ll I itt obtail a solutiont which will satisfy the

LqPui-LL Id lift ai d dray, then1 it will adJust AiS arid BHS toredauvu thy rnOnig1jts.

56. Lti b=15- RB

V.i tablet; 161 - 175 ass;i,jgn thL proper airfoil data

tu blad' s'gp,.nts one through fifteen. The -:rogram will

72.

Us-

uI

JRC-OsAt + UTsin-At,

UT =UTcos-L Mi-A,

Figure 13- Tip Sweep Calculat'on Diagram

accept up to five airfoil data decks. The RB array is

initialized with the value of one for all 15 segments. if

only one blade airfoil deck is used, there is no need to

enter any values for RB. Only enter values for the segmentswhich will use airfoil data sets two through five. If no

spar data are entered, SPAR equals zero, segment number one

is considered the cut out segment and is assigned the value

of RB(1) equals one for calculation purposes and the valueof RB(1) equals zero for printout indentification. In this

case, segment one produces only drag but no lift. If SPAR

is greater than zero, RB(I), I = 1, SEAR, will be assignedthe value of zero for printout indentification. These

segments will use the spar data entered and will calculate

both lift and drag.

I. CASE INPUT FORMAT

Below is located the format that is used to input case

data to the program. All input data cardb are cf the format(12, 14, 5F12.0). The input cards contain se.vvrs flalds

which are called NNUM, NLOC, C(1), C(2), C(3), C(4) ard

C(5). An example of a data input card is as follow.5 1 700. 26.8 1148.6 .002246 4.0

1. NNUM is the number of inputa on this card,where C(N) are the inputs. NNUM must appear in

either cclumn one or two. NNUM has a minimum

of one and a maximum of fivoq. In tho abovoexample NNJM is five.

2. NLOC is the item or variable numbtL of inputC(1) This refers to the item numbers that. ar'

found in the section on CamE Input Lintitgmj.

NLOC is in 14 format and mut;t be right.

justified in columns four thr-u9h nix. In thqexample NLOC L'ifera to itum nu tinbar ouni, tIk

Rlotor Tip Speed.

74.

3. C(1) through C(5) are the values correspondingto the input items NLOC through NIOC + NNUM.

Each value must contain a decimal point and be

in columns 7 - 18 for C(1), 19 - 30 for C(2),

31 - 42 for C(3), 43 - 54 for C(4) and 55 - 66

for C(5). In this example, values are entered

for variables one through five, the Tip Speed,Radius, Speed of Sound, Air Density and Numberof Blades.

4. Omision of NNUM will cause program terminationwith an error explanation statement. NNUM

greater than five will cause unknown problems.

Omission of NLOC will cause the present card'svalues of C(N) to be entered into the items

indicated by the previous card's NLOC. Failure

to right justify NLOC, or NLOC greater than

200, will cause unknown problems. Failure toproperly locate correctly any input value

within its own field on the card will cause

o*troz in both that 1,nput and tne number whose

field it encroaches on.

J. CASE OP'T'IONAL OUTPUT INDICATORS

Tho program haa two variables that control the outputprLntout, variable 91, SKIPIN, and 98, PRINT. The program

will automatically produco a printout of the case input data

for each case and a one-page summary of the inital

confld•on'4 antd iteration limitations the user placed upontha |prorjLam. It will also produce a one--page summary of the

rsulting forcus, moments and calculated rotor horsepower

tor thi rinial con!verged solution. The user can also receive

an acho pLintout of the airfoil and spar data decks. If

thin acho printout in desired, the user will find the

cc;rrqct pr.rintout indicators described in sections C and D

7 :5.

entitled Blade and Spar Data Deck Requirements.

The main optional printout variable is variable 98,

PRINT. PRINT can be a number from 1 to 1,111,111 depending

upon the option desired. If no value is enter for PRINT,the user will receive the printout described above. Below,

is listed the PRINT Options. If, fcr example, PRINT is

assigned a value of 111, the user will receive printout

options 1, 10 and 100.

OPTIONAL OUTPUT INDICATORS

Option 1 Angle of attack, Mach number,

section lift and drag

coefficients, inflow angle, lift,

and sweep angle at each azimuthal

position for each radial blade

segment. Only for converged

flapping solution.Option 10 Converged flapping angle, rate,

and acceleration at each

azimuthal positicn.

Option 100 Converged integrated forces on

blade at each azimuthal station.

Option 1000 Harmonic analysis of blade forces

for converged case.

Option 10000 Harmonic analysis of air loads

for ccnverged case.

DEBUGGING OR TRANSIENT OPTIONS

Option 100000 Transient flapping angle, rate,

and acceleration at each

azimuthal station.Optj 1000000 Option 100000 plus blade

velocities, angles, Mach number,

section coefficients, and lift

for each blade segment at each

azimuthal station.

76.

~ ,-- - -- - - - - - -

The user is cautioned that the debugging optionscan give a huge amount of output data.

The second printout option is the variable SKIPIN,number 91. This variable controls the summary fcrce, momentand horsepower output discussed in the first paragraph. IfSKIPIN is greater than zero, this summary will be printedonly for the final converged solution. It SKIPIN is equalto zero or a negative number, this summary will be printedfor each loop through the major iteratlon (force summation)routine. SKIPIN has a default value of one. If the programis not converging to a solution, the user can sezimmediately, with very little extra printout, exactly whatintermediate solutions the program is producing by settingSKIPIN egual to zero. This may help the user in decidingwhether or not to use the Iteration Gain Factor, IGC,variable 123.

K. IBM 360 EXECUTION CONTROL CARDS

This section illustrates the control cards required toexecute the GRP program using the IBM 360 at the NavalPostgraduate School. The program may b4 run under Os orCP/CMS. There are two wayj of running th* program undar 0.The first is to run the entire program and data through thecomputer at the same time. The seccnd way in to gtoLe th,@main program on a disk as a library program and anteL onlythe data through the card rea4er for each deuirtd case. Thesecond method has the two advantagAs of (1) not reqtjirinqthe user to enter the entire 1100 card main prcgrarm througjtithe card reader for each run and (2) the amount of CPU time

reguired can be reduced nsince tho main protgram does not. hav.vto be recompiled for' each run. 7he J)L oqtam LQUirL;Q

approximately one minute and forty secon,!s of C1,U ti mv tocompile. The normal run time for 4ach cai• it |,pipoxlwatvly

77.

VA:

20 CPU seconds. This can vary with the amount of printout

data requesttd. If the program is compiled on the CP/CMS,

the user will have to request 344K bytes core size on the

login message. The standard 256K bytes core size is not

large enough for compiling. However, once the program has

been compiled, it can be executed within the 256K normally

available.

The following cards are required to execute the entire

program through the card Leader at one time.Standard Job Card

// EXEC FOfTCLGPEGION.FORT=150K

// REGION.GO-180K

//FORT.SYSIN DD *

Main GRP Program/"

i/GO.SYSIN DD 0

Came Input Data

The following two programz are used to reserve space and A

load the progrrm onto a .isk.

Standard Job Card

// EXC GMZEFBRl14

//LOAD DD DSNaS1395.HELOIJNIT-3330,VCL=SER-DISO01// DISPm(NEWKEEP),LABELURETPD-15O,SPACEu(CYL,(1,1,1))/" !

Standard Job Car&-

// EXEC F08TCL,REGI0N.FOHTw180K

//FOPT.SYSIN DD 0

Maim GRP Program/0

// LIK.SYSLMOD DD UNIT13330,VOLwS5.RuDISK01,

// VSN-31395.11EL0(GRP) ,DISP-11ETho 51395 unoed abovo and below must be change to S for

atudg4rt oL F for faculty with the a[propiate user number

instead of i395. The fuilvwii.j c a Li mus t be u c to t

76.

, !-

execute the program once stored on a disk.

Standard Job Card

//GO EXEC PGM=GRP,REGION=ISOK

//STEPLIB DD UNIT=3330,VOL=SER=DISK01,DISP=SHR,

// DSN=S1395.HELO

//FT06F001 DD SYSOUT=A,DCB=(RECFM=FBA,LRECL=133,BLKSIZE=3325

//FT05F001 DD

Case Input Data/*

J. Sample Program Output

This section describes in deta.l the output available

from the GRP program. In addition, a sample computer output

is included with each describtion. The program will printup to ten different tables. Seven of these tables are

optional and are not automatically printed. The ten tables

are as follows.

1. Echo Printout of Rotor Blade and Spar Data

2. Case Input Data Card Listings

3. Summary of Input Data

4. Debugging or Transient Information (Options

100,000 and 1,000,000)

5. Summary of Forces, Moments and Horsepower

6. Converged Flapping Sclution (Option 10)

7. Converged Integrated Forces (Option 100)

8. Harmonic Analysis of Z Force (Option 1000)

9. Converged Blade Analysis (opticn 1)

10. Harmonic Analysis of Air Loads (Option 10,000)

The input variable PRINT, item 98, controls the output

of items four and six through ten. Items two, three and

five are always outputed. Item one is controlled by the

first data card on the rotor blade and spar section input

decks.

79.

1. Echo_2otor Blade Printout

The next page contains a partial sample Echo

Printout of Rotor Blade Input Data. This printout

illustrates that (1) the printout was requested (WBLADE =

10.), (2) there are two airfoil decks to be entered, (3)

there are nine Mach numbers for which CL's are to be read in

and (4) the remaining portion of the Printout is the values

of Mach numbers, angles of attack and lift coefficients for

the first airfoil deck.

VI

I.-_

I-

eo.-

-----------

W1 vv"". -4 pa ' ýn j k -f d WW..,:It,,RWI.Ji~cflC.Jt'tJ U1Q)4'JL .LJDWIL u't'.L tVLQt rio 0004JW U'JCCA CtCC

Q^C3UQ tUQCOU OgaC-tar.Mwran .Cwufl rJnh C'4NW1 '.JN(N7 W -SW 04' en'4Ju' ffN 'UO$IILt~t .~.J'fl~i'w 't'J i fiJ geIeJ v~sJ uU ' s uia e.ec, r4 4s

LJOOJ~tJUO UaC.LLJ.JJOO 0tf Lwur U#01J0 QJij Oflw 5104J cJtut tW iwin

I I i l 0 f 1 1 i i lLLI0 (WeU~b LIýC-aiepC 4L.Mryflpq Ct-iiJ4C4tNfbflt 'c-N~N~uIJ

I4mea If - I , t'J i I of 11~.~ 11e- 1. it- N

LJ UUJ?11444 J ýIzv Ulf, (C~a V10M .a m 0'41 I ,*-etj1 IfIr .11 .4I"1-4-r V rl-S W4e- eAJ-w I .cM.JZ'401s1'W

41 004 'i'JOpUý ~ W e0Plr~ uJu.eti.-alf "AJW4wip

mamma-- ý-fi

2. jThe next page contains a sample Case Input Data CardLicting. This is one of the automatic printouts. IT is anecho printout of the input data cards

III

I

82.

Qj OjuO "r N-J'A 0r a-CJ tpU-d aNPrf~ri~pu~ON

-fut "1140J4t 41fe~w

OUO -4-UlU -4.2.q 30-

o kJLJ rl--'ljp0-4#r ' UUk ot

* c 4*P *..* tpp- t

4- r3y"~

ENIL Lr O'"j o",

ru k. i' (3M'.. , CO

U- I W' .. . d -.1 u um"j -,u j'Ot

I -- f

3. Summar 21 Z Input__2412

The next page contains the automatic Summary ofInput Data printout. The foll.oving information can be seenon this sample output.

a. The blade was divided into 15 segments startingfrom the hinge offset and proceeding outward.

b. No values were entered for the local massdesnsity, i;uput variables 38 - 52. Instead, values for theFirst Moment, M(M) - 85, and the Second Moment, ROM -INERTIA - 1450, about the Flapping Hinge, inpuV items 78 and79, were entered.

c. The I row indicates the calculated centers ofeach segment expressed in a pe-centage of the distance outthe rotor blade.

d. The BLADE DECK row indicates that thei firstblade segment was considered a Spar aegmont. 5egments twothrough five and twelve through fifteen belong to airfoildata deck number one, while segmenta six through eleven

belong to airfoil data deck numb*r two.a. The rest of the information, with one uxcoption,

is a summary of the cabe input data. The exception is theterm THRUST FACTOR. This in the value umed to

nondimensionalize all the calculated forces In the trogi#m.

The THRUST FACTOR equals v •. - oments arenonditmensionalized by the THRUST YACTOP times thai radius.

f. Thp 1jJo9LaI check.: to s -@ i f all tho bladetegi 's, delta X's, plus thk, diatancv from the shaft to the

hinc- cbit, /R, -_5d up to ono. If, orn the In:.intout,SUMID), * i/ d does not equal one, tho user ha. me,1# amihta|& iam~ iwhsi-re with thei delta X's ol in thO Xl/R 1lUlIjuL.

c (

s Uwi WA LI

.1 t1

6"e• I

V. A, .. . ... . 01*I v r" a. ns *i S1 I IilIid l

, -: I " U. * d

A -% ((w - m4

on 41 48

Ad tv gll

* LiP - i ee uI L

Ad "I * . I es

* -f I I• 9~ Cq -, VI VI - - , tutu • cv il

IIS LIl iiiif 0 IA - *t ii•Siit5 bi . AS

: a a'" = <, ":_ -n,,

. U , , Is ,..

• P '' i a 4 h v -S. U * - i

aI a AL

_ _ I ii vi I.a, _. : . .. .. . . I, .. A *,, ,

* fw, a..& a e h

F l- e e, U -,, ', U ,- "Q"'IU . ,- ti ,.

* P.I I C I C 9 i I l l

* IilU 1 4! -, *

F 59 - 4 A A. f * -

S. -" A '* I- A* *-- -vl 0 C flu

* 4 I teI ii ,

a. d 4. t i j l4 m diit f

£ - he Li U C L

L,,

a~~~~A 0 . tA q a i - a

S.~I a1 *-% i L ME ~ . S . 5a* a U w w a U * LU ftd

* 0'3 L a tIA .4 4 hhLICW*WU a o

* P.- t iF - U ~ aI,. iF ~ ~ ~ ~ . to - do A - - 4i 5

F~ I A#I J

*4 r. so, W~

t- ' 11 j e t CO~ j . 6

F~6 U S C i

ml I- IV

As 4i

do up IN F& U

F~~ do ft.

- 1e~FI

a~C 0AU w u n w

.0 CA tia I

P~w dl. 49 cJW c l1

W, i I ne . ig -. tAO Is, Li'A U -

4. e2_&bJUn T_ ansient Information

The next page illustrates the Debugging or TransientPrintout. This is PRINT Option 1,000,000. The informationavailable includes flapping angles, rates and accelerationsat each az.uutfal position with a radial position display ofUP, UT, U, PHI, Angle-of-Attack, Hach Number, CL, CD andLift Per Inch produced.

This option is generally outputed only when the useris experiencing unknown difficulties with the GRP program.The program will output all of the above information forevery revolution and iteration until a ccnverged solution isobtained or the program runs out of allowable computer time.If desired, the variable SKIPIN, item 91, will provide aprintout of the Forces, Moments and Horsepower Summary aftereach major iteration.

k96.

S - Z

MI0Nr r 104 ý f 1 L r f

-~~~~ t N4r - N~.u~J "-'J ,in

ww C.Jr '. Z -EN II flf,

o ~ v ca C-3 I.' I- u4 O~ Q~.

~ -~oc~o ~ naao~*oouo

-r fli LA M e

QOJ* C .** * *.. *. ** ** ** CCD

O~cOOOO~ COCJOQOOOO onoOOunoco

o ~~~o-'~'r'- - Z Nt m7aQ'(.n - olouNo-.MO.D0I-

'r -P a-- . I IW

u4 .--0l OO L* "M U.'IJw'-r '

~~~~" UOJ ~ \~~ VC

4.) Ir. 4-UO Luj U N L -r vr)

C- ff I **f CUQ 4 Jr4C.*r*4 % - r ý

0 -14-'-'I'm '2~ I.m r-4 r,4 Ne, N. ' - L.Lr-r CO

' C, 0, **.. r co .C* C. ,

VI !~ ~ .c441 NrJNIN.d4 V` N N r~j7- rl-c'

II II P-

II o rr F-~ L -o (1c em > fl* uU, 7 U-- L;r'--. .. .0 > ~ ~ J- 4n.-, c7 On

m- d' L~n UNU" N V V W,-'rU- I- pC .1411%0 n ()7j I 0j~~r 10 M.WC -7 r~- Nm0

C J.UM4 ((iLr&. o. tA te. UN 10LuVIud'.',u-m.,

k^ ý4M.?Q'IWU I/ qJ4,O IN LIA F.-PyF-0 ,or-N .Im

a a

The next printout illustrates the Summary of Forces,

Moments and Horsepower. This page is automatically outputed

for the converged solution. A printout of this summary can

be obtained for each loop through the major iteration

routine by the use of the input variable SKIPIN, item 91.

The following information can be observed from thic samplo

printout.

a. The cyclic lateral and longitudinal inputa, A15#

D1S, A2Z and B2S are printed at the tcp of the page. bhis

exaskle indicates that there were no second harmonic inEuts

for 42$ and b/5, which is normal.

b. V$ETA .75 is the pitch angle of the blade at the

75 percent radius ctation at the PSI equals zero azimuthal

posi tio", in moat options the program will iterate upon tho

values for Tnrn .75 in its search for a converged solution.

c. LAti:! refers to the converged value for the

uriform Inflow ratio. If a rat. of climb o 4dencvut we'

used in• thy G~an, that rate divided by the tip speed woild

have to le subtractd tiLO or added to this value of LAMODD,

respectivviy.

4. MU (Y) S ailid MV (Y) t, are the advance LatIhs ill the

longitudinal and 1at6'Ldl dxteWtio nS vt1Ath xnp|eut to th1

shaft axis.

e. CT, CQ, CII, CL and C or'* th: ca¢lculate•d verall.

coefficient; of ThLust, TVr9uQ, 1I Pet', Lift and NLag, All

of those items, are nondlmonuionaliz'i4 by th, value of the

Thrust Factor.

f. Lift oad lnrag forcus are Galculated with r'4sjuct

to the tqlativm winl axim. Thruwt ai0 It I ft;ceN SI

calcu-Iatei with respuct to the contiol axis. Z forleIs ae

calcuated with rebpact !) the shaft axit, X and Y foti.iw

are calculat ed pVei pordiculat to th a hf t. axis.

g. Th. Equivalent DIg Iq thi total dioa fcrcv

created by thu fuuelage, flat plate irea tiuuu dynamicý

pressure, plus the profile drag created by the turning rotorsystem.

h. The Equivalent P. A., or Equivalent Flat Platearea is obtained by dividing the Equivalent Drag by the

dynamic prossure.

I.. Alpha(S) is the shaft axis orientation whileAlpha(c) is the control axis orientation. The program usesmomentum theory to calculate Alpha(S) . Alpha (C) isdetermined from the following relationship, AiFha(S) =

Alpha(C) + DIS.1. LAT. DLS. and LONG. DIS. are the lift vector

offset as a percentage of the rotor radius frcm the rotorshaft to create the program's rolling and pitching moments.

K. PM and RM are the calculated aerodynamic pitchand roll moments about the shaft axis. The SHEARS Hub Pitchand Roll Moments are calculated from summing theaerodynamic, inertia and elastic hub restraint moments.

•1 ¶.

x U.

"M CJ U LU USI

o i 4~L 0., AI Li 10

rft LI ' - ~ f

m't UN 6. UI 7 0

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it .- L1 . I U .- S1 LI - U 2 I £ I

14 w 11*1

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6. LEa£Ljl.

The next page contains a sample printout of thq

converged flapping solution. The first item tc appear arethe flapping values at the PSI equal zero azimuthal position

for each revolution prior to the converged revolution on thefinal iteration. The values for flapping angle and its

first two derivatives are expressed in radians.the second part of this printout is the actual

flapping values for the converged revoltion. The differencebetween the flapping angles and rates between the PSI equalzero and 360 degree position must be less than the

tolerances entered in variables 101 and 102. The numbershere are also in radians. The last itom is a rourietrcoefficient series for the flapping angle and it iscalculated in degree.s. All calculations are done in

respects to the shaft axis. This is PRINT Option 10.

91.

q1 ar e ywde 4.ie e tdee

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Ic, t .. tl I l' .

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a it / .' , 'q't ,e,,I i ,-i , bdll !

El '-El S • ! ll!l -, -

* ,4,,• / ,7 . . . •,• l

7. Force In t ecation

The next table is a sample Force Integration Output.

This is PlaINT Option 100. The forces are a printout for one

blade only at a particular azimuthal position. CQ, CQL andCQD are the Coefficients of Torque, Torque due to Lift andTorque due to Drag. CQ is calculated from CQ CQD - CQL.

CX, CY and CZ are all related to forces in the shaft axis

reference system. CMHS is the Coefficient of PitchingMoient due to aerodynamic, inertia and hub elastic restraint

moments about the Shaft Axis and CLHS is the Coefficient of

Rolling Moment due to these same forces. In the printoutthe (B) character is the number of tlades, which in this

example is four. SIG is the solidity of the rotor system.MAX B*CQD/SIGMA is a blade stall indicatcr. It is

calculated by determining the first azimuthal stationbetween the PSI equal 180 degree and 360 degree positionthat has a value of CQD greater than the value of CQD at the

P:SI eyual 180 degrees azimuthal position. j

vi

4i

t~ ~,.

I ,a a I I a i 1 1 1 6 i

.........................*... .

.4 . A A. . A 4A ai. 4 A .. . A I A AwI'all U111L. l l-- 1.111 ill6. II-Lj- L

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8. Harmoni Analysis of Z Force

The following is a harmonic analysis of the forces

in the Z or shaft axis direction. Z force is positive in

the downward direction. This is PRINT Option 1000.

95.

Q* 0nN L)PUN 'Oct 9=o

* m

ZH 00

M04 IflP LADtn 0 CM

000

04 IH 0Q

=ntn Y C14 tntn '-~

0-U,4

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C4 14~ -

op

9. converqe'9_Blade An a lysis

The next four pages contain the sample overallsummary of events occuring on the blade at each azimuthal

and radial position. This is PRINT Option 1. Variable 121,

PPSI, controls the azimuthal intervals that are printed out.

The output items are as follows.

a. X - Center location of the blade segment

b. ALPHA- Angle-of-Attack

c. MACH - Local Mach numberd. CL - Local Coefficient of tift

e. CD - Local Coefficient of Drag

f. PHI - Local Inflow Angle

g. L(LB/IN) - Lift produced per inch on segment

h. Sweep - Sweep Angle of airflow

9

97.

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U Li 42L Li'O~i~O~iLJU OLJ U 4UU~t.LiUO%4b..JaU~i Li i u U~aa fl~JI~~t .U *~4 I 4 44*~.4 9 4 *44.449

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'A ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ = ol No uNVC'i..tLIW~ll (i ' iLC41.LLJWW4~4L 41 C.u14IN¶

10. Harmonic of Air Loads IThe next printout contains the harmonic analysis of

the lift generated per inch on each rotor blade segment.

This is PRINT Option 10000. The airloads are computed in

two harmonic forms, both containing terms out to and

including the tenth harmonic. The forms are

Lift per Inch = AO - Alcosv - Blsink- A2cos2'e -

B2sin2'q - A3cos3V - B3sin34 ..... - A1Ocos10Y -

B1Osin1OV and

Lift per Inch = AO - C1sin(+4,) - C2sin(2'f4÷)

-C3sin(3Y+V ..... - ClOsin(10O+.÷6).

The Ratio column is determined from the coefficients

in Column C. From this ratio, one can immediately determine

which airload harmonic is dominant and the relative

relationship of this to the other harmonic coefficients.

102.

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I--

107.

IV. JRP SAMPLE ANALYSIS

The GRP program was executed using data representing a

relatively new rotor blade. The results were compared %ith

results predicted by the blade's manufacturer. The rotor

blade was an unsymmetrical blade. The rotor radius was

diiided into three sections. Sections one and three were

made of the same type airfoil. The blade included a sweep

tip design. The blade and helicopter configuration analyzed

are typical of a helicopter that could be used by the U.S.

Navy in a LAMPS type mission.

For the analysis, the blade was divided into 15

segments. The program used the manufacturer's values for

the First and Second Moment of Inertia about the Flapping

Hinge, vice local blade mass densities. The GRP assumed

uniform inflow for all flight velocities . It used a rigid

blade analysis while the actual blade does have live twist.

The program was run at five different flight weights,

ranging from 16,359 to 20,829 pounds. A flat plate area of

35.8 square feet was used at all speeds. The GRP was run

for forward flight speedz of 40 to 160 knots at ten knot

increments. The program was executed at sea level, tropical

day condition. The manufacturer's predicted rotor

horsepower was obtained from his Shaft Horsepower versus

True Airspeed curves and was corrected to rotor horsepower

by using the manufacturer's Mechanical Efficiency curves.

The results of the analysis are shown in the next

several tables. Table VI illustrates a comparison of the

GRP required rotor horsepower divided by the manufacturer's

required rotor horsepower. The GR. agreed within an average

of two percent on the entire range frcm 40 tc 160 knots.

The GRP agreed within an average of one percent for the

cruise range between 70 and 140 knots. It can be seen that

between 40 to 60 knots there is a much larger difference

10.

between the two required horsepowers. It is felt that the

inflow in this region is not uniform as assumed, but highly

mixed and irregular. Also, the GRP results were less thanthe manufacturer's horsepower in the 150 to 160 knot range.

This area represents the region of top speed for the

helicopter, and much of the retreating blade is in the stall

region. Also, it is expected that there is a change in

fuselage attitude at this high speed, which would increasethe flat plate area above what was used in the program. TheGRP program's maximum endurance velocities agreed exactly

with those predicted by the manufacturer.

TABLE I WEIGHT = 16359 LBSVELOCITY GRP RHP MANUFACTURER'S RHP RATIO(GRP/MAN)

40 1105 1152 .96

50 1006 1041 .97

60 960 973 .99

70 955 958 1.00

80 984 971 1.01

90 1047 1034 1.01

10V 1142 1109 1.03

110 1273 1237 1.03

120 1438 1396 1.0J

130 1643 1589 1.03

140 1893 1852 1.02

150 2300 2216 1.04

160 2567 2745 .94

109.

TABLE II WEIGHT = 17321 LBS

VELOCITY GRP RHP MANUFACTURER'S RHP RATIO(GRP/MAN)

40 1185 1247 .95

50 1075 1122 .96

60 1018 1038 .98

70 1008 ;009 1.00

80 1028 1028 1.00

90 1086 1078 1.01

100 1176 1153 1.02

110 1303 1264 1.03

120 1465 11445 1.01

130 1669 1633 1.02

140 1921 1902 1.01

150 2227 2257 .99

160 2598 2791 .93

TABLE III WEIGHT = 19246 LBS

VELOCITY GRP RHP MANUFACTURER'S RHP RATIO(GRP/MAN)

40 1366 1449 .94

50 1229 1301 .94

60 1148 1186 .97

70 1118 1142 .98

80 1126 1153 .98

90 1172 1199 .98

100 1255 1273 .99

110 1375 1387 .99

120 1530 1557 .98

130 1731 1746 .99

140 1983 2038 .97

150 2290 2440 .94

160 2665 - -

II0.

-. -- 't't t

TABLE IV WEIGHT = 19658 LBS

VELOCITY GRP RHP MANUFACTURER'S RHP RATIO(GRP/MAN)

40 1407 1496 .94

50 1264 1340 .94

60 1178 1231 .96

70 1142 1173 .97

80 1149 1179 .98

90 1193 1221 .98

100 1274 1299 .98

110 1392 1414 .98

120 1546 1566 .99

130 1747 1773 .99

140 1999 2048 .98

150 2305 2477 .93

160 2682 -

TABLE V WEIGHT = 20829 LBS

VELOCITY GRP RHP MANUFACTURER'S RHP RATIO(GRP/MAN)

40 1532 1627 .94

50 1370 1457 .94

60 1269 1338 .95

70 1222 1267 .96

80 1219 1258 .97

90 1256 1297 .97

100 1330 1375 .97

110 1445 1477 .98

120 1598 1638 .98

130 1798 1866 .96

140 2045 2157 .95

150 2354 2589 .91

160 2747 -

2-11.

TABLE VI RATIO COMPABISON

WEIGHT RATIO

VELOCITY 16359 17321 19246 19658 20829 AVERAGE

40 .96 .95 .94 .94 .94 .95

50 .97 .96 .94 .94 .94 .95

60 .99 .98 .97 .96 .95 .97

70 1.00 1.30 .98 .97 .96 .98

80 1.01 1.00 .98 .97 .97 .99

90 1.01 1.01 .98 .98 .97 .99

100 1.03 1.02 .99 .98 .97 1.00

110 1.03 1.03 .99 .99 .98 1.00

120 1.03 1.01 .98 .99 .98 1.00

130 1.03 1.02 .99 .99 .96 1.00

140 1.02 1.01 .97 .98 .95 .99

150 1.04 .99 .94 .93 .91 .96

160 .94 .93 - - - .94

AVERAGES FOR ENTIRE SPEED RANGE

AVERAGE 1.00 .99 .97 .97 .96 .98

AVERAGES FOR CRUISE RANGE

70 - 140 KNOTS

AVERAGE 1.02 1.01 .98 .98 .97 .99

1i21

V. CONCLUSIONS

The logic and theory used in the GRP program was

investigated and found to be sound. However, there were three

discrepancies in the Navy's version of the program that did

require attention. The calculations in the reverse flow section

on the rotor were incorrect, the calculation of the chord at the

75 percent radius position was incorrect and the original Trim

Option for reducing moments would not work. All of the above

discrepancies were corrected.

Three desirable features were added to the GRP program.

First, the ability to analyze a rotor blade composed of more than

one airfoil type was added. The program will now accept up tofive different airfoil data input decks for use in analyzing arotor system. Secondly, the program would only calculate performance

in level flight. The ability to calculate performance in climbs

and descents has been added. Lastly, the program will now calculate

Lhe aerodynamics for a swept tip rotor blade design.

The results of the sample analysis described in Section IV

indicates that the program does produce highly accurate performance

predictions. The averaged GRP rotor horsepower was within two

percent of the manufacturer's data. The results were within one

percent when compared in the area of normal crv.ise flight. The i

GRP, in this analysis, assumed uniform inflow, constant flat plate

area and a rigid rotor blade. It is felt that while complicated,

computer-time-comsuming procedures can be taken to reduce these

assumptions, they are not warranted if the GRP is to be used

strictly as a helicopter performance prediction program.

113~.---.---

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LIST OF REFERENCES

1. NACP4 Report 487, g ArdnmnAi~yi gteAl

Resals y John B. Wheatley, 197

2. NACA Report 716, A 4implifit- method pf ngtprmiiringt

by F. J. Bailey, 1941.

3. United Aircraft Research Labortories Report R-0725-3 A

EX12 0 y Nelson H. Kemp, 1tb

L4. Gessow, A. and tivers, G. C Ardng~ c no. %9 9 he.Hlcotr Fredecýick Ungar Pilishin-o, 97

5. Sikorsky Aircraft Roport 50355 GapgýYd gtPerformance M4ýhnA,~ by W. H. Ger~ee, September 1963.~

6. Army Air Mobility Research and Development LaboratorReport TR-74-1 0A,ýtrýf A ~~tSmlto

M. Davis, et al, C lapter 3, June194

7. Bramvelli A.R.S., Helicopter Dynamics, John Wiley andSons, 176

137.

INITIAL DISTRIBUTION LIST

No. opies

1. Defense Documentation Center 2Cameron StationAlexandria, Virginia 22314

2. Library, Code 0142 2Naval ostgraduate SchoolMonterey, California 93940

3. Department Chairman, Code 67 1Department of AeronauticsNaval Post raduate SchoolMonterey, California 93940

4. Professor L.V. Schmidt, Code 67SX 1Department of AeronauticsNaval Postgraduate SchoolMonterey, California 93940

5. Professor M.P. Platzer, Code 67PL 1Department of AeronauticsNaval Postgraduate SchoolMonterey, California 93940

6. Commauding Officer 2Attn: Mr. George Unger• AIR-530132BNaval Air System CommandWashington, D.C. 20361

7. Lt James W. Loiselle, USN 1251ý Granada CircleSpring Valley, California 92077

138.