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NASA TECHNICAL NOTE -8-

EFFECTS OF ROTOR MODEL DEGRADATION ON THE ACCURACY OF ROTORCRAFT REAL-TIME SIMULATION

Jucob A. Houck und Roland L. Bowles . I ? , I . ( I

Langley Research Center ' I : ,

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N A T I O N A L AERONAUTICS A N D SPACE A D M I N I S T R A T I O N W A S H I N G T O N , D. C. DECEMBER 1976

https://ntrs.nasa.gov/search.jsp?R=19770009078 2020-05-01T03:35:04+00:00Z

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TECH Ll0RARY KAFB,NM

IIllill11111llllllllll11111lllllllll1lllIll 0134095 -

1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.

NASA TN D-8378 I 4. Title and Subtitle 1 5. Report Date

EFFECTS OF ROTOR MODEL DEGRADATION ON THE A d C U R A C Y December 1976 OF ROTORCRAFT REAL-TIME SIMULATION 6. Performing Organization Code

7. Author(s) 1 8. Performing Organization Report No.

Jacob, A . Houck and Roland L . Bowles L-1108310. Work Unit No.

9. Performing Organization Name and Address !745-01-01-01 N A S A Lang ley Resea rch C e n t e r Hampton, V A 23665 t -11. Contract or Grant No.

13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address

i-T e c h n i c a l Note

N a t i o n a l A e r o n a u t i c s and S p a c e A d m i n i s t r a t i o n - _ ~~

Washington , DC 20546 14. Sponsoring Agency Code

..~ 1 - ~~

15. Supplementary Notes a r t o f t h e i n f o r m a t i o n p r e s e n t e d h e r e i n was i n c l u d e d i n a t h e s i s e n t i t l e d "Computa t iona l A s p e c t s o f Real-Time S i m u l a t i o n o f R o t a r y Wing Aircraf t" s u b m i t t e d by Jacob A l b e r t Houck i n p a r t i a l f u l f i l l m e n t o f t h e r e q u i r e m e n t s f o r t h e d e g r e e of Master of S c i e n c e , The George Washington U n i v e r s i t y , Hampton, V i r g i n i a , May 1976.

. - . ~ . -~ ~. . . ~ - -.. ~ 16. Abstract

A s t u d y was conduc ted t o d e t e r m i n e t h e e f f e c t s o f d e g r a d i n g a r o t a t i n g b l a d e e l e m e n t r o t o r m a t h e m a t i c a l model t o meet v a r i o u s real-time s i m u l a t i o n r e q u i r e m e n t s o f r o t o r c r a f t . T h r e e methods o f d e g r a d a t i o n were s t u d i e d : r e d u c t i o n o f number of b l a d e s , r e d u c t i o n o f number o f b l a d e segmen t s , and i n c r e a s i n g t h e i n t e g r a t i o n i n t e r v a l , which h a s t h e c o r r e s p o n d i n g e f fec t o f i n c r e a s i n g b l a d e a z i m u t h a l advance a n g l e . The t h r e e d e g r a d a t i o n methods were s t u d i e d t h r o u g h s t a t i c t r i m compar i sons , t o t a l r o t o r f o r c e and moment c o m p a r i s o n s , s i n g l e b l a d e f o r c e and moment compar i sons o v e r one comple t e r e v o l u t i o n , and t o t a l v e h i c l e dynamic r e s p o n s e compar i sons . Recommend a t i o n s are made c o n c e r n i n g model d e g r a d a t i o n which s h o u l d s e r v e a s a g u i d e f o r f u t u r e u s e r s o f t h i s m a t h e m a t i c a l model , and i n g e n e r a l , t h e y a r e i n o r d e r of mini mum impac t on model v a l i d i t y : ( 1 ) r e d u c t i o n o f number o f b l a d e segmen t s , ( 2 ) reduct i o n of number of b l a d e s , and ( 3 ) i n c r e a s e o f i n t e g r a t i o n i n t e r v a l and . a z i m u t h a l advance a n c l e . Ext reme l i m i t s a r e s p e c i f i e d beyond which t h e r o t a t i n : b l a d e e l e ment r o t o r m a t h e m a t i c a l model shou ld n o t be used .

- - . - . 7. Keywords (Suggested by Authoris1 ) 18. Distribution Statement

Roto r m a t h e m a t i c a l model d e g r a d a t i o n U n c l a s s i f i e d - U n l i m i t e d Real-time s i m u l a t i o n Ro ta ry wing H e l i c o p t e r s

S u b j e c t C a t e g o r y 05 9. Security Classif. (of this report) 20. Security Classif. (of this page1

.-.~ U n c l a s s i f i e d I Uncla ssif i e d-~- - . . . ...

*For sale by the National Technical Information Service, Springfield. Virginra 22161

EFFECTS OF ROTOR MODEL DEGRADATION ON THE ACCURACY

OF ROTORCRAFT REAL-TIME SIMULATION*

Jacob A. Houck and Roland L. Bowles Langley Research Center

SUMMARY

A study was conducted to determine the effects of degrading a rotating blade element rotor mathematical model to meet various real-time simulation requirements of rotorcraft. Three methods of degradation were studied: reduction of number of blades, reduction of number of blade segments, and increasing the integration interval, which has the corresponding effect of increasing blade azimuthal advance angle. The three degradation methods were studied through static trim comparisons, total rotor force and moment comparisons, single blade force and moment comparisons over one complete revolution, and total vehicle dynamic response comparisons. Recommendations are made concerning model degradation which should serve as a guide for future users of this mathematical model, and in general, they'are in order of minimum impact on model validity: ( 1 ) reduction of number of blade segments, (2 ) reduction of number of blades, and ( 3 ) increase of integration interval and azimuthal advance angle. Extreme limits are specified beyond which the rotating blade element rotor mathematical model should not be used.

INTRODUCTION

In view of the expanding interest in helicopter research and development in the past decade at the Langley Research Center, several real-time man-in-the-loophelicopter siinulaticn programs have been developed for the Langley real-time simulation (RTS) facility. (See ref. 1.) The simulations have been used as analytical tools,forman and vehicle performance evaluation and in support of flight-test programs. These studies have included flight director development, development and evaluation of advanced heads-up computer-generated displays, route structure development for intercity transportation, evaluation of terminal area navigation and approach procedures, motion-visual research, and mathematical model research and development. The rotorcraft mathematical models employed in these studies have varied from simple linear perturbation models to full force and moment models including rotating blade element rotor models.

An effort to adapt and evaluate a helicopter mathematical model for real-time simulation, including a state-of-the-art rotor model, was initiated in 1974

*Part of the information presented herein was included in a thesis entitled "Computational Aspects of Real-Time Simulation of Rotary Wing.Aircraft" submitted by Jacob Albert Houck in partial fulfillment of the requirements for the degree of Master of Science, The George Washington University, Hampton, Virginia, May 1976.

to support the rotor research planned at the Langley Research Center. This model (ref. 1 ) represents the Rotor Systems Research Aircraft (RSRA). The RSRA is a flying test platform to be used in the evaluation of advanced rotor and control systems. The RSRA mathematical model was derived from a general helicopter simulation model which has been used extensively in nonreal time in the design of helicopters and compound vehicles. linear large-angle representation in six rigid body degrees of freedom. In addi-

The model is basi.cally a total-force non

tion, rotor blade flapping, lagging, and hub rotational degrees of freedom are represented. Each simulated component of the aircraft (fig. 1 ) is modularized within the mathematical model. This characteristic allows easy manipulation of the aircraft's configuration.

The main problem encountered when using this mathematical model for real-time man-in-the-loop simulation studies is obtaining real-time operation while using the full rotating blade element rotor model (actual number of blades and a representative number of segments for each blade). Real-time operation is reached when the computer execution time for the active portion of the program is less than or equal to the prescribed integration time interval. In the past this has been accomplished by degrading the model in various ways. The purpose of this paper is to investigate the various methods of rotor and total vehicle model degradation. This investigation includes the effects due to reducing the number of blades, reducing the number of blade segments, varying the integration interval in size which, in turn, varies the azimuthal advance angle, and combinations of the three methods. This study provides data on the effects of degrading the rotating blade element model to meet real-time computer requirements for the simulation of rotorcraft.

SYMBOLS AND ABBREVIATIONS

Measurements and calculations were made in the U.S. Customary Units. They are presented herein in the International System of Units (SI) with the equivalent values in the U.S. Customary Units given parenthetically except in computer-generated figures. When two symbols f o r a concept are given, the second symbol denotes the computer symbol.

b blade

integers

n total number of data points

PB body-axis roll rate, degrees per secondpb

;b PBD body-axis roll acceleration, radians per second2

S blade segment

At integration interval, seconds

xa XA lateral cyclic control position, percent of total travel

2

i

‘b

XC

xi

XP

Yi

a

B

5

e

v

U

4

AY

n

XB longitudinal cyclic control position, percent of total travel

xc collective control position, percent of total travel

value of ith data point, dimensionless

XP ~ tail rotor pedal control position, percent of total travel

nondimensional distance along blade at ith station

angle of attack, degrees

BR blade flap angle, degrees

XLAG blade lag angle, degrees

THET pitch attitude, degrees

mean value

standard deviation

PHI roll attitude, degrees

azimuthal advance angle, degrees

rotor rotational speed, radians per second or revolutions per minute

Subscripts:

INB(n> inboard end

OUTB( n) outboard end

Abbreviations:

CDC Control Data Corporation

IFR instrument flight rules

NASA National Aeronautics and Space Administration

VTOL vertical take-off and landing

ROTOR MODEL DESCRIPTION

The following brief description of a blade element model for an articulated rotor system is provided. The total rotor forces and moments are developed from a combination of the aerodynamic, mass, and inertia loads acting on each simulated blade. For each blade simulated, the flapping and lagging

“ 3

degrees of freedom are represented. In addition, when the rotor speed governor is disengaged, the rotor shaft degree of freedom is released. Blade element theory is used to determine the aerodynamic loads. Blade inertia, mass, and weight effects are fully accounted for in the model. The elemental aerodynamic loads, dependent on local blade angle of attack and velocity vector, are calculated for each blade segment and are summed along each blade. The blade segmentdefinition is based on the assumption of equal annuli area and is defined in the appendix. The mass and ineria loads, dependent on blade and aircraft motion, are added to the aerodynamic loads for each blade. This summation gives the shear loads on the blade root hinge pins. Total rotor forces are obtained by summing all the blade hinge pin shears with regard to azimuth. Rotor moments result from the offset of the hinge shears from the center of the shaft. Blade flapping and lagging motion are determined from aerodynamics and inertia moments about the hinge pins. Interference effects from the rotor onto the fuselage and empennage are accounted for. The following basic assumptions 2re made in the rotor representation:

( 1 ) No account is taken of rotor blade o r airframe flexibility.

(2) Air mass flow degree of freedom through the rotor can be represented by applying a simple lag to the calculation of downwash. The only nonuniform flow is due to increases in forward speed which cause a redistribution of the uniform flow from the front to the back of the disk.

( 3 ) Simple sweep theory can be applied to determine the aerodynamic effects of yawed flow on the blade element, from unyawed flow blade data.

(4) Quasi-static aerodynamic loads are assumed.

(5) The flapping and lagging blade hinges were assumed coincident.

For a more detailed treatment of the rotor model and computer program structure, the reader is directed to the appropriate sections in reference 1.

PROBLEM DESCRIPTION

This section provides an insight into the difficulties encountered with the use of this rotor mathematical model, and the methods utilized to overcome these problems. This section and the next describe the detailed study covered in this paper to evaluate these methods.

Problems have arisen in the past when using the rotating blade element model for man-in-the-loop real-time simulation applications primarily because of the inadequate computing bandwidth of current computers. In order to use this model at all, gross degradation of the rotor representation and/or integration interval size were required. This was done so that the computer execution time fo r the active part of the program would be less than o r equal to the desired integration interval and, in other words, would achieve real-time execution. The mathematical model was developed in a nonreal-time mode to insure validity of the rotor model (see ref. I ) . In the nonreal-time mode, the rotor is represented by the actual number of blades, a representative number of blade segments

to represent the blade loading adequately, the actual rotor rotational rate, and a sufficiently small azimuthal advance angle, that is, a small integration interval size. Since this type of representation precludes the achievement of real-time execution, various combinations of reduced blades, blade segments, rotor rotational rate, and increased azimuthal advance angle are required to achieve real time while still retaining "satisfactory" static and dynamic comparisonswith data from the nonreal-time model. The steps necessary to reduce the mathematical model of the rotor, which accounts for a large part of the computationtime, for real-time operation are not routine; however, some general guidelines do exist, and they are used as a starting point for this study. The guidelines are a minimum number of three blades, three blade segments, and a maximum of.55O of azimuthal advance as presented in reference 2.

Figure 2 shows the effect of rotor rotational speed and azimuthal:update on allowable program execution time in order to prevent computational divergence of the flapping equation of motion. This is represented by the relationship, AY = 57.3Q At. Several present and future helicopters are provided for comparison. As can be seen the maximum program execution time available for the RSRA vehicle, if given a 30° azimuthal advance, is approximately 25 milliseconds. This program execution time must now be matched to the computational speed of the digital computer which is used for the simulation study.

Figure 3 presents a rotorcraft vehicle with a five-bladed rotor model with a rotational rate of 200 rpm. An azimuthal advance angle of 30' was chosen for illustrative purposes. Program execution time has been normalized to unity for the CDC 6600 computer with the ICOPS RUN'compiler. The CDC CYBER 175 with the NOS FTN compiler (optimization enabled) is represented at its tested bandwidth of 3.5 times faster than the CDC 6600. Minimum blade and blade-segment boundaries are presented. The cross-hatched area represents the combinations of blades and blade segments which can be modeled on the CDC 6600. Note that the five-blade five-blade-segment representation normally used (5b x 5 s ) is on the borderline of achieving real time for this azimuthal advance angle and leaves no execution time for additions to the program. It can be seen that the CDC CYBER 175 would be able to handle the representation easily.

The problem at hand is that of representing a rotorcraft vehicle on a digital computer in real time with an adequate rotor mathematical model for both objective and subjective tests and with the knowledge that additional computer requirements such as visual systems, complete cockpit requirements, landing-gear models, electronic flight control computer interfacing, etc., are normally included in man-in-the-loop simulations.

The purpose of this paper is to present an expanded systematic parametric study consisting of blade reduction, blade-segment reduction, integration interval increase (that is, azimuthal advance angle increase), and combinations of the three methods of degradation. Effects on both static and dynamic solutions are presented; thus, the best method of representing the rotor system under the constraints of real-time computer duty cycle time and accuracy of solution required is determined.

5

TECHNICAL APPROACH

I n o r d e r t o s t u d y the effects of r e d u c i n g number of blades, number o f b l a d e segments, and i n c r e a s i n g i n t e g r a t i o n i n t e r v a l s i z e , f o u r t y p e s o f tes ts were conducted. They c o n s i s t e d o f v e h i c l e s t a t i c t r i m compar i sons , t o t a l r o t o r f o r c e and moment comparisons (mean and s t a n d a r d d e v i a t i o n ) , b lade parameter comparisons f o r a 360° sweep, and dynamic r e s p o n s e comparisons for the t o t a l v e h i c l e .

The r o t o r mathematical model was set up w i t h f i v e b l a d e s and t e n b l a d e segments. A r o t o r speed o f 200 rpm was chosen so t h a t f o r t he i n t e g r a t i o n i n t e r v a l s s t u d i e d , t h e r o t o r b lades would always assume t h e same az imutha l l o c a t i o n s f o r each r o t a t i o n . F i n a l l y , a n i n t e g r a t i o n i n t e r v a l o f 1/240 second w a s chosen which, i n t u r n , caused t h e blade az imutha l advance a n g l e t o be 5'. T h i s d e s c r i p t i o n c o n s t i t u t e s t h e b a s e l i n e c o n f i g u r a t i o n and was used as a base t o which a l l o t h e r c o n f i g u r a t i o n s cou ld be compared; however, t he f i v e - b l a d e f ive -b lade segment r o t o r i s c o n s i d e r e d t h e s t a n d a r d Langley c o n f i g u r a t i o n .

The r o t o r mathematical model was degraded s y s t e m a t i c a l l y . The number o f b l ade segments was h e l d c o n s t a n t a t f i v e and i n t e g r a t i o n i n t e r v a l c o n s t a n t a t 1/240 second as t h e number o f b lades was reduced . The number o f b lades was t h e n held c o n s t a n t a t f i v e as t h e number o f b l a d e segments was r educed . A separate c o n f i g u r a t i o n which c o n s i s t e d o f three b l a d e s and three b l a d e segments was used t o s t u d y combinat ion effects. I n o r d e r t o s t u d y t h e e f fec t o f i n t e g r a t i o n i n t e r v a l s i ze , a r o t o r mathematical c o n f i g u r a t i o n w i t h f i v e b l ades and f i v e b l a d e segments and a n o t h e r w i t h three b l a d e s and three b lade segments were selected. I n t e r v a l s o f 1/240 second ( A Y = 5'1, 1/30 second ( A Y = 40'1, and 1/20 second (AY = 60') were chosen. The 1/30- and 1/20-second i n t e r v a l s were chosen f o r t h e f o l l o w i n g , r e a s o n s : 1/30 second i s a p p r o x i m a t e l y t h e i n t e g r a t i o n i n t e r v a l used a t t h e Langley Research C e n t e r f o r real-time s i m u l a t i o n and 1/20 second i s approx i ma te ly t h e i n t e g r a t i o n i n t e r v a l used by o t h e r real-time s i m u l a t i o n f a c i l i t i e s . The appendix describes t h e methods used i n d e g r a d i n g t h e r o t o r model. Tab le I d e t a i l s a l l c o n f i g u r a t i o n s used i n t h i s s t u d y .

The s t a t i c t r i m tests were set up i n t h e f o l l o w i n g manner. The v e h i c l e was started a t an a l t i t u d e o f 152.4 m (500 f t ) w i t h t h e r o t o r mathematical model s e t f o r t he d e s i r e d C o n f i g u r a t i o n . The v e h i c l e was t h e n trimmed a t v a r i o u s a i r speeds over t h e airspeed r a n g e . Two airspeeds are p r e s e n t e d f o r t h i s comparison , t h a t o f 0 knot ( h o v e r ) , and 120 k n o t s ( c r u i s e ) . While t h e v e h i c l e was s t a b i l i z e d a t each t r i m airspeed, a l l c o n t r o l p o s i t i o n s and a i rc raf t a t t i t u d e s were r e c o r d e d . These c o n s i s t e d o f c o l l e c t i v e p o s i t i o n , l o n g i t u d i n a l and l a t e ra l c y c l i c p o s i t i o n s , p e d a l p o s i t i o n , v e h i c l e p i t c h and r o l l a t t i t u d e s , and v e h i c l e a n g l e o f a t t a c k .

The t o t a l r o t o r f o r c e and moment t e s t s were se t up by s t a r t - i n g t h e v e h i c l e a t t h e t r i m c o n d i t i o n s determined i n t h e s t a t i c t r i m tes ts . The v e h i c l e was flown i n s t r a i g h t and l e v e l tr immed f l i g h t f o r 5 seconds d u r i n g which a l l f o r c e s and moments from t h e r o t o r were r eco rded t o d e t e r m i n e t h e i r mean s t e a d y - s t a t e v a l u e ,

n l l = 1 z x in

i=1

6

The standard deviation

In

of each was also computed to give some insight into how steady the forces and moments were for the trim condition. A run of 5 seconds was chosen so that enough data would be available for the calculations, whereas the inherent total vehicle instabilities would not have had enough time to contaminate the results.

To determine the individual blade parameter data with respect to azimuthal position, the 360° blade sweep tests were set up by once again starting the vehicle at the previously determined trim conditions. The vehicle was then flown in straight and level trimmed flight long enough, approximately 0.3 second, for the index blade to make one complete revolution. During the 360° revolution, data were taken at each azimuthal position achieved by the blade. Data recorded consisted of the index blade forces and moments, and blade flapping and lagging motions.

The totai vehicle dynamic response tests were set up by again starting the vehicle at the trim conditions determined previously in the static trim tests. The vehicle was then flown in straight and level trimmed flight. At 1 second into the flight, a 5-percent right lateral cyclic ( 0 percent equals full left stick, 100 percent equals full right stick) pulse was applied for 1 second and then the stick was returned to the trim position. The run continued until either 20 seconds had elapsed o r the vehicle flew into the ground, pitched up to exactly 90° (this condition causes the computer program to sustain a fatal error due to the method used for computing the Euler angles), o r the rotor blade flapping went unstable (this condition also causes a fatal error in the computer program). During the 20 seconds of the test run, all pertinent vehicle states were recorded. These consisted of the total vehicle body linear and angular accelerations, linear and angular velocities, body attitudes, and blade flapping and lagging motions. The lateral axis was chosen for the dynamic tests because of the low r o l l inertia relative to the pitch axis, and therefore it would show more sensitivity to changes in the integration interval.

RESULTS AND DISCUSSION

Static Trim Comparisons

The vehicle was started at an altitude of 152.4 m (500 ft) and was trimmed over the speed range for the various rotor configurations. The five-blade tenblade-segment model is presented as the baseline for all comparisons. The five-blade five-blade-segment configuration presented in each case is the standard real-time rotor configuration used at Langley Research Center. Table I1 presents the effect of blade reduction on static trim. As can be seen, the largest

7

error occurs between the five-blade ten-blade-segment configuration and the five-blade five-blade-segment configuration which can be attributed to a bladesegment-reduction effect.

Table I11 presents the effect of blade-segment reduction on static trim. For the airspeeds considered, the largest errors occur in collective position (xc) and roll attitude ( 4 ) when compared with the baseline configuration. These errors can be attributed to a degradation in blade aerodynamic definition.

Table IV presents the effect of combination blade and blade-segment reduction on static trim for a fixed integration interval of 1/240 second. The two cases chosen were the configuration used for the man-in-the-loop simulation at Langley and a worst possible case. Comparing these results with those of tables I1 and I11 indicates that the error in the combination reduction case comes from the blade-segment-reduction effect.

Table V presents the effect of increasing integration interval, and therefore blade azimuthal advance angle, on static trim for two rotor configurations.The integration intervals presented and their corresponding azimuthal advance angles are 11240 second ( Ai = 5 O ) , 1/30 second ( AY = 40°), and 1/20 second ( by 60').trol position error and r o l l angle is the largest attitude error. However, for

As in the previous cases, collective position is the largest con-

integration interval of 1/20 second, the rotor model has deteriorated to such an extent at 120 knots that obvious er rors appear in all parameters.

Total Rotor Force and Moment Comparison

The total rotor force and moment tests were set up by starting the vehicle at the trim conditions determined for each rotor configuration in the static trim tests. The vehicle was then flown in straight and level flight for 5 seconds during which the rotor forces and moments were recorded so that a mean value and standard deviation could be determined for each; thus, a performance measure is provided to compare the rotor configurations. Figures 4 and 5 present the effect of blade reduction on total rotor forces and moments for the hover and 120-knot cases, respectively. The figures show the rotor thrust, horizonal force, side force, pitching moment, rolling moment, and torque plotted against number of blades. Reducing the number of blades has little effect on the mean values; however, differences in the standard deviations can be seen, the larger differences occurring for 120 knots. In general, as forward velocity increases and the number of blades is decreased, the standard deviation increases and indicates that larger internal oscillations are occurring in the rotor which is an n/rev amplitude amplification because of the reduced number of blades.

Figures 6 and 7 present the effect of blade segment reduction on total rotor forces and moments for the hover and 120-knot cases, respectively. The figures show the forces and moments plotted against number of blade segments. For both velocities, small differences occur in the standard deviation as the number of b'lade segments is reduced. Differences in mean value are small, the largest error occurring in torque at 120 knots. Thus, the primary effect of reduction of blade segments on the total rotor forces and moments is a corre

8

sponding r e d u c t i o n i n r o t o r t o r q u e which becomes more s i g n i f i c a n t a t each speed as v e l o c i t y i n c r e a s e s .

F i g u r e s 8 and 9 p r e s e n t t he effect of i n c r e a s i n g t h e i n t e g r a t i o n i n t e r v a l , and t h e r e f o r e t h e a z i m u t h a l advance a n g l e , on t o t a l r o t o r f o r c e s and moments f o r t h e hover and 120-knot cases, r e s p e c t i v e l y . Two r o t o r c o n f i g u r a t i o n s are p l o t t e d , f ive-blade-f ive-blade segments and three-blade-three-blade segments. For bo th v e l o c i t y cases some d i f f e r e n c e s occur i n t h e mean v a l u e s o f t h e r o t o r f o r c e s and moments f o r t h e five-blade-five-blade-segment c o n f i g u r a t i o n when i n c r e a s i n g t h e i n t e g r a t i o n i n t e r v a l from 1/240 second ( A Y = 5 O ) t o 1/30 second ( A Y = 40°), t h e largest d i f f e r e n c e b e i n g i n t h e r o t o r t o r q u e . Although there are s e v e r a l n o t i c e a b l e d i f f e r e n c e s i n t h e s t a n d a r d d e v i a t i o n s , t h e i r magnitude is r e l a t i v e l y small. When i n c r e a s i n g t o a n i n t e g r a t i o n i n t e r v a l o f 1/20 second ( B Y 60'1, obv ious e r r o r s o c c u r i n t h e mean v a l u e s , ex t r eme ly l a r g e e r r o r s o c c u r r i n g i n t h e r o t o r moments, e s p e c i a l l y f o r 120 k n o t s . Cor re spond ing ly ,ex t r eme ly large i n c r e a s e s i n t h e s t a n d a r d d e v i a t i o n s occur . The three-bladethree-blade-segment c o n f i g u r a t i o n t e n d s t o f o l l o w t h e same t r e n d s ; however, i n g e n e r a l , t h i s c o n f i g u r a t i o n shows larger d i f f e r e n c e s i n t h e s t a n d a r d d e v i a t i o n v a l u e s , t h e s e larger d i f f e r e n c e s b e i n g due t o t h e a d d i t i o n a l combined e f fec ts o f r e d u c i n g t h e number o f b l a d e s and b l a d e segments.

360° B lade Sweep Comparison

The 360° b l a d e sweep tes t was s e t up by s t a r t i n g t h e v e h i c l e a t t h e p r e v i o u s l y determined t r i m c o n d i t i o n s . The i n d e x b l a d e was al lowed t o make one comp l e t e r e v o l u t i o n . During t h i s r e v o l u t i o n , data were t a k e n a t each a z i m u t h a l p o s i t i o n ach ieved by t h e b l a d e . Data r eco rded c o n s i s t e d o f b l a d e f o r c e s and moments, and b l a d e f l a p p i n g and l a g g i n g motions. S i n c e r e d u c i n g t h e number o f blades has no meaning i n t h i s t e s t ( t h e i n d e x b l a d e i s always p r e s e n t i n anyc o n f i g u r a t i o n ) , no data are p r e s e n t e d .

F i g u r e 10 p r e s e n t s t h e e f fec t of b lade segment r e d u c t i o n on b l a d e parame te rs f o r one r e v o l u t i o n a t 120 k n o t s . Only v e r y s l i g h t d i f f e r e n c e s are appare n t , t h e most prominent d i f f e r e n c e be ing a f a i r l y c o n s t a n t r e d u c t i o n i n t o r q u e o v e r t h e e n t i r e 360' r e v o l u t i o n . The data gathered f o r t h e hover case show no d i f f e r e n c e i n any o f t h e parameters e x c e p t f o r a s l i g h t change i n t o r q u e .

F i g u r e 11 p r e s e n t s t h e effect o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l t o 1/30second and 1/20 second on b lade parameters f o r one r e v o l u t i o n a t 120 k n o t s . For t h e case of i n t e g r a t i o n i n t e r v a l o f 1/30 second , s l i g h t d i f f e r e n c e s e x i s t i n f l a p a n g l e ( B ) , t h r u s t , p i t c h i n g moment, and r o l l i n g moment. A c o n s t a n t b i a s e x i s t s i n lag a n g l e o f a p p r o x i m a t e l y 0 . 5 O . The largest e r r o r e x i s t s i n t o r q u e which shows a r e l a t i v e l y c o n s t a n t lower v a l u e e x c e p t f o r a v e r y small p o r t i o n o f t h e sweep where i t assumes a h igher v a l u e . For t h e hover case, o n l y a s l i g h t d i f f e r e n c e i n t o r q u e was a p p a r e n t .

For t h e case o f i n t e g r a t i o n i n t e r v a l o f 1/20 second , e r r o r s are a p p a r e n t i n a l l t h e p a r a m e t e r s , o n l y s i d e f o r c e and h o r i z o n t a l f o r c e r ep r . e sen t ing t h e basel i n e s o l u t i o n . The largest e r r o r s e x i s t i n l ag a n g l e which shows a relat i v e l y c o n s t a n t 4' b i a s , and t o r q u e which shows a v a l u e ex t r eme ly lower t h a n t h a t of t h e b a s e l i n e s o l u t i o n . For t h e hover case, s l i g h t d i f f e r e n c e s appea r i n

9

a l l pa rame te r s e x c e p t f o r lag a n g l e which shows a bias o f l o , and t o r q u e which, i n g e n e r a l , is below t h e b a s e l i n e s o l u t i o n .

Dynamic Response Comparison

The t o t a l v e h i c l e dynamic r e s p o n s e t es t s were se t up by s t a r t i n g t h e vehicle a t t h e t r i m c o n d i t i o n s determined p r e v i o u s l y i n t h e s t a t i c t r i m tes ts . A t 1 second i n t o t he f l i g h t , a 5-percent I-second la teral c y c l i c p u l s e was a p p l i e d .During t h e t es t r u n , a l l p e r t i n e n t v e h i c l e states, such as l i n e a r and a n g u l a r a c c e l e r a t i o n s and v e l o c i t i e s , body a t t i t u d e s , and b lade f l a p p i n g and l a g g i n g motions, were reco rded . O f these s ta tes , body r o l l a c c e l e r a t i o n ( b b ) , body r o l l ra te ( p b ) , r o l l a n g l e ( @ I , and b lade f l a p a n g l e ( 8 ) are p r e s e n t e d f o r i l l u s t r a t i o n of t h e effects of t h e v a r i o u s mathematical model d e g r a d a t i o n s . Each case p r e s e n t e d is compared w i t h t h e b a s e l i n e r o t o r t o i l l u s t r a t e these effects.

Time-history comparisons ( r e f . 1 ) o f a five-blade-five-blade-segment r o t o r and f i v e - b l a d e three-blade-segment r o t o r w i t h t h e five-blade-ten-blade-segment b a s e l i n e r o t o r a t t h e hover and 120-knot c o n d i t i o n s show no d i f f e r e n c e .

F i g u r e 12 p r e s e n t s t h e effect o f b lade r e d u c t i o n on v e h i c l e dynamic r e sponse a t 120 k n o t s . The case r e p r e s e n t e d i s t h a t o f a three-blade f ive -b lade segment r o t o r compared w i t h t h e b a s e l i n e r o t o r . The e f fec t of t h e r e d u c t i o n o f number of b l a d e s is v e r y a p p a r e n t i n t h e r o l l a c c e l e r a t i o n and r o l l r a te c u r v e s . The mathematical model a m p l i f i e s t h e r o t o r r e sponse when it scales t h e r o t o r f o r c e s and moments f o r t h e c o r r e c t number o f b l a d e s i n t h e r o t o r system. The high-frequency o s c i l l a t i o n i n r o l l a c c e l e r a t i o n affects t h e r o l l r a t e b u t has a n i n s i g n i f i c a n t effect on r o l l a t t i t u d e because of t h e small i n t e g r a t i o n i n t e r v a l . The hover case shows t h e same effects , however, t o a smaller degree.

F i g u r e 13 p r e s e n t s t h e effect o f combinat ion blade and blade-segment reduct i o n on v e h i c l e dynamic r e s p o n s e f o r 120 k n o t s . It can be s e e n by comparing t h i s f i g u r e w i t h f i g u r e 12 and data i n r e f e r e n c e 1 t h a t most of t h e d i f f e r e n c e between t h e three-blade-three-blade-segment r o t o r and t h e b a s e l i n e r o t o r comes from t h e r e d u c t i o n i n number o f b l a d e s , and t h a t blade-segment r e d u c t i o n has r e l a t i v e l y no e f fec t .

F i g u r e 14 p r e s e n t s t h e effect of i n c r e a s i n g i n t e g r a t i o n i n t e r v a l from 1/240 second t o 1/30 second on v e h i c l e dynamic r e sponse a t 120 k n o t s . The r o t o r conf i g u r a t i o n cons ide red is the five-blade-five-blade-segment r o t o r and a g a i n , i t is compared w i t h t h e b a s e l i n e r o t o r . There is now a n o s c i l l a t i o n i n roll accele r a t i o n which approaches 0.3 rad/sec2. T h i s effect i s due t o a combinat ion o f aerodynamic d e f i n i t i o n i n t he r o t o r and numer i ca l s o l u t i o n and i s ana logous t o a c o n t i n u a l c o n t r o l i n p u t i n t o t h e r o t o r model. As t he v e h i c l e goes t o h i g h e r f o r ward v e l o c i t i e s , t h e ampl i tude o f t h i s o s c i l l a t i o n w i l l i n c r e a s e and more o f i t s effect w i l l f i l t e r th rough t h e body numer i ca l i n t e g r a t o r s t o r o l l r a te and r o l l a n g l e . The o s c i l l a t i o n shown i n f i g u r e I4 i s a p p a r e n t i n r o l l ra te and can j u s t b a r e l y be s e e n i n t h e r o l l ang le : The ave rage v a l u e o f r o l l ra te has been affected, and t h e v e h i c l e has r o l l e d t o 15O some 2 seconds fas te r t h a n t h e basel i n e v e h i c l e . The b l a d e f l a p p i n g t e n d s t o smooth i t s e l f o u t i n ampl i tude i n s t e a d o f f o l l o w i n g t h e a n t i c i p a t e d r e s p o n s e as d e p i c t e d by t h e b a s e l i n e r o t o r .

10

F i g u r e 15 p r e s e n t s t h e da t a f o r t h e hover case. The same t r e n d s are a p p a r e n t , b u t t o a smaller e x t e n t .

F i g u r e 16 p r e s e n t s t h e effect o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l from 1/240 second t o 1/30 second on v e h i c l e dynamic r e sponse a t 120 k n o t s f o r t he threeblade-three-blade-segment r o t o r . I n t h i s case the effect o f reduced number o f b l a d e s , reduced number o f b l a d e segments , and i n c r e a s e d i n t e g r a t i o n i n t e r v a l can be s e e n . A s t h e r o t o r l o a d s up a t h i g h s p e e d s , t h e e f fec ts o f b l a d e r e d u c t i o n and i n t e g r a t i o n i n t e r v a l c o n t i n u e t o a m p l i f y each o t h e r u n t i l a n o s c i l l a t i o n i n r o l l a c c e l e r a t i o n o c c u r s which i s as l a r g e as the effect o f t h e p u l s e i n p u t , and i n t h e f i n a l seconds o f t h e r u n , t h e o s c i l l a t i o n is doub le t h e p u l s e i n p u t . Thus, t h e v e h i c l e e q u a t i o n s o f motion are c o n t i n u a l l y e x p e r i e n c i n g p u l s e i n p u t s . Again, t h e a v e r a g e roll ra te v a l u e i s h i g h e r t h a n i t shou ld be , and the v e h i c l e r o l l s 15' approx ima te ly 3 seconds fas te r t h a n t h e b a s e l i n e v e h i c l e . The f l a p a n g l e a g a i n t e n d s t o smooth o u t i n s t e a d of f o l l o w i n g t h e a n t i c i p a t e d r e sponse . F i g u r e 17 p r e s e n t s t h e data f o r t h e hover case. The same t r e n d s are a p p a r e n t ,bu t t o a smaller e x t e n t .

F i g u r e 18 p r e s e n t s t h e effect o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l from 1/240 second t o 1/20 second on v e h i c l e dynamic r e sponse a t 120 k n o t s f o r t h e f i v e blade-five-blade-segment r o t o r . One can e a s i l y see t h a t t h e numer i ca l s o l u t i o n and aerodynamic d e f i n i t i o n of t h e t o t a l model have broken down and are i n c o r r e c t . Both t h e v e h i c l e and r o t o r are h i g h l y u n s t a b l e . The reader shou ld n o t e t h a t t h e run was s topped a t approx ima te ly 16 seconds t o keep t h e computer program from s u s t a i n i n g a f a t a l e r r o r due t o numer i ca l d i v e r g e n c e . F i g u r e 19 p r e s e n t s t h e data f o r t h e hover case. Again, t h e reader can see t h e obv ious breakdown o f t h e

.numer i ca l s o l u t i o n . T h i s case l o o k s somewhat bet ter t h a n t h e 120-knot case s i n c e t h e r o t o r aerodynamics a t hover are less complex, and t h e r e f o r e are n o t as s e v e r e l y affected by t h e numer i ca l i n s t a b i l i t i e s .

F i g u r e 20 p r e s e n t s t h e e f fec t of i n c r e a s i n g i n t e g r a t i o n i n t e r v a l from 1/240 second t o 1/20 second on v e h i c l e dynamic r e sponse a t 120 k n o t s f o r t h e threeblade-three-blade-segment r o t o r . Again, as w i t h t h e f ive-blade-f ive-bladesegment r o t o r , t h e v e h i c l e and r o t o r both become u n s t a b l e under these c o n d i t i o n s , and t h e computer program s u s t a i n e d a f a t a l e r r o r a t approx ima te ly 9 seconds i n t o t h e run because the blade f l a p a n g l e became u n s t a b l e . F i g u r e 21 p r e s e n t s t h e data f o r t h e hover case. Although t h i s case l o o k s b e t t e r t h a n t h e 120-knot case, it is obv ious t h a t t h e s o l u t i o n i s a g a i n i n c o r r e c t .

RECOMMENDATIONS FOR USING ROTATING BLADE ELEMENT MODEL

FOR REAL-TIME SIMULATION

The f o l l o w i n g recommendations are made f o r u s i n g a r o t a t i n g b l a d e element model f o r t h e s i m u l a t i o n o f a n a r t i c u l a t e d r o t o r system.

( 1 ) Do n o t r educe the number of b l a d e segments t o less t h a n three.

( 2 ) Do n o t r educe t h e number o f b l ades t o less t h a n three. A r e d u c t i o n t o less t h a n t h e a c t u a l number of b l ades when s t u d y i n g c o n t r o l sys t ems o r r o t o r systems should be c r i t i c a l l y examined.

11

(3) Do not increase integration interval to larger than 1/30 second.

( 4 ) Do not increase azimuthal advance angle to larger than 40'. (Note that 1/30 second integration interval and 40° azimuthal advance angle correspond to each other exactly only at a rotor speed of 200 rpm.)

( 5 ) If the model must be degraded, reduce blade segments first, blades second, and integration interval and azimuthal advance angle last.

(6) If any one of these factors must be degraded beyond the stated limits to reach real time, it is recommended that either a different rotor mathematical model be used o r a faster computer be used.

CONCLUSIONS

This paper describes the results of a series of tests designed to examine the effects of degrading a rotating blade element rotor mathematical model of an articulated rotor system in order to fit the model within set computer timing constraints. The three methods of degradation studied were those of reduction of number of blades, reduction of number of blade segments, and increase of integration interval size and thus increase of blade azimuthal advance angle. The tests conducted consisted of static trim comparisons; total rotor force and moment comparisons; index blade force, moment, and angle comparisons for one 360' blade sweep; and, finally, total vehicle dynamic response comparisons for a lateral cyclic control pulse input.

Recommendataions are made concerning model degradation which should serve as a guide for future users of this mathematical model, and in general, they are in order of minimum impact on model validity: (1) reduction of number of blade segments, (2) reduction of number of blades, and ( 3 ) increase of integration interval and azimuthal advance angle. Extreme limits are specified beyond which the rotating blade element rotor mathematical model should not be used.

From the data collected the following conclusions were reached:

( 1 ) Reducing blade segments does not appear to influence the solution to any great extent. This is due to the method used in determining the blade-segment locations which are based on equal annuli area.

( 2 ) Reducing the number of blades has no effect on static trim data; however, the amplification of n/rev (n blades per revolution) amplitude affects dynamic response because of the rotor forces and moments oscillating over a wider band. Two effects result: (1 ) The rotor, and therefore the body, tends to respond to the number of blades simulated and to the tip path plane formed by them, and (2) the numerical integration formulas used in the integration of the body accelerations and rates tend to respond to the increased amplitude of the high-frequency content contained in the accelerations and rates.

(3) The worst single effect is that of increasing integration interval. This has the double effect of increasing blade azimuthal advance angle and affecting the numerical integrators. When the azimuthal advance angle is

12

increased, the number of points used to define the rotor forces and moments as the blades sweep out the tip path plane is reduced. This condition causes an unrepresentative definition of the aerodynamic data to occur. The second problem arises with the numerical integration formulas one must use for solution of the body equations of motion in real-time man-in-the-loop simulations. The faster formulas, computationally, are the ones which cannot stand large interval sizes and still retain their numerical accuracy and stability. And as would be expected, these two effects tend to couple and amplify each other.

( 4 ) Because of the lack of rotor aerodynamic definition, these effects tend to amplify as the vehicle forward velocity increases and the rotor loads up. Thus, although some degradation might be acceptable at low speeds, it may not be suitable at all at higher speeds.

Langley Research Center National Aeronautics and Space Administration Hampton, VA 23665 November 24, 1976

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APPENDIX

ROTOR MODEL DEGRADATION METHODS

The f o l l o w i n g s e c t i o n s describe t h e methods used i n t h e d e g r a d a t i o n o f t h e r o t o r mathematical model.

Blade Reduct ion

Once t h e number of b l a d e s t o be s i m u l a t e d is chosen, t h e b l a d e s are e v e n l y d i s t r i b u t e d around t h e d i s k by u s e of t h e f o l l o w i n g e q u a t i o n :

where

"b a n g l e between a d j a c e n t b l a d e s , s t a r t i n g w i t h b l a d e 1 a t 0'

bS number of b l a d e s s i m u l a t e d

Then t h e f o r c e s and moments are c a l c u l a t e d f o r t h e b l a d e s s i m u l a t e d . Next , t h e t o t a l r o t o r f o r c e s and moments f o r t h e r o t o r system are c a l c u l a t e d i n a manner r e p r e s e n t e d by t h e f o l l o w i n g g e n e r a l e q u a t i o n s :

b=b.

where

FT t o t a l r o t o r f o r c e s

Fb i n d i v i d u a l b l a d e f o r c e s

MT t o t a l r o t o r moments

Mb i n d i v i d u a l b l a d e moments

a c t u a l number o f b l a d e s

bS number o f b l a d e s s imula t ed

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APPENDIX

Blade-Segment Reduction

First, the number of blade segments (n ) to be simulated is chosen. The positioning of these segments along the blases is determined by the followingequations from reference 1 whose derivation is based on the assumption of equal annuli area. The reader is directed to sketch (a) for the definitions of the various blade-segment variables.

Distance to segment center of lift for first segment:

where

nondimensional distance from center of rotation to hinge

5 ’ nondimensional distance from hinge to start of blade

Distance to segment center of lift for subsequent segments:

Distance to inboard end of segment from center line:

Distance to outboard end of segment from center line:

15

5

APPENDIX

Segment width:

Mean chord of segment:

Integration Interval or Azimuthal Advance Angle Relationship

The user can determine either an integration interval or an azimuthal advance angle. Defining one uniquely defines the second by assuming a constant rotor speed.

where

BY azimuthal advance angle

At integration interval

rotor rotational speed

16

I

52

I

REFERENCES

1. Houck, Jacob A l b e r t : Coi?putat ional Aspec ts o f Real-Time S imula t ion of Rotary Wing Aircraft. M.S. T h e s i s , George Washington Univ. , 1976. ( A v a i l a b l e as NASA CR-147932.)

2. Cooper, D. E . ; and Howle t t , J . J.: Ground Based H e l i c o p t e r S imula t ion . Symposium on S t a t u s o f T e s t i n g and Model Techniques f o r V/STOL A i r c r a f t (Ess ing ton , P a . ) , American H e l i c o p t e r SOC., O c t . 1972.

17

4;1

TABLE I.- ROTOR MATHEMATICAL MODEL CONFIGURATIONS UTILIZED

~~ .- -

lotor speed, Number of blades

Number of blade segments

Integration interval, sec

Azimuthal advance angle, deg

- _ _ .- - __.

*200 "5 *lo *l 240

*5

200 5 5 1 240

5

200 4 5 1 240

5

200 3 5 1 240

5

200 5 4 1 240 - 5

200 5 3 1 240

5

200 3 3 1 240

5

200 5 5 1 30 - 40

200 3 3 1 30 - 40

200 5 5 1 20 - 60

200 3 3 1 20 - 60

~.~~

.. . - .-

*This is the baseline configuration.

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TABLE 11.- EFFECT OF BLADE REDUCTION ON STATIC T R I M

[ I n t e g r a t i o n i n t e r v a l of 1/240 second]

Rotor X C’ 1 1 ‘at I 9 1 ‘ 9

conf igu ra t ion* pe rcen t pe rcen t pe rcen t pergent deg 1 5b x 10s 53.046 58.420 54.458 55.901 6.765 -4.137 -89.967 5b x 5 s 52.467 58.428 54.407 55.756 6.764 -4.086 -89 967 4b 5 s 52.466 58.428 54.407 55.756 6.764 -4.086 -89 - 967 3b x 5 s 52.466 58.428 54.406 55.756 6.764 -4.086 -89.967

5b x 10s 45.035 81.096 48.581 75.612 1.658 -2.039 1.319 5b x 5 s 44.465 80.895 48.594 75.492 1.657 -1.992 1.316 4 b x 5 s 44.466 80.895 48.595 75.493 1.656 -1 -992 1.317 3b x 5 s 44.465 80.895 48.598 75.493 1.657 -1.992 1.344

*b denotes blade and s denotes b l a d e segment; t h u s , 5 b x 10s denotes a . five-blade-ten-blade-segment c o n f i g u r a t i o n .

TABLE 111.- EFFECT OF BLADE-SEGMENT REDUCTION ON STATIC T R I M


Rotor xc’ Xb9 xa9 x g 9 degconf igu ra t ion* I percen t 1 percen t 1 percen t 1 p e r e n t 1 deg r Tdeg

Hover ‘ 5b x 10s 53.047 58.420 54.458 55.901 6.765 -4.137 -89.967 5b x 5 s 52.467 58.428 54 .40.7 55.756 6.764 -4.086 -89.967 5b x 4 s 52.086 58.436 54.362 55.622 6.767 -4.093 -89.967 5b x 3 s 51 .304 58.443 54.283 55.393 6.771 -4.073 -89.967

120 k n o t s

5b x 10s 45.035 81.096 48.581 75.612 1.659 5b x 5 s 44.465 80.895 48.594 75.492 1.657 5b x 4 s 44.088 80.825 48.568 75.439 1.661 5b x 3 s 43.275 80.576 48.442 75.295 1.677

*b denotes blade and s denotes blade segment; t h u s , 5b x 10s denotes a five-blade-ten-blade-segment c o n f i g u r a t i o n .

19

I

TABLE 1 V . -

Rotor conf igu ra t ion*

5b x 10s 5b x 5 s 3b x 3 s

EFFECT OF COMBINATION BLADE AND BLADE-SEGMENT REDUCTION

ON S T A T I C TRIM


x C 9 'b 9 xa 9 percen t pe rcen t pe rcen t

I I I

120 k n o t s

9 , a, deg deg

~~

-

-4.137 -89.967 -4.086 -89.967 -4.073 -89.967

1.311 -

*b deno tes b l a d e and s deno tes blade segment; t h u s , 5 b x 10s denotes a f ive-blade- ten-blade-segment c o n f i g u r a t i o n .

20

TABLE V.- EFFECT OF INCREASING INTEGRATION INTERVAL

FOR TWO ROTOR CONFIGURATIONS

Rotor I n t e g r a t i o n c o n f i g u r a t i o n * i n t e r v a l , s e c

L .

5 b x 10s 1/240 5b x 5s 1/240 5b x 5s 1 /30 5b x 5s 1/20 3 b x 3s 1/240 3b x 3s 1/30 3b x 3s 1/20

1/240 1/240 1/301/20 1 /240 1/30 1/20

x ~ ’I ‘b, xa X ’ p e r c e n t pe rcen t p e r c e n t pe rgen t

Hover

53.047 58.420 54.458 55.901 52.467 58.428 54.407 55.756 52.650 58.475 54.462 55.765 52.739 58.539 54.500 55.797 51.304 58,443 54.283 55.392 51 -417 58.481 54.364 55.356 51.547 58.563 54.368 55.419

120 kno t s

45.035 81.096 48.581 75.612 44.465 80.895 48.594 75.492 42.276 79.661 49.117 76.037 28.305 70.212 49.144 80.964 43.276 80.576 48.438 75.295 41.023 79.344 48.993 76.102 35.761 1 76.347 45.576 78.605

*b deno tes b l ade and s deno tes b l ade segment t h u s , five-blade-ten-blade-segment c o n f i g u r a t i o n .

6.765 -4.137 -89.967 6.764 -4.086 -89.967 6.801 -4.587 -89.967 6.787 -4.622 -89.967 6.771 -4.073 -89.967 6.789 -4.582 -89.967 6.795 -4.586 -89.967

5b x 1 0 s deno tes a

21

I

Mai'n rotor11

Tail r o t o r

P i 1o t Control system > i

r r

l-f--+ Em pen na g e

A V V . SimulatorEquations c o c k p i to f mot ion

c mot ion iv i sua1

A c

~~ ~

Figure 1.- Simplified block diagram of mathematical model.

*Human factors such as visual flicker, motion base stepping,instrument flicker, etc. which are noticeable to pilot.

**Mathematical convergence and divergence.

10

WW**Divergent flapping v) \ v) Q)

W - 12.5a, v) 5E v)

m mW aJE *r E

-7 c,

S *Human factors limit c,

0 I I L / I I 1 1-2 / / 1 / / / 16.7 -r .r c, c,3 3u uW aJX xaJ Q)

Ei 5L L0, - 2 5 0”0L La a

- 50

- - 1 I I L 0 100 200 300 400 500

Rotor speed, rpm Figure 2.- Effect of rotor speed and azimuthal update on allowable program

execution time for flapping convergence.

23

2

.:r: y / 5

0 LaRC r o t o r model ;Ay=30°

-1 I - =- I I 1 4 6 8

Number of b l a d e segments F igure 3.- Comparison of program execut ion time f o r a CDC 6600 computer wi th

a CDC CYBER 175 computer f o r a f ive -b lade , 200-rpm r o t o r model.

24

0

1

Standard deviat ion

Standard deviat ion

76287 -62212 (17150)-I M e a n 1 (-1400

c)

Y

z

76064 -6450 I I I ( 17100) 3 4 5 ( - 1 450, 3 4 5

llumber o f blades liumber o f blades

Figure 4.- E f f e c t of b l a d e r educ t ion on t o t a l r o t o r f o r c e s and moments a t hover. I n t e g r a t i o n i n t e r v a l , 11240 second; f i v e b l ade segments.

L

u

U

0

587 -10169 (130 (-7500

Standard d e v i a t i o n

c

356 -10508 ( 80 Mean E (-7750 0 Mean c.?

c c)

.m L0 c)OL

133 I I I -10846 I (30) 3 4 5 (-8000 3

P E

I 1

4 5 Number o f blades Number of blades

Figure 4.- Continued.

- 0

L

-3152 I ^___.

4.2E

-3322 8 (-2450) 0)

.-0

I0 c)w

-3491

-- Standard devia t ion

Standard dev ia t ion

- 57622IMean

-5

0

5 (42500)-Mean

I I 1 56266 I I 1 I (41500) 3 4 5

tlumber of blades

Figure 4.- Concluded.

0

D

77843 (17500

Standard deviation

-LI Y,.. z +3 Mean4 7561'$ (1700[ L

I.'

P

73395 I I I

(16500 j 3 4 5

Number o f blades

Figure 5.- Effect o f b l a d e r e d u c t i o n I n t e g r a t i o n i n t e r v a l ,

. Standard deviation

- -3781I Mean

(-850).-L .c

L 1 I Vcc

-i

-4893 1 1 I I (-1100) 3 4 5

Number o f blades

on t o t a l r o t o r f o r c e s and moments a t 120 kno t s . 1/240 second; f i v e b l a d e segments.

I

-601 3 (-6500) -- Standard deviat ion

1779 14001

-r Standard deviat ion

3-2P

-10500(-77501- IMean -4 PPar

890 (2001

1

-12202 I

3 4 5 Eiumber o f blades

Figure 5. - Continued.

2

I

W 0

1356 (1000

c

-339 m (-250)

.-c

-2034 I

(-1500)

41 352 (30500

Standard deviat ion

Standard deviat ion

Mean

2 39996 .(29500 al=mMean P P +l0

I I I 38640 I I I 3 4 5 (28500) 3 4 5

Number of blades. ltumber of blades

Figure 5 . - Concluded.

1 - -

0

0 0

76509 -6005 (1 7200 ( - 1 350)'

-II-z

Standard deviation al .a - L

Lz- 76287 - -62272 (17150 (-1400

.cL Mean

.-N L

I II

s CL *

L PT

a

76061 I I I 1 -6450 (17100 3 4 5 10 ( -1450)

3 4 5 10 llumber o f blade segments llumber o f blade segments

Figure 6.- Effect o f b l a d e segment r educ t ion on t o t a l r o t o r f o r c e s and moments a t hover. I n t e g r a t i o n i n t e r v a l , 1/240 second; f i v e blades.

9

0

wN

-10169 (130) (-7500

l i 8 Standard deviat ion

c 1T 5 I HeanI m

1 3 3 , I 1 I I 1 -10846 I I I I i (33) 3 4 5 10 (-3000)' 3 4 5 10

Number o f blade segments Number of blade segments


?

0

E

0

(-2325)

-3152 ~ !

1Standard deviat ion

Mean-0

L04J w

I

-3491 I I I I(-2575 3 4 5 10

llumber of blade segments

Standard deviat ion

l l + m

--; 57622 g (42500 I0+ L.

+w

56266 I I 1 1 (41500,

3 4 5 10 Number of blade segments


W W

I

0

0

w J=

77843 -2669 (-600)( 175001

Standard deviat ion

I- l p [ i-4;a;dard deviat ion

c

2 -3781=: ..(-850)-N I P.-

I

73395 I I 1 I -4893 .I I I 1 I(16500) 3 4 5 10 (-1100) 3 4 5 10 f!umber o f blade segments Number of blade segments

Figure 7.- Effect o f b l a d e segment r e d u c t i o n on t o t a l r o t o r f o r c e s and moments a t 120 kno t s . I n t e g r a t i o n i n t e r v a l , 1/240 second; f i v e t j lades.

5 coo, 3 4 5 (-9000) 3I 11.1 10

‘.12_Il 4I I 10

llumber o f blade segments llumbrr o f blade segments

Figure 7. - Continued.

0 0

W m

1356 41 352 ( 1000 (30500)

Standard d e v i a t i o n

a -'c* Mean-9 z -339

(-250

P P $ Idean PStandard deviat ion

8I cn .r 7 -0 I* p:

-2034 38640 1 I I 1

(28500) 3 4 5 10

Number o f blade segments Number o f blade segments


--

c

4440 (1000)

0 3b X 3s r o t o r

0 5b X 5s r o t o r Standard dev ia t ion

AY = 5' At=1/240 sec

AY = 40' At-1/30 sec

A 1 = 60' At'1/20 sec

D E ni

w L E -6672 n31- (-1500)e,0

.0'

L0 e,x0

Standatd dev ia t ion-

+ "'1Mean

I

( 1 6 5 0 0 ~ 0 i n 20 10 40 50 60

Azimuthal advance angle, deg

-17793 I 1 I I I . I I

Figure 8.- Effect of increasing integration interval on total rotor forces and moments at hover.

I

73395 I

- Iu

0

W (x,

(2000) (-7000)

3b X 3s r o t o r

0 5b X 5s r o t o r

A'? = 5" At=1/240 sec Standard devia t ion-.n- AY = 40" At=1/30 sec -L

AY = 60" At=1/20 s e c [ ; ;

u

(k IZS a ' 2 -10846 Standard devia t ion P (-8000)-

L0 Y .x

-8896 I I I I I / -12202 I I I I I( -2ooo j b 10 20 30 40

I

50 60 (-9000) 6 10 20 30 40

I

50 60 Azimuthal advance angle , deg Azimuthal advance angle , deg


0

c)

0

0

-2034 (-1500)

Y

5 -3051 (-22505

0)

.7 cL

L0

0=

-4067

0 3b X 3s rotor

0 5b X 5s rotor Standard deviation

A1 = 5' At=1/240 sec

AY = 40" At=1/30 sec Standard deviation

T I Mean T

AP = GO" At.1120 sec -7 L

I 50043 f% (37500 L.T +

'I I0 Y w

I I I I I I I 40674 I 8 I I I I

F i g u r e 8.- C o n c l u d e d .

1

0

0

F 0

80068 2224 (18000) ( 500 )

68947 - LQ-0- -5560

I Mean 0 .Z c

(-1250)-

0 3b X 3s rotor Y

Standard deviation- III-.., T--m

z

CY 0 5b X 5s rotor L0

4.3CY

57827 I I 1 I 1 I _ _ -1 3345 I 1 I I I I I (13000) 0 10 20 30 40 50 60 (-3000) 0 10 20 30 40 50 60

Azimuthal advance angle, deg Azimuthal advance angle, deg

Figure 9.- Effect of increasing integration interval on total rotor forces and moments at 120 knots.

0

#

81196 -2712 (2000) I (-2OOO)i

0 3b X 3s r o t o r Standard devation 0 5b X 5s r o t o r Mean

A'Y 40' At.1130 sec

AY = 60' At=1/20 sec Standard devia t ion

T.v)

I0YLL

-4448 -16270 I I I I I 1(-1000 I

m

0

42030 (31 000)

0 3b X 3s rotor

0 5b X 5s rotor

AY 5' At=1/240 s e c

AY = 40' At=1/30 s e c

AY = 60' At=1/20 s e c

H

p i

Standard deviation

Mear

-.?

Standard deviat ion

Mean 1L0 c) Pm!

-_ I 8135 , I I I I I I I

(6000) 0 10 20 30 40 50 60

Azimuthal advance angle , deg

Figure 9 . - Concluded. P

c

0

-7-10

-5

0 - CIJ -W 2 0: M

7--5

-10

i 2 t S S e16 e88-15 0 1 1 1 1 1 1 1 1 l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1360 Azimuthal a n g l e , deg

- 5b X 10s r o t o r

5b X 3s r o t o r

Azimuthal a n g l e , deg

F i g u r e 10.- Effect o f blade-segment r e d u c t i o n on one blade r e v o l u t i o n data a t 120 k n o t s . I n t e g r a t i o n i n t e r v a l , 1/240 second.

43

31000L


4- 5b X 3s r o t o r

t!

t--322500* I I 1 1 1 1 1 1 l l . L L I . I I 1 1 I I I I 1 1 1 J 1 1 U-1

44

325000(

-325000L I 1 I I I I 1 I I I 1 1 1 1 1 1 1 1 1 l l 1 1 1 1 l l 0 72 L Y Y 216 288 360


- 5b X -10s rotor

5b X 3s rotor

-F -500z -n

c, s Q)E 0E -5000(5, S -7

c u c, 'F

a -3500

- l l l D 0 0 t 1 _ 1 U 4 . 1 1 1 1 1 I I I 1 I I 1 I 1 I 1 1 I 1 1 1 I I 1 I 0 7e L Y Y 216 288 360


F i g u r e 10.- Continued.

45

11500

5750

0

--5750

t

1 7 0 0 0 r

13250

F z J -

2000 0 72


-l-5b X 3s r o t o r

1 1 1 1 l l 1 1 1 1 1 1 1 1 1 1 I I 1 1 1 1 360~~ 1Y Y 216 e88

Azimuthal angl e, deg F i g u r e 10.- Concluded.

46

- 5b X 10s r o t o r ; 1/240 sec

-k 5b X 5s r o t o r ; 1 /30 s e c

o 5b X 5s r o t o r ; 1 /20 s e c

aJ U

- 0

x 0 0 0 0 0 0

4. + + + + 4. ' 4 . +

i :I

L - I I I I I I 0 I 1 72 I I I I I 1YY 1 I I I I e16 e66 360I::E11_1

Azimuthal ang le , deg F igu re 11.- Effect o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l t o 1/30 second and

1/20 second on one blade r e v o l u t i o n data a t 120 kno t s .

47

I

c,

11000---22260

-z .)

(r '3 13600 -L -I I- -

-Y760

--

4

0

-4000 I 1 I I I 1 1 1 1 1 I 1 1 1 1 I l l 1 1 1 1 1 1 1 1 0 7 2 1 Y Y 216 266 360

- 5b X 10s r o t o r ; 1/240 s e c

-k 5b X 5s r o t o r ; 1/30 sec

o 5b X 5s r o t o r ; 1/20 sec

72 1Y Y elti 360

Azimuthal a n g l e , deg F i g u r e 11.- Cont inued.

48

325000 �

-325000 0tL1 1..1 1 . I J 1 J 1 1 1-1 1 1 J 1 I I l 1 1 J 1 I 1 I 1 I I 1 72 1Y Y 216 266 360

Azimuthal a n g l e , deg - 5b X 10s r o t o r ; 1/240 s e c

d- 5b X 5s r o t o r ; 1 /30 s e c

0 5b X 5s r o t o r ; 1/20 sec

I



49

t

-1 1500 ~ I l i i l ~ l l l l J ~ 2 1 6 - 266 360

Azimuthal angl e , deg

- 5b X 10s r o t o r ; 1/240 s e c

4- 5b X 5s r o t o r ; 1/30 s e c

0 5b X 5s rotor; 1/20 s e c

13260

m

- 0 0 + +

0

I I I l l 0 72 1Y Y

Azimuthal angl e , deg Figure 11.- Concluded.

50

I

(a> 3b x 5s r o t o r .

x 10-1

I 4

( b ) 5 b x 1 0 s r o t o r .

F igu re 12.- Effect o f blade r e d u c t i o n on v e h i c l e dynamic r e s p o n s e a t 120 k n o t s . I n t e g r a t i o n i n t e r v a l , 1/240 second.

51

I-//

0

-10

( c ) 3b x 5s rotor.

L / \ L3 1w o5 ~ F

-101 c

(d) 5b x 10s rotor.


52

L11 6 W 0

c\ U

I L

-10

-15 6-E ( e > 3b x 5s rotor.

- 1 5 h 1 I . I I 1 I - I I I0 4 8 I 16 I I I 20

T

(f) 5b x 10s r o t o r .


53

10

m 6 W 0

- 0 ar.=

-10-i-15

0 4 8 16 20 T I M E , SEi2

( g ) 3b x 5s r o t o r .

10

f 3 6 LIJ 0

- 0 CE m

-10

I - 1 5 I l l I I I 1 I 1 1 1 1 I

16 20

1

( h ) 5 b x 10s r o t o r .

F igu re 12.- Concluded.

54

?

n m a

L

(a> 3b x 3s rotor.

x 10-1

(b) 5b x 10s rotor.

Figure 13.- Effect of combination blade and blade-segment reduction on vehicle dynamic response at 120 knots. Integration interval, 1/240 second.

55

-D --m' -LL -5 ------10 ----15- I 1 I I 1 I 1 1 1 I l l 1 1 1

16

( c ) 3b 3s r o t o r .

( d ) 5 b x 1 0 s r o t o r .

F igu re 13.- Continued.

56

20

-10-L - 1 5 h 1 I I I I I I I I I I I I I I I I I J

0 Y 8 12 16 20 TIME, SEC

(e> 3b x 3 s r o t o r .

-10-1 -1st-A.

0

(f) 5 b x 1 0 s r o t o r .


57

-15 I l l I l l I l l I I I I I I I 0 (t 8 12 16

T I M E ] S E C '

(g) 3b x 3s r o t o r .

( h ) 5b x 10s r o t o r .

F igure 13.- Concluded.

58

20

I

(b ) 5b x 10s r o t o r ; 1/240 second.

F i g u r e 14.- Effect of i n c r e a s i n g i n t e g r a t i o n i n t e r v a l t o 1/30 second on v e h i c l e dynamic r e sponse a t 120 k n o t s f o r a f ive-blade-f ive-blade-segment ro to r compared w i t h t h e five-blade-ten-blade-segment b a s e l i n e r o t o r w i t h a n i n t e g r a t i o n i n t e r v a l of 1/240 second.

59

n l

-10

( c ) 5 b x 5s r o t o r ; 1/30 second.

1

I

m' -Q - 5--t-

- 1 0---- 1 5 - 1

0 I 1 I l l4 e 12 16 20

T I M E , SfC ( d ) 5 b x 10s r o t o r ; 1/240 second.

F i g u r e 14.- Cont inued.

60

a w 0

? w I e

-10

(e) 5b x 5s r o t o r ; 1/30 second.

-10

-15 I 0 4

(f) 5b x 10s r o t o r ; 1/240 second.


61

(g ) 5b x 5s r o t o r ; 1/30 second.

( h ) 5 b x 10s r o t o r ; 1/240 second.

F igu re 14 . - Concluded.

62

-10

-15 0

( a ) 5b x 5 s r o t o r ; 1/30 second.

x t o - '

( b ) 5 b x 10s r o t o r ; 1/240 second.

F i g u r e 15.- Effect o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l t o 1\30 second on v e h i c l e dynamic r e s p o n s e a t hover f o r a f ive-blade-f ive-blade-segment r o t o r compared w i t h t h e f ive-blade-ten-blade-segment b a s e l i n e r o t o r w i t h an i n t e g r a t i o n i n t e r v a l o f 1/240 second.

63

(c) 5b x 5 s r o t o r ; 1/30 second.

mi a -5

- l o b , -15

0

( d ) 5b x 1 0 s r o t o r ; 1/240 second.


64

-10

-15 0 ~- I-F, ( e > 5b x

10

m W n-I

I a OW

-15 .. I 0 Y

5 s r o t o r ; 1/30 second.

/T/ \ I20

(f) 5 b x 10s r o t o r ; 1/240 second.


65

(g) 5 b x 5s rotor; 1/30 second.

10

u3 5 W n

7tx m

O F - - - - -

-10-E ( h ) 5 b x 10s rotor; 1/240 second.


66

,n IO-'

151

( a ) 3b x 3s r o t o r ; 1/30 second.

F i g u r e 16.- Effect o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l t o 1/30 second on v e h i c l e dynamic r e sponse a t 120 k n o t s f o r a three-blade-three-blade-segment r o t o r compared wi th t h e f ive-blade-ten-blade-segment b a s e l i n e r o t o r w i t h an i n t e g r a t i o n i n t e r v a l o f 1/240 second.

67

I

15

10

u 5 W tn \ E.3 w o 0

mi LL

-5

-10

-15 0 Lt 8 12 16 20

TIME, SEC

( c ) 3b x 3s r o t o r ; 1/30 second.

-w 0 -D --m" a c--5 ------10 ---

-15- I I I I l l 1 1 1 I I I I I I J 20

( d ) 5b x 10s r o t o r ; 1/240 second.


68

'E -10- f I d20

T I M E , SEC ( e ) 3b 3s r o t o r ; 1/30 second.

-10g-r1 1 I-.1 1 ..I 1 I_ljlllJ20- I 5 k . l - 1 ~ _ 1I

0 't 8 12 16TIMEl S E C

5b x 10s r o t o r ; 1/240 second.


69

-10-E (g) 3b x 3s r o t o r ; 1/30 second.

10

W D

? O CY m

-10 L

16 20

( h ) 5b x 10s r o t o r ; 1/240 second.

F igu re 16.- Concluded.

70

x t o - ’

!t Ill

L

- 1 5 k I 1 I I I 1 I I 1 I I I I 1 I I I I I I 0 + 8 1? 16 20

T I M E , SEC

( a > 3b 3s rotor; 1/30 second.

x 10-1

( b ) 5b x 10s r o t o r ; 1/240 second.

F i g u r e 17.- E f f e c t o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l t o 1/30 second on v e h i c l e dynamic r e sponse a t hover f o r a three-blade-three-blade-segment r o t o r compared w i t h t h e five-blade-ten-blade-segment b a s e l i n e r o t o r w i t h an i n t e g r a t i o n i n t e r v a l of l /240 second.

71

-15 0 Lt 8 12 16

T I M E , S E C

( c > 3b 3s rotor; 1/30 second.

(d) 5 b x 10s ro to r ; 1/240 second.


72

20

( e ) 3b x 3s r o t o r ; 1/30 second.

-IS 0-i (f) 5 b x 10s rotor; 1/240 second.


73

-5

-15 F I I I . - . I I I I . 1. 1_ -1_ . I .I_I_1_-_1 I I 1 1 I 0 9 8 . 12 16 20

TIME, SEC ( g ) 3b 3s r o t o r ; 1/30 second.

I I I 20


F i g u r e 17. - Concluded.

74

l o t - “ n

( a ) 5b 5s r o t o r ; 1/20 second.

( b ) 5b x 10s r o t o r ; 1/240 second.

F i g u r e 18.- Effect o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l t o 1/20 second on v e h i c l e dynamic r e sponse o f 120 k n o t s f o r a f ive-blade-f ive-blade-segment r o t o r compared wi th t h e five-blade-ten-blade-segment b a s e l i n e r o t o r w i t h an i n t e g r a t i o n i n t e r v a l o f 1/240 second.

75

W D

m’ (L -E10

(c) 5 b x 5s r o t o r ;

‘“Ec

l o +

c3 W D t-

I

1/20 second.

I* / I //

1

I

(d) 5b x 10s r o t o r ; 1/240 second.


76

r

-10

-150 1 1-1I I 1

5b x 5s r o t o r ; 1/20 second.



77

( g ) 5 b x 5s r o t o r ; 1/20 second.

-10-i ( h ) 5 b x 10s r o t o r ; 1/240 second.

F i g u r e 18.- Concluded.

78

I

x 10-1

10

0 r 8 12 20 T I H E , SEC

( a > 5b x 5s r o t o r ; 1/20 second.

- 1 5 k - I 1 1 I 1 1 I I I I I I I I I I I I I 0 Lt 8 12 16 20

TI f lE , SEC


F i g u r e 19.- Effect o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l t o 1/20 second on v e h i c l e dynamic r e sponse a t hover f o r a f ive-blade-f ive-blade-segment r o t o r compared w i t h t he f ive-blade-ten-blade-segment b a s e l i n e r o t o r w i t h an i n t e g r a t i o ni n t e r v a l of 1/240 second.

79

- l o b

v K

"TIME, S E i L

(c) 5 b x 5s rotor; 1/20 second.

'"E

-1s I l l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 9 6 12 16 20

TIME, SEC

(d) 5 b x 10s rotor; 1/240 second.


80

~

( e ) 5b x 5s rotor; 1/20 second.

10*E n -- 0

-10

~-~~-15 0 1 I 1 ~ I ~ - J J ~ . l 1 1 1 I I I Lt 6 12 16 20TIME, SEC (f) 5b x 10s rotor; 1/240 second.


81

-10-E (g> 5b x 5s r o t o r ; 1/20 second.

1 16


F i g u r e 19. - Concluded.

82

. I d - 1 I I ' 1 I I I I I I Id20 0 4 12 168

TI1 SEC

( a ) 3b x 3s rotor; 1/20 second.

x 10-1

- 1 g \ -15 L 1 1 I 1 1 1 I _1 1 1 1 -1-I 1 I I 1

0 Lt 8 12 16 20TIHE, SEC


F i g u r e 20.- Effect o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l t o 1/20 second on v e h i c l e dynamic r e sponse a t 120 knots for a three-blade-three-blade-segment rotor compared w i t h t h e f ive-blade-ten-blade-segment b a s e l i n e r o t o r w i t h an i n t e g r a t i o n i n t e r v a l of 1/240 second.

83

10

(c> 3b 3s r o t o r ; 1/20 second.

(d) 5 b x 10s r o t o r ; 1/240 second.


84

I

t

( e ) 3b 3 s rotor; 1/20 second.

-10-5i -15 F I I I ii I

0

5b x 10s ro to r ; 1/240 second.


85

I

-LO-E I

I

( g ) 3b x 3 s rotor; 1/20 second.



86

1

N

u W m \ D U C Y

7 n m e

- 1

- 1 s h I ' 0 8 12 16

TIME, SEC ( a ) 3b x 3s r o t o r ; 1/20 second.

x IO-'

- 1 5 h 1 I 1 1 1 I I 1 1 1 I 1 1 I

9

d20

I 1 I I 0 Lt 8 12 16 20

T I I I E , SEC ( b ) 5b x 10s r o t o r ; 1/240 second.

F i g u r e 21.- Effect o f i n c r e a s i n g i n t e g r a t i o n i n t e r v a l t o 1/20 second on v e h i c l e dynamic r e sponse a t hover f o r a three-blade-three-blade-segment r o t o r compared w i t h t h e f ive-blade-ten-blade-segment b a s e l i n e r o t o r w i t h an i n t e g r a t i o n i n t e r v a l o f l /240 second.

a7

u W Lo \ W W 0

m’ a

1.I 20

(c) 3 b x 3s rotor; 1/20 second.

u wtn \ /W W n mi a

-IS 1 1 I I I 1 1 1- 1 1.- 1 1 J 1 1. 1 I 0 Lt 8 12 16 20

T I H E , SEC

( d ) 5 b x 10s rotor; 1/240 second.


aa

(e> 3b x 3s r o t o r ; 1/20 second.

n

T/ / \

S t C3



I

10

r r , sW L3

-15 1 1 1 I l l 1 1 1 I l l I l l 0 Y 6 ' 12 16 20

TIHE, SEC (g) 3 b x 3s r o t o r ; 1/20 second.

- l o b --


F i g u r e 21.- Concluded.

90 NASA-Langley, 1976 L-1 1083

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON. D.C. 20546 POSTAGE A N D FEES P A I D

N A T I O N A L AERONAUTICS A N D OFFICIAL BUSINESS SPACE A D M I N I S T R A T I O N

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