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EXPERIMENTAL AND ANALYTICAL INVESTIGATION OF PROPELLER WHIRL FLUTTER OF A POWER PLANT O N A FLEXIBLE WING

by Robert M . Bennett and Samuel R. Bland

Langley Research Center Langley Station, Hampton, Vd.

NATIONAL AERONAUTICS A N D SPACE A D M I N I S T R A T I O N WASHINGTON, D. C. AUGUST 1964

https://ntrs.nasa.gov/search.jsp?R=19640017748 2020-05-12T05:29:05+00:00Z

TECH LIBRARY KAFB, NM

EXPEFUMENTAL AND ANALYTICAL INVESTIGATION O F

PROPELLER WHIRL FLUTTER O F A POWER PLANT

ON A FLEXIBLE WING

By Robert M. Bennett and Samuel R. Bland

Langley Research Center Langley Station, Hampton, Va.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For s a l e by t h e O f f i c e of T e c h n i c a l Serv ices, D e p a r t m e n t of Commerce,

Washington, D.C. 20230 -- P r i c e $1.75

ExpERIMEPTI!AL AND ANALYTICAL INVESTIGATION O F

PROPELLER W H I R L FLUPI" O F A POWER PLAN"

ON A FLEXIBLE WING

By Robert M. Bennett and Samuel R. Bland Langley Research Center

SUMMARY

Results of an invest igat ion of propel ler whirl f l u t t e r a r e presented f o r a model consisting of a s ingle propel ler and simulated power p lan t mounted with f l e x i b i l i t y i n p i t ch and yaw t o a cantilevered, semispan wing. Several config- urations d i f fe r ing i n t h e power-plant parameters were investigated ana ly t ica l ly and experimentally. An analysis i s presented employing four uncoupled modes: wing bending, wing tors ion, power-plant p i tch , and power-plant yaw. Experi- mental r e su l t s a r e compared with t h i s four-mode analysis and with an analysis including only power-plant freedoms.

The experimental and ana ly t i ca l r e su l t s ind ica te t h a t t he e f f ec t s of t he wing on the whir l f l u t t e r boundary were la rge f o r some cases, depending on the system parameters, and were generally s t ab i l i z ing f o r t he configurations con- sidered. The four-mode analysis generally gave b e t t e r r e su l t s than the two- mode analysis , but only gave fa i r agreement i n some instances. The analyses indicated tha t i n cases of large-amplitude wing motion, wing aerodynamics can have s igni f icant s t ab i l i z ing e f fec ts .

INTRODUCTION

A design consideration f o r propeller-driven a i r c r a f t i s t he prevention of a dynamic i n s t a b i l i t y , generally termed propel ler whir l f l u t t e r o r autopreces- sion, i n which the propel ler hub wobbles or executes a whirling motion. The f lu t te r f o r an i so l a t ed power plant has been t r ea t ed ana ly t ica l ly i n refer- ences 1 t o 7. Comparisons of measured and calculated f l u t t e r boundaries f o r a windmilling propel ler ( r e f . 8) indicated t h a t t he whi r l - f lu t te r boundary could be s a t i s f a c t o r i l y predicted i f ' accurate aerodynamic parameters were used. Cal- culat ions f o r th rus t ing propel lers i n t h e cruise condition ( r e f s . l t o 4) have indicated l i t t l e e f f ec t of t h rus t on whirl f l u t t e r . However, t h e question of t he e f f ec t s of t he wing has been considered only b r i e f l y i n these investiga- t ions . Furthermore, comparison of t h e theo re t i ca l r e s u l t s of reference 1 f o r an i so la ted power p lan t with the data of reference 9 f o r a four-engine model indicated large differences t h a t might be a t t r i bu ted t o the wing and complete

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model. Since further investigation of the effects of the wing was desirable, the present investigation was undertaken with the following objectives:

(1) To determine the whirl-flutter boundaries of a power plant (with wind- milling propeller) on a cantilever, semispan wing, which not only has realistic parameters but is also more readily amenable to analysis and physical interpre- tation than a complicated four-engine model.

(2) To investigate the system of reference 8 on the semispan wing and eval- uate the range of applicability of the theory for an isolated power plant in a limited manner.

( 3 ) To analyze combined wing and power-plant flutter, evaluate the analysis, and explore the dynamics of wing-power-plant coupling.

For this investigation the model of reference 8 was mounted on a canti- levered semispan wing having an aspect ratio of 6.97 (based on area of exposed wing panel) and a taper ratio of 0.430. Flutter boundaries were measured over a range of power-plant stiffnesses and damping levels and propeller blade angles. In addition, three other configurations differing in power-plant param- eters were investigated over a range of blade angles for a single value of stiffness and of damping. Following the list of symbols in appendix A, the equations of motion are developed in appendix B for coupled wing-power-plant flutter by using an uncoupled-mode analysis and by omitting aerodynamic coupling between the wing and propeller. The results of this analysis are compared with the results of the experiment and with the theory for an isolated power plant. The effects of gyroscopic coupling on the natural vibration modes are'briefly considered in appendix C.

APPARATUS AND TESTS

Tunnel

The investigation was made in the Langley transonic dynamics tunnel which is a slotted-throat, variable-pressure, single-return wind tunnel having a test section 16 feet square (with cropped corners). Mach numbers up to 1.2 and at stagnation pressures from near vacuum to slightly above atmospheric. Either air or freon can be used as a test medium. Large windows are provided f o r close, unobstructed viewing of the model. The dynamic pressure in the test section can be rapidly reduced when flutter occurs by the operation of a bypass valve in a channel which connects the plenum chamber sur- rounding the test section to the return passage of the tunnel.

It is capable of operation at

The present investigation was conducted in air at near atmospheric condi- tions and at Mach numbers less than 0.3.

2

Model

General - arrangement .. - . - - . and mounting.- The model consisted of a semispan wing with a -single nacelle-power-plant combination with windmilling propel ler . Although no attempt was made t o sca le the model from a spec i f ic airplane, t he model parameters are believed t o be r e a l i s t i c since t h e model w a s adapted from t h e four-engine model of reference 9. The model was cantilevered from t h e tun- n e l w a l l as shown i n an ove ra l l view i n f igure 1. This was accomplished by sandwiching t h e root por t ion of t h e wing spar with heavy p l a t e s (see f i g . 21, bo l t ing them together, and clamping t h e assembly i n a remotely operable turn- table , which was used t o adjust t h e angle of a t tack of t h e model (see sect ion e n t i t l e d "Tests"). I n order t o minimize e f f ec t s of the w a l l boundary layer and t o fair around t h e r i g i d inboard port ion of t h e wlng, a simulated fuselage w a s bol ted t o t h e tunnel w a l l . pl ished by means of a f l e x i b l e foam mater ia l .

Sealing around the wing-fuselage juncture was accom-

Simulatzd power pla.a.- Four configurations, d i f f e r ing i n t h e parameters of t h e s i m l a t e d power plant , were t e s t e d and a r e denoted herein as configurations A, B, C, and D. The system consisted of a simulated engine and a propel ler attached t o an aluminum mounting beam through a gimbal with spring-restrained p i t ch and yaw freedoms. A variable-speed motor with an eccent r fca l ly mounted weight on i t s shaf t formed p a r t o f t h e engine mass and served as a shaker device f o r configurations A and B. The schematic f o r each configuration i s given i n f igure 3 . Configuration A w a s t he symmetrical model of reference 8. Configura- t i o n B d i f fe red from configuration A only i n the locat ion of t h e lead weights, which were moved from t h e rear of t h e engine t o the mounting beam at t h e same distance from the e l a s t i c ax is ( f i g . 3 ( a ) ) . engine mass was a lso removed from the gimbal s t ruc ture and mounted on the mounting beam at t h e same dis tance from the e l a s t i c ax is ( f ig . 3 (b ) ) . t i o n D d i f fe red from configuration C i n t h e spring and c l i p arrangements which permitted unequal p i t ch and'yaw s t i f fnesses . f igura t ion a r e given i n t a h e I.

For configuration C y t he simulated

Configura-

The mass parameters f o r each con-

The aluminum beam of t h i s model on which t h e gimbal was mounted was similar t o t h a t of reference 9, but was s t i f fened t o r a i s e i t s frequenhies considerably above t h e whi r l - f lu t te r frequencies by t h e addltion of L s e c t i o n doublers ( f i g . 3 ) . propel ler configuration A were 29 cps i n p i t ch and 24 cps i n yaw.

For example, can t i lever frequencies of t h e mounting beam with engine-

The nacel le was made of ba lsa construction and w a s f ixed t o t h e wing beam with a stiff mounting bracket. Clearance f o r whirling motions was le f t between the nacel le and spinner.

The four-blade aluminum propel le r was t h a t used f o r t he invest igat ion of references 8 and 9. The va r i a t ion of windmilling advance r a t i o V/nD with reference blade angle i s given i n f igure 4 (from ref. 8).

Wing.- The wing, shown i n f igure 2, consisted of a built-up aluminum spar f o r s t i f f n e s s and ba lsa pods f o r a i r f o i l contour. were sealed with f l ex ib l e , foam p l a s t i c s t r i p s . For analysis purposes, t he root was considered t o be f ixed a t t h e b o l t s thruugh t h e outboard edge of t he sand- wiching p la tes , 1.7 inches inboard of t h e edge of t h e f irst pod.

The gaps between t h e pods

The rounded

3

t i p w a s t r e a t e d as a streamwise t i p ending a t t h e t i p of t h e wing spar ( f ig . 2) . The aspect r a t i o was 6.97 (based on area of t h e exposed wing panel) and the taper r a t i o was 0.430. was swept forward 4.9'. e l a s t i c ax i s i s shown i n f igure 3 .

The center l i n e of t he wing spar, which was t h e e l a s t i c axis, The nondimensional dis tance from t h e midchord t o the

Although t h e e l a s t i c ax is was swept forward, t h e pods were formed as stream- wise s t r i p s ( f ig . 2 ) . axis of each s t r i p w e r e measured i n the streamwise d i rec t ion a t t h e spanwise pos i t ion of t he center of grav i ty of t h e s t r i p , and were considered t o be t h e average values over t h e width of t he s t r i p . (The measurements f o r t h e spar were made on a similar wing spar cut i n t o s t r i p s . ) these parameters a r e presented i n figure 6 i n which the nondimensional parameters a r e based on the streamwise semichord at the spanwise s t a t ion of t h e center of gravi ty of t h e s t r i p . The parameters f o r t h e propeller, power plant , nacelle, and mounting beam (not included i n f i g . 6 ) are presented i n table I.

The m a s s , s t a t i c unbalance, and i n e r t i a about t h e e l a s t i c

The spanwise d i s t r ibu t ions of

The spanwise d i s t r ibu t ions of bending and t o r s i o n a l s t i f f n e s s a r e presented i n f igure 7. The s t i f f n e s s values were obtained by d i f f e ren t i a t ing parabolic approximations of the slope of t he s t a t i c def lec t ion curve f o r a concentrated load at t h e t i p . The slopes, which were measured with a mirror system and were averaged f o r severa l values of loading magnitude, a r e a l so presented i n f igure 7.

Vibration Modes

I n order t o check t h e mathematical descr ipt ion of t h e wing by using t h e measured physical propert ies , a series of v ibra t ion measurements were made and compared with calculat ions. For t h e bare wing spar, which has small s t a t i c unbalance coupling, measured first and second bending frequencies were 17.6 cps and 59.5 cps, respectively. Corresponding calculated uncoupled frequencies (obtained by t h e method of r e f . 10) were 17.0 cps and 66.4 cps. The calculated first to r s iona l frequency of t h e bare wing spar was 267 cps; coupled modes were measured a t 255 and 270 cps, t he l a t t e r being more near ly a pure to r s ion mode.

With t h e pods added t o the wing spar, but t he nacelle omitted, t h e f i r s t t h ree measured modes were 10.6 cps (f irst bending), 39.0 cps (second bending), and 47.0 cps (f irst to r s ion ) . two uncoupled bending modes and one uncoupled t o r s i o n a l mode, a l l coupled by s t a t i c unbalance i n a Rayleigh-Ritz type ana lys i s ) were 11.2 cps, 43.1 cps, and 53.4 cps, respectively. Therefore, t h e mathematical descr ipt ion of t h e wing using measured physical propert ies s a t i s f a c t o r i l y pred ic t s the measured frequen- c i e s of t he bare spar or of the wing without t h e nacelle.

Corresponding calculated frequencies (considering

The measured node l i n e s of t h e f i rs t three modes, which involve pr imari ly wing motion, a r e sketched i n f igure 8 f o r t h e complete model of configuration A with shape and f i rs t t o r s i o m l (17.1 cps) mode shape a r e presented i n f igure 9. these two modes a r e coupled by s t a t i c unbalance and the engine p i t ch mode, the calculated coupled frequencies compare with measurements as follows:

f e = fq = 9.3 cps. The calculated uncoupled f i rs t bending (10.1 c p s ) mode '&en

4

I I

I I II II

-~

Predominant response

- .. ~~ -

Engine pi tch, 8 9.3

Engine yaw, I$

I F i r s t wing bending, hl

Wing tors ion, a

Second wing bending, h2

1 Measured I frequency, cps Calculated coupled

frequency, cps

9.2

9.3

10.1

17.4

9.2

9.3

10.1

17.4

9-3

9.3

9-3

10.4

14.3

Good agreement i n frequency i s obtained f o r these modes with t h e exception of t h e 14.3-cps mode which i s pr imari ly wing tors ion . The poor agreement f o r t h i s mode may be a r e s u l t of considering only two wing modes i n the calculat ions. Although the next coupled mode involving pr imari ly wing motion has a frequency of 39.1 cps, t h e calculated uncoupled second bending mode frequency i s 23.7 cps, and i t s inclusion could possibly a f f ec t t h e r e su l t s .

Instrument a t ion

An automatic d i g i t a l readout system was used t o record t h e tunnel s t a t i c and stagnation pressures and stagnation tmpera ture .

The accelerat ions of t h e simulated engine mass were detemined by l i nea r accelerometers mounted on t h e nonrotating pa r t of t h e pivot ing mass. The fre- quency and a qua l i t a t ive indicat ion of t he amplitudes of t h e wing bending and to r s iona l o sc i l l a t ions were determined by s t r & i n gages near the wing root . These data were recorded on a d i r e c t wri t ing oscil lograph recorder f o r real-t ime monitoring of the test . pickup, mounted a t the propel ler shaf t , which drove an e lec t ronic counter.

Propel ler ro t a t iona l speed was measured by a magnetic

Tests

I n t h e tests of configurations A and B, t he tunnel was brought up t o a low speed, t h e model was i n i t i a l l y disturbed s inusoidal ly by t h e shaker, and t h e nature of t h e f r e e v ibra t ions was observed v i sua l ly and on t h e oscil lograph record. A i r s t r e a m ve loc i ty was then increased by a small increment, and the model again disturbed. mately constant amplitude o s c i l l a t i o n was produced. The tunnel ve loc i ty was then rapidly reduced t o prevent model damage. t i o n s C and D was e s s e n t i a l l y t h e same; however, since t h e shaker was ineffec- t i v e f o r these configurations random tunnel turbulence served a s t h e f l u t t e r exc i ta t ion . "he absence of t h e shaker d id not appear t o -a i r t he se lec t ion of t h e f l u t t e r condition by t h e observer. The wing w a s maintained a t nearly zero s t a t i c t i p def lec t ion i n bending by varying t h e angle of a t tack of t h e wing with t h e mounting turn tab le .

This procedure was continued u n t i l a sustained, approxi-

The procedure f o r configura-

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I n some cases t h e s t a b i l i t y of t h e f l u t t e r mode was Fndependent of ampli- tude as indicated by linear theory. However, as i n t h e tests reported i n refer- ence 8, cases w e r e encountered i n which t h e amplitude of f l u t t e r was l imited at a given velocity, and increased with increasing velocity, indicat ing a nonlinear e f f ec t . The wind-off power-plant damping f o r these cases increased with ampli- tude. The per t inent damping coeff ic ient was assumed herein, as i n reference 8, t o be that derived from t h e records of the decay of free vibrat ion at t h e ampli- tude of f l u t t e r . Also, t he re were some instances where unstable osc i l l a t ions were produced i f su f f i c i en t ly large amplitudes of motion were excited without an increase i n velocity. amplitude; hence, t h e appropriate damping coef f ic ien t was assumed t o be that derived f’” t h e decay records at t h e amplitude where t h e motion began t o be unstable. given s t i f f n e s s o r damping level . during a series of runs, the accuracy of

The damping f o r these cases decreased with increasing

Damping records were taken before and after each series of runs at a A s t h e power-plant damping varies somewhat

2% i s estimated t o be about kO.003.

Power-plant s t i f f n e s s and damping levels and propel ler blade angle were varied separately f o r configuration A. i n se r t ing sponge rubber between the gimbal r ings. ) Several blade angles, but only one l e v e l of s t i f f n e s s and damping, were investigated f o r configurations B, C, and D. A summary of tes t conditions and data i s given i n t a b l e 11.

(The damping l e v e l was adjusted by

A l l t h e f l u t t e r t e s t s were made in air at near atmospheric conditions and at Mach numbers less than 0.3.

DESCRIPTION OF ANALYSIS

An analysis of t h e whirl f l u t t e r of a wing-propeller-power-plant combina- t i o n including wing f l e x i b i l i t y i s developed i n detail i n appendix B and i s applied herein t o t h e model of t h i s report . The analysis i s based on a Lagrangian formulation of t h e equations of motion employing def lect ions i n four uncoupled vibrat ion modes a s generalized coordinates. i c a l l y i s depicted i n figure 10. system moun-bed with spring-restrained gimbals t o a r i g i d mounting beam which i s mounted on the f l ex ib l e wing. sion, are considered. s t ra ight , unswept e l a s t i c axis. included; however, aerodynamic interference e f f ec t s between t h e wing and pro- pe l l e r , and nacel le aerodynamic forces a r e neglected. The aerodynamic forces ac t ing on t h e propel ler are expressed i n terms of s t a b i l i t y der ivat ives a s i n references 1, 2, and 8. employed t o represent t h e wing aerodynamic forces .

The system t r ea t ed analyt- It consis ts of a single, r i g id engine-propeller

Two vibrat ion modes of t h e wing, bending and t o r -

Both propel ler and wing aerodynamic forces are These a r e described as bending and twist ing about a

T k modified-strip analysis method of reference ll i s

From t h i s analysis t h e speed for neut ra l s t a b i l i t y (boundary between s tab le and unstable motion) i s de temimd. This f l u t t e r speed, expressed i n nondimen- s iona l form as V/Rwg, i s determined a s a f’unction of the damping of t h e engine p i t c h mode, 258, by specifying t h e following damping parameters: h, gh, and 2{ ,~ /250. The method of solution and input quant i t ies required are discussed i n appendix B. A discussion of t h e determination of t h e input quant i t ies f o r t h e

6

model of t h i s report i s a l so given i n appendix B. of t he appl icat ion of t h i s analysis with experiment f o r t he configuration of t h i s report i s discussed i n t h e following section.

The comparison of t h e r e s u l t s

RESUITTS AND DISCUSSION

Remarks on Wing F l u t t e r

A conventional wing f l u t t e r . .analysis as obtained from the f l u t t e r determi- nant (eq. (B43)) by assuming complete r i g i d i t y of the power-plant mounting i n p i t ch and yaw and by using t h e f i rs t t o r s i o n a l and bending modes of the wing indicated no f l u t t e r so lu t ion with o r without propel ler aerodynamics. be pointed out t h a t s tud ies of references I 2 and 13 indicate t h a t , i n general, more than two uncoupled modes would be required t o predict f l u t t e r accurately when la rge concentrated weights a r e attached t o the s t ruc ture . f l e x i b i l i t y i n t h e mounting of a concentrated mass can have an important inf lu- ence on f l u t t e r (ref. 14). located wel l ahead of t h e e l a s t i c axis - a condition which has been shown t o have a strong s t a b i l i z i n g influence on wing f l u t t e r (see r e f . 171 - it i s f e l t t h a t t h e present predict ions based on a two-mode analysis a r e a t l e a s t indicat ive t h a t c l a s s i c a l wing f l u t t e r occurs at ve loc i t i e s well above those considered herein. Hence, t he primary emphasis i n t h i s invest igat ion w i l l be on the e f f e c t s of t he wing on propel le r whir l f l u t t e r .

It should

Furthermore,

However, since t h e concentrated nacel le mass i s

Whirl F l u t t e r

The experimental f l u t t e r data f o r configuration A a r e presented i n f igure 11 i n the form of viscous damping of t h e power-plant modes required f o r s t a b i l i t y as a function of reduced ve loc i ty i n terms of t he average wind-off f re -

quency, 6 = e; t h e stable region i s above t h e boundary. Such a boundary 2 i s unique f o r a given blade angle f o r f l u t t e r of a power plant f l ex ib ly mounted on a r i g i d s t ructure , but becomes a f'unction of t h e mass, na tu ra l frequencies, and damping of t he wing f o r the power p lan t mounted on a f l e x i b l e wing. Thus, t he power-plant s t i f f n e s s l e v e l i s indicated ( f i g . 11) by t h e r a t i o of uncoupled frequencies z/%. A s densi ty varied s l i g h t l y from one t e s t point t o another and the damping i n p i t ch and damping i n yaw were not widely d i f f e ren t , the damping

parameter 25 =

reference densi ty value of 0.0022 slug/cu f't. one f o r the power p lan t mounted on a r i g i d s t ruc ture ( r e f . 8 ) . The power-plant damping i s t r ea t ed as viscous damping, as b e t t e r agreement was obtained i n re fer - ence 8 with viscous damping than with s t r u c t u r a l damping. However, t he damping of t h e wing modes i s considered t o be s t r u c t u r a l damping herein.

V/% c D +

has been used t o adjust t h e experimental data t o a 3 2!5e i- 259 PO

2 P -

This r e l a t i o n i s an approximate

The data f o r configuration A are compared with t h e theory including only power-plant p i t c h and y a w freedoms (from re f . 8 ) and with t h e theory including four modes: power-plant p i t c h and yaw, wing bending, and wing to r s ion f o r a

power-plant r e s t r a i n t s t i f f n e s s r e su l t i ng i n a frequency r a t i o A comparison of the data with the theory f o r two modes shows good agreement f o r some cases, but only fair f o r others. It should be noted t h a t t h e results of reference 8 f o r an i so l a t ed power p lan t with two degrees of freedom indicated very good agreement between theory and experiment. The theory f o r four modes ( f o r Os/% = 1.13) shows a s t ab i l i z ing e f f e c t i n comparison with the theory f o r two modes a t the lower damping l eve l s and i s i n very good agreement with the data ( fo r that the regions of la rge coupling between t h e whirl ing modes and t h e wing modes, as indicated by frequency s h i f t s , are near t he coincidences of the frequencies of the whirling modes with those of t he couplea modes involving primarily wing motion. than those of the f i r s t coupled mode; however, f o r values of bending frequency the whir l f l u t t e r frequencies were as much as 0.9 of the fre- quency of t he f i r s t coupled mode. frequencies i s sham i n the data ( f i g . 11).

G/% of 1.13.

a/% = 1.13) except a t a blade angle of 5 8 O . Appendix C indicates

The w h i r l f l u t t e r frequencies f o r configuration A ( tab le 11) were lower above the f i r s t 5

N o consis tent e f f e c t of the higher whirling

The f lu t te r boundaries calculated f o r configurations B, C, and D including only power-plant p i t ch and yaw modes a re compared with the boundaries calculated including wing bending and to r s iona l modes f o r th ree blade angles i n f igure 12. For configuration D, the presentation i s based on V / R q as q and WJ/ are widely d i f fe ren t ( tab le 11). dic ted f o r configurations B and D, but a la rge s t ab i l i z ing e f f e c t i s predicted f o r configuration C a t t h e higher blade angles. A l s o , f o r configuration C where t h e wing i s s ign i f i can t ly s tab i l iz ing , the f l u t t e r boundary i s l e s s sens i t ive t o damping ( f i g . 12(b) ) .

A s m a l l s t ab i l i z ing e f f e c t of the wing i s pre-

The theo re t i ca l f l u t t e r speed boundaries employing e i the r two or four modes are compared with experimental data f o r a s ingle value of power-plant r e s t r a i n t s t i f f n e s s and damping f o r each configuration i n f igure 13. Corresponding values of flutter-frequency r a t i o are a l s o presented i n f igure 13. The theo re t i ca l values f o r configuration A, taken from f igure 11, show l i t t l e e f f ec t of t he wing on the f l u t t e r speeds ( f i g . l3(a)) ; however, some e f f ec t i s indicated by the experimental f l u t t e r speed a t a blade angle of 5 8 O . Calculated f l u t t e r frequen- c i e s f o r two- and four-mode analyses d i f f e r l i t t l e and a r e i n good agreement with experiment ( f ig . l3 (a) ) . speed boundary employing four modes i s higher than t h a t obtained using two modes, but, as f o r configuration A, i s i n good agreement with experiment only a t the lower blade angles ( f ig . l 3 ( b ) ) . Calculated f l u t t e r frequencies show l i t t l e e f f e c t of the wing, but are higher than experiment by about 10 t o 15 percent ( f i g . 13 (b) ) . four modes i s considerably higher than that calculated with two modes and i s i n good agreement with experiment over the range of blade angles investigated experimentally ( f i g . l 3 ( c ) ) . F l u t t e r frequencies a re a l so i n b e t t e r agreement from the four-mode analysis ( f i g . l 3 ( c ) ) , but are high by about 10 t o 25 percent. Only a small s t ab i l i z ing e f f e c t of the wing i s indicated by the four-mode anal- y s i s f o r configuration D ( f ig . l 3 ( d ) ) . higher than experiment. Calculated f lu t te r frequencies a re i n good agreement f o r the two-mode calculations, bu t a r e low when four modes are employed

For configuration B, the theo re t i ca l f lu t te r

For configuration C, the theo re t i ca l f l u t t e r boundary employing

However, both theo re t i ca l boundaries a re

( f ig . 13(d> 1

8

Thus, f o r configuration A t he wing had a small e f fec t ; however, fo r con- f igurat ions B and C t h e wing had a s ignif icant s t ab i l i z ing e f fec t . The addi t ion of t he two wing modes t o t h e theory of reference 1 generally improved t h e agree- ment of t h e theory with experiment, pa r t i cu la r ly f o r configuration C, but did not f u l l y predict t h e s t ab i l i z ing e f f ec t s or trends. The r e s u l t s of reference I 2 showed t h a t t h e f l u t t e r mode shape of a wing with a large concentrated mass could be qui te d i f fe ren t from t h e first bending or t o r s iona l mode. I n par t icu lar , t h e r e l a t ive amplitude a t f l u t t e r a t t h e pos i t ion of t h e concentrated mass was higher than t h a t f o r t h e f i rs t bending mode and lower than t h a t f o r t h e f irst to r s iona l mode. Such a r e s u l t could lead t o t h e observed t rends herein, and suggests t h a t inclusion of addi t ional wing modes m a y be required f o r b e t t e r correlat ion.

I n reference 1 a comparison of t h e theory f o r t he i so la ted power plant with data of reference 9 indicated t h e p o s s i b i l i t y of large s t ab i l i z ing e f f e c t s of the wing and complete model. Also, unpublished ana ly t ica l s tudies by R. J. Zwaan and H. Bergh of t he National Aero-Astronautical Research I n s t i t u t e , N.L.R., Amsterdam,.show tha t generally the wing has a s t ab i l i z ing e f f ec t on whir l f l u t - t e r , but under cer ta in conditions wing f l e x i b i l i t y can give a reduction i n the f l u t t e r speed.

Although t h e theory including four modes was generally conservative f o r configurations A, B, and C, f o r configuration D, t h e four-mode theory predicted higher speeds than t h e two-mode theory and was higher than experiment (or uncon- servat ive) . For configuration D, the r a t i o of yaw s t i f f n e s s t o p i t ch s t i f f n e s s was approximately 9:l. However, if t h e f l e x i b i l i t y of t h e nacelle mounting beam, which w a s not considered i n t h e analysis, w e r e taken in to account t h i s r a t i o would be reduced somewhat; t h i s may possibly account fo r t h e measured f l u t t e r speeds being somewhat lower than the predicted f l u t t e r speeds.

The Effect of Wing Aerodynamics

The e f f ec t of wing aerodynamics has been considered by omitting t h e wing aerodynamic terms, equation (B391, i n a four-mode analysis fo r configuration C . The resu l t ing f l u t t e r charac te r i s t ics a re compared with t h e two-degree-of-freedom calculations and four-degree-of-freedom calculat ions including wing aerodynamics i n f igure 14 f o r three propel ler blade angles. The asymptotes (taken from f i g . 151 which t h e coupled frequencies approach f o r large values of no wing or propel ler aerodynamics) are included f o r comparison. These asymptotes represent boundaries which t h e coupled frequencies cannot cross (see appendix C ) . For t h e Bo blade angle, which i s f o r large values of region of high coupling, there i s l i t t l e difference among the calculated t rends. However, fo r t he other blade angles, two f l u t t e r solut ions were obtained when wing aerodynamics were omitted. These two solutions w e r e obtained i n t h e region of high coupling, as indicated by t h e approach of two coupled frequencies t o one another. However, when wing aerodynamics are included ( f i g . 141, only one of t h e f l u t t e r solutions i s obtained. Thus, it i s indicated t h a t t h e inclusion of wing aerodynamics i s qui te important for cases t h a t involve large coupling of wing and propel ler motion; t h e omission of wing aerodynamics could lead t o an unnecessarily conservative f lu t te r speed.

fi/E (with

R / c u ~ , and beyond t h e

9

I Il1l111l1l1l II I I

Propel ler whi r l f l u t t e r boundaries were measured f o r a s ingle power plant mounted on a cantilevered semispan wing f o r severa l configurations d i f f e r ing i n power-plant parameters. An uncoupled-mode ana lys i s of t h e system was made employing the following four degrees of freedom: power-plant pi tch, and power-plant yaw. Comparisons of experimental data, r e s u l t s of t h e four-mode analysis, and t h e r e s u l t s of an analysis including only power-plant freedoms indicate t h e following conclusions :

wing bending, wing tors ion,

1. The e f fec t of t h e wing on t h e whir l f l u t t e r boundary can be la rge depending on the system parameters. These e f f e c t s were generally s t ab i l i z ing f o r the configurations considered.

2. The four-mode analysis generally gave b e t t e r r e s u l t s than the two-mode analysis, but only gave f a i r agreement i n some cases.

3 . I n cases of la rge coupling between t h e wing and the propel ler whir l modes, t he inclusion of wing aerodynamics i n t h e analysis can have la rge s ta - b i l i z i n g e f f ec t s .

Langley Research Center, National Aeronautics and Space Administration,

Langley Stat ion, Hmpton, V a . , Apr i l 28, 1964.

10

APPENDIX A

SYMBOLS

A i j

a

b

b r

Cm

Cn

CY

CZ

C 1,

D

EI

f l u t t e r determinant element (see eq. (€342))

nondimensional dis tance from wing midchord t o e l a s t i c ax is , measured perpendicular t o e l a s t i c ax i s , pos i t ive rearward, f r a c t i o n of semichord b

nondimensional dis tance from wing midchord t o l o c a l aerodynamic center ( f o r steady flow), measured perpendicular t o e l a s t i c ax is , pos i t ive rearward, f r ac t ion of semichord b

dis tance from wing midchord t o e l a s t i c axis a t propel ler s t a t ion , pos i t ive rearward, f t

r a t i o of wing l o c a l semichord t o reference semichord, measured perpendicular t o e l a s t i c ax is , b/br

wing semichord, measured perpendicular t o e l a s t i c ax is , f t

wing reference semichord, 0.6344 f t

MY,P propeller pitching-moment coef f ic ien t , pV2S 'R

propeller yawing-moment coef f ic ien t , %,p pV2S 'R

FY,P

FPV 5 ' propel le r side-force coef f ic ien t ,

1 2

FZ,P , n

propeller ver t ica l - force coef f ic ien t , +vcs ' 2

wing l o c a l l i f t - c u r v e slope f o r a streamwise sect ion i n steady flow, per radian

d iss ipa t ion function ( see eq. (B21)) ; a l s o propeller diameter, 2R, f t

wing bending s t i f f n e s s , l b - f t 2

11

I

ea

e6

F

FY,P

i-’z , P

f

fi

G

G J

g i

h

Ia, W

I a , P

10,P

12

dis tance from wing e l a s t i c axis t o plane of propeller-blade quarter-chord poin ts a t three-quarter radius , pos i t ive rearward, ee + Z e f (ab)p, f t

dis tance from gimbal p i t c h axis t o plane of propeller-blade quarter-chord poin ts at three-quarter radius , pos i t ive rearward, f t

dis tance from gimbal yaw axis t o plane of propeller-blade quarter- chord poin ts at three-quarter radius , pos i t ive rearward, f t

real p a r t of complex c i rcu la t ion function based on kr

propel le r aerodynamic force along Y-axis, l b

propel le r aerodynamic force along Z-axis, lb

vibra t ion frequency, cps

v ibra t ion frequency i n i t h mode, cps

imaginary p a r t of complex c i r cu la t ion function based on

wing t o r s i o n a l s t i f f n e s s , l b - f t 2

kr

s t r u c t u r a l damping coef f ic ien t f o r i t h v ibra t ion mode

l o c a l v e r t i c a l bending displacement of wing e l a s t i c ax is , pos i t ive down, f t

v e r t i c a l bending displacement h of wing e l a s t i c a x i s a t t h e propel le r s t a t ion , f t

mass moment of i n e r t i a of propeller about a x i s of rotat ion, s lug-f t2

generalized mass moment of i n e r t i a of wing about e l a s t i c axis ( see eq. ( B 6 ) ) , s lug-f t2

generalized m a s s moment of i n e r t i a of engine-propeller about e l a s t i c axis (see eq. (B12)), slug-f t2

generalized t o t a l mass moment of i n e r t i a about e l a s t i c axis, I,J + I,,P, s lug-f t2

generalized mass moment of i n e r t i a of engine-propeller about gimbal p i t c h a x i s (see eq. (B12)), s lug-f t2

_-._ ,,,,. ., 111-.1,,,, .,I .I I II 11.1.11111 1 1 1 1 . 1 1 - 1 1111 1111 111 111.1.11111 ll1.11111.1 I I 11.11

i = d - 1

i0

J

K i

kr

l e

generalized m a s s moment of i n e r t i a of-engzne-propeller about gimbal yaw ax i s (see eq. ( B 1 2 ) ) , s lug-ft2

generalized cross m a s s moment-of-inertia term (see eq. ( B 1 2 ) ) , s lug-ft2

generalized m a s s moment of inertia of propel ler about ro ta t ion axis ( see eq. ( ~ 1 6 ) ) , slug-ft2

m a s s moment of i n e r t i a of engine-propeller about gimbal p i t ch axis (see eq. (B13)), slug-ft2

m a s s moment of i n e r t i a of engine-propeller about gimbal yaw axis (see eq. (B13)), slug-f t2

wing m a s s moment of i n e r t i a about e l a s t i c axis per un i t length (see eq. (B7) ) , slug-f t

windmilling propel ler advance r a t i o , V/nD

dimensional s t i f f n e s s i n i t h vibrat ion mode

reduced frequency based on wing reference semichord, b+/V

distance from wing midchord t o gimbal p i t ch axis , posi t ive rearward, f t

distance from wing midchord t o gimbal yaw axis, posi t ive rearward, f t

engine-propeller generalized m a s s (see eq. (B12)), slugs

wing generalized mass (see eq. ( B 6 ) ) , s lugs

t o t a l generalized m a s s % + Mp, slugs

nondimensional generalized m a s s (see eq. ( B 2 7 ) )

propel ler aerodynamic moment about Y-axis, f t - l b

propel ler aerodynamic moment about Z-axis, f t - l b

osc i l l a to ry wing aerodynamic moment about e l a s t i c axis per un i t length, pos i t ive leading edge up, lb

engine-propeller m a s s (see eq. ( B 1 3 ) ) , s lugs

Qi

Qi,P

Q i , W

Qij ,P

Qi j ,W

9

q i

si R

r

ra

S '

Sa, P

Sa, w

Sa, WP

engine-propeller mass per un i t length, s lugs / f t

wing mass per u n i t length (see eq. ( B 7 ) ) , s lugs / f t

wing mass per un i t area, slugs/sq ft

propel ler ro ta t iona l speed, r p s

osc i l l a to ry wing aerodynamic l i f t per u n i t length, posi t ive down, l b / f t

generalized force i n i t h vibrat ion mode

generalized propel ler aerodynamic force i n i t h vibrat ion mode

generalized wing aerodynamic force i n i t h vibrat ion mode

nondimensional generalized propel ler aerodynamic force (see eq- (B3W

nondimensional generalized wing aerodynamic force (see eq. (B39))

propel ler p i t ch r a t e , 6R/V

i t h generalized coordinate (see eqs. (B4) and (B10))

complex amplitude of qi (see eq. (€324))

propel ler radius , 0.8438 f t

propel ler yaw ra t e , $R/V

nondimensional radius of gyration about wing e l a s t i c axis ,

propel ler disk area, sR2, sq f t

engine-propeller generalized m a s s unbalance about e l a s t i c axis (see eq. (B12)) , slug-f t

wing generalized mass unbalance about e l a s t i c axis (see eq. ( B 6 ) ) , s lug - f t

t o t a l generalized mass unbalance about e l a s t i c axis, SaYw + Sa,p, s lug-f t

se ,P

S

e S

T

TP

TW

Til

P

YP

zP

engine-propeller generalized m a s s unbalance about p i t ch axis (see eq. (B12)), slug-f t

wing semispan, f t

engine-propeller mss unbalance about p i t ch axis (see eq. ( B l 3 ) ) , s lug - f t

wing mass unbalance about e l a s t i c axis per un i t length (see eq. 63711, slugs

k ine t i c energy, f t - l b

k ine t i c energy of engine-propeller system (see eq. ( B 8 ) ) , f t - l b

k ine t i c energy of wing (see eq. ( B ? ) ) , f t - l b

k ine t i c energy of propel ler rotat ion, including gyroscopic pre- cession energy (see eq. (SI&)), f t - l b

time, sec

poten t ia l energy, f t - l b

free-stream veloci ty , f t / s ec

coordihate axis system (see f i g s . 10 and 16)

coordinate distances i n X,Y,Z coordinate system, f t

nondimensional dis tance from wing e l a s t i c axis t o l o c a l center of gravi ty , measured perpendicular t o e l a s t i c axis, posi t ive rear- ward, f r ac t ion of semichord b

l a t e r a l displacement of m a s s element of engine-propeller system, pos i t ive r igh t , f t

v e r t i c a l displacement of m a s s element of engine-propeller system, pos i t ive down, f t

v e r t i c a l displacement of mass element of wing, posi t ive down, f t

l o c a l wing angle of a t tack, posi t ive leading edge up, radians

wing angle of a t t ack a a t propel ler s ta t ion , radians

propel ler blade angle a t three-quarter propel ler radius, deg

v i r t u a l displacement i n i t h generalized coordinate

15

6W

fi

7

0

- e

"P

A

h

P

PO

If

n

cu

cui

- cu

v i r t u a l work, f t - l b

viscous-damping coeff ic ient , r a t i o of viscous damping t o c r i t i c a l damping f o r i t h vibrat ion mode

+ f : Po average viscous-damping coeff ic ient , -

2 P

nondimensional spanwise var iable , f r ac t ion of semispan s

p i t ch angle of propeller-shaft axis about gimbal axis , measured from wing chord l i n e a t propel ler s ta t ion , posi t ive propel ler up, radians

e f f ec t ive engine-propeller p i t ch angle, measured from r e l a t i v e wind d i rec t ion ( see eq. ( B 3 3 ) ) , radians

3 2 J ~ P R b r dens i ty- iner t ia r a t i o , -

sweep angle of wing e l a s t i c axis, deg

frequency rat i o , L U ~ / L U

a i r density, slugs/cu f t

nominal a i r density, 0.0022 s h g / c u f t

vibrat ion mode shape i n i t h mode

vibrat ion mode shape i n i t h wing mode a t propel ler s ta t ion

yaw angle of propeller-shaft axis, radians

e f fec t ive engine-propeller y a w angle, measured from re l a t ive wind d i rec t ion (see eq. ( B 3 3 ) ) , radians

propel ler ro t a t iona l speed, 2m, radians/sec

vibrat ion frequency, radians/sec

vibrat ion frequency of i t h vibrat ion mode , radians/sec

+ cuIf 2 ,

average wind-off power-plant vibrat ion frequency,

radians/sec

R

16

Subscripts:

f condition a t f l u t t e r

i, j modal designation; i , j = 0 , $, h, or a

P a r t i a l der ivat ives a re denoted by double subscripts; f o r example :

acm, and so fo r th . 3% cme = - a0

Dots over symbols indicate der ivat ives with respect t o time.

111lll IIIIII I II I

APPENDIX B

ANALYSI s

The ana ly t i ca l model considered here i s pictured i n f igures 10 and 16. The system t r e a t e d has four degrees of freedom: p i t c h and yaw of a s ingle r i g i d engine-propeller system and wing bending and to r s ion . y s i s i s employed t o obtain s t a b i l i t y boundaries f o r t h e system. The two elas- t i c modes of t h e wing a r e described as bending and twis t ing about a s t r a igh t , unswept e l a s t i c axis. Propeller and wing aerodynamic forces are included; how- ever, t he aerodynamic in te r fe rence e f f e c t s between t h e wing and propeller and t h e nace l le aerodynamic forces a r e neglected.

A modal-type anal-

Equations of Motion

A Lagrangian formulation of the equations of motion i s employed i n which def lec t ions i n t h e four uncoupled v ibra t ion modes serve as generalized coordinates.

Kinetic energy.-

where Tw represents porting s t ruc ture ; Tp

The t o t a l ' k i n e t i c energy of t h e system i s given by

T = TW + Tp +

t h e contribution of t h e wing represents t h e contribution

and r i g i d power-plant sup- of t he f l e x i b l y mounted,

nonrotating propeller-engine system; To l e r , including t h e k i n e t i c energy of gyroscopic precession.

i s the'energy of t he ro t a t ing propel-

The k i n e t i c energy of t he wing i s

where, f o r bending of and twis t ing about an e l a s t i c ax is ,

z(x,y,t) = h + (x - ab)u (B3)

The bending h and tors ion u def lec t ions a r e given i n terms of mode shapes and generalized coordinates by

18

- . I ..... ....,......-.,,..-- I

d

Subst i tut ing the expressions i n equations (B3) and (B4) i n to equation (B2) leads t o t h e r e s u l t

where

are generalized mass parameters and

are wing-section mass parameters.

The k ine t ic energy of t h e nonrotating propel ler i s

where the mass of t h e system i s assumed t o be concentrated along the propeller axis and the integrat ion i s taken over t h e pivoting system, and where

19

(B9) 1 zp(xy t ) = hp + ( x - ab)+ + (x - 2

Y,(X,t) = -(x - 2Jr)Jr

The p i t ch and yaw def lec t ions a r e given i n terms of generalized coordinates by

Subs t i tu t ing these r e s u l t s into equation (B8) l eads t o the r e s u l t

where

are generalized m a s s parameters and

20

I I

II"1.11. I 1 I 1 1 1 111 I.

a r e engine-propeller m a s s parameters. The integrat ions are carr ied out over the pivoting engine-propeller system.

The k ine t i c energy of t he ro ta t ing propel ler i s given by an obvious exten- sion of the r e s u l t i n reference 1 as

TQ = 1 2 I.[.' + a(6 + +)d I n terms of t he generalized coordinates i n equations (B4) and (B10) equa- t i o n (B14) becomes

where IQ rota t ion and

i s the mass moment of i n e r t i a of t h e propel ler about i t s ax is of

a r e generalized m a s s parameters of t he engine-propeller system.

The t o t a l k ine t i c energy i s seen t o be

21

IIl1111llIlll I I

where

-.- The po ten t i a l energy may be expressed i n the form

where K i s generalized s t i f fnes s . I n terms of uncoupled frequencies and gen- eral ized masses t h e po ten t i a l energy i s expressed as

Dissipation function.- The s t ruc tu ra l damping concept, i n which a diss ipa- t i v e force proportional t o displacement i s employed, will be used i n the deriva- t i o n herein. For such damping, t h e d iss ipa t ion function i s

where 0 i s the frequency of vibrat ion of the system and g i s the s t ruc tu ra l damping coeff ic ient .

Lagrange's equations.- The equations of motion of t h e system a re obtained .from Lagrange * s equation

where i = 0, 9, h, or a. Subst i tut ing t h e expressions from equations (BlT). , (B20), and (B21) in to equation (B22) and performing the indicated operations y ie lds the equations of motion

22

d

Assume simple harmonic motion with

where

i n equation (B23) and dividing the 0 , Jr, and a equations by -Ie,pm e

'eiwt (R being the propel ler radius) y ie lds and the h equation by - E IO,? t he nondimensional equations of motion

98, e t c . , a r e complex amplitudes of motion. Making these subs t i tu t ions 2 i u k

1

7

23

where

J = - IrV ilR

are nondimensional frequency parameters, br is the wing reference semichord, and

are nondimensional generalized masses.

24

Generalized Aerodynamic Forces

The aerodynamic forces ac t ing on the propel ler are expressed i n terms of propel ler s t a b i l i t y der iva t ives as w a s done i n references 1, 2, and 8 . The modified-strip analysis method of reference 11 i s employed t o represent t h e wing aerodynamic forces . The d i rec t ions of the aerodynamic forces and moments a re shown i n f igure 16.

Vir tual work.- The expression f o r t he work done on the system by v i r t u a l displacements of the coordinates i s

where the generalized forces have the following representations i n terms of pro- p e l l e r and wing aerodynamics

Q+ = Q+,P

‘h ‘h,P + ‘h,W

~a = %,p + ~ , w J Propel ler aerodynamic forces . - The aerodynamic forces act ing on the propel-

l e r are-Fe-e f i g . 16)

Qe,p = (MY,P + eeFz,p)qe

Q$,P = (%,P - e$Y,P)’P+

Qh,P = FZ,Pqh,P

J %,P = (MY,P + euFZ,P)(Pu,P

1

where

ea = eo + l e + (ab)p

A s i n references 1 and 8, My,p and %,p a re moments about the Y- and Z-axes i n the propel ler plane and Fy,p and FZ,P are forces act ing i n t h e Y and

Z These forces and moments may be expressed i n terms of propel ler s t a b i l i t y der ivat ives as

d i rec t ions on the propel ler axis i n the propel ler plane.

7

J are p i tch and yaw angles with respect t o the r e l a t i v e wind direct ion. t h a t i n equation (B32) a l l possible force and moment der ivat ives tha t depend on angle and angular rate are included; angular accelerat ion der ivat ives (which account f o r v i r t u a l mass e f f ec t s ) have been neglected, as i n reference 1. subst i tut ion of equation (B33) i n t o equation (B32), the use of generalized coordinates (eqs. (B4) and ( B l O ) ) , and the assumption of harmonic motion (eq. (B24)) lead t o the f i n a l expressions f o r the nondimensional propeller aerodynamic forces

Notice

The

26

where

i s an i n e r t i a r a t io parameter and

27

The aerodynamic forces ac t ing on t h e wing introduced i n eq. a r e given i n general as

01 I

where P(7) i s t h e wing-section lift force and G(q) i s t h e sect ion pitching moment about t he e l a s t i c axis. Any su i tab le unsteady aerodynamic theory may be used t o evaluate these expressions. method of reference 11 i s employed as follows:

I n t h i s report , t h e modified-strip analysis

where, for an unswept e l a s t i c axis,

i

The c i rcu la t ion functions f o r unsteady flow, F and G, depend, i n general, on & and Mach number. The c i r c l ed numbers a re the i n t e g r a l s given i n appendix A of reference 11. They are , i n t h e present notat ion,

28

@ = s,' cZUB2(.- 2s + a, - dq

F l u t t e r Determinant

Subst i tut ing equations (B34) and ( ~ 3 8 ) i n t o equation (B25) f i n a l form for the equations of motion. I n matrix notation

y ie lds the

29

I

- p = O

where

( ~ 4 2 )

I n order f o r t h i s s e t of equations (eq. (B41)) t o have a non t r iv i a l solu- t i on , t h e determinant of t h e coef f ic ien t matrix must vanish. This f l u t t e r determinant i s

= o

Application of Analysis

Solution of determinant.- The f l u t t e r determinant has been programed f o r a high-speed e lec t ronic d i g i t a l computer. I n t h i s program, f o r a given case, a l l input quan t i t i e s a re f ixed except kr, 4, and A2. A sequence of values of reduced frequency kr and propeller s t r u c t u r a l damping i n p i t c h ge i s used. For each combination of values of kr and ge the f l u t t e r determinant i s expanded t o give a four th degree cha rac t e r i s t i c polynominal i n Points of neu t r a l s t a b i l i t y of t h e system are obtained f o r t he combination of values of kr and ge t h a t makes one of t he four roots of the cha rac t e r i s t i c polynominal a pos i t ive real quant i ty . The values of G, go, and A2 at t h i s point of neu t r a l s t a b i l i t y are used here in t o compute t h e f l u t t e r frequency r a t i o

A2.

t h e f l u t t e r speed r a t i o

and t h e equivalent viscous damping (see ref. 1) required f o r neut ra l s t a b i l i t y

Whirl f l u t t e r boundaries are known t o be qui te sens i t ive t o t he amount of s t r u c t u r a l damping i n t h e system (see r e f s . 1, 2, and 5 , f o r example) ; hence, it w a s not des i rab le t o assume equal damplng i n a l l four modes of t h i s ana lys i s . This i s the reason f o r t h e choice of t he somewhat lengthy method of solut ion outlined. T h i s procedure allows the wing and propeller-engine damping t o be varied independently.

I I

With t h e solut ion of t he f l u t t e r determinant obtained as out l ined here the elements i n the square matrix i n equation (B41) a r e completely specif ied. choosing one of t h e four generalized coordinates, say possible t o solve t h i s set of equations f o r t h e r a t i o s of t he other generalized coordinates t o &,. The r e l a t i v e magnitudes and phases of these complex ampli- tude r a t i o s give some indica t ion of t h e nature of t h e f l u t t e r mode.

By &, a r b i t r a r i l y , it i s

Determination of input quant i t ies . - The input quan t i t i e s required t o solve the f l u t t e r determinant are l i s t e d i n equation (B42). obtained as outlined i n t h i s sect ion.

These quan t i t i e s were

The nondimensional generalized m a s s parameters M i j are l i s t e d i n equa- t i o n (B27). The required in t eg ra l s were evaluated by using t h e measured mass proper t ies of t h e model with the computed, uncoupled wing bending and tors ion mode shapes ( f i g . 9 ) . The propeller-engine mode shapes, Cpe and cp,,,, were taken as uni ty . The wing bending and tors ion frequencies employed were those calculated f o r t h e computed modes. The engine-propeller frequencies, % and 9, were uncoupled, measured frequencies. The propel le r yaw-to-pitch damping r a t i o , damping coef f ic ien t for t he f irst coupled mode w a s measured and found t o be 0.003. It w a s assumed t h a t t he damping coef f ic ien t f o r both uncoupled wing modes w a s a l so t h i s value.

g$/ge, w a s taken as uni ty i n a l l cases. The wing s t ruc tu ra l

A s indicated i n t a b l e 11, the re was a s m a l l va r i a t ion i n density among t h e da t a points . I n t h e analysis , a nominal value of po = 0.0022 slug/cu f t w a s used.

The propel le r nondimensional generalized aerodynamic forces a re l i s t e d i n For no aerodynamic interference between t h e wing and the pro- equation ( ~ 3 6 ) .

p e l l e r , t he following symmetry r e l a t ions between propeller s t a b i l i t y der iva t ives may be assumed:

A l l o ther propel le r der iva t ives were neglected. The values of t h e s t a t i c der iva t ives ( Cme Cz8, hq,, CZq) used were measured values reported i n

reference 8 f o r a windmilling propel ler . employed were those computed by the method of reference 16. These deriva-

t i v e s a re l i s t e d i n t ab le I11 f o r t h e f i v e propeller blade angle se t t ings used i n these tests. The values of windmilling-propeller advance r a t i o , J, and the distances from the gimbal axes t o the plane of propeller-blade quarter-chord points a t three-quarter radius, ee and e,,,, are a l so l i s t e d i n t ab le 111. I n a l l cases considered

A s i n this reference, the values of

c"q

ee = e+.

The wing nondimensional generalized aerodynamic forces are given i n equa- t i o n (B39) . The aerodynamic in t eg ra l s (eq. (B40)) were evaluated numerically by using t h e calculated bending and tors ion mode shapes. The c and a,

values f o r t he exposed panel were computed from three-dimensional l i f t i n g - surface theory (kernel.function method) f o r the planform of this model t rea ted as a r i g i d wing a t a steady-state angle of a t tack a t zero Mach number. These parameters are shown i n f igure 17. It should be noted t h a t these aerodynamic in t eg ra l s (eq. (B40)) remain constant throughout t he analysis.

2,

33

.. . . . .

APPENDIX c

EFFECT OF GYTIOSCOPIC COUPLING ON THE FREQUENCIES

OF VIBRATION MODES

Derivation of t h e Frequency Equation

The e f f e c t s of gyroscopic coupling on the v ibra t ion mode frequencies are examined a t zero airspeed by determining the coupled v ibra t ion frequencies over a wide range of propeller ro t a t iona l speed. The frequency equation fo r zero airspeed i s obtained from the f l u t t e r determinant (eq. (B43) ) by equating the generalized aerodynamic forces Qi j t o zero i n the determinant elements (eq. (B42)) and by expanding t h e r e su l t i ng determinant. The e f f ec t of damping on t h e v ibra t iona l frequencies i s neglected, such t h a t g i s zero i n a l l modes. I n addi t ion, t h e f a c t o r fl/krJ i n the off-diagonal mass terms, which i s based on the assumption of a windmilling propel ler , i s put i n the form

- A . The determinant f o r t he coupled frequencies then becomes: br'ue R1;2

1 - A2 MOh

= 0 (c1)

Mea -%a m A Mha Ma[ - h2 $) where i s defined by equation (B27). Values of A = (I) are desired f o r

values of R u+. Expansion of t h e determinant gives a polynomial i n A2 and I which can be wr i t ten as:

34

where

and

Discussion and Application

Some observations can be made about the behavior of t h e mechanical system from inspection of equation (C2). For any value of A, only one posi t ive value of mode cannot be the same as the frequency of any other mode a t any value of a / % .

varies , the two curves of the frequency p lo t s p l i t and do not cross. Such frequency s p l i t s are a l so encountered i n other v ibra t ing systems ( fo r example, see r e f . 1.7). The numerator of equation (C2) gives t h e values of t he coupled frequencies a t a/% of zero. (Note t h a t t h e yawing mode i s uncoupled f o r of zero.) The denominator of equation (C2) gives three asymptotes:

poles, and uniqueness of f o r a given value of A, the frequencies of t h e modes can be readi ly t raced f o r var ia t ions of

i s given by equation (C2); thus, t he frequency of any one vibrat ion

Hence, i f t he frequencies of two vibrat ion modes approach each other as

A2 = 0, and the two roots of BIA 4 + B2A2 + B3 = 0. Thus, with the zeros,

a/%.

35

Five cases a r e considered f o r i l l u s t r a t i o n : two cases f o r configuration A, and one each f o r configurations €3, C, and D. The r e s u l t s are presented i n f i g - u re 15 f o r four modes and f o r only two modes, 8 and $. These f i v e cases vary widely i n t h e loca t ion of the uncoupled engine frequencies r e l a t i v e t o the wing frequencies. Figure 13 shows t h a t t h e e f f e c t s of increasing a/% may r e s u l t i n some cases i n considerable d i f fe rences i n t h e values of t h e coupled frequencies from those a t a/% system. A similar ana lys i s showing the e f f e c t of gyroscopic coupling involving wing motion f o r one example i s given i n reference 18.

of zero, depending upon t h e parameters of the

I

1. Reed, Wilmer H., 111, and Bland, Samuel R. : An Analy-tical Treatment of Aircraf t Propeller Precession Ins tab i l i ty . NASA TN D-659, 1961.

2. Sewall, John L.: An Analytical Trend Study of Propeller W h i r l Ins tab i l i ty . NASA 'I" D-996, 1962.

3. Ravera, Robert 5.: Effects of Steady State Blade Angle of Attack on Pro- pe l le r W h i r l F lu t te r . Rep. No. ADR 06-01-63.1, Grumman Aircraf t Erg . COTE)., Ju7y 1963.

4. Reed, Wilmer H., 111, and Bennett, Robert M. : Propeller W h i r l F lu t te r Considerations for V/STOL Aircraft . Cal/Trecom Symposium Proceedings V o l I1 - Dynamic Load Problems Associated With Helicopters and V/STOL Aircraft , June 1963.

5. Houbolt, John C., and Reed, Wilmer H., 111: Propeller-Nacelle Whirl F lu t te r . Jour. Aerospace Sci., vol. 29, no. 3, Mar. 1962, pp. 333-346.

6. Head, A. L., Jr.: A Review of the Structural Dynamic Characteristics of the XC-142A Aircraft . Problems Associated With Helicopters and V/STOL Aircraft , June 1963.

Cal/Trecom Symposium Proceedings Vol I1 - Dynamic Load

7. Taylor, E. S., and Browne, K. A.: Vibration Isolat ion of Aircraf t Power Plants. Jour. Aero. Sci., vol. 6, no. 2, Dec. 1938, pp. 43-49.

8. Bland, Samuel R., and Bennett, Robert M.: Wind-Tunnel Measurement of Pro- pel le r Whirl-Flutter Speeds and Stat ic-Stabi l i ty Derivatives and Com- parison With Theory. NASA TN D-1807, 1963.

9. Abbott, Frank T. , Jr., Kelly, H. Neale, and Hampton, Kenneth D.: Investi- gation of Propeller-Power-Plant Autoprecession Boundaries for a Dynamic- Aeroelastic Model of a Four-Engine Turboprop Transport Airplane. NASA TN D-1806, 1963.

10. Houbolt, John C., and Anderson, Roger A.: Calculations of Uncoupled Modes and Frequencies i n Bending or Torsion of Nonuniform Beams. 1948.

NACA TN 1522,

11. Yates, E. Carson, Jr.: Calculation of F lu t te r Characteristics fo r Finite- Span Swept or Unswept Wings a t Subsonic and Supersonic Speeds by a Modified S t r i p Analysis. NACA RM L37L10, 1958.

12. Runyan, H a r r y L., and Watkins, Charles E. : Flu t t e r of a Uniform Wing With an Arbi t rar i ly Placed Mass According t o a Differential-Equation Analysis and a Comparison With Experiment. NACA Rep. 966, 1930. (Supersedes NACA TN 1848.)

37

Illllll

13. Woolston, Donald S., and Runyan, Harry L.: Appraisal of Method of Flutter Analysis Based on Chosen Modes by Comparison With Experiment for Cases of Large Mass Coupling. NACA TN 1902, 1949.

14. Yates, E. Carson, Jr.: Theoretical Investigation of the Subsonic and Supersonic Flutter Characteristics of a Swept Wing Employing a Tuned Sting-Mass Flutter Suppressor. NASA TN D-542, 1960.

13. Runyan, Harry L., and Sewal1,’John L.: Experimental Investigation of the Effects of Concentrated Weights on Flutter Characteristics of a Straight Cantilever Wing. NACA ‘I!N 1594, 1948.

16. Ribner, Herbert S.: Propellers in Yaw. NACA Rep. 820, 1945. (Supersedes NACA ARR 3LO9.)

17. Silveira, Milton A., and Brooks, George W.: Analytical and Experimental

NASA MEMO 11-5-38~, 1958. Determination of the Coupled Natural Frequencies and Mode Shapes of a Dynamic Model of a Single-Rotor Helicopter.

18. Scanlan, R. H., and Truman, John C.: The Gyroscopic Effect of a Rigid Rotating Propeller on Engine and Wing Vibration Modes. Jour. Aero. Sci., vol. 17, no. 10, Oct. 1950, pp. 633-659, 666.

s

0.00838 .00858 .00858 .00838

/

0.03801 .009128

- -02849 - .02849

Configuration

0.1814 .2224 .2823 .2823

A B C D

0.2237 * 2783 .3667 - 3674

TABLE I.- MASS PAFL4METERS

Propeller and power p lan t (pivoted)

MP slugs

0.2388 - 1.978 - 1379 - 1379

~

io o r i,,,,

0.0634

slug-f t 2

.04148

.01772

.01709

I I

(b) Nacelle and mounting beam (nonpivoted m a s s )

I n e r t i a , Configuration s lugs

I I I A B C D

71 S t a t i c unbalance,

I

-0.1496 - .1962 - .2689 - .2689

aValues given are r e l a t i v e t o the e l a s t i c axis of t h e wing.

39

TABLE 11.- MODEL TEST DATA

25 35 46 52 58 35 25 35 46 52 58 52 25 35 46 52 58 52 25 35 46 58 25 35 46 52 52 58 35 46 52 58

25 35 40 46 52

__

25 30 35 46 52

25 35 46 52 58 58

9.21

i I I

9.62 9.76

10.17 10.55

10.35 10.45

1 i

I

11.0

11.25 J,

11.6

14.4

I I 1

~~ ~

19.4

7-95

7.64

fv CPS

1' 9.67 9.60

I I

10.01 10.47

10.1 10.5

1 I 1

11.0

11.25 3.

11.7

14.6 I 20i0

23.5

1 23.3

2%

0.0072

1 .0074

.1 1

.oogo

.0089 I

.0091

.0149

1 i

I

.0084

. O l d

. 0290 .l,

0.0351 I I I

0.0289

0.0367

.OB2

P I slugs/cu ft I 2% I

Configuration A

0.0132 I *r 'P" .0107 O r

.0120 .I .0112

*:"' L .0324

Configuration B I

0.0299 I Configuration C

0.0266 I Configuration D

0.0310 I .Ox99

O.(

.oozy3

. 00230

.00228

\1 .0023l

.0023j

0.00226 I I 0.00233

0.00233

nl rps

61.5

29.0 25.0 21.5 47.0 67.5 48.0 33.5 28.0 23.5 28.5 67.5 50.5 34.5

26.0 65.5

22.8 68.0 45.0 35-0 31.0 30.5 26.0 63.5 46.5

36.0

41.0

28.0 24.0

42.0 31.0

40.0

100 68.0 60.5 53.0 49.0

90.5 72.0 62.0 52.0 50.0

66.0 48.5 37-0 32.5 29.0 27-5

Vf f t /sec

125 121 126 135 148 138 138 143 144 151 162 154 143 149 149 151 168 141 128 125 134 156 132 128 143 162 163 177 188 202 216 251

201 201 218 232 274

185 177 181 229 281

130 138 155 171 199 189

ff, CPS

5.70 6.58 7.24 7.40 7.67 6.71 5.81 6.66 7.40 7- 70 7.89 8.28 6.38 7.30 7-92 8.40 8.46 8.40 6.47 7.78 8.29 8.78 7.6 8.00 8.68 8.79

9.20 7.60

9.09

8.41 8.60 8.94

6.58 7.97 8.46 8.80 8.96

6.16 7.11

8.44 8.26

7.96

3.95 4.69 5-09 5.w 4.86 5.03

~ - . .. .. .... .

TABLE 111.- PROPEILER AERODYNAMIC QUANTITIES USED I N THE ANALYSIS

J

1.25 1.80 2.66 3-31 4.21

~

%€I

0.0447 .0382 .0296 * 0253 .0213

cZ9

-0.232 - .295 - .364 -.402 - -440

cmlf

0.0985

.0866

.0801 - 0734

.0$4

I

41

~61-291 Figure 1.- Photograph of model i n the tes t section of the Langley transonic dynamics tunnel.

Wing beam

(a) Typical section through wing.

(b) Plan view.

Figure 2.- Schematic of wing showing layout of spar and balsa pods. A l l . dimensions are given in inches.

. t i o n of lead weights

Shaker - simulated engine m a s s

Round spring Clip (pos i t ion adjustable)

_ ~ ~ _ . -

I =Position of lead weights

f o r configuration B I I L-section doubler1

(a) Configurations A and B.

Nonrotating shaft housing

%Wing beam (elastic axis)

Mounting beam

(b) Configuration C.

Pitch spring I ’

r Simulated engine mass 2

(c) Configuration D.

Figure 3.- Schematic representation of power-plant arrangement for each configuration. A l l dimensions in inches.

44

J

4 . 6

4.2

3.8

3.4

3.0

2.6

2.2

1 .8

1.4

1.0 20 24

0

28

0

r"

32 36 40

i

44

/

48

,'

52

,/

56 60

Figure 4.- Experimental variation of advance ratio with geometric blade angle at three-quarter blade radius for windmilling propeller (ref. 8).

45

7)

Figure 5.- Nondimensional position of elastic axis along the w i n g span.

.24 .- ?

.04

0

.20 l- i

r\ v W

r\ n v

I l----1 n

I I

La-,

r-U-1 I

-u---1

* La-J L*--a-d L + - 1 -

1 I I I I 1 I I I

mw' slugs/ft I

.16 !-

L - 8

0

4 1

5 Wing beam and wing pods

- - a-- Wing beam

(a) mss.

Figure 6.- Distribution of wing mass parameters along the wing span. Symbols indicate position along the span of the center of gravity of each strip. Parameters f o r nacelle and mounting beam are given in table I(b).

.28

.24

.20

.16

2 ra

.12

.08

.04

0

L-

U

r- -J

--L-

0

(b) Nondimensional i ne r t i a parameter.

Figure 6.- Continued.

a X

?

n n I ”

t: I I 01 L A 1

I 1 I I 0 .1 .2 . 3 . 4 . 5 . 6 .7 .a .9 1 .o

I I I I 1 I I

rl

(e) Nondimensional static unbalance parameter.

Figure 6.- Concluded.

deflection,

EI, lb-f t2

2.8

2 . 4

2.0

1.6

1.2

.8

0 .1 . 2

c

.3

0

V

. 4

0

T

a

8 . 9

0

-1

1.0

(a) S t i f f n e s s and slope of s t a t i c def lec t ion curve i n bending.

Figure 7.- Spanwise d i s t r i b u t i o n of s t i f f n e s s e s and s lopes of s t a t i c def lec t ion curves. Square symbols ind ica te values obtained by numerical d i f f e r e n t i a t i o n of t h e slopes between da ta points ; diamonds correspond t o values obtained by numerical d i f f e r e n t i a t i o n a t da ta poin ts .

-0- -__ 1 1.0

‘1

1.2

1 . 0

.8 Torsional

deflection radians/ft-ib

. 6

.4

. 2

0

2.4

2.0

0, lb-ft’

1.6

1.2

.8

. 4

0

8 - w -

7 I 7 I I

i .7 $ .5 ‘1

b

. 4 . 6 . 8 .9 1

Tr

T 4z .8

‘1

(b) S t i f f n e s s and slope of s t a t i c def lec t ion curve i n tors ion .

Figure 7.- Concluded.

.

c Tunnel w a l l

6.0- I f l = 10.4 cps

~ ~ - Measured node l i n e s

Predominant response: f l - f i r s t bending f 2 - f i r s t t o r s i o n f g - second bending

Figure 8.- Sketch of layuut of model i n tunnel showing measured node l i nes and frequencies of modes t h a t involved primarily wing motion f o r configuration A, inches unless otherwise specified.

f g = f q = 9.3 cps. All dimensions i n

52

I

1.0

.9

.a

.7

.6

'Ph Or 'Pa

.5

.4

. 3

.2

.1

0

I I I I + 4 I 1

.2

i I

~

I I I

dz /T I I I .4

rl

Figure 9.- Calculated uncoupled mode shapes of vibration for the model.

Kh

h

(a) Section a t propel ler s ta t ion .

Airstream I Midchord

Elastic axis

X

(b) Plan view.

Figure 10.- Sketch of model as t rea ted i n analysis.

54

. . . . . -. . ..

.04

.03

2z

.02

-

-

- I Ilns t ab1 e

.02

. 01

4 4.0 4.4

m e o r y

__ Four modes: e,$,h and a

Two modes: e and q - -~

Experiment

/

...

. 9 0

1.2

S t a b l e

1.6 2.0 2 . 4

V

RU

- - 2.8 .4

. 4

.4

.8

.8

.8

= 25O. '0.75R

f?

2.8 1 able

0

Un s t ab1 e IK 3.2 :

4 7

4 1 4.0 4.4

E 4.0 4.4

.04

.03

22

.02

. 0 1

0

f

0 1 'l

2.8

s t a b l e

2.0 I: u n s t ab1 e

I 3.2 3.6 1 . 2 1 . 6

(c)

Figure 11.- Comparison of experimental and theoretical whirl flutter speeds for configuration A.

55

.04

.03

2z

.02

0 01

0

.04

.03

.02

. 01

~- i Theory

Four modes: e , q , h , and a

Two modes: e and $I ---

0 .4 .8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0

V RU -

( e ) = 5 8 O .

Figure 11. - Concluded.

4.4

4.4

56

.03

2z

.02

, , , . I . . . I .

---Four modes: e , @ , h , and n

- ~ - Two modes: e and t

Stable 1 1

/ / ~

.8 1 .2

1

1.6 2.0 2 . 4 2.8 3.2

V R W - -

3.6

( a ) Configuration B.

I .8

0.75R

Stab1

I

1.6

,

2.0 2 . 4 / 2 . 8 3

V

R W

- -

(b) Configuration C.

tat

4

4.0 4.4 4 . 8

Unstable

i 1 -

b 4.0 4.4 4 . 8

( e ) Configuration D.

Figure l.2.- Comparison of theore t ica l w h i r l f l u t t e r speeds with and without wing f l e x i b i l i t y .

57

1.0

.9

.a

.7

.6

.5

0 Experiment

Theory

- Four modes: ---- Two modes:

e

4

*

S

I I

& l e 4a

/ i

I =I= t 1 I I

52

I I

i + 1 I I I I

I I I ,I/ 7

I I I

‘ I I 56

P0.75R’ deg

(a) Configuration A, 2E = 0.029.

Figure 13.- Comparison of experimental and theoretical whirl flutter speeds and frequencies.

I I I I I I I

I I I I I

I I 60

I

1 .o

.9

.a

Of - - .7 w

.6

.5

. 4

rd - n W

0

5= I .--

Experiment

(b) Configuration B, 2c = 0.032.

Figure 13. - Continued.

59

I

I l l / I IM 4x1 I

' H I I / I l l I I I I 4-4-11 I l l / I I I I I I I I

24 28 32 36 40 44 48 52 56 60

P0.75R2 deg

- ( c ) Configuration c, 25 = 0.026.

Figure 13.- Continued.

60

1.0

.9

. a

- wf .7 %

.6

.5

. 4

, , . . . . - Theory

FOUX modes: e ,$ ,h , and a -___-- TWO modes: 0 and $

0

I

I c =T

1 I

table

40

Experiment

-1 I I I x. -P

I

I f I I

I Stat I 44 48

I LY & t

52 !- I I I

- (a) Configuration D, 25 = 0.032.

Figure 13. - Concluded.

61

.10

.08

.06

z .04

.02

0

1.0

.8

.6

- "_f

.4

.2

0

s t a b l e

1 - 1.2 1.4 1.6

Uns t ab1 e

1.- . 2.0

. ~ I . . ~ . I I 2.2 2.4 2.6

V - RZ

I 2.8

1 I 3.0 3.2

Theory

F O ~ + modes with wing aerodynamics Two modes ( e and $) I n f i n i t e asymptotes Four modes without wing aerodynamics

- - - - - - - - - -__-

0 Experiment

. . . .

I I u . l--l-- -1. I L.- ~ - 1 - .- 1 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 1.2 1.4

V - RE

7 I 1 . I .~~ -.

5 6 8 3

= 2 5 O . '0.75R Figure 14.- Effect of wing aerodynamics on flutter characteristics for configuration C.

.10

.08

.06

2E

. 04

.02

0

1 .o

.8

.6

W f - - W

.4

.2

C

. . 1.2 1.4 1.6

stable

/ - /

I 1.8

Theory

Four modes with wing aerodynamics - - - - - - - - - - TWO modes ( e and 0) - _ _ ~ Infinite asymptotes -___.__ Mode 1 Four modes without wing _ _ _ _ .__-- Mode 2) aerodynamics

0 Experiment

_ I I I I . . . 1 I . ~ I I 1.4 1.6 1.8 2.0 2.2 2.4 2.6 1.2 V

R w - -

I .. I I I I I I 1.0 1.2 1.4 1.6 1.8 2.0 2.2

n - - W

= 46'. (b) 75R Figure 14.- Continued.

1 - 2.8 3.0 3.2

1 L 2.4 2.6

.10

.08

.06

2E

-04

.02

0

1.0

.a

.6

Of - - W

. 4

.2

0

, /, ,

stable

RZ Theory

Four modes w i t h wing aerodynamics Two modes ( e and 0) I n f i n i t e asymptotes Mode 1 Four modes without wing Mode 2) aerodynamics

- - - - - - - - - -~~

1.2 1.4 1.6 1.8 2.0 2.2 2.4 V - RZ

I I I I I 1 .o 1.2 1.4 1.6 1.8

n - - W

I 2.6

I 2.0

I I I

2.8 3 . 0 3 . 2

I I 2.2 2.4

Figure 14.- Concluded.

64

. .. I

2.0

.2

Four modes: @ , $ , h , and a Two modes: e and 0 - - __ - - -

--- Infinite asymptotes

--- .2

1.8- Uncoupled frequencies 0 e 0 1

0 I 2 3 4 5 6 7 0

i a /

0 a

2 3 4 5 6 7

1 /'

1.2

.6 .6

n - w 6

(a) Configuration A, fg = 11.65 cps. (b) Configuration A, fg = 9.27 cps.

Figure 15.- The e f fec t of propeller rotational speed on natural frequencies of vibration.

Four modes: B , $ , h , and a Two modes: 9 and ly - - - - - - - -

- - - Infinite asymptotes

1.8 Uncoupled frequencies

A

(c) ConfigurationQB, fg = 14.5 cps. (d) Configuration C , f g = 19.7 cps.

Figure 15.- Continued.

3.6

3.2

2.8

2 .4

2.0

0 - *e

1.6

1.2

.8

. 4

0 .1

Four modes: 8,S,h, and a - - - - - - - Two modes: 0 and . - __ - ~ n f i n i t e asymptotes

Uncoupled f r e q u e n c i e s

0 e * 0 h A a

(e) Configuration D, fg = 7.80 cps.

Figure 15. - Concluded.

.9 1.0

Airstream

Figure 16.- Sketch showing directions of aerodynamic forces and moments.

3

3: 5 Y

w 03 03 Ln

Figure 17.- Distribution of static aerodynamic parameters.

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