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A S A E C H N I C A L

T R A N S L A T I O N

ELICOPTERS - CALCULATION AND

olume 11. Vibrations and Dynamic Stability

y M . L. M Z , et dZ.

Mashinostroyeniye Press

oscow, 1967

. . -I

*T iF-494v.2C .I

N A S AT T F - 5 1 9

DESIGN

ATIONALA E R O N A U T I C SA N D S PA C EADMINISTRATION WASHINGTON:;a:

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TECH LIBRARY KAFB, NM

Illlllll11 lllllllll111IllIllI10068972

NASA TT F-519

HELICOPTERS - CALCULATION AND DESIGN

Volume II. Vibrations and Dynamic Stability

By M. L. Mil ' , et al.

Trans l a t ion of "Vertolety . Raschet i proyekt irovaniye.2 . Kolebaniya i dinamicheskaya prochnost ' .""Mashinostroyeniye" Press, Moscow, 1967.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION- ~ - ~ -~ _ _

F or s a l e b y t h e C l ea r i ngho use f o r Fede ra l Sc i en t i f i c and Techn i ca l I n fo rma t ionSpr in g f i e l d , Vi rg in i a 22151 - CFSTI p r i c e 3.00

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I I Il11111111111111111III

M O TATION

The work "H eli co pt ers , Ca lc ul at io n and Design11 i s p ub li sh ed i n t h r e evolumes :

Vol.1 - Aerodynamics;Vo L I I - Vibrat ions and Dynamic Strength;Vol . 111- Design.

The second volume g i v e s a n ac co u nt o f c e r t a i n pr ob le ms of t h e t h e o r y ofvi br a t io ns and methods of ca l cu la t i ng s t resses s e t up d u r in g s u ch v i b r a t i o n s i nh e li co p te rs i n f l i g h t , and, i n p a r t i c u la r , i n t h e r o t o r b la de .

Methods a re p r es en t ed f o r c a l c u l a t i n g t h e s e r v i c e l i f e of a s t r u c t u r e a n df o r c a l c u l a t i n g h e l i c o p t e r v i b r a t i o n s wh ich pe r m i t d e t er m in i ng t h e a m p li t ud e sof t he se v ib ra t i on s and comparing them wi th th e nom s of comfort . For t h ef i r s t t i m e i n S ov ie t l i t e r a t u r e , t h e p rob lem of coupled v i br a t io ns o f ro to r andf u s e l a g e i s examined.

The t h e o ry o f s e l f - e x c i te d o s c i l h t i o n s of a sp ec ia l type known as ' groundresonance" i s d i sc u ss e d i n d e t a i l . The c h a r a c t e r i s t i c s of the occur rence ofs uc h v i b r a t i o n s i n a h e l i c o p t e r o n t h e gr ou nd , d u r i n g t a k e o ff an d l a n d i n g r un ,a n d u n d e r f l i g h t c o n d i t i o n s are examined.

S p e c i a l c a s e s , l i t t l e e l u ci d a te d i n t h e g e n e ra l Litera ture , o f c a l c u l a t i n gb e a r i n g s t h a t o p e r a t e un de r s p e c i f i c c o n d i t i o n s of r o l l i n g a r e e xamined i n as e p a r a t e c h a p t e r. The same c h a p t e r g i v e s a n a c co u nt of t h e the ory and method ofc a l c u l a t i n g a new t y p e of thrust b e a r i n g of h i g h l o a d c a p a c i t y an d b e a r i n g sunder compound loads.

The book i s i n t en d e d f o r en g i n e er s of d e s i g n o f f i c e s , s c i e n t i f i c wo rk er s,g r a d u a t e s t u d e n t s , an d t e a c h e r s o f h i g h e r i n s t i t u t e s of l e a r n i n g . It might beu s e f u l t o e n g i ne e r s of h e l i c o p t e r m a n u fa c tu r er s and t o s t u d e n t s f o r f u r t h e r i n gt h e i r knowledge of t h e v i b r at i o ns and d y d c s t r e n g t h of h e l i c o p t e r s . C e r t a i ns e c t i o n s o f t h e book w i l l be u s ef u l a k o t o f l i g h t and t e c h n ic a l s t a f f s of he l ic o p t e r f l i g h t units.

There a re 35 t a b l e s , i l l u s t r a t i o n s , a n d 47 r e f e r e n c e s .

Cand. Tech. S c i . R.A.Mikheyev, R e v i e w e r

% Numbers i n t h e ma rg in i n d i c a t e p a g i n a t i o n i n t h e f o r e i g n t e x t .

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PREFACE L2The f i r s t volume of t h e work "Helicop ters , Ca lcu lat io n and Designl , pub

l i s he d i n 1966, w a s devoted t o aerodynamics: th eor y and methods of ca lc u l a t in gth e a er odyn am ic ch a r a c t e r i s t i c s o f r o t o r s and an aerodynamic c a l c u l a t i o n ofh e l i c o p t e r s o f v a r i o u s c o n f i g u r a t i o n s .

T h a t volume in clud ed a n account of t h e t h e o r y of r o t o r f l u t t e r which u su all y b el on gs t o t h e c a t eg o ry of a e r o e l a s t i c i t y - a n area between aerodynamics andmechanica l s t rength .

The present, second volume i s a l o g i c a l c o n t i n u a t i o n of t h e f i r s t and i s

dev ot ed t o v ib r a t i on s an d dynamic s t r e n g th s of he l i cop t e r s .The problems of t he s t a t i c s t r e n gt h of h el ic op te rs con-prise no fundamental

l y new a s pe c t s i n compar ison G t h wha t i s known i n a i r c r a f t c o n s t r u c t i o n . Withr e s p e c t t o v i b r a t i o n s and dynamic s t r e n g t h , h e l i c o p t e r s e x h i b i t a number of pec u l i a r i t i e s wh ich were r ec ogn i zed when t hey f i r s t appeared as a new type of flyin g machine. These pe c u l i a r i t i e s loomed l a rge du r ing - i f one may us e t h e exp r e s s i o n - t h e "s t ruggle for ex is tence11 of t h i s new t y p e of c r a f t i n t h e o v e r a l lsystem of a i r t r a n s p o r t m e a n s no t r e q ui r i n g a i r f i e l d s .

The r e c i t a l o f t h e p ro bl ems o f v ib r a t i on and dynaxdc s t r e ng t h of t he he l ic o p t e r b e g i n s w i t h a d e s c r i p t i o n of a method of c a l c u l a t i n g t h e e l a s t i c v i b r at i o n s o f i t s r o to r b l a de , wh ic h a r e similar i n fundamental eq uat i ons and methods

of s o l u t i o n t o t h o s e use d i n t h e t h e o r y of f l u t t e r bu t have a d i f f e r e n t t r e n ds i n c e u l t i m a t e l y t h e c a l c u l a t i o n r ed uc es m ainly t o a s o l u t i o n of t h e p u r e l y mecha n i c a l s t r e ng th p robl em, namely t o a de t e rmina t i on o f var i ab le s t r e s s e s a c t i n gi n t h e bl ad e, and th e n , w i t h t h e u s e of d a t a on t h e f a t i g u e l imits of a s p e c i f i cs t r u c t u r e , t o a d e t e r m i n a t i o n o f s e r v i c e l i f e , i .e . , b l ade l i f e .

Problems of vi br at io ns and dynamic s t r en g th a r e impor tan t no t o n ly f rom th ev ie wp oi nt of r e l i a b i l i t y of t h e c r a f t . Also t h e s e r v i c e l i f e of machines, andhence t h e i r economy, depends on t h e s ol ut io n of th es e problems.

I n p a r t i c u l a r , t h i s volume examines cur r en t methods of ca lc u l a t in g e l a s t i cv i b r a t i o n s o f a bla de, performed on high-speed el e ct ro n ic computers which permits de t e r min i n g t he var iab le s t r e s s e s s e t up i n t h e b l a d e .

In ve st ig at io ns of t h e I tground resonancel t mode of #br at io n, j u s t as a s t u d yof t h e v i b r a t i o n s o f a s t r u c t u r e , c o n s t i t u t e t h e p r i n c i p a l theme o f t h e t h e o r yo f h e l i c o p t e r v i b r a t i o n s .

Elim inat ion o f Ifground resonance" Vi br at i on s which, i f t h e y a r i s e and dev e l o p f u r t h e r , l e ad t o d e s t r u c t i o n of t h e c r a f t on t h e ground and, i n t h e c a seof m u l t i r o t o r c o nf i gu r at i on s , a l s o i n t h e air, has always been one of t h e mainp robl em s con f ron t i ng t h e d e s ign e r. The p,roblem of Vi b r a t i o n s of h e l i c o p t e rp a r t s , examined from t h e viewpoint of crew and pass en ge r comfort, i s a l s o qui tei n p o r t a n t . It i s n ot d i f f i c u l t t o estimate t h e a c u t en e s s of t h i s problem when

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t h i n k ing o f t h e power o f t h e cons t an t s o u rce o f such v ib r a t i on s - a huge ro to ro pe ra ti ng i n a h i g h l y variable v e l o c i t y f i e l d .

The las t chap t e r o f th is volume i s d ev ot ed t o a c a l c u l a t i o n of s p e c i a lb e a r i n g s , a n e c e ss i t y i n d es ig ni ng many of t h e he l i co pt er components and th usr e p r e s e n t i n g a t r a n s i t i o n a l c ha pt er t o t h e t h i r d volume on IIHelicopter Designrv.

The volume "Design11 w i l l g i v e a b r i e f s tud y o f t h e main p robl ems i n l ayou tof h e l i c o p t e r s , s e l e c t i o n of t h e b a s i c p a r a m e te r s of h e l i c o p t e r s i n c l u d i n gwinged types, and auxi l i a ry p rop u l s i on units such as t r a c t o r p r o p el l e rs o r supplementary j e t engines. Economic co nsi de rat ion s of av ia t i on engineering, of imp o r t a n c e i n d e si g ni n g, w i l l also be p re sen t ed .

T h i s volume a l s o p r e sen t s a discuss ion of p roblems of ba l anc ing , con t ro l l ab i l i t y , and s t a b i E t y f ro m t h e v ie wp oi nt o f s e l e c t i n g p a r a m e te r s f o r t h e c o n t r o l

system, as w e l l as problems of des ign i ng in d i v i du a l components of t he he l ic op ter .

The second volume W i b r a t i o n s and Dynamic Strength11 w a s w r i t t e n by:In t roduc t ion , M.L.Mil, ; Chapter I , A.V.Nekrasov, Chaipters I1 and 111, L.N.Grodko; Chapter IV, M.A.Leykand. S e c t i o n l l of Chapter I w a s w r i t t e n by A.V.Nekrasov i n c o ~ b o r a t i o nw i t h en gi ne er Z.Ye .Shnurov.

I n p r e p a r i n g t h e m a nu sc ri pt t h e a u t h o r s were a s s i s t e d by engineers F.L.Zarzhevskaya, V.M .Kostromin, and I .V.Kurov.

I n t h i s volume, we made use of t h e r e s u l t s of ca l cu l a t i on s pe rformed byen gi ne er s Yu .A .Myagkov, 0 . P .Bakhov, V.F. Khvostov, S. A. Gohb t sov, V.M .Pchelkin,S.Ye .Sno, V.G.Pashkin, N.F.Shevnyakova, N.M.Kiseleva, L.V.Artamonova, V.F.Semina,N.A.Matske#ch, V. I.Kiryushkina, and A.G .Orlova.

The reviewer, R.A.Mikheyev, of fe re d many va lu ab le co me nt s .Engineer L.G.Rudnitskiy w a s i n ch ar ge o f t h e f i n a l pr epa ra t ion of t he manu

s c r i p t f o r p u b li c at i on .

The a u t h o r s e x p r es s t h e i r s i n c e r e g r a t i t u d e t o t h e s e c o wo rk er s.

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TABLE OF CONTENTSPage

P r e f a c e .......................................................... iiiI n t r o d u c t i o n ..................................................... 1CHAPTER I ELASTIC VIBRATIONS AND BLADE STRBGTH ................. 11

S e c t i o n 1. Problems of Calculation, Basic Assumptions, andDer i va t i on of Di ff e r en t i a l Equations of BladeBending Deformations .............................. 121. U l t imat e Purpo se o f Ca l cu l a t i n g E l a s t i c

Blade Vibra t ions ............................... 122. Calc ula t ion of Blade S t re ngt h

..................13

3 . Fl i g h t Regimes De t r imen t a l t o t he Fa t i gueS t r e n g t h o f t h e S t r u c t u r e ...................... u,

4 . Assumption of a Uniform Induced Velocity Field .. 15 5. Assumptions i n Cal cul at ing Aerodynamic b a d s

on t he B la d e P r o f i l e ........................... 1 56. Rela t ion of Deformat ions due t o Bending i n

Two Mutual ly Perpendicu la r Di rec t ions andCorrespo nding A s s q t i o n s f o r C a l c u l a t i o n ...... 16

7. Conside ra t ion of Torsional Deformation of aB l a d e i n C a lc u la t io n s of F l e x u r a l Vi b r a t i o n s ... 17

8 . Two C a l c u l a t i o n Steps i n Blade Design:C a l cu l a t i o n o f Natura l Vi br a ti o n f i e quencyand Ca l cu l a t i on o f S t r e s se s

....................18

9. Idea l ize d Blade Models Used i n C alcu la t i on ..... 1810. D e r i v a t i o n o f t h e D i f f e r e n t i a l E q ua t io n o fBlade Bending i n a Cen t r i f u ga l Force F i e lda t Vi b r a t i o n s i n t h e F la pp in g P l a n e ............ x)

11. D i f f e r en t i a l Equ at ion of B lade Bending i n t h eRotor P lane of Rota t ion ........................ 22

S e c t i o n 2. Free Vibr a t io ns of th e Blade of a Nonrota t ingRotor ............................................. 221. Method of Ca l cu l a t i on f o r So lu t i on o f t he

In t e g r a l Equa t i on of B lade Vib ra t i ons .......... 222. C a l c u l a t i o n of t h e Natural Vi br at io n Modes and

Frequencies of a Blade Model wi th Dis cre te l yDi s t r i b u t ed Pa ra m e t e r s

......................... 243. Condi t ion of Or t hogonal i t y and Calc u la t ion of- Success ive Natural Vi br at io n Harmonics ......... 26 4.. C h a r a c t e r i s t i c s of C a l c u l a t i o n o f Natural

Vi b ra t i on F requen c i e s ami Modes of a E n g e dBlade .......................................... 27

5. C a l c u l a t i o n o f t h e Natural Vi br at io n Modes andFrequencies of a Blade as a Simply Supported Beam ........................................... 28

S e c t i o n 3 . Approximate Method of Determining t h e Natural Blade

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Vi b r at i o n F r eq ue nc ie s i n a Cen t r i f u ga l ForceField ............................................ 291. Use of B.G.GalerkinTs Method f o r Determ ining

the N a tu r a l B l ade Vi b ra t i on F requenc i e s .. ..... 292. Resonance Diagram of Blade Vibrations ... ... ... 313. Se l ec t i on of B la de P a rame te r s t o E l im ina t e

Resonance du r i ng Vib ra t i on i n t h e F l app ingP lane ......................................... 32

4. Se l e c t i o n o f B lade P arame ter s t o E l im ina t eRes onances i n t h e P l ane o f Ro t a t i on ........... 34

S e c t i o n 4. Ca lc u l a t i on o f Natural Blade Vibrat ionModesand Frequencies i n a Cen t r i f u ga l Fo rce Field ..... 361. Purpose and Problems of C al cu la ti on ........... 362. k n i t s of A p p l i c a b i l i t y of C a l c u l a t i o n

Methods Reducing t o a S o l u t i o n of t h eIn t e g r a l Equa t ion of B lade Vib ra t i ons ..... . .. 37

3. Possible Methods of C a l c u l a t i n g Free BladeVib ra t i on s i n a C en t r i f u ga l Force f i e l d ....... 39

4 . Three-Moment Method f o r C a l c u l a t i n g NaturalBlade Vibra t io n Modes and Frequencies i n aC e n t r i f u g a l F or ce f i e l d ....................... 39

5 . De te rm ina tio n of Bending Moments on t h eBasis of Known Forces ......................... k6

6 . De te rm in at io n of D ispla ceme nts fro m KnownBending Moments ............................... 48

7 . Case of a Blade Rigidly Attached a t the Root .. 508 . P o s s ib l e S i m p l i f i c a t i o n s i n C a l c u l a ti n gt h e C o e f f i c i e n t s .............................. 509 . Certa in R e s u lt s o f C a l c u l a t i n g t h e N a t u r a l

Blade Vi br at io n Modes and Frequ enc ies ......... 5 1S e c t i o n 5. To r s i o n a l Vi b ra t i o n s o f a Blade .................. 57

1. P rob lems S ol ved i n Ca l cu l a t i ng To r s iona lVi b r a t i o n s .................................... 57

2. Di ff e r en t i a l E qu at io n of To r s iona l B ladeVi b r a t i o n s .................................... 58

3. Determina t ion of t h e Natural To r s i o n a lBlade V ib ra ti on Modes and Fre que ncie s ......... 60

4 . Determina t ion of th e Natural Vi br at io n Modesand Frequencies of a Rotor as a Whole ......... 64

S e c t i o n 6. Combined Flexural and Torsional Blade Vibrat ions .. 651. Coupling of Flex u r a l and To r s iona l Vib ra t i ons .. 652. Method of C alc ula t in g Binary Vib rat ion s ..... .. . 663. E f f e c t of Coup%ng between Bending and

Tor s ion a t Natural Vibrat ion Frequency .. . . ... 71S e c t i o n 7 . Forced Blade Vibrat ions .......................... 77

1. Use of B.G.GalerkinTs Method f o r Ca lc ul at in gBlade Defomnations. Det erm inat ion of S t a t i cDeformations of a Blade ....................... 77

2. Determina t ion of Blade Deformations wi th Per io di cAp pli cat ion of a n ESrternal Load ............... 79

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3. Simp l i f i ed Approach t o Ca l cu l a t i on o fForced Blade 'Vibrat ions

.......................81

4. Amplitude Diagram of Blade Vibrations ......... 835 . C a l c u l a t i o n o f Vi b r a t i o n s a t Appl i ca t i onPhase of M e r n a l b a d Variable o v er t h eBlade Length .................................. 84

6. Aerodynamic Load on a Rigid Blade . , ... .... .... 857. Determina t ion of the Blade F lapping

C o e f f i c i e n t s ................................. 898. S i m p l i f i e d C a l c u l a t i o n o f E l a s t i c B la de

Vi b r a t i o n s ..................................... 93S e c t i o n 8. Ca lc ul at ion of Bending Stresses i n a .... ..........ladea t Low and Moderate Flying Speeds 97

1. C h a r a c t e r i s t i c s D i s t i ng u i s h in g F l y i ng

Regimes a t Low and Moderate Speeds .. ......... 972. Method of C al cu la t in g S t r e s s e s ............... 973. A s s q t i o n s i n De term inin g InducedVe l o c i t i e s ................................... 98

4. Mathematical Formulas f o r InducedVe l o c i t y F i e l d D e t er m t na t io n ................. 100

5 . Transformat ions of Mathematical Formulasi n P a r t i cu l a r Cases .......................... 102

6. Numer ica l Determina t ion of t h e I r r te gra l sJ(F,) and J(F,) .... .......................... 103

7. Assumptions Adopted i n AerodynamicForce Determinations ......................... 107

8. Mat hem atic al Formulas ............ ...... ...... 1079 . Convers ion t o an Equiva len t Rotor ............ 11110. Basic Assumptions Used i n Ca lc ula t i onof Bending Stresses ...........6.............. 111

EL. D i f f e r e r r t i a l Equat ion of Blade Vib ra t i onsand i t s S o l u t i o n ... ..... ......... ........ .... 112

12. D e t er m i na t io n of t h e C o e f f i c i e n t s o n t h eLeft-Hand Side of th e Equat ions i n Table 1.8 ..

13. D e t er m i na t io n o f t h e C o e f f i c i e n t s o n t h eRight-Hand Si de of t h e Equat ion ofTable 1.8 .................................... 116

14. S y s t e m of Equations a f t e r S u b s t i t u t i o n o fEqs,(8.34) and (8.38) ........................ 117

15. General Computational Scheme .....,. .. ........ 11816. Determina t ion of Deformat ion Coe ff ic i en ts .... 11817. Computational Program ......... .... ... .... .... 12018. Comparison of Ca lc ul at ion wi th Ekperiment

a t Low F l y i n g Speed .......................... 12219. Comparison of Ca lcu la t i on w it h Ekperiment

a t Moderate-Speed Mode .......................20. P o s s i b i l i t i e s o f F u r t h e r R efi ne men t

of Ca l cu l a t i o n R es u l t s ....................... 125S e c t i o n 9. Ca lcu l a t i on o f B l a d e B e n d i n g Stresses, w i t h

Cons ide r a t i o n o f t h e N on l inea r Dependence of

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I I I I1111111111 ll1IlllllIlllllI

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Aerodynamic Co ef f i c i en ts on P ro f i l e Angleof At tack an3 Mach Nmber ........................ 1271. Flight Regimes ................................ 1272. Determination of Aerodynamic Loads ............ 1273 . Method of Blade Calculat ion as a System

whose Motion i s Coupled by P r e s c r i b e dVibration Modes ............................... 129

4 . Mathematical Formulas f o r a Blade Modelw i t h D i s c r e t e P a r a m e te r s ...................... 133

5. Cons ide r a t i o n o f a Variable InducedVe l o c i t y f i e l d ................................ 134

6. C h a r a c t e r i s t i c s o f N um er ic al I n t e g r a t i o no f D i f f e r e n t i a l Eq ua ti on s of E l a s t i c

B lade Vib ra t i ons .............................. 13 57. Numerical In te gr a t io n Method Proposed byL.N.GrodJso and O.P.E akhov ..................... uc38 . Sequence of O p er a ti on s i n R e c a l c u l a t i on

and P r a c t i c a l E v a l u at i o n of DifferentI n t e g r a t i o n Steps .............................

9 . Comparison of Results by NumericalI n t e g r a t i o n M e th o d s w i t h C a l c u l a t i o nof Harmonics .................................. uc7

10. some CaIcukt ion ResuI t . s ...................... v18S e c t i o n 10. C a l c u l a t i o n of F l e x u r a l Vi b r a t i o n s w i t h D i r e c t

De t e rmina t io n o f t h e Pa th s o f Mo t ion of P o in t sof t h e Blade ...................................1. P r in c i p l e of t h e Method of Ca l cu l a t i on .......2. Dete rmina t ion o f E l a s t i c Fo rce s App li ed

t o a Po in t o f t h e Bl ade by AdjacentSegments ..................................... 153

3 . C h a r a c t e r i s t i c s o f N um er ic al I n t e g r a t i o no f Eqs . (m . l ) ................................ 156

4. Equat ions of Motion fo r a Mul t i h ingeAr t i cu la te d Blade Model ...................... 158

5. S equence o f Op e r a t i ons i n Ca l cu l a t i ngE l a s t i c Vi b r a t i o n s b y t h e N um er ic alI n t e g r a t i o n Met hod ........................... 161

6 . Method o f Ca l c u l a t i o n w i th I nve r s e Orderof Determining Variables i n N um er ic alI n t e g r a t i o n .................................. 1637. Comparative Eva lua t io n of Vari ous Methodso f C a l cu l a t i ng F l e x u r a l B lade Vib ra t i ons ..... 165

S e c t i o n 11. Fa t i g u e S t r eng th an d B lade Ufe ................. 16%1. Te s t i n g a S t r u c t u r e t o D ete rm in e i t s

S e r v i c e Life ................................. 16%2. D i s pe r si o n o f t h e C h a r a c t e r i s t i c s o f

M u r a n c e i n F a t i gu e Tests ................... 1693. B as ic C h a r a c t e r i s t i c s of t h e F a t ig u e

S t r e n g t h o f S t r u c t u r e ........................ 1704. Stresses S e t Up i n t h e Blade S t r u c tu r e

i n F l i g ht .................................... 173v i i i

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5. m o t h e s i s o f E n e a r Summation of DamageP o t e n t i a l a & Average Equivalent Amplitudeo f A l t e rna t i ng Stresses ...................... 1756. Disper s ion of th e Ampli tudes of A l te rna t in gStresses i n an Assigned Fl ig ht Regime ........ 179

7. Method of Cal cu la t ing Ser v ice E f e w i t h t h eUse o f R e l i a b i l i t y C o e f f ic i e n ts .............. 180

8 . Method of A.F.Selikhov f o r Ca lc ula t in g t h eRequired S a f e t y F a c t o r w i t h R es pe ct t o t h eNumber of Cycles llN .......................... u3

9. Determina t ion of S l og a t Given F iduc i a lP r o b a b i l i t y .................................. 187

10. D i sp e rs i on i n t h e S t r e s s L ev el s f o r Var io usSt ruc tura l Spec imens and R e l i a b i l i t y Margin

wi th Respec t t o th e Ampli tude of Al t e rna t ingStresses vo .................................. 189l l . M ethod o f D e te rmin ing t h e R e l i a b i l i t yMargin I,, roposed by A.F.Selikhov ........... 194

12. Example of C alc ul at i on of S erv ice Life ....... 19813. Po ss ib le Ways of Detemnining t h e M i n i m u m

Endurance L i m i t of a S t r u c t u r e ............... 202a. Advantages and Disadvantages of Various

Approaches i n Determining t h e NecessaryRe l i ab i l i t y M argins , a nd Es t ima t i on o fthe i r Accuracy .............................. 20 5

15. Blade S t ren gth Requi rements i n Des ignS e l e c t i o n .................................... 206

1 6 . S t r e n g t h o f a Blade wi th Tubular S tee lSpar ......................................... 20717. S t r e ng th of a Blade with Duralumin Spar ...... 21118. Effe c t o f Ser v ice Condi t ions on Fa t igu e

S t r e n g t h of Spars ............................ 212CHAPTER I1 HELICOPTEE VIBRATIONS ............................. 2vc

S e c t i o n 1. Forces Caus ing Hel icopter Vibra t ions ............. 2%1. Exc i t a t i on F requenc i e s ........................ 2vc2. Dependence of t h e Frequency Spectrum of

Ex cit in g Forces on th e Harmonic Contentof Blade Vi br at i on s ........................... 217

S e c t i o n 2. Fle xu ra l Vi b ra t i o n s o f t h e Fuse l age as anElastic Beam ..................................... 2271. Cal cul a t io n of Forced Vib ra t i ons of a nE l a s t i c Beam by t h e Method of m a n s i o n i nNatural Modes ................................. 228

2. Dynamic Rigidity of a Beam. Resonance andAntiresonance ................................. 234

3. App lic at io n of t h e Method of Dynamic Ri gi di tyt o t h e Vib ra t i on Analysis of Side-by-SideH e l i c o p t e r s ................................... 237

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4.. Method of A u x i l i a r y Mass ...................... &l5. U f e c t of Damping Forces. Vib ra t i ons a tResonance .....................................

Sec t ion 3. Vibr a t i on Ana lys is w i th Cons ide ra ti on o fFuse l age Cha rac t e r i s t i c s .........................1. F u s e l a g e c h a r a c t e r i s t i c s . L a t e r a l and

Ve r t i c a l Vi b ra t io n s ...........................2. C a l c u l a ti o n of F us el ag e Vi b r a t io n s i n t h e

Plane of Symmetry by t h e Method of Resid ues ..* 2513. Cons idera t ion of t h e EXfect of Shearing

Deformation ................................... 258Sec t ion 4. Combined Vi br at io ns of t h e System Fuselage-Rotor .. 259

1. Vibrat ions of the System Fuselage-Rotor ....... 2592 Calcu l a t i on o f t h e Natural Rotor Blade

Vi b r a t i o ns i n t h e P la n e of R o ta t io n , w i t hCons ide ra ti on of E l a s t i c i t y o f t h e RotorShaft and Attachment t o t h e Fuselage ............ 263

S e c t i o n 1. S t a b i l i t y of R o to r on an E l a s t i c Base ............ 27 51. Statement of Problem and Equations of

Motion .......................................... 27 52. S t a b i l i t y Analysi s and Bas ic Results .......... 2813. Phys i ca l P i c t u r e o f Rotor Behav5or i n t he

Presence of Ground Resonance ......... ..... .... 292 4. Rotor on an I s o t r o p i c Elas t ic Base ............ 298S e c t i o n 2. Lateral Vibra t ions of a Single-Rotor Helicopter .. 2991. Preliminary Comments .......................... 2992. Lateral and Angular St i ff n e s s of Landing Gear.

Flexura l Center ............................... 3003. Natural La t e ra l Vib ra t i ons of a Hel i cop t e r .... 304 4. Determination of Damping Coefficients ...... ... 3085. Combined Ac tio n of t h e System Shock

Strut-Pneumatic Tire ..........................6. Reduction of t h e Problem t o Cal cul a t i on of a

Rotor on an E la s t ic Base ...................... 31 27. Analys is of th e Resul t s of Ground Resonance

Ca lcu l a t i ons .................................... 3ucS e c t i o n 3. C h a r a c t e r i s t i c s of Damping of k d i n g Gear andBlade. In fl ue nc e on Ground Resonance ...... ..... 3151. Determination of the Damping Coefficient

of t h e Landing Gear Shock Absorber ............ 3152. E ff ec t of Locking of th e Shock Absorber as a

Consequence of Fr ic t i on a l Res is tance of t h eGland and Self-Excited Vi bra t io ns of th eHe l i cop t e r ....................................

3. Ch ar ac te ri s t ic s of Blade Dampers andt h e i r A n a l y s i s ...................................... 322

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4 . Ef fe ct of Flappin g Motion of Rotor onGround Resonance .............................. 325

S e c t i o n 4 . Ground Resonance of a Hel i cop t e r du r ingGround Run ....................~.................... 3301. Stiffness and Damping of a Wobbling T i r e ...... 3302. C a l c u l a t i o n of Ground Resonance and Results ... 3353. Ground Resonance on Breaking Contac t o f the

Tires with the Ground ......................... 337S e c t i o n 5. Ground Resonance of Helicopters of Other

Co n f igu ra t i ons ................................... 3391. General Comments .............................. 3392. C a l c u l a t i o n of L a t e r a l Natural Vi b r a t i o n s

wi th Cons ide r a t i o n of Three Degrees ofFreedom ....................................... 340

3. C a l cu l a t i o n o f Natural H e l i c o p t e r Vi b r a t i o n si n t h e P l a n e of Symmetry ( b n g i t u d i n a lVi b r a t i o n s ) ................................... 3474 . Reduct ion of t h e Problem t o Ca lcu l a t io n of aRotor on an El as t i c Base ...................... 352

5 . S e lf - Ex c it e d Vi b r a t i o n s i n F l i g h t o f aH e l i c o p t e r w i t h a n E l a s t i c F u se l ag e ........... 3 54

S e c t i o n 6 . S e l e c t i o n of Basic Parameters of Landing Gearand Blade Dampers . Design Recommendations ....... 3561. Se l e c t i on of B lade Damper C h a r a c t e r i s t i c s ..... 3572. Rotor w i t h I n te r bl a de a s t i c Elements

and Dampers ................................... 3613 . Sel ec t io n of S t i f fn es s and Damping

C h a r a c t e r i s t i c s f o r l a n d i n g Gears ............. 364.4 . C e rt ai n Recommendations f o r LandingGear Design ................................... 367

CHAPTER IV THEX)RETICALPRINCIPLES OF CAEULATING BEXRIJTGSOF MAIN KELICOPTER COMPONENTS . . . ... 369

S e c t i o n 1. Equat ions of S t a t i c Equi l ib r ium of Rad ia l andRadial-Thrust Bal l Bearings under Combined Load .. 370

S e c t i o n 2. Calc u la t ion of Rad ia l and Radial -Thrus t B a l lBearings under Combined Loads, .... .. .. ....... ........o r Absence ofMis a l i g me n t o f t h e Races 3771. P r e s s u r e on B a l l s ............................. 3772. Reduced Loads ................................. 3843 . S t a t i s t i c a l T h e o r y of Dynamic had-Carrying

Capac i ty ...................................... 3854. Effec t o f Axial b a d . on Bearing Performance ... 3935 . Approximate Solutions of Equations (2.1)

and (2.2) ..................................... 3976 . Relative Displacements of Races ... .. .......... 403

S e c t i o n 3. C e r t a i n Problems i n C a l cu l a t i ng Radi al -Th rus tB a l l Bear ings wi th Cons idera t ion of Misalignmentof t h e i r Races un de r h a d ........................ 405

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1. B as i c Re l a t i o n sh i p s ........................................... ........

4Q52. Case of Pure * Moment 4093. Simultaneous Action of Moment and Axial

Force ......................................... 4-134 . Limit Dependences on Small Loads .............. 4.55 . D i s t r i b u t i o n o f h a d b etw een Rows of E a&

of Double-Row Radial-Thrust Bal l Bearings ..... 4216 . Exanples of C alc u la t io n ....................... 424

S e c t i o n 4. Cal cul a t i on of Tapered Ro l l e r Bear ings underCombined Loads ................................... 4261. Ca lc ul at io n of Single-Row Tapered Ro ll er

Bearings ....................................... 4262. Remarks on Ca lc ul at io n of Bearing Assemblies

of Two Tapered Rol le r Bear ings ................ 431S e c t i o n 5 . Ca lc u l a t i o n of Vib ra t i ng Bea rings ... ............. 4331. Ch ar ac te r i s t ic s of t h e Mechanism of Wear ofA n t i f r i c t i o n B e ar in g s un de r Vi b r a t i o nCondit ions .................................... 434

2. Lubr ica t ion of Highly Loaded VibratingB ea ri ng s i n t h e P r e se n c e o f Small Vi b r a t i o nAmplitudes .................................... 435

3. C a l c u l a t i o n................... ..... . ........... ..f Hub Bea rings i n Main and T a i lRotors 4414. C a l c u l a t i o n of B e ar i ng s f o r t h e P i t c h C o n t r o l

and Co nt ro l Mechanisms ........................ 450S e c t i o n 6 . Theory and S e l e c t i o n of Basic Parameters of Thrus t

Bear ings wi th llSlewed l R ol le rs .. ..... ........ .... 4531. Determination of the Time T, .... ........ ...... 4542. Se le ct io n of Angles of Slope of Cage Sea ts .... 4593. F r i c t i o n L os se s ............................... 4634 . Add i t i ona l Con s i d e r a t i ons of Opt imal Thrus t

Bearing Design wi th lSlewedll Ro ll er s ..... .... 4655 . Example of Calculating a Thrust Bearing

with lSlewedl Rollers ... ........ .......... ....References .....................................................

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INTFDDUCTION Lz

A s soon as a n a i r c r a f t e n gi n e of s u f f i c i e n t power a nd l i g h t i n w ei gh t h adbeen c rea t ed and t he f i r s t he li co pt er took of f from th e ground, problems ofb al an ci ng , c o n t r o l l a b i l i t y, and s t a b i l i t y of t h i s c r a f t a rose . These were mainl y aerodynamic problems. If we c o n s i d e r t h e f i r s t f l i g h t o f d e l a Cierva,tsa u t o g i r o s i n 1925-1926 t o be t h e s t a r t of f l i g h t of r ot ar y- wi ng a i r c r a f t , t h e nw e c a n say t h a t t h e s t a t e d pro ble ms w e r e m a in ly s o lv e d i n t h e f i r s t decade(1926-1936) of th e i r development. The new ty pe of f l y i n g machine w a s thus curedof i t s 1lchildhood diseases11 .

However, as soon as t h e f i r s t series-produced machines appeared and theyw e r e p l ac e d i n s e r v ic e , more ser iou s de f i c i en c i es of he l ico pte rs became apparentsuch as, f o r example, f a t i g u e due t o i n s u f f i c i e n t dynamic s t r e n g t h of c e r t a i ns t r u c tu ra l members.

New dynamic problems aro se wi th th e wider p r ac t i ca l use of au tog i ros andesp ec i a l l y of h e l i co p t e r s , which en t e r ed t h e s cene a t the end of t h e T h i r t i e sand beginning of th e Fo rt ie s on a new improved technical basis . These probhmsp e r t a i ne d p r i m a r i l y t o o s c i l l a t i o n s a nd v i b r a t i o n s o f i n d i v i d u a l s t r u c t u r a l e l ements and of the h e l ic op te r as a whole, which a r e har m ul owing t o t h e s t r e s s e ss e t up i n t h i s case , o r a r e imperm issible from th e viewpoint of necessary crewand pas senge r comfor t, and a l s o i nc lude t h e p robl em o f s e r v i c e l i f e o f s t ruct u r a l e lements oper a t ing under h igh va r ia b l e s t re ss es . The l a t t e r problem,namely t h e i n c r e a s e i n s e r v i c e l i f e , i s c o n s t a n t l y ga in ing i n importances i n c e t h e a m o r ti z a ti o n and o v e r a l l l i f e of a hel ico pte r, de te rmined by th e l i f etimes of i t s components, has an e f fe c t on th e cos t -e ffec t i vene ss i n i t s use asa means of tr a n s p o r t a t i o n . The s e r v i c e l i f e , i n t u r n , i s determined mainly byt h e l e v e l of t h e v a r i a b l e s t r e s s e s s e t up i n t h e s t r u c t u r e ; t h e r e f or e , t h e a ccuracy w i th wh ich t he se are c a l c u l a t e d i s one of t h e ba si c problems of a dynamics t r e n g t h analysis of he l i cop t e r s .

A t r a c t o r p r o p e l l e r of a c on ve nt io na l a i r c r a f t o p er a te s p r a c t i c a l l y i n a na x i a l flow and, l i ke an eng ine , s e t s up no no t i ceab l e va r i ab l e s t r e s s e s i n t h es t r u c tu r a l members. Only tak eof f , l anding , and f l i g h t under condi t ions of atmos p h e r i c t u rb u l e n c e (a nd , o n m i l i t a r y a i r c r a f t , m aneuvers) c r e a t e a p p r e c i a b l edynamic loads on t h e a i r c r af t s t ru c t ur e , 'but a t r e l a t i v e l y f e w l oad cyc l e s (o fth e orde r of t en s and hundreds of thousands of cyc les ) dur ing th e l i f e t i m e of

t h e a i r c r a f t . I n t h i s ca se , one can speak about r epea t ed s t a t i c l oads .The l oads on t h e he l i co p t e r are @ t e d i f f e r e n t . I t s m a i n s t r u c t u r a l

members are loaded dynamically, t h e number of load ings o ft en exceeding t e n s ofmi l l i ons o f cyc l e s du r ing i t s l i f e t ime . T h i s i s due p r im ar i l y t o t h e a symmet ri cf l ow pa s t t h e ro to r , which r o t a t e s and s imu l t aneous ly advances. I n s o doing, /6t h e b l a d e i s s u bj ec t t o var iable aerodynamic loads as a consequence of the changei n r e la t iv e f l ow ve lo c i t y and ang l e s of a t t a ck of i t s s e c t i o n s . A l l forces andmoments a c t i n g on t h e b l ad e are t r ans mi t t ed t o t h e hub and ro to r con t ro l s . Theforces and moments a r r i v i n g fr om d i f f e r e n t b l a d e s a r e mutually compensated, witht h e e x c e p t io n of l o a ds a c t i n g w i t h f r e q u e n ci e s whose r a t i o t o t h e r o t o r rpm i s

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a mu l t ip l e of t h e b la de number. These loa ds are t r a n sm i t t e d t o t h e f u s e l a g eand t o t h e n o n r o ta t i n g p a r t of t h e r o t o r c o n t r o l s ys te m and t h e r e s e t up n o t i c e

able variable stressesThus, t h e pr ob lem o f v i b r a t i o n s and dynamic s t r en g t h s i n he l i c op t e r con

s t r u c t i o n i s n o t o n l y much br o ad e r t h a n i n a i r c r a f t c o n s t r u c t i o n b ut , i n manycases , has no d i r e c t a na lo gy i n t h e l a t t e r.

Recogni t ion of t h e impor tance of t h e problems of dynamic s t r en gt h w a s notimmediately obvious. Thus, even t h e cause s of t h e f i r s t a c c i d e n t s o f a u t o g i r o si n 1936-1937, d u ri n g w hic h t h e s e c r a f t o v er t ur n ed i n t h e a i r , were a t t r i b u t e df o r long t o i n s u f f i c i e n t dynamic s t a b i l i t y . In t k i s r e sp e ct , i n p a r t i c u l a r , inv es ti ga ti on s of t h e d y d c s of a r o t o r w i t h h in ge d b l a d e s a t c u r v i l i n e a r m ot io no f t h e c r a f t w e r e undertaken (s ee Sect .2, Chapt.11 of Vol.1). T h i s t h e o r y l a t e rf ou nd w ide a p p l i c a t i o n i n t h e e l ab o ra t i o n o f prob lems o f dynamic s t a b i l i t y andc o n t r o l l a b i l i t y of helicopters. However, i t never uncovered th e t r u e cause oft h e above-mentioned a cc id en ts . A s w a s s u b s e q ue n t l y r e a l i z e d , t h e c a us e w a s i ns u f f i c i e n t d ynamic s t r e n g t h of t h e r o t o r b l a d e s .

These problems w e r e r ec og ni ze d l i t e r a l l y by hit and miss. The f i r s t experim e n t a l a u t o g i r o s and h e l i c o p t e r s w e r e small and thus had a r a t h e r h igh s t r u c t u r a lr i g i d i t y . However, t h e f i r s t i n c r ea s e i n s i z e immed ia t ely encoun te r ed conside rable d i f f i c u l t i e s . F or i n s t a n c e , o n t h e A-4 autogiro, which had a d i ame te rsomewhat l a r ge r t ha n i t s p r e d e c e s s o r t h e 2EA a u to g ir o , s e r i o u s d i f f i c u l t i e s a r os eowing t o i n s u f f i c i e n t t o r s i o n a l r i g i d i t y of t h e bl ad e. The b la d e a ng le , i n t h ef i r s t f l i g h t , i n c r e a s e d so much due t o t o r s i o n a l d e f or m a ti o n t h a t a u t o r o t a t i o nwas i m p os s ib l e and t h e f l i g h t a h o s t ended i n cr ac ku p.

The i n v e s t i g a t i o n o f t h i s phenomenon was c om pl et ed w i t h t h e p u b l i c a t i o n o fa pap e r on t h e dynam ic tw i s t i ng o f a r o t o r b l ad e i n f l i g h t [ see (Ref .2) 1, i nwhich the f i r s t s u g g e s t i o n s were made as t o t h e n e c e s s i t y o f m atching t h e c e n t e ro f g r a v i t y and t h e c e n t e r o f p r e s s u r e , and i n w hi ch c o n s i d e r a t i o n s of t h e e f f e c tof b la de p r o f i l e o n s t a t i c s t a b i l i t y and c o n t r o l l a b i l i t y o f t h e c r a f t were examined. T h i s i n v e s t i g a t i o n r e s u l t e d i n a s y m e t r i c p r o f i l e s , e ns ur in g a l a r g er e s e rv e of a u t o r o t a t i o n , which were a do pt ed i n t h e e n g i n e e r i n g p r a c t i c e o f S o v ie th e l i c o p t e r c o n s t ru c t i on . A se t of d i f f e r e n t p r o f i l e s w a s u se d f o r t h e b l a d earrangement. The recommendations i n t h e ab ove p ap e r w er e s u f f i c i e n t t o p r e v e ntf l u t t e r i n t h e f i r s t Sovie t he l icopters which had a ro tor span of about I- +m.

The development of t h e Sovie t he l ic op ter i nd us t r y i s cha rac t e r i z ed byl a r g e r s t e p s t h a n tha t of t h e h e l i co p t e r in d u s t ry i n o t h e r c ou n tr i es ( t h i s a l s oenabled Sovie t des ign ers , who had s t a r t ed l a t e r t o bu i ld h e l i c op t e r s, t o c r e a t em achines v a s t l y s u p e r i o r t o modern f o r e i g n h e l i c o p t e r s w i t h r e s p e c t t o l i f tc a p a c i t y and s i z e ) . Whereas, a f t e r t h e f i r s t s u c c e s s f u l f l i g h t o f t h e S ik or sk yS - 5 l w i t h a r o t o r s p a n o f I- +m b u i l t i n 1947, th e Americans , i n 1950-1951, beganworking on a c r a f t w i t h a r o t o r o f 15.5 m d i ame te r ( S - 5 5 ) , t h e S o v i e t d e s i g n e r s ,a f t e r c r e a t i n g t h e M i - 1 h e l i c o p t e r w i t h a &m r o t o r , c o n s t r u c t e d as e a r l y as1952 t h e Mi-4 and Yak-& h e l i c o p t e r s w i t h 21-m r o t o r s . It i s n o t s u r p r i s i n gt h a t suc h a jump i n s i z e l e d t o a new pr ev io us ly unencountered phenomenon: I nb ot h c r a f t , t h e r o t o r began t o f l u t t e r d u ri ng t h e f i r s t t a k e o f f . We r e a d i l ycoped with t h i s p robl em, bu t p robl em s of t h e t he o ry o f f l u t t e r had t o w a i t along t i m e f o r s o l u t i on .

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We f i r s t encountered t h i s new phenomenon i n A p r i l 1952 w i th t h e Mi-4 he=-copter, when i t w a s r e a d y f o r i t s maiden take off . A t ons et of overspeeding,t h e b l a de s began t o f l a p i n a random manner, b en di ng t o a n e v e r i n c r e a s i n g ex

t e n t and t hr e at e ni n g t o s t r i k e t h e airframe. The t e s t crew r e a l i z e d t h a t t h i sw a s a new phenomenon never before encountered. T h i s c o n s t i t u t e d s o - c a l l e df l u t t e r of t h e r o t o r b l ad es . A t t h a t t h e , no o ne th o ug h t o f t h e f a c t t h a t t h i sw a s t h e very same t y p e o f f l u t t e r under s t u d y by many s c i e n t i s t s i n t h e USSR ando t h e r c o u n t r i e s . A cco rdin g t o t h e d a t a ava i l ab le a t t h a t time, f l u t t e r w a s note x pe c te d s i n c e i t w a s t ho ug ht t o a r i s e a t about 500 r p m r a t h e r t h a n a t a r o t o rr p m of 100-110, as ac tu a l l y happened i n t h e Mi-& h e l i c op t e r. The dec i s i vef a c t o r f o r t h e o ccur rence of f l u t t e r i n t h i s c a s e w a s t h e f a c t t h a t t h e l a r gef o r c e s g e n e ra t e d o n a r o t o r of s u c h a diameter produced appr ec ia b le deformat iono f t h e swash p la t e of t h e a u toma t i c p i t c h con tro1 ,wh ich i s e q u i v a l e n t t o a decrease i n t o r s io n a l r i g i d i t y of t h e b la de s, and also t h e f a c t t h a t a l a r g ev a l u e o f t h e c o e f f i c i e n t o f t h e f l a p p i n g c om pe ns at or ( c l o s e t o u n i t y ) had b ee ns e l e c t e d f o r t h e s e m ach ine s; t h i s p o i n t had b ee n di s re g ar d ed i n e a r l i e r i n v e s t i

g a t i on s of f l u t t e r . A s a r e s u l t t h e r e w a s no r e as o n t o t h i n k o f h e l i c o p t e rf l i gh t s , s in ce f l u t t e r s e t i n a p p re c i a bl e b ef o r e t h e o p e r at i n g r p m of t h e r o t o rw a s reached ,

It became c l e a r , i n s t u d y i n g t h e p a t t e r n o f f l u t t e r ( f l a p p i n g , b en di ng , andt w i s t i n g of t h e b la d es ) t h a t t h i s phenomenon could be prevented only by ut i l izi n g t h e t o r q u e s f ro m t h e i n e r t i a f o r c e s g e n e r at e d d ur i n g d is pl ac em en t of t h eb l a d e s e c t i o n s on f l a p pi n g . Without a s s o c i a t i n g r o t o r f l u t t e r w i t h wing f l u t t e rwhere - as known fo r long - t h e m u tu al p o s i t i o n o f c e n t e r of g r a v i t y, f l e x u r a laxis , an d c en t e r of p r e s s u re i s of prime importance, we s imply a t tached counterw e ig ht s t o s e v e r a l p o i n t s a l on g t h e b l a de l e n g t h t o c r e a t e moments o f i n e r t i aof o p p o s i t e s i g n d u r i n g v i b r a t i o n s a nd t h e n r e p e a t e d s t a r t - up of t h e r o t o r ; w eh e d i a t e l y understood t h a t we had i n hand a r e l i a b l e means o f s t opp ing f l u t t e r .

Thus, within a s h o r t t i m e th i s problem w a s p r a c t i c a l l y so lved , and by May1952 t h e f i r s t f l i g h t s w i t h t h e Mi-4 h e l i c o p t e r w e r e made.

A t t h e same t im e, f l u t t e r oc cu rr ed a l s o o n t h e Yak-& he l i co pt er which hadt h e same hub an d a u to m a ti c p i t c h c o n t r o l a s d i d t he Mi -4 h e l i c o p t e r b u t t h eb l a d e s w e r e of d i f f e r e n t d e s i g n ( wi t h l a r g e r f l e x u r a l and t o r s i o n a l r i g i d i t y ) .However, a s a consequence of t h e f a c t t h a t t h e r i g i d i t y of t h e a u to ma ti c p i t c hco n t r o l and t h e p a rame te r s o f t h e f l a p p i ng compensator were d ec i s i ve i n t h e occ u rr e nc e of f l u t t e r , w e a l s o e nc ou nt er ed f l u t t e r of t h e v e r y same form and a tt h e same r p m o n t h e b la d e s of t h e Yak-& h e l i c o p t e r as w e l l a s on t h e Mi-& h el icop t e r.

Thus, wi th in s ev er a l weeks a p r a c t i c a l s o l u t i o n w a s f ou nd f o r p r e v e n t i n gf l u t t e r , which i s used even now. The s c i e n t i f i c theory, however, t o de te rminew h e t h e r f l u t t e r w i l l o r Will not occur and - i f i t d i d OCCUT - a t what rpm andi n w h a t form, w a s developed by us du r ing t he subsequen t f ou r yea r s .

It should be s t a t e d tht, a f t e r comple t ion of s t ud ie s of f l u t t e r o n t h eground (by sh i f t ing t h e c e n t e r i n g o f t h e b l a d e fo rw ar d it w a s pos s ib l e t o t ld ri ve f1i t beyond the M t s of t h e work ing r p m and even beyond t h e maxi p e r m i s s i b l erpm of t he engine on t h e ground) , t he re s t i l l e x i s t e d t h e p o s s i b i l i t y o f i t so cc ur re nc e i n f l i g h t . T h i s l ed t o de t r im en t a l happenings. I n J a n u a ry 1953,

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II1 I I1I111111111IIll

c r a s h o f a M i - 4hel ico pte r took p la ce , whose causes were n o t s a t i s f a c t o r i l y /8d e f i n e d f o r a l m os t t h r e e y e a r s . I n s p e c t i o n r e v e a l e d t r a c e s of im pa ct of t h e

b l a d e s o n t h e c o c k p i t . T h i s had never been observed before. W e should no tet h a t , dur ing normal f la pp ing mot ion, th e b lade does no t come i n t o conta c t wi tht h e c a b i n u n l e s s t h e l ow er r e s t r i c t o r s o f b l a d e ov er ha ng o r coning s tops a r er up tu re d i n t h e air.

It i s obv ious t h a t ou r s ea r c h f o r t h e cause of t h i s acc iden t w a s d i l i g e n t ,when r e a l i z i n g t h a t t h e c r a s h d i d n ot s t op e i t h e r a c t u a l f l i g h t s o r s e r i e s product ion of t h i s p r o t o t y p e .

During 1954, a number of p i l o t s observed an unusu a l phenomenon i n f l i g h t ,which came t o be known as I X al i be r ny y e f f e ct ( a f t e r t h e p i l o t who w a s t h e f i r s tt o n ot ic e i t ) . Kaliberqyy e s t a b l i s h e d t h a t i n a power descent a t a b l ade sett i n g a n g l e o f a b o ut 6 - yo, t h e b l ad e s began t o f l a p o u t of t h e i r c o n i n g a n g l e .T h i s s topped a f t e r r e s e t t i n g t h e b l ad e s t h a t had a somewha t d i f f e r en t t r an sve r sec e n t e r i n g . However, two y e a r s l a t e r , w h e n f l i g h t - t e s t i n g a s e t o f b l a d e s f o rabsence of t h e I lKal ibernyy e f fe c t " , i . e . , dur ing a power d escent wi th an ang leof p i t c h o f 6 - 703', t h i s phenomenon became s o predominant, a t s u ch s t r o n g f l a pp i n g of t h e b l a d es , t h a t i t w a s d i f f i c u l t t o make a f o r c e d l a n d i n g w i t h t h ec r a f t . It should be mentioned h ere th a t c lo se t o t h e ground, upon changing t oanothe r reg ime, t h e b lade f lap pi ng s topped and th e c r a f t behaved normal ly. Av i s u a l i n s p e c t i o n o f t h e h e l i c o p t e r a f t e r t h e f l i g h t r e ve a le d r u pt u re d b l a def o o t i n g s ( s o - c a l l ed movable s l o t t e d t r a i l i n g e dge of t h e b l a d e ) , w hic h i n d i c a t e db en di ng of t h e b l a d e i n t h e p l a n e of r o t a t i o n . E v er y th i ng e l s e w a s i n goodworking order. It w a s dec ided t o make a de ta i l ed checkout of t h i s h e l i c o p t e rw i t h t h e same se t o f b l ade s . F l i g h t t e s t s were c a r r i e d o u t t o c heck a nd s t u d y

t h i s phenomenon.Measurements of t h e b l a des showed t ha t t h e c en t e r i ng had s h i f t ed by abou t

l$ of t h e chord more rearward tha n i t s p o s i t i o n a t t h e t h e o f t h e b la de s leavi n g t h e m a n uf a ct u re r. T h i s can be expla ined as f o l l ows : The blades w e r esheath ed wi th plywood. The ce nt er of g ra vi ty of plywood i s about a t 50% of t h echord. Therefore, as soon as t h e wood s w e l l s and i n c r e a s e s i n w ei gh t d ue t o t h ea bs or be d m o i st u re , t h e c e n t e r o f g r a v i t y o f t h e e n t i r e b l a d e Will s h i f t t o w a r dth e t r a i l i n g edge . The above happen ing w i t h t h e he l i cop t e r occu rr ed i n autumnwhen the humidity w a s high.

Dur ing these t e s t s i t w a s a l s o c o nc l u s iv e l y e s t a b l i s h e d t h a t t h e c h a r a c t e ro f b l a d e f l a p p i n g and t h e m ot io ns o f t h e c o n t r o l s t i c k d u r i n g f l i g h t i n al%l ibernyy e ffec t11 reg ime a r e complete ly ana logous t o th e f lap pin g and mot iono f t h e s t i c k r e c or de d i n ground t e s t s where blades a r e caused t o f l u t t e r by a na r t i f i c i a l l y c r e a t e d t a i l - h e a v i n e s s . T h i s complex pro ce du re made i t p o s s i b l e t oe s t a b l i s h t h a t t h e phenomenon o c c u rr i ng i n f l i g h t w a s i d e n t i c a l w i th t h a t no tedon the ground. Thus i t w a s e s t a b l i s h e d t h a t t h e IIK alib ern yy e f f e c t " i s noneo t he r t h a n t h e o ns et of f l u t t e r i n f l i g h t . On t h e basis of t h i s conc lu s ion i tw a s c o n j e c t u r e d t h a t t h e e a r l i e r u ne xp la in ed f l i g h t a c c i d e n t i n wh ich t h e b l a d ess t r u c k t h e c o c k p i t w a s a l s o n o t h i n g e l s e t h an f l u t t e r of t h e b la des i n f l i g h ta r i s i n g a t a r o to r r pm a t which i t did no t appear when ope ra t ing on th e ground.

4; On t h e Mi-& h e l i c o p t e r , f l u t t e r s e t s i n p r i m ar i l y i n t h i s regime.

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The v ibra t ions of a hinge d b l ad e i n f l u t t e r , u n l i ke t h e v i b r a t i o n s o f aconve n t i o n a l a i r c r a f t Wing, are capable of a f l ap pi ng motion whose ampli tudebu i lds up u n t i l t h e b l a d e i m p a ct s o n t h e c on in g s t o p s a nd, a f t e r b reak ing t he se ,s t r ikes t h e c a b i n .

That t h i s phenomenon w a s not uncovered for a l ong t ime can be a t t r i b u t e d Lat o t h e e r roneous a ssump tion based o nm o d e l t e s t s t h a t , if f l u t t e r o n t h e groundi s e l i m i n a t e d , i t c an no t o c cu r i n air dur ing forwar d motion. However, p r a c t i c a lexperience and, l a t e r , more r i go rou s e xpe r imen t s w i th he l i cop t e r s and , f i na l l y,co r r espo n d in g t h e o r i e s have shown t h a t t h e r e a r e f l i g h t regim es i n which f l u t t e ra t t h e o p e r a t i n g rpm of t h e r o t o r Will not occur on t h e ground but may occur i nf l i g h t .

It should be s t a t e d t h a t , as e s t a b l i s hed i n i nve s t i g a t i o ns , t h e phenomenono f f l u t t e r a l s o had been ob se rved e a r l i e r o n h e l i c o p t e r s . A lr ea dy i n 1949, t h eMi-1 h e l i c o p t e r w a s equipped with a r o t o r w i t h w id er b l ad es t o i n c re a s e t h es a f e t y f a c t o r r e l a t i v e t o f lo w s ep ar at io n. I n f l i g h t , t h i s ro tor p roduced buffe t in g which could no t be e l i m i na t e d . A f t e r t h e t h e o r y of f l u t t e r had b ee nworked out and a l l a spe c t s o f t h i s phenomenon had been clar if ied, i t became poss i b l e n ot o nl y t o a t t r i b u t e t h e j o l t i n g of t h e M i - 1 he l i cop t e r w i th wide b l adest o a n a pp ro ac h of t h e re gim e t o f l u t t e r bu t a l s o , a n d wi th ou t f u r t h e r d i f f i c u l t y,t o d e si g n and c o n st r u c t ( i n 1956) a 35-m eter r o t o r f o r t h e Mi-6 and Mi-10 h e l ic o p t e r s . P e r f e c t i o n o f t h i s r o t o r w a s con fi rmed by t h e f a c t t h a t a week a f t e rt h e i n i t i a l takeoff the new heavy Mi-6 h e l i c o p t e r w a s ab l e t o c om plete t h e t r a i ni ng f l i g h t f o r p a r t i c ip a t io n i n t he Ai r Parade on Aviation Day a t Tushino.N e it h er t h e n no r l a t e r d i d a ny th in g d e t ri m e n t al , a s s o c i a t e d w i t h f l u t t e r , o cc urw i th t h e s e c r a f t . T h i s c o n s t i t u t e s t h e h i s t o r i c a l a s p e ct o f t h e f l u t t e r p roblem .

Of no l e s s importance i s t h e p ro bl em of d e te r m in i ng v a r i a b l e s t r e s s e s i nblades, which i s s o l v e d by s t u d y in g t h e i r f o r c e d v i b r a t i o n s .

During the f i r s t decade of t h e i r development , h e l i c o p t e r rotors were ac tua ll y d e si gn ed w i th o u t p r e c a l c u l a t i o n o f var i ab le s t r e s s e s a r i s i n g i n f l i g h t . A tt h a t t i m e , c a l c u l a t i o n w a s cumbersome and ina cc ur at e and o f t e n completed onl ya f t e r t h e c r a f t w a s a t t h e a i r f i e l d . It w a s on ly th e development of computat i o n a l m ethods f o r v a r i a b l e s t resses , a l l o wi n g t h e u se of hi gh-s pee d d i g i t a lco mpute rs, t h a t p e r m i t t e d d e s i gn i n g b l a d es w i t h d e l i b e r a t e s e l e c t i o n of r i g i d i t yand mass d i s t r i b u t i o n s o as t o avoid harmful resonance , r e d u c e t h e s t r e s s l e v e l ,and t hus ensu re l ong s e rv i ce l i f e a n d b l a d e r e l i a b i l i t y .

It should be no t ed t ha t r e f i ne me n t of s t r e s s a n a ly s i s f o r b la de s l e d t ofu r t h e r devel opment i n dep th an d e l a bo ra t i on o f t h e ae rodynamic t heo ry. Ass h m n t h e f i r s t volume, r e fi n em e n t o f t h e c a l c u l a t i o n o f f l i g h t d a t a d i d n otmake i t ne ce s sa ry t o de v e l o p t h e c o q l e x and cumbersome vo r t e x t heo ry o f a r o t o r .Never the less , i t i s o n l y t h e v o r t e x t h e o r y t h a t p e r m i t s d e te r m in i ng t h e nonunif o r m i t y o f t h e i nd uc ed v e l o c i t y f i e l d , caus ing var iable b l ade l oad ing a t f r equencies t h a t e x c i t e f l e x u r a l v i b r a t i o n s of t h e b l ad e s of second, t h i r d , a n dhigher harmonics . Therefore , i n s t ress analysis, o n l y t h e v o r t e x th e o r y c a ng i ve r e s u l t s c l o se t o t h o se o bserved i n r e a l i t y.

Vi b r a t i o n s c o n s t i t u t e d a n o t h e r no l e s s imp orta nt problem. This problem hasalways been one of t h e most d i f f i c u l t i n t h e deve lopmen t o f r o t a r y wing a i r c r a f t .

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Dozens of Sovie t and fo re ig n des igns , in t e re s t in g f rom t h e Viewpoint o f concept i o n and f l i g h t d a t a , n ev er came t o c o m p le ti o n owing t o t h e h ig h l ev e l of vib r a t i o n .

I n c on v en t io n al a i r c r a f t , t h e s o u r c es of v i b r a t i o n a r e not as powerfu l asi n h e l i c op t er s . Furthermore , bo t h engines and pr op e l le rs which are t h e ma inv i b r a t i o n e x c i t e r s i n c o nv en ti on al a i r c r a f t ca n be a d e q u a te l y i s o l a t e d fro m t h es t r u c t u r e by means of sp e c ia l shock abs orb ers . High- frequency resonancep ro du ce d by suc h ex c i t e r s c a n be e l i m i na t ed quite e a s i l y by comparat ively minor

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m o d i f i c a ti o n s of t h e s t r u c t u r e . I n a h el ic op te r , i n a d d i t io n t o t h e f a c t t h a tt h e p e r t u r b i n g f o r c e s p ro du ced by t h e r o t o r s are a p p re c ia b ly g r e a t e r t h a n i n ac o nv e nt io n al a i r c r a f t , t h e f r e q u e n c ie s fr om t h e s l o w l y r o t a t i n g r o t o r are r a t h e rl o w and, i n co mb in at io n w i t h t h e n a t u r a l o s c i l l a t i o n f r eq u e n c i e s of t h e f u s e l a ge ,engine, wing, or t a i l u n i t , g i v e r i s e t o r es on an ce l e a d i n g t o a p p r e c i a b l e vibrat i o ns w i th a n amp l it ude o f d i sp lacem en t which , i n s t e ad y f l i g h t regimes, r e achesmagni tudes of t h e ord er of 0.3 - 0.4 r mn and i n s h o rt - ti m e r eg im es , p r i o r t ol a n d i n g of t h e h e l i c o p t e r , even 1 - 2 mm i n t h e crew c ab in .

Resonance wi t h fundamenta l tone s of t h e natural f u s e l a g e v i b r a ti o n s o f t e nare p r a c t i c a l l y i m p o s s i b l e t o damp o u t by c ha ng in g t h e r i g i d i t y of t h e s t r u c t u r ei n a n a l re a dy b u i l t machine. Therefore, i t i s w o r t a n t t o make a c o r r e c t e s t imate o f t h e natural v i b r a t i o n fr eq ue nc y o f t h e f u s e l a g e and t o c a l c u l a t e t h ev i b r a t i o n a mp li tu de i n d e si gn in g t h e c r a f t .

I n overcoming v i br a t io ns , main emphasis m u s t be on reducing the magnitudesof var iable forces produced by t h e r o t o r and a c t i n g o n t h e f u s e l a ge . Thesef o r c e s are caused by b lade v ibra t ion . 31 t u r n , such b lade v i br a t io ns may bel a r g e r or smaller d ep en di ng o n t h e c l o s e n e s s of t h e i r natural f re qu e nc ie s t o t h e

f r e q u e n c i e s o f t h e external e x c i t a t i o n s o ur c es .

I n a l l cases , c l o s e n e s s t o r e s on a nc e w i l l l e ad t o a n i n cr e as e i n b la destresses. However, i f t h e v ib r a t i o n s occu r w i t h t h e ha rmon ic f r equency z b + 1( o r z b - 1 for v i b r a t i o n s i n t h e p la n e of r o t a t i o n of t h e r o t o r ) o r w i t h t h eharmonic frequency zb f o r v i b r a t i o n s i n t h e f l a p p in g p la n e (where zb i s t h enumber o f b l a de s ) , t h en t h e fo r ce s a r e s m e d a nd t r a n s m i t t ed o ve r t h e h in ge st o t h e h ub and t h r o u g h ' i t t o t h e f u s e l a ge , c au si ng v i b r a t i o n .

Ve r t i c a l v i b r a t i o n s , w hi ch a r e t h e t yp e mos t d i sa gr ee ab ly perce ived by man,are l a r g e l y caused by f o r c e s a c t i n g i n t h e p l a n e of r o t a t i o n of t h e r o t or , s i n c et h e s e f o r c e s , a p p l i e d h i g h ab ov e t h e c e n t e r o f g r a v i t y o f t h e h e l i c o p t e r , c r e a t eappreciable moments t h a t e x c i t e f l e x u r a l v i b r a t i o n s of t h e f u s e l a g e . I n t h i scase, i t i s na tura l t h a t t h e g r e a t e s t Vi b r a t io n a m pl it ud es ( a n t i n od e s ) arereached a t t h e e nd s of t h e f u s e l a g e and h ence i n t h e c o c k p i t .

It w a s found t h a t , i n d et er m in in g t h e na tura l v i b r a t i o n f r e q u e n c i e s o fh e l i c o p t e r b l a d e s , i t must be c o n s ide r ed t h a t t h e ro to r hub does not r ema inf i x e d d u ri ng t h e v i b r a t i o n s i n c e i t i s a t ta c he d t o a n e l a s t i c f u se la g e. Thus,i n a n a n a l y s i s of v i b r a ti o n s , t h e c r a f t sh ould be t r e a t e d as a s i n g l e dynamics y s t e m w i t h e l a s t i c b l ad e s hinged t o a hub a t t a c he d t o a n e l a s t i c f u se l ag e .

It i s obvious tha t i t i s o n l y l a t e l y tha t such a ca l c u l a t i o n scheme cou ldbe developed and made avai lable f o r s tu d y. A s f a r as we know, we are t h e f i r s t ,

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i n th i s book, t o p r e s e n t a method of c a l c u l a t i n g h e l i c o p t e r v i b r a t i o n s i n t h ed e s i g n s t a g e ,

L a t e r i n t h i s volume, we w i l l d i s c u s s s e l f - e x c i t e d o s c i l l a t i o n s o f a h e l ic o p t e r, general ly known as Itground resonance"

Des igners f i r s t encountered the phenomenon of ground resonance more than30 years ago when one of t h e f i r s t S o v i e t a u t o g i r o s , t h e A-6 (des igned by V.A.Kuznetsov), w a s equipped w it h low-pressure t i r e s which w e r e new a t t h a t t i m e .The o l e o s t r u t s were removed from th is he l ic op te r . An unexpec ted v i br a t io n occu rre d i n t h e f i r s t t a k e o f f attempt. The h e l ic op te r rocked f rom wheel t o wheela t c o n s t a n t l y i n c r e a s i n g a m p l i t u d e , f i n a l l y jumping upward so t h a t t h e w he elsb r o k e c o n t a c t w i t h t h e g r o u n d . The t a k e o f f en de d i n f a i l u r e .

S in c e t h e t e s t s w e r e recorded by a motion-picture camera, i t w a s p o s s i b l e /11t o e s t a b l i s h t h a t t h e blades h ad e x ec u te d i n c r e a s i n g l y s t r o n g e r v i b r a t i o n s a bo utt h e d r a g hinge. T hese v i b r a t i o n s , w hi ch oc c u r r ed i n a c e n t r i f u g a l f o r c e f i e l d ,produced a p e r i o d i c d i sp l ac em e nt of t h e c e n t e r o f g r a v i t y of t h e e n t i r e l i f t i n gs ys te m r e l a t i v e t o t h e c e n t e r o f t h e hub and t h u s e x c i t e d v i b r a t i o n s o f t h e h e l ic o p t e r s t a n d i n g o n t h e g r o u n d . It i s obvious t h a t , if t h e f r e q u e nc y o f d i sp la ce me nt s o f t h e r o t o r c e n t e r o f g r a v i t y c o i n c i d e s w i t h t h e f re qu en cy of m t ur a l v i b r a t i o n s o f t h e h e l i c o p t e r o n p ne um at ic t i r e s , s uc h v i b r a t i o n s a r e ab let o i n c r e a s e. It would seem t h a t t h e p h y s i c a l a s p e c t of t h e phenomenon i s c l e a r .The energy t h a t f e d t h e s e i n c r e as i n g v i b r a t i o r s w a s e i t h e r t h e e nerg y of t h ee ng in e t u r n i n g t h e r o t o r o r, w i t h t h e e ng i ne c ut o u t , t h e k i n e t i c e ne rg y of t h er o t a t i n g r o t o r .

However, numerous inves t iga t ions , which a re s t i l l i n pro gres s , were neededt o d e ve lo p t h e t h e o r y o f g ro un d re so n an c e a nd t o s t u d y i t s new manifestat ions,p o s s i b l y i n new b a s i c a l l y d i f f e r i n g c o n f i g u r a ti o n s and s t r u c t u r e s .

The f i r s t t h e o r e t i c a l work e x p l a i n in g t h e nature o f s e l f - o s c i l l a t i o n s o fthe "ground resonancel l type w a s done as e a r l y as 1936 by 1.P.Bratukhin and B.Ya.Z he re bt so v. I n p a r t i c u l a r , t h e r e s u l t s of t h e i r i n v e s t i g a t i o n s made i t p o s s i b l et o eliminate g ro un d r e so n a nc e i n t h e w or ld 's l a r g e s t a u t o g i r o , t h e A-15 w i t h ar o t o r s p a n o f 18 m which w a s c o ns t ru c te d i n 1936 from the design by V.A.Kuznetsovand M.L.Milf . In t h e d e s i g n o f t h e h ub of t h i s a u t o g i r o , s p r i n g s m ounted t o t h eb h d e - v i b r a t i o n r e s t r i c t o r a ro un d t h e d ra g hin ge were used. The spr ings wereg i v en t h e n a t u r a l v i b r a t i o n f r eq u e nc y o f t h e b l a d es i n t h e p l an e of r o t a t i o n ,which el iminated I tground resonancel t .

There i s no doubt tha t , a t t h e t i m e , t h e phenomenon of ground resona nce w a sa l s o known i n th e Western Cou nt r ies and had undergone some s tu dy the re , s i nc ee v e n t h e f i r s t s u c c e s s f u l d e l a Cierva a u t o g i r o s , f o r example t h e C-19, hade l a s t i c c o u p l in g s ( sh oc k a b s o r b e rs ) c on ne cte d t o t h e b l a d e s o v e r f r i c t i o ndampers.

However, many des ig ner s con t inued f o r some t ime t o p roduce au tog i ros wi tho u t d amp ers i n t h e d r ag hinges. A model of such a machine w a s t h e A-7 a u t o g i r od ev el op ed i n 1937 by N.I.Kamov. It made suc ces s fu l f l i g h t s wi thout dampers ont h e r o t o r hub. The s e c r e t of t h e s u c c es s w a s t h e f a c t t h a t t h i s w a s t h e f i r s tt i m e a t r i c y c l e l an d in g ge a r w a s used, which ens ure d a p r a c t i c a l l y v e r t i c a l

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I1 I l111 IIIIII

p o s i t i o n of t h e r o t o r a x i s d u r i ng e n g in e r e w i n g b e f o r e t a k e o f f and a f t e r t h el and i n g s t op . Thi s caused small i n i t i a l p e r t u r b a t i o n s due t o d e f l e c t i o n of t h eb la de s i n t h e p l an e of r o t a t i o n , s i n c e t h e i n i t i a l d e f l e c t i o n s of t h e b l ad e s a re

p ro du ce d by t h e p r o j e c t i o n of t h e f o r c e o f g r a v i t y o n t o t h e p l a n e o f r o t a t i o n .Another impor tan t po in t w a s t h e f r i c t i o n f or c e i n t h e h inges ( a t t h a t t i n e ,bronze bushings were used i n th e h inges) , which cannot be d is re ga rd ed i n t h ep r es e nc e of a p p re c i a bl e c e n t r i f u g a l f o r c e s ; i n th i s case th ey produced suff ic i en t l y l a r ge damping. On one occa sion , t h e p i l o t S.A.Korzinshchikov a f t e r oneof t h e f l i g h t s f o r go t t o pu sh t h e c o n t r o l s t i c k i m me di at el y a f t e r landing andthu s d id no t ch ange t h e c r a f t f rom a t h r ee -po in t l and ing ( t a i l skid and mainl a nd i ng g e a r ) t o a s t a n da r d p o s i t i o n ( w i t h s u p p or t o n t h e f r o n t l e g ) ; g roundresonance occurred a f t e r subsequ en t dec r ea se i n r o to r rpm aJing t o t h e l a r gei n i t i a l d i st u rb a nc e i n bl ad e d e f l e c t i o n s i n t h e p l a ne of r o t a t i o n ( t h e a x i s ofr o t o r r o t a t i o n w a s i n c l i n e d a t an angle of 14" t o t h e g r ou nd ), c a us i ng t h eb l a d es t o b r ea k and t h e he l i co p t e r t o be damaged.

Thus, th e problem assumed con st an t l y newer a sp ec ts from one expe rim enta lm ode l t o ano the r.

Since, a t t h a t time , no exac t ca l cu la t i on of t h e requ i red damping of b ladev i b r a t i o n s e x i s t e d ( i n th e presence of g round resonance v ibr a t i ons , t h e dampingof v i b r a t i o n s of t h e c r a f t by shock absorbers on the landing gear i s of e q u a l /12i m po r ta n ce ) , d e s i g n e r s a t t e mp t ed t o s e l e c t a minimum va lue of th e f r i c t i o nmoment of t h e hub damper. T h i s w a s d i c t a t e d by t h e d e s i r e t o r e d u c e v a r i a b l ebending moments s e t up i n t h e p r e s e n c e o f a damper dur ing forced v ibra t ions oft h e b l ad es i n f l i g h t .

A s i s known, f r i c t i o n dampers cause v i br a t io ns a t t h r e s h o l d e x c i t a t i o n .If t h e e x c i t a t io n i s small, %.e., t h e ex c i ta to ry moment i s smaller t h a n t h ef r i c t i o n m me nt , no v i b r a t i o n s Will appear . However, v i b r a t i o n s may sudde nlya r i s e i n a hel icopter which i s fa i l -pro of wi t h res pec t t o ground resonance andhad a l r e a d y b ee n i n a c t u a l s e r v i c e . Thi s can be a t t r i b u t e d t o t h e f a c t t h a t ,i n a g ive n c a se , t h e i n i t i a l per tur ba t io ns may be g r e a t e r t h a n u s u al . Thi s ca seoccur red i n t h e M i - 1 he l ic opt er when ta x i in g obl iqu e ly acro ss deep r u t s made bya t r u c k . I n t h i s c ase , a random disturbance of t i l t s t r o n g l y ro cked t h e c r a f ton i t s pneumatic t i r e s , caus ing i t t o a c q u ir e s uc h l a r g e v i b r a t i o n a m pl it ud est h a t th e a va i l ab le damping i n t h e hub became inade quate and ground resonancea rose . The p i l o t G.A.Tinyakov remedied th i s i n a simple manner by ta ki ng o f f ;t h i s s to pp ed t h e v i b r a t i o n s s i n c e t h e e l a s t i c c ou pl in g, i .e . , t h e c o u p En g w i t ht h e ground, w a s broken.

Thi s ca s e sugge st e d t h e n eed f o r making u se o f v i s cous f r i c t i o n , i . e . , i ns t a l l i n g hydr au l ic b lade v ib ra t i on dampers i n t h e hub, f o r which th e moment off r i c t i o n d oe s no t r em ai n c o ns t a n t b u t i n c r e a s e s w i t h t h e v ib ra t ion ampl i tude .

However, pr a c t i c e co ns ta nt ly re qu ir ed improvement and development of t h et he or y i n t h i s a r e a . One merely need r e c a l l t h e gene ra t i on of g round r esonancewhe n t he h e l i co p t e r i s a t ta c he d t o i t s moorings, w i th t h e eng ine ope ra t i ng .

Se ver al ca se s of ground resonance were observed a l s o when th e wheels of t h eh el ic op te r, i n t d i n g du rin g ta keo ff o r l anding , had on ly s l i gh t g round con t ac t ,so that t h e p r o p u ls i v e f o r c e of t h e r o t o r cane c l o s e t o t h e we ig ht of t h e c r a f t

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a nd t h e shock s t r u t s w i t h t h e u s u a l p r e t i g M e n i n g w e r e f u l l y ext ended. The d i ff e r e nce be tw een t he weigh t a nd t he p r o p u l s i ve fo r c e of t h e c r a f t w a s absorbedonly by the pneumatic t i r e s .

It i s obv ious t h a t , i n t h i s case , no t on ly W i l l t h e v i b r a t i o n f r e q u e n c ie so f t h e c r a f t c hange b ut t h e r e a l s o W i l l be no damping of t h e s t r u t s . Thus,ground resonance occurre d her e which had never been observed i n a h e l i c o p t e rt h a t w a s not moored o r w a s not taxi ing, a t ve ry small wheel loading.

To avoid such cases, w e began us in g so-c al led two-chamber landing-gears t r u t s , w h i c h were sh oc k s t r u t s p r ov i de d w i t h a second low-pressure chamber f o ra bs o r bi ng t h e v i b r a t i o n ene rgy of t h e c r a f t when i t made only slight ground cont a c t w i t h t h e p ne um at ic t i r e s w h i l e t h e main s t r u t s were no t ope ra t i ve .

Problems of t h e theo ry of ground resonance are e s p e c i a l l y i m p o r t a n t f o rtw i n - r o to r con f igu ra t i ons when t h e e l a s t i c sy st em coup l i ng bo th ro to r s , be i tt h e f u s e la g e i n t h e f or e- an d- af t o r t h e Wing i n t h e s ide-by -s ide con f igu ra t i on ,

has low n a t u r a l v i b r a t i o n f r e q u e n ci e s . I n t h e p r e sence of such v ib r a t i o ns , app r e c i a b l e d i sp l ac e m en t s of t h e r o t o r hub may t a k e p l a c e , c r e a t i n g t h e p o s s i b i l i t yof e ne rg y t r a n s f e r b etween bl ad e o s c i l l a t i o n s and o s c i l l a t i o n s of t h e l i f t i n gs t r u c t u r e . Vi b r a ti o n s of t h i s t y p e a r e pos s ib l e no t on ly on t he g round bu t a l s oi n f l i g h t .

A similar problem a r i s e s i n d es ig ni ng t a i l ro tors wi th drag h inges mountedt o a f l e x i b l e t a i l boom.

The development of harmonic and improved c r a f t i s p o s s i b l e o n l y if t h e d es i g n e r i s s u f f i c i e n t l y co mp ete nt n ot o n l y i n g e n e r a l p ro bl em s o f d e s i gn b u t a l s oi n s p e c i a l p roble ms h av in g t o d o w i t h t h e t h e o r y and c a l c u l a t i o n of t h e i n d i - ,&vidua l e lements .

A modern he l i co pt er con ta in s many e ss en t i a l h igh l y loaded mechanica l comp o ne n ts whose r e l i a b i l i t y a nd s e r v i c e l i f e d epend i n many r e s p e c t s on t h e p e rformance of th e bear ing assembl ie s . Consequently, he l i co pt er des igne rs shouldbe f a m i l i a r w i t h t h e t h e o r y and c a l c u l a t i o n o f r ol l er bea r ings . Thi s p e r t a i n ss p e c i f i c a l l y t o c as es of t he work of r o l l e r bea r ing s i n complex combinat ions ofe x t e r n a l l o a d s a nd i n t h e p r e s en c e o f r o ck i ng m ot io n of low amplitude.

For t h i s reason , we i n c lude d a cha pter g iv ing answer t o problems of t h et h e o r y and c a l c u l a t i o n of bear in g assembPies of hubs, cy c l ic p i t c h con t ro l , ando th e r u n i t s . One o f t h e m ost i n t e r e s t i n g p robl ems de sc r i bed i n Chapt er I V i st h e t h e o ry of s p e c i a l t h r u s t r o l l e r b e ar i ng s i n which, owing t o t h e p o s i t io n i n gof t h e r o l l e r s a t a n a n gl e t o t h e r a d i a l d i r e c t i o n , t h e cage - du r ing t h e rocking mot ion - n ot o n l y v i b r a t e s a l o n g w i t h t h e movable c o l l a r b ut a l s o c on ti nuo u s l y r o t a t e s i n one d i r e c t i o n . Thi s prev ents l o c a l wear of t h e raceways andi n c re a s e s t h e l i f e t i m e o f t h e b e a r i n g.

It shou ld be n o t e d t h a t t h e u s e of such bearings i n t h e f e a th e r i ng hin geo f r o t o r hubs r e s u l t e d i n a n a p pr e ci a bl e i n c re a s e i n s e r v i c e Life.

He l l c o p t e r eng i nee r i ng requires a h i g h g e n e r a l l e v e l of t h e o r e t i c a l ands c i e n t i f i c training of t h e des i gn engineer, s i nc e dynamic problems are of much

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g r e a t e r im po rt an ce f o r h e l i c o p t e r s ( r o t a r y wing a i r c r a f t ) t h a n f o r r e g u l a r a i rc r a f t ( p r o t o t es w i t h f i x e d wing, a l th o u gh l a t e l y a l s o i n c l ud i n g t i l t wings andvariable sweq?. T k i s i s confirmed by the f a c t t h a t t h e f e w designers who made

n o t a b l e c o n t r i b u t i o n s t o t h e de ve lo pm en t o f h e l i c o p t e r e n g i n e e r i n g an d e s p e c i a ll y tho se who had p r a c t i ca l success , were s i m u l t a n e o u s l y o u t s t a n d i n g s c i e n t i f i ct h e o r i s t s . These in cl u d e B.N.Yurtyev, P ro f. A.M.Cheremukhin, and Pr o f. I.P.b a t u k h i n who, i n t h e Thi r t i e s , w e r e t h e de v e l op e r s of t h e f i r s t S o v i e t h e l ic o p t e r s f r o m t h e l EA t o t h e I l E A p r o t o t y p e s ; P r o f . Fo cke, t h e d e s i g n e r o f t h eFW-61 and FA-223 h el ic op te rs i n Germany; one of t h e p i o n e e r s of a v i a t i o n LouisBreguet; Pro f . Doran who cre at ed th e f i r s t f i en ch he l ic op te r s ; and many o th ers .

It should be n ot ed t h a t t h e p r e s e n t l e v e l o f t h e o r e t i c a l t r a i n i n g o f des i gn er s work ing for t h e f o re mo st h e l i c o p t e r e n g i n e e ri n g f irms of th e world i svery high, as f a r as c a n be jud ged fro m t h e l i t e r a t u r e . F or t h i s r e a s o n , n e i t h e rt h e e n g i n ee r - ca l c ul a t o r n or t h e d e s i g n e r wo rk in g i n h e l i c o p t e r e n g i n ee r in g s h o u l dhave any d i f f i c u l t y i n a s s i m i l a t i n g t h e mater ia l presented below.

The authors hope that th i s second volume w i l l f i n d re ade rs and be foundu s e f u l .

+$ +$K

The i n s e r t s show p h ot o gr ap h s o f t h e m ain S o v i e t h e l i c o p t e r s i n se r ies prod u c t i o n . These are t h e f i r s t S o v i e t s er i es - pr o d uc e d h e l i c o p t e r s w i t h p i s t o nengines M i - 1 and Mi-4, d ev elo ped i n 1949 and 1952. H a v i n g b ee n p ro du ce d i nla rg e numbers, th es e pr ot oty pes range now among t h e mos t widespread va r i an t s o fh e l i c o p t e r s .

Other photographs show t h e Mi-6 h e l i c o p t e r w i t h t wo t u r b o p r o p e n g i n e s /ucd ev el op ed i n 1957 and t h e Mi-10 he l i co pt er (1962) which i s a f l y i n g c r an e w i t ha h ig h l an d in g ge ar ,a da pt ed f o r l i f t i n g and t r a n s p o r t i n g heavy s t o r e s r i g i d l ymounted on the underbel ly. I n 1965, a world l i f t i n g re co r d f o r h e l i c o p te r s w a se s t a b l i sh e d w i t h t h i s c a rg o c r a f t : 25 t o n s were l i f t e d t o a height of 2830 m.

The next pi ct u re s g iv e t h e Mi-2 and Mi-$ h e l i c o p t e r s w h ich a re a secondgen era t i on of Sovie t l i g h t and medium vers ion s . The l i f t i n g sy st em s of t h eM i - 1 and Mi-4 were r e t a i n e d on t h e se , b u t t h e s i n g l e p i s t o n e ng in e w a s r e p l a c e dby two tu rboprop engines .

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C H A P T E RI /15ELASTIC VIBRATIONS ANDBLADE STFENGTH

C a l c ul a t io n of e l a s t i c Vi b r at i o n s i s a n e c e ss a r y e lem en t i n t h e p r o c e s s o fdeveloping new blade designs. It forms an i n s e p a r a b l e p a r t of t h e c a l c u l a t i o no f b l a d e s t r e n g t h .

To d ev e lo p h e l i c o p t e r b l a d e s it i s n e ce s sa r y t o s o l v e many p r e s e r r t l y qui tecomplex technolog ica l and des ignproblems . I n t h e i r s o l u t i o n , a cc ou nt must bet a k e n o f t h e mo st d i v e r s e r e q u i re m e n t s an d p r i m a r i l y o f t h e r eq u ir em e nt o f h i g hf a t i g u e s t r e n g t h o f t h e s t r u c t u r e .

The work o f d e s i g n i n g b l a d e s u s u a l l y i n v o l v e s t h e f o l l o w i n g b a s i c s t e p s :S e l e c t i o n o f materials f o r i n d i v i d u al s t r u c t u r a l members , d e t e r m i n a t i o nof op t imal parameters , and des ign of th e b lade .S e l e c t i o n o f t h e bes t t e c h n o l o g i c a l p r o c e s s e s e n su r i n g h i g h es t f a t i g u es t r e n g t h of i t s main s t re ss ed elements, and manufacture of th e blade.F l i g ht t e s t s w i t h a n a l y s i s o f s t r e s s e s set up i n f l i g h t .Dynamic t e s t s and e v a l u a t i o n o f t h e b l a de s e r v i c e l i f e .Performance of t h e complex of f i na l i z i n g , inc l ud i ng work on redu c t io nof a c t i v e s t r e s s e s a nd i n c r e a s e i n f at i g u e s t r e n g t h of t h e s t r u c t u r e .Acceptance t e s t s and s t a r t of s e r i e s p r o d u c t i o n .A n a l y s i s o f o p e r a t i o n of ser ies-produced bl ad es under var iou s high-loadand endurance condit ions and layout of f i n a l d e s i gn s f o r b l ad e s e r i e sb ase d o n t h e a n a l y t i c a l d a t a .

C a l c ul a t io n s of e l a s t i c b l a d e v i b r a t i o n s a re r e q u i r e d a t many stages oft h i s work, b u t p r i m a r i l y a t t h e i n i t i a l s t a g e w hich t e r m i n a t e s w i t h t h e a c t u a lb l a d e d e s i g n .

I n s e l e c t i n g t h e b l ad e p a ra m e t er s and i t s s t r u c t u r a l m a t e r i a l s , one of t h em a i n c r i t e r i a i s the magnitude of a l t e r n a t i n g s t r e s s e s s e t q~ i n f l i g h t and t h ec o r r e l a t i o n between t h e s e s t r e s s e s a nd o t h e r s c h a r a c t e r i z in g t h e f a t i g u e s t r e n g t hof t h e s t r u c t u r e . I t i s o n l y b y c a l c u l a t i o n s t h a t t h e mag nitu de o f t h e s es t r e s s e s c an be determined and an estimate made of t h e s t r e n g t h o f t h e s t r u c t u r ea t th i s s t a g e . To d e s i g n t h e b l a d e w i t h i n t h e r e q u ir e d - u s u a l l y r a t h e r s h o r t -pe r io d , th e des i gne r should have av a i la b l e modern methods and computa t iona lmeans t o o b t a i n a r a p i d s o l u t i o n t o a n y number o f p o s s i b l e p ro bl em s.

O f si ml ar importance i s t h e c a lc u l at i o n i n t h e f i n a l i z i n g s ta g e. A s ar u l e , i n new b l a d e d e s i g n s t h e var iab le s t r e s ses a r e e x c e ss i v e , c o n f ro n t i n g t h ed e s i g n e r w i t h t h e p r o b l e m of t h e i r r ed u ct i on . For this , t h e o c c u r re n c e p a t - /16t e r n o f s t r e s s e s measured i n f l i g h t m ust be conf i rmed by ca lcu la t ion , fo l lowedb y d e v i s i n g means f o r t h e i r r e d u c t i o n by v a r y in g sme of t h e parameters . Toa t t e m p t a s o l u t i o n o f t h i s p ro bl em w i t h o u t c a l c u l a t i o n g e n e r a l l y means e x c e s s i v el o s s o f t i m e i n check ing un ve r i f i ed assumpt ions and w a s t e of cons id erab le fundsi n m a nu fa ct ur in g a b l a d e t h a t might be r e j e c t e d a f t e r f l i g h t t e s t i n g .

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A r e d u c t i on o f a l t e r n a t i n g stresses i s extremely important and permits noto nl y a n i n cr ea s e i n t h e r e l i a b i l i t y and s e rv i ce l i f e of t h e b la d e b u t a l s o a nimprovement inm ec ha ni ca l and f l y i ng cpa l i t i es of a h e l i c o p t e r s u c h as, f o r

example, f l y i n g speed and l i f t capac i t y, which i n modern he l i co p t e r s are o f t e nl im i t e d because o f s t r e ng th cond i t i ons .

So lu t i on o f a l l t h e s e problems would not be e x c es s iv e l y d i f f i c u l t i f t h ec a l c u l a t i o n r e s u l t s would s u f f i c i e n t l y w e l l coinc ide With tho se observed du r ingi n - f l i g h t s t r e s s a n a l y s i s . Unfortunately, t h i s i s no t a lways t he ca se s i ncec a l c u l a t i o n does n ot n e c e s s a ri l y gi v e r e s u l t s s a t i s f a c t o r y f o r p r a c t i c e .

C a l c u l a t io n s f o r d et er m in in g t h e n a t u r a l v i b r a t i o n f r e q u e n ci e s a r e m ostr e l i a b l e . Usually, an accuracy of th e order of *2% i s achieved. Therefore, a l lc a l c u l a t i o n s on t h e e x c l u s i o n of r es on an ce y i e l d h i g h r e l i a b i l i t y. C a l c u l at i o n s of a l t e r n a t i n g s t r e s s e s a t c ru i s i ng and maxi" f l y ing speeds are not icea b l y l e s s r e l i a b l e . The s t r e s s v a l ue s o b t a in e d i n t h e s e c a l c u l a t i o n s u s u a l l y

a r e 15-25% lower than s t resses measured i n f l i g h t . Consequen tly, t h e s t r e s sana ly se s i n t he se regimes do no t always s a t i s fy t h e de s igne r. Neve r the le s s, t h ee r r o r c a n be compensated t o a c e r t a i n e x t e nt by i n tr o d u c i ng i n t o t h e c a l c u l a t i o na c o r r e c t i o n a l l o w i n g f o r a constant divergence from experiment.

A s t i l l g r e a t e r e r r o r i s p o s s i b le i n c a lc u l a t i on s of a l t e r n a t i n g s t r e s s e sa t low flying speeds.

It i s obvious from the above t h a t t h e c a l c u l a t i o n methods f o r a l t e r n a t i n gb l ade s t r e s s e s require f u r t h e r e l a b o r a t i o n . Nevertheless, pr ac t i ce has shownth a t pa ramete r s e l e c t i o n and b l ade f i n i sh in g w i thou t even t he se impe r f ec t methodsi s r a t h e r i n e f f e c t i v e . Therefore, t h i s Chapter W i l l g i v e a de t a i l ed accoun t of

va r ious ca l c u l a t i o n methods. I nour opini