AD-774-836

SUMARY OF PROPELLER DESIGN PROCEDURES AND DATA. VOLUME II. STRUCTURALANALYSIS AND BLADE DESIGN

William Amatt, et al

Henry V. Borst and AssociatesRosemont, Pennsylvania

Nov 73

I. D~wpArt of CeifsctljmaI Todg i ImUmtion Set$

:: M

Unclassified

S.. .. . .. . .. n .. Al : 03U L, +

Secutyll C5.ILfIlet~on

DOCUMENT CONTROL DATA . R & DOAIgSI*GIN rIvi ITV (C.

9l ... sJ 5* ACAOAT SUCURIIVY CUSS,, VIASIOS

Henry V. Borst & Associates Unclassified353 Yorkshire Road js. ..o..Rosemont, Pennsylvania

SUMMARY OF PROPELLER DESIGN PROCEDURES AND DATAVOLUME 11 - STRUCTURAL ANALYSIS AND BLADE DESIGN

Final Report

Ijilliars AmartWilliam E. BaetesHenry V. Borst

November 1973-.. ISI T oC ON s0. NO. 0m51I.*SsrS Re.T IUbflSJ

DAA0O-72-C-0033D.AJOZ72cTo USAAMRDL Technical Report 73-34B

S. Task 1G162207MA7203 O n.so si.o ,.n...,,-- a,.., s.sa,.

Approved for public release; distribution unlimited.

, y NOT.I. 11N2OPIN-11 MIIITAA t ACTIVITY

Eustis DirectorateVolume II of a 3-volume report U.S. Army Air Mobility R&D Laboratory

Fort Eustie, Virginia

The technology needed for the design and installation of propellers is presented andsumsmarized in three volumes.

Volume II (Structural Analysis and Blade Design) contains the theory and data for thedetailed structural and vibration analysis of propellers. Included are estimating

procedures for initial design purposes; the details for designing solid, hollow, and

composite blades; and manufacturing techniques used. t

I02--ucSo orNATIONAL TECHNICALINFORMATION SERVICE

53 WARWtR,?Il of co0eRCE

DD To.?..1473 .N::: :F4 S A..... Unclassified

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DISCLAIMERS

The findings in this report are not to be construed as an officialDepartment of the Army position unless so esignated by other authorizeddocuments.

When Government drawings, specifications, or other data are used for anypurpose other than in connection with a definitely related Governmentprocurement operation, the United States Government thereby incurs noresponsibility nor any obligation whatsoever: and the fact that theGovernment may have formulated, furnished, or in any way supplied thesaid drawings, specifications, or other data is not to be regarded byimplication or otherwise as in any manner licensing the holder or anyother person or corporation, or conveying any rights or permission, tomanufacture, use, or sell any patented invention that may in any way be 8related thereto.

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DISPOSITION INSTRUCTIONS

Destroy this report when no longer needed. Do not return it to theoriginator.

- i-- I

Task 1G162207AA7203Contract DAAJ02-72-C-0033

USAAMRDL Technical Report 73-34BNovember 1973

SUMMARY OF PROPELLER DESIGNPROCEDURES AND DATA

VOLUME I1STRUCTUR21, ANALYSIS AND BLADE DESIGN

By

William AmattWilliam E. BatesHenry V. Borst

Prepared by

HENRY V. BORST & ASSOCIATESRosemont, Pennsylvania

for

EUSTIS DIRECTORATEU.S. ARMY AIR MOBILITY RESEARCH AND DEVELOPMENT LABORATORY I -

FORT EUSTIS, VIRGINIA

Approved for public release;ldistribution unlimited.

SUKRARY

This volume presents the structural design and analysis of thehigh-performance aircraft propeller. The various types ofblade structures used, including solid, hollow, and compositeblades are analyzed. The effects of changes of the variousblade parameters such as diameter, thickness ratio and solidityare considered.

So that an accurate analysis can be conducted, a complete dis-cussion of loads that the propeller will encounter is pre-sented. Included ace the loads generated when operating athigh inflow angles, aerodynamic loads and centrifugal loads.

Complete methods of analysis are presented along with esti-mating procedures for the early design phases. The estimatingprocedures given are particularly important during the aero-dynamic design and analysis phase so that the proper trade-offs can be made. The methods of analysis given include thebasic blade stress calculation, propeller vibration andresonant frequencies, blade and flexural resonance, bladeflutter, stall flutter, etc.

The structural section includes material for analyzing the huband blade retention, and discussion of the various types ofmaterials is also presented.

Propeller blades are discussed, including the basic designcriteria, material considerations and details of construction.The basic methods for designing a blade are presented as wellas the details of section layout blade integral characteristicsfor the hollow types and the shank fairing details.

"The methods for blade manufacturing are discussed for solidaluminum, hollow steel and compos.te blades, along with therelative costs of each.

Blade ice control methods and also other environmental problemsassociated with blades are covered.

-- Ii--

--_ I i llllmlllll Ill... |111 ~

TABLE OF CONTENTS

PageSUMMARY ......................................... i i

LIST OF ILLUSTRATIONS ........................... Xii" OF, TABES ............. ...................... x

LIST OF SYMBOLS ................................. Xxi

INTRODUCTION .................................... 1

STRUCTURAL ELEMENTS ............................. 1

THE PROPELLER BLADE ............................. 3

Material ..................................... 3Types of Structure ........................... 3Solid Section ................................ 3Hollow Section ............................... 6Spar Section ................................. 7Aerodnamic Parameters ....................... 7

Airfoil Shape ............................. 7Activity Factor, Thickness Ratio andPitch Distribution ......................... 9

BLADE RETENTION ................................. 9

The Blade Shank .............................. 99Blade Bearing ................................ .11The Blade Nut ............................... 16Hub Barrel Thread ............................ 16Retention Variation .......................... 16The Dural Shank .............................. 21Integral Race Retention ...................... 21Flanged Hub Retention .................. , ..... 24

PROPELLER HUB ................................... 24

Shaft-Mounted Hub ............................ 28Nose-Mounted Hub ............................. 30

ESTIMATING PROCEDURES ........................... 32

Preceding page blank

TABLE OF CONTENTS (Continued)

PaqeESTIMATING PROCEDURE - WEIGHT

Solid Blade ................................... 33Hollow Blade .................................. 33Hollow Fiber Glass Blade (Foam Filled) ........ 33Total Propeller Weight ........................ 33

ESTIMATING PROCEDURES - BLADE SECTION DATA ...... 34

Solid Sections ............................... 36Hollow Sections .............................. 37

ESTIMATING PROCEDURES - PROPELLER LOADS ......... 37

Blade Centrifugal or Mass Loads .............. 39Blade Aerodynamic Loads - Steady State ....... 41Blade Aerodynamic Forces - Harmonic, lxP ..... 44Shaft Forces - lxP ........................... 46lxP Blade Moments ............................ 51lxP Forces - Nonconventional Aircraft ........ 51

ESTIMATING PROCEDURES - RESONANT FREQUENCIES .... 54

PROPELLER LOADS ................................. 56

PROPELLER LOADS - AERODYNAMIC ................... 57

Airplane Aq Factor ........................... 57First-Order Propeller Loads .................. 63Higher Order Loads ........................... 74Steady-State Aerodynamic Loads ............... 75Propeller Mass Loads ......................... 78Blade Centrifugal Force ...................... 78Blade Centrifugal Restoring Moments .......... 81Blade Tilt .................................... 82Blade Centrifugal Twisting Moment............. 88Centrifugal Straightening Moment ............. 93Gyroscopic Forces ............................ 95Miscellaneous Factors ......................... 98

Polar Moment of Inertia ................... 98

Blade Inertia, Pitch Change Axis .......... 100

METHODS OF ANALYSIS ............................. 100

Blade Section Properties ..................... 100Blade Section Properties - CompositeStructures ................................... 103

vi

i ..,-. _ _ _-_,_ _"' ... _ /~i"'ilii'ij

TABLE OF CONTENTS (Continued)

Pa|e

Blade Section Properties - Effect of Twiston Stiffness ................................. 107Blade Steady-State Force and MomentDistribution ................................. 112Blade Steady-State Moments ................... 113Blade First-Orde? Haii6nic Moments ........... 118Retention Flexibility - Effects on BladeForces and Moments ........................... 120Solution of Blade Moment Equations ........... 122Basic Blade Stress Calculation ............... 125Propeller Vibration and Resonant Frequencies 130Blade Resonance_.............................. .133Flexural Resonance .............................. 135Reactionless and Nonreactionless ShaftModes .,.................... 139Torsional Resonance .......................... 145Engine vibrations ............................ 148Vibrations Caused by Whirls .................. 150Blade Flutter ................................ 155Stall Fiutter ................................ 156Wake Flutter .................................. 168Miscellaneous Factors ........................ 171Blade Buckling ............ ................... 171Torsional Deflection ......................... 173Internal :Blde Pressure ...................... 176

RETENTION LOADS AND ANALYSIS .................... 177

Equivalent Centrifugal Force .................. 178Blade Flange Analysis ........................ 180Hub Thread Relief Analysis ................... 182Thread Bending Stress ........................ 189Combined Fillet Stress ....................... 189Blade Bearing Analysis ....................... 192Bearing Overhang ............................. 195..1Bearing Friction Alleviation of Load........... 197Bearing KT Factor ............................ 199Bearing Overturning .......................... 201Hub Loads and Analysis ....................... 201Propeller Materials .......................... 204

PROPELLER BLADES ................................ 215

] vii

TABLE OF CONTENTS (Continued)

Paqe

INTRODUCTION .................................... 215

BLADE DESIGN CRITERIA ........................... 215

Aerodynamics ................................. 215Reliability .................................. 215Survivability ................................ 216Interchangeability ........................... 217Producibility ................................ 218Inspectability ............................... 219Maintainability .............................. 219Materials.................................... ..... 220Design ....................................... 221Structural and Testing Considerations ........ 221Facilities ................................... 222

BLADE MATERIAL CONSIDERATIONS ................... 222

Material .onsiderations - Design ............. 223Material Considerations - Blade Retention .... 226Material Considerations - Operational ........ 227Material Considerations - Processing ......... 229....2

METHODS OF BLADE CONSTRUCTION - ALUMINUM ALLOY .. 229

Blade Finish ................................. 231Balance Provision .......................... .. 231Fairings ..................................... 231Development Time ............................... 231Survivability ................................ 231

METHODS OF BLADE CONSTRUCTION - EXTRUDEDHOLLOW STEEL .................................... 232

Extruded Tube ................................ 232Planform Shapes .............................. 232Cross Sections ............................... 235Outboard Construction ........................ 238Environmental Protection ..................... 238Root Ends .................................... 240Balance Provisions ........................... 240Fairings ..................................... 242Development Time ............................. 244..2Survivability .................................... 244

viii

TABLE OF CONTENTS (Continued)

Pacge

METHODS OF BLADE CONSTRUCTION - WELDEDHOLLOW STEEL .................................... 244

Description - Side-Welded Blades ............. 244Description - Edge-Welded Blades ............. 245Inboard Weld Geometry ........................ 248General - Welded Blades ...................... 248Development Time ............................. 348

if

METHODS OF BLADE CONSTRUCTION - COVEREDSPAR HOLLOW STEEL ............................... 248

Description .................................. 251Fairing ...................................... 251Root End ..................................... 253Materials .................................... 253General - Spar Type Blades ................... 253Development Tinte ............................. 253

METHODS OF BLADE CONSTRUCTION - FIBER-REINFORCEDCOMPOSITE COVERED SPAR TYPE ..................... 253..2

METHODS OF BLADE CONSTRUCTION - FIBER-REINFORCED COMPOSITE - MONOCOQUE TYPE ........... 254

Variations ................................... 254Erosion Protection ........................... 256Environmental and Lightning Protection .... 256Survivability ................................ 256Development Time ............................. 257

RELATIVE BLADE WEIGHTS .......................... 257

Solid Aluminum Versus Hollow Steel ........... 258Relative Weights ... ......................... 258

RELATIVE BLADE COSTS ............................ 258

BLADE DESIGN .................................... 259

Blade Planform - Design Limitations .......... 260The Number of Blades ......................... 260Shank Size Selection ......................... 260

ix

TABLE OF CONTENTS (Continued)

Pane

BLADE CHARACTERISTIC DATA ........................... 261

Geometric Layout ............................. 261Estimating Section Properties - Solid andHollow Blades ................................ 261Optimizing Mass Distribution and ProfileThickness ..................................... 272Final Characteristic Data .................... 279Calculation of Ordinates ....................... 279Faired Intermediate Profile Ordinates ........ 284

LAYOUT OF OVERSIZE SECTIONS ..................... 287

Structural Geometry .......................... 287Data for Integrated Section Properties ....... 288Integrated Section Properties and StressAnalysis ....................................... 288Weight and Balanco/Volumes and SurfaceAreas/Polar Moment ............................. 290Blade Drawings ............................... 290Design Report ................................ 291

BLADE ICE CONTROL ............................... 292

Effects of Ice on Propeller B)ad=; ............ 292Surface TreatmenL and Compounds .............. 292Fluid Anti-Icing ............................. 293Hot Air Ice Control System ................... 29.Electrical Ice Control. ........................ 297Typical Electrical Deicing Circuit ........... 298

LIGHTNING CONSIDERATIONS ........................ 3.06

Metal Propellers................................. 306Composite Propellers ......................... 306

MANUFACTURING METHODS - SOLID ALUMINUMALLOY BLADES .................................... 307

Forgings ..................................... 307Master Blade ................................. 307Production Blade Fabrication ................. 308Finishing .................................... 308Final Finishing and Balance .................. 309

x

TABLE OF CONTENTS (Continued)

Paqe

MANUFACTURING METHODS - WELDED HOLLOWSTEEL BLADES .................................... 309

Plate Machining .............................. 310Welding and Weld Grinding .................... 315Shank Upset .................................. 317Pressure Die Form and Stress Relief .......... 317Clean and Braze Edges ......................... 319Pressure Die Quench and Draw ................. 320Metal Finishing .............................. 320Blade Finish Operations ....................... 321Final Inspection, Operations andAerodynamic Matching ......................... 322Zinc Plate, Balance and Final Inspections .... 323

MANUFACTURING METHODS - EXTRUDED HOLLOW STEELBLADES .............................................. 323

Extrusion ..................................... 324Tube Preparation ............................. 328Tube Partial Hot Flattened and T.E.Solid Edge Added ............................. 329Finishing Operations ......................... 330

MANUFACTURING METHODS - COVERED SPAR HOLLOWSTEEL BLADE ..................................... 330

Processing Tubular Spar ...................... 331Shell ........................................ 331Assembly ..................................... 331Finishing ....................................... .. 332

MANUFACTURING METHODS - MONOCOQUE FIBERGLASS BLADES .................................... 332

Procurement of Materials ........................ 332Fabrication .................................. 332

SIZE CONSIDERATIONS IN MANUFACTURING ............ 334

RELATIVE BLADE COSTS ............................ 334

BLADE RELIABILITY ............................... 335

xi

TABLE OF CONTENTS (Continued)

PaBe

Blade Life....................................... 335Blade Durability ............................. 335

BLADE MAINTENANCE .................................. 337

Daily Inspections................................ 337Damage Allowance Determination ............... 337Damage Allowances and Rework - SolidAluminum Alloy Blades ........................ 338Damage Allowances and Rework - HollowSteel Blades .................................... 339Damage Allowance and Rework - Monocoque FiberGlass Blades ................................. 342Base Overhaul of Blades ...................... 344Factory Overhaul of Blades ................... 345

TESTING ......................................... 345

Basic Materials Testing ...................... 346..3Core Material Testing ........................ 346Pairing or Cuff Materials Testing .............. 347Component Tests ................................. 347Full.-Scale Laboratory Testing ................ 347Whirl Testing Blades ............................ 348Engine Testing Blades ........................ 348Gyro Rig Testing Blades ...................... 349Flight Testing Blades ........................ 349

CONCLUSIONS ........................................ 352

LITERATURE CITED ................................ 353

DISTRIBUTION ....................................... 354

xii

LIST OF ILLUSTRATIONS

i ure

1 Typical Four-Bladed Propeller System ........ 2

2 The Blade Beam .............................. 4...

3 Blade Types .................................. .... 5

4 Thickness Distribution for Series 16and 65 Airfoils .............................. 8

5 Typical Section Standard Blade Retention .... 10

6 Typical Flanged Blade Shank ................. 12.,1

7 Typical Blade Bearing ...................... 14

8 Typical Blade Nut Section ....................... 18

9 Typical Hub Barrel Section .................. 19

10 Flanged Shank - Dural .......................... 22

11 Typical Section - Integral Race Retention ... 23

12 Typical Section - Flanged Hub Retention ..... 25

13 Spring Type Retention Preload ............... 26

14 Schematic Hub Load System ................... 27

15 Shaft-Mounted Propeller Schematic ........... 29

16 Hose-Mounted Propeller Schematic ............ 31

17 Basic Airfoil Dimensions ....................... 35

18 Ratio of Torsional Stiffness - HollowBlade to Solid Blade ............................. 38

19 Normalized Blade Force and MomentDistribution ................................... 40

20 Aerodynamic Loads ............................... 42

21 Variation of Upwash Along Semi-Span ......... 45

22 Infinite Aspect Ratio Slope of the Lift Curva-ture Variation With Mach Number ............. 49

xiii

LIST OF ILLUSTRATIONS (Continued)

Figure Pag~e

23 Lock-Goldstein Correction Versus Sin 0at 70 Percent Blade Station ................. 50

24 Normalized Blade Modes - Fundamental ........ 52

25 Airplane in Climbing Attitude ............... 58

26 Components of Inflow Angle .................. 58

27 Nominal Aq Diagram .......................... 61

28 Airplane in Yaw ............................. 62

29 Resultant Aq ................................ 64

30 Development of Periodic Forces .............. 66

31 Blade Aero-Torsion .......................... 68

32 Nominal Blade Forces ..... . .................. 76

33 Typical Blade Pitch Distribqtion ............ 79

34 Blade Centrifugal Force .... ................ 83

35 Blade Tilt................................... .84

36 Centrifugal Twisting Moment ................. 90

37 Blade Straightening ......................... 94

38 Gyroscopic Forces ........................... 96

39 Basic Blade Diaensions ...................... 101

40 Typical Composite Section ................... 104

41 Equivalent Structural Section ............... 104

42 Equivalent Section of Figure 40 Fiber Glass . 106

43 Blade-Induced Effects ....................... 110

44 Blade Loads and Deflection ................... 114

xiv

LIST OF ILLUSTRATIONS (Continued)

Fiqure Pace

45 Blade Stress Distribution ................... 126

46 Blade Transverse Stress ..................... 129

47 Fillet Stress Concentration Factor .......... 131

48 Blade Damage - Rework ....................... 132

49 Factors Related to Plate Grind-Outs ......... 132

50 Blade Frequency Diagram ..................... 138

51 Resolution of Periodic Torque Force ......... 141

52 Resonance of the Complete System ............ 151

53 The Dynamics of Whirls ...................... 153

54 Lift Variation - Oscillating Airfoil ........ 157

55 Flutter Boundary ............................ 157

56 Variation of Flutter Amplitude With rpm,Blade-Angle Constant ........................ 158

57 Angle of Stall Versus ..................... 160

58 V.80 Versus blo~x for Various Xo'S ........... 162

59 Average CL~ma.,x Versus M for FlutterCalculation ................................. 164

60 Slope of the Lift Curve versus ThicknessRatio ....................................... 164

S61 Static Inflow Angle .i ..................... 165

62 Average Angle of Zero Lift .................. 165..1

63 Typical Flutter Boundaries Showing Effectof Live Blade Deflection .................... 166

64 Wake Flutter Factor "a" Versus Blade Angle .. 170

65 Blade Buckling .............................. 172

xv

LIST OF ILLUSTRATIONS (Continued).

Figureae

66 Retention Loads ............................. 179

67 Blade Flange ................................ 181

68 Ezb Thread Relief ............................ 183

69 Hub Thread Flange ........................... 183

70 Straight Thread System ...................... 185

71 Tapered Thread System ....................... 186

72 Typical Straight Hub Thread LoadDistribution ................................ 187

73 Equivalent Flange - Outboard Thread ......... 190

74 Thread Bending .............................. 190

75 Nondimensional Distribution of Thread Reliefand Thread Bending Stresses Around BaseFillet ...................................... 191

76 Bearing Geometry ............................ 193

77 Bearing Deflection Constants(C bo + C 81) ............................... 194

78 Bearing Stress Constant ..................... 196

79 Bearing Overhang ............................. 197

80 Bearing Contact Area Constants .............. 198

81 Conparison of Theoretical and ExperimentalLoad Transmissibility Factors, Q', forTwo No. 2 Bearings .......................... 200

82 Basic Hub Loads ......................... ... 203

83 Typical Blade Fillet Endurance Test ......... 205

84 Working Goodman Diagram - solid AluminumBlades (Including Notch Effects and Factorof Safety) .................................. 206

Xvi

LIST OF ILLUSTRATIONS (Continued)

Figjre Paqe

85 Working Goodman Diagram - Blade CurvesInclude Notch Effect and Factor of Safety .... 207

86 Variation of Material Strength WithHardness Steel - 4340 Type .................. 208

87 Working Goodman Diagram Includes NotchEffects and Factor of Safety ................ 213

88 Mixed Fiber Composites, Specific Modulus

Envelope ..................................... 224

89 Aluminum Alloy Blade Details ................ 230

90 Extruded Hollow Steel Blade Tubular Blank ... 233

91 Extruded Hollow Steel Blade DevelopedPlanforms ................................... 234

92 Eztruded Hollow Steel Blade Sections ........ 236

93 Extruded Hollow Steel Blade Sections ........ 237

94 Extruded Hollow Steel Blade Tip Details ..... 239

95 Typical Hollow Steel Blade Root End ......... 241

96 Shank Fairings .............................. 243

97 Side-Welded Hollow Steel Blade Sections ..... 244

98 Side-Welded Hollow Steel Blade Sections ..... 247

99 Edge-Welded Hollow S'eel Blade Sections ..... 249

100 Inboard Geometry on Hollow Steel Blades ..... 250

101 Covered Spar Hollow Steel Blade ............. 250

102- Fiber-Reinforced Composite Blade ............ 255

103 Blade Design Characteristic Curves .......... 260

104 Illustrative Section........................ 263

105 Geometric Layout ....... ...... 264

LIST OF ILLUSTPATIONS (Continued)

FicrurePane

106 Solid Section Properties .................... 266

107 NACA Series 16 Solid Sections -Form Factors . 267

S108 NACA Series 65 Solid Sections -Form Factors . 268

109 NACA Series 65 Solid Sections -Form Factors . 269

110 NACA Series 65 Solid Sections -Form Factors . 270

111 Typical Extruded Hollow Steel Blade WallThickness ................................... 271

112 Form Factor Curve - Hollow Sections forEstimating Area ............................. 273

113 Form Factor Curve - Hollow Sections forEstimating IMinor ........................... 274

114 Form Factor Curve - Hollow Sections forEstimating IMajor ........................... 275

115 Form Factor Curve - Hollow Sections forEstimating Z ................................ 276

116 Form Factor Curve - Hollow Sections forEstimating cThrust ........................... 277

117 Estimating New Values of IMinor and IcWhen h is Varied ............................ 280

118 Estimating New Values of IMinor and I/cWhen tp is Varied ........................... 281

119 Early Propeller Airfoils .................... 282

120 Construction of NACA Series Sections ........ 283

121 Typical Layout of Leading-EdgeStructural Geometry ......................... 289

122 Typical Fluid Anti-Icing System ............. 294

123 Typical Hot Air Ice Control Blade ........... 296

xviii

LIST OF ILLUSTRATIONS (Continued)

124 Propeller Ice Control Schematic - FourEngine Airplane ............................. 299

125 28 Volt DC Ice Control Cyclic Time Schematic. 300

126 Phase A = Deicing .......................... 301

127 Phase B = Anti-icing ........................ 302

128 Phase C = Deicing .......................... 303

129 Deicing Boot Installation .................. 305

130 Taper Milling Operation ..................... 311

131 Finish Milling of Plate ..................... 313, 3

132 Plate Blanking .............................. 314

133 Two Methods of Welding Blades ............... 316

134 Shank Upset ................................. 318

135 First Step - Extrusion ...................... 325

136 Second Step - Expansion ..................... 326

137 Third Step - Extrusion ...................... 327

138 Blade Damage Due to 40mn Shell ............. 336

139 Gyro Rig for Blade Testing .................. 350

xiX

= LIST OF TABLES

Table Paqe

I Shank Size - Diameter ....................... 13 1 -1I Flanged Shank Dimensions, Nominal, Steel

Blades, Reference Figure 6 ...................

III Typical Blade Retention Bearing DataStdel Blades ................................. 17

IV Typical -Hub Barrel Dimensions,Refcrtence Figure 9 .......................... 20....2

V Di-mensional Comparison Between Dural andSteel Blade Shank, Reference Figure 6For Dimensions .............................. 23

VI Blade Torsional Stiffness Factors ........... 108

VII 5ummary of Shot Peening Specifications ...... 210

VIII Summary of Rolling Specifications ........... 212

IX Physical Propn'rties Fiber GlassReinforced Plastic Blade Material ........... 214

X For Calculating NACA Series 16Profile Ordinates ........................... 285

XI For Calculating NACA Series 65 ProfileOrdinates (Modified Trailing Edge) .......... 286

'cc _

LIST OF SYMBOLS

A Angularity of air inflow, deg

A Area of section, in.2

Aq The IxP excitation factor, degree lb/in.2

AF Activity factor per blade

AR Aspect ratio of the wing

a Slope of lift curve vs. apparent angleof attack, radians- 1

ao Slope of lift curve for infinite aspect ratio

B Number of blades

b Blade chord, in.

b Wing span, ft

b, Semi-chord, ft

be Effective chord length, in.

C Torsional stiffness of untwisted bladesection, in.-lb/rad per inch of length

C' Torsional stiffness due to bladetwist, in.-ib/rad per inch of length

CL Coefficient of lift

CT Total torsional stiffness of

section, in.-lb/rad per inch of length

CF Centrifugal force at any blade station r, lb

D Propeller diameter, ft

d Bearing ball diameter, in.

e The base of the natural logarithms

E Modulus of elasticity, psi

F Vertical force on shaft, lb

xxi

LIST OF SYMBOLS (Continued)

F Blade shear force, lb

f Intensity of stress, psi

G Shear modulus of elasticity, psi

g Gravitational constant, ft/sec2

- h Maximum thickness of section, in.

he Effective section thickness, in.

I Moment of inertia of section, in.4

ih Hub polar moment of inertia, slug-ft2

Imax Major moment of inertia of section, in.4

Immn Minor moment of inertia of section, in.4

:Ip Polar moment of inertia, slug-ft2

J Polar moment of inertia of section, in.4

k Radius of gyration, in.

k Section constant for torsional stiffnessformula, in.

-k Constant of an elastic spring

L Length, in.

L Lift, lb

M Bending moment, in.-lb

M The lxP couple, in.-lb

M Mach number

SMG Gyroscopic moment, in.-lbMy(x) Bending moment (vibratory)

in fore-aft plane, in.-lb

Mz(x) Bending moment (vibratory)in torquewise plane, in.-lb

xxii

LIST OF SYMBOLS (Continued)

m Mass, slugs

N Propeller rotational speed, rpm

n Propeller rotational speed, rps

P Load on beam or column or spring, lb

Py Loading component, vibratory,fore-aft plane, lb/in.

Pz Loading component, vibratory,torquewise plane, lb/in.

Friction transmissibility (retentions) lb/in.

Qa Aerodynamic twisting moment, in.-lb

Centrifugal twisting moment, in.-lb

Ou Centrifugal untwisting moment, in.-lb

q Dynamic pressure, psi

R Tip radius of blade, in.

r Radius to any given blade section, in.

r Radius of gyration, in.

S Wing area

S.F. The lxP side force, l.I

s Blade radius, variable, in.

* se Endurance limit stress, lb/in. 2

SSO Steady stress, lb/in.2

-! sAlternating stress,, lb/in.2

T Torque, in.-lb

St Plate thickness, in.-t Time, sec

____ _ _ _ _ _ _ _ _ _

LIST OF SYMBOLS (Continued)

V Velocity, fps

.10 Apparent wind velocity, fps

wL Blade solid lead edge width, in.

WT Blade solid trailing edge width, in.

x Blade radius, fixed, in.

y Blade deflection, fore-aft plane, in.

Ymax Deflection of beam

z Blade deflection, torquewise plane, in.

a Angle of attack, apparent, deg

"ai Inflow angle, deg

Ox Blade angle at any given radius, deg

Sp Contact angle (bearing races)

Pref Blade angle at the blade reference radius, deg

8 Weight density, lb/in. 3

Z Induced torsional stiffness factor, in.6

Unit strain, in./in.

Local upwash angle

i0 o Average upwash angle

Reduced frequency parameter(flutter), Poisson's ratio

A Coefficient of friction

p Mass density, slugs/in. 3

Intensity of stress, psi

Solidity factor (flutter)

C.F. Centrifugal stress, psi

xxiv

-" ,, _ ..

LIST OF SYMBOLS (Continued)

Intensity of shear stress, lb/in.2

True wind angle

36o Apparent wind angle, deg

Phase angle

n Rotational speed, circular frequency ofvibration, rad/sec

Rotati.onal speed, rad/sec

Torsionel frequency of oscillation, rad/sec

'ccv

INTRODUCTION

The purpose o! this volume is to present the structural detailswhich must be considered in the basic design of the high-per-formance aircraft propeller. As such, the presentation willbe concerned with the primary structural elements, loads andmethods of analysis. It is intended that these discussionswill be general yet inclusive; however, no attempt is made tocategorize specific procedures with a specific type of pro-peller. ThereZore, some discretion will be required on thepart of the analyst to determine whether a given criterion isor is not applicable to this particular problem. An obviousexample: secondary plate stresses can be important in P. mono-coque type blade, but is of no concern in a blade having solidsections. It must be further advised that the structural anal-ysis should not be designated as, "a quick check to be surethat things will hold together." Structural detail requiresan in-depth study and is an equal party.er with aerodynamics,weight and cost in achieving the optimum design.

STRUCTURAL ELEMENTS

In an elementary structural sense, the aircraft propeller'sbasic function is to transmit the aerodynamic thrust and torqueto the airframe. The propeller is also a rotating mass, andits components are subjected to centrifugal effects which areself-balancing within the structure. A sketch of this ele-mentary system is shown in Figure 1. There are three basiccomponents to be considered: (1) the blade, (2) the bladeretention, and (3) the hub. A general discussion of theseelements is presented on the following pages. A more detaileddiscussion of analysis is given in the section Methods ofAnalysis.

It should be noted that some treatments of propeller designlist only two basic components - the blade and the hub - withthe retention considered as part of either or both. Morerecent experience has shown that a better understanding can beachieved by considering the retention separately. It is alsoobvious that there are other components in the total propellerthat require structural analysis. Generally, however, theseare associated with the control system and their analysis isrelatively straightforward.

1T

-Blade CentrifugalN= Propeller Force

Normal ForceBlade

Blade Thrust ForceBlade Torque Force

M M=Yaw

Reaction Thrust

INI r Reaction

SM =PitchingShnatfetrln / Moment

Sotation

Figure 1. Typical Four-Bladed Propeller System.

2

THE PROPELLER BLADE

From a structural viewpoint the propeller blade can be con-sidered as a cantilever beam fixed at the hub or shank end.This beam is subjected to nonuniformly distributed axial andlateral loads. Further, this beam has a varying cross-sectionand a geometric twist along its length. Figure 2 illustratesthis beam configuration. Three basic parameters contribute tothe structural design: material, types of structure and aero-dynamic parameters.

Material

A primary requirement of propeller material is a high strength-to-weight ratio and a high fatigue strength. In the largerhigh performance propeller, steel (SAE 4330, 4340; UTS=140000psi and up) and aluminum alloys (2025S & 7075S) have been ex-tensively used. The major background and experience is withthese materials, and their continued use would be recommendedfor conventional application and where a material evaluationprogram is not practical. Both magnesium and titanium alloyshave been considered in blade design studies. These materialshave not been extensively evaluated by practical application,but they offer a potentially good material from a structuralstandpoint. For future applications, composites appear tooffer the blade designer the best opportunity to achieve theoptimum weight-strength design. Fiber glass reinforced plas-tic blades have been designed and built for prototype appli-cations at a significant weight saving, as compared to theirmetal equivalent. An extensive material evaluation programwould be required to select an optimum material from the com-posites currently available.

Types of Structure

There are three primary types of blade structure: solid, hol-low or monocoque, and spar type. Typical cross sections areillustrated in Figure 3

Solid Section

The solid blade is the easiest to manufacture, and the struc-tural analysis is relatively straightforward. The normalservice damage, nicks, scratches, etc., can be easily removedby grinding or polishing without a significant loss of mate-rial. As with all solid section structures alarge portion ofthe material is not being used to its full potential and there-fore, this type of blade is not an optimum weight design.

3-- _____________

Torque Loadin

2_ Trailing Edge-\ 1 -S~Centrifugal Force

Shanjk ransiti 2 Leading Edge- 3

(a) Planform View

Thrust Loading

Camber Si de

(b) Edgewise View

2

(c) Sections

Figure 2. The Blade Beam.

4

.-, =7__ 1

(a) Solid Section

(b) Hollow Section

(c) Spar Section

Figure 3. Blade-Types.t5

1 I

However, in small diameters (up to 13-14 ft) the weight dif-ferential between the solid and other types of constructioncan be small. In fact, up to about 13 ft, a solid aluminumblade is lighter than a comparable hollow steel design. There-fore, because of its simplicity the solid section type has beenalmost exclusively used in the smaller sized propellers on con-ventional aircraft installations. As the propeller size in-creases, the weight of the solid blade can become prohibitive.Also in special application, VTOL for example, where an opti-mum weight is essential, the solid blade would generally beundesirable.

Hollow Section

The hollow or monocoque design gives the designer considerablefreedom to utilize the material to its design potential andtherefore achieves a good strength-weight balance. The fabri-cation cycle of this type structure is more involved than inthe solid design, and the various techniques (welding or form-ing in metal and filament orientation in composites) can in-fluence material properties and therefore, the structuraldesign. Manufacturing processes also have a significant in-fluence in determining the size of the internal cavity: sharp-hess (stress concentration) at internal fillets, and minimumplate thickness, thereby imposing some limits on the designpotential.

Structural analysis of the hollow type structure is considerablymore complex than in the case of the solid design. The beamsection is a plate or shell structure, and the associated prob-lems of deformation, buckling and plate vibration must be con-sidered by the analyst. In blades having a large plate span,these effects can be controlled by the use of longitudinalribs, or by using a lightweight filler such as foam. Thetransition area, Figure 2, of this blade must also be care-fully designed. An abrupt fairing from the airfoil to theround shank section can induce very high local stresses.

Hollow blades also contain a volume of trapped air which underrotation develops a centrifugal head, subjecting the plates toa high internal pressure in the tip region. For this reason"the tip of the hollow blade is usually vented. Further, ifthe blade material is prone to atmospheric corrosion, the in-ternal surfaces, which are not readily accessible for inspec-tion, must be protected to prevent such corrosion and resultingstress concentration. Service damage becomes a significantfactor in the hollow type blade design. In the thin plate thenormal nicks and scratches can become serious stress-raisersand it is essential to provide extra plate thickness so thatnormal damage can be safely blended out.

6

Spar Section

The spar type of blade is a modification of the monocoque de-sign. It consists of a main spar or structural member whichis a simpl- round or elliptical tube. The airfoil is thenformed by bonding a thin sheath to the spar structure; stain-less steel and composites have been used in prototype designs.Generally, the cavities are filled to provide plate stability.The design and analysis of the spar is essentially the same asin the case of the hollow design. However, the simpler tubeshape is generally easier to fabricate than in the case offorming a tube airfoil section. A significant advantage isthat the primary structure is protected from service damageand a design allowance for the resulting stress concentrationand material removal for repair are not required. Therefore,the only limitations imposed on designing the spar to an opti-mum strength-weight ratio are those that may be imposed bymanufacturing processes. Another unique feature of this typeof design is that when the sheath is damaged beyond acceptablelimits, it can be removed and replaced without loss of themain structural element. The major design criteria with thistype of construction is to provide adequate bond between thespar and the sheath. If a sheath is lost due to bond failure,the aerodynamic and mass balance of the propeller is destroyed.The resulting force unbalance could be catastrophic.

Aerodynamic Parameters

The aerodynamic parameters are, for purposes of this discussion,defined as those blade dimensions or characteristics which areestablished primarily to obtain the specified propeller per-formance. These factors influence the structural design ofthe blade and include airfoil section, activity factor, thick-ness ratio, and pitch distribution.

Airfoil Shape

The selection of the blade cross section or aerodynamicshape is the primary responsibility of the aerodynamicist.However, it does define the structural characteristics,area, moments of inertia, etc. In present-day propellersthe NACA series 16 or series6S sections are commonly used.A representative plot of the thickness vs. the chord ofthese two sections is shown in Figure 4 . It is obviousthat the fuller trailing edge of the 16 series will pro-vide slightly higher structural characteristics. In hol-low sections particularly, small changes in stiffness andmoments of inertia can be obtained without a significantmass increase by using the 16 section rather than the 65.The choice of section can therefore become a compromise.

7-

.- 40

*0 0

r, . G'

0%

--- , -..- ' I/ I,4

- 0

i i Al

Cl9

. i2..LL 4 -,ff-

-I I

., 4-L 00

-- 4.q%-q

8=

Activity Factor, Thickness Ratio and Pitch Distribution

The blade activity factor, thickness ratio and pitch dis-tribution define the basic blade dimensions and rate oftwist along the blade span. While being primarily de-.fined by aerodynamic performance requirements, these samedimensions are fundamental to the strength and stiffnesscharacteristics of the design. It is therefore essentialthat the aerodynamic and structural requirements arefully coordinated early in the design phase. It has beencommon experience to require some compromise of aero-dynamic ideals, particularly with respect to thicknessratio, in order to satisfy structural integrity.

BLADE RETENTION

The blade retention is basically a joint by which the propel-ler blade is attached to the propeller hub. The function oithis joint is to transmit the blade loads to the hub structurewhile permitting the blade freedom of rotation about the pitchchange axis. There are a multitude of retention designs anda detailed discussion of all types would be impractical inthis report. Rather, a basic retention is presented in somedetail and other versions which have been used are shown toillustrate various approaches to the design problem. Theessential components of any retention involve the blade shank,a bearing, a hub attachment and the hub barrel. These elementswill be discussed primarily with respect to structural require-ments. There are,however, numerous details such as provisionfor lubrication, seals, etc., that must be considered in theoverall design. It must also be noted that provision must beincorporated to permit a good preload to the bearing stack toensure an essentially fixed end restraint on the blade.

The so-called flanged shank type of retention, referred to asthe standard retention, has been one of the more popular de-signs, and a typical configuration is shown in Figure 5.This type of retention is a simple design and has the advan-tage of an extensive developmental history. It is easy toassemble and provides for an easily accessible means of apply-ing a solid preload to the bearing stack.

The Blade Shank

The typical flanged blade shank is shown in Figure6 as a free body subjected to the blade forces. The bladeshank can be considered as a fundamental element in the over-all design of the system. With the loads known, it is a rela-tively simple procedure to establish a structurally adequatetube section. However, the analyst has a considerable choice

__ 9

L

.- ' -

Blade Nut

/ +

BladeSh, ank

V -Hub-. m Barrel

BladeBearing +I F

ZPreloadBearing

FigUre 5. Typical SectionStandard Blade Retention.

10

between diameter and wall thickness. It is apparent from Fig-ure 5 that the shank diameter establishes the bearingdiameter and the hub barrel diameter, and is thus a primaryfactor in determining the hub size or a significant portionof the overall propeller weight. It may therefore be desir-able in preliminary design phases to select more than oneshank size and lay out the resulting bearing-hub structure toevaluate the most desirable configuration. For interchange-ability reasons, the propeller industry has established stan-dard shank sizes. These are measured by the outside diameter,diameter m, as in Figure 6, and these standard sizes aregiven in Table I . Generally a standard size is selected butthis is not mandatory and bastard sizes have been used. The4 shank (146.3 mm) diameter has been quite common. The in-side diameter and the flange dimensions are selected to satisfystructured requirements. The high stress area of the shank isthe flange fillet. Standard flange analysis methods are gen-erally used, and extensive testing and photoelastic study haveindicated a stress concentration in the order of 1.1 to 1.2for the usual shank proportions. Typical shank dimensions forsteel shanks are given in Table II.

As an added safety precaution, the fillet area is usually sub-jected to cold work, rolling and short peening to improve thefatigue strength of the material. With this type of desigti itshould be noted that the inner bearing race is made in twohalves. There is therefore, a high bearing stress against theblade shank and the under vibratory loads relative motion be-tween race and shank. The shank bearing surface is thereforesubject to galling,and the associated high stress concentra-tion. To alleviate this condition, that area of the shank isusually short-peened. The short-peened surface is less suscep-tible to galling and further, the small pockets retain lubri-cant, giving further protection.

Blade Bearing

The blade bearing transmits loads from the blade shank to theblade nut while permitting rotation around the blade axis.For this type of retent-on a special angular contact thrustbearing is required. A typical bearing is shown in Ficure 7.Figure 7a shows a- single piece race, while Figure 7b showsa similar design with individual races; both types have beenused. Generally, the single piece race provides a more rigidjoint and minimizes galling on the blade shank and hub barrel.The individual race is,however, easier to install. The reten-tion bearings are made as matched sets and the inner race isthen split diametrically, so that it can be installed over theblade flange. The entire bearing is built up in place on theblade shank.

Blade Loads

Blado

m

I 01

E~ ~ r eaction-__ke - Cnter of Bearing.Coitact

p

Figure 6. Typical Flanged Blade Shank.

12

77-7

TABLE I. SHANK SIZE - D1IMETER

Shank 5ize Nominal DiameterNo. (Millimeters)

2 1153 1304 1405 1656 1757 1908 205

TABLE II. FLANGED SHANK DIMENSIONS,NOMINAL, STEEL BLADES, REFERENCE FIGURE 6

Shank DimensionSize m 1 d p h hl r

2 4.5262 3.236 5.1160 5.25 .625 .644 .140

4 5.5101 4.220 6.1004 6.23 .625 .644 .140

4 5.7713 4.220 6.2353 6.36 .750 .774 .140

5 6.5056 4.546 7.062 7.22 .850 .979 .140

6 6.8880 5.000 7.584 7.688 1.150 .941 .250

8 8.0720 6.401 8.880 8.880 1.165 .834 .250

13

-_ __ __ __ __--_

M eCF :-CF

Ball

Equations (37) and (38) may at first glance seem formidaDbe,but the various integrations are readily evaluated by tubularintegration using standard calculating machines. However, themode shape for the respective mode must first be estimated,and as a first approximation the curves given in Figure 24can be used.

The effect of rotation on the fundamental frequency can beestimated from the Southwell equation,

fR = fo + (c)(n) (39)

where fR = fundamental blade frequency at n, cps

fO = fundamental blade nornrotating frequency, cps

C = constant

n = propeller speed, rps

Values of C are:

first flapping mode C = 1.5 - 2.0

first edgewise mode C = 0.8 - 1.0

The frequency of first torsion mode is for all practical pur-poses unaffected by rotational speed.

PROPELLER LOADS

The ultimate structural design of the aircraft propeller re-quires an accurate evaluation of the various loads generatedby the propeller system. These loads have two basic sources:(1) the aerodynamic components developed by the airfoil sec-tions of the blade, and (2) the components developed by themass of the propeller subjected to various dccelerations.

It is the purpose of this section to present in some detailprocedures by which these various loads can be computed. Inmany instances the complete mathematical development is notpresented. Howevwr, in all cases the initial steps and finalresults are given and the intermediate steps can be easilydeveloped, or in tVae case of the more complex problems, ade-quate references are indicated.

56

PROPELLER LOADS - AERODYNAMIC

The first requirement in establishing the aerodynamic loadsis the definition of the structural design conditions. Basic-ally, two areas of propeller operation must be considered.First, the static takeoff condition, i.e., just prior to brakerelease, the aircraft velocity is zero, the propeller is oper-ating at maximui, horsepower and generating maximum thrust.The magnitudes of the loads and stresses is generally a maxi-mum at this takeoff condition, but they are essentially steadystate. The second -rea of interest is during flight. Thetotal load or stress magnitude iL generally lower than at the3tatic condition, but a large component of the total load isharmonic at predominantly lxP frequency (one load cycle perpropeller revolution). Fatigue, therefore, becomes a designcriterion, and long life fatigue requirements at flight con-ditions generally dictate the propeller design.

As will be shown later, the 1xP harmonic loading, at least onconventional aircraft. is approximately proportional to theso-called airplane Aq factor, where A is the propeller angle-of-attack and q is the airplane dynamic pressure. The criti-cal flight design conditions, therefore, occur at the maximumAq values and this quantity must be evaluated over the flightenvelope. There are ge-nerally two flight conditions that areof concern: (1) Early climb at maximum gross weight, wherethe propeller inflow angle A, is relatively high and q is low,(2) Minimum qioss weight at maximum velocity which results ina small value of A but a maximum value of q. Highly maneuver-able aircraft may require the evaluation of loads at otherhigh load factor conditions defined by the airplane V-n dia-gram. Special aircraft such as propeller driven V/STOL typesmay also require load studies at conditions of propeller oper-ation peculiar to the specific type of application.

Airplane Ag Factor

In order to evaluate the structured design condition for agiven propeller, the airplane Aq factor must be established.This factor is a characteristic of a given aircraft configur-ation, and therefore an accurate evaluation requires that cer-

tain basic aircraft data be made available. The data requiredwill be enumerated later in this section. Referring to Figure25, it is apparent that the airflow enters the propeller discat io.e angle with respect to the direction of flight. Theangle between the local velocity and the thrust line is thepropeller angle of attack or Angle A. Figure 26 shows theconventional propeller wing layout and the angle A is seen tobe:

A = a- 8 + (40)

57

Direction of Thrust

I Plane cf Rotation

Direction of .E-Flight

Figure 25. Airplane in Climbing Attitude.

/Propeller DiscAbsolute Angle of Atteck

= Upward Angle -

Direction of Flight

Figure 26. Components of Inflow Angle.

58

where a = the wing angle of attack - degrees*

8= angle between the propeiltzrthrust line and the zerolift line - degrees*

: upwash angle - dcrees*

The wing angle of attack, a , can be found from elementaxyaerodynamics noting that airplane lift = the airplane weig 1t.

L :W = Pv2 insda

W- q(dCL/d, ) (41)

W aircraft weight -lb

S -wing area - ft 2

q PV 2 = (Vi) 2 /29 - lb/ft 2

Vi airplane indicated velocity -kn

d L- slope of the aircraft lift curve withda

resp2-t to the zero lift line.

The angle 8 is a fixed quantity and is determined by the air-craft geometry.

The angle c is dependent upon the proximity of the propellerto the wing leading edge, proximity to the fuselage and otheraircraft characteristics. Evaluation of this angle obviouslyrequires a detailed evaluation of the airflow characteristicsin the proximity of the propeller disc. It is also obviousthat this angle will vary at different points in the propellerplane. Experience has shown that an averaged value taken atthe 0.70 propeller radius is a good representative value forpurposes of estimating the airplane Aq factor.

Fran the foregoing, it is apparent that the evaluation of theAq factor is a simple calculation, but the following data

* Aq values as generally referred to, i.e., 1200 Aq, are indeg-lb/ft 2 units. When used in load equations the value isconverted to radian units.

59

must be supplied:

1. Velocity range

2. Gross weight and Wing Area (or wing loading, W/S)

3. Slope of the aircraft lift curve dCL/da including theeffects of flaps, etc.

4. Average upwash angle as a function of airplane angleof attack and/or other pertinent parameters.

It has been common practice to evaluate the Aq factor for the -maximum and minimum gross weight over the speed range of theairplane. A typical plot of the computed results will give acurve such as is shown in Figure 27a. Experience has shownthat due to extraneous factors, a value of zero Aq seldomexists, and An fact the computed values are generally in theorder of 150 to 200 Aq too low. Further, the change in signindicates a 1800 phase 2hift in the harmonic load. Since onlythe magnitude of Aq is important, the change in sign is neg-lected. Therefore, ignoring the phase shift and applying theexperience correction factor, the resulting Aq curve for agiven aircraft will have the form shown in Figure 27b.

The Aq curve illustrated is essentially balanced, i.e., themaximum value at low speed is approximately the same as themaximum value at high speed, with a minimum value at cruise.This would be an optimum design which is not usually obtained.The shape of the curve can be adjusted by small changes in theangle a of Equation (40) . A low value of 6 will give higherAq climb. increasing the value of 6 will result in the maxi-mum value of Aq Z; high speed. Therefore, it is usuallynecessary for the airframe and propeller designers to co-ordinate the thrust line orientation in order to establish thebest Aq for the given installation.

Examination of Equations (40) and (41) shows that the Aqcurve, thus established, represents the lxP excitation factorthat exists on the propeller during normal 1-g flight. It istherefore usually referred to as the nominal Aq. The effectsof aircraft maneuvers such as yaw, sideslip, high-g pull outs,etc., must be combined with this nominal. Referring to Figure28, the ai):craft in a yaw attitude will generate a lxP excita-tion factor equal to * q where 4 is the given yaw angle.Ideally, the harmonic forces generated by this factor willhave a 900 phasing with respect to the forces attributable tothe nominal Aq and the magnitude of the resultant is the simplesquare root of the sum of the squares. Since the forces areproportional to ]Lq and 4,'q it is easily shown that for the caseof yaw, the magnitude of the effective Aq is:

60

SFlapsisd DownFlpRercd

V440d Flaps Retrace

-H (aaH

W500 I P3

S 0 Indicated AirspeedNS-500- I Max. Gross Wt. 1

-1000-- Min. Gross Wt.

n(a) Calculated Da Aq

61500

C44

S. 10a0

() 500ete - A

Figure 27. Nominal Aq Diagram. '61I

Figure 28. Airplane in Yaw.,

62

N N . ..1-7I 'i L , .|. .. I L a

Aq eff~ (JA4m)+ (#~q

q A 2 + ~2(42)

It should be noted that the nominal Aq is not in a pure pitchdirection, nor is the value * pure yaw, and the phasing be-tween the harmonic force vectors is not 900. Equation (42)is therefore an approximation, but one which experience hasproven to be adequate for design purposes.

For illustrative purposes, a simple development of the forcefactor resulting from the combinations of the nominal Az atsome assumed phase with yaw is shown in Figure 29. The cor-rect phasing can only be obtained from extensive flow studies,either analytical or experimental, and such studies have notbeen carried out extensively in the past.

The Aq resulting from high -g pull outs or similar maneuverscan be computed by the use of Equations (40) and.(41);Equation (41) is modified to

nW (43)S q(dD1 /d.

where n applicable load factor

Special conditions can exist on unconventional aircraft. Ona tilting propeller type V/STOL, for example, during thetransition of the propeller from hover attitude to conven-tional flight position, the flow is quite complex and a simpleAq evaluation such as given previously has not proven too re-liable. This special type of propeller for VTOL airplaneswill be covered in more detail in the next section.

First-Order Propeller Loads

One of the major design parameters on the present-day high-pur-formance aircraft propellers is the magnitude of the firstorder or commonly called IxP aerodynamic loading. This loadis produced when the airstream enters the propeller disc atsome angle with respect to the thrust line, in effect a pro-peller angle of attack. Under such a condition a harmonicaerodynamic force is generated by the blades with a frequencyequal to the propeller rotational speed. Hence, the namefirst-order-propeller or lxP.

63

Maim&

V a

LW LW

Left Yaw Right Yaw

LA = Blade Harmonic Force Vector Due To Aq

Lp = Blade Harmonic Force Vector Due To 4 q

LA= KAqI L* = Kfq

L1 = LA sin (flt+8); L2 = L* sinut

0= Assumed Phase Angle

Resultant (Left Yaw)

LR = (LA2 + L*2 + 2LAL cos

L-Aq- q*

Aqeff = (A2

+ *2 + 2A 1 cos )q

Similarly For Right Yaw:

Aqeff = (A2 +q2 - 2A cos8 )q

Figure 29. Resoltant Aq.

64

Referring to Figure 30 and considering a conventional air-craft conigucation where the angle A is small, the followingassumvti-n can be made:

1. The velocity components are uniform over the disc.

2. Cos A = 1.0; Sin A = A (44)

3. 0= o; W=Wo

Then, the basic lift on a section at radius r is given as:

Lr = PWo 2 CL b Ar

where CL = Section lift coef

b = Section chord - ft

Wo = Resultant velocity - ft/sec

P= Mass density of air - slugs/ft3

This basic lift will change with changes in velocity andchanges in lift coefficient. The change in lift can be writ-ten

ALr = p2 W.AWo+?+WO 2 -C- Aa bAr A45)

From Figure 30 and the foregoing assumptions,

AW VA cos~o sinat

Aa =VA sin~o sinntWo

Substituting these values in (45) the maximum value of theharmonic lift is

___k I b (46)4 = Aq 2 CL cot + da b (46)

Equation (46)shows that the periodic IxP force is proportionalto the excitation factor Aq.

The periodic lift can be evaluated for each station along theblade for any given flight condition. Aq must be known frompage 61, and appropriate values of the CL, 0 and dCL/da mustbe established. Ideally these data should be determined by

65

PropellerDisc

V VCos

A-- \ II / sin AV

Section at Radius rf y Ji L '

~~fz

V cos A

2wn rV sin A sin nt

Figure 30. Development of Periodic Forces.

66

aerodynamic strip analysis.

The development of Equation (46) is obviously based on aquasi-steady flow assuming a rigid blade. The periodic forcewill produce corresponding periodic blade deflection in bothbending and torsion. These will in turn influence both AW and

*a , Equation (45). Considering all of these effects canbecome quite complex and there are several classic studies ofthe forces generated by oscillating airfoils, for example,Theodorsen, Reference 3 . Also the helicopter industry hasstudied the problem extensively with respect to rotor design.

Fortunately, however, propeller experience to date has indi-cated that the effects of blade deflection on the lxP harmonicload are negligble with the exception of the torsion.

Referring to Figure 31, the torque acting on the blade elementat radius r is

4Q A LL(a) + hydmr + I-a - ACF dy b (47)

dr

where y = flexural acceleration dy2 /dt 2 in./sec

2

a = torsional acceleration d2 a/dt 2 rad/sec

2

dm= section mass

I = section polar moment of inertia - in.lb/sec2

A CF = elemental centrifugal force - lb

The lxP forced vibration on a propeller blade is well removedfrom resonance, and the inertia terms ydm and I a are rela-tively small. Further, in the more highly flexible tipregions the blade deflection is such that the centrifugal com-ponent is approximately equal to the lift components

ACF d A A L

Therefore, neglecting the inertia terms and using the equalityexpressed abovethe effective torque on the section isessentially a couple and can be expressed as

r xcp (48)

67

/r

LiCF

/Y

Blade Deflection

AL

SCenter of

Aerodynamic c g.

Section c.g. dC

Figure 31. Blade Aero-Torsionr

68

where (AQ/ Ar)ixp = blade torcque due to the lxPharmonic lift in.-lb/in.

Xcp = distance from the aero-dynamic center to thesection mass center, in.

The location of the center of pressure is often taken at the chord, but a more accurate location can be obtained fromFigures 70 to 73 in the aerodynamics section, Volume I.

The blade torque at any station r is thenR

-fIAilx dr (49)

and the corresponding deflection is

(ar)IxP - fi(Qr)lxP/CTldr (53)

0Then the change in harmcnic lift is

I-l1 w2 dCL(=AL 2 1 2-- (ixP) (51)Ar2 da X

where (AL/ Ar)0 = change in harmonic lift due toblade twist - lb/in.

lxP = angular deflection of the bladesection at radius r due to har-monic lift - rad

CT = blade section torsional stiffnessconstant, see Equation (6),in.-lb/rad/in.

W = the resultant velocity, ft/secand is obtained from aerodynamicstrip analysis or it can be esti-mated from the relation

W = V/sin0 (see Figure 30)

R = tip radius

The tntal harmonic lift on the given section ic therefore

( 6 e q . (46 1 ) (5 2 )69

It is usually necessary to take the results of (52) andrepeat the calculations of Equations (49) and (51) and thusconverge on a final value. Experience htLs shown that in themore recent blade developments, this torsional influence in-creases the basic lift of Equation (46) in the order of 15%in the more flexible outboard sections of the blade.

The final value of harmonic lift as computed by Equation (52)at the various radii along the blade can be resolved intothrust and torque components.

Thrust force = fY = r cos )Torque force = f= ALI in0J

These components can then be integrated to obtain the bladeshear and moments distribution in the thrust and torquewisedirections.

R

(F) = f cos dr thrust force

(My) =ffi L cosodrdr thrustwise moment

R x b(54)

(Fz) a =LI sin.dr torque force

(MZ) Rrf L) sin0 drdr torquewisc m..ent(MzAr t

These aerodynamic forces and moments represent the forcingfunction on a blade spring mass system. In addition, inertiaand centrifugal effects contribute to the total blade load.These additional influences are considered on pages 75 to 80.It should be noted at this point, however, that on a rotatingblade subjected to a simple harmonic vibration at lxP frequency,the effects of centrifugal and inertia loading cancel eachother out at the center of rotation, i.e., the blade zeroradius. Therefore, if Equations (54) are integrated to thezero radius, the resulting aerodynamic loads are essentiallythe total load at station zero due to the lxP excitation.

Remembering that =quations (54) are actually hkarmonic forces,the force or moment on a given blade with respect to time is

70

Ft = F sin nt (55)

Mt = M sinrRt I

where the subscript t denotes the instantaneous value of agiven force or moment.

Assuming Equation (55) represents the force or moment on agiven blade, designated blade No. 1, then the force on thenext adjacent blades at the same instant are

Ft (Blade 2) = F sin at + )

Ft (Blade 3) = F sin (at + 2) 5

etc.

where )L= spacing between blades i.e., 1200 on a 3-way,900 on a 4-way

if the instantaneous forces, Equations (55) and (56), areeva!iated on each blade for the zero radius, the summation ofthese individual blade forces will give the shaft loads pro-duced by the lxP harmonic blade load. The results give thefollowing for a propeller having more than two blades. Theseforces are illustrated .ai Figure 14.

.(Fy}O = Thrust = 0

E(Fz)o = Nonharmonic, i.e., steady-state radialnormal force

B(Fz)o (57)2

where B = No. blades(Y0o = Nonharmonic steady-state yawing moment, My.

B=y)o _ (58)=My- 2

(Mz)o = 0 = Shaft torque Qs

On a two-way propeller, N = 2(Fz)O, M = 2(My)O and the shaftforces and moment are harmonic with a frequency of twice pro-peller speed.

71

Equations (57) and (58) give 4.he shaft forces for a purepitch attitude, if the attitude is pure yaw, then

(Fz)o = Side force, S - (Fz)o B2 0(59)

(My)o = Pitching moment, M -(MY) B_2

In those cases where the effective Aq is a combination ofpitch and yaw, the resultant can be found by vector additionsimilar to Equation (42) . The direction of the shaft forceswill depend upon the airplane attitude, pitch-up, pitch-down,left yaw or right yaw,and the direction of propeller rotation.A right-hand-rule and other schemes can be expressed to givethese shaft force directions; however, once a basic under-standing of the generations of these forces is attained, thedirections are quite obvious.

In large propellers, and in propellers having high excitationfactors, Aq, the foregoing shaft forces can be a significantfactor in the design of the propeller shaft.

Throughout the previous development, only the lift vector hasbeen considered. Obviously there will be a cyclic drag vari-ation, and an expression similar to Equation (46) can be devel-oped expressing the lxP harmonic drag vector. However, experi-ence has shown that in the case of the conventional aircraft,the cyclic drag is relatively small with respect to the liftcomponent, and that it can be safely neglected for structuraldesign purposes.

The previous devalopment applies specifically to the propellerinstallations where the propeller's angle of attack is small,and correlation of calculated and flight test data on con-ventional aircraft has been very good. Such correlations indi-cate a maximum error in the order of 7%.

When the angle A exceeds the 10-15 degree range, the accuracyof Equation (46) decreases rapidly and for A values above 200the use of that equation would be unreliable. Several studieshave been made to develop a universal lxP loading equation.A preliminary approach is given in detail in Reference 4.In that development the lxP lift corresponding to Equation(46) is given as

72

AL sin 2 Atb CL cot0(2 V0 coso)+ ba]

+ 3. ab - cod~ A sin0] sin at

- qba sin A (cot% - 2.cosAt (60)

where the terms not previously defined are (see Figure 30)

Vo, Vi = components of induced velocity definedin Reference 4 - ft/sec

W resultant velocity - ft/sec

V = free stream velocity - ft/sec

e = propeller section angle of attack - rad

The sin at and the cos at terms of the above equation can beconsidered separately. Considering the aircreft attitude tobe pure pitch and integrating Equation (60) to the zero radius,it is easily shown that the following shaft force and momentsare produced (See Figure 14):

Force component Force Moment

sin at Normal Yawcos at Side Pitch

Loads computed on the basis of Equation (60) have not givensatisfactory results. The major problem appears to be in thedetermination of correct values for the induced velocity com-ponents, Vo and Vi. 1'ne more classical type studies have alsobeen initiated where the flow is expressed as functions of pro-peller radius, and rotational angle, forces computed and thevarious harmonics obtained by Fourier Analysis. All of thesestudies have shown that the induced velocity distribution dur-ing high angle-of-attack operation of a propeller has a largeeffect on the loading. A major problem in propeller analysishas been the analytical and/or experimental evaluation of thisinflow velocity distribution.

The lack of reliable theoretical loading, makes it necessaryfor the designer to resort to a semiempirical approach toestablish propeller LxP harmonic loads for the high angle-of-attack conditions.

73ir-

Blade load distributions and resulting shaft loads can be com-puted on the basis of Equations (57), (58) and (60) . Thecorresponding shaft loads are also obtained from data such asgiven in Reference 4 or other model test data. The calcu-lated blade load distributions can then be proportioned by theratio of test loads to calculated loads. This technique hasbeen proven very satisfactory in correlations calculated andflight test data on prototype V/STOL propellers,

SHiner Order LoadsThe relatively simple development of Reference 5 and the morecomplex flow studies show that in addition to lxP the aero-dynamic load on the propell'r blade contains harmonic ccapon-ents at frequencies cirresponding several multiples of therotational speed, i.e., 2xP 3xP, 4xP etc., and the existenceof such a loading has been verified in flight test. Indica-tions are, at least on conventional aircraft, that these higherorder components are small and have generally been neglectedin evaluating design loads, but it has been common practice tomake allowance for such extraneous factors when establishing adesired factor of safety.

The higher harmonics can become significant if the propelleroperating speed is such that the exciting 2xP, 3xP, etc.,frequency is in the proximity of a natural blade frequency.Further, the resulting propeller shaft forces due to the higherfrequency loads are generally harmonic and a correspondingvibration will be transmitted to the propeller supportingstructure. A more detailed discussion of these aspects willbe: covered in a latar section. However, it might be noted atthis point, that the imposed shaft forces will be of the fol-lowing form:

T = A1 cos rdlt n = KB

T =0 n KB

N = A2 cos (n + 1) at n + 1 = KB

= A2 cos (n - i)St n - = KB

N = 0 n 1 / KB (61)

* = A3 cos (n + 1) at n + 1 = KB

M = A4 cos (n - 1) mt n - I = KB

n +l KB

74....-...

where T, N, M are harmonic components of the shaftthrust, radial force and moment.

Al, A2 , A3 = Represent tht max. values ofthrust, normal force,moment

n = ordar of vibration

f = propeller rotational speed, radians

K integers, 0, 1, 2, 3, etc.

B =the number of blades in the propeller.

There have been some flight test data on V/STOL aircraft withthe propeller operating at high angles-of-attack, which indi-cated a significantly high 2xP harmonic load. The evidence,however, has not been conclusive and the preliminary attemptsto define the airflow through the disc and the subsequent har-monic analysis of the resulting loads, have not shown com-parable high values of excitation due to higher orders. Thesefactors indicate a need for better definition of the propellerinflow velocity distribution, particularly at the higher pro-peller angles of attack.

Steady-State Aerodynamic Loads

The steady-state aerodynamic loads are those force componentsassociated with the nominal velocity of the blade airfoilsection, i.e.,. the resultant of the rotation speed and the for-ward velocity of the aircraft. This in effect represents themean velocity through the disc, and the resulting loads are themean loads upon which the lxP and other harmonics are super-imposed. Further, it is the steady-state loads which providethe net thrust to satisfy the velocity requirements of a givenflight condition and alsc provide the reaction for the horse-power input to the propeller.

Referring to Figure 32, the basic lift and drag forces on ablade of elemental length, Ar, is

L =pW2 CL b dr (} (62)D = pw2 CD b dr

75

TorqueRotation fz Force

,fY Thrust

- Force 1

AL

SCenter of Twist

V

f2 ,,n r f

Section at r

V = Free-Stream Velocity G = Blade Angle

V2 = Induced Velocity 6 = Wind Angle

2 nr = Rotational Velocity a = Angle of Attack

W = Resultant Velocity

Figure 32. Nominal Blade Forces.

76

The thrust and torque forces are

fy = AL_ cosj6 - sinO

Ar Ar

fz = A D COSO + AL sino (63)&-r -- r

where fy = thrust loading ib/in.

fz = torque loading ib/in.

The resulting blade shears forces and moments can be obtained

by integrating Equations

Thrust force = (FY)r =jRf &-

Thrust bending = ('M)r =I f fydrdr (64)r r (64) _Torque force = (Fz)r= fz dr

Torque bending = (Mz)r = f fzdrdr

It should be obvious that the total propeller thrust is equalto the thrust per blade times the number of blades, and thatthe torque moment per blade times the number of blades must becompatible with the input horsepower. That is,

Bjfy dr = Propeller thrust

r f r 63025 (input horsepower) (65)B :Propeller

rpm

The accurate evaluation of the section thrust and torque,Equation (63), for a given condition of propeller operation,requires an aerodynamic strip analysis. The results of such astudy are often provided in terms of an incremental thrustcoefficient ACT, and an incremental torque coefficient ACQ.The corresponding forces can be evaluated from the followingexpression:

77

Thrust: = (fy)rPn2 D4

(66)

Torque: CQ - z rpn2z D5=r fd

'With reference to Figure 32, it is apparent that aerodynamicforces will produce a twilting of the blade section. Thiswill in turn change the angle of attack, a , thus increasingthe aerodynamic forces in a manner similar to that discussedpreviously, Equations (49) and (52). However, in the case ofthe steady-state loadi na it must be remembered that the aero-dynamic analysis by which the loads are determined is performedto satisfy given conditions of operation. Therefore, the re-quired blade angle distributions as established by aerodynamicanalyses must represent the final torsionally deflected posi-tion or the so-called live blade angle distribution.

In order to allow for the torsional deflection, the bladetwist is subtracted from the aerodynamic pitch distribution,and the resulting angle is specific for manufacturing.

$mfg = Paero - 0twist (67)

The two-blade angle distribution is illustrated in Figure 33.By this technique,'the blade will under load;deflect torsion-ally to the angle necessary to produce the required aerodynamicloads. However, it should be obvious that this process canonly be optimized for one flight condition. Therefore, onlyone condition can be iilected,such as takeoff, cruise, etc.,as the most desirable for optimdm performance of the giveninstallation. The procedures for calculating the blade twistunder steady-state loads are given in detail in the section onMethods of Analysis.

Propeller Mass Loads

The rotating mass of the propeller generates inertia forceswhich must be evaluated in establishing the total force systemon the propeller. The development of these forces is presentedin some detail in the following sections.

Blade Centrifugal Force

Consider a blade element as shown at some radius r,Figure 34;the centrifugal force on that element is

78

i ~80"

70-

60]

:50- Live or" 5O Aerodynamic

Angle

40- Blade Angle ToiolTorsional

DeflectionUnder Load

301! 10 .2 .4 .6 .8 1.0

Blade Proportionate Radius, I

Figure 33. Typical Blade Pitch Distribution.

79

I

ACF = dmrl &2 (68)

where dm = element mass considered concentrated at thec.g., lb-sec 2 /in.

rI = radius from the center of rotation to the

section centroid, in.

S= rotational velocity, rad/sec

"w= 2 rN/60, N = prop rpm

in propeller blades, the angle x is very small, and rI can

be taken equal to r.

Also, dm = a Ar dxg

S= material density, lb/in.3

(Ar) = st:ction area @ radius r, in.2

dr = elemental length, in.

Substituting in Equation (68)

(ACF)r = 32 Arrdi ( e~r = (a~r dr

and the total centrifugal force at any giver radius r1

R

CFrl = Ardr (69)ri

Noting that w= 2 rN/60 where N is the propeller rpm, andg equals 386 in/sec2 , Equation (69) can be written

CFrl = 28.4 (dr (70)

Evaluating the above integral to the blade butt, rI rb, givesthe total blade centrifugal force that must be reacted by the

-hub-blade retention.

80

Blade Centrifugal Restorinq Moments

Since the propeller blade is essentially a cantilever beam, itdeflects in the direction of the resultant load. The deflec-tion therefore has components in both the thrust, y, andtorque, z, directions, and the deflected blade axis ib illus-trated in Figure 34. This deflection provides amoment arm for the centrifugal force and the sense of themoment is such as to tend to return the axis to its undeflec-ted position.

Considering the centrifugal force of the elemental mass dm at

radius r, the angle X , is small and therefore

r = rI, cosX= 1.0, sin X =

&CF 1 = ACF cos X = ACFACF2 = ACF sin A = ACF Z-r

r

In the thrust direction, the moment at rI due to the forceSCF is

(AMYCF)rl = ACF(yr -yr) (71)

and the total moment due to the centrifugel Force on all bladeelements outboard of rI is

(MyCF)rl f= (Yr - Yrl) dCF (72)

Integrating the above by parts, it can be shown that

R

AM~YCF)rl f ' (CF) r dr (73)r rr

where (CF) is the total centrifugal force at any station r

(d y/dr)r = the slope of the thrustwise deflectioncurve at any station r.

In the torquewise direction,Figure 34,

(6MCFz)rl ACF(zr - Zr 1 ) - CF 2 (r - rl) (74)

81 -

Integrating the total centrifugal moment at rl is

I R RR

(MC11z)rl (dz/dr)r (CF)r dr - -- 1 z rrdr2

r~ ~ -rr"

where z and (az/dr) are the magnitude and slope ofthe torquewise deflection curve.

The first term is similar to the thrust bending component, andthe second term is in the same form as Equation (69). It will'be noted that in Figure 34 the force A cr passes through thecenter of rotation. Therefore,

r R 2rRrR(MCFz)O = 0 or, (dz/ciz) (CF)dr J- w zA dtdr

0 OrIt should be apparent that these centrifugal momft't cannot beevaluated unless the blade deflection is known. They cannont,,therefore, be readily computed as an isolated blade load com-ponent. Techniques for evaluating these components will bepresented in the Methods of Analysis section.

Blade Tilt

The principle involved above provides a potential means bywhich the bending moments on the blade can be controlled to adesired magnitude. As previously shown the moment caused bythe centrifugal force acting through a deflection of the bladeaxis produces a moment tending to restore the blade to its un-deflected position. In other words, a moment which subtractsfrom the aerodynamic moments, and is particularly effectivo inthe thrust-wise direction. Therefore, if an artificial deflec-tion or tilt is manufactured into the blade, the result can bea significant reduction in the net moment as felt by the blade-beam. This elementary concept is illustrated in Figure 35a,which shows the blade axis tilted forward, i.e., in the thrustdirection by the anglee . Usually a is a very small angle andtherefore, r, = r coso = r.

The slope of the tilted blade axis is

dy/dr tan 8 = constant

82

z centroid

H dr Blade

axis~~ axisatis

bCrr

D)eflected bladeaxis toruaw stwse d

CC

axisz toor 1sed

rI

Figure 34. Blade centrifugal Force.

83

y

~axi

CF

- -0Tilt angle Normal

rm axis

rl

(a)

YA

CF

0 (Normal atisrt itch change

A

r

(b)y

( r .- rt ) t a n 9

9

'Pitch change axis

Figure 35. Blade Tilt.

84

Blade ref. axis

Centroid

y (a) N4r+3

CF; (r+1)

(r

__________(r + 1)

(e)

Figure 35. Continued.

85

Therefore, from Equation (73)

(Mtilt)rl = tan a f CF dr (7C)r

This elementary consideration would be very desirable if thetilt is pure thrust and the pitch change axis and the bladeaxis are coincident. This condition can be achieved by in-corporating the tilt angle in the hub barrel. This, however,presents some difficulty in manufacturing and can influence"the design of the pitch change system. It is therefore morecommon to introduce tilt into tho blade by kinking the shankat some radius rt just outboard of the hub barrel. This con-figuration is illustrated in Figure 35b.

For tilt in a pure thrust d~rection,Equation (76) is valid,but the maximum value of the tilt moment will be

Mtiltmax. = tan0 f CF dr (77)rt

and the value is constant from rt inboard.

Figure 35c shows the geometric relationships of a blade sec-tion at radius r. It should be quite apparent that for thistype of tilt there is only one blade angle setting at whichthe tilted blade axis is in the pure pitch direction, and itis designated P I in Figure 35c. At this blade angle setting,Equation (77) is valid.

Now assuming that the flight condition changes and a new bladeangle setting I82 is required, the section rotates about thepitch change axis by A$ , where AL = R1 - 82. The sectionis then tilted in both the thrust and torque direction.

Y2 = (r - rt) tanecosap

z2 = (r - rt) tanG sin dp

and

dy/dt = tans cosa

dz/dt =-tane sinA

Substituting in Equations (73) and (75), the tilt momentsbecome

86

tilt R(My) tilt = tang cosAf CF dr

owz tilt =-) an8 sinA CF drF

g tanesin f f({r - rt)A drdr](78)

where the (-) sign indicates that for the case shown, i.e., aewdecrea; ,in pitch from the optimun, position 8.l' the torquewisetilt moment will add to the torque aerodynamic moment.

It is also to be noted that in a condition of reverse thrust,the moments due to forward tilt will also add to the aero-dynamic moments. Obviously, therefore, with tilt incorporatedin the blade shank, the analyst must c6nriider the effects ovcrthe entire pitch range of the propeller in order to achievethe most practical benefit.

The foregoing development has assumed that the blade axis whichis a manufacturing reference and also defines the tilt, isessentially coincident with the centroidal axis. For the con-ventional propeller blade this is approximately true, and theprevious equations can be used without significant error.However, there have been designs where this is not the case,as illustrated in Figure 35d . This is essentially thesame section as shown in Figure 35c, but with the centroiddisplaced from the reference axis. The radius (a) to th'-centroid and its angle 0 with respect to the tilt plane canbe determined from section layouts or

a = (Xcg cos 1)2+ [+ r - rl) tana - Xcg sing1]2)

sin--1 Xcg cos8 1 /a

where the distance Xcg must be taken from section designdetails.

The tilted position of the centroidal axis now becomes

y= a cos (n - 0

z= a sin ( p- 0)if the blade geometry is such that the displacements as com-puted at the various blade radii form a reasonably straightline, Equations 78a and 78b can be easily modified to

87

incorporate the average centroidal slope. It is possible how-ever for the centroidal displacements to form a very irregularcurve when plotted against the blade radius as illuscrated inFigure 35e. In such a case the displacements are notsmooth curves and may have significant discontinuities, there-fore the foregoing equations could be quite inaccurate. Forsuch cases it is desirable to compute the tilt moments by thebasic Equations (71) and (72). The general equation forthe tilt moment at any radius r' is then

R

~~Y~r'tilt r - Yr')(ACF)r R(Mz)rtilt (zr - Zr)(t&CF)r E (r - r')(ACF2)r

r r

where ACF = B/g w2 (Ar)r A r (79)

ACF 1 = ACF z/r = 3/g w2 (Ar)zAr

4r - the incremental blade length betweenselected 2adii.

Obviously, the more radii used to evaluate