Table of Contents

Acknowledgements Notation 1. Introduction 1-1 The Helicopter 1-1.1 The Helicopter Rotor 1-1.2 Helicopter Configuration 1-1.3 Helicopter Operation 1-2 History 1-2.1 Helicopter Development 1-2.2 Literature 1-3 Notation 1-3.1 Dimensions 1-3.2 Physical Description of the Blade 1-3.3 Blade Aerodynamics 1-3.4 Blade Motion 1-3.5 Rotor Angle of Attack and Velocity 1-3.6 Rotor Forces and Power 1-3.7 Rotor Disk Planes 1-3.8 NACA Notation 2. Vertical Flight I 2-1 Momentum Theory 2-1.1 Actuator Disk 2-1.2 Momentum Theory in Hover 2-1.3 Momentum Theory in Climb 2-1.4 Hover Power Losses 2-2 Figure of Merit 2-3 Extended Momentum Theory 2-3.1 Rotor in Hover or Climb 2-3.2 Swirl in the Wake 2-3.3 Swirl Due to Profile Torque 2-4 Blade Element Theory 2-4.1 History of the Development of Blade Element Theory 2-4.2 Blade Element Theory for Vertical Flight 2-4.2.1 Rotor Thrust 2-4.2.2 Induced Velocity 2-4.2.3 Power or Torque 2-5 Combined Blade Element and Momentum Theory 2-6 Hover Performance 2-6.1 Tip Losses 2-6.2 Induced Power Due to Nonuniform Inflow and Tip Losses 2-6.3 Root Cutout 2-6.4 Blade Mean Lift Coefficient 2-6.5 Equivalent Solidity 2-6.6 The Ideal Rotor 2-6.7 The Optimum Hovering Rotor 2-6.8 Effect of Twist and Taper 2-6.9 Examples of Hover Polars 2-6.10 "Disk Loading, Span Loading, and Circulation" 2-7 Vortex Theory 2-7.1 Vortex Representation of the Rotor and Its Wake 2-7.2 Actuator Disk Vortex Theory 2-7.3 Finite Number of Blades 2-7.3.1 Wake Structure for Optimum Rotor 2-7.3.2 Prandtl's Tip Loading Solution 2-7.3.3 Goldstein's Propeller Analysis 2-7.3.4 Applications to Low Inflow Rotors 2-7.4 Nonuniform Inflow (Numerical Vortex Theory) 2-7.5 Literature 2-8 Literature 3. Vertical Flight II 3-1 Induced Power in Vertical Flight 3-1.1 Momentum Theory for Vertical Flight 3-1.2 Flow States of the Rotor in Axial Flight 3-1.2.1 Normal Working State 3-1.2.2 Vortex Ring State 3-1.2.3 Turbulent Wake State 3-1.2.4 Windmill Brake State 3-1.3 Induced Velocity Curve 3-1.3.1 Hover Performance 3-1.3.2 Autorotation 3-1.3.3 Vortex Ring State 3-1.4 Literature 3-2 Autorotation in Vertical Descent 3-3 Climb in Vertical Flight 3-4 Vertical Drag 3-5 Twin Rotor Interference in Hover 3-6 Ground Effect 4. Forward Flight I 4-1 Momentum Theory in Forward Flight 4-1.1 Rotor Induced Power 4-1.2 "Climb, Descent, and Autorotation in Forward Flight" 4-1.3 Tip Loss Factor 4-2 Vortex Theory in Forward Flight 4-2.1 Classical Vortex Theory Results 4-2.2 Induced Velocity Variation in Forward Flight 4-2.3 Literature 4-3 Twin Rotor Interference in Forward Flight 4-4 Ground Effect in Forward Flight 5. Forward Flight II 5-1 The Helicopter Rotor in Forward Flight 5-2 Aerodynamics of Forward Flight 5-3 Rotor Aerodynamic Forces 5-4 Power in Forward Flight 5-5 Rotor Flapping Motion 5-6 Examples of Performance and Flapping in Forward Flight 5-7 Review of Assumptions 5-8 Tip Loss and Root Cutout 5-9 Blade Weight Moment 5-10 Linear Inflow Variation 5-11 Higher Harmonic Flapping Motion 5-12 Profile Power and Radial Flow 5-13 Flap Motion with a Hinge Spring 5-14 Flap Hinge Offset 5-15 Hingeless Rotor 5-16 Gimballed or Teetering Rotor 5-17 Pitch-Flap Coupling 5-18 "Helicopter Force, Moment, and Power Equilibrium" 5-19 Lag Motion 5-20 Reverse Flow 5-21 Compressibility 5-22 Tail Rotor 5-23 Numerical Solutions 5-24 Literature 6. Performance 6-1 Hover Performance 6-1.1 Power Required in Hover and Vertical Flight 6-1.2 Climb and Descent 6-1.3 Power Available 6-2 Forward Flight Performance 6-2.1 Power Required in Forward Flight 6-2.2 Climb and Descent in Forward Flight 6-2.3 D/L Formulation 6-2.4 Rotor Lift and Drag 6-2.5 P/T Formulation 6-3 Helicopter Performance Factors 6-3.1 Hover Performance 6-3.2 Minimum Power Loading in Hover 6-3.3 Power Required in Level Flight 6-3.4 Climb and Descent 6-3.5 Maximum Speed 6-3.6 Maximum Altitude 6-3.7 Range and Endurance 6-4 Other Performance Problems 6-4.1 Power Specified (Autogyro) 6-4.2 Shaft Angle Specified (Tail Rotor) 6-5 Improved Performance Calculations 6-6 Literature 7. Design 7-1 Rotor Types 7-2 Helicopter Types 7-3 Preliminary Design 7-4 Helicopter Speed Limitations 7-5 Autorotational Landings after Power Failure 7-6 Helicopter Drag 7-7 Rotor Blade Airfoil Selection 7-8 Rotor Blade Profile Drag 7-9 Literature 8. Mathematics of Rotating Systems 8-1 Fourier Series 8-2 Sum of Harmonics 8-3 Harmonic Analysis 8-4 Fourier Coordinate Transformation 8-4.1 Transformation of the Degrees of Freedom 8-4.2 Conversion of the Equations of Motion 8-5 Eigenvalues and Eigenvectors of the Rotor motion 8-6 "Analysis of Linear, Periodic Systems" 8-6.1 "Linear, Constant Coefficient Equations" 8-6.2 "Linear, Periodic Coefficient Equations" 9. Rotary Wing Dynamics I 9-1 Sturm-Liouville Theory 9-2 Out-of-Plane Motion 9-2.1 Rigid Flapping 9-2.2 Out-of-Plane Bending 9-2.3 Nonrotating Frame 9-2.4 Bending Moments 9-3 In-plane Motion 9-3.1 Rigid Flap and Lag 9-3.2 In-Plane Bending 9-3.3 In-Plane and Out-of-Plane Bending 9-4 Torsional Motion 9-4.1 Rigid Pitch and Flap 9-4.2 Structural Pitch-Flap and Pitch-Lag Coupling 9-4.3 Torsion and Out-of-Plane Bending 9-4.4 Nonrotating Frame 9-5 Hub Reactions 9-5.1 Rotating Loads 9-5.2 Nonrotating Loads 9-6 Shaft Motion 9-7 Coupled Flap-Lag Torsion Motion 9-8 Rotor Blade Bending Modes 9-8.1 Engineering Beam Theory for a Twisted Blade &nbs 10-8.2 Finite-Length Vortex Line Element 10-8.3 Rectangular Vortex Sheet 11. Rotary Wing Aerodynamics II 11-1 Section Aerodynamics 11-2 Flap Motion 11-3 Flap and Lag Motion 11-4 Nonrotating Frame 11-5 Hub Reactions 11-5.1 Rotating Frame 11-5.2 Nonrotating Frame 11-6 Shaft Motion 11-7 Summary 11-8 Pitch and Flap Motion 12. Rotary Wing Dynamics II 12-1 Flapping Dynamics 12-1.1 Rotating Frame 12-1.1.1 Hover Roots 12-1.1.2 Forward Flight Roots 12-1.1.3 Hover Transfer Function 12-1.2 Nonrotating Frame 12-1.2.1 HoverRoots and Modes 12-1.2.2 Hover Transfer Functions 12-1.3 Low Frequency Response 12-1.4 Hub Reactions 12-1.5 Two-Bladed Rotor 12-1.6 Literature 12-2 Flutter 12-2.1 Pitch-Flap Equations 12-2.2 Divergence Instability 12-2.3 Flutter Instability 12-2.4 Other Factors Influencing Pitch-Flap Stability 12-2.4.1 Shed Wake Influence 12-2.4.2 Wake-Excited Flutter 12-2.4.3 Influence of Forward Flight 12-2.4.4 Coupled Blades 12-2.4.5 Additional Degrees of Freedom 12-2.5 Literature 12-3 Flap-Lag Dynamics 12-3.1 Flap-Lag Equations 12-3.2 Articulated Rotors 12-3.3 Hingeless Rotors 12-3.4 Improved Analytical Models 12-3.5 Literature 12-4 Ground Resonance 12-4.1 Ground Resonance Equations 12-4.2 No-Damping Case 12-4.3 Damping Required for Ground Resonance Stability 12-4.4 Two-Bladed Rotor 12-4.5 Literature 12-5 Vibration and Loads 12-5.1 Vibration 12-5.2 Loads 12-5.3 Calculation of Vibration and Loads 12-5.4 Blade Frequencies 12-5.5 Literature 13. Rotary Wing Aerodynamics III 13-1 Rotor Vortex Wake 13-2 Nonuniform Inflow 13-3 Wake Geometry 13-4 Vortex-Induced Loads 13-5 Vortices and Wakes 13-6 Lifting-Surface Theory 13-7 Boundary Layers 14 Helicopter Aeroelasticity 14-1 Aeroelastic Analyses 14-2 Integration of the Equations of Motion 14-3 Literature 15 Stablity and Control 15-1 Control 15-2 Stability 15-3 Flying Qualities in Hover 15-3.1 Equations of Motion 15-3.2 Vertical Dynamics 15-3.3 Yaw Dynamics 15-3.4 Longitudinal Dynamics 15-3.4.1 Equations of Motion 15-3.4.2 Poles and Zeros 15-3.4.3 Loop Closures 15-3.4.4 Hingeless Rotors 15-3.4.5 Response to Control 15-3.4.6 Examples 15-3.4.7 Flying Qualities Characteristics 15-3.5 Lateral Dynamics 15-3.6 Coupled Longitudinal and Lateral Dynamics 15-3.7 Tandem Helicopters 15-4 Flying Qualities in Forward Flight 15-4.1 Equations of Motion 15-4.2 Longitudinal Dynamics 15-4-2.1 Equations of Motion 15-4-2.2 Poles 15-4-2.3 Short Period Approximation 15-4-2.4 Static Stability 15-4-2.5 Example 15-4-2.6 Flying Qualities Characteristics 15-4.3 Lateral Dynamics 15-4.4 Tandem Helicopters 15-4.5 Hingeless Rotor Helicopters 15-5 Low Frequency Rotor Response 15-6 Stability Augmentation 15-7 Flying Qualities Specifications 15-8 Literature 16 Stall 16-1 Rotary Wing Stall Characteristics 16-2 NACA Stall Research 16-3 Dynamic Stall 16-4 Literature 17 Noise 17-1 Helicopter Rotor Noise 17-2 Vortex Noise 17-3 Rotational Noise 17-3.1 Rotor Pressure Distribution 17-3.2 Hovering Rotor with Steady Loading 17-3.3 Vertical Flight and Steady Loading 17-3.4 Stationary Rotor with Unsteady Loading 17-3.5 Forward Flight and Steady Loading 17-3.6 Forward Flight and Unsteady Loading 17-3.7 Thickness Noise 17-3.8 Rotating Frame Analysis 17-3.9 Doppler Shift 17-4 Blade Slap 17-5 Rotor Noise Reduction 17-6 Literature Cited Literature Index