Wayne Johnson

The history of the helicopter may be traced back to the Chinese flying top (c. 400 B.C.) and to the work of Leonardo da Vinci, who sketched designs for a vertical flight machine utilizing a screw-type propeller. In the late nineteenth century, Thomas Edison experimented with helicopter models, realizing that no such machine would be able to fly until the development of a sufficiently lightweight engine. When the internal combustion gasoline engine came on the scene around 1900, the stage was set for the real development of helicopter technology.
While this text provides a concise history of helicopter development, its true purpose is to provide the engineering analysis required to design a highly successful rotorcraft. Toward that end the book offers thorough, comprehensive coverage of the theory of helicopter flight: the elements of vertical flight, forward flight, performance, design, mathematics of rotating systems, rotary wing dynamics and aerodynamics, aeroelasticity, stability and control, stall, noise and more.
Wayne Johnson has worked for the U.S. Army and NASA at the Ames Research Center in California. Through his company Johnson Aeronautics, he is engaged in the development of software that is used throughout the world for the analysis of rotorcraft. In this book, Dr. Johnson has compiled a monumental resource that is essential reading for any student or aeronautical engineer interested in the design and development of vertical-flight aircraft.

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
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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…