Rotorcraft Aeromechanics

Contents
Preface
1 Introduction
1.1 The Helicopter
1.1.1 The Helicopter Rotor
1.1.2 Helicopter Configuration
1.1.3 Helicopter Operation
1.2 Design Trends
1.3 History
1.4 Books
Bibliography
2 Notation
2.1 Dimensions
2.2 Nomenclature
2.2.1 Physical Description of the Blade
2.2.2 Blade Aerodynamics
2.2.3 Blade Motion
2.2.4 Rotor Angle-of-Attack and Velocity
2.2.5 Rotor Forces and Power
2.2.6 Rotor Disk Planes
2.3 Other Notation Conventions
2.4 Geometry and Rotations
2.5 Symbols, Subscripts, and Superscripts
Subscripts and Superscripts
Abbreviations
2.6 References
Bibliography
3 Hover
3.1 Momentum Theory
3.1.1 Actuator Disk
3.1.2 Momentum Theory in Hover
3.1.3 Momentum Theory in Climb
3.2 Hover Power
3.3 Figure of Merit
3.4 Extended Momentum Theory
3.4.1 Rotor in Hover or Climb
3.4.2 Swirl in the Wake
3.5 Blade Element Theory
3.5.1 History of Blade Element Theory
3.5.2 Blade Element Theory for Vertical Flight
3.5.3 Combined Blade Element and Momentum Theory
3.6 Hover Performance
3.6.1 Scaling with Solidity
3.6.2 Tip Losses
3.6.3 Induced Power due to Nonuniform Inflow
3.6.4 Root Cutout
3.6.5 Blade Mean Lift Coefficient
3.6.6 Equivalent Solidity
3.6.7 The Ideal Rotor
3.6.8 The Optimum Hovering Rotor
3.6.9 Elementary Hover Performance Results
3.7 Vortex Theory
3.7.1 Vortex Representation of the Rotor and Wake
3.7.2 Actuator Disk Vortex Theory
3.7.3 Finite Number of Blades
3.8 Nonuniform Inflow
3.8.1 Hover Wake Geometry
3.8.2 Hover Performance Results from Free Wake Analysis
3.9 Influence of Blade Geometry
3.9.1 Twist and Taper
3.9.2 Blade Tip Shape
3.10 References
Bibliography
4 Vertical Flight
4.1 Induced Power in Vertical Flight
4.1.1 Momentum Theory for Vertical Flight
4.1.2 Flow States of the Rotor in Axial Flight
4.1.3 Induced Velocity Curve
4.2 Vortex Ring State
4.3 Autorotation in Vertical Descent
4.4 Climb in Vertical Flight
4.5 Optimum Windmill
4.6 Twin Rotor Interference in Hover
4.6.1 Coaxial Rotors
4.6.2 Tandem Rotors
4.7 Vertical Drag and Download
4.8 Ground Effect
4.9 References
Bibliography
5 Forward Flight Wake
5.1 Momentum Theory in Forward Flight
5.1.1 Rotor Induced Power
5.1.2 Climb, Descent, and Autorotation in Forward Flight
5.1.3 Rotor Loading Distribution
5.2 Vortex Theory in Forward Flight
5.2.1 Actuator Disk Results
5.2.2 Induced Velocity Variation in Forward Flight
5.3 Twin Rotor Interference in Forward Flight
5.3.1 Tandem and Coaxial Configurations
5.3.2 Side-by-Side Configuration
5.4 Ducted Fan
5.5 Influence of Ground in Forward Flight
5.5.1 Ground Effect
5.5.2 Ground Vortex
5.6 Interference
5.6.1 Rotor-Airframe Interference
5.6.2 Tail Design
5.6.3 Rotor Interference on Horizontal Tail
5.6.4 Pylon and Hub Interference on Tail
5.6.5 Tail Rotor
5.7 References
Bibliography
6 Forward Flight
6.1 The Helicopter Rotor in Forward Flight
6.1.1 Velocity
6.1.2 Blade Motion
6.1.3 Reference Planes
6.2 Aerodynamics of Forward Flight
6.3 Rotor Aerodynamic Forces
6.4 Power in Forward Flight
6.5 Rotor Flapping Motion
6.6 Linear Inflow Variation
6.7 Higher Harmonic Flapping Motion
6.8 Reverse Flow
6.9 Blade Weight Moment
6.10 Compressibility
6.11 Reynolds Number
6.12 Tip Loss and Root Cutout
6.13 Assumptions and Examples
6.14 Flap Motion with a Hinge Spring
6.15 Flap-Hinge Offset
6.16 Hingeless Rotor
6.17 Gimballed or Teetering Rotor
6.18 Pitch-Flap Coupling
6.19 Tail Rotor
6.20 Lag Motion
6.21 Helicopter Force and Moment Equilibrium
6.22 Yawed Flow and Radial Drag
6.23 Profile Power
6.24 History
6.24.1 The Beginning of Aeromechanics
6.24.2 After Glauert
6.25 References
Bibliography
7 Performance
7.1 Rotor Performance Estimation
7.1.1 Hover and Vertical Flight Performance
7.1.2 Forward Flight Performance
7.1.3 D/L Formulation
7.1.4 Rotor Lift and Drag
7.1.5 P/T Formulation
7.1.6 Rotorcraft Performance
7.1.7 Performance Charts
7.2 Rotorcraft Performance Characteristics
7.2.1 Hover Performance
7.2.2 Power Required in Level Flight
7.2.3 Climb and Descent
7.2.4 Maximum Speed
7.2.5 Ceiling
7.2.6 Range and Endurance
7.2.7 Referred Performance
7.3 Performance Metrics
7.4 References
Bibliography
8 Design
8.1 Rotor Configuration
8.2 Rotorcraft Configuration
8.3 Anti-Torque and Tail Rotor
8.4 Helicopter Speed Limitations
8.5 Autorotation, Landing, and Takeoff
8.6 Helicopter Drag
8.7 Rotor Blade Airfoils
8.8 Rotor Blade Profile Drag
8.9 References
Bibliography
9 Wings and Wakes
9.1 Rotor Vortex Wake
9.2 Lifting-Line Theory
9.3 Perturbation Solution for Lifting-Line Theory
9.4 Nonuniform Inflow
9.5 Wake Geometry
9.6 Examples
9.7 Vortex Core
9.8 Blade-Vortex Interaction
9.9 Vortex Elements
9.9.1 Vortex Line Segment
9.9.2 Vortex Sheet Element
9.9.3 Circular-Arc Vortex Element
9.10 History
9.11 References
Bibliography
10 Unsteady Aerodynamics
10.1 Two-Dimensional Unsteady Airfoil Theory
10.2 Lifting-Line Theory and Near Shed Wake
10.3 Reverse Flow
10.4 Trailing-Edge Flap
10.5 Unsteady Airfoil Theory with a Time-Varying Free Stream
10.6 Unsteady Airfoil Theory for the Rotary Wing
10.7 Two-Dimensional Model for Hovering Rotor
10.8 Blade-Vortex Interaction
10.9 References
Bibliography
11 Actuator Disk
11.1 Vortex Theory
11.2 Potential Theory
11.3 Dynamic Inflow
11.4 History
11.5 References
Bibliography
12 Stall
12.1 Dynamic Stall
12.2 Rotary-Wing Stall Characteristics
12.3 Elementary Stall Criteria
12.4 Empirical Dynamic Stall Models
12.5 References
Bibliography
13 Computational Aerodynamics
13.1 Potential Theory
13.2 Rotating Coordinate System
13.3 Lifting-Surface Theory
13.3.1 Moving Singularity
13.3.2 Fixed Wing
13.3.3 Rotary Wing
13.4 Boundary Element Methods
13.4.1 Surface Singularity Representations
13.4.2 Integral Equation
13.4.3 Compressible Flow
13.5 Transonic Theory
13.5.1 Small-Disturbance Potential
13.5.2 History
13.6 Navier-Stokes Equations
13.6.1 Hover Boundary Conditions
13.6.2 CFD/CSD Coupling
13.7 Boundary Layer Equations
13.8 Static Stall Delay
13.9 References
Bibliography
14 Noise
14.1 Helicopter Rotor Noise
14.2 Rotor Sound Spectrum
14.3 Broadband Noise
14.4 Rotational Noise
14.4.1 Rotor Pressure Distribution
14.4.2 Hovering Rotor with Steady Loading
14.4.3 Vertical Flight and Steady Loading
14.4.4 Stationary Rotor with Unsteady Loading
14.4.5 Forward Flight and Steady Loading
14.4.6 Forward Flight and Unsteady Loading
14.4.7 Doppler Shift
14.4.8 Thickness Noise
14.5 Sound Generated Aerodynamically
14.5.1 Lighthill's Acoustic Analogy
14.5.2 Ffowcs Williams-Hawkings Equation
14.5.3 Kirchhoff Equation
14.5.4 Integral Formulations
14.5.5 Far Field Thickness and Loading Noise
14.5.6 Broadband Noise
14.6 Impulsive Noise
14.7 Noise Certification
14.8 References
Bibliography
15 Mathematics of Rotating Systems
15.1 Fourier Series
15.2 Sum of Harmonics
15.3 Harmonic Analysis
15.4 Multiblade Coordinates
15.4.1 Transformation of the Degrees of Freedom
15.4.2 Matrix Form
15.4.3 Conversion of the Equations of Motion
15.4.4 Reactionless Mode and Two-Bladed Rotors
15.4.5 History
15.5 Eigenvalues and Eigenvectors of the Rotor Motion
15.6 Analysis of Linear, Periodic Systems
15.6.1 Linear, Constant Coefficient Equations
15.6.2 Linear, Periodic Coefficient Equations
15.7 Solution of the Equations of Motion
15.7.1 Early Methods
15.7.2 Harmonic Analysis
15.7.3 Time Finite Element
15.7.4 Periodic Shooting
15.7.5 Algebraic Equations
15.7.6 Successive Substitution
15.7.7 Newton-Raphson
15.8 References
Bibliography
16 Blade Motion
16.1 Sturm-Liouville Theory
16.2 Derivation of Equations of Motion
16.2.1 Integral Newtonian Method
16.2.2 Differential Newtonian Method
16.2.3 Lagrangian Method
16.2.4 Normal Mode Method
16.2.5 Galerkin Method
16.2.6 Rayleigh-Ritz Method
16.2.7 Lumped Parameter and Finite Element Methods
16.3 Out-of-Plane Motion
16.3.1 Rigid Flapping
16.3.2 Out-of-Plane Bending
16.3.3 Non-Rotating Frame
16.3.4 Bending Moments
16.4 In-Plane Motion
16.4.1 Rigid Flap and Lag
16.4.2 Structural Coupling
16.4.3 In-Plane Bending
16.4.4 In-Plane and Out-of-Plane Bending
16.5 Torsional Motion
16.5.1 Rigid Pitch and Flap
16.5.2 Structural Pitch-Flap and Pitch-Lag Coupling
16.5.3 Torsion and Out-of-Plane Bending
16.5.4 Non-Rotating Frame
16.6 Hub Reactions
16.6.1 Rotating Loads
16.6.2 Non-Rotating Loads
16.7 Shaft Motion
16.8 Aerodynamic Loads
16.8.1 Section Aerodynamics
16.8.2 Flap Motion
16.8.3 Flap and Lag Motion
16.8.4 Non-Rotating Frame
16.8.5 Hub Reactions in Rotating Frame
16.8.6 Hub Reactions in Non-Rotating Frame
16.8.7 Shaft Motion
16.8.8 Summary
16.8.9 Large Angles and High Inflow
16.8.10 Pitch and Flap Motion
16.9 References
Bibliography
17 Beam Theory
17.1 Beams and Rotor Blades
17.2 Engineering Beam Theory for a Twisted Rotor Blade
17.3 Nonlinear Beam Theory
17.3.1 Beam Cross-Section Motion
17.3.2 Extension and Torsion Produced by Bending
17.3.3 Elastic Variables and Shape Functions
17.3.4 Hamilton's Principle
17.3.5 Strain Energy
17.3.6 Extension-Torsion Coupling
17.3.7 Kinetic Energy
17.3.8 Equations of Motion
17.3.9 Structural Loads
17.4 Equations of Motion for Elastic Rotor Blade
17.5 History
17.6 References
Bibliography
18 Dynamics
18.1 Blade Modal Frequencies
18.2 Rotor Structural Loads
18.3 Vibration
18.4 Vibration Requirements and Vibration Reduction
18.5 Higher Harmonic Control
18.5.1 Control Algorithm
18.5.2 Helicopter Model
18.5.3 Identification
18.5.4 Control
18.5.5 Time-Domain Controllers
18.5.6 Effectiveness of HHC and IBC
18.6 Lag Damper
18.7 References
Bibliography
19 Flap Motion
19.1 Rotating Frame
19.1.1 Hover Roots
19.1.2 Forward Flight Roots
19.1.3 Hover Transfer Function
19.2 Non-Rotating Frame
19.2.1 Hover Roots and Modes
19.2.2 Hover Transfer Functions
19.3 Low-Frequency Response
19.4 Hub Reactions
19.5 Wake Influence
19.6 Pitch-Flap Coupling and Feedback
19.7 Complex Variable Representation of Motion
19.8 Two-Bladed Rotor
19.9 References
Bibliography
20 Stability
20.1 Pitch-Flap Flutter
20.1.1 Pitch-Flap Equations
20.1.2 Divergence Instability
20.1.3 Flutter Instability
20.1.4 Shed Wake Influence
20.1.5 Forward Flight
20.1.6 Coupled Blades
20.2 Flap-Lag Dynamics
20.2.1 Flap-Lag Equations
20.2.2 Articulated Rotors
20.2.3 Stability Boundary
20.2.4 Hingeless Rotors
20.2.5 Pitch-Flap and Pitch-Lag Coupling
20.2.6 Blade Stall
20.2.7 Elastic Blade and Flap-Lag-Torsion Stability
20.3 Ground Resonance
20.3.1 Ground Resonance Equations
20.3.2 No-Damping Case
20.3.3 Damping Required for Ground Resonance Stability
20.3.4 Complex Variable Representation of Motion
20.3.5 Two-Bladed Rotor
20.3.6 Air Resonance
20.3.7 Dynamic Inflow
20.3.8 History
20.4 Whirl Flutter
20.4.1 Whirl Flutter Equations
20.4.2 Propeller Whirl Flutter
20.4.3 Tiltrotor Whirl Flutter
20.5 References
Bibliography
21 Flight Dynamics
21.1 Control
21.2 Aircraft Motion
21.3 Motion and Loads
21.4 Hover Flight Dynamics
21.4.1 Rotor Forces and Moments
21.4.2 Hover Stability Derivatives
21.4.3 Vertical Dynamics
21.4.4 Directional Dynamics
21.4.5 Longitudinal Dynamics
21.4.6 Response to Control and Loop Closures
21.4.7 Lateral Dynamics
21.4.8 Coupled Longitudinal and Lateral Dynamics
21.5 Forward Flight
21.5.1 Forward Flight Stability Derivatives
21.5.2 Longitudinal Dynamics
21.5.3 Short Period Approximation
21.5.4 Lateral-Directional Dynamics
21.6 Static Stability
21.7 Twin Main Rotor Configurations
21.7.1 Tandem Helicopter
21.7.2 Side-by-Side Helicopter or Tiltrotor
21.8 Hingeless Rotor Helicopters
21.9 Control Gyros and Stability Augmentation
21.10 Flying Qualities Specifications
21.10.1 MIL-H-8501A
21.10.2 Handling Qualities Rating
21.10.3 Bandwidth Requirements
21.10.4 ADS-33
21.11 References
Bibliography
22 Comprehensive Analysis
22.1 References
Bibliography
Index