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