ROTORCRAFT AEROMECHANICS

Transcription

1 ROTORCRAFT AEROMECHANICS Rotorcraft is a class of aircraft that uses large-diameter rotating wings to accomplish efficient vertical takeoff and landing. The class encompasses helicopters of numerous configurations (single main rotor and tail rotor, tandem rotors, coaxial rotors), tilting proprotor aircraft, compound helicopters, and many other innovative concepts. Aeromechanics includes much of what the rotorcraft engineer needs: performance, loads, vibration, stability, flight dynamics, and noise. These topics cover many of the key performance attributes and many of the often encountered problems in rotorcraft designs. This comprehensive book presents, in depth, what engineers need to know about modeling rotorcraft aeromechanics. The focus is on analysis, and calculated results are presented to illustrate analysis characteristics and rotor behavior. The first third of the book is an introduction to rotorcraft aerodynamics, blade motion, and performance. The remainder of the book covers advanced topics in rotary-wing aerodynamics and dynamics. worked at the U.S. Army Aeromechanics Laboratory from 1970 to 1981, at the NASA Ames Research Center. He was with NASA from 1981 to 1986, including several years as Assistant Branch Chief. Dr. Johnson founded Johnson Aeronautics in 1986, where he developed rotorcraft software. Since 1998, he has worked at the Aeromechanics Branch of NASA Ames Research Center. Dr. Johnson is the author of the comprehensive analysis CAMRADII and the rotorcraft design code NDARC and of the book Helicopter Theory (1980). He is a Fellow of the American Institute of Aeronautics and Astronautics (AIAA) and the American Helicopter Society (AHS) and received a U.S. Army Commander s Award for Civilian Service, NASA Medals for Exceptional Engineering Achievement and Exceptional Technology Achievement, the AHS Grover E. Bell Award, the Ames H. Julian Allen Award, the AIAA Pendray Aerospace Literature Award, and the 2010 AHS Alexander Nikolsky Honorary Lectureship.

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3 Cambridge Aerospace Series Editors Wei Shyy and Vigor Yang 1. J. M. Rolfe and K. J. Staples (eds.): Flight Simulation 2. P. Berlin: The Geostationary Applications Satellite 3. M. J. T. Smith: Aircraft Noise 4. N. X. Vinh: Flight Mechanics of High-Performance Aircraft 5. W. A. Mair and D. L. Birdsall: Aircraft Performance 6. M. J. Abzug and E. E. Larrabee: Airplane Stability and Control 7. M. J. Sidi: Spacecraft Dynamics and Control 8. J. D. Anderson: A History of Aerodynamics 9. A. M. Cruise, J. A. Bowles, C. V. Goodall, and T. J. Patrick: Principles of Space Instrument Design 10. G. A. Khoury (ed.): Airship Technology, Second Edition 11. J. P. Fielding: Introduction to Aircraft Design 12. J. G. Leishman: Principles of Helicopter Aerodynamics, Second Edition 13. J. Katz and A. Plotkin: Low-Speed Aerodynamics, Second Edition 14. M. J. Abzug and E. E. Larrabee: Airplane Stability and Control: A History of the Technologies that Made Aviation Possible, Second Edition 15. D. H. Hodges and G. A. Pierce: Introduction to Structural Dynamics and Aeroelasticity, Second Edition 16. W. Fehse: Automatic Rendezvous and Docking of Spacecraft 17. R. D. Flack: Fundamentals of Jet Propulsion with Applications 18. E. A. Baskharone: Principles of Turbomachinery in Air-Breathing Engines 19. D. D. Knight: Numerical Methods for High-Speed Flows 20. C. A. Wagner, T. Hüttl, and P. Sagaut (eds.): Large-Eddy Simulation for Acoustics 21. D. D. Joseph, T. Funada, and J. Wang: Potential Flows of Viscous and Viscoelastic Fluids 22. W. Shyy, Y. Lian, H. Liu, J. Tang, and D. Viieru: Aerodynamics of Low Reynolds Number Flyers 23. J. H. Saleh: Analyses for Durability and System Design Lifetime 24. B. K. Donaldson: Analysis of Aircraft Structures, Second Edition 25. C. Segal: The Scramjet Engine: Processes and Characteristics 26. J. F. Doyle: Guided Explorations of the Mechanics of Solids and Structures 27. A. K. Kundu: Aircraft Design 28. M. I. Friswell, J. E. T. Penny, S. D. Garvey, and A. W. Lees: Dynamics of Rotating Machines 29. B. A. Conway (ed): Spacecraft Trajectory Optimization 30. R. J. Adrian and J. Westerweel: Particle Image Velocimetry 31. G. A. Flandro, H. M. McMahon, and R. L. Roach: Basic Aerodynamics 32. H. Babinsky and J. K. Harvey: Shock Wave Boundary-Layer Interactions 33. C. K. W. Tam: Computational Aeroacoustics: A Wave Number Approach 34. A. Filippone: Advanced Aircraft Flight Performance 35. I. Chopra and J. Sirohi: Smart Structures Theory 36. W. Johnson: Rotorcraft Aeromechanics 37. W. Shyy, H. Aono, C. K. Kang, and H. Liu: An Introduction to Flapping Wing Aerodynamics 38. T. C. Lieuwen and V. Yang: Gas Turbine Engines

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5 Rotorcraft Aeromechanics NASA Ames Research Center

6 cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY , USA Information on this title: / This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2013 Printed in the United States of America A catalog record for this publication is available from the British Library. ISBN Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

7 Preface page xvii 1 Introduction The Helicopter The Helicopter Rotor Helicopter Configuration Helicopter Operation Design Trends History Books 25 2 Notation Dimensions Nomenclature Physical Description of the Blade Blade Aerodynamics Blade Motion Rotor Angle-of-Attack and Velocity Rotor Forces and Power Rotor Disk Planes Other Notation Conventions Geometry and Rotations Symbols, Subscripts, and Superscripts References 38 3 Hover Momentum Theory Actuator Disk Momentum Theory in Hover Momentum Theory in Climb Hover Power Figure of Merit Extended Momentum Theory 45 vii

8 viii Rotor in Hover or Climb Swirl in the Wake Blade Element Theory History of Blade Element Theory Blade Element Theory for Vertical Flight Combined Blade Element and Momentum Theory Hover Performance Scaling with Solidity Tip Losses Induced Power due to Nonuniform Inflow Root Cutout Blade Mean Lift Coefficient Equivalent Solidity The Ideal Rotor The Optimum Hovering Rotor Elementary Hover Performance Results Vortex Theory Vortex Representation of the Rotor and Wake Actuator Disk Vortex Theory Finite Number of Blades Nonuniform Inflow Hover Wake Geometry Hover Performance Results from Free Wake Analysis Influence of Blade Geometry Twist and Taper Blade Tip Shape References 91 4 Vertical Flight Induced Power in Vertical Flight Momentum Theory for Vertical Flight Flow States of the Rotor in Axial Flight Induced Velocity Curve Vortex Ring State Autorotation in Vertical Descent Climb in Vertical Flight Optimum Windmill Twin Rotor Interference in Hover Coaxial Rotors Tandem Rotors Vertical Drag and Download Ground Effect References Forward Flight Wake Momentum Theory in Forward Flight Rotor Induced Power Climb, Descent, and Autorotation in Forward Flight Rotor Loading Distribution 129

9 ix 5.2 Vortex Theory in Forward Flight Actuator Disk Results Induced Velocity Variation in Forward Flight Twin Rotor Interference in Forward Flight Tandem and Coaxial Configurations Side-by-Side Configuration Ducted Fan Influence of Ground in Forward Flight Ground Effect Ground Vortex Interference Rotor-Airframe Interference Tail Design Rotor Interference on Horizontal Tail Pylon and Hub Interference on Tail Tail Rotor References Forward Flight The Helicopter Rotor in Forward Flight Velocity Blade Motion Reference Planes Aerodynamics of Forward Flight Rotor Aerodynamic Forces Power in Forward Flight Rotor Flapping Motion Linear Inflow Variation Higher Harmonic Flapping Motion Reverse Flow Blade Weight Moment Compressibility Reynolds Number Tip Loss and Root Cutout Assumptions and Examples Flap Motion with a Hinge Spring Flap-Hinge Offset Hingeless Rotor Gimballed or Teetering Rotor Pitch-Flap Coupling Tail Rotor Lag Motion Helicopter Force and Moment Equilibrium Yawed Flow and Radial Drag Profile Power History The Beginning of Aeromechanics After Glauert References 241

10 x 7 Performance Rotor Performance Estimation Hover and Vertical Flight Performance Forward Flight Performance D/L Formulation Rotor Lift and Drag P/T Formulation Rotorcraft Performance Performance Charts Rotorcraft Performance Characteristics Hover Performance Power Required in Level Flight Climb and Descent Maximum Speed Ceiling Range and Endurance Referred Performance Performance Metrics References Design Rotor Configuration Rotorcraft Configuration Anti-Torque and Tail Rotor Helicopter Speed Limitations Autorotation, Landing, and Takeoff Helicopter Drag Rotor Blade Airfoils Rotor Blade Profile Drag References Wings and Wakes Rotor Vortex Wake Lifting-Line Theory Perturbation Solution for Lifting-Line Theory Nonuniform Inflow Wake Geometry Examples Vortex Core Blade-Vortex Interaction Vortex Elements Vortex Line Segment Vortex Sheet Element Circular-Arc Vortex Element History References 363

11 xi 10 Unsteady Aerodynamics Two-Dimensional Unsteady Airfoil Theory Lifting-Line Theory and Near Shed Wake Reverse Flow Trailing-Edge Flap Unsteady Airfoil Theory with a Time-Varying Free Stream Unsteady Airfoil Theory for the Rotary Wing Two-Dimensional Model for Hovering Rotor Blade-Vortex Interaction References Actuator Disk Vortex Theory Potential Theory Dynamic Inflow History References Stall Dynamic Stall Rotary-Wing Stall Characteristics Elementary Stall Criteria Empirical Dynamic Stall Models References Computational Aerodynamics Potential Theory Rotating Coordinate System Lifting-Surface Theory Moving Singularity Fixed Wing Rotary Wing Boundary Element Methods Surface Singularity Representations Integral Equation Compressible Flow Transonic Theory Small-Disturbance Potential History Navier-Stokes Equations Hover Boundary Conditions CFD/CSD Coupling Boundary Layer Equations Static Stall Delay References Noise Helicopter Rotor Noise 493

12 xii 14.2 Rotor Sound Spectrum Broadband Noise Rotational Noise Rotor Pressure Distribution Hovering Rotor with Steady Loading Vertical Flight and Steady Loading Stationary Rotor with Unsteady Loading Forward Flight and Steady Loading Forward Flight and Unsteady Loading Doppler Shift Thickness Noise Sound Generated Aerodynamically Lighthill s Acoustic Analogy Ffowcs Williams-Hawkings Equation Kirchhoff Equation Integral Formulations Far Field Thickness and Loading Noise Broadband Noise Impulsive Noise Noise Certification References Mathematics of Rotating Systems Fourier Series Sum of Harmonics Harmonic Analysis Multiblade Coordinates Transformation of the Degrees of Freedom Matrix Form Conversion of the Equations of Motion Reactionless Mode and Two-Bladed Rotors History Eigenvalues and Eigenvectors of the Rotor Motion Analysis of Linear, Periodic Systems Linear, Constant Coefficient Equations Linear, Periodic Coefficient Equations Solution of the Equations of Motion Early Methods Harmonic Analysis Time Finite Element Periodic Shooting Algebraic Equations Successive Substitution Newton-Raphson References Blade Motion Sturm-Liouville Theory 582

13 xiii 16.2 Derivation of Equations of Motion Integral Newtonian Method Differential Newtonian Method Lagrangian Method Normal Mode Method Galerkin Method Rayleigh-Ritz Method Lumped Parameter and Finite Element Methods Out-of-Plane Motion Rigid Flapping Out-of-Plane Bending Non-Rotating Frame Bending Moments In-Plane Motion Rigid Flap and Lag Structural Coupling In-Plane Bending In-Plane and Out-of-Plane Bending Torsional Motion Rigid Pitch and Flap Structural Pitch-Flap and Pitch-Lag Coupling Torsion and Out-of-Plane Bending Non-Rotating Frame Hub Reactions Rotating Loads Non-Rotating Loads Shaft Motion Aerodynamic Loads Section Aerodynamics Flap Motion Flap and Lag Motion Non-Rotating Frame Hub Reactions in Rotating Frame Hub Reactions in Non-Rotating Frame Shaft Motion Summary Large Angles and High Inflow Pitch and Flap Motion References Beam Theory Beams and Rotor Blades Engineering Beam Theory for a Twisted Rotor Blade Nonlinear Beam Theory Beam Cross-Section Motion Extension and Torsion Produced by Bending Elastic Variables and Shape Functions Hamilton s Principle Strain Energy 688

14 xiv Extension-Torsion Coupling Kinetic Energy Equations of Motion Structural Loads Equations of Motion for Elastic Rotor Blade History References Dynamics Blade Modal Frequencies Rotor Structural Loads Vibration Vibration Requirements and Vibration Reduction Higher Harmonic Control Control Algorithm Helicopter Model Identification Control Time-Domain Controllers Effectiveness of HHC and IBC Lag Damper References Flap Motion Rotating Frame Hover Roots Forward Flight Roots Hover Transfer Function Non-Rotating Frame Hover Roots and Modes Hover Transfer Functions Low-Frequency Response Hub Reactions Wake Influence Pitch-Flap Coupling and Feedback Complex Variable Representation of Motion Two-Bladed Rotor References Stability Pitch-Flap Flutter Pitch-Flap Equations Divergence Instability Flutter Instability Shed Wake Influence Forward Flight Coupled Blades Flap-Lag Dynamics Flap-Lag Equations 798

15 xv Articulated Rotors Stability Boundary Hingeless Rotors Pitch-Flap and Pitch-Lag Coupling Blade Stall Elastic Blade and Flap-Lag-Torsion Stability Ground Resonance Ground Resonance Equations No-Damping Case Damping Required for Ground Resonance Stability Complex Variable Representation of Motion Two-Bladed Rotor Air Resonance Dynamic Inflow History Whirl Flutter Whirl Flutter Equations Propeller Whirl Flutter Tiltrotor Whirl Flutter References Flight Dynamics Control Aircraft Motion Motion and Loads Hover Flight Dynamics Rotor Forces and Moments Hover Stability Derivatives Vertical Dynamics Directional Dynamics Longitudinal Dynamics Response to Control and Loop Closures Lateral Dynamics Coupled Longitudinal and Lateral Dynamics Forward Flight Forward Flight Stability Derivatives Longitudinal Dynamics Short Period Approximation Lateral-Directional Dynamics Static Stability Twin Main Rotor Configurations Tandem Helicopter Side-by-Side Helicopter or Tiltrotor Hingeless Rotor Helicopters Control Gyros and Stability Augmentation Flying Qualities Specifications MIL-H-8501A Handling Qualities Rating 906

16 xvi Bandwidth Requirements ADS References Comprehensive Analysis References 919 Index 921

17 Preface Rotorcraft is a class of aircraft that uses large-diameter rotating wings to accomplish efficient vertical takeoff and landing. The class thus encompasses helicopters of numerous configurations, tilting proprotor aircraft, compound helicopters, and many other innovative concepts. Defining aeromechanics is more difficult. Today s dictionaries do not capture what the term means for the rotorcraft community. The definitions are not broad enough, and they do not reflect the multidisciplinary facet of the word as applied to rotorcraft. In my 2010 Nikolsky Lecture for the American Helicopter Society, I proposed the following definition: Aeromechanics: The branch of aeronautical engineering and science dealing with equilibrium, motion, and control of elastic rotorcraft in air. Aeromechanics covers much of what the rotorcraft engineer needs: performance, loads, vibration, stability, flight dynamics, and noise. These topics cover many of the key performance attributes and many of the often encountered problems in rotorcraft designs. As with my previous book Helicopter Theory (written in 1976, published in 1980 by Princeton University Press, republished in 1994 by Dover Publications), this text is focused on analysis, with only occasional reference to test data to develop arguments or support results, and with nothing at all regarding the techniques of testing in wind tunnels or flight. Calculated results are presented to illustrate analysis characteristics and rotor behavior. Generally these results were obtained using computer programs that I have developed: the rotorcraft comprehensive analysis CAMRAD II and the sizing code NDARC. I aim to be comprehensive in coverage, presenting in as much depth as possible what engineers need to know about modeling rotorcraft aeromechanics. Although connections to Helicopter Theory are apparent throughout this text, many significant advances in the theory have occurred since I assume the reader has a general knowledge of aeromechanics fields, such as classical dynamics, beam theory, liftingline theory, and two-dimensional airfoils. I do provide introductory material where needed as a foundation for rotary-wing developments. Unlike Helicopter Theory,no attempt is made to be comprehensive in the bibliography. Sources are cited where considered historically important or the development is associated with a specific xvii

18 xviii Preface work and also to direct the reader to expanded coverage of a subject. Several topics conclude with an outline of the history of their theoretical development. The scope of this text is still a subset of rotorcraft aeromechanics. A shaft-driven helicopter is the primary focus, but other rotorcraft configurations are discussed, and most of the analysis is relevant to all configurations. Based mainly on my experience and interests, important areas such as structures, materials, and propulsion are not covered. The topic of flight dynamics here encompasses the aircraft behavior, but not handling qualities, which would require treatment of the control system and the pilot. Computational fluid dynamics is covered with an emphasis on identifying the unique aspects of the methods as applied to rotary wings. The equations are developed with an emphasis on rotating wings, but nothing is presented regarding solution procedures. Chapter 1 describes the helicopter rotor and helicopter configurations. Design trends are shown to illustrate the aircraft characteristics. A brief history of helicopter invention is presented. Chapter 2 summarizes the notation used in this book, providing an important overview of the description of a rotor for analysis. Chapters 3 to 8 constitute an introduction to rotorcraft aerodynamics and performance. Chapters 9 to 14 cover advanced aerodynamic topics, and Chapters 15 to 20 cover advanced topics in dynamics. Chapter 3 begins the analysis of aeromechanics by considering hover, which is the key to helicopter effectiveness; it presents the first description of momentum theory, blade element theory, and vortex theory. Chapter 4 extends the analysis to vertical flight, both climb and descent. Chapter 5 examines the wake in forward flight in terms of momentum and vortex theories. Aerodynamic interference is covered, including rotor-to-airframe and rotor-to-tail effects. Chapter 6 on the edgewise flight of a rotor is the longest chapter in the book, dealing with blade element theory calculation of rotor forces and power and beginning the analysis of blade motion, particularly flapping. Chapter 7 summarizes performance analysis for the isolated rotor and for the complete aircraft. Chapter 8 discusses rotor and rotorcraft configurations further, as well as special topics related to design. Chapter 9 deals with wings and wakes, as the start of advanced aerodynamics. Lifting-line theory and nonuniform inflow from a vortex wake are covered, including free wake geometry. Chapter 10 presents unsteady aerodynamic theory, beginning with the classical two-dimensional analysis, and covers special models and problems for the rotary wing. Chapter 11 is on actuator disk models, concluding with dynamic inflow theory. Chapter 12 describes dynamic stall of airfoils, stall of rotor blades, and stall effects on rotor performance and loads. Chapter 13 on computational aerodynamics focuses on the unique aspects of applications to rotorcraft. Chapter 14 deals with the theory of rotor-generated noise. Chapter 15 introduces advanced dynamics by describing the mathematics of rotating wings, including multiblade coordinates and Floquet theory, and the solution of equations of motion. Chapter 16 on blade motion is a long chapter that derives the equations of motion for blade flap, lag, and pitch degrees of freedom, including hub reactions and shaft motion, and the aerodynamic loads. Solutions of these equations are found in subsequent chapters. Chapter 17 derives linear and nonlinear beam theory for rotor blades. Chapter 18 on rotor dynamics covers blade frequencies, structural loads, vibration, and higher harmonic control. Chapter 19 is on the stability and response of the blade flap motion, which is fundamental to the behavior of helicopter rotors. Chapter 20 solves the equations of motion for several stability

19 Preface xix problems, including pitch-flap flutter, flap-lag dynamics, ground resonance, and whirl flutter. Chapter 21 analyzes rotorcraft flight dynamics, including hover and forward flight operation, single main rotor and multi-rotor configurations, control gyros, and flying qualities specifications. Chapter 22 concludes the book with a discussion of rotorcraft comprehensive analyses. My thanks go to my wife Juliet for her support. The time I spent on this book was as much hers as mine. I am indebted to Michael P. Scully, William Warmbrodt, Gloria K. Yamauchi, Franklin D. Harris, Anubhav Datta, Christopher Silva, and Gareth D. Padfield for reviewing the draft manuscript. Their numerous suggestions have resulted in a much improved work. I got into helicopter research through some interesting thesis topics at the Massachusetts Institute of Technology. I stay in the field because I like the multidisciplinary part of aeromechanics and because we have not run out of problems to solve. Since graduating from MIT, I have been associated with NASA and the U.S. Army at Ames Research Center, even during the 12 years I spent working alone as Johnson Aeronautics. My first assignment at Ames was with the 40- by 80-Foot Wind Tunnel branch and the latest is with the Aeromechanics Branch, always doing rotorcraft research. I have enjoyed collaboration with many capable people, both at Ames and around the world. I am fortunate that they are good friends as well as good engineers. Palo Alto, California August 2012