Rotorcraft Aeromechanics (Cambridge Aerospace Series, Series Number 36)

"A rotorcraft is a class of aircraft that uses large-diameter rotating wings to accomplish efficient vertical take-off 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 configuration 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 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"-- Provided by publisher Contents 9 Preface 19 1 Introduction 23 1.1 The Helicopter 23 1.1.1 The Helicopter Rotor 25 1.1.2 Helicopter Configuration 28 1.1.3 Helicopter Operation 29 1.2 Design Trends 30 1.3 History 36 1.4 Books 47 Bibliography 47 2 Notation 49 2.1 Dimensions 49 2.2 Nomenclature 49 2.2.1 Physical Description of the Blade 49 2.2.2 Blade Aerodynamics 51 2.2.3 Blade Motion 51 2.2.4 Rotor Angle-of-Attack and Velocity 52 2.2.5 Rotor Forces and Power 52 2.2.6 Rotor Disk Planes 53 2.3 Other Notation Conventions 53 2.4 Geometry and Rotations 54 2.5 Symbols, Subscripts, and Superscripts 55 Subscripts and Superscripts 9 Abbreviations 60 2.6 References 60 Bibliography 60 3 Hover 61 3.1 Momentum Theory 61 3.1.1 Actuator Disk 62 3.1.2 Momentum Theory in Hover 62 3.1.3 Momentum Theory in Climb 64 3.2 Hover Power 65 3.3 Figure of Merit 66 3.4 Extended Momentum Theory 67 3.4.1 Rotor in Hover or Climb 68 3.4.2 Swirl in the Wake 70 3.5 Blade Element Theory 74 3.5.1 History of Blade Element Theory 74 3.5.2 Blade Element Theory for Vertical Flight 76 3.5.3 Combined Blade Element and Momentum Theory 81 3.6 Hover Performance 82 3.6.1 Scaling with Solidity 83 3.6.2 Tip Losses 83 3.6.3 Induced Power due to Nonuniform Inflow 85 3.6.4 Root Cutout 86 3.6.5 Blade Mean Lift Coefficient 86 3.6.6 Equivalent Solidity 87 3.6.7 The Ideal Rotor 88 3.6.8 The Optimum Hovering Rotor 89 3.6.9 Elementary Hover Performance Results 90 3.7 Vortex Theory 92 3.7.1 Vortex Representation of the Rotor and Wake 95 3.7.2 Actuator Disk Vortex Theory 96 3.7.3 Finite Number of Blades 99 3.8 Nonuniform Inflow 105 3.8.1 Hover Wake Geometry 106 3.8.2 Hover Performance Results from Free Wake Analysis 108 3.9 Influence of Blade Geometry 109 3.9.1 Twist and Taper 109 3.9.2 Blade Tip Shape 112 3.10 References 113 Bibliography 113 4 Vertical Flight 114 4.1 Induced Power in Vertical Flight 114 4.1.1 Momentum Theory for Vertical Flight 115 4.1.2 Flow States of the Rotor in Axial Flight 118 4.1.3 Induced Velocity Curve 120 4.2 Vortex Ring State 124 4.3 Autorotation in Vertical Descent 127 4.4 Climb in Vertical Flight 131 4.5 Optimum Windmill 133 4.6 Twin Rotor Interference in Hover 133 4.6.1 Coaxial Rotors 134 4.6.2 Tandem Rotors 137 4.7 Vertical Drag and Download 139 4.8 Ground Effect 141 4.9 References 143 Bibliography 143 5 Forward Flight Wake 145 5.1 Momentum Theory in Forward Flight 145 5.1.1 Rotor Induced Power 145 5.1.2 Climb, Descent, and Autorotation in Forward Flight 150 5.1.3 Rotor Loading Distribution 151 5.2 Vortex Theory in Forward Flight 155 5.2.1 Actuator Disk Results 156 5.2.2 Induced Velocity Variation in Forward Flight 158 5.3 Twin Rotor Interference in Forward Flight 159 5.3.1 Tandem and Coaxial Configurations 160 5.3.2 Side-by-Side Configuration 162 5.4 Ducted Fan 163 5.5 Influence of Ground in Forward Flight 166 5.5.1 Ground Effect 166 5.5.2 Ground Vortex 166 5.6 Interference 168 5.6.1 Rotor-Airframe Interference 168 5.6.2 Tail Design 169 5.6.3 Rotor Interference on Horizontal Tail 169 5.6.4 Pylon and Hub Interference on Tail 170 5.6.5 Tail Rotor 171 5.7 References 172 Bibliography 172 6 Forward Flight 174 6.1 The Helicopter Rotor in Forward Flight 174 6.1.1 Velocity 175 6.1.2 Blade Motion 176 6.1.3 Reference Planes 183 6.2 Aerodynamics of Forward Flight 187 6.3 Rotor Aerodynamic Forces 190 6.4 Power in Forward Flight 195 6.5 Rotor Flapping Motion 200 6.6 Linear Inflow Variation 206 6.7 Higher Harmonic Flapping Motion 208 6.8 Reverse Flow 210 6.9 Blade Weight Moment 214 6.10 Compressibility 215 6.11 Reynolds Number 216 6.12 Tip Loss and Root Cutout 217 6.13 Assumptions and Examples 217 6.14 Flap Motion with a Hinge Spring 223 6.15 Flap-Hinge Offset 230 6.16 Hingeless Rotor 234 6.17 Gimballed or Teetering Rotor 235 6.18 Pitch-Flap Coupling 237 6.19 Tail Rotor 241 6.20 Lag Motion 242 6.21 Helicopter Force and Moment Equilibrium 245 6.22 Yawed Flow and Radial Drag 248 6.23 Profile Power 252 6.24 History 259 6.24.1 The Beginning of Aeromechanics 259 6.24.2 After Glauert 261 6.25 References 263 Bibliography 263 7 Performance 265 7.1 Rotor Performance Estimation 267 7.1.1 Hover and Vertical Flight Performance 267 7.1.2 Forward Flight Performance 268 7.1.3 D/L Formulation 270 7.1.4 Rotor Lift and Drag 271 7.1.5 P/T Formulation 272 7.1.6 Rotorcraft Performance 273 7.1.7 Performance Charts 275 7.2 Rotorcraft Performance Characteristics 276 7.2.1 Hover Performance 276 7.2.2 Power Required in Level Flight 281 7.2.3 Climb and Descent 283 7.2.4 Maximum Speed 284 7.2.5 Ceiling 285 7.2.6 Range and Endurance 286 7.2.7 Referred Performance 288 7.3 Performance Metrics 288 7.4 References 292 Bibliography 292 8 Design 293 8.1 Rotor Configuration 293 8.2 Rotorcraft Configuration 297 8.3 Anti-Torque and Tail Rotor 303 8.4 Helicopter Speed Limitations 305 8.5 Autorotation, Landing, and Takeoff 307 8.6 Helicopter Drag 313 8.7 Rotor Blade Airfoils 316 8.8 Rotor Blade Profile Drag 320 8.9 References 323 Bibliography 323 9 Wings and Wakes 325 9.1 Rotor Vortex Wake 325 9.2 Lifting-Line Theory 328 9.3 Perturbation Solution for Lifting-Line Theory 334 9.4 Nonuniform Inflow 340 9.5 Wake Geometry 348 9.6 Examples 357 9.7 Vortex Core 361 9.8 Blade-Vortex Interaction 367 9.9 Vortex Elements 372 9.9.1 Vortex Line Segment 373 9.9.2 Vortex Sheet Element 375 9.9.3 Circular-Arc Vortex Element 378 9.10 History 381 9.11 References 385 Bibliography 385 10 Unsteady Aerodynamics 388 10.1 Two-Dimensional Unsteady Airfoil Theory 388 10.2 Lifting-Line Theory and Near Shed Wake 398 10.3 Reverse Flow 402 10.4 Trailing-Edge Flap 403 10.5 Unsteady Airfoil Theory with a Time-Varying Free Stream 404 10.6 Unsteady Airfoil Theory for the Rotary Wing 409 10.7 Two-Dimensional Model for Hovering Rotor 413 10.8 Blade-Vortex Interaction 425 10.9 References 434 Bibliography 434 11 Actuator Disk 436 11.1 Vortex Theory 436 11.2 Potential Theory 446 11.3 Dynamic Inflow 454 11.4 History 458 11.5 References 461 Bibliography 461 12 Stall 464 12.1 Dynamic Stall 465 12.2 Rotary-Wing Stall Characteristics 470 12.3 Elementary Stall Criteria 473 12.4 Empirical Dynamic Stall Models 479 12.5 References 482 Bibliography 482 13 Computational Aerodynamics 484 13.1 Potential Theory 484 13.2 Rotating Coordinate System 486 13.3 Lifting-Surface Theory 488 13.3.1 Moving Singularity 488 13.3.2 Fixed Wing 490 13.3.3 Rotary Wing 491 13.4 Boundary Element Methods 493 13.4.1 Surface Singularity Representations 493 13.4.2 Integral Equation 495 13.4.3 Compressible Flow 496 13.5 Transonic Theory 499 13.5.1 Small-Disturbance Potential 499 13.5.2 History 502 13.6 Navier-Stokes Equations 503 13.6.1 Hover Boundary Conditions 504 13.6.2 CFD/CSD Coupling 505 13.7 Boundary Layer Equations 507 13.8 Static Stall Delay 509 13.9 References 510 Bibliography 510 14 Noise 515 14.1 Helicopter Rotor Noise 515 14.2 Rotor Sound Spectrum 518 14.3 Broadband Noise 521 14.4 Rotational Noise 524 14.4.1 Rotor Pressure Distribution 525 14.4.2 Hovering Rotor with Steady Loading 527 14.4.3 Vertical Flight and Steady Loading 532 14.4.4 Stationary Rotor with Unsteady Loading 533 14.4.5 Forward Flight and Steady Loading 534 14.4.6 Forward Flight and Unsteady Loading 536 14.4.7 Doppler Shift 538 14.4.8 Thickness Noise 538 14.5 Sound Generated Aerodynamically 540 14.5.1 Lighthill's Acoustic Analogy 540 14.5.2 Ffowcs Williams-Hawkings Equation 541 14.5.3 Kirchhoff Equation 544 14.5.4 Integral Formulations 544 14.5.5 Far Field Thickness and Loading Noise 549 14.5.6 Broadband Noise 554 14.6 Impulsive Noise 557 14.7 Noise Certification 562 14.8 References 564 Bibliography 564 15 Mathematics of Rotating Systems 567 15.1 Fourier Series 567 15.2 Sum of Harmonics 569 15.3 Harmonic Analysis 570 15.4 Multiblade Coordinates 571 15.4.1 Transformation of the Degrees of Freedom 571 15.4.2 Matrix Form 574 15.4.3 Conversion of the Equations of Motion 575 15.4.4 Reactionless Mode and Two-Bladed Rotors 579 15.4.5 History 582 15.5 Eigenvalues and Eigenvectors of the Rotor Motion 584 15.6 Analysis of Linear, Periodic Systems 586 15.6.1 Linear, Constant Coefficient Equations 588 15.6.2 Linear, Periodic Coefficient Equations 590 15.7 Solution of the Equations of Motion 595 15.7.1 Early Methods 595 15.7.2 Harmonic Analysis 597 15.7.3 Time Finite Element 598 15.7.4 Periodic Shooting 599 15.7.5 Algebraic Equations 600 15.7.6 Successive Substitution 601 15.7.7 Newton-Raphson 601 15.8 References 602 Bibliography 602 16 Blade Motion 604 16.1 Sturm-Liouville Theory 604 16.2 Derivation of Equations of Motion 606 16.2.1 Integral Newtonian Method 607 16.2.2 Differential Newtonian Method 607 16.2.3 Lagrangian Method 608 16.2.4 Normal Mode Method 608 16.2.5 Galerkin Method 610 16.2.6 Rayleigh-Ritz Method 611 16.2.7 Lumped Parameter and Finite Element Methods 612 16.3 Out-of-Plane Motion 612 16.3.1 Rigid Flapping 612 16.3.2 Out-of-Plane Bending 614 16.3.3 Non-Rotating Frame 618 16.3.4 Bending Moments 619 16.4 In-Plane Motion 621 16.4.1 Rigid Flap and Lag 621 16.4.2 Structural Coupling 624 16.4.3 In-Plane Bending 625 16.4.4 In-Plane and Out-of-Plane Bending 626 16.5 Torsional Motion 628 16.5.1 Rigid Pitch and Flap 628 16.5.2 Structural Pitch-Flap and Pitch-Lag Coupling 632 16.5.3 Torsion and Out-of-Plane Bending 635 16.5.4 Non-Rotating Frame 640 16.6 Hub Reactions 641 16.6.1 Rotating Loads 641 16.6.2 Non-Rotating Loads 646 16.7 Shaft Motion 650 16.8 Aerodynamic Loads 655 16.8.1 Section Aerodynamics 655 16.8.2 Flap Motion 661 16.8.3 Flap and Lag Motion 663 16.8.4 Non-Rotating Frame 666 16.8.5 Hub Reactions in Rotating Frame 671 16.8.6 Hub Reactions in Non-Rotating Frame 675 16.8.7 Shaft Motion 677 16.8.8 Summary 682 16.8.9 Large Angles and High Inflow 686 16.8.10 Pitch and Flap Motion 688 16.9 References 692 Bibliography 692 17 Beam Theory 693 17.1 Beams and Rotor Blades 693 17.2 Engineering Beam Theory for a Twisted Rotor Blade 694 17.3 Nonlinear Beam Theory 702 17.3.1 Beam Cross-Section Motion 703 17.3.2 Extension and Torsion Produced by Bending 707 17.3.3 Elastic Variables and Shape Functions 707 17.3.4 Hamilton's Principle 709 17.3.5 Strain Energy 710 17.3.6 Extension-Torsion Coupling 716 17.3.7 Kinetic Energy 716 17.3.8 Equations of Motion 718 17.3.9 Structural Loads 719 17.4 Equations of Motion for Elastic Rotor Blade 721 17.5 History 725 17.6 References 728 Bibliography 728 18 Dynamics 732 18.1 Blade Modal Frequencies 732 18.2 Rotor Structural Loads 737 18.3 Vibration 739 18.4 Vibration Requirements and Vibration Reduction 744 18.5 Higher Harmonic Control 751 18.5.1 Control Algorithm 752 18.5.2 Helicopter Model 753 18.5.3 Identification 754 18.5.4 Control 758 18.5.5 Time-Domain Controllers 759 18.5.6 Effectiveness of HHC and IBC 762 18.6 Lag Damper 762 18.7 References 768 Bibliography 768 19 Flap Motion 771 19.1 Rotating Frame 771 19.1.1 Hover Roots 772 19.1.2 Forward Flight Roots 774 19.1.3 Hover Transfer Function 780 19.2 Non-Rotating Frame 780 19.2.1 Hover Roots and Modes 782 19.2.2 Hover Transfer Functions 783 19.3 Low-Frequency Response 788 19.4 Hub Reactions 791 19.5 Wake Influence 795 19.6 Pitch-Flap Coupling and Feedback 804 19.7 Complex Variable Representation of Motion 805 19.8 Two-Bladed Rotor 806 19.9 References 809 Bibliography 809 20 Stability 810 20.1 Pitch-Flap Flutter 810 20.1.1 Pitch-Flap Equations 810 20.1.2 Divergence Instability 812 20.1.3 Flutter Instability 813 20.1.4 Shed Wake Influence 817 20.1.5 Forward Flight 818 20.1.6 Coupled Blades 818 20.2 Flap-Lag Dynamics 819 20.2.1 Flap-Lag Equations 820 20.2.2 Articulated Rotors 822 20.2.3 Stability Boundary 824 20.2.4 Hingeless Rotors 824 20.2.5 Pitch-Flap and Pitch-Lag Coupling 827 20.2.6 Blade Stall 830 20.2.7 Elastic Blade and Flap-Lag-Torsion Stability 830 20.3 Ground Resonance 832 20.3.1 Ground Resonance Equations 833 20.3.2 No-Damping Case 835 20.3.3 Damping Required for Ground Resonance Stability 840 20.3.4 Complex Variable Representation of Motion 843 20.3.5 Two-Bladed Rotor 844 20.3.6 Air Resonance 849 20.3.7 Dynamic Inflow 849 20.3.8 History 850 20.4 Whirl Flutter 854 20.4.1 Whirl Flutter Equations 854 20.4.2 Propeller Whirl Flutter 856 20.4.3 Tiltrotor Whirl Flutter 858 20.5 References 863 Bibliography 863 21 Flight Dynamics 866 21.1 Control 866 21.2 Aircraft Motion 868 21.3 Motion and Loads 871 21.4 Hover Flight Dynamics 875 21.4.1 Rotor Forces and Moments 875 21.4.2 Hover Stability Derivatives 878 21.4.3 Vertical Dynamics 881 21.4.4 Directional Dynamics 882 21.4.5 Longitudinal Dynamics 883 21.4.6 Response to Control and Loop Closures 889 21.4.7 Lateral Dynamics 894 21.4.8 Coupled Longitudinal and Lateral Dynamics 896 21.5 Forward Flight 900 21.5.1 Forward Flight Stability Derivatives 901 21.5.2 Longitudinal Dynamics 902 21.5.3 Short Period Approximation 905 21.5.4 Lateral-Directional Dynamics 908 21.6 Static Stability 910 21.7 Twin Main Rotor Configurations 911 21.7.1 Tandem Helicopter 912 21.7.2 Side-by-Side Helicopter or Tiltrotor 916 21.8 Hingeless Rotor Helicopters 917 21.9 Control Gyros and Stability Augmentation 917 21.10 Flying Qualities Specifications 923 21.10.1 MIL-H-8501A 924 21.10.2 Handling Qualities Rating 928 21.10.3 Bandwidth Requirements 929 21.10.4 ADS-33 931 21.11 References 935 Bibliography 935 22 Comprehensive Analysis 937 22.1 References 941 Bibliography 941 Index 9 1107606918,9781107028074 A rotorcraft is a class of aircraft that uses large-diameter rotating wings to accomplish efficient vertical take-off 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 configuration concepts. Aeromechanics covers much of what the rotorcraft engineer needs: performance, loads, vibration, stability, flight dynamics, and noise. These topics include many of the key performance attributes and the often-encountered problems in rotorcraft designs. This comprehensive book presents, in depth, what engineers need to know about modelling rotorcraft aeromechanics. The focus is on analysis, and calculated results are presented to illustrate analysis characteristics and rotor behaviour. 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.