The Pennsylvania State University The Graduate School ELASTICALLY TAILORED COMPOSITE ROTOR BLADES FOR STALL ALLEVIATION AND VIBRATION REDUCTION A Thesis in Aerospace Engineering by Matthew W. Floros cfl Matthew W. Floros Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December
We approve the thesis of Matthew W. Floros. Date of Signature Edward C. Smith Associate Professor of Aerospace Engineering Thesis Adviser, Chair of Committee Farhan Gandhi Assistant Professor of Aerospace Engineering George Lesieutre Professor of Aerospace Engineering Charles Bakis Professor of Engineering Science and Mechanics Dennis K. McLaughlin Professor of Aerospace Engineering Head of the Department of Aerospace Engineering
Abstract An aeroelastic analysis has been developed to study the effects of elastic couplings on blade response, loads, and dynamic stall. Low and high speed conditions are examined at high thrust and cruise-level thrust for both hingeless and articulated rotor configurations. The blade cross-sectional model is based on Vlasov theory for multi-cell closed sections with thick walls. The structural model includes non-classical effects of transverse shear, torsion-related warping, and two-dimensional inplane elasticity. For the aeroelastic analysis, the blade is modeled as an elastic beam undergoing deflections in flap, lag, and torsion. The blade governing equations are approximated by a finite element in space model. A twelve degree of freedom specialized finite element is employed in the structural model which accounts for torsion-related warping. The analysis includes both a quasisteady and a time-domain unsteady aerodynamic model including the effects of non-linear separation and dynamic stall. The nonlinear periodic response is calculated through a finite element in time procedure with displacement and velocity continuous elements. The blade and hub loads are calculated using the force summation method and the reaction force method. The two loads calculation methods are compared for articulated and hingeless rotor configurations. The reaction force method is shown to more accurately predict blade bending moments in articulated rotors. The blade is modeled as a NACA airfoil section consisting of a D-spar and skin. Elastic couplings are introduced by anisotropy of plies in the D-spar to produce pitch-flap couplings. Results indicate that by coupling elastic twist to the second flap mode, the angle of attack on the retreating side of the rotor disk can be reduced up to two degrees, resulting in a significant reduction in blade stall for both articulated and hingeless rotors. Sensitivity studies are conducted to determine the effects of flap frequency on the induced twist. The third harmonic of twist is shown to be sensitive to flap frequency and can be tuned to reduce certain 4/rev vibratory loads. iii
Contents List of Figures List of Tables List of Symbols Dedication Acknowledgements ix xvii xix xxv xxvi 1 Introduction 1 1.1 Background and Motivation... 1 1. Summary of Previous Research... 5 1..1 AeroelasticallyConformableRotor... 5 1.. ModelingofCompositeBlades... 8 1..3 Stability of Composite Rotor Blades...... 1 1..4 OptimizationofCompositeBlades... 19 1..5 Fixed-Wing and Tilt-Rotor Applications of Composite Tailoring... 5 1..6 Aerodynamic Modeling... 8 1.3 Summary and Objectives of Current Research..... 33 Analytical Formulation 35.1 BasicConsiderations... 36.1.1 CoordinateSystems... 36 iv
.1. Orderingscheme... 4.1.3 Non-dimensionalization... 41.1.4 Definition of Flight Condition.... 4. BladeKineticEnergy... 44..1 Integral Expressions... 44.. IntegrationwithGaussQuadrature... 51..3 HybridAxialForceCalculation... 5..4 FiniteElementDiscretization... 54.3 Expansions of common expressions in the kinetic energy..... 61.4 CompositeCross-SectionModel... 63.4.1 Overview of Classical Laminated Plate Theory...... 64.4. VlasovTheoryforClosedSections... 67.4.3 IntegrationintoBladeFiniteElementModel... 7.4.4 Geometry-Specific Issues for Symmetric NACA XX Airfoils... 74.5 Linear Aerodynamics... 75.5.1 VelocityDuetoVehicleandBladeMotion... 76.5. Quasi-Steady Airloads... 81.5.3 ModificationsforReverseFlow... 87.5.4 InflowCalculation... 87.6 Unsteady Aerodynamics and Dynamic Stall...... 9.6.1 UnsteadyNormalForce... 91.6. UnsteadyPitchingMoment... 98.6.3 Airloads with Nonlinear Separation...... 1.6.4 Airloads with Dynamic Stall..... 15.6.5 BladeSectionForces... 111.7 Aerodynamics Implementation in Finite Element Analysis.... 11.7.1 Linear Aerodynamics Implementation..... 11.7. Non-Linear Aerodynamics Implementation... 116.8 Non-LinearSolution... 119 v
.9 NormalModesAnalysis... 11.9.1 TransformationofDifferentialEquationstoModalSpace... 11.9. ModalIdentification... 13.1FiniteElementinTime... 14.1.1 DiscretizationwithHamilton sprinciple... 15.1. Velocity-Continuous Shape Functions..... 16.1.3 Matrix Assembly..... 13.11 Lag Spring and Damper for Articulated Rotors.... 136.11.1 DerivationofSpringandDamperRates... 137.11. PitchLinkStiffness... 139.11.3 ImplementationintoFiniteElementMethod... 14.1LoadsCalculation... 14.1.1 ReactionForceMethod... 144.1. ForceSummationMethod... 149.1.3 CalculationofBladeRootLoadswithConstraintEquations... 157.1.4 Calculation of Blade Velocities and Accelerations..... 158.1.5 HubLoadsCalculation... 159.1.6 Rotor Response and Load Harmonics..... 16.13VehicleTrim... 16.13.1 Helicopter Propulsive Trim Calculation.... 165.13. Helicopter Wind Tunnel Trim.... 168 3 Comparison of Analytical Models 17 3.1 Importance of Nonlinear Aerodynamics Model.... 17 3.1.1 Aerodynamics Comparison at High Thrust... 171 3.1. Aerodynamics Comparison at Low Thrust... 179 3. Comparison of Loads Calculation Methods...... 186 3..1 LoadsComparisonforaHingelessRotor... 186 3.. Loads Comparison for an Articulated Rotor... 189 3..3 LoadsCalculationSummary... 198 vi
4 Elastic Tailoring Results and Discussion 4.1 GeneralDesignConsiderations... 4. StallReductionforaHingelessHelicopterRotor... 3 4..1 HingelessRotorBaselineDesignProcedure... 3 4.. HingelessRotorElasticallyTailoredDesign... 5 4..3 Hingeless Rotor Uncoupled and Coupled Modes..... 7 4..4 Hingeless Rotor High Speed Flight Condition... 1 4..5 Hingeless Rotor Low Speed Flight Condition... 18 4.3 Stall Reduction for an Articulated Helicopter Rotor... 7 4.3.1 Articulated Rotor Baseline Design Procedure... 7 4.3. Articulated Rotor Elastically Tailored Design... 9 4.3.3 Articulated Rotor Uncoupled and Coupled Modes..... 9 4.3.4 Articulated Rotor High Speed Flight Condition...... 31 4.3.5 Articulated Rotor Low Speed Flight Condition...... 44 4.4 SensitivityStudies... 51 4.4.1 SensitivitytoRotorFrequencies... 51 4.4. SensitivitytoMagnitudeofCouplings... 65 5 Conclusions and Recommendations 71 5.1 AnalyticalModel... 7 5. EffectsofElasticCouplings... 7 5.3 CalculationofBladeandHubLoads... 73 5.4 RecommendationsforFutureWork... 74 Bibliography 77 Appendix Composite Cross Section Stiffness Matrix Coefficients 83 vii
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List of Figures 1.1 Elastic twist with and without pitch-flap coupling, μ =.35, C T =ff =:7... 16 1. Lag damping as a function of blade pitch angle for negative pitch-flap coupled configuration (Ω=8RPM)... 19 1.3 Comparison of measured and predicted symmetric wing bending mode stability boundaries for the 1/5 scale baseline and tailored wings in the off-the-downstop condition... 7 1.4 Blade tip twist with different aerodynamics models, μ=.35, C T =ff=.7, M tip =.7 3.1 Transformation between the hub-fixed, non-rotating and rotating coordinate systems 37. Euler angles for transformation from deformed to undeformed coordinate system. 38.3 Movement of a point from the undeformed frame to the deformed frame... 39.4 Blade principal axes and definitions of offsets of center of gravity and aerodynamic centerfromtheelasticaxis... 48.5 Finiteelementdegreesoffreedomandintegrationintoblademodel... 55.6 Definitions of longitudinal and transverse directions and ply angle in a composite plate... 65.7 Relationship between n,s,ο coordinate systems for Vlasov theory and ο,, system forblade... 68.8 Force and moment resultants and displacements for a composite plate.... 69.9 Blade displacements and forces... 69.1Aplatesegmentinthebladecross-section... 71.11 Velocities and blade airloads at an airfoil section.... 79.1 Center of Pressure variation for UMARC and GCHAS implementations... 17 ix
.13 Finite element discretization for an eight element, three node per element model.. 18.14 Graphical Depiction of Unconstrained Finite Element in Time Matrix.... 135.15 Graphical Depiction of Constrained Finite Element in Time Matrix...... 136.16 Forces and moments on the helicopter for propulsive trim..... 163 3.1 Trim controls for quasi-steady and full non-linear aerodynamics models, μ=.35, C T =ff=.1... 171 3. Blade response for quasi-steady and full non-linear aerodynamics models, x=.99r, μ=.35, C T =ff=.1... 17 3.3 Lift, drag, and moment airloads for quasi-steady and full non-linear aerodynamics models, x=.99r, μ=.35, C T =ff=.1... 173 3.4 Angle of attack for baseline hingeless rotor with quasi-steady aerodynamics, μ=.35, C T =ff=.1... 175 3.5 Angle of attack distribution for baseline hingeless rotor with non-linear aerodynamics model, μ=.35, C T =ff=.1... 175 3.6 Lift distribution for baseline rotor with quasi-steady aerodynamics, μ=.35, C T =ff=.1... 176 3.7 Lift distribution for baseline rotor with non-linear aerodynamics, μ=.35, C T =ff=.1... 176 3.8 Lift coefficient vs. angle of attack for quasi-steady and non-linear aerodynamics, μ=.35, C T =ff=.1,x/r=.99r... 177 3.9 Pitch link loads for quasi-steady and full non-linear aerodynamics models, μ=.35, C T =ff=.1... 178 3.1 Comparison of rotor power for quasi-steady and non-linear aerodynamics, C T =ff=.1... 178 3.11 4/rev vibratory loads for quasi-steady and full non-linear aerodynamics models, μ=.35, C T =ff=.1... 179 3.1 Control settings for quasi-steady and full non-linear aerodynamics models, μ=.35, C T =ff=.7... 18 x
3.13 Rotor response for quasi-steady and full non-linear aerodynamics models, x=.99r, μ=.35, C T =ff=.7... 181 3.14 Section airloads for quasi-steady and full non-linear aerodynamics models, x=.99r, μ=.35, C T =ff=.7... 18 3.15 Angle of attack distribution for baseline hingeless rotor with quasi-steady aerodynamics model, μ=.35, C T =ff=.7... 183 3.16 Angle of attack distribution for baseline hingeless rotor with non-linear aerodynamics model, μ=.35, C T =ff=.7... 183 3.17 Lift coefficient vs. angle of attack for quasi-steady and non-linear aerodynamics, μ=.35, C T =ff=.7,x/r=.99r... 184 3.18 Pitch link loads for baseline hingeless rotor with quasi-steady and full non-linear aerodynamics models, μ=.35, C T =ff=.7... 184 3.19 4/rev vibratory loads for baseline hingeless rotor with quasi-steady and full nonlinear aerodynamics models, μ=.35, C T =ff=.7... 185 3. Rotating frame hub loads calculated by the force summation method and reaction force method for hingeless rotor in physical space and modal space, μ =.35, C T =ff=.7... 188 3.1 Hingeless rotor torsion moments at ψ = ffi and ψ = 18 ffi using the force summation method and the reaction force method, μ=.35, C T =ff=.7... 19 3. Hingeless rotor flapwise bending moments at ψ = 9 ffi and ψ = 7 ffi using the force summation method and the reaction force method, μ=.35, C T =ff=.7... 191 3.3 Hingeless rotor chordwise bending moments at ψ = ffi and ψ = 18 ffi using the force summation method and the reaction force method, μ=.35, C T =ff=.7... 19 3.4 Rotating frame hub loads calculated by the force summation method and reaction force method for an articulated rotor in physical space and modal space, μ =.35, C T =ff=.7... 194 3.5 Articulated rotor torsion moments at ψ =9 ffi and ψ = 7 ffi using the force summation method and the reaction force method, μ=.35, C T =ff=.7... 196 xi
3.6 Articulated rotor flapwise bending moments at ψ = 9 ffi and ψ = 7 ffi using the force summation method and the reaction force method, μ=.35, C T =ff=.7... 197 3.7 Articulated rotor chordwise bending moments at ψ =9 ffi and ψ = 7 ffi using the force summation method and the reaction force method, μ=.35, C T =ff=.7... 199 4.1 Layup of the spar and skin portions of the NACA 15 cross section..... 4. Flapmodesforbaselinehingelessblade... 8 4.3 Lagmodesforbaselinehingelessblade... 9 4.4 Torsionmodesforbaselinehingelessblade... 1 4.5 Flapmodesforelasticallytailoredhingelessblade... 11 4.6 Lagmodesforelasticallytailoredhingelessblade... 1 4.7 Torsionmodesforelasticallytailoredhingelessblade... 13 4.8 Response of baseline and elastically tailored hingeless rotors, x=.99r, μ=.35, C T =ff=.1... 13 4.9 Trim settings for baseline and elastically tailored hingeless rotors, μ=.35, C T =ff=.1... 14 4.1 Angle of attack distribution for baseline and elastically tailored hingeless rotors, μ=.35, C T =ff=.1... 15 4.11 Dynamic stall shed vortex distribution for baseline hingeless rotor, μ=.35, C T =ff=.1... 16 4.1 Dynamic stall shed vortex distribution for elastically tailored hingeless rotor, μ=.35, C T =ff=.1... 16 4.13 Comparison of shed vortex strength at tip of baseline and elastically tailored hingeless rotor blades, x=.99r, μ=.35, C T =ff=.1... 17 4.14 Comparison of moment coefficient at tip of baseline and elastically tailored hingeless rotor blades, x=.99r, μ=.35, C T =ff=.1... 18 4.15 Comparison of pitch link loads of baseline and elastically tailored hingeless rotor blades, μ=.35, C T =ff=.1... 19 4.16 Comparison of vibratory loads of baseline and elastically tailored hingeless rotor blades, μ=.35, C T =ff=.1... 19 xii
4.17 Response of baseline and elastically coupled hingeless rotors, x=.99r, μ=.5, C T =ff=.1... 4.18 Trim settings for baseline and elastically tailored hingeless rotors, μ=.5, C T =ff=.1... 1 4.19 Angle of attack distribution for baseline hingeless rotor, μ=.5, C T =ff=.1... 3 4. Angle of attack distribution for elastically tailored hingeless rotor, μ=.5, C T =ff=.1... 3 4.1 Dynamic stall shed vortex distribution for baseline hingeless rotor, μ=.5, C T =ff=.1... 4 4. Dynamic stall shed vortex distribution for elastically tailored hingeless rotor, μ=.5, C T =ff=.1... 4 4.3 Comparison of shed vortex strength at tip of baseline and elastically tailored hingeless rotor blades, x=.99r, μ=.5, C T =ff=.1... 5 4.4 Comparison of pitch link loads of baseline and elastically tailored hingeless rotor blades, μ=.5, C T =ff=.1... 5 4.5 Comparison of vibratory loads of baseline and elastically tailored hingeless rotor blades, μ=.5, C T =ff=.1... 6 4.6 Flap modes for baseline articulated blade... 3 4.7 Lag modes for baseline articulated blade... 31 4.8 Torsion modes for baseline articulated blade...... 3 4.9 Flap modes for baseline articulated blade... 33 4.3 Lag modes for baseline articulated blade... 34 4.31 Torsion modes for baseline articulated blade...... 35 4.3 Trim controls for baseline and elastically tailored articulated rotors, μ=.35, C T =ff=.9... 35 4.33 Blade tip response for baseline and elastically tailored articulated rotors, x=.99r, μ=.35, C T =ff=.9... 36 4.34 Angle of attack distribution baseline articulated rotor, μ=.35, C T =ff=.9... 37 xiii
4.35 Angle of attack distribution for elastically tailored articulated rotor, μ=.35, C T =ff=.9... 37 4.36 Dynamic stall shed vortex strength for high speed baseline articulated rotor, μ = :35, C T =ff=.9... 38 4.37 Dynamic stall shed vortex strength for high speed elastically tailored articulated rotor, μ=.35, C T =ff=.9... 38 4.38 Dynamic stall shed vortex strength for baseline and elastically tailored articulated rotors, x/r =.91, μ=.35, C T =ff=.9... 39 4.39 Lift, drag, and moment airloads for baseline and elastically tailored articulated rotors, x/r =.91, μ=.35, C T =ff=.9... 39 4.4 Dynamic stall shed vortex strength for baseline and elastically tailored articulated rotors, x/r =.91, μ=.35, C T =ff=.9... 4 4.41 Pitch Link Loads for baseline and elastically tailored articulated rotors, x/r =.91, μ=.35, C T =ff=.9... 41 4.4 Nonlinear separation point for baseline articulated rotor, μ=.35, C T =ff=.9... 4 4.43 Nonlinear separation point for elastically tailored articulated rotor, μ=.35, C T =ff=.9... 4 4.44 4/rev vibratory loads for baseline and elastically tailored articulated rotors, μ=.35, C T =ff=.9... 43 4.45 Trim controls for baseline and elastically tailored articulated rotors, μ=.5, C T =ff=.115... 44 4.46 Blade tip response for baseline and elastically tailored articulated rotors, x =.99R, μ=.5, C T =ff=.115... 46 4.47 Lift, drag, and moment airloads for baseline and elastically tailored articulated rotors, μ=.5, C T =ff=.115, x/r =.91.... 46 4.48 Angle of attack distribution baseline articulated rotor, μ=.5, C T =ff=.115.... 47 4.49 Angle of attack distribution for elastically tailored articulated rotor, μ=.5, C T =ff=.115... 47 xiv
4.5 Dynamic stall shed vortex strength for baseline articulated rotor, μ=.5, C T =ff=.115... 48 4.51 Dynamic stall shed vortex strength for elastically tailored articulated rotor, μ=.5, C T =ff=.115... 48 4.5 Dynamic stall shed vortex strength for baseline and elastically tailored articulated rotors, μ=.5, C T =ff=.115, x/r =.91... 49 4.53 Dynamic stall shed vortex strength for baseline and elastically tailored articulated rotors, μ=.5, C T =ff=.115, x/r =.91... 49 4.54 4/rev vibratory loads for baseline and elastically tailored articulated rotors, μ=.5, C T =ff=.115... 5 4.55 Variation in magnitude and phase of elastic twist with second flap frequency for baseline and coupled hingeless rotors, μ =.35, C T =ff=.7... 5 4.56 Higher bending modes for hingeless blade with highly coupled third flap mode.. 54 4.57 Variation in magnitude and phase of elastic twist with second flap frequency for baseline and coupled hingeless rotors (tip up, nose up couplings), μ =.35, C T =ff=.7... 56 4.58 Variation of 4/rev hub shears with second flap frequency for baseline and elastically tailored hingeless rotors with highly coupled third flap mode (tip up, nose up couplings), μ=.35, C T =ff=.7... 57 4.59 Variation in magnitude and phase of elastic twist with second flap frequency for baseline and coupled hingeless rotors (tip up, nose up couplings), μ =.35, C T =ff=.7... 58 4.6 Variation of 4/rev hub shears with second flap frequency for baseline and elastically tailored hingeless rotors with highly coupled third flap mode (tip up, nose up couplings), μ=.35, C T =ff=.7... 59 4.61 Elastic bending modes for coupled articulated blade... 61 4.6 Variation in magnitude and phase of elastic twist with second flap frequency for baseline and coupled articulated rotors, μ =.35, C T =ff=.7... 63 xv
4.63 Variation of 4/rev hub shears with second flap frequency for baseline and elastically tailored articulated rotors, μ=.35, C T =ff=.7... 64 4.64 Variation in magnitude and phase of elastic twist with magnitude of coupling for hingeless rotor, μ =.35, C T =ff=.7... 66 4.65 Variation of 4/rev hub shears with coupling magnitude for hingeless rotor, μ=.35, C T =ff=.7... 67 4.66 Variation in magnitude and phase of elastic twist with magnitude of coupling for articulated rotor, μ =.35, C T =ff=.7... 69 4.67 Variation of 4/rev hub shears with coupling magnitude for articulated rotor, μ=.35, C T =ff=.7... 7 xvi
List of Tables 1.1 Namingconventionforpitch-flapandpitchlagcouplings... 8.1 Orderoftermsusedinaeroelasticanalysis... 4. Basic non-dimensionalization parameters... 41.3 Complex non-dimensionalization factors... 41.4 Numberingconventionforcompositestiffnesscoefficients... 64.5 Unsteady Aerodynamics Deficiency Constants..... 94.6 Linear Coefficients of U R... 114.7 Linear Coefficients of U P... 114.8 Linear Coefficients of U T... 114.9 Matrix sizes for finite element in time analysis with example 8 elements, 8 modes, and 3 nodes per element..... 136 4.1 Composite blade material properties..... 1 4. Hingeless Rotor and Vehicle Properties... 4 4.3 Laminate Properties for Baseline Hingeless Blade... 5 4.4 Natural Frequencies for Baseline and Elastically Tailored Hingeless Blades.... 5 4.5 Blade Properties for Coupled Hingeless Blade..... 6 4.6 Elastic twist harmonics for baseline and coupled rotors, x=.99r, μ=.35, C T =ff=.1... 13 4.7 Elastic twist harmonics for baseline and coupled rotors, x=.99r, μ=.5, C T =ff=.1... 1 4.8 Laminate Properties for Articulated Rotor Blades... 8 4.9 Articulated Rotor and Vehicle Properties... 8 xvii
4.1 Natural Frequencies for Baseline and Elastically Tailored Articulated Blades.... 3 4.11 Elastic twist harmonics for baseline and coupled articulated rotors, μ=.35, C T =ff=.9... 35 4.1 Properties for Hingeless Rotor Blades with Highly Coupled Third Flap Mode... 54 4.13 Properties for Articulated Rotor Blades... 6 4.14 Natural Frequencies for Baseline and Elastically Tailored Articulated Blades.... 61 xviii
List of Symbols A 1 A 5 [A]; [B]; [D] [C] C P C Q C T C c C d C l C m C n C v C [E]; [F ]; [H] ff g F A F ;F ;G ;G H H 1 H 4 H v ;H w ;H^ffi ^I I ; ^J I ; ^K I Unsteady aerodynamics constants Classical laminated plate theory constitutive matrices Damping matrix Power coefficient Torque coefficient Thrust coefficient Chord force coefficient Drag coefficient Lift coefficient Moment coefficient Normal force coefficient Vortex lift coefficient Lag damper rate Higher order laminated plate constitutive matrices Force vector Axial force Higher order shear forces in composite model Rotor longitudinal force Hermitian polynomial shape functions Lag, flap, and torsion shape functions (respectively) Unit vectors in the inertial coordinate system xix
^I H ; ^J H ; ^K H Unit vectors in the hub-fixed, non-rotating coordinate system ^I; I b I I J ^J; ^K Unit vectors in the hub-fixed rotating coordinate system Blade flap inertia Blade lag inertia Blade torsion inertia Jacobian of transformation for Gauss quadrature [K] K ij K K L L 1 L 3 L u L () () M Stiffness matrix Composite stiffness coefficient Torsion spring stiffness Lag spring stiffness Element Length Lagrange polynomial shape functions Axial shape function Lift Mach number M () () Moment [M ] Mass matrix M ;M M! N N 1 N 6 N b P ο ;P οs Q ο fqg R R ο S Flap and lag moments in composite model Warping moment in composite model Axial blade force for composite model Time shape functions Number of blades Higher order shear force resultant in composite model Higher order moment resultant in composite model Vector of reaction forces Rotor radius Higher order moment resultant in composite model Aerodynamic time xx
St T T I T TR T DU T RH T UH T UR T f T prop. T s T v T vl U R ;U T ;U P U x ;U y ;U z V T * V =V * V b * V w W W g X I ;Y I ;Z I X H ;Y H ;Z H Strouhal number for vortex shedding in unsteady aerodynamics model Rotor thrust Time constant for lift deficiencies Tail rotor thrust Rotation matrix between the deformed coordinate system and undeformed coordinate system Rotation matrix between the hub-fixed, rotating coordinate system and the hubfixed, non-rotating coordinate system Rotation matrix between the undeformed coordinate system and the hub-fixed, nonrotating coordinate system Rotation matrix between the undeformed coordinate system and the hub-fixed, rotating coordinate system Time constant for nonlinear separation Propeller thrust Torsion moment in composite model Time constant for vortex shedding Aerodynamic time for shed vortex to reach trailing edge Relative wind velocity at a point, expressed in the deformed coordinate system Relative wind velocity at a point, expressed in the undeformed coordinate system Rotor tip speed (= ΩR) Total relative wind velocity (vector/scalar) Relative wind velocity due to blade motion Relative wind velocity due to blade rotation and vehicle forward speed Vehicle weight Weight for Gauss quadrature Displacements in the inertial coordinate system Displacements in the hub-fixed, non-rotating coordinate system X; Y; Z Displacements in the hub rotating coordinate system X (1) X (5) a Lift deficiency functions Reference lift curve slope xxi
b 1 b 5 c c c 1 c d d e d e g e f f f 1 ^ ; ^ ; ^k h k ;k 1 ;k k m1 ;k m m m q Unsteady aerodynamics constants Chord length Constant lift coefficient Linear lift coefficient Nonlinear lift coefficient Constant drag coefficient Nonlinear drag coefficient Offset of blade elastic axis and aerodynamic center (positive ac aft) Offset of blade elastic axis and center of gravity (positive cg forward) Lag hinge offset Flow separation point along chord Constant moment coefficient Linear moment coefficient unit vectors in the undeformed blade coordinate system Rotor height above vehicle cg Unsteady pitching moment constants Radii of gyration of the blade cross-section in the and directions Blade mass per unit length Blade reference mass per unit length Generalized coordinates x; y; z Displacements in the undeformed blade coordinate system * r Generic position vector _* r r Λ Ω ff Derivative of position vector relative to rotating coordinate system Radial skew angle for yawed flow Rotational speed Angle of attack ff 1 Angle of attack at which flow separation is at.7c ff E ff s Effective angle of attack accounting for lift deficiencies Longitudinal tilt of rotor shaft (positive forward) xxii
fi fi; ; fi 1c fi 1s fi p Prandtl-Glauert correction factor Euler angles for rotation between deformed and undeformed coordinate systems Longitudinal flapping Lateral flapping Blade precone angle r Offset between elastic axis and aerodynamic computation point (positive toward leading edge) t Finite element in time degrees of freedom ffl () ffl ο ;ffl οs ;ffl nο ffl ο ;ffl ο ;ffl ο ;ffl fl Indicator of order for ordering scheme Plate inplane strains in composite model, contour coordinates Plate inplane strains in composite model, orthogonal coordinates Lock number» x ;» y Drees inflow parameters» ο ;» οs ;» nο Plate inplane strains in composite model Rotor inflow (total) i Rotor induced inflow T Warping deformation μ ^ffi ffi s ffi ;ffi ;ffi xi ψ ρ ff ff 1 ;ff ;ff 3 fi v Advance ratio Blade elastic twist about deformed elastic axis Lateral tilt of rotor shaft (positive toward advancing side) Rotations about flap, lag, and torsion axes in composite model Azimuth angle Air density Rotor solidity Modification factors for aerodynamic decay rates Aerodynamic time since vortex detachment Blade collective pitch 1c Longitudinal cyclic pitch 1s Lateral cyclic pitch xxiii
T Built-in blade twist c Angle between contour coordinate and horizontal in composite model fp Vehicle flight path angle d Damping ratio of single degree of freedom system ο g ο; ; Gauss point location for Gauss quadrature Displacements in the deformed coordinate system ο; n; s Displacements in the contour coordinate system (composite model) () A Aerodynamic quantity () C Circulatory quantity () F Quantity associated with the fuselage () I Inertial quantity () R Quantity associated with the rotor () V Vehicle quantity () i Non-circulatory (impulsive) quantity () i Quantity at current iteration () n Quantity at current azimuth/time () _ Derivative with respect to time Λ () μ() Derivative with respect to azimuth Non-dimensional quantity () Derivative with respect to space; in unsteady aerodynamics differentiates quantities modified by different deficiency functions, see accompanying text xxiv
Dedication This thesis is dedicated to my loving wife Sabrina. xxv
Acknowledgements As is customary with PhD students, there is a long list of people and entities whom I would like to thank for helping me along in my graduate work. Without these people, the road to my dissertation would have been considerably rockier. Hopefully I will not leave anyone out since there are so many. First, I would like to thank the Army Research Office and technical monitor Gary Anderson for funding the initial stages of my work. Likewise, I would like to thank my adviser, Dr. Edward Smith, for finding creative ways to pay my salary after the Army Research Office contract ran out. Specifically, I would like to thank the people at the Leonhard Center and the ECSEL Project at Penn State, Boeing Helicopters, technical monitor Steve Glusman, and Sikorsky Aircraft, technical monitor Bob Goodman, for funding the remaining years of my study. Second, there are many co-workers who helped me figure out answers to difficult technical and operational questions. First and foremost, Jon Keller, for answering all of my research-related and goofball questions for several years now. The others I refer to as the elders since for a long time, it seemed like none of us would ever graduate, Chris Brackbill, Anna Howard, and Patty Stevens. These people helped me numerous times when I was fighting with MATLAB, Microsoft Word (before my conversion to LATEX) and various difficult mathematical or helicopter-related concepts, as well as proofreading my documents and presentations. Third, I would like to thank those who gave me computer help. Specifically, Eric Hathaway, Anirudh Modi, and Anurag Agarwal for answering my endless Linux questions, and Eric Hathaway and Dr. Gary Gray for helping me with numerous L A TEX issues. Using L A TEX has truly made my thesis writing experience much more enjoyable than it would otherwise have been. I also feel I should thank the numerically intensive computing group at Penn State who helped me get past xxvi
those nagging FORTRAN issues and progress in the endless process of debugging my code. Fourth, I must thank all of the support staff in the aerospace department, specifically Debby Mayes and Debbie Jacobs for doing all of those little day to day tasks that need to be done. It was nice having someone to look out for me when I miss a deadline by one day. Finally, I would like to thank my wonderful wife Sabrina for leaving her nice job in Harrisburg to return to State College and live with me for over a year after I had hoped to finish and leave. Her constant encouragement was a great help in the years before we were married. Having her around makes it that much easier to get up in the morning come to work. xxvii
Chapter 1 Introduction Composite blades have been used on many helicopters for the past thirty years. The advantages of using composite materials in the rotor system are now well known. Chapter 1 of Reference [1] provides a comprehensive history of the introduction of composite materials into rotor systems. Therefore, only a brief overview leading to the motivation for the current work is presented in the first section. The second section describes more recent work in rotor dynamics. Specifically, three main topics are covered. The first is the passive modification of elastic twist, followed by analysis of composite wings and blades, and finishing with the advanced aerodynamics model which is used in the present work. The last section specifically lists the objectives of the current study. 1.1 Background and Motivation The unique properties of advanced composite materials, such as graphite/epoxy and glass/epoxy make them ideal for use in the rotor environment. When first introduced, composite materials were predominantly used as black aluminum, meaning that the composite blades were designed to perform identically to metal blades, only made out of composite materials. Even now, although blades are designed with the properties of composites in mind, their full potential is far from being realized. The primary reason for this is that analysis methods to predict the behavior of composites have not been available or have not been validated to the point where the industry is comfortable using them. Because of this, composite designs tend to be extremely conservative. 1
CHAPTER 1. INTRODUCTION The main reason for the switch to composite rotor blades over metal blades is their superior fatigue properties. For the same weight, composite materials are much stiffer and/or stronger than metals. Because of this, the high vibration and high fatigue environment of the helicopter rotor is ideally suited for their use. Using composite materials, rotor blades can be constructed to last much longer than metal blades or even be designed for infinite life meaning that they will last as long as the helicopter. Other important considerations relate to fabrication methods for composite blades versus metal blades. Typically, to manufacture a metal blade or propeller, the blade is molded without any builtin twist, then mechanically twisted to produce a certain linear twist. Producing non-linear twist or other complex geometric features is difficult and expensive. Composites are normally manufactured by either placing them inside molds or wrapping them around plugs or mandrels. Because of this, manufacturing of complex shapes can be much less costly. There is a high start-up cost developing the tooling (i.e. the mold or plug), but then the same tooling can be used to manufacture many composite parts before wearing out. This has allowed for variation in thickness, airfoil shape, and built-in twist with span to be practical. An important property of composites that has not been fully exploited is their anisotropy. Onedimensional (fiber) and two-dimensional (flake) composites have stiffness properties in one or two directions which are very different than properties in the other direction(s). This allows the designer substantial freedom to tailor the stiffness properties of the structure. A structure can be built to be very stiff in bending but soft in torsion or vice versa. The designer has the ability to design in any ratio he or she wishes rather than being constrained to the 1/3 rule of metals. Additionally, these properties can be exploited to produce elastic couplings. In the fixed-wing community, bending-torsion coupled wings allowed the experimental X-9 aircraft to fly to supersonic speeds with forward swept wings without encountering a divergence instability. Elastic couplings have not been used on any production helicopters to date. More recently, composite materials have allowed for hingeless and bearingless rotor designs. In such a design, the complex sequence of hinges that accommodate the flap and lag motion, and possibly the pitch motion, are replaced with a single torsionally soft flexbeam and a torque tube. This reduces hub drag, complexity, maintenance, and cost. There is an additional advantage in
1.1. BACKGROUND AND MOTIVATION 3 corrosive environments such as shipboard operation, or even in extremely cold environments, where the composite flexures are not subject to the same problems as hinges. While the use of advanced composite materials has allowed the blades to survive the high vibration environment without failing, the problem still remains that the rotor vibrations are transmitted to the fuselage, shortening the life of the mechanical and electronic components and causing discomfort to passengers and crew. This problem is unique to rotorcraft due to the size and rotating frequency of the main rotor. While other vehicles, such as propeller-driven fixed-wing aircraft, also have large rotating propellers, they are much smaller relative to the size of the vehicle and rotate at a much higher speed, which is less annoying to the passengers. Vibration in the cabin remains a major problem in helicopter design and operation. With identical blades, many of the vibrations in each blade are canceled by other blades. The rotor is carefully tracked and balanced to remove much of the vibration that would otherwise be transmitted. Tracking is normally performed by adjusting a trailing edge tab so that all of the blades develop the same lift for the same collective pitch. Balancing involves adjusting small weights in the blade tip so that the large centrifugal forces are equal for each blade. Despite careful tracking and balancing, vibration at integer multiples of the rotating frequency is still transmitted to the fuselage. For example, three-bladed helicopter transmits the most vibration to the fuselage at 3/rev, with successively less vibration at 6/rev, 9/rev and so on. These vibrations represent dynamic variations which the blades experience simultaneously, and cannot be eliminated by tracking and balancing. Another unique problem associated with rotorcraft is that of dynamic stall. The rotor blades are constantly in dynamic flap, lag, and pitch motion, to maintain steady rotor forces in forward flight. At high speed and/or high thrust, this can cause a phenomenon called dynamic stall. In dynamic stall, a combination of high angle of attack and a positive pitch rate temporarily keeps the airfoil from stalling, and a vortex forms at the leading edge. At a critical condition, the airfoil does stall, and the vortex detaches and is shed into the wake, producing abrupt changes in the lift and moment airloads. Dynamic stall can be felt in fixed-wing aircraft when a whip stall is performed where the angle of attack of the vehicle is increased rapidly to stall. The vortex shedding can be felt as a bump just as the wings stall. While this is rarely experienced in a fixed-wing aircraft, in a helicopter
4 CHAPTER 1. INTRODUCTION at high thrust, each blade can experience dynamic stall at every revolution. One solution to the vibration problem is to use a large number of blades. The higher frequency vibration is both smaller in magnitude and physiologically less disturbing to passengers. Dividing the total thrust between more blades also causes dynamic stall of one blade to have less influence on the total system. Helicopters with six or seven blades are much more comfortable to occupants than those with two or three blades. There has been a trend toward more blades, and currently there are few helicopters in production with less than four blades. Unfortunately, increasing the number of blades further increases the complexity and cost of the helicopter, and is not a preferable solution. A different solution which has received significant research attention in the past ten years is the canceling of vibration at the source by active control. Strategies such as higher harmonic control or individual blade control have been proposed. In these applications, the blades receive pitch inputs from either devices in the control system, high frequency oscillations of the swash plate, or aerodynamic devices such as trailing edge flaps. In Reference [], Wilkie et al recently proposed using active fibers in the rotor blade laminates to induce dynamic twist. While active control has been promising in simulation, there are still many challenges in its implementation. It has been very difficult to design actuators which are capable of producing the forces or moments required to allow the active devices to work, fit in a small area (such as inside a rotor blade), and withstand the high loads of the rotor environment. Equally important is the issue of safety. When using any active system, the designer must also address the question of failure of the active system. Helicopter manufacturers have resisted placing actuators for vibration control in the primary flight control system for this reason. Third is the issue of power. Actuators must receive either electrical or mechanical power which must be supplied by a power supply or some type of linkage to the engine power, increasing vehicle weight and decreasing payload. Another possibility which avoids some of these problems is that of passive vibration control. Passive treatments can be divided into discrete devices, such as absorbers, or more integral treatments, such as constrained layer damping or more integral still, structural optimization or elastic tailoring. In many ways, a passive method is superior to an active method. Rather than placing active components in the structure, the structural design is instead modified to produce the desired charac-
1.. SUMMARY OF PREVIOUS RESEARCH 5 teristics. An example is structural optimization, where a computer simulation attempts to calculate a design meeting certain criteria iteratively. The passive approach is more fail-safe because there are no discrete actuators or power sources which can fail. With passive devices such as absorbers, the possibility of failure is present, but for elastic tailoring, the vibration treatment fails only when the major structural component itself fails. As with active methods, there are difficulties with passive methods. The first is that the authority is normally much less than that of an active system. It is easier to input energy to a system and have an actuator force the system to behave as desired than it is to design the structure to behave correctly by itself. Second, the passive design cannot be turned off like an active system. The design must be robust to changes in flight condition and not perform poorly at off-design conditions. Despite these challenges, all else being equal, a passive design is preferable to an active design. This dissertation explores the possibility of using composite couplings to passively alleviate blade stall and reduce vibration of rotor systems. 1. Summary of Previous Research This section describes previous research pertinent to the current study. The section starts with the aeroelastically conformable rotor study, where permutations on blade properties were examined in an attempt to improve rotor performance. The next section addresses the progress in modeling composite rotor blades. Then, two sections describe several studies examining stability of coupled composite rotors and optimization studies attempting to reduce vibration through composite tailoring. Next, there is a section discussing some recent advances in composite tailoring for fixed wing and tilt rotor aircraft. The last section is a review of previous research concerning the advanced aerodynamic model used in the current study. 1..1 Aeroelastically Conformable Rotor In the late 197 s and early 198 s, a study was conducted to explore the possibility of improving performance in helicopters by passively shaping the elastic twist of the blades. The project was termed the aeroelastically conformable rotor. In the study, the objective was to optimize the blade
6 CHAPTER 1. INTRODUCTION elastic twist to achieve maximum lift at each airfoil section without exceeding the drag divergence angle of attack. In an analytical study (Refs. [3] and [4]), researchers attempted to define an ideal rotor twist distribution and quantify the effects of such a distribution on the rotor performance and loads. Methods to produce favorable twist were also analyzed. The rotor they examined was similar to that of a Sikorsky UH-6 Blackhawk helicopter. Variations in the tip sweep, airfoil section camber, locations of the center of gravity and aerodynamic center relative to the elastic axis, and the torsional stiffness were examined. A low speed design point of μ =:1 and C T =ff =:1, and a high speed design point of μ =:3 and C T =ff =:85 were examined. One of the important observations was that tip sweep reduces the angle of attack on the advancing tip in high speed flight. This is because the advancing tip is often at a negative angle of attack, and the tip sweep means the center of lift is behind the pitch axis, causing a nose-up moment. Conversely, in hover and low speed, when the entire rotor is at positive angles of attack, the swept tip produces a nose-down moment, causing increased elastic windup, in addition to the built-in negative twist along the span. Effects of camber were also examined. To increase the effectiveness of the camber, the torsional stiffness of the outer half of the blade was reduced by a factor of four (GJ :5R R = GJ =4). The combination of a torsionally soft blade and reverse (nose-up) camber tended to reduce the pitch link loads and significantly reduced the flap bending loads, but caused an increase in total power required. Moving the center of gravity had little effect on the loads and power, but the movement of the aerodynamic center produced substantial twist. A 1/rev elastic twist variation was produced outboard, with a /rev variation inboard. Based on these observations, two optimum designs were selected. The first featured a swept tip and the torsion stiffness of the outboard half of the blade was 1/4 of the baseline stiffness. The second design was similar to the first, only the aerodynamic center was also moved 15% of the chord ahead of the elastic axis from 5% span to the tip. This was accomplished by increasing the chord in this region but keeping the trailing edge straight. Some important guidelines on modifying rotor parameters emerged from the study. First, it
1.. SUMMARY OF PREVIOUS RESEARCH 7 is beneficial for blade loads to produce 1/rev twist which reduces the advancing blade twist and increases the retreating blade twist. Second, the combination of tip sweep and reduced torsional stiffness outboard is one way of achieving such a twist distribution. It produced a power savings of about /3 of that predicted for the ideal rotor twist. Negative camber reduces the flap bending moments but produces /rev and 3/rev twist of undesirable phase which degrades the rotor performance. Placing the aerodynamic center ahead of the elastic axis from 6% to 9% span produces beneficial /rev twist, but results in a flap-torsion instability, requiring the addition of tip sweep to stabilize the rotor, and the vibratory loads increase. Finally, the swept tip improves the hover performance by creating large elastic windup to beneficially redistribute airloads. The effects of camber, swept tips, and movement of the aerodynamic center were subsequently tested experimentally in the Transonic Dynamics Tunnel (TDT) at NASA Langley Research Center [5, 6]. A 9-foot diameter articulated rotor model with interchangeable tips, also based loosely on the Sikorsky UH-6 Blackhawk rotor was used in the test. The baseline and modified blades had the same flap frequencies, but different torsion frequencies due to the torsional softening of the outer half of the modified blades. The baseline blade had a swept tip which produced 1 ffi - ffi of steady nose-down twist and about 1 ffi of 1/rev twist. Variations included an adjustable trailing edge tab to introduce reverse camber, a rectangular tip, and a swept tapered tip with anhedral. The analytical results were mostly supported by the wind tunnel test. The reflex camber/swept tip combination produced a 1%-1% power savings over the baseline, which was not predicted, but reduced the flap bending moments by 6% without increasing the torsion loads as the analysis showed. The mechanism for this reduction is the untwisting of the advancing blade. In the baseline configuration, the advancing tip is bent downward due to the negative angle of attack at the tip. The reverse camber reduces the downward bending and thus the blade bending moment. Tip sweep also reduced the flap bending moments and torsion moments. Both tip sweep and reflex camber tended to reduce the vibratory hub loads. In general, untwisting the advancing blade was beneficial for reducing the flap bending and root torsion moments. Tip sweep provides nose-down steady twist which increases with thrust and a nose-up 1/rev twist near the second quadrant. The anhedral swept tip produced nearly the same results as the
8 CHAPTER 1. INTRODUCTION Table 1.1: Naming convention for pitch-flap and pitch lag couplings Coupling type Analysis Properties Fibers point toward... Description Positive pitch-flap ffi 3 >, K 5 > trailing edge pitch down, flap up Negative pitch-flap ffi 3 <, K 5 < leading edge pitch up, flap up Positive pitch-lag ffi 4 <, K 35 < lower surface pitch down, lag back Negative pitch-lag ffi 4 >, K 35 > upper surface pitch up, lag back other swept tip configurations. Reflex camber produces nose-up steady twist, nose-up 1/rev twist on the advancing side, and nose-up /rev on the advancing and retreating sides. 1.. Modeling of Composite Blades The next several sections address elastic couplings in composite blades. Many different researchers from several different universities are cited, with different methods for describing the elastic couplings. A key is provided in Table 1.1 for convenience. The first work with the contour-based cross-section analysis methods used in this study was conducted by Vlasov (Ref. [7]). The two-dimensional plate equations for a thin-walled cross section were related to the one-dimensional beam equations through geometric considerations. A more modern textbook on the theory of thin-walled structures was written by Gjelsvik (Reference [8]). The theory could be used to model generally shaped, thin-walled, isotropic sections. Theory for both open and closed sections was included. These early works addressed issues such as modeling different geometries, non-classical effects such as out-of-plane warping, and different boundary conditions. Vlasov theory was initially extended to composites by Bauld and Tzeng [9]. Only symmetric laminates were considered, and transverse shear was not taken into account. The symmetry assumption simplified the analysis by removing many of the elastic constants in classical laminated plate theory. The inplane forces are decoupled from the moments in symmetric laminates, i.e. the composite [B] matrix is all zeroes. Additional work on static stability of the composite beams (buckling) and nonlinear equilibrium was also included in the study. Most composite structures used in practice to this day have symmetric laminates or some other type of simplifying layup schedule because of the difficulty of modeling generally anisotropic struc-
1.. SUMMARY OF PREVIOUS RESEARCH 9 tures. In addition, a higher degree of anisotropy results in more design variables in an already complex system. Couplings are introduced slowly as predictive methods improve. Chandra and Chopra further extended Vlasov theory to include transverse shear effects of the beam, although transverse shear of the individual laminates was not considered, see Refs. [1] [1]. Several analytical and experimental studies were conducted. In the first study, bending-torsion coupling was introduced into I-beams through anti-symmetric layups between the the top and bottom flanges. The laminates themselves were symmetric. The impetus of the study was to develop an advanced cross-section model to capture the effects of elastic couplings and non-classical effects, such as torsion-related warping and transverse shear in a direct analytical method. Closed-form solutions for I-beams under tip bending and torsional loadings were developed and tested against experimentally determined results. The study showed that unlike isotropic beams, composite beams with large aspect ratios (on the order of 6) were strongly affected by warping restraints. The effects of elastic couplings were successfully captured. Closed-form solutions for twist under a tip bending load and twist under a tip torque compared well with experimental results. The applicability of the closed-form solutions is somewhat limited, but they are convenient to validate the cross-section analysis and help correlate coupled cases with experiments. Only uniform I-beams with individual tip bending or torsional loadings were considered. Later, a dynamic analysis of rotating I-beams was conducted, see Ref. [13]. Analytical predictions of natural frequencies of I-beams with warping restraint effects were compared to experimental results. This analysis from Reference [1] was extended further to thick walled sections by Centolanza and Smith in References [14] and [15]. A thick-wall correction based on the work of Reddy (Ref. [16]) was included in the analysis. Although no experiments were performed, the analysis was compared with three-dimensional finite element models and the closed-form solutions and experimental data in Reference [1] for validation. An advanced finite element model to capture the warping restraint effects in composite beams was developed by Floros and Smith in References. In that study, the closed-form results from Reference [1] were examined using the finite element method. A specialized torsion-with-warping beam element was developed and integrated into a static finite element analysis. Results using the new element compared very well to the closed-form solutions. More complex geometries and load-