The Pennsylvania State University The Graduate School College of Engineering DESIGN AND ANALYSIS OF ROTOR SYSTEMS WITH

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1 The Pennsylvania State University The Graduate School College of Engineering DESIGN AND ANALYSIS OF ROTOR SYSTEMS WITH MULTIPLE TRAILING EDGE FLAPS AND RESONANT ACTUATORS A Thesis in Aerospace Engineering by Jun-Sik Kim c 2005 Jun-Sik Kim Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2005

2 The thesis of Jun-Sik Kim was reviewed and approved by the following: Edward C. Smith Professor of Aerospace Engineering Thesis Co-Advisor Co-Chair of Committee Kon-Well Wang William E. Diefenderfer Chaired Professor in Mechanical Engineering Thesis Co-Advisor Co-Chair of Committee Farhan S. Gandhi Associate Professor of Aerospace Engineering Joseph F. Horn Assistant Professor of Aerospace Engineering Mary I. Frecker Associate Professor of Mechanical Engineering George A. Lesieutre Professor of Aerospace Engineering Head of the Department of Aerospace Engineering Signatures are on file in the Graduate School.

3 Abstract The purpose of this thesis is to develop piezoelectric resonant actuation systems and new active control methods utilizing the multiple trailing-edge flaps configuration for rotorcraft vibration suppression and blade loads control. An aeroelastic model is developed for a composite rotor blade with multiple trailing-edge flaps. The rotor blade airloads are calculated using quasi-steady blade element aerodynamics with a free wake model for rotor inflow. A compressible unsteady aerodynamics model is employed to accurately predict the incremental trailing edge flap airloads. Both the finite wing effect and actuator saturation for trailing-edge flaps are also included in an aeroelastic analysis. For a composite articulated rotor, a new active blade loads control method is developed and tested numerically. The concept involves straightening the blade by introducing dual trailing edge flaps. The objective function, which includes vibratory hub loads, bending moment harmonics and active flap control inputs, is minimized by an integrated optimal control/optimization process. A numerical simulation is performed for the steady-state forward flight of an advance ratio of It is demonstrated that through straightening the rotor blade, which mimics the behavior of a rigid blade, both the bending moments and vibratory hub loads can be significantly reduced by 32% and 57%, respectively. An active vibration control method is developed and analyzed for a hingeless rotor. The concept involves deflecting each individual trailing-edge flap using a compact resonant actuation system. Each resonant actuation system could yield high authority, while operating at a single frequency. Parametric studies are conducted to explore the finite wing effect of trailing-edge flaps and actuator saturation. A numerical simulation has been performed for the steady-state forward flight (µ = ). It is demonstrated that multiple trailing-edge flap configuration with the resonant actuation system can reduce the required trailing-edge flap hinge moments by 37% to 61% in each individual actuator compared to single-flap iii

4 configuration for high speed flight conditions. A novel resonant actuation concept is developed to efficiently realize the helicopter vibration and blade loads control. The resonant actuation system (RAS) is achieved through both mechanical and electrical tailoring. With mechanical tuning, the resonant frequencies of the actuation system (includes the piezoelectric actuator and the related mechanical and electrical elements for actuation) can be adjusted to the required actuation frequencies. This obviously will increase the authority of the actuation system. To further enhance controllability and robustness, the actuation resonant peak can be significantly broadened and flattened with electrical tailoring through the aid of an electric network of inductance, resistance, and negative capacitance. A piezoelectric resonant actuation system model is derived for active flap rotors. The optimal values of the electrical components are explicitly determined. An equivalent electric circuit model emulating the physical actuation system is derived and experimentally tested to investigate the initial feasibility of the piezoelectric resonant actuation system. It is demonstrated that the proposed resonant actuation system can indeed achieve both high active authority and robustness. It is shown that the actuator authority is significantly increased from 1.25 to 4.5 degrees as compared to the static value, with wide operating bandwidth of 8 Hz. In addition to this, the RAS is compared to an equivalent mechanical system to provide better physical understanding. Design guidelines of the RAS are derived in dimensionless forms. Feed-forward controllers are developed to realize the electric network dynamics and to adapt the phase variation. The control strategy is then implemented via a digital signal processor (DSP) system. Performance of the resonant actuation system is analyzed and verified experimentally on a full-scale piezoelectric tube actuator for helicopter rotor control. Promising results are illustrated that the actuator stroke is increased 2 to 3.5 times compared to its static value with bandwidth of 5 to 10 Hz. iv

5 Table of Contents List of Figures List of Tables List of Symbols Acknowledgments ix xv xvi xxvi Chapter 1 Introduction Background and Motivation Literature Review Helicopter Vibration Reduction Active Trailing-edge Flaps Smart Actuation System Development Piezoelectric Networks Summary of Literature Review Problem Statement and Objectives Overview of Dissertation Chapter 2 Helicopter Model Preliminary Background Vehicle Kinematics and Coordinate Systems Blade Deformed Kinematics and Coordinate Systems Nondimensionalization and Ordering Scheme Variational Formulation Structural Model v

6 2.2.1 Strain Energy of Rotor Blade Kinetic Energy of Rotor Blade Aerodynamic Blade Loads Quasi-steady Airloads Noncirculatory Airloads Quasi-steady Aerodynamics Implementation Inflow and Free Wake Model Linear Inflow Free Wake Model Aeroelastic Analysis Aeroelastic Response Coupled Propulsive Trim Chapter 3 Trailing Edge Flap Formulation Inertial Contribution Aerodynamic Models Incompressible Model Compressible Model Active Trailing Edge Flap Control Algorithm Feedback Form of Global Controller Active-Passive Hybrid Design Chapter 4 Active Loads Control Using a Dual Flap Configuration Introduction Description of Analytical Models Numerical Results and Discussions Baseline Articulated Rotor Analysis Rigid Blade vs. Elastic Blade A Single Flap for Moment Reduction Dual Flap Performance Multicyclic Control for Moment and Vibration Reduction Active-Passive Hybrid Design Summary Chapter 5 Helicopter Vibration Suppression via Multiple Trailing-Edge Flaps with Resonant Actuation Concept Introduction vi

7 5.2 Description of Analytical Models Results and Discussions Baseline Hingeless Rotor Analysis Flap Effect to Free-Wake Geometry Determination of Trailing-Edge Flap Locations Finite Wing Effects Effectiveness of Multiple-Flap Configuration Vibration Reduction with Multicyclic Control Summary Chapter 6 Piezoelectric Actuation System Synthesis Introduction Piezoelectric Actuation System Model Piezoelectric Tube Actuator Inertial and Aerodynamic Loads Coupled Actuator-Flap-Circuit System Mechanical Tuning and Electrical Tailoring Mechanical Tuning Electrical Tailoring Equivalent Electric Circuit Model Van Dyke Model Analysis and Experimental Verification Summary Chapter 7 Design and Test of Resonant Actuation Systems Design Guidelines of the RAS Resonant Actuators with R-L elements Resonant Actuation Systems with Additional Capacitance Summary of Design Guidelines for the RAS circuitry Dynamic Characteristics of the RAS in Forward Flight A Perturbation Method Analysis of Time Responses Vibration Reduction Within Available Actuation Authority Experimental Realization of the RAS Controller Design Bench Top Testing Power Consumption of Piezoelectric Resonant Actuation Systems Piezoelectric resonant actuators without circuitry vii

8 7.4.2 Piezoelectric resonant actuators with circuitry Summary Chapter 8 Conclusions and Recommendations Summary of Research Efforts and Achievements Recommendations for Future Work Appendix A Derivation of Strain Measure 238 A.1 Coordinate Transformation A.2 Foreshortening Term A.3 Deformed Coordinate System Appendix B Rotor System Matrices and Force Vectors 246 B.1 Strain Energy of Rotor Blades B.1.1 Stiffness Coefficients of Composite Beam B.1.2 Stiffness Matrices and Force Vectors B.2 Kinetic Energy of Rotor Blades B.2.1 Stiffness Matrix B.2.2 Mass Matrix B.2.3 Damping Matrix B.2.4 Force Vectors Appendix C Formulations using Mathematica 254 C.1 Rotor Strain Energy C.2 Rotor Kinetic Energy C.3 Rotor Quasi-steady Aerodynamic Loads C.4 Trailing-Edge Flap s Inertial Loads Bibliography 279 viii

9 List of Figures 1.1 Various sources of a helicopter vibration Helicopter vibration variation vs. forward flight speed Blade vortex interaction [4] Aerodynamic environment in forward flight Vibratory loads transmitted to fuselage [5] Schematic of a dynamic vibration observer [5] Composite tailoring of helicopter rotor blades [49] Active Control of Structural Response (ACSR) systems: (a) engine platform (b) cabin [20] Schematic of Higher Harmonic Control (HHC) Schematic of Individual Blade Control (IBC) Schematic of Active Trailing-edge Flap (ATF) Schematic of Active Twist Rotor (ATR) Schematic of semi-active actuators located at blade root region [4] Karman SH-2 Seasprite helicopter with servo-flaps Rotor with on-blade elevons in the NASA Ames Wind Tunnel [30] Elevon motion over one rotor revolution with 4/rev voltage excitation (760 RPM, µ = 0.2) [30] Actuator and flap dynamic system model [83] Blade-pitch indexing for the swashplateless rotor configuration [91] Single and dual flap configurations [94] AFC being inserted at active blade assembly [3] Macro-Fiber Composite (MFC) actuator Piezoelectric bender actuators [107] MD900 Helicopter Double X-frame actuators for MD900 Helicopter [112] BK117 Helicopter Flap unit assembly for Eurocopter BK117/EC145 [114] ix

10 1.27 Piezoelectric tube actuator for ATF [117] Passive piezoelectric vibration absorber [121] Active-Passive Piezoelectric Network (APPN) [121] Experimental setup (a) Beam with APPN and negative capacitance, (b) Circuit diagram of negative capacitance [125] Performance comparison: voltage driving response : shunt circuit without negative capacitance; : shunt circuit with negative capacitance [125] Scaling of aerodynamic hinge moment and actuator block torque Conceptual diagram of a resonant actuation system Vehicle and rotating coordinate systems Undeformed blade coordinate systems Deformed blade coordinate systems Cross-section coordinates before and after deformation Deformations in terms of Euler angles Schematic of the wake, discretized in space and time [131] Flow chart of an aeroelastic analysis with a free wake Finite elements for composite rotor blades Finite element discretization in time Vehicle configuration for propulsive trim Schematic of blade cross-section incorporating a trailing edge flap Nomenclature for a thin airfoil with a flap Dual flap configuration for active loads control Conceptual sketch of dual flap mechanism for active loads control Articulated blade coupled flap mode shapes Articulated blade coupled lag mode shapes Articulated blade torsion mode shapes Control settings of articulated rotor, µ = Blade tip response of articulated rotor, µ = Comparison of vibratory hub loads for active loads control Dual flap profile for moment reduction with 1/rev control Control settings of baseline and actively controlled rotors Harmonics of flapwise bending moment along the radial station Flapwise bending moment distribution before control Flapwise bending moment distribution after control Dual flap profile with 1 and 2/rev control inputs Dual flap profile with 1, 2 and 3/rev control inputs x

11 4.16 Comparison of vibration index and maximum flapwise bending moment with different control inputs Vibratory hub shears comparison for active loads control Comparison of vibration index and maximum flapwise moment of hybrid designed rotor Dual flap profiles for retrofit and hybrid designs Control settings of baseline, retrofit and hybrid designed rotors Blade non-structural mass and pitch-flap composite coupling stiffness, K 25, distribution Various configurations of the rotor with trailing-edge flaps Hingeless blade coupled flap mode shapes Hingeless blade coupled lag mode shapes Hingeless blade torsion mode shapes Control settings of hingeless rotor, µ = Control settings of hingeless rotor, µ = Blade tip response of hingeless rotor, µ = Blade tip response of hingeless rotor, µ = Vertical wake geometry with the number of beam elements and the presence of the flap for µ = Vertical wake geometry with the number of beam elements and the presence of the flap for µ = Blade tip responses with the number of beam elements and the presence of the flap for µ = Blade tip responses with the number of beam elements and the presence of the flap for µ = Control settings with the number of beam elements and the presence of the flap for µ = Control settings with the number of beam elements and the presence of the flap for µ = Vibration reduction vs. radial locations of trailing-edge flaps Lift curve slope vs. aspect ratio for elliptical lift distribution Vibration reduction by multi-flap configuration with lift flap Flap deflection harmonics of multi-flap configuration with the lift flap, advance ratio: µ = 0.15, actuator saturation: δ sat f = 4 o Polar diagram of flap motion for single-flap configuration, advance ratio: µ = 0.15, actuator saturation: δ sat f = 2 o Polar diagram of flap motion for dual-flap configuration, advance ratio: µ = 0.15, actuator saturation: δ sat f = 2 o xi

12 5.21 Flap deflections of dual-flap configuration with 4/rev control input, advance ratio: µ = 0.15, actuator saturation: δf sat = 2 o Hinge moments in single- and dual-flap configurations with 4/rev control input, advance ratio: µ = 0.15, actuator saturation: δ sat f = 2 o Comparison of vibratory hub loads, advance ratio: µ = 0.15, actuator saturation: δ sat f = 4 o Flap deflections of single-flap configuration, advance ratio: µ = 0.15, actuator saturation: δ sat f = 4 o Flap deflections of multiple-flap configuration, advance ratio: µ = 0.15, actuator saturation: δ sat f = 4 o Comparison of vibratory hub loads, advance ratio: µ = 0.35, actuator saturation: δ sat f = 4 o Flap deflections of single- and multiple-flap configuration, advance ratio: µ = 0.35, actuator saturation: δ sat f = 4 o Hinge moments in single- and multiple-flap configuration, advance ratio: µ = 0.35, actuator saturation: δ sat f = 4 o A piezoelectric tube actuator configuration Forces and moments acting on the trailing-edge flap Schematic of the PZT tube with R-L circuit and negative capacitance Fulcrum amplification mechanism for the PZT tube actuator Equivalent electric circuit model of the resonant actuation system Trailing-edge flap deflections of the resonant actuation system for Mach-scaled rotor Realization of the equivalent electric circuit for the resonant actuation system Comparison of analytical and experimental results of the RAS for a Mach-scaled rotor A piezoelectric network with a series R-L circuit Actuator strokes with the optimal tuning ratios and various coupling coefficients Actuator stroke and bandwidth variations with coupling coefficients Actuator strokes with the optimal inductance tuning and various resistance tuning values, ξ = Schematic of an equivalent mechanical system to a piezoelectric actuation system Electric charges with the optimal tuning ratios for various coupling coefficients xii

13 7.7 A piezoelectric network with a series R-L circuit and an additional capacitor Schematic of an equivalent mechanical system to a piezoelectric actuation system with an additional capacitor Actuator strokes with additional capacitance for ξ = Relative actuator stroke and bandwidth variations with modified coupling coefficients Peak-to-peak flap deflections of a resonant actuator with various flight speeds Variations of instantaneous frequencies along the azimuth Time history of flap motions of the actuation system without circuitry with 4/rev voltage excitation, µ = Peak-to-peak flap deflections of the RAS with various flight speeds Time history of flap motions of the RAS with 4/rev voltage excitation, µ = Time history of flap motions of the nominal actuation system with 4/rev voltage excitation, µ = Comparison of vibratory hub loads for an advance ratio of 0.15 within the available actuator authority Comparison of vibratory hub loads for an advance ratio of 0.35 within the available actuator authority Controller diagram of the resonant actuation system Diagram of an adaptive phase controller based on Matlab/Simulink Experimental set-up for the resonant actuation system Equipments used in the experiment Frequency responses of the PZT tube actuator before and after mechanical tuning Analytical predictions of a resonant actuation system:, tuned system w/o the voltage signal function;, RAS with ˆξ = 0.5;, RAS with ˆξ = Experimental results of a resonant actuation system:, tuned system w/o the voltage signal function;, RAS with ˆξ = 0.5;, RAS with ˆξ = Time responses of flap deflection signal before/after phase control with 24Hz input signal Time responses of flap deflection signal before/after phase control with 26Hz input signal Frequency response of current and phase variation for ξ = 0.5 with different resistances Frequency response of current and its phase with circuitry for ξ = xiii

14 7.30 Apparent electric power of a piezoelectric actuator with circuitry for ξ = Actuator strokes under the optimal tuning ratios for ξ = A.1 Thin-walled cross section of rotor blade xiv

15 List of Tables 2.1 Nondimensionalized parameters Order of terms used in aeroelastic analysis Vehicle properties Order of terms for trailing edge flaps Constraints and bounds for design variables Baseline articulated rotor properties for active loads control Trailing-edge flap properties for active loads control Natural frequencies of baseline articulated rotor Reduction of maximum bending moments maximum moments Reduction of vibration and moment reductions with different control inputs Comparisons of maximum moment, vibration index and control efforts Hingeless rotor and trailing-edge flap properties Natural frequencies of baseline hingeless rotor Control input sequences and flap locations for multiple-flap configurations Peak-to-peak hinge moments in single- and dual-flap configurations with 4/rev control input, µ = Peak-to-peak hinge moments in single- and multiple-flap configurations, µ = Piezoelectric material properties of PZT-5H for a Mach-scaled rotor Design parameters for the RAS circuitry xv

16 List of Symbols a A a A f A ij, B ij, D ij A M A s Lift curve slope Cross-section area of blade Acceleration vector Cross-section area of the flap Laminate stiffness matrices Amplification ratio in the actuation system Surface area of the PZT electrode b Airfoil semi-chord, b = c/2 c c f c o Cp e Cp s C p C add c D 55 Blade chord The flap chord Constant lift coefficient Total capacitance Piezoelectric material capacitance Structural modal damping of the PZT tube actuator Added capacitance in the electric network Elastic constant (open-circuit) xvi

17 c 1 C l, C d, C m Ĉ ij C C T C N, C M, C H C(k) d d o d 1 d 2 d 15 D i e e d e g E i f o f 1 F A Linear lift coefficient Lift, drag, and moment coefficients Reduced transformed stiffness for laminated composites Damping; Modal damping Thrust coefficient Unsteady normal, pitching and hinge moment coefficients of trailingedge flap Theodorsen s lift deficiency function Offset from blade elastic axis to the flap hinge Constant drag coefficient Linear drag coefficient Quadratic drag coefficient Charge constant in piezoelectricity Electric displacement component Offset between mid-chord and the flap hinge Offset of blade elastic axis and aerodynamic center Offset of blade elastic axis and c.g. Electric field component Constant moment coefficient Linear moment coefficient Centrifugal force in Appendix B F, {F } Force vector f e Excitation force in equivalent mechanical system to the piezoelectric actuation system xvii

18 F x, F y, F z g 15 H h H 1,, H 4 H t1,, H t6 h 15 I I b I f J k K K ij K 22, K 33, K 55 K 25 K c K n K Q KQ a L Resultant blade shear forces; N b /rev vibratory hub shear forces (longitudinal, lateral and vertical) Voltage constant in piezoelectricity Electric enthalpy density function Rotor height above vehicle c.g. Beam element shape functions Temporal element shape functions Piezoelectric constant Identity matrix Blade flap inertia Trailing-edge flap mass moment of inertia Objective function Reduced frequency; Spring stiffness Stiffness; Modal stiffness; Spring stiffness Composite stiffness coefficient Flap bending, lag bending, and torsion stiffness Pitch-flap composite coupling stiffness Electro-mechanical coupling stiffness Blade loads vector containing the flapwise curvature harmonics Inverse of the PZT tube actuator capacitance Inverse of the added capacitance Inductance xviii

19 L u, L v, L w L C, D C, M C L NC, M NC l p L el L LV L m, R m, C m M m m f M H M M, C M, K M m m, c m, k m mkm1, 2 mkm2 2 m o M x, M y, M z M LV N N b N el N s N t Blade sectional shear forces Circulatory lift, drag and moment per unit length Noncirculatory lift and moment per unit length Length of the PZT tube actuator Length of beam element Vibration index Inductance, resistance and capacitance in Van Dyke circuit Mach number; Modal damping Blade mass per unit length The flap mass per unit length The flap hinge moment Mass, damping and stiffness of the primary system in equivalent mechanical system to the piezoelectric actuation system Mass, damping and stiffness of the secondary system in equivalent mechanical system to the piezoelectric actuation system Flapwise and lagwise mass moment of inertia per unit length Blade reference mass per unit length Resultant blade sectional moments; N b /rev vibratory hub moments (rolling, pitching and torque) Blade moment index Axial force Number of blades Number of beam elements Number of PZT segments Number of temporal elements xix

20 n, s Contour coordinates in the cross-section of rotor blades p i q q R q t Q Q s r r r p r I r II R R i R o S s S f S ij t T 1,, T 14 Applied traction in the piezoelectric actuator Generalized displacement Normal mode displacement Actuator stroke (tip displacement) Electric charge Applied charge density Radial blade station; Resistance tuning ratio Position vector Radial coordinate in the polar coordinate system Trailing-edge flap chordwise c.g. from hinge Flap radius of gyration about the flap hinge Rotor radius; Resistance Inner radius of the PZT tube Outer radius of the PZT tube Nondimensional time in unsteady aerodynamics of the flap Natural coordinate in the finite element The flap first sectional moment Mechanical strain component in piezoelectricity Time Theodorsen coefficients T HI Rotation matrix between x H i and x I i T RH Rotation matrix between x R i and x H i xx

21 T uh Rotation matrix between x u i and x H i T ur Rotation matrix between x u i and x R i T φ T ω T ij St. Venant torsion Valsov torsion Mechanical stress component in piezoelectricity u, v, w Displacements in the undeformed coordinate system U x, U y, U z V V b V w V x, V y, V z V a V c W Relative wind velocity components in the deformed coordinate system Total relative wind velocity Relative wind velocity due to blade motions Relative wind velocity due to vehicle forward speed and rotor inflow Relative wind velocity components in the undeformed coordinate system Voltage across the PZT Control voltage in the actuation system Weighting matrices x, y, z Positions in the undeformed coordinate system x CG, y CG x e, y e x f i, ef i x F i, e F i x H i, e H i x I i, e I i x R i, e R i Longitudinal and lateral distances from vehicle c.g. to rotor hub Degrees of freedom in equivalent mechanical system to the piezoelectric actuation system The flap deformed coordinate system Vehicle coordinate system Hub-fixed coordinate system Ground-fixed inertial coordinate system Hub-rotating coordinate system xxi

22 x u i, e u i x Z n α α s α w, β w, γ w β β p β11 S γ γ ij δ δ f δ n δ sat δu δv δw ɛ ε ε ij ε T 11 Undeformed blade coordinate system Vector containing passive design variables Hub loads vector containing the N b /rev harmonics Angle of attack Longitudinal shaft tilt angle (positive nose down) Scalar weighting parameters used in weighting maxtrices Prandtl-Glauert correction factor Precone angle of the undeformed blade Impermittivity in piezoelectricity Lock number Engineering shear strain component Variational operator; Inductance tuning ratio Trailing-edge deflection angle Control vector of trailing-edge flap harmonics Actuator saturation angle First variation of strain energy First variation of kinetic energy First variation of virtual work done by external forces Incremental quantity Small parameter in perturbation method Small parameter in ordering scheme Normal strain component Dielectric constant xxii

23 ζ η η t θ θ o θ tw θ tr θ 1 θ 75 θ 1c θ 1s θ x, θ y, θ z Θ κ ij κ x, κ y Λ λ λ i λ T Coordinate in the deformed coordinate system; Nondimensionalized damping coefficient Coordinate in the deformed coordinate system Temporal nodal displacement Coordinate in the polar coordinate system Collective pitch angle (rigid blade pitch) Linear pretwist angle of a rotor blade Tail rotor collective pitch Total blade pitch angle Rigid blade pitch angle at 75% radial location of blade Lateral cyclic pitch angle Longitudinal cyclic pitch angle Euler angles Vector of control settings Curvature component Drees linear inflow parameters Axial skew angle Rotor inflow Induced rotor inflow Warping function of blade cross-section µ Advance ratio (non-dimensionalized forward flight speed) ξ ˆξ ρ Coordinate in the deformed coordinate system; Generalized electro-mechanical coupling coefficient Modified electro-mechanical coupling coefficient Air density xxiii

24 ρ p ρ s σ σ ij φ s ˆφ φ Φ φ s ψ Ψ ω ω ω D ω E ˆω E ω p, ω s Ω Piezoelectric material density Material density of rotor blade Rotor solidity Stress component Lateral shaft tilt angle (positive advancing side down) Elastic twist angle of blade Elastic twist angle with respect to undeformed elastic axis Modal matrix used in normal mode transformation Applied electric potential Azimuth angle Mode shape function of the PZT tube actuator Angular velocity vector Nondimensionalized frequency, ω/ω E Open-circuit frequency Short-circuit frequency Short-circuit frequency with added capacitance Parallel and series frequencies in Van Dyke circuit Rotor rotational speed Superscripts () A Aerodynamic quantities () E Short-circuit properties in piezoelectricity () D Open-circuit properties in piezoelectricity () fi Trailing-edge flap inertial contribution () fa Trailing-edge flap aerodynamic contribution xxiv

25 () H Terms associated with hub () I Inertial quantities () L Linear terms in energy () NL Nonlinear terms in energy () o Baseline value; Degree with numbers () S Properties at constant strain in piezoelectricity () T Transpose of matrix or vector; Properties at constant stress in piezoelectricity () V Vehicle quantities () 1 Inverse of matrix () Derivative with respect to a coordinate x. () Derivative with respect to time t; Derivative with respect to azimuth in Appendix B () Derivative with respect to azimuth ψ; Optimal tuning ratios in the electric network ˆ() Terms and responses associated with added capacitance Subscripts () r Main rotor contributions () f Trailing-edge flap contributions () ST Static response xxv

26 Acknowledgments It is a pleasure to thank the many people who made this thesis possible. It is difficult to overstate my gratitude to my Ph.D. supervisors, Dr. Edward C. Smith and Dr. Kon-Well Wang. With their enthusiasm, their inspiration, and their great efforts to explain things clearly and simply, they helped to make research fun for me. They provided encouragement, sound advice, good teaching, and lots of good ideas. I would have been lost without them. I would also like to express my appreciation to Dr. Farhan S. Gandhi, Dr. Joseph F. Horn and Dr. Mary I. Frecker for their helpful comments and being advisory committee members. I would like to thank the many people who have taught me aerospace engineering: my undergraduate teachers at Inha University (especially Dr. Si-Yoong Ryu, Dr. Ki-Ook Kim and Dr. Jin-ho Kim) and my graduate teachers (especially Dr. Maenghyo Cho). I am indebted to my many student colleagues for providing a stimulating and fun environment in which to learn and grow. I am especially grateful to Dr. Jianhua Zhang, Dr. Joseph Szefi, Dr. Phuriwat Anusonti-Inthra and Jose Palacios at Rotorcraft Center of Exellence, and to Dr. Ronald Morgan, Dr. David Heverly, David Belasco and Dr. Michael Philen at Structural Dynamics and Control Lab. David and Michael were particularly helpful to conduct experiments, patiently teaching me how to use an apparatus. I wish to thank my korean friends Dr. Dooyong Lee, Youngtae Ahn (especially for providing a ride) and Seongkyu Lee for their friendship. I musk thank all of the support staff, specifically Debbie Mottin and Robin Grandy in the aerospace department, and Karen Thal in the mechanical department. I would like to express my deepest appreciation to my family for their support and encouragement. I wish to thank my parents, my parents-in-law and my sisters. Lastly, and most importantly, I would like to thank my wonderful wife Eun-Ae for her love, patience, and the many sacrifices she has made. xxvi

27 Dedication This thesis is dedicated to my lovely wife Eun-Ae Ayu. xxvii

28 Chapter 1 Introduction Many methodologies have been explored to reduce helicopter vibration. Trailing edge flaps for such a purpose have been studied for the past twenty years. A brief overview of the introduction of active vibration controls using trailing edge flaps and smart actuators is presented in the first section. In the second section, the previous works in rotorcraft vibration controls and smart actuator development are described. In the third section, the problem statement and research objectives of the thesis are presented. The overview of this thesis is presented in the last section.

29 2 1.1 Background and Motivation A rotorcraft has been a very important mode of aerial transportation due to its capability of vertical take-off and landing, enabling many unique missions such as rescue operation at sea. It has, however, also been under several serious constraints such as poor ride quality due to high levels of vibration [1] and noise, restricted flight envelope, low fatigue life of the structural components, and high operating cost. Unlike the wings of conventional fixed wing aircraft, helicopter rotor blades experience periodic motions that result in 1/rev variation of aerodynamic and inertial loads along the azimuth. They also undergo a significant load fluctuation along the rotor spanwise direction. These operations will cause two major issues. One is the vibratory load of the rotor hub, coming from higher harmonic components of aerodynamic and inertial loads of rotor blades (Figure 1.1). This is the principal source of helicopter fuselage vibration and has an impact on helicopter performance, fatigue life of onboard equipment, and passenger comfort. The other issue is the bending moment of rotor blades, which is a primary source of blade fatigue stresses. Figure 1.1. Various sources of a helicopter vibration

30 3 Vibratory loads in helicopters arise from a variety of sources such as the main rotor system, the aerodynamic interaction between the rotor and the fuselage, the tail rotor, the engine and transmission, and atmospheric turbulence, which generally has low frequencies. However, the most significant source of vibration in a helicopter is the main rotor because of the unsteady aerodynamic environment acting on highly flexible rotating blades. The vibration level is generally low in hover and increases with higher forward speed. There are two regimes: low speed flight (transition) and high-speed flight, where the vibration levels are critical (Figure 1.2). At low forward speed (µ = 0.1), the blade tip vortices in the wake stay close to the rotor disk, causing a severe blade vortex interaction, which results in a substantially higher harmonic loading (Figure 1.3). At high forward speed, the rotor disk tilts forward and the wake is swept away from the disk plane; the wake-induced vibrations become small at high advance ratio. Figure 1.2. Helicopter vibration variation vs. forward flight speed Figure 1.3. Blade vortex interaction [4]

31 4 A typical aerodynamic environment of the helicopter main rotor during forward flight is depicted in Figure 1.4, where helicopter flight velocity adds to the blade element rotating velocities on the advancing side (0 o < ψ < 180 o ), and subtracts from it on the retreating side (180 o < ψ < 360 o ) [2, 3]. The resulting aerodynamic environment may be characterized as follows: high tip Mach number on the advancing side, and blade stall effects on the retreating side. A reverse flow region is also generated at the inboard on the retreating side. Such a complicated environment results in an instantaneous asymmetry of the aerodynamic loads acting among the blades at different azimuthal locations. This results in a vibratory response of a flexible blade structure, adding more complexity to the air loads asymmetry. Figure 1.4. Aerodynamic environment in forward flight The rotor hub acts as a filter, transmitting to the pylon and then to the cabin only harmonics of the rotor forces at multiples of N b /rev, where N b is the number

32 5 of blades. It has been shown that only the N b /rev and N b ± 1/rev rotating frame loads contribute to the N b /rev fixed frame loads (see Figure 1.5). More specifically, the N b /rev thrust and torque are caused by corresponding N b /rev rotating frame loads, whereas the N b /rev rotor drag- and side forces, and pitching- and rolling moments are caused by the corresponding N b ± 1/rev rotating frame loads. This result is based on the assumption that all the blades are identical and have the same periodic motion. This is strictly true only if the rotor blades are tracked and the helicopter is in trimmed flight. If the blades are not perfectly tracked, the blade to blade dissimilarity will result in a 1/rev vibration transmitted to the fuselage [6]. Figure 1.5. Vibratory loads transmitted to fuselage [5] Various methodologies for vibration reduction have been proposed in literature. There are several types of active strategies, such as HHC (Higher Harmonic Control) [7 11], IBC (Individual Blade Control) [12 15], ACSR (Active Control of Structural Response) [16 21], ATR (Active Twist Rotor) [22 25], and ATF (Active Trailing-edge Flap) [26 31]. It has been shown that improvements in helicopter vibration reduction can be achieved through the implementation of active control technology by smart material. One of the most promising methods is the IBC using active trailing-edge flaps. Therefore, active trailing-edge flaps to reduce the rotorcraft vibration have been given considerable attention in the research com-

33 6 munity [32, 33]. It has been shown that improvements in helicopter vibration reduction can be achieved by smart materials, such as piezoelectric materials. Piezoelectric actuation systems are expected to be compact, light weight, low actuation power, and high bandwidth devices that can be used for multi-functional roles such as to suppress vibration and noise, and increase aeromechanical stability. While piezoelectric materials-based actuators have shown good potential in actuating trailing edge flaps, they can only provide a limited stroke. This limitation can be critical in cases where large trailing-edge flap deflections are required or with large size rotor blades. The efforts to improve the piezoelectric actuator performance have been made by researchers in developing amplification mechanisms of various types [34]. Recently, a piezoelectric stack actuator with a double X-frame amplification device to deflect a full-scale flap on a MD Explorer helicopter has been developed and tested in whirl tower facility [112]. Flap deflection angles of ±3.5 o (at 450 V) were achieved during whirl tower test. A multiple piezoelectric actuator configuration has been considered and tested in Eurocopter to adjust the required control power and surface [114]. In this work, a single flap is segmented into three parts, and all actuator is controlled by the same command. On the other hand, multiple trailing-edge flap configurations have been studied to reduce the vibration of helicopter rotor system, in which each actuator operates independently. Several research works have shown that dual flap configuration is superior over a single flap in vibration reduction [78, 94]. In general, a single trailing-edge flap works well for the purpose of vibration reduction of helicopter rotor. With typical control inputs 3,4,5/rev, it has been demonstrated via numerical simulations that vibration level can be reduced by about 80%. As mentioned earlier, however, piezoelectric actuators provide a limited stroke. For the same trailing edge flap deflections, the actuator design specification in multiple-flap configuration is more relaxed than that in single-flap configuration, because the required hinge moment is less due to the small control surface area.

34 7 1.2 Literature Review This section describes previous studies relevant to the present subject. For convenience, the section is divided into five parts. The section starts with the overview of helicopter vibration reduction, where various methodologies were examined in attempt to reduce the vibration. In Section 1.2.2, previous studies using active trailing-edge flaps are highlightened. The review of smart actuation system development is given in Section 1.2.3, followed by piezoelectric networks that can help to enhance the actuator performance. The last subsection is a summary of the literature review Helicopter Vibration Reduction There have been considerable efforts to reduce the vibration in helicopter [35, 36], and vibration alleviation methodologies may be categorized into the four groups: 1. Passive means such as vibration absorbers, isolation devices and blade tailoring 2. Active vibration absorbing devices in fuselage 3. Direct modification of the excitation frequencies on the rotor blades 4. Active-passive and semi-active vibration reduction technologies In the past, numerous vibration control devices have been proposed and developed. The most common passive devices are the dynamic vibration absorber and the isolation mount. A typical dynamic absorber is a single degree of freedom system with a relatively small mass on a spring (Figure 1.6). If tuned to the excitation frequency, it can generate opposing oscillating force in resonance with the excitation to enforce a node on the support structure. For example, the dynamic hub absorber such as a simple [37] or bifilar pendulum has been successfully applied to S-76 helicopter [38]. The vibration isolation mount is another typical passive control device, such as pads of rubber or springs are placed between the vibrating system, to reduce the transmitted force from the vibrating system to the support structure. In helicopters, the conventional transmission mounting to

35 8 the airframe is replaced by elastomeric supports [39 41]. The blade structural and aerodynamic properties are tailored using automated optimization techniques to reduce the vibration [42 46]. Composite tailoring (Figure 1.7) has also been studied extensively since composite material provides excellent opportunities for developing light weight/high stiffness structures, as well as providing elastic couplings for potential optimal designs [47 51]. A passive approach has merit in that it does not need additional power. Passive methods, however, usually exhibit limited performance and cannot adapt to system and operating condition changes. Figure 1.6. Schematic of a dynamic vibration observer [5] Figure 1.7. Composite tailoring of helicopter rotor blades [49] In active vibration absorbing devices, active control actions are directly applied on the airframe. Two successfully flight tested airframe-based active controls are

36 9 active vibration suppression [16, 17] and Active Control of Structural Responses (ACSR) tested on a modified S-76B helicopter [18 20]. In ACSR system, vibration sensors are placed at key locations in the fuselage, where minimal vibration is desired (Figure 1.8). Depending on the vibration levels from sensors, a controller calculates proper actions for actuators, such as electro-hydraulic, piezoelectric and inertial force actuators, to reduce the vibration. The ACSR system has successfully made into production on the helicopter such as the Westland EH101 and the Sikorsky S-92 Helibus [20]. Figure 1.8. Active Control of Structural Response (ACSR) systems: (a) engine platform (b) cabin [20] Passive and active absorbing devices are still used in most of the rotorcraft flying today although they also bring unavoidable penalties in terms of weight and tend to affect vibrations only at discrete points. Therefore, efforts to modify directly the excitation forces have been sought by modifying unsteady aerodynamic forces acting on the rotor blades. Among them, Higher Harmonic Control (HHC) systems have received the most attention [7 11]. In this approach, servo-actuators are used to excite the conventional swashplate in the collective, longitudinal cyclic and lateral cyclic modes at the frequency of N b, resulting in blade pitching oscillations at three frequencies of (N b 1), N b and (N b +1) of HHC in the rotating frame (Figure 1.9). These higher harmonic blade pitch motions can generate additional

37 10 unsteady aerodynamic and oscillatory inertial loads with the right amplitudes and phases to alleviate hub vibration. Therefore, the vibration can be suppressed at the source before it is propagated into the fuselage. Even though HHC is shown to be highly effective, several drawbacks have impeded the implementation of the HHC concept on production helicopters. One of drawbacks is that HHC system uses the primary control system (swashplate and pitch links) to transfer higher harmonic pitch inputs to the rotor blades. Thus, considerable power is required to operate the actuators. Figure 1.9. Schematic of Higher Harmonic Control (HHC) An alternative to HHC is the Individual Blade Control (IBC) [12 15], in which each blade is individually controlled in the rotating frame over a wide range of desired frequencies. This control concept is a more general approach that removes some of the limitations of HHC such as the fixed excitation frequency. Usually, hydraulic actuators are mounted in conjunction with the blade pitch links in the rotating frame. The control inputs to the actuators are based on the feedback signals from the sensors mounted on the blades. A hydraulic slip ring unit is required to transmit the hydraulic power to the actuators in the rotating frame (Figure 1.10). Numerous analytical studies, wind tunnel tests and flight tests of HHC or IBC have demonstrated their potential for the substantial vibration reduction of up

38 11 Figure Schematic of Individual Blade Control (IBC) to 90 percent. However, both approaches have limitations on their practical implementation. Apart from the considerable weight penalties and high cost, large actuation power is needed to pitch the entire blades. The complexity of their actuation systems as well as their adverse impacts on vehicle reliability and maintainability has hampered their availability. Furthermore, since both HHC and IBC introduce control through the conventional swashplate, which is the primary flight control system of the helicopter, it will influence the airworthiness of the helicopter. Another IBC concept using a trailing-edge flap has been explored extensively for vibration reduction. This concept uses a small flap on each blade to generate the desired unsteady aerodynamic loads (Figure 1.11). This concept can be equally effective, but uses less power than the conventional IBC system. In this approach, a partial span trailing edge flap is located in the outboard region of the blade. The active flap control inputs affect the blade inertial loads and rotor dynamics as well as the unsteady aerodynamic loads. Since only a very small portion of the blade, about 4 5 percent [52], is actuated, this approach needs much less control power compared to HHC or IBC. Furthermore, this active flap control system is totally separated from the conventional swashplate; thus, it has little influence on airworthiness. A number of analytical simulations [26 29, 31], some wind tunnel tests [30, 67] and full-scale whirl tower test [112, 114] of the active flap have demonstrated that it has the potential to significantly reduce the vibratory loads, alleviate noise, enhance the rotor performance and handling qualities. Additional

39 12 information on vibration reduction using active trailing-edge flaps can be found in two recent survey papers [32, 33]. The detailed literature review will be described on the active trailing-edge flaps in Section Figure Schematic of Active Trailing-edge Flap (ATF) With the emergence of smart materials, such as Active Fiber Composites (AFC) [53], Macro-Fiber Composite (MFC) [54] and piezoelectric materials, the Active Twist Rotor (ATR) concept [22 25] has been proposed (Figure 1.12). One of the advantages is simplicity of its actuation mechanism compared to the active trailingedge flap actuation. The ATR concept has also a merit in that it does not increase the profile drag of the blade just as discrete flap does. While the ATR technology can produce a significant vibration reduction, power requirements are expected to be much higher than those for active trailing-edge flaps. Other vibration reduction technologies are active-passive or semi-active concepts developed to combine the advantages of both purely active and passive concepts. Although active means can more effectively reduce helicopter vibration and can be adaptive to system operating condition changes, their performances are often limited by the authority of the actuators. Active-passive approach has been proposed and investigated to make up for the weak point (actuator authority) in active flap system. A hybrid design approach can reduce the required flap deflections via active-passive optimization while retaining the same vibration level as that of the conventional active flap control [55 57]. A semi-active approach using

40 13 Figure Schematic of Active Twist Rotor (ATR) cyclic variation of the effective flap, lag, and torsion stiffness or damping variations has been proposed for helicopter vibration reduction [58, 59]. This approach involves evaluating sensitivity of hub variation to cyclic changes in stiffness and damping of the blade root region (Figure 1.13). Figure Schematic of semi-active actuators located at blade root region [4] Active vibration reduction systems are comprised of similar basic components; sensors, actuators, and a controller. In most active vibration control methodologies, the major issue is the actuator itself. Thus numerous smart actuators are under development. These concept designs of smart actuators have been either wind tunnel tested or bench tested for helicopter vibration reduction. The key to on-blade vibration control has been the advent of smart structures, in particular those incorporating electrically driven piezoceramic materials, exhibiting high en-

41 14 ergy density and high bandwidth. The on-blade smart systems open a new domain for vibration control, aeromechanical stability augmentation, handling qualities enhancement and noise reduction. The detailed literature review of smart actuation system developments will be described in Section Active Trailing-edge Flaps The trailing edge flaps were used to control the 1/rev rotor primary controls (collective and cyclic) for earlier helicopters,such as Pescara, d Ascanio and Kaman. The extension of this concept for providing control at higher harmonics was identified in a early work on multicyclic control [60]. In 1970 s, the first investigation of the trailing edge flaps (servo-flaps, see Figure 1.14) for multicyclic vibration control was conducted,called as Kaman Multicyclic Controllable Twist Rotor (MCTR). The MCTR utilized the first harmonic servo flap inputs incorporated with conventional pitch control, and the multicyclic control was generated by the electrohydraulic actuators mounted in the rotating frame. The wind tunnel testing with the maximum flap inputs up to ±6 degrees at frequencies up to 4/rev were conducted, and showed significant reduction in vibratory hub loads with appropriate 2/rev inputs [61]. McCloud III has studied the feasibility of reducing both vibration and blade loads using a single servo-flap, where he applied the scheduled multi-cyclic control inputs at 1/rev, 2/rev, 3/rev, and 4/rev to the MCTR [26]. His results have shown that multi-cyclic control can achieve both vibration and bending moment reductions with a large 1/rev control input. With the advances in the development of smart material-based actuators, active vibration controls using trailing-edge flaps are revisited for helicopter vibration reduction in 1990 s. An analytical study on vibration reduction in a four-bladed hingeless rotor using an actively controlled servo flap was conducted by Millott and Friedmann [27, 62 65]. A time-domain simulation to reduce 4/rev hub loads was implemented, and the reduction in vibratory hub loads levels around 90 % were reported. It was noted that spanwise flap position and blade torsional stiffness were key factors governing the performance in vibration reduction. The vibration

42 15 Figure Karman SH-2 Seasprite helicopter with servo-flaps level for rotor with a single trailing-edge flap using both the fully elastic and rigid blade models was investigated. It was reported that the predicted vibration level using the rigid blade model is much less than that of using the elastic blade model. An extensive proof-of-concept investigation of plain trailing edge flap was conducted by McDonnell Douglas Helicopter Systems. A 12-foot diameter Active Flap Rotor (AFR) model was tested in NASA Langley wind tunnel. The trailing edge flap was actuated by a cam-follower and cable arrangement that could provide various flap inputs by interchanging and rotating the programming cams. The reductions in hub loads up to 80 % were demonstrated during testing [66 68]. Milgram and Chopra [28, 69 71] carried out an analytical study on the effectiveness of plain trailing edge flaps for vibration suppression. This analysis incorporated an unsteady aerodynamic model based on indicial response functions of a flapped airfoil [72, 73] as well as a free wake model [74, 75] that are implemented into the UMARC [76]. The reductions in vertical hub shear loads up to 98% was reported using an open-loop controller with actuation frequencies of 3/rev and 4/rev. The validation of the analysis with test results from McDonnell Douglas AFR test was carried out,and correlation between predicted and measured results

43 16 was generally fair. It was reported that varying the phase angle of the flap motion had a significant effect on the blade 4/rev flatwise and inplane bending loads. Straub and Hassan [29] conducted a conceptual sizing and design study for a full scale demonstration trailing edge flap system. Structural parameters were investigated to determine a feasible flap/actuator combination with the consideration of the blade-flap-actuator dynamics. Based on the previous researches [27, 62 65], Mytle and Friedmann developed a new two-dimensional compressible unsteady aerodynamic model using a rational function approximation approach for the blade-flap combination [77, 78]. It was found that the control power and active flap deflections are significantly higher and larger as compared to those predicted with quasi-steady aerodynamics. They also examined and compared various flap configurations, such as servo-flap, plain-flap and dual-flap. Fulton and Ormiston conducted a hover test on a two-blade, 7.5 ft diameter dynamic rotor model with 10% on-blade elevons driven by piezoceramic bimorph actuators in the U.S. Army Aeroflightdynamics Directorate hover test chamber [79]. The test was successful to provide an encouraging basis for wind tunnel testing. Subsequently, a wind tunnel test [30] was conducted at NASA Ames Research Center (Figure 1.15). The test was performed at advance ratios from 0.1 to 0.3 at two different tip speeds (450 RPM and 760 RPM). Two important test, the elevon phase sweeps and frequency sweeps, were carried out to provide a measurement of elevon effectiveness and the rotor /elevon dynamic response characteristics. It was observed that the azimuthal time history of the elevon motion with 4/rev voltage excitation includes not only 4/rev elevon motion but also moderate 1/rev content, due in part to the azimuthal variation of elevon aerodynamic stiffness opposing the PZT actuator stiffness (Figure 1.16). de Terlizzi and Friedmann investigated Blade Vortex Interaction (BVI) effects on advanced geometry rotors [80]. The rotor wake model used in this study was extracted from the comprehensive rotor analysis code CAMRAD/JA [81]. The results indicated that the active trailing edge flap input angles of 15 o are required

44 17 Figure Rotor with on-blade elevons in the NASA Ames Wind Tunnel [30] Figure Elevon motion over one rotor revolution with 4/rev voltage excitation (760 RPM, µ = 0.2) [30] to alleviate the BVI induced vibratory hub loads at the advance ratio of A phase sweep simulation of an active flap was carried out and correlated with the wind tunnel test results at NASA Ames by Fulton and Ormiston [30], and the correlation was found to be good in most cases. Straub and Charles [31] have simulated vibration reduction by active flap

45 18 trailing-edge flaps using the comprehensive analysis code CAMRAD II [82]. This study was in support of the development of a full scale rotor test with smart actuators. The composite, bearingless rotor of MD900 with trailing-edge flaps was studied for active control of vibration, noise and aerodynamic performance. The coupled blade/flap/actuator dynamics and their effect on flap motions and system stability were investigated. Recently, Shen and Chopra [83, 84] developed a comprehensive aeroelastic analysis of a fully coupled blade-flap-actuator system based on UMARC (Figure 1.17). Parametric study was conducted to examine the coupling effect on the vibration reductions. It was shown that actuator dynamics can not be neglected, especially for a torsionally soft smart actuation system. This work was extended to consider the stability of active flap rotors [85]. Figure Actuator and flap dynamic system model [83] Other, recent, studies have addressed the issue of individual blade control of a helicopter with dissimilar rotor blades. The blade control is implemented using a conventional HHC algorithm coupled with a refined Kalman filter approach [86,87]. The controller was shown to reduce successfully the vibratory hub loads due to blade dissimilarities. In addition to offering improved vibration, acoustic noise, and rotor aerodynamic performance, the on-blade control concept may also perform the primary flight control function of the rotor and thereby replace the swashplate, pitchlinks, and hydraulic actuators of the traditional helicopter control system [88]. A few studies of advanced rotor control systems for both primary flight control

46 19 and rotor vibration control using active trailing-edge flaps have been performed. Army-sponsored studies by Bell and McDonell-Douglas Helicopter Company to explore the potential benefits of active control while eliminating the need for portions of the conventional helicopter flight control system [89]. Recently, Shen and Chopra [90 92] developed a comprehensive rotor code to analyze the swashplateless rotor configuration. A multicyclic controller was implemented, and the feasibility of trailing-edge flap performing both primary control and active vibration control was examined. With a large blade pitch index angle of 16 deg (Figure 1.18), the required half peak-to-peak values of trailing-edge flap deflections were below 6 deg. Among other technical issues, either conventional electromechanical and hydraulic actuators or a large blade pitch index angle introduced a number of practical drawbacks. Figure Blade-pitch indexing for the swashplateless rotor configuration [91] On the other hand, multiple trailing-edge flap configurations have been studied to reduce the vibration of helicopter rotor system, in which each actuator operates independently. Myrtle and Friedmann [78] have shown that the dual flap configuration (Figure 1.19) is almost completely unaffected by the change of torsional stiffness of rotor blade. Recently, Cribbs and Friedmann [93] have developed the flap deflection saturation model through an automated approach to reduce the required maximum flap deflection. They have shown that the imposition of saturation of flap deflection could result in the different profile and reduced magnitude of the active flap while maintaining almost the same vibration level as models

47 20 without actuator saturation. The actuator saturation model has been extended to reduce the vibration due to dynamic stall using a dual flap configuration [94]. In this work, the effect of dynamic stall was incorporated by using the ONERA dynamic stall model, and the drag due to the flap deflection was also considered. They showed that dual flap is superior over a single flap in vibration reduction. Recently, Liu et. al conducted a study of the combined helicopter noise and vibration problem using a dual flap configuration [95, 96]. An acoustic prediction based on WOPWOP [97] was combined with the aeroelastic analysis based on a flexible blade. It was observed that a dual flap configuration is more effective than a single flap configuration when the actuator saturation is considered. It was also reported that the noise penalty is mainly due to the large flap deflections obtained when saturation limits are not imposed. Figure Single and dual flap configurations [94] Smart Actuation System Development There appeared an opportunity of having multiple light weight sensors/actuators embedded or surface-mounted at several locations in rotor blades and optimally distribute actuation with the aid of modern control algorithm [34, 98, 99]. By employing active materials for such sensors/actuators in order to implement individual blade control, one can potentially obtain advantages in terms of weight and power consumption when compared to traditional hydraulic systems. These new

48 21 actuators only requires electrical power to operate. Two main concepts have been under development for the active material application: rotor blade flap actuation and integral blade twist actuation [100]. For the integral blade twist actuation concept, the actuators may be embedded throughout the structure, which provides redundancy in operation. A major challenge with integral blades is to develop a design that presents sufficient twist authority while providing the torsional stiffness required for the aeroelastic performance of the blade. Chen and Chopra [22], based on the piezoelectric actuator presented in Barrett [101], built and tested a 6-ft diameter two-bladed Froudescaled rotor model with banks of piezoceramic crystal elements in ±45 o embedded in the upper and lower surfaces of the test blade. Rodgers and Hagood [102] manufactured and hover tested a single one-sixth Mach scaled model blade of CH-47D where the integral twist actuation was obtained through the use of Active Fiber Composite (AFC) [53]. An intentional reduction by 50% on the baseline torsional stiffness was imposed and regarded to improve twist actuation. Hover testing on the MIT Hover Test Stand Facility demonstrated tip twist performance between 1 o and 1.5 o in the rotating environment. Another example of an integral blade twist concept has been studied as part of a NASA/Army/MIT Active Twist Rotor cooperative agreement program [103]. The structural design of the ATR prototype blade employing embedded AFC actuators (Figure 1.20) was conducted based on a newly developed analysis for active composite blade with integral anisotropic piezoelectric actuators [104]. The Macro-Fiber Composite (MFC) has been recently developed at NASA Langley based on the same idea as the AFC in using the piezoelectric fibers under interdigitated electrodes [54]. In this actuator, shown in Figure 1.21, the piezoelectric fibers are manufactured by dicing from low-cost monolithic piezoceramic wafers. Thus, it retains most advantageous features of the AFC with a potentially lower fabrication cost. This actuator is currently being tested for its basic characteristics, and it has been considered for use in many aerospace applications. The controllable twist rotor approach makes it easy to embed smart materials

49 22 Figure AFC being inserted at active blade assembly [3] Figure Macro-Fiber Composite (MFC) actuator into a rotor blade and results in an aerodynamically clean blade. However, since the entire blade must be twisted, it is very difficult to achieve the targeted control authority of ±2 degrees of blade tip twist with the current state of the art smart materials [34]. Another approach is the discrete active trailing edge flap driven by smart material actuators. The primary concerns in this approach are to obtain high actuation force and stroke with minimal weight penalty, and that the actuation mechanism must be designed to fit into the geometric confines of the blade structure. Over the past decade, several actuators have been developed to deflect a rotor blade trailing edge flap. The ideal trailing edge flap actuator would generate enough torque and stroke to overcome the aerodynamic, inertial loads, friction, and all

50 23 other restricting moments. Piezoelectric actuators must be used in conjunction with an amplification mechanism to be effective in rotorcraft trailing edge flap applications because of the small-induced strain capability of the smart material actuators. Flap actuator designs that incorporate piezoelectric bender actuators have a low force output and in general, are restricted to small scale wind tunnel models [ ]. One of the piezoelectric bender actuators developed by Koratkar and Chopra [107] is shown in Figure Figure Piezoelectric bender actuators [107] Piezoelectric stack actuators, typically used in larger scale applications, have a larger force output than benders but produce a relatively smaller stroke. In order to achieve the required flap deflections, stack flap actuator designs require more complex amplification mechanisms than benders. Trading the actuation force with actuation displacement using mechanical amplification makes it possible to use the piezostack to actuating the trailing edge flap of a full-scale blade. Lee and Chopra designed an actuator to drive a full-scale trailing edge flap using two piezostacks with double-lever [109] and bi-directional double-lever [110] amplification mechanism. In a parallel study, Prechtl and Hall designed a X-Frame trailing edge servo flap stack actuator [111]. The Mach-scaled rotor model with the trailing edge flap servo-flap actuation was hover tested in wind tunnel. Straub et. al [112, 113] used a piezostack with a Double X-frame amplification device (Figure 1.24) to

51 24 deflect a full scale flap on a MD Explorer helicopter (Figure 1.23). Flap deflection angles of ±3.5 (at 450 V) were achieved during whirl tower test [112]. Figure MD900 Helicopter Figure Double X-frame actuators for MD900 Helicopter [112] Eurocopter [114] has designed a full-scale actuator trailing edge flap for the BK117/ EC145 helicopter (Figure 1.25). A multiple piezoelectric actuator configuration has been considered and tested to adjust the required control power and surface. In this work, a single flap is segmented into three parts, and all actuator is controlled by the same command. A 16% span flap was deflected ±5 in whirl tower testing [114]. In this test, the units of the trailing edge flap of the blade was

52 25 cut out and the foam used as support between the upper and lower blade skin was substituted by a flat box made from carbon fiber (Figure 1.26). Figure BK117 Helicopter Figure Flap unit assembly for Eurocopter BK117/EC145 [114] There are many alternative concepts proposed for actuating trailing edge flaps. Clement et al designed a piezoceramic C-Block actuators for active flap system and a bench-top test was conducted [115]. Recently, a new piezoelectric induced shear tube actuator (Figure 1.27) was developed and bench tested by Centolanza and Smith [116]. By exploiting the high energy density of the d 15 mode of the piezoceramic material, along with the well-matched shape factor of the piezoelectric tube, this new actuation system can generate higher force and stroke than the

53 26 current piezo bender actuators or piezo stack actuators. Moreover, it has a clearer and simpler amplification mechanism. Figure Piezoelectric tube actuator for ATF [117] Piezoelectric Networks Because of their electro-mechanical coupling characteristics, piezoelectric materials have been explored extensively for both active and passive vibration control applications. In a passive situation (Figure 1.28), the piezoelectric materials are usually integrated with an external shunt circuit [118,119]. Electrical field/current will then be generated in the shunt circuit because of the electro-mechanical coupling feature. It has been shown that with proper design of shunt components (inductor, resistor, or capacitor), on can achieve the electrical damper or electrical absorber effects. The integration of an inductively shunted piezoelectric element with an active source was first studied by Agnes [120]. It was observed that the active-passive hybrid configuration retains the passive damping ability while allowing additional performance using active control. Because of the inductive shunt, this configuration is best suited for narrow-band applications. It is important to note that shunting the piezoelectric in a passive manner does not preclude the use of shunted

54 27 Figure Passive piezoelectric vibration absorber [121] piezoelectric materials as active actuators. Indeed, the integration of the passive and active approach, such as the one illustrated in Figure 1.29 often referred to as an active-passive hybrid piezoelectric network (APPN), has shown promising results [121]. This APPN configuration not only preserves the passive damping ability of the shunt circuits, but also, has been found to amplify the active control authority around the tuned circuit frequency [ ]. A few studies have been performed on vibration control using APPN concept, which includes simultaneous optimal control/optimization for determining passive parameters and active gains in APPN, multi-input multi-output applications [126], and integration of APPN with traditional viscoelastic treatments [127]. Figure Active-Passive Piezoelectric Network (APPN) [121]

55 28 Figure Experimental setup (a) Beam with APPN and negative capacitance, (b) Circuit diagram of negative capacitance [125] On the other hand, the performance of piezoelectric material-based actuators can be augmented with electric networks that include inductor, resistor and negative capacitor [125, ]. Tang and Wang performed analysis and experiment for active-passive hybrid piezoelectric networks [125]. It was shown that the electro-mechanical coupling of the integrated system is increased by introducing negative capacitance that is realized by a negative impedance convert circuit with an operational amplifier (Figure 1.30). The overall control authority was significantly improved since the structure can be driven to a higher amplitude given the same level of voltage input. The bandwidth of the amplification effect was greatly increased due to the negative capacitance (Figure 1.31).

56 Figure Performance comparison: voltage driving response : shunt circuit without negative capacitance; : shunt circuit with negative capacitance [125] 29

57 Summary of Literature Review Literature review shows that a great deal of research has been conducted in designing passive and active devices for helicopter vibration reduction. Several topics were addressed including helicopter vibration reduction methodologies, active trailing-edge flaps, smart actuation system development, and piezoelectric networks to enhance the actuator performance. The traditional passive approaches to vibration reduction, such as absorbers and isolation mountings have generally not proven to be effective and/or efficient enough to realize the desired comfort level of jet-smooth rotorcraft flight. Vibration reduction using active control has been extensively explored. Despite the promising performance of the HHC and IBC concepts for vibration reduction, many practical concerns have to be addressed for these systems can be used in a production helicopter. They use the primary control system (swashplate and pitch links) to transfer pitch inputs to the rotor blades. In the event of failure of control systems, the pilot may not have full control of the helicopter (airworthiness). Although the ACSR approach offers significant vibration reduction performance for modern helicopters, the vibration reductions obtained may be localized at the sensor locations, regardless of vibration levels elsewhere in the airframe, and the vibration levels in the rotor system are left unaltered. The active twist rotor approach (ATR) makes it easy to embed smart materials into a rotor blade and results in an aerodynamically clean blade. However, since the entire blade must be twisted, it is very difficult to achieve the targeted control authority Active trailing-edge flaps have been explored extensively for vibration reduction, since this concept uses a small flap on each blade to generate the desired unsteady aerodynamic loads. This can be equally effective, but uses less power than the conventional IBC system. A number of analytical simulations and wind tunnel tests of active trailing-edge flaps have demonstrated that it has the potential to significantly reduce the vibratory hub loads. In addition to offering improved vibration and acoustic noise, active trailing-edge flaps could be used as the primary control flight control system (i.e., swashplateless helicopter) and active loads control system with the low frequency inputs (e.g., 1/rev and 2/rev).

58 31 With the emergence of smart materials, such as AFC, MFC and piezoelectric materials, the ATR convept was proposed. This has a merit in that it does not increase the profile drag of the blade. However, it is difficult to achieve the targeted control authority with the current state of the art smart materials. On the other hand, many actuators based on piezoelectric materials have been developed for active trailing-edge flaps. Among them, recently, two piezoelectric actuators were tested in whirl tower facilities. One is a Double X-frame actuator for MD Explorer helicopter, and the other is a DWARF actuator for Eurocopter BK117 helicopter. Flap deflection angles of ±3.5 o 5 o were achieved during whirl tower tests. On the other hand, the performance of piezoelectric material-based actuators can be augmented with electric network. Piezoelectric materials with electrical networks have been utilized to create the shunt damping for structural vibration suppression. It was also recognized that such networks not only can be used for passive damping, they can also be designed to amplify the actuator authority around the tuned circuit frequency. It was demonstrated that the electro-mechanical coupling of the integrated system can be increased by introducing electric components such as inductor, resistor and negative capacitor. Based on the literature review, in the problem statement and objective section that follows, the problem associated with current state-of-the-art actuators and active trailing-edge flap controls are described, then the objectives of the current study are specifically listed.

59 Problem Statement and Objectives From the review of smart actuators for trailing edge flaps, it is observed that active material actuated rotor blade controls are emerging as a viable solution. Active trailing edge flap experiments with piezoelectric material based actuators have shown more remarkable results than actively twisted blades with embedded piezo elements. However, the major drawback of smart material based actuators is their low stroke. Although many actuation schemes have been examined to amplify the strokes by trading actuation force with displacement, so far only ±3 5 degrees peak to peak active flap deflections are achievable in whirl tower test (hover condition). The achieved active flap deflections are far away from the required control authority for the large size full-scale rotor blades, since the aerodynamic hinge moment will obviously increase as the size of rotor blades and forward flight speed increase. The performance of the smart actuators with varying scale of rotor blade is depicted in Figure 1.32 showing the aerodynamic hinge moment and actuator block torque as a function of blade chord. It can be seen that the aerodynamic moment increases with blade chord at a larger rate than the block torque available. The state-of-the-art actuators for trailing edge flap could be able to generate a useful combination of flap deflection and block torque for a light class helicopter application, such as MD900 and BK117, (blade chord of 10 inches), however, the performance of the actuators will be drastically reduced if the blade chord length is increased further. Therefore, the actuator stroke remains a critical design limit in practical implementation of active flaps for large size rotors. In other words, actuator authority for heavy lift class helicopters presents a major technical barrier. The goal of this research is to address the current issues and advance the state of the art in helicopter rotor vibration control by exploring new actuation ideas and using multiple trailing-edge flaps. To achieve this goal, general tasks involve the development of rotor systems with multiple trailing edge flaps and resonant actuation systems. A resonant actuation system is introduced to enhance the effectiveness of piezoelectric actuators. Through mechanical tailoring, the resonant frequencies of the actuation system (including the piezoelectric actuator and

60 33 Figure Scaling of aerodynamic hinge moment and actuator block torque the related mechanical and electrical elements for actuation) can be tuned to the required actuation frequencies that are 3, 4, and 5/rev for typical four-bladed helicopters. This will increase the authority of the actuation system; however, it could be hard to control because of the narrow operating bandwidth and dramatic change in phase. These issues can be resolved through electric circuit tailoring. Then the actuation resonant peak can be significantly broadened and flattened, and the changes in phase can be much gradual. In this case, one can achieve a high authority actuation system without the negative effects of resonant problems (Figure 1.33). A resonant actuation system, however, may not cover the entire operating frequencies (3,4,5/rev). This can be resolved using the rotor blades with multiple trailing-edge flaps, in which each flap is designed to operate at a single frequency that is one of the operating frequencies. In this way, one can achieve the ultimate solution to overcome the barrier of actuator authority for large size full-scale rotor blades, and a lighter actuation system could be achieved for MD900 or BK117 class

61 34 Figure Conceptual diagram of a resonant actuation system helicopters that are currently considered for flight test. The specific research goals for this new concept, as well as some of the proposed approaches, are described below: 1. Investigate the feasibility of using multiple trailing edge flaps for vibration reduction and blade loads reduction. In pursuing this objective, the initial goal is the development of an comprehensive rotor aeroelastic analysis incorporating multiple trailing edge flaps. The unsteady compressible model developed by Hariharan and Leishman [73] is used for trailing-edge flaps, which is able to predict reliable flap hinge moments. The free-wake model developed by Tauzsig and Gandhi [131, 132] is incorporated to predict realistic vibratory hub loads. Following this, a multicyclic control algorithm for multiple trailing edge flaps with the resonant actuation concept will be developed to reduce either the blade loads or the vibration. An actuator saturation will be also considered to reflect the available actuator authority and to avoid the additional aerodynamic interference to the vehicle trim, which is based on the bisection method. Active loads control for an articulated composite rotor. An active loads control method will be developed to reduce the blade loads as well as the vibration. This can be achieved by dynamically straightening

62 35 the blade, which mimics the behavior of a rigid blade, via dual flap configuration. An aeroelastic analysis will be performed for the high speed flight condition. A parametric study will be performed to determine the flap location along the rotor spanwise direction for the vibration reduction. The finite wing effect of trailing-edge flaps will be also examined since a flap span is smaller than one in a single flap configuration. Finally, vibratory hub loads will be compared among various trailing-edge flap configurations. 2. Develop a resonant actuation system (RAS) for active trailing-edge flaps that will be used in multiple trailing-edge flap configurations Develop an analytical model of the resonant actuation system. Up to date, no systematic methods have been derived for tailoring the electrical parameters, such that a desired actuator authority can be achieved. An equivalent mechanical system will be examined to provide better physical understanding. Investigate the time-varying characteristics of resonant actuation system in forward flight. The time-varying characteristics of a RAS will be analyzed via a perturbation method and frequency response functions between peak-to-peak flap deflections and control voltage input. Realize the resonant actuation system, especially under high voltage operations. A method of implementing the electric networks will be realized via a digital signal processor (DSP). A controller will be designed to adapt the change in phase at the vicinity of resonant frequency (one of operating frequencies). Through this effort, the performance of a RAS will be validated experimentally via a bench-top test emulating hover condition.

63 Overview of Dissertation This thesis consists of eight chapters, which are organized as follows: 1. The first chapter introduces the background and motivation for the current research. A comprehensive review of literature related to the present research topics is included, and the research objectives are stated. 2. The second chapter describes the physical and mathematical models of the helicopter rotor system used in the current research. The coordinate systems of blades and vehicle, ordering schemes, and variational formulation is described. The composite blade and quasi-steady aerodynamic models are derived, and the free-wake model is also presented. Finally, an aeroelastic analysis that includes finite element formulations and vehicle trim is discussed. 3. The trailing-edge flap formulation and optimal controller for both vibration and blade loads are described in Chapter 3. The inertial loads of trailing-edge flaps are derived based on the previous work [69]. Followed by the discussion of available aerodynamic models to predict the aerodynamic loads generated by flap motions. Finally, the optimal controller is described based on the minimization of an objective function. 4. Chapter 4 investigates the feasibility of multiple trailing-edge flaps for simultaneous reductions of vibration and blade loads. The concept involves straightening the blade by introducing dual trailing edge flaps in a conventional articulated rotor blade. Numerical simulation is performed for the steady-state forward flight of an advance ratio of In Chapter 5, vibration reduction using the multiple trailing-edge flap configuration is investigated. The concept involves deflecting each individual trailing-edge flap using a compact resonant actuation system that is described in Chapter 6. Each resonant actuation system could yield high authority, while operating at a single frequency. Numerical simulation is performed for the steady-state forward flight (µ = ) condition.

64 37 6. Chapter 6 describes the development of the resonant actuation system, which is utilized for the blade loads and vibration controls presented in Chapter 4 and 5. A piezoelectric actuation system model is derived for active flap rotors, and then followed by mechanical tuning and electrical tailoring, where the optimal tuning parameters for electric networks are explicitly determined. An equivalent electric circuit model emulating the physical actuation system is derived and experimentally tested. 7. Chapter 7 extends the new resonant actuation system concept to provide design guidelines and better physical understandings. Dynamic characteristics of the RAS is examined for the case of forward flights. Vibration reduction performance of various flap configurations is evaluated within the available actuation authority. Bench top tests are conducted by utilizing the designed controller with electric network dynamics and phase compensation. Finally, the power requirement of the actuation system is characterized. 8. Finally, the research efforts and achievements in this thesis are summarized in the last chapter. Recommendations for future work are also discussed.

65 Chapter 2 Helicopter Model One of the objectives of current study is to investigate the feasibility of multiple trailing-edge flaps for vibration reduction. It is required to have a fully coupled flap-lag-torsion model that can be used for an analysis of both hingeless and fully articulated hubs. The formulation presented in this chapter is adapted from several sources [76,133,134] and is included here for completeness. Much of the theory has been re-derived for the current work, and hence some of the equations are slightly different from the cited sources. This chapter describes the physical and mathematical models used in the present investigation. In the first section, the coordinate systems of blade and vehicle, ordering schemes, and variational formulation will be described as the preliminary background. In following two sections, the composite blade and quasisteady aerodynamic models will be derived. Additional aerodynamic loads generated by trailing-edge flaps will be described in the following chapter. In the fourth section, main rotor inflow and free wake models will be presented. Finally, an aeroelastic analysis that includes finite element formulations and vehicle trim will be discussed.

66 Preliminary Background In order to understand the following derivations, the preliminary backgrounds, such as the coordinate systems of vehicle and blade, ordering schemes and variational formulation, will be described. This will be carried throughout the rest of analysis Vehicle Kinematics and Coordinate Systems Figure 2.1 shows the coordinate systems for describing the motion of the helicopter system. Rigid body motion of the vehicle is defined relative to a ground-fixed inertial coordinate system, superscript I. The vehicle coordinate system, superscript F, are fixed to the vehicle center of gravity. The hub-fixed coordinate system, superscript H, is defined parallel to the vehicle coordinate system. z H y H x H z I z F C.G. y F x F y I s s x I y H y R x R V x H Figure 2.1. Vehicle and rotating coordinate systems

67 40 Position vectors based on each coordinate systems are defined as: r = x I î I + y I ĵ I + z IˆkI = x I 1e I 1 + x I 2e I 2 + x I 3e I 3 = x I i e I i = x H i e H i = x F i e F i (2.1) where the boldface denotes the vector. For convenience, the tensorial formulation is adopted. The transformation between the inertial system and the hub-fixed nonrotating systems is defined as e H 1 e H 2 e H 3 = 1 0 α s 0 1 φ s α s φ s 1 e I 1 e I 2 e I 3, e H i = T HI ij e I j, (2.2) in which α s is the longitudinal shaft tilt angle (positive nose down), and φ s is the lateral shaft tilt angle (positive advancing side down). Their amounts are assumed to be small. The hub-rotating coordinate system, superscript R, is rotating at constant angular velocity, ω = Ω e H 3, with respect to the hub-fixed nonrotating frame (see Figure 2.1). The transformation between hub-fixed and rotating systems can be obtained as e R i = Tij RH e H j, T RH = where ψ is the azimuth angle and equals Ωt. cos ψ sin ψ 0 sin ψ cos ψ , (2.3) The undeformed blade coordinate system, superscript u, are attached to the undeformed blade. The undeformed blade is at a precone angle of β p as shown in Figure 2.2. The coordinate transformation between the rotating coordinate system, superscript R, and the undeformed blade coordinates system, superscript u, is given by e u i = Tij ur e R j, T ur = cos β p 0 sin β p sin β p 0 cos β p. (2.4)

68 41 z z H y R p y H x p x R x H Figure 2.2. Undeformed blade coordinate systems Letting Tij uh = Tik ur Tkj RH, the transformation between the hub-fixed nonrotating system and the undeformed blade system is written as e u i = Tij uh e H j, T uh = cos β p cos ψ cos β p sin ψ sin β p sin ψ cos ψ 0 sin β p cos ψ sin b p sin ψ cos β p. (2.5) Blade Deformed Kinematics and Coordinate Systems Accurate modeling of elastic rotor blades requires that deflections are considered to be moderately large. Although deflections are assumed to be moderately large, strains are still assumed to be small. These assumption are so called deflections with small strains and moderately large rotation. In view of solid mechanics, it is similar to von Karman partial nonlinearity. The strain-displacement relationships are developed based on the deformed configuration (Eulerian formulation) so that the same reference is used for both the stress and strain tensors. To second order which was defined by Reference [134], however, the two results, Lagrangian based on the undeformed configuration and Eulerian formulation, are equivalent. Details of this procedure and the strain

69 42 z y e e p w e p o u v x Figure 2.3. Deformed blade coordinate systems measure are given in Appendix A. The deformed blade is characterized by the deformed blade coordinate system, superscript d. As shown in Figure 2.3, e d 2 and e d 3 are aligned with the principal cross-section axis. A point P o on the undeformed elastic axis undergoes deflections u, v and w in the e u 1, e u 2 and e u 3 directions and moves to a point P as shown in Figure 2.3. Then the blade cross section containing P undergoes a rotation θ 1 about the deformed elastic axis. The total blade pitch, θ 1, is defined as θ 1 = θ o + ˆφ, (2.6) where θ o is the rigid pitch angle due to control pitch and pretwist. In general, the pretwist can be an arbitrary function of radial location, i.e., θ o = θ o (r). For a blade with linear a pretwist, the rigid pitch angle can be expressed by θ o = θ 75 + ( x.75)θ tw + θ 1c cos ψ + θ 1s sin ψ. (2.7) as Total blade pitch, θ 1, also includes elastic twist ˆφ. The elastic twist is defined ˆφ = φ x 0 w 2 v dx. (2.8) x x2 The φ is the elastic twist about the undeformed elastic axis, while the ˆφ can be viewed as the elastic twist about the deformed elastic axis. This is a nonlinear

70 43 u z e3, e d e g A.C. E.A. C.G. o u y e 2, Undeformed Cross Section u z e3, d, e 3 d, e 2 ^ 1 o w Deformed Cross Section v u y e 2, Figure 2.4. Cross-section coordinates before and after deformation kinematic effect arising from moderately large rotation [134], see Figure 2.4. The coordinate transformation between the undeformed and deformed blade coordinate systems is given by e d i = T du ij e u j, (2.9) where T du = cos θ x cos θ y cos θ y sin θ z sin θ y cos θ z sin θ x sin θ y cos θ x sin θ z cos θ x cos θ y sin θ x sin θ y sin θ z cos θ x cos θ z sin θ y + sin θ x sin θ z cos θ z sin θ x cos θ x sin θ y sin θ z cos θ y sin θ x cos θ x cos θ y. (2.10) As shown Figure 2.5, the Euler angles, θ x, θ y, and θ z are defined in terms of blade

71 44 e e z y 1 e ds w' ds dw dx dv dx v' ds dx 1 ( ) ( dv 2 dx ) ds ds dw 2 dx x Figure 2.5. Deformations in terms of Euler angles deformations as θ x = θ 1, θ y dw dx = w, θ z dv dx = v, (2.11) and the sequence of body fixed rotation (w.r.t. the undeformed axis) is θ z θ y θ x. (2.12) Substituting the above relations (Equation 2.11) and simplifying to second order terms yield the transformation between deformed and undeformed blade positions T du = 1 v 2 2 w 2 2 v w v cos θ 1 w sin θ 1 (1 v 2 2 ) cos θ 1 v w sin θ 1 (1 w 2 2 ) sin θ 1 v sin θ 1 w cos θ 1 (1 v 2 2 ) sin θ 1 v w cos θ 1 (1 w 2 2 ) cos θ 1. (2.13) For small elastic twist angle, the trigonometric terms, cos θ 1 and sin θ 1, can be

72 45 approximated by cos θ 1 cos θ o ˆφ sin θ o, sin θ 1 sin θ o + ˆφ cos θ o, (2.14) By substituting the above equation into T du, we can decompose of it as following constant, linear, and nonlinear matrices T du (T du ) o + (T du ) L + (T du ) NL : O(ε 2 ), (2.15) where (T du ) L = (T du ) o = cos θ o sin θ o 0 sin θ o cos θ o 0 v w v cos θ o w sin θ o ˆφ sin θ o ˆφ cos θo, (2.16), (2.17) v sin θ o w cos θ o ˆφ cos θ o ˆφ sin θ o 1 2 (v 2 + w 2 ) 0 0 (T du ˆφ(v 1 sin θ o w 2 cos θ o ) v 2 cos θ o 1 ) NL = v w 2 sin θ w 2 sin θ o o. (2.18) 1 ˆφ(v cos θ o + w 2 sin θ o ) v 2 sin θ o 1 v w 2 cos θ w 2 cos θ o o Nondimensionalization and Ordering Scheme Many of the derivations contain complex expressions that are multiplied together. In order to reduce the total number of terms in the formulation, and ordering scheme is applied. This provides a method for systematically neglecting terms based on their relative magnitude. The larger terms are kept, while the smaller terms are neglected, or ordered out. In developing the analysis, the physical quantities are nondimensionalized by the reference parameters given in Table 2.1. Nondimensional quantities are only used in the subsequent analysis.

73 46 Table 2.1. Nondimensionalized parameters Physical Quantity Reference Parameter Length R Time 1/Ω Mass/Length m o Velocity ΩR Acceleration Ω 2 R Force m o Ω 2 R 2 Moment m o Ω 2 R 3 Energy or Work m o Ω 2 R 3 In formulating Hamilton s principle, it is important to neglect higher order terms to simplify the analysis. Terms up to second order are retained in the analysis by introducing the nondimensional quantity ɛ, such that ɛ 1. Some third order terms related to elastic torsion are also retained in the energy expressions. The order of various terms is given in Table 2.2. The Mathematica programs, which are provided in Appendix C for reference, are used to derive the equations that will be presented in Chapter 2 and to apply the ordering scheme. Table 2.2. Order of terms used in aeroelastic analysis Term List Order Ĉ ij O(ε 2 ) x, h, x cg, y cg, m, O(1) ψ x µ, cos ψ, sin ψ, θ o, θ tw, θ 75, θ 1c, θ 1s, c 1, d 2 O(1) v, w, ˆφ, β p, η, ζ, n, α s, φ s O(ε) λ, c o, d 1, f o O(ε) u, d o, f 1, λ T O(ε 2 ) e d, e g O(ε 3/2 ) Special notice is taken to the torsion terms km 2 1 and km 2 1 and the wall thickness of blade n. These terms normally would be order ε 2, but are treated as being ε in this analysis. The reason is that in the torsion equations, most of the terms are small compared to the flap and lag bending terms. The wall thickness n is retained to capture the effect of wall thickness. Therefore, km 2 1, km 2 2 and n have

74 47 been treated as O(ε) and O(ε), respectively. The definition of the reference mass per unit length, m o, requires special consideration. It is defined as the mass per unit length of a uniform blade with the same flap inertia as the blade being considered. Thus, the reference mass per unit length of a non-uniform blade is calculated by m o = 3I b R = 3 R 0 mr2 dr. (2.19) 3 R 3 Also, time derivatives are slightly different when non-dimensionalized. They are related by ψ = Ωt, so derivatives are given by d() dt = d() dψ ψ = Ωt, (2.20) dψ dt = Ωd() dψ = Ω(), (2.21) d 2 () dt 2 = d2 () dψ 2 d 2 ψ dt 2 = Ω2 d2 () dψ 2 = Ω(). (2.22) Variational Formulation Hamilton s variational principle is used to derive the system equations of motion. For a conservative system, Hamilton s principle states that the true motion of a system, between prescribed initial conditions at time t 1 and final conditions ate time t 2, is that particular motion for which the time integral of the difference between the potential and kinetic energies is a minimum. For an aeroelastic system, e.g., the rotor, there are nonconservative forces which are not derivable from a potential function. The generalized or extended Hamilton s principle, applicable to nonconservative systems, is expressed as δπ = t2 t 1 (δu δt δw ) dt = 0, (2.23) where δu is the first variation of strain energy, δt is the first variation of kinetic energy, and δw is the virtual work done by external forces.

75 48 The motion of rotor blade results in the elastic deformation and the acceleration of blade. Thus, the main source of the strain and kinetic energies is a main rotor blade. External forces are caused by aerodynamic forces and moments. These variations have contributions from the rotor, the trailing edge flap, and the actuators. The contributions from each terms can be expressed as the sum of contributions from each blade. The variations can be written as N b δu = (δu r + δu a ), (2.24) b=1 N b δt = (δt r + δt f + δt a ), (2.25) b=1 N b δw = (δw r + δw f + δw a ), (2.26) b=1 where the subscript r denotes contribution from the rotor, the subscript f denotes contribution from the trailing edge flap (TEF), and the subscript a denotes contribution from the actuator for TEF. The main rotor s structural and aerodynamic contributions are described in following two sections, Section 2.2 and Section 2.3, respectively. The trailing-edge flap s inertial and aerodynamic contributions will be described in Chapter 3. In this study, the contribution from actuator is neglected in the analysis of helicopter dynamics. Coupled actuator and trailing-edge flap dynamics will be discussed in Chapter 6.

76 Structural Model The strain and kinetic energies of rotor blades are formulated by deriving the integral equations in the blade deformed coordinate ξ and undeformed coordinates y and z. They are discretized for use with the finite element method. The strain measure used in the strain energy section and framework in the kinetic energy section are given in Appendix A Strain Energy of Rotor Blade Assuming that each rotor blade is a long slender anisotropic laminated composite beam, the constitutive relationships between stress and strain in an orthotropic ply are given by σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 (k) Q 11 Q 12 Q Q 12 Q 22 Q Q 13 Q 23 Q = Q Q Q 66 (k) ε 11 ε 22 ε 33 γ 23 γ 13 γ 12 (k), (2.27) where 1, 2, 3 denotes the material coordinate system. The subscript (k) denotes the k th lamina. Let us assume a cylindrical geometry that has a curvilinear coordinate system (ξ, s, n). The constitutive equation (Equation 2.27) can be transformed into a physical coordinate system as follows: σ ξξ σ ss σ nn σ sn σ ξn σ ξs (k) = Q 11 Q12 Q Q16 Q 12 Q22 Q Q26 Q 13 Q23 Q Q Q44 Q Q45 Q55 0 Q 16 Q26 Q Q66 (k) ε ξξ ε ss ε nn γ sn γ ξn γ ξs (k), (2.28) where Q ij denotes the transformed stiffness. These transformation relations can

77 50 be found in Reference [140]. For a long slender geometry, the uniaxial stress assumption (σ nn = θ ss = θ sn = 0) is valid. In addition, it can be assumed that the transverse shear strain θ ξn is small, because the warping of thin wall due to flexure can be neglected. These assumption allow us to use the classical lamination theory [140]. By applying these assumption to Equation 2.28, the corresponding relationship between stresses and strains can be obtained by { σξξ σ ξs where Ĉij are defined as } [ ] Ĉ11 Ĉ 16 = Ĉ 16 Ĉ 66 (k) (k) { εξξ } γ ξs (k), (2.29) Ĉ 11 = Q 11 + Q 2 13 Q 22 2 Q 12 Q13 Q23 + Q 2 12 Q 33 Q 2 23 Q 22 Q33, (2.30) Ĉ 16 = Q 16 + Q 13 Q23 Q26 + Q 12 Q26 Q33 + Q 13 Q22 Q36 Q 12 Q23 Q36 Q 2 23 Q 22 Q33, (2.31) Ĉ 66 = Q 66 + Q 2 26 Q 33 2 Q 23 Q26 Q36 + Q 22 Q2 36 Q 2 23 Q 22 Q33. (2.32) Substituting Equations A.33 and A.34 into Equation 2.29 and integrating with respect to n, which is a thickness direction of the thin wall, give the following relationship between resultant forces and strains of thin wall N ξξ N ξs M ξξ M ξs A 11 A 16 B 11 B 16 = A 16 A 66 B 16 B 66 B 11 B 16 D 11 D 16 B 16 B 66 D 16 D 66 ε o ξξ γξs o k ξξ k ξs, (2.33) where (A ij, B ij, D ij ) = n/2 n/2 Ĉ (k) ij (1, n, n 2 ) dn. (2.34)

78 51 Then the first variation of strain energy of the blade can be expressed by δu r = R 0 s ( Nξξ δε o ξξ + N ξs δγ o ξs + M ξξ δk ξξ + M ξs δk ξs ) ds dx, (2.35) where stress resultant N ij, moment resultant M ij, axial strain ε o ξξ, and in-plane shear strain γξs o can be decomposed into linear and nonlinear terms N ξξ = N L ξξ + N NL ξξ, M ξξ = M L ξξ + M NL ξξ, (2.36) N ξs = N L ξs + N NL ξs, M ξs = M L ξs + M NL ξs. (2.37) The strain energy can be also decomposed as δu r = δu L r + δu NL r, (2.38) where δu L r = R 0 R δur NL = + 0 R 0 s s s ( N L ξξ δε L ξξ + N L ξsδγ L ξs + M L ξξδk ξξ + M L ξsδk ξs ) ds dx, (2.39) ( N NL ξξ ( Nξξ δε NL ξξ δε L ξξ + N NL ξs δγ L ξs + M NL ξξ ) δk ξξ + Mξs NL δk ξs ds dx ) + N ξs δγξs NL ds dx. (2.40) The first variation of linear strains and curvatures (Equations A.33 and A.34) is given by δε L ξξ = δu e y o δv z o δw λ T δ ˆφ, (2.41) δγξs L = γ ξs t δ ˆφ, (2.42) δk ξξ = cos θ δw sin θ δv + q δ ˆφ, (2.43) δk ξs = 2 δ ˆφ. (2.44) Substituting the above equations into Equation 2.39 gives another form of linear strain energy. The variation of linear strain energy of composite beam can be

79 52 rewritten by δu L r = R 0 ( N x δu e M y δw + M z δv + T ω δ ˆφ + T φ δ ˆφ ) dx, (2.45) where N x = N L ξξ ds, (2.46) M y = ( zo N L ξξ + M L ξξ cos θ ) ds, (2.47) M z = T ω = T φ = ( yo Nξξ L Mξξ L sin θ ) ds, (2.48) (q M L ξξ λ T Nξξ) L ds, (2.49) (2 M L ξs + Nξs L γ ξs) t ds, (2.50) in which the direction of moments is based on the undeformed coordinate system e u i and the direction of torque and force is based on the deformed coordinate system e d i. N x denotes the axial force resultant, M y is the flapwise bending moment and M z represents the lagwise bending moment. T ω is the torque due to warping, which is so called Valsov torsion, and T φ is the St. Venant torque. These force, moment and torque can be expressed in terms of degrees of freedom, such as u e, v, w and ˆφ. The constitutive equation of composite beam for linear strain can be written as follows: N x M y M z T ω T φ where K ij are defined as = K 11 K12 K13 K14 K15 K 12 K22 K23 K24 K25 K 13 K23 K33 K34 K35 K 14 K24 K34 K44 K45 K 15 K25 K35 K45 K55 u e w v ˆφ ˆφ, (2.51) K 12 = K 12 cos θ o + K 13 sin θ o, (2.52) K 13 = K 13 cos θ o K 12 sin θ o, (2.53)

80 53 K 22 = K 22 cos 2 θ o + K 33 sin 2 θ o + K 23 sin 2θ o, (2.54) K 23 = (K 33 K 22 ) sin θ o cos θ o + K 23 cos 2θ o, (2.55) K 24 = K 24 cos θ o + K 34 sin θ o, (2.56) K 25 = K 25 cos θ o + K 35 cos θ o, (2.57) K 33 = K 33 cos 2 θ o + K 22 sin 2 θ o K 23 sin 2θ o, (2.58) K 34 = K 34 cos θ o K 24 sin θ o, (2.59) K 35 = K 35 cos θ o K 25 sin θ o. (2.60) The other K ij terms are defined to equal to K ij which denotes K ij of which the rigid pitch angle θ o is set to zero, i.e., K ij = K ij θo 0. (2.61) The K ij are defined in Appendix B.1.1. The stiffness coefficients K 11, K 22, K 33, K 44, and K 55 are related to in-plane stiffness EA, flap bending rigidity EI y, lag bending rigidity EI z, warping rigidity EI ωω, and torsional rigidity GJ in the isotropic beam, respectively. The coupling terms K 25 and K 35 denote the flaptorsion coupling and the lag-torsion coupling stiffness, respectively. The coupling term K 13 is also related to EA ea EA ea etc. can be found in Reference [76]. of the isotropic beam. Definite of EA, GJ, and It is assumed that the other coupling terms except K 13, K 15 K 25, and K 35 can be vanished or neglected, because the purpose of present study is to reduce the helicopter rotor vibration and blade loads using the tension-torsion, the flaptorsion, or the lag-torsion coupling. The terms related to warping, ζ A 11 ds and ζη A11 ds, are therefore discarded. For such a configuration, the stiffness matrices can be rewritten, in terms of shape function, as follows: [K uu ] = [K uw ] = [K uv ] = K 11 Hu T Hu dx, (2.62) K 13 sin θ o Hu T Hw dx, (2.63) K 13 cos θ o Hu T Hv dx, (2.64)

81 [K up ] = [K ww ] = [K wv ] = [K wp ] = [K vv ] = [K vp ] = [K pp ] = 54 K 15 Hu T Hp dx, (2.65) (K 22 cos 2 θ o + K 33 sin 2 θ o ) Hw T Hw dx, (2.66) (K 33 K 22 ) cos θ o sin θ o Hw T Hv dx, (2.67) (K 25 cos θ o + K 35 sin θ o ) Hw T Hp dx, (2.68) (K 33 cos 2 θ o + K 22 sin 2 θ o ) Hv T Hv dx, (2.69) (K 35 cos θ o K 25 sin θ o ) Hv T Hp dx, (2.70) (K 44 Hp T Hp + K 55 Hp T Hp ) dx. (2.71) by The first variations of nonlinear strains (Equations A.33 and A.34) are given δε NL ξξ = (η 2 + ζ 2 )θ o (δ ˆφ ) + w δv + v δw ( + (η 2 + ζ 2 ) ˆφ δ ˆφ + w v δ ˆφ ) + w ˆφ δv + v ˆφ δw + (z o v y o w )δ ˆφ + ˆφ (z o δv y o δw ), (2.72) δγ NL ξs = γ t ξsv δw + γ t ξsw δv. (2.73) Substituting Equations 2.72 and 2.73 into Equation 2.40 yields another form of a strain energy. Then the variation of nonlinear strain energy of composite beam is rewritten by R ( δur NL = 0 + N NL w N NL x δu e M NL y δw + N NL δ ˆφ ˆφ δw + Mz NL δv + Tω NL δ ˆφ + Tφ NL δ ˆφ ) dx, (2.74) where N NL x = N NL ξξ ds, (2.75)

82 M NL y = M NL z = ( z o N NL ξξ { yo N NL ξξ 55 + Mξξ NL cos θ ) y o N ξξ ˆφ ds, (2.76) Mξξ NL sin θ ) + N ξξ (z o ˆφ + (ζ 2 + η 2 )(w θ o + w ˆφ + w 2 v ) + N ξs γ tξsw } ds, (2.77) T NL ω = T NL φ = (q M NL ξξ { 2 M NL ξs ) λ T Nξξ NL ds, (2.78) } + N NL ξs γ t ξs + N ξξ (η 2 + ζ 2 )( ˆφ + θ o + w v ) ds, (2.79) N NL { w = N ξξ (η 2 + ζ 2 )( ˆφ } v + θ ov + w v 2 ) + N ξs γ ξsv t ds, (2.80) Nφ NL = N ξξ (y o w + z o v ) ds. (2.81) Note that some linear terms are appeared in Equations , because they are treated as the nonlinear terms (see Appendix A.1). Nonlinear strain energy terms are complicate, so the ordering scheme up to order O(ε 2 ) based on Table 2.2 is applied to reduce the complexity. The laminate stiffness B ij and D ij have been discarded, because the warping through the wall thickness is very small, when compared to the beam cross-section warping. Then the following reduced nonlinear terms are obtained. N NL x M NL y M NL z = Ak 2 11( 1 2 ˆφ 2 + θ ov w ) + Ak 2 11 ˆφ θ o + ˆK 13 (w ˆφ cos θo v ˆφ sin θo ) + ˆK 15 v w, (2.82) = ( ˆK 33 ˆK 22 )(v ˆφ cos 2θo + w ˆφ sin 2θo ) + ˆK 13 cos θ o u e ˆφ + ˆK 25 (v w cos θ o ˆφ ˆφ sin θ o ) + ˆK 35 (v w sin θ o + ˆφ ˆφ cos θ o ) + (AB 1 sin θ o + AB 2 cos θ o ) ˆφ θ o, (2.83) = ( ˆK 33 ˆK 22 )(w ˆφ cos 2θo v ˆφ sin 2θo ) ˆK 13 sin θ o u e ˆφ + ˆK 55 w ˆφ + Ak11θ 2 ou ew + ˆK 15 u ew + ˆK { 25 (w w ˆφ ˆφ } ) cos θ o 2w v sin θ o + ˆK { 35 (w w ˆφ ˆφ } ) sin θ o + 2w v cos θ o + (AB 1 cos θ o AB 2 sin θ o ) ˆφ θ o, (2.84) T NL ω = ˆK 44 w v, (2.85)

83 56 T NL φ = ˆK 35 (w ˆφ cos θo v ˆφ sin θo ) ˆK 25 (w ˆφ sin θo + v ˆφ cos θo ) + ˆK 55 w v + Ak 2 11u e ˆφ + Ak 2 11u eθ o, (2.86) N NL w = ˆK 25 (v w cos θ o v 2 sin θ o ) + ˆK 35 (v w sin θ o + v 2 cos θ o ) + ˆK 55 v ˆφ + Ak11θ 2 ou ev + ˆK 15 u ev, (2.87) Nφ NL = ( ˆK 33 ˆK 22 ) { } v w cos 2θ o + (w 2 v 2 ) sin θ o cos θ o + ˆK 13 (w u e cos θ o v u e sin θ o ) + ˆK 35 (w ˆφ cos θ o v ˆφ sin θ o ) ˆK 25 (w ˆφ sin θ o + v ˆφ cos θ o ),(2.88) where Ak11 2 = AB 1 = AB 2 = A 11 (η 2 + ζ 2 ) ds, (2.89) A 11 η(η 2 + ζ 2 ) ds, (2.90) A 11 ζ(η 2 + ζ 2 ) ds, (2.91) the terms ˆK ij are defined as K ij of which the stretching bending coupling B ij and bending rigidity D ij of thin wall are set to zero, i.e., ˆK ij = K ij Bij,D ij 0, (2.92) and the underline terms are linear terms. The nonlinear terms except the underline terms are treated as the nonlinear force vector. The stiffness matrices due to the linear terms can be expressed as [K up ] = Ak11 2 Hu T H p dx, (2.93) [K wp ] = (AB 1 sin θ o + AB 2 cos θ o )θ o Hw T H p dx, (2.94) [K vp ] = (AB 1 cos θ o AB 2 sin θ o )θ o Hv T H p dx, (2.95) and the nonlinear forces in terms of shape functions are given by {F u } = N NL x H T u dx, (2.96)

84 57 ( {F w } = My NL {F v } = {F p } = M NL z ( T NL ω H T v H T p ) Hw T + N NL w HT w dx, (2.97) dx, (2.98) + T NL φ H T p + N Ṋ φ HT p ) dx, (2.99) where the linear terms are discarded. Total stiffness matrices of strain energy and explicit force vectors are given in Appendix B Kinetic Energy of Rotor Blade The kinetic energy of the rotor blade, δt r, depends on the blade velocity. This velocity is generally due to: (1) blade motion relative to the hub, as well as (2) the motion of the hub itself. This relationship is expressed mathematically as V = V b + V f, (2.100) where V b is the velocity of the blade relative to the hub and V f is the velocity (at the blade) induced by the motion of the fuselage. In the present analysis, the hub is assumed to be rigidly attached to the fuselage. The velocity due to the fuselage motion has been neglected in this study. The position vector, r, of an arbitrary point on the blade after deformation is given, in the mixed coordinate system, as r = (s + u e λ T ˆφ )e d 1 + ηe d 2 + ζe d 3, = (x + u λ T φ )e u 1 + ve u 2 + we u 3 + ηe d 2 + ζe d 3, (2.101) where the first equations are based on the deformed coordinate system and the second equations are based on the undeformed coordinate system. Note that the transformation relationships between the undeformed and deformed coordinate system is valid in sense of the infinitesimal quantities. The infinitesimal length along the deformed axis, ds, may be equal to the

85 58 infinitesimal length along the undeformed axis, dx. The difference between two incremental length can be expressed by (dx + du) = 1 (v 2 + w 2 )(ds + du e ), { 1 1 } 2 (v 2 + w 2 ) (ds + du e ), (2.102) and du du e 1 2 (v 2 + w 2 )dx, (2.103) u = u e 1 2 x 0 (v 2 + w 2 )dx, (2.104) where u e is very small, so we can usually neglect this term. This relationships can be explained in that the infinitesimal displacement, du, shall be shrunk as the beam rotates rigidly. Assuming that the beam is inextensible, the length of beam along the deformed axis, s, in terms of x is given by se d 1 = { x 1 2 x 0 } (v 2 + w 2 )dx e u 1 + ve u 2 + we u 3. (2.105) Referring to Equation 2.14, the relationship between the deformed and undeformed coordinate system is given by e d 2 (v cos θ 1 + w sin θ 1 ) e u 1 + cos θ 1 e u 2 + sin θ 1 e u 3, (2.106) e d 3 (v sin θ 1 w cos θ 1 ) e u 1 sin θ 1 e u 2 + cos θ 1 e u 3. (2.107) By substituting Equations and into Equation 2.101, the deformed position vector can be rewritten, in the undeformed coordinate system, as follows: r = x 1 e u 1 + y 1 e u 2 + z 2 e u 3, (2.108) where x 1 = x + u e 1 2 x 0 (v 2 + w 2 )dx λ T φ v ȳ w z, (2.109) y 1 = v + ȳ, z 1 = w + z, (2.110)

86 59 ȳ = η cos θ 1 ζ sin θ 1, z = η sin θ 1 + ζ cos θ 1. (2.111) Then the velocity of an arbitrary point on the blade can be calculated as V b = r t + ω r, ω = Ω eh 3 Ω(β p e u 1 + e u 3), (2.112) and the velocity components are given by V b = V bx e u 1 + V by e u 2 + V bz e u 3, (2.113) where V bx = ẋ 1 Ωy 1, (2.114) V by = ẏ 1 + Ωx 1 Ωz 1 β p, (2.115) V bz = ż 1 + Ωy 1 β p. (2.116) The derivatives of the arbitrary point on the blade with respect to time are given as ẋ 1 = u e x 0 (v v + w ẇ )dx λ T φ ( v + w θ1 )ȳ (ẇ v θ1 ) z, (2.117) ẏ 1 = v θ 1 z, (2.118) ż 1 = ẇ + θ 1 ȳ, (2.119) and the kinetic energy variation can be expressed by δt r = R 0 A ρ V δ V dadx. (2.120) Substituting Equations to into Equation and integrating by parts yield δ T r = δt r m o Ω 2 R 3 = 1 0 A ρ s (T x1 δx 1 + T y1 δy 1 + T z1 δz 1 ) dadx, (2.121)

87 60 where ρ s is the material density of structure and in which x 1 = u e x T x1 = x 1 + 2y 1 + x 1 z 1 β p, (2.122) 0 T y1 = y 1 2x 1 + y 1 + 2z 1β p, (2.123) T z1 = z 1 2y 1β p x 1 β p + z 1 β 2 p, (2.124) ( v v + v v + w w + w w ) dx λ T φ ( v + w θ 1 + 2θ 1w ) ȳ ( w v θ 1 2θ 1v ) z, (2.125) y 1 = v θ 1 z θ 12ȳ, (2.126) z 1 = w + θ 1 ȳ θ 12 z. (2.127) The variations of the deformed position vector components, x 1, y 1 and z 1, are given by δx 1 = δu e λ T δφ ȳδv v δȳ zδw w δ z x 0 (v δv + w δw )dx, (2.128) δy 1 = δv + δȳ, (2.129) δz 1 = δw + δ z, (2.130) where δȳ = z δ ˆφ, δ z = ȳ δ ˆφ. (2.131) where Now the variation of kinetic energy (Equation 2.121) can be rewritten by δ T r = 1 0 ( Vx I δu + Vy I δv + Vz I δw + Mz I δv My I δw + Mxδ I ˆφ ) dx, (2.132) [ Vx I = m x + u e 1 2 x 0 (v 2 + w 2 )dx u e

88 61 V I y V I z + x 0 ( v v + v v + w w + w w ) dx + 2v wβ p e g cos θ o (v v + 2θo ˆφ + θo 2 v θo w 2θow ) ( e g sin θ o 2θ 1 + β p + w w + θo v + θ o2 w + 2θov )], (2.133) [ x = m v v + 2w β p 2u e + 2 (v v + w w )dx 0 + e g cos θ o (1 + 2v + 2 ˆφ θo + θo 2 + θo ˆφ ) + 2θoβ p + 2θow ( + e g sin θ o θ 1 ˆφ + 2w θo [ = m w (x + 2v )β p e g cos θ o (θ1 θo2 ˆφ) + e g sin θ o (2θo ˆφ + θo 2 + θo ˆφ )] + 2θoβ p Mz I = m [e g {( x 2v ) cos θ o + x ˆφ } sin θ o + (km1 2 sin 2 θ o + km2 2 cos 2 θ o ) ( 2θo ˆφ + v v + θo 2 v θo w 2θow ) 2 ˆφ 2θ o v )], (2.134), (2.135) (k 2 m2 sin 2 θ o + k 2 m1 cos 2 θ o )2θ o ˆφ (2.136) + cos θ o sin θ o (km2 2 km1) 2 ( w w + 2θ1 + β p + θo v + θ o2 w + 2θov )], My I = m [e g {( x 2v ) sin θ o x ˆφ } cos θ o + (cos 2 θ o k 2 m1 + sin 2 θ o k 2 m2)β p + sin θ o cos θ o (km2 2 km1)(v 2 v + 4θo ˆφ + θo 2 v θo w 2θow ) + (km1 2 cos 2 θ o + km2 2 sin 2 θ o ) (2.137) ( w + 2θ1 w + θo v + θ o2 w + 2θov )], ) Mx I = m [e g sin θ o ( v + v + xv ) e g cos θ o (w + xβ p + xw +2v β p + sin θ o cos θ o (km1 2 km2)( v +2θ1w ) (2.138) kmθ ˆφ(k ] m1 2 km2) 2 cos 2θ o 2w (km1 2 cos 2 θ o + km2 2 sin 2 θ o ), where mk 2 m1 and mk 2 m2 represent the flapwise and lagwise mass moments of inertia per unit length, respectively, and are given by m(e g, km1, 2 km2) 2 = A ρ s (η, ζ 2, η 2 )da, (2.139)

89 62 and mk 2 m = mk 2 m1 + mk 2 m2. (2.140) By substituting Equation into Equation 2.132, the variation of kinetic energy can be also expressed by δ T r = 1 0 ( V I x δu e + V y I δv + V z I δw + Mz I δv My I δw + Mxδ I ˆφ ) dx, (2.141) where V I x = V I x + m 1 2 x 0 (v 2 + w 2 )dξ V I x, (2.142) 1 V y I = Vy I m(x + 2v )v + v V I y mv 2 x 0 x m(ξ + 2v )dξ (2.143) m v v dξ, (2.144) 1 V z I = Vz I m(x + 2v )w + w V I z mw 2 x 0 x m(ξ + 2v )dξ (2.145) m w v dξ, (2.146) in which the underline terms denote the curvature effect due to the geometric nonlinearity. This effect may be neglected under assumption of a moderately large rotation. A helicopter rotor system is under a strong centrifugal force, so that the deformed geometric shape is almost a straight line except near root of the blade. Here we developed two types of the variation of a kinetic energy. One is the forces and moments based on the undeformed coordinate system, and the other is based on the mixed coordinate system. In previous section, the variation of a strain energy was derived in the deformed coordinate ξ that corresponds to u e. The variation of a kinetic energy given in Equation 2.141, therefore, should used to be consistent with the strain energy formulation. Equation 2.132, however, may

90 63 be used to get the blade root loads with the force summation method, since the hub loads are defined in the hub-fixed coordinate system. The matrix form of kinetic energy is implemented using the weak form of strain energy. For example, the weak form of a double integral term can be expressed as follows: m(x + 2v ) x 0 [ 1 (v δv + w δw )dξ = x ] m(ξ + 2v )dξ (v δv + w δw ). (2.147) The system matrices, such as stiffness, damping and mass matrices, and force vectors due to the kinetic energy of rotor blades are given in Appendix B.2.

91 Aerodynamic Blade Loads The linear aerodynamic model is based on a quasi-steady strip theory. The blade velocity is derived in the blade undeformed frame and then translated to the deformed frame to calculate the airloads. In the present analysis, motion of the fuselage is not considered. The aerodynamics only take into account the wind velocity, including the helicopter forward and climb speed and the rotor rotation, and the motion of the blades relative to the hub. V = V w + V b, (2.148) where V w is the wind velocity with contributions from the vehicle forward speed and the rotor inflow and V b is the blade velocity relative to the hub fixed frame resulting from blade rotation and blade motions. The blade velocity due to the helicopter forward speed and the rotor rotation is expressed as V w = µωr e H 1 λωr e H 3, (2.149) where µ is the rotor advance ratio, λ is the rotor non-dimensional inflow, and Ω is the rotor angular velocity. The rotor inflow ratio, λ, for small longitudinal shaft tilt angles, consists of two components and is expressed by λ = µ tan α s + λ i µα s + λ i. (2.150) These velocities, V w and V b, must be transformed through the blade precone, β p, and around the rotor azimuth, ψ. This can be done through direction cosine matrices about body fixed axes, first about the rotor azimuth, then through precone. From Equation 2.5, the transformation matrix between the hub coordinates, e H i, and the blade coordinates that are the undeformed coordinates, e u i, is given by e u i = T uh ij e H j. (2.151) The wind velocity on the undeformed coordinate systems with a small angle

92 65 assumption for precone, β p, is expressed as V w = V wi e u i = (µωr cos ψ λωrβ p ) e u 1 µωr sin ψ e u 2 (µωr cos ψβ p + λωr) e u 3. (2.152) The blade velocity, Vb, with respect to the rotating undeformed frame can be written as V b = r t + ω r = V bi e u i. (2.153) From Equations , the blade velocity components are given by V b1 = u λ T φ ( v + w θ1 )ȳ (ẇ v θ1 ) z Ω(v + ȳ), (2.154) V b2 = v θ 1 z + Ω(x + u λ T φ v ȳ w z) Ω(w + z)β p, (2.155) V b3 = ẇ + θ 1 ȳ + Ω(v + ȳ)β p, (2.156) where u = u e 1 2 x (v 2 + w 2 )dx, u = u e x 0 0 (v v + w ẇ )dx, (2.157) φ = ˆφ + w v, θ 1 = θ o + ˆφ, (2.158) ȳ = η cos θ 1 ζ sin θ 1, z = η sin θ 1 + ζ cos θ 1. (2.159) For quasi-steady aerodynamics, the rotor blade aerodynamic loads are calculated using a blade section strip analysis based on the angle of attack at the threequarter chord location. In the cross section, this point is at η = η r and ζ = 0, simplifying the blade velocity expressions. Then the total velocity, V = V b V w, can be written as V = V i e u i, (2.160)

93 66 in which V x = u λ T ˆφ Ω(v + η r cos θ 1 ) + λωrβ p µωr cos ψ η r ( θ 1 w + v ) cos θ 1 η r (ẇ θ 1 v ) sin θ 1, (2.161) V y = v + Ω(u + x) + µω sin ψ η r θ1 sin θ 1 β p Ω(w + η r sin θ 1 ) Ωη r (v cos θ 1 + w sin θ 1 ), (2.162) V z = ẇ + λωr + µωrβ p cos ψ + Ω(v + η r cos θ 1 )β p + η r θ1 cos θ 1. (2.163) The blade section loads are calculated using the resultant velocity and aerodynamic angle of attack in the deformed blade. Therefore, these velocities should be transformed to the blade deformed frame. The transformation matrix is given in Equation e d i = Tij du e u j, V = Vi e u i = V i Tij du 1 e d j = U j e d j. (2.164) Thus the velocity components, U j, using the transformation matrix property, Tij du 1 = Tji du, are given by V i T du ji = T du ji V i = U j. (2.165) The velocities in the deformed frame are used to find the blade angle of attack and to determine the radial flow along the blade. While performing the transformation, θ 1 is replaced with θ o + ˆφ to modify cos θ 1 and sin θ 1, and assuming ˆφ is a small angle. sin θ 1 = sin(θ o + ˆφ) sin θ o + ˆφ cos θ o, (2.166) cos θ 1 = cos(θ o + ˆφ) cos θ o ˆφ sin θ o. (2.167) After transformation to the deformed frame using Equation 2.165, the resultant blade velocity in the deformed rotating frame neglecting the warping term and the higher order terms referring to Table 2.2 is expressed as

94 67 U x ΩR = u v + λ(β p + w ) + µv sin ψ + +xv + v v + w w { + µ cos ψ 1 + β p w + 1 } 2 (v 2 + w 2 ) η r (1 + v ) cos θ o + η r ( ˆφ w ) sin θ o, (2.168) { U y ΩR = cos θ o u + x + v + (λ + w ) ˆφ wβ p + vv λβ p v 1 2 xv 2 + µ sin ψ(1 v 2 } 2 ) + µ cos ψ(β ˆφ p + w ˆφ + v ) { + sin θ o λ(1 β p w 1 2 w 2 ) + w x( ˆφ + v w ) v ˆφ + vβp + vw + µ cos ψ(β p ˆφv + w ) µ sin ψ( ˆφ } + v w ), (2.169) U z ΩR = η r( ˆφ + θ o + β p + w ) + sin θ o { u x v + λ(β p v ˆφ) w ˆφ + wβp vv xv 2 + µ sin ψ( 1 + v 2 } 2 ) µ cos ψ(β ˆφ p + w ˆφ + v ) { + cos θ o λ + w x ˆφ v ˆφ + vβp + vw λ(β p w + w 2 2 ) xv w + µ cos ψ(β p v ˆφ + w ) µ sin ψ( ˆφ } v w ), (2.170) where U x, U y, and U z denote the radial, tangential, and perpendicular velocities, respectively. These velocities are also referred to as U R, U T, and U P. The displacements, u, v, w, and the radial position, x, are the normalized quantities with the blade length, R. The superscript, (), represents the derivative with respect to the azimuth ψ.

95 Quasi-steady Airloads In this section, the blade loads are modeled as quasi-steady. This means that although the loads change with time, at a particular instant, they are assumed to be only a function of conditions at that instant. In other words, for a certain section of the blade at a certain azimuth, the circulatory lift is only dependent upon the angle of attack of that section at that time. The blade circulatory airloads per unit length in the rotating deformed frame, e d i, can be written as L C = 1 2 ρu 2 cc l, (2.171) D C = 1 2 ρu 2 cc d, (2.172) M C = 1 2 ρu 2 c 2 C m, (2.173) where U is the incident velocity and C l, C d, C m are the section lift, drag, and moment coefficients, respectively. c is the chord of blade section. The aerodynamic coefficients are expressed, in terms of the angle of attack, as C l = c o + c l α, (2.174) C d = d o + d 1 α + d 2 α 2, (2.175) C m = f o + f 1 α = c mac + f 1 α, (2.176) in which c o is the lift coefficient when the airfoil is at a zero angle of attack, c 1 is the lift curve slope, often given by C lα or C nα, and d o is the profile drag of the blades due to viscous effects and the nonlinear term, d 2, causes a drag rise with large angles of attack. f o or c mac is the zero angle pitching moment coefficient about aerodynamic center and f 1 is the slope of the moment curve. These relations are restricted to the incompressible attached flow conditions. Compressibility effects are accounted by modifying the lift curve slope, c l, as c l = c l M= β, β = 1 M 2, (2.177)

96 69 where β is the Prandtl-Glauert factor. This correction factor works quite well at low Mach numbers, M 0.85, characteristic of a rotor blade. For the most part of rotor disk, the rotor blade sees normal flow conditions. In other areas, where the angle of attack is extremely high or the flow is reversed, the dynamic pressure is low, so there is not much effect on the overall response. Near this region, particularly near the boundary forward flow and reverse flow, where the tangential velocity U T is near zero, the small angle approximation breaks down. The detail description of reverse flow model can be found in References [6,76,133]. The axial force, chord force, normal force, and pitching moment about the elastic axis are given by L C u = D C sin Λ, (2.178) L C v = L C sin α D C cos α, (2.179) L C w = L C cos α + D C sin α, (2.180) M Cˆφ = M C e d L C w, (2.181) where L C i are the external loads along the deformed frame, e d i, M Cˆφ is the pitching moment about the deformed elastic axis, Λ is the axial skew angle due to the radial component of incident velocity, U R, acting on the blade, and e d is the chordwise offset of the aerodynamic center behind the elastic axis. The equations are non-dimensionalized by the Lock number, γ, the blade inertia, I b, and the standard non-dimensionalization factor Ω and R. The Lock number and uniform blade inertia are given by γ = ρacr4, I b = m or 3, (2.182) I b 3 where a is the lift curve slope, C lα. All forces, moments, and velocities are nondimensionalized using the scheme described in Table 2.1. The section lift and drag forces, L C and D C, are non-dimensionalized as follows: 1 2 ρcu 2 = 1 2 ( ρr 2 m o ) ( c ) ( ) U 2 = γ ( ) 2 U = γ R Ω 2 R 2 6a ΩR 6a U 2. (2.183)

97 70 Then, The airloads, Equations , are nondimensionalized by Equation L C u = γu 2 6a ( C d sin Λ), (2.184) L C v = γu 2 6a (C l sin α C d cos α), (2.185) L C w = γu 2 6a (C l cos α + C d sin α), (2.186) M Cˆφ = γu 2 ( c ) 6a R C m e d L C w. (2.187) Substituting the expression for C l,c d, and C m from Equations , and using the following approximations yield sin α α, cos α 1, U U T, α U P U T, sin Λ U R U T, (2.188) L C u = γ { } do U R U T, (2.189) 6a L C v = γ { } d o U 2 T (c o U P d 1 U P )U T + (c 1 d 2 )U 2 P, (2.190) 6a L C w = γ } {c o U 2T (c 1 + d o )U T U P + d 1 U P U P, (2.191) 6a M Cˆφ = γ { c ) } (f o (U 2T + U 2P ) f 1 U P U T e d L C w. (2.192) 6a R The nondimensionalized aerodynamic forces and pitching moment acting on a blade section can be expressed by the following vector form. L C = L C i e d i, M C = M Cˆφ e d 1. (2.193) It is consistent with the velocity in the deformed frame, V = U i e d i. The aerodynamic forces in the undeformed frame, e u i, are obtained by using the orthogonal coordinate transformation. The transformation matrix between the deformed and undeformed coordinate systems, e d i = T du ij e u j, has already developed in Equation

98 on Page 44. The aerodynamics forces in the undeformed frame are given by L C = L C i e d i = L C i T du ij e u j = L C i e u i, M C = M Cˆφ e d 1 M Cˆφ e u 1, (2.194) L C i = L C j T du ji = T du ij L C j, M Cˆφ M Cˆφ, (2.195) where the pitching moment, M Cˆφ, is assumed small, so that higher order terms can be neglected as follows: M Cˆφ e d 1 = M Cˆφ (1 v 2 2 w 2 2 )eu 1 + M Cˆφ v e u 2 + M Cˆφ w e u 3, M Cˆφ e u 1. (2.196) The virtual work done by aerodynamic forces can be written by δw C aero = 1 0 ( L C u δu + L C v δv + L C wδw + M Cˆφ δ ˆφ ) dx. (2.197) This variational form should be expressed in terms of the elastic axis displacement, u e, to be consistent with the previous formulations for strain and kinetic energy of rotor blades. This can be achieved by using the relationship between the elastic displacement and the axial displacement in the mixed coordinate system, Equation A.23 on page 242. where δw C aero = 1 0 (ˆLC u δu e + ˆL C v δv + ˆL C wδw + M Cˆφ δ ˆφ ) dx, (2.198) ˆL C u L C u, (2.199) 1 ˆL C v = L C v L C u v + v L C u dξ, (2.200) x 1 ˆL C w = L C w L C u w + w L C u dξ, (2.201) and another form is available for finite element formulation, which is called the x

99 72 weak form. The weak form of Equation is given by δw C aero = 1 0 { L C u δu e + L C v δv + L C wδw + M Cˆφ δ ˆφ } + (v δv + w δw ) 1 x L C u dξ dx, (2.202) where the underline term could be neglected, because it is the higher order term, L C u the foreshortening term. Thus Equation can be simplified as: δw C aero 1 0 ( L C u δu e + L C v δv + L C wδw + M Cˆφ δ ˆφ ) dx. (2.203) Noncirculatory Airloads The airloads acting on the rotor blade can be classified into two categories that are circulatory and noncirculatory. The derivation of the circulatory loads has been performed in the previous section. The noncirculatory loads, which are so called apparent or virtual forces, will be derived in this section. When the airfoil has a general motion, the lift and moment of the noncirculatory origin (the apparent mass forces) must be added. For a airfoil section undergoing plunge motion, h, and pitch motion, α, the noncirculatory lift and pitching moment [139] are given by, L NC w = L 2 + L 3 = ρπb 2 (ḧ a hb α) + ρπb 2 U α), (2.204) M ˆφ NC = a h bl 2 ( 1 2 a h)bl 3 + M a, M a = ρπb4 α, (2.205) 8 in which L 2 is a lift force with center of pressure at the mid-chord, of amount equal to the apparent mass, ρπb 2, times the vertical acceleration at the mid-chord point. L 3 is a lift force with center of pressure at the three-quarter chord point, of the nature of a centrifugal force, of amount equal to the apparent mass, ρπb 2, times U α. M a is a nose-down moment equal to the apparent moment of inertia,

100 73 ρπb 2 (b 2 /8), times the angular acceleration, α. U = ΩR(x + µ sin ψ), a h b = (e d + c ), ḧ = ẅ (2.206) 4 α = θ 1 = θ o + ˆφ, b = c 2 (2.207) where U denotes the free stream tangential velocity, a h b is the distance from midchord to the elastic axis (positive aft), ḧ is the plunge acceleration (positive down), α is the pitch angle (positive nose up) and b is the airfoil semi-chord. The noncirculatory airloads are assumed to act directly on the blade undeformed section. The virtual work done by the noncirculatory airloads that are normalized is given by where L NC w δw NC aero = 1 0 (L NC w δw + M NC ˆφ δ ˆφ)dx, (2.208) = γπc { w + (c/4 + e d )(θo + 12a ˆφ ) + (x + µ sin ψ)(θo + ˆφ } ), (2.209) M NC ˆφ = γπc { (c/4 + e d )w (c/4 + e d ) 2 (θo + 12a ˆφ ) (c/2 + e d )(x + µ sin ψ)(θ o + ˆφ ) c2 32 (θ o + ˆφ ) }. (2.210) Quasi-steady Aerodynamics Implementation The virtual work by the total airloads, which includes both the circulatory and noncirculatory airloads contributions, is given by δw aero = δwaero C + δwaero, NC 1 { = L C u δu e + L C v δv + (L C w + L NC w )δw 0 + (M Cˆφ + M ˆφ NC )δ ˆφ } dx. (2.211)

101 74 This equation is discretized using the finite element method. Because many terms in Equation contain displacements, the discretization can be carried out to produce a complex forcing vector, treating the displacements as known. In order to improve the efficiency of the numerical implementation, the terms of velocities (Equations ) are divided into constant, linear, and non-linear groups as follows: Ū R = ŪRC + ŪRL + ŪRNL, (2.212) Ū T = ŪT C + ŪT L + ŪT NL, (2.213) Ū P = ŪP C + ŪP L + ŪP NL, (2.214) where Ū RC = λβ p µ cos ψ η r cos θ o, (2.215) Ū RL = u v + xv + λw + µ (v sin ψ + β p w cos ψ) v η r cos θ o + η r ( ˆφ w ) sin θ o, (2.216) Ū RNL = v v + w w µ (v 2 + w 2 ) cos ψ, (2.217) Ū T C = (x + µ sin ψ) cos θ o + (λ + µβ p cos ψ) sin θ o, (2.218) Ū T L = (u + v + λ ˆφ ) wβ p + µβ p ˆφ cos ψ + µv cos ψ λβ p v cos θ o + (w x ˆφ µ ˆφ ) sin ψ + vβ p + w µ cos ψ λβ p w sin θ o, (2.219) Ū T NL = {w ˆφ + vv 12 } v 2 (x + µ sin ψ) + µw ˆφ cos ψ cos θ o + ( v ˆφ + vw xv w µv ˆφ cos ψ µv w sin ψ 12 ) λw 2 sin θ o, (2.220) Ū P C = (θo + β p ) η r + (λ + µβ p cos ψ) cos θ o (x + µ sin ψ) sin θ o, (2.221) ( ) { Ū P L = ˆφ + w η r u + v + λ( ˆφ } β p v ) β p w + µ cos ψ(β p ˆφ + v ) sin θ o { + w x ˆφ + vβ p λβ p w + w µ cos ψ ˆφµ } sin ψ cos θ o, (2.222) Ū P NL = ( w ˆφ vv + 12 xv ) v 2 µ sin ψ w ˆφµ cos ψ sin θ o ( + v ˆφ + vw 1 ) 2 λw 2 xv w v ˆφµ cos ψ + v w µ sin ψ cos θ o. (2.223)

102 75 By substituting Equations into Equations and using Equation 2.15 on Page 45 that was also broken into constant, linear, and nonlinear terms, the stiffness, damping, and mass matrices are made. The detailed derivation of these matrices are omitted here for the sake of brevity.

103 Inflow and Free Wake Model There are several types of inflow models, such as uniform or linear inflow, prescribed wake, free wake, and finite state wake models. Accurate modeling of the induced inflow is important for analysis of flying qualities, detailed power calculations, noise. For the rotor response, loads and vibrations, a simple model is normally sufficient. At low speed and decent flight conditions, however, the wake stays close to the rotor disk and has a dominating influence on the rotor inflow. In this study, two models are used to calculate the rotor inflow. First, a simple linear flow model will be described, and followed by a sophisticated wake model Linear Inflow The induced velocity in the rotor plane is the most non-uniform, it being strongly affected by the presence of discrete tip vortices that sweep downstream near the rotor plane. In forward flight, the time-averaged longitudinal and lateral inflow can be approximately represented by the variation λ = µ tan α + λ i (1 + k x x cos ψ + k y x sin ψ), (2.224) where λ i is the induced inflow ratio, r represents the radial location along the rotor blade, and k x = 4 3 which is the Dree s model. C T λ i = 2 µ 2 + λ, (2.225) 2 ( ) 2 λ (1 1.8µ 2 ) 1 λ, (2.226) µ µ k y = 2µ, (2.227)

104 Free Wake Model The free wake model (Figure 2.6) used in this study was developed by Tauszig and Gandhi [131,132] following the free wake methodology initially developed by Bagai and Leishman [74, 75]. This wake model was implemented at Penn State and has been used to study new rotor design to alleviate Blade Vortex Interaction. In the present study, two modules were extracted from Tauszig and Gandhi s model. One module determines the free wake geometry; the other module performs the induced inflow distribution calculation. Once the vorticity strength and wake geometry are known, the induced velocity (rotor disk inflow) is evaluated using the Biot-Savart Law. Figure 2.6. Schematic of the wake, discretized in space and time [131] A general equation describing the positions of the vortex filaments in the rotor wake can be derived from Helmholtz s law (the vorticity transport theorem) by assuming that for each point in the flow the velocity is convected at the local velocity, V loc. Considering a single element of a trailed vortex filament, the fundamental equation describing the transport of the filament is d r dt = V loc ( r, t), (2.228)

105 78 where r = r(ψ w, ψ b ) is the position vector of a point on the filament at a time or wake age ψ w that was trailed from the blade when it was at an azimuth angle ψ b. By assuming that every vortex filament is convected through the flow field at the local velocity, a governing equation the geometry of a single element of the vortex filament can be written as: r + r V = ψ w ψ b Ω + 1 V Ω ind [ r(ψ b, ψ w )], (2.229) where the summation is carried out over the total number of trailed vortex filaments, which is the number of azimuthal discretization, that contribute to the induced velocity field at any given point. The induced velocity V ind, can be determined using the Biot-Savart law. The numerical solution of Equation can be accomplished by discretizing the wake and using finite difference to approximate the derivatives. Landgrebe [145] developed the explicit time finite difference schemes to solve the Equation The convergence was achieved by obtaining the periodic distorted wake geometry. Time stepping schemes, however, have been found to be rather susceptible to numerical instabilities, particularly at low advance ratios, often necessitating the use of artificial numerical damping. Bagai and Leishman [75] developed the more stable method by enforcing the wake periodicity as the boundary condition. This method is called the Pseudo-Implicit Predictor-Correct (PIPC) method by its developers. In this method, the computational domain was defined as a discretized grid in time ψ b and space ψ w. An initial wake geometry with a linear inflow is used to start the free-wake calculations. The convergence of this method was achieved when the RMS change in wake geometry (L 2 norm), between two successive iterations is within some prescribed tolerance. Some modifications have been made by Tauszig and Gandhi [132] to the wake calculation methodology. For instance, instead of using one straight segment with a constant vortex core to represent the trailing vortices, multiple segments that match an arc in a piecewise linear manner with increasing a vortex core were used to accurately determine the release tip vortex strength and near wake.

106 79 Figure 2.7. Flow chart of an aeroelastic analysis with a free wake The flow chart of an aeroelastic analysis with a free wake is presented in Figure 2.7. The procedure is started by specifying a desired advance ratio and thrust level. The iterative procedure is carried out to achieve the convergence. Two steps are used to get the final results that are the wake geometry and vehicle trim. The first step is to provide the initial condition to the free wake analysis using an aeroelastic analysis with a linear flow and a rigid wake geometry. Then the wake geometry calculation is carried out until the RMS change in wake geometry becomes sufficiently small. Based on new wake geometry, the induced velocity over the rotor disk is obtained. With this, an aeroelastic analysis is again carried out to provide the bound circulation to the wake geometry routine. This process is repeated until both the wake geometry and vehicle trim meet their convergence criterion.

107 Aeroelastic Analysis In this section, a nonlinear periodic system equation is described based on previous formulations presented in Section 2.2, Section 2.3 and 2.4. The solution of an aeroelastic response problem is carried out in two steps. First, spatial discretization based on the finite element method is used to eliminate the spatial dependence, and subsequently the combined structural and aerodynamic equations are discretized to find the solution via the finite element method in time. To reduce the computational efficiency, the normal mode transformation is applied to the blade finite element equations. This reduces the number of spatial degrees of freedom from that of blade degrees of freedom to a smaller number of modes. The force summation method is applied in the calculation of the blade loads and hub loads. A coupled trim procedure is used to obtain the vehicle trim condition. The inertial and aerodynamic loads due to the presence of the trailingedge flaps are treated as an additional load vector for the blade response. This will be discussed in Chapter 3 on Page Aeroelastic Response The blade finite element discretization is shown in Figure 2.8. The blade is discretized into 5 to 10 beam elements, each consisting of 12 degrees of freedom. The shape functions based on Hermitian polynomials are used within the elements, which are the slope continuous functions. For example, the flapping displacement w i for the i-th beam element is interpolated by where w i = H 1 H 2 H 3 H 4 w 1 w 1 w 2 w 2 i, (2.230) H 1 = 1 4 (s + 2) (s 1)2, (2.231)

108 81 H 2 = L(i) el 8 (1 s2 ) (1 s), (2.232) H 3 = 1 4 (2 s) (s + 1)2, (2.233) H 4 = L(i) el 8 (s2 1) (s + 1), (2.234) in which L (i) el is the length of the i-th beam element. Trailing edge flaps S Figure 2.8. Finite elements for composite rotor blades The extended Hamilton s principle presented in Section can be rewritten in the discretized form as [ 2π Nel 0 i=1 δ{q} T i ([M] i {q } i + [C] i {q } i + [K] i {q} i {F (q)} i ) ] dψ = 0, (2.235) in which subscript () i indicates the i-th beam element, N el is the number of element, and the elemental nodal displacement vector, {q} i, is given by {q} T i = v 1 v 1 w 1 w 1 ˆφ 1 v 2 v 2 w 2 w 2 ˆφ 2 i, (2.236) where w and v represent flap and lag displacements, respectively. ˆφ denotes the elastic twist of a blade. Derivatives with respect to the azimuth ψ are placed in the damping and mass

109 82 matrices. First derivatives are damping terms, and second derivatives are mass terms. They are not physically stiffness, mass and damping, but mathematically behave similarly to stiffness, mass, and damping. Then elemental contributions are assembled into the global governing differential equation of motion that is given by [M]{q } + [C]{q } + [K]{q} = {F (q)}, (2.237) where [M], [C], and [K] represent the global mass, damping, and stiffness matrices, respectively. {F } denotes the global force vector that includes nonlinear terms and inertial and aerodynamic contributions from trailing-edge flaps. {q} is the global displacement vector. In order to reduce the computational cost, the finite element equations in terms of physical nodal displacements are transformed into the modal space. The natural frequencies and modes of the blade are calculated through an eigenvalue problem solving procedure. The frequencies are calculated from an undamped system, such that, [M]{q } + [K]{q} = {0}. (2.238) This equation is solved by an conventional eigenvalue solver. Then the global blade displacement vector {q} can be represented by a linear combination of the eigenvectors. The blade equation is converted to the normal mode equation via a modal transformation. {q} = [Φ] {q R }, (2.239) where [Φ] is the matrix of eigenvectors or madal matrix, and {q R } represents the vector of normal mode coordinates. The size of the system is reduced by only picking certain modes to represent the blade motion. Typically four flap bending modes, three lag bending modes and two torsion modes are selected. Then the modal equation of motion is expressed as follow: [m]{q R } + [c]{q R} + [k]{q R } = {f(q R )}, (2.240) where [m], [c], [k], and {f} are the modal mass, damping, and stiffness matrices

110 83 and force vector, which are given by [m] = [Φ] T [M] [Φ], (2.241) [c] = [Φ] T [C] [Φ], (2.242) [k] = [Φ] T [K] [Φ], (2.243) [f] = [Φ] T {F }. (2.244) The nonlinear periodic differential equation presented in Equation can be numerically solved using the finite element method. This equation can be rewritten as: 2π 0 δ{q R } T ([m]{q R } + [c]{q R} + [k]{q R } {f(q R )}) dψ = 0. (2.245) This can be discretized to eliminate the temporal dependence. The time period (one revolution) is divided into a number of equally spaced temporal elements, as shown in Figure 2.9, where each element has three nodes. For instance, one of modal amplitudes, q Ri, for the i-th temporal element can be interpolated by η t1 q Ri = H t1 H t2 H t3 H t4 H t5 H t6 ηt1 η t2 ηt2 (2.246) η t3 ηt3 i where H t1 = s s3 1 2 s s5, (2.247) H t2 = ψ i 8 ( s 2 s 3 s 4 + s 5), (2.248) H t3 = s s3 1 2 s4 3 4 s5, (2.249) H t4 = ψ i 8 ( s 2 s 3 + s 4 + s 5), (2.250)

111 84 H t5 = 1 2s 2 + s 4, (2.251) H t6 = ψ i ( s 2s 3 + s 5), (2.252) 2 which are the velocity continuous shape functions based on Hermitian polynomials, where ψ i represents the time-interval of the i-th temporal element, and {η t } i is the i-th temporal nodal displacement vector. The number of degrees of freedom of a temporal element depends on the number of selected modes S Figure 2.9. Finite element discretization in time Substituting Equation into Equation yields N t ψi+1 i=1 ψ i δ{η t } T i ( [K t (ψ)] i {η t } i {F t (η t, ψ)} i ) dψ = 0, (2.253)

112 85 where [K t (ψ)] i = {H t } T i [m(ψ)]{h t } i + {H t } T i [c(ψ)]{h t } i + {H t } T i [k(ψ)]{h t } i, (2.254) {F t (η t, ψ)} i = {H t } T i {f(η t, ψ)}, (2.255) and assembling the temporal elements, Equation takes the form [K t ]{η t } = {F t (η t )}. (2.256) This can be solved by using a Newton-Raphson method that is a gradient-based method. In this type of iteration scheme, the solution is found by calculating a gradient between the current point and the solution point. Performing a first-order Taylor series expansion to Equation about the current condition, {η t } (i), yields {η t } (i) = ( [K t ] F t η t ηt =η (i) t ) 1 ( ) [K t ]{η t } (i) {F t (η (i) t )}, (2.257) where {η t } (i) = {η t } (i+1) {η t } (i). (2.258) The initial guess of the response {η t } (0) is solved by {η t } (0) = [K t ] 1 {F t (0)}. (2.259) When the error vector given in Equation 2.258, {η t } (i), is sufficiently close to zero, the response is considered converged Coupled Propulsive Trim Once the aeroelastic response has been obtained, the next step is to compute the blade loads, from which the hub loads can be obtained. Then the vehicle trim

113 86 condition can be obtained using the hub steady loads together with the forces and moments acting on the fuselage. The force summation method is used to calculate the blade loads, which is based on Newton s law, F = m a, the sum of the inertial loads must equal the applied forces. The inertial loads are calculated from closed-form expressions derived in Section 2.2.2, Equation on Page 62. The applied loads to a section of the blade are the aerodynamic loads, and the elastic forces within the blades. If the inertial loads and aerodynamic loads are known, then the elastic forces can be derived as well. Aerodynamic loads are calculated using Equation on Page 73 in Section Then the resulting blade section loads are expressed by L u = L A u + V I x + L f u, L v = L A v + V I y + L f v, (2.260) L w = L A w + V I z + L f w, where superscripts () f M u = M Aˆφ + M I x + M f u, M v = v M Aˆφ + M I y + M f v, (2.261) M w = w M Aˆφ + M I z + M f w, represent trailing-edge flap contributions that include the inertial and aerodynamic loads due to trailing-edge flaps described on Page 101 in Chapter 3. These section loads are in the rotating frame and are also in the undeformed blade coordinate system. To calculate the rotating frame hub loads, additional integration is required. Equations and only represent the loads per unit length. To obtain the blade shear forces and moments at the hub, the spanwise integration is carried out from the hub center to the blade tip. Then, F x F y F z = 1 0 L u L v L w dx, (2.262)

114 87 M x M y M z = 1 0 L v w + L w v + M u L u w L w x + M v L u v + L v x + M w dx. (2.263) The hub loads are obtained by summing the blade root loads between the N b blades at each azimuth position. The rotating frame hub loads are translated to the fixed frame in the vehicle principal directions. The fixed frame hub loads are expressed by FX H (ψ) = FY H (ψ) = FZ H (ψ) = MX H (ψ) = MY H (ψ) = MZ H (ψ) = N b i=1 N b i=1 N b i=1 N b i=1 N b i=1 N b i=1 ( F i x cos ψ i F i y sin ψ i β p F i z cos ψ i ), (2.264) ( F i x sin ψ i + F i y cos ψ i β p F i z sin ψ i ), (2.265) ( ) F i z + β p Fx i, (2.266) ( M i x cos ψ i M i y sin ψ i β p M i z cos ψ i ), (2.267) ( M i x sin ψ i + M i y cos ψ i β p M i z sin ψ i ), (2.268) ( ) M i z + β p Mx i, (2.269) where Fx, i Fy, i and Fz i are the blade root shear forces from Equation due to the i-th blade, and Mx, i My, i and Mz i are the blade root moments due to the i-th blade from Equation The time histories of the hub forces and moments presented in Equations are not useful by themselves. These forces and moments are broken up into harmonics of the rotor frequency Ω, 1/rev. For a balanced rotor, only integer multiples of N b /rev are transmitted to the hub. Otherwise, for a dissimilar rotor, the large 1/rev forces would remain. The steady components of the hub loads are the rotor thrust, longitudinal and side forces, rolling and pitching moments, and the rotor shaft torque. The steady loads are therefore used in the helicopter trim

115 88 process. For propulsive trim of a helicopter, there are three forces and moments which must be zero for the vehicle to be equilibrium. These are controlled by the vehicle controls and orientation. The controls and vehicle orientation states, such as collective pitch θ o, cyclic pitch θ 1c and θ 1s, tail rotor collective pitch θ tr, and the shaft tilt angles α s and φ s, are referred to as control settings. It is assumed that the engine can supply all the power needed to maintain the flight condition. The vehicle equilibrium equations to be solved are given by {F V (Θ)} = 0, (2.270) where {F V (Θ)} = F V x F V y F V z M V x M V y M V z T, (2.271) Θ = θ 75 θ 1c θ 1s α s φ s θ tr T (2.272) in which Fx V, Fy V, and Fx V are the vehicle force residuals, and Mx V, My V, and Mz V are the moments. The detailed formulations can be found in References [76, 133]. The nonlinear vehicle equilibrium equations given in Equation can be solved to find the control settings Θ via a first order Taylor s series expansion with respect to the control settings. Θ i = ( {F V } Θ 1 {F Θ=Θi) V (Θ i )}, (2.273) where Θ i = Θ i+1 Θ i, (2.274) the Jacobian matrix is approximated by a forward finite difference, {F V } Θ {F V (Θ i+1 )} {F V (Θ i )}, (2.275) Θ=Θi Θ i and the initial guess Θ 0 is estimated by a rigid blade trim analysis based on blade flap dynamics.

116 89 The vehicle properties used in this research (see Figure 2.10), which are based on Reference [135], are listed in Table 2.3. Figure Vehicle configuration for propulsive trim

117 90 Table 2.3. Vehicle properties Total vehicle weight 5800 lbs Vehicle longitudinal c.g. offset, x CG /R 0 Vehicle lateral c.g. offset, y CG /R 0 Hub location above vehicle c.g., h/r 0.2 Vehicle flat plate area, f/πr Tail rotor radius 3.24 ft Tail rotor solidity, σ tr 0.15 Tail rotor location, x tr /R 1.2 Tail rotor location above vehicle c.g., h tr /R 0.2 Tail rotor lift coefficient, c 0tr, c 1tr 0, 6 Horizontal tail location, x ht /R 0.95 Horizontal tail planform area, S ht /πr Horizontal tail lift coefficient, c 0ht, c 1ht 0, 6

118 Chapter 3 Trailing Edge Flap Formulation This chapter describes the trailing-edge flap formulation and optimal controller for both vibration and blade loads. In the first section, the inertial loads of trailingedge flaps are derived based on the previous work [69]. Followed by the discussion of available aerodynamic models to predict the aerodynamic loads generated by flap motions. Finally, in the third section, the optimal controller is described based on the minimization of an objective function. This objective function includes three quadratic functions related to vibratory hub loads and blade loads, and active flap control inputs.

119 Inertial Contribution The trailing-edge flap coordinate system is presented in Figure 3.1 showing that flap coordinates x f i are attached on the trailing edge flap and move along with the flap by the deflection angle δ f. z w d f v y z f y f x f d Figure 3.1. Schematic of blade cross-section incorporating a trailing edge flap The position vector of a mass element on the trailing-edge flap in the flap coordinate system e f i is given by r f = 0 x f 2 x f 3 e f (3.1) This position vector can be transformed into the blade deformed coordinate system e d i presented in Section r f = x fd i e d i, (3.2)

120 93 in which x fd 1 = 0, x fd 2 = x f 2 δ f x f 3 d, x fd 3 = x f 3 + δ f x f 2, (3.3) where δ f is the flap deflection, and d denotes the offset from the trailing-edge flap hinge line to the blade elastic axis. This is further transformed into the blade undeformed coordinate system using Equation 2.9 on Page 43: r f u = r f + (x + u) v w e u, (3.4) where and r f = x fu i e u i, (3.5) x fu i = x fd j T ji du. (3.6) The position vector r f u presented in Equation 3.4 is used in the formulation of flap inertia forces. The position vector r f presented in Equation 3.5 is needed to calculate the flap sectional moments with respect to the blade deformed elastic axis. To calculate the sectional loads, the acceleration of a trailing-edge flap particle with respect to the blade undeformed coordinate system is derived first. Based on motion of a particle in a moving coordinate system [69, 142], the acceleration is expressed as: a f = R + Ω r f u + Ω (Ω Ω) + r f u + 2Ω r f u, (3.7) where R represents the hub position vector with respect to an inertial frame, and Ω β p e u 1 + e u 3. (3.8) The first term in the underline terms could be dropped out, since the steady state flight is assumed in this study, which implies that the hub does not accelerate. The second terms can be also dropped out, if the blade rotational speed is assumed

121 94 constant. The inertial contribution to the blade sectional loads associated with a trailingedge flap mass element dm f can be written as: where A f L fi = a f dm f, A f (3.9) represents the flap cross section, and the trailing-edge flap s inertial contribution to the sectional moments about the blade deformed elastic axis, in the undeformed basis e u, are calculated by M fi = r f a f dm f. A f (3.10) Table 3.1. Order of terms for trailing edge flaps Term List Order δ f O(ε 1/2 ) δ f, δ f, d, x f O(ε) y f O(ε 3/2 ) z f O(ε 5/2 ) The acceleration terms that are independent of trailing-edge flap motions are already included in the blade inertial loads. Terms associated with trailing-edge flap deflections, such as δ f, δ f and δ f, therefore needs to be considered as the flap inertial contributions. By applying the small angle assumptions and the ordering scheme up to O(ε 3 ) (Tables 2.2 and 3.1), which is performed using the Mathematica program presented in Appendix C.4, the resulting normalized inertial sectional forces are given by L fi u = 0, (3.11) L fi v = S f ( δf sin θ o 2δ fθ o cos θ o + δ f θ 2 o ) sin θ o δf sin θ o δ f θo cos θ o, (3.12)

122 95 L fi ( w = S f 2δ f θo sin θ o δ f θo 2 cos t o ) + δf cos θ o δ f θo sin θ o, (3.13), and the inertial sectional moments are M fi ( u = S f δf v cos θ o d δ f cos 2 θ o xδ f β p sin θ o ) δ f v cos θ o δ f w sin θ o d δf If δf, (3.14) M fi v = S f xδ f cos θ o, (3.15) M fi w = S f xδ f sin θ o, (3.16) where S f and I f represent the flap first sectional moment and flap hinge moment of inertia, respectively, and they are defined by S f = x f 2 dm f, (3.17) A f I f = (x f 2) 2 dm f. (3.18) A f The inertial trailing-edge flap hinge moment is calculated by MH I = (r f h af ) e h 1 dm f, (3.19) A f where r f h = 0 (xf 2 δ f x f 3) (x f 3 + δ f x f 2) e h, (3.20) in which e h is the coordinate system that has its origin on the trailing edge flap hinge line with an offset d to the blade elastic axis. Here the acceleration a f is expressed in the deformed coordinate system e d by transforming Equation 3.7 using Equation 2.9. Applying the ordering scheme up to O(ε 3.5 ), one can obtain the inertial contribution to the trailing-edge flap hinge moment. MH I = S f { 1 2 d sin 2θ o + (v x v v ) sin θ o } + (xβ p + xw + w ) cos θ o d θo

123 96 I f { 1 2 sin 2θ o + δ f } + θo, (3.21) where the symmetry of trailing-edge flap about x f 1 x f 2 plane is assumed, A f x f 3 dm f = 0, (3.22) A f x f 2 x f 3 dm f = 0. (3.23) Note that the inertial hinge moment, Equation 3.21, includes the blade motion as well as the trailing edge flap motion.

124 Aerodynamic Models One of the present study is to reduce the helicopter vibration or the blade loads using trailing-edge flaps. It is important to accurately model the aerodynamic loads generated by trailing-edge flap motions. In order to accurately predict the flap performance and flap hinge moments, it is essential to use an unsteady aerodynamic model. In this section, the widely used classical unsteady aerodynamic model derived by Theordorsen [143] and the subsonic compressible flow model developed by Hariharan and Leishman [72, 73] are described Incompressible Model An unsteady airfoil theory with an oscillating trailing edge flap has been developed by Theodorsen [143]. The most important assumptions behind this approach are that thin airfoil theory holds and flow is incompressible. Although the incompressible assumption may not be accurate enough, this theory is a good starting for understanding the aerodynamics due to trailing-edge flaps. Considering a large aspect ratio wing in incompressible and inviscid flow, a thin airfoil with a flap is undergoing two degrees of freedom: heave motion h(t) and pitch motion α(t) about the blade elastic axis, as shown in Figure 3.2, where a and e are the normalized distance from the mid-chord to the blade elastic and to the flap hinge, respectively. The aerodynamically unbalanced trailing-edge flap rotates about the flap hinge by the angle δ f (t) relative to the chord line. Elastic axis Flap hinge -b ab eb +b U h c f Figure 3.2. Nomenclature for a thin airfoil with a flap

125 98 The unsteady lift coefficient, C N, of a thin airfoil with a flap undergoing oscillatory motion resulting from trailing-edge flap deflection δ f and rate δ f is given by C N = b ( ) ( UT U 2 4 δ T10 f bt 1 δf + 2πC(k) π δ f + bt ) 11 δ 2πU f, (3.24) where U is the free stream velocity and b is the semi-chord of he airfoil. C(k) is Theodorsen s lift deficiency function with reduced frequency, k = ωb/u, which represents the integrated effect of shed vortex sheet extended from trailing-edge flap all the way minus infinity. The coefficients T 4, T 1, T 10, and T 11 are geometric terms, which depend on the size of the flap relative to the airfoil chord. The underline terms in Equation 3.24 represents non-circulatory or apparent mass terms due to the inertia of the fluid. When the incompressible fluid assumption is used, these terms are proportional to instantaneous displacements. The remaining terms are the circulatory components due to creation of circulation and the effects of the shed wake vorticity. Similar expressions can be derived for the pitching moment, C M, and hinge moment, C H, generated by trailing-edge flap motions. The detailed derivation can be found in References [72, 135, 143] Compressible Model In the subsonic compressible flow, both the magnitude and phase of the aerodynamic loads are affected by compressibility effects, thus the simple correction by the Prandtl-Glauert factor cannot address the phase issue. Unlike the incompressible flow, both the circulatory and non-circulatory loads are subject to time history effects. The time dependency, however, originates in different phenomena. The non-circulatory loads display time delays as a result of the finite-speed propagation of an acoustic wave disturbance created by the initial perturbation, while in case of the circulatory loads the delays are caused by the finite velocity at which shed circulation is convected downstream away from the airfoil. The non-circulatory loads dominate initially and the circulatory loads dominate as time progresses. The compressible unsteady aerodynamic model used in the present study, which

126 99 was developed by Hariharan and Leishman [73], captures the unsteady effects in the time domain via an indicial function representation. All derivations and definitions in detail, such as the unsteady normal force, pitching moment, flap hinge moment and drag, can be found in Reference [72]. Here a brief description of the recursive formulation used in unsteady aerodynamics of a flapped airfoil is presented. It is convenient to express the unsteady lift coefficient C N in terms of coefficient due to flap deflection δ f and coefficient due to the flap rate δ f. Then, C N (S) = C Nδ (S) + C N δ(s), (3.25) where S is the non-dimensional time defined as the distance traveled by airfoil in semi-chord, which is given by S = 2Ut c = 2Ūψ, (3.26) c where () represents the non-dimensional quantity, and Ū x + µ sin ψ. by Considering an arbitrary flap motion, δ f, the lift coefficient due to δ f is given C Nδ (S) = 2T 10 β 2(1 e) δeff f (S) + M T ( ) N δ K n Nδ K N n δ, (3.27) where β is the Prandtl-Glauert compressibility factor, β = 1 M 2, and δ eff f the effective flap deflection and is given by δ eff f (S) = δf n X1 n Y1 n, (3.28) in which δf n is the geometry flap deflection at the given instant in time step n. KN n δ, K N n δ, X1 n, and Y1 n are the deficiency functions, which account for the time history between the forcing and the aerodynamic response, and these functions can be expressed as one step recursive formulae. T N δ constant. is is the non-circulatory time

127 100 For an arbitrary flap rate δ f, the lift coefficient is given by C N δ(s) = T 11 2β ( ) eff δ f c (S) + U (1 e)2 2M T N δ ( ) KN n δ K N n δ, (3.29) where the first term represents ( the circulatory term, and the second term indicates ) eff the non-circulatory term. (S) is the effective flap rate, which is given by, δ f c U ( ) eff ( δ f c (S) = U δ f c U ) n X n 2 Y n 2, (3.30) in which ( δ f c U ) n is the geometry flap deflection at the given instant in time step n, and K n N δ, K n N δ, X n 2 and Y n 2 are the deficiency functions. T N δ is the non-circulatory time constant. Similar expressions can be derived for the pitching moment, C M (S), and hinge moment, C H (S), due to the flap deflection δ f and flap rate δ f. The detailed derivation can be found in References [72]. With the trailing edge flap lift, pitching moment and hinge moment coefficients, additional forces and moment generated by trailing-edge flaps based on the blade deformed coordinate system can be obtained, in the non-dimensionalized form using Equation on Page 69, L fa u = 0, (3.31) L fa v = γū 2 6a C N sin α, (3.32) L fa w = γū 2 6a C N cos α, (3.33) M fa ˆφ = γū 2 6a c C M, (3.34) and the flap hinge moment in the blade deformed coordinate system is given by MH A = γū 2 6a c C H. (3.35)

128 101 These forces and moment should be transformed into the blade undeformed coordinate system to be consistent with the formulation of rotor blade, except the flap hinge moment. Then, via the transformation matrix Tij du 2.9, L fa i = Tij dut LfA j, M fa ˆφ presented in Equation M fa ˆφ. (3.36) The trailing-edge flap s inertial and aerodynamic loads are treated as an additional loads for the aeroelastic analysis. The inertial contribution of a flap was presented in Equations Now the total force contribution of trailing-edge flaps to the blade sectional loads can be written as: and the moment contribution L f u = L fa u L f v = L fa u L f w = L fa u M f u = M fa ˆφ M f v = v M fa ˆφ M f w = w M fa ˆφ + L fi u, + L fi v, (3.37) + L fi w, + M fi u, + M fi v, (3.38) + M fi w. These forces and moments are added into the blade sectional forces and moments, which are presented in Equations and on Page 86 in Section

129 3.3 Active Trailing Edge Flap Control Algorithm 102 In this section, the optimal controller is developed based on the minimization of an objective function to reduce the rotor induced vibration or rotating frame blade loads. The objective function includes two quadratic functions related to vibratory hub loads and blade loads, and active flap control inputs. To limit the trailing-edge flap deflections, the actuator saturation is considered, which is another optimization loop. An active-passive approach developed by Zhang et. al [55 57] is briefly reviewed, which will be used in an active loads control to reduce the control efforts and blade loads Feedback Form of Global Controller This section describes a frequency domain control algorithm used in this study. The control problem can be transformed from the time domain to the frequency domain because of the periodic nature of blade response in forward flights. This periodic assumption is, however, only valid for steady state conditions. Thus a frequency domain controller is only applicable to steady state flight conditions. In the present study, a multicyclic controller developed in Reference [144] is modified and implemented. This approach is based on minimization of an objective function, such that J = Z T n W Z Z n + K T n W K K n + δ T n W δ δ n, (3.39) where Z n is a hub loads vector containing the N b /rev cos and sin harmonic components of the fixed hub loads at time step n (three hub shear forces and three hub moments). K n is a blade loads vector containing the flapwise curvature harmonics, K n = κ (i) 1c κ (i) 1s κ (i) 2c κ (i) 2s T, i = 1, 2,, n K, (3.40) in which κ (i) represents the flapwise curvature 1/rev and 2/rev harmonics at the i-th radial station. n K is the number of curvature sensing locations along the rotor spanwise direction. δ n represents the harmonics of the control inputs. For a typical

130 103 four-bladed rotor, δ n = δ f1c δ f1s δ f5c δ f5s T, (3.41) where subscripts c and s denote the cosine and sine components. With this control input vector, the trailing-edge s flap time history can be expressed by δ f (ψ) = 5 [ δ fic cos(iψ) + δ fis sin(iψ) ]. (3.42) i=1 The matrices W contain penalty weights for the harmonics of the curvature W K, the vibration W Z and the control inputs W δ, they are given by W K = (1 β w ) γ w I, (3.43) [ ] αw I 0 W Z = (1 β w ), (3.44) 0 (1 α w )I W δ = β w I, (3.45) where I and 0 represent identity and null matrices, respectively. α w, β w and γ w are scalar weighting parameters. By changing α w, the controller is instructed to weight more or less on vibratory hub shears or moments. A controller may be based on a local linearization assumption. Then, first order Taylor series expansions of Z n and K n with respect to a current control input vector δ n 1 yield Z n = Z n 1 + T Z (δ n δ n 1 ), (3.46) K n = K n 1 + T K (δ n δ n 1 ), (3.47) where the sensitivity matrices T relate the linearized system response to multicyclic control inputs and need not be square (however, it will in general have more rows than columns). These matrices are numerically calculated by perturbing the control harmonics individually, T Z = Z n 1 δ n 1, T K = K n 1 δ n 1, (3.48)

131 104 Substituting Equations 3.46 and 3.47 into Equation 3.39 yields J = [Z n 1 + T Z (δ n δ n 1 )] T W Z [Z n 1 + T Z (δ n δ n 1 )] + [K n 1 + T K (δ n δ n 1 )] T W K [K n 1 + T K (δ n δ n 1 )] + δ T n W δ δ n. (3.49) Then minimizing J by solving yields J δ n = 0, (3.50) δ n = D [ T T ZW Z T Z + T T KW K T K ] δn 1 D [ T T ZW Z Z n 1 + T T KW K K n 1 ], (3.51) where D [ T T ZW Z T Z + T T KW K T K + W δ ] 1. (3.52) This can be further simplified by defining C Z D T T Z W Z, (3.53) C K D T T K W K. (3.54) Then, Equation 3.51 can be rewritten as: δ n = C Z Z n 1 + C K K n 1 (C Z T Z + C K T K ) δ n 1. (3.55) The update law of the optimal controller that minimizes the objective function J is expressed by δ n = δ n δ n 1, = C Z Z n 1 + C K K n 1 (C Z T Z + C K T K + I) δ n 1. (3.56)

132 105 From the definitions presented in Equations , one can find that DW δ = C Z T Z + C K T K + I, (3.57) C δ. (3.58) Then, the update law of the controller given in Equation 3.56 can be concisely expressed as: δ n = C Z Z n 1 + C K K n 1 C δ δ n 1. (3.59) Either a local controller or a global controller based on the current derivation or Reference [144] can be implemented. For a linear system, the global controller is appropriate. If the system has strong nonlinear properties, the local controller is needed to avoid error. Since the transfer matrices, T, have to be updated constantly in a local controller, it will become very time consuming during an optimization process. In this study, a so-called feedback form of global controller is implemented, since the system is only moderately nonlinear. In this controller, the transfer matrices given in Equation 3.48 are assumed to be constant over the entire range of the control input. This controller, however, is a closed-loop form when the control input during each step is determined by the feedback of the measured vibration levels of the previous control step [135]. One important issue associated with the implementation of active trailing-edge flap systems to the vibration control and blade loads control involves actuator saturation. Saturation can be due to limitations associated with piezoelectric actuation which can provide flap deflections of 4 o or less. Alternatively, when larger flap deflections are possible, for practical reasons, it is desirable to limit flap authority to 3 4 degrees, so as to avoid interfering with the helicopter handling qualities. An effective way of limiting saturation without loss of control effectiveness can be achieved by applying the auto weight approach [93]. In this study, an algorithm based on the bisection method is implemented to

133 106 find an optimal weighting value of β w. This can be expresses as: minimize [ ˆδ ] 2 n (β w ) 2 δ sat subject to 0 β w < 1 (3.60) where ˆδ n represents the optimal control gain vector, and δ sat is the prescribed limiting value of flap deflections or actuator saturation angle. In this algorithm, if the flap deflection is overconstrained or underconstrained, the weighting matrix W δ is appropriately modified to relax or tighten the flap deflection constraint by adjusting β w. Then the new weighting matrix is input into the optimal control calculation routine and the process repeats until the flap is properly constrained Active-Passive Hybrid Design A hybrid active-passive optimization process, which was developed by Zhang et al. [55 57], is implemented by combining an optimal control law with nonlinear optimization programming for a composite rotor blade model. The passive optimization is solved with a gradient-based nonlinear constrained minimization program, the modified feasible direction method [146]. The final optimal control/optimization results are obtained when both passive design parameters and active control actions are optimized. For an active-passive approach, the constrained optimization problem can be formulated as: minimize min [f( x, δ n )] δ n subject to g i ( x) i = 1, 2,, n c (3.61) x L x x U in which x represent the vector containing passive design variables, such as the blade mass and stiffness distributions. n c is the number of constraints. f( x, δ f ) is

134 107 the objective function given in Equation The design variable vector is given by x = m K 22 K 33 K 55 K 25 K 35 T (i), i = 1, 2,, N el (3.62) where K 22, K 33 and K 55 represent the flap bending, lag bending and torsional bending stiffness, respectively. K 25 and K 35 denote the pitch-flap and pitch-lag composite coupling stiffness. The constraints g i ( x) are given by g j = 1 ω j, j = 1, 2, 3 (3.63) ωj L g k = ω k ω U k 1, k = 4, 5, 6 (3.64) where ω j represent the first natural frequencies of each elastic flap, lag, and torsion modes. Superscripts L and U denote lower and upper bound, respectively. The frequency placement constraints will prevent blade resonance as well as unrealistic blade design. Side constraints are also used to prevent the design variables from reaching impractical values during the optimization process. For the optimization tool, a commercial optimization package (DOT [146]) is used. After the optimization is completed, the trim condition of the vehicle should be checked, because large pitch-flap coupling stiffness K 25 could result in the severe change of primary control settings. In this case, the vehicle should be re-trimmed to ensure the proper final results after solving the optimization problem of Equation The converged trim solutions are usually obtained after three or four iteration. Detailed description of an active-passive hybrid design method can be found in References [135].

135 Chapter 4 Active Loads Control Using a Dual Flap Configuration The purpose of this chapter is to investigate the feasibility of multiple trailing-edge flaps for the simultaneous reductions of vibration and blade loads. The concept involves straightening the blade by introducing dual trailing edge flaps in a conventional articulated rotor blade. A classical incompressible theory presented in Section is employed to predict the incremental trailing edge flap airloads. The objective function, which includes vibratory hub loads, bending moment harmonics and active flap control inputs, is minimized by an integrated optimal control/optimization process described in Section 3.3. Numerical simulation has been performed for the steady-state forward flight of an advance ratio of It is demonstrated that through straightening the rotor blade, which mimics the behavior of a rigid blade, both the bending moments and vibratory hub loads can be significantly reduced.

136 Introduction An active control using trailing-edge flaps can significantly reduce the vibration level of helicopter. It may, however, result in the increase of blade loads. McCloud III [26] has studied the feasibility of reducing both vibration and blade loads using a single servo-flap. His results have shown that multi-cyclic control can achieve both vibration and bending moment reduction with large 1/rev control input. The vibration level for rotor with a single trailing-edge flap using both the fully elastic and rigid blade models has been investigated by Millot [63]. His results have shown that the predicted vibration level using the rigid blade model is much less than that of using the elastic blade model. This implies that if one can straighten the rotor blade, the vibratory hub loads can be reduced. The simplest way to achieve this condition is to make the rotor blade very stiff. This will, however, result in high bending moments although the vibration level will become low. The other possibility is to straighten the blade through active actions, in which one can reduce both the bending moment and vibration. The use of trailing-edge flaps for vibration suppression has been studied carefully in past decade. However, studies focused on the use of trailing-edge flaps for blade loads control are rare. The objective of this research is to explore the feasibility of utilizing active control for both rotor blade bending moment and vibration reductions. The dual trailing-edge flaps are introduced and applied for such purposes. One of the active flaps is located at the out-board region and the other one at mid-span. A typical articulated rotor blade is selected as a test bed for this investigation. A control law for dual flap configuration is developed and verified through numerical results for a steady-state forward flight condition. An objective function, which includes the flap-wise curvature harmonics, is defined at five different radial stations so that minimizing the objective function results in the straightened blades. Composite tailoring via a pitch-flap coupling is also evaluated utilizing an active-passive approach.

137 Description of Analytical Models The trailing-edge flap model, which is based on the incompressible theory developed by Theodorsen [143], is formulated and then integrated into the blade model. Two partial span trailing-edge flaps, which are plain type flaps, are used to straighten the blade. One is located at the outboard region of the blade, 0.9R 1.0R, and the other is located at around the mid-span, 0.4R 0.6R, as shown in Figure 4.1. Flap span size and location are determined through a parametric study. In-board flap 0.4~0.6R Out-board flap 0.9~1.0R Figure 4.1. Dual flap configuration for active loads control In the present analysis, motion of the fuselage is not considered. The aerodynamics only takes into account the wind velocity, including the helicopter forward speed and the rotor rotation, and the motion of the blades relative to the hub. A linear inflow distribution, which is the Drees model presented in Section 2.4.1, is assumed in this study. Sophisticated wake models are important for analysis of flying qualities, detailed power calculations, and noise. For the rotor response, loads, and vibration, a simple model is normally sufficient, especially for high-speed forward flight conditions. The aerodynamic model for trailing-edge flap, which is based on the incompressible theory derived by Theodorsen [143], is used in the present study. This quasi-steady model may underestimate the hinge moment of trailing-edge flap when compared to the compressible theory developed by Hariharan and Leishman [72], especially at high Mach number. However, the hinge moment is not the scope of this chapter. A composite rotor blade is discretized into five finite elements. A coupled propulsive trim scheme is implemented to simultaneously determine the blade non-

138 111 linear steady response, vehicle orientation and control setting. The vibratory hub loads are calculated by integrating the blade and active trailing-edge flap inertial and aerodynamic loads along the blade using the force summation method, while bending moments at each radial station are calculated by the curvature method for the sake of computational efficiency. For the purpose of bending moment reduction, an objective function that retains the curvature information of the blade is introduced, so that minimizing the objective function results in the straightened blade. A conceptual sketch of dual flap mechanism is depicted in Figure 4.2. For a four-bladed articulated rotor, the flap is typically actuated at combinations of 3, 4, and 5/rev to reduce the vibratory hub loads. The objective of the present study is to minimize bending moments of the rotor blade. The 1/rev component of flapwise bending moment harmonics is dominant, and its maximum occurs at around mid-span. For this reason, 1/rev control input is considered. The control weighting parameters α w, β w and γ w are set to 0.2, 0.0 and 1.0, respectively, since the study of this chapter focus on the bending moment reduction. The optimal control algorithm was described in Section 3.3 in detail. Figure 4.2. Conceptual sketch of dual flap mechanism for active loads control A hybrid active-passive optimization process described in Section is implemented by combining an optimal control law with nonlinear optimization programming for a composite rotor blade model. The passive optimization is solved with

139 112 a gradient-based nonlinear constrained minimization program. Both constraints and bounds for the design variables are presented in Table 4.1. The frequency placement constraints will prevent blade resonance as well as unrealistic blade design. Table 4.1. Constraints and bounds for design variables Constraints Lower limit Upper limit ω 1 / ω1 o (flap) ω 2 / ω2 o (lag) ω 3 / ω3 o (torsion) Design variables Lower bound Upper bound m/m o K ii / Kii o K Numerical Results and Discussions For numerical studies, a four-bladed articulated composite rotor with two plain active flaps is investigated. Baseline blade properties and trailing edge flap data are given in Tables 4.2 and 4.2, respectively. As mentioned in the previous section, the active flaps are set to activate in 1/rev or 1, 2 and 3/rev. The prescribed flap motion is considered in this study. Results are obtained at a forward flight speed of an advance ratio of All the forces and moments are nondimensional quantities (see Table 2.1 on Page 46 in Section 2.1.3). Three cases are computed for comparison among different approaches: 1. Baseline: A generic, uniform, articulated rotor blade. 2. Actively controlled blade or Retrofit design: Add the active dual flaps to the tuned blade, determines the optimal control inputs, and then trim the blade. 3. Hybrid design: Simultaneously redesign the blade pitch-flap composite coupling stiffness K 25 and active control inputs.

140 113 Table 4.2. Baseline articulated rotor properties for active loads control Hub type articulated rotor Number of blades, N b 4 Rotor radius, R 16.2 ft Hover tip speed, ΩR 650 ft/sec Hover tip Mach number, M tip 0.58 Airfoil NACA 0015 Lift coefficients, c o, c 1 0, 5.73 Drag coefficients, d o, d 1, d , 0, 0.2 Blade chord, c/r 0.08 Solidity, σ 0.1 Thrust coefficient over solidity, C T /σ 0.07 Blade linear twist, θ tw 8 o Precone, β p 0 Lock number, γ 6.34 Flap bending stiffness, K22 o Lag bending stiffness, K33 o Torsional stiffness, K55 o Flapwise mass moment of inertia, mkm Lagwise mass moment of inertia, mkm Blade mass/length, m o slugs/in Advance ratio, µ 0.35 For the purpose of comparison, the vibration and blade moment indices are defined: V LV = F 2 x + F 2 y + F 2 z + M 2 x + M 2 z, (4.1) M LV = max [M flap (r, ψ)], (4.2) where F x, F y and F z are 4/rev vibratory hub shear forces, and M x, M y and M z are 4/rev vibratory hub moments. M flap represents the flapwise bending moment in the rotating frame Baseline Articulated Rotor Analysis The baseline rotor blades are generic, uniform, articulated rotor blades. Blade natural frequencies and mode shapes are calculated based on the vehicle trim. In

141 114 Table 4.3. Trailing-edge flap properties for active loads control In-board flap location Out-board flap location Flap chord ratio, c f /c 0.2 Flap mass per unit length, m f /m o Flap chordwise c.g. (after flap hinge), r I /c f Flap radius of gyration about flap hinge, rii 2 /c2 f Offset from blade elastic axis to flap hinge, d/c f 0.55 this chapter, four flap modes, three lag modes and two torsion modes are used for the purpose of modal reduction. The baseline blade natural frequencies are presented in Table 4.4. The blade first four flap mode shapes, the first three lag mode shapes and the first two mode shapes are shown in Figures Table 4.4. Natural frequencies of baseline articulated rotor Mode Frequency, /rev Flap Flap Flap Flap Lag Lag Lag Torsion Torsion The rotor thrust coefficient C T is assumed to be 0.007, and quasi-steady blade element aerodynamics with a linear inflow model is used to obtain the blade response. The rotor trim control settings and blade tip response are shown in Figures 4.6 and 4.7, respectively, for an advance ratio of 0.35.

142 Flap Lag Torsion 1st mode Flap Lag Torsion 2nd mode Radial position, x rd mode Radial position, x Flap Lag Torsion 4th mode Flap Lag Torsion Radial position, x Radial position, x Figure 4.3. Articulated blade coupled flap mode shapes

143 Flap Lag Torsion 1st mode Flap Lag Torsion 2nd mode Radial position, x Radial position, x 3rd mode Flap Lag Torsion Radial position, x Figure 4.4. Articulated blade coupled lag mode shapes Flap Lag Torsion 1st mode nd mode Flap Lag Torsion Radial position, x Radial position, x Figure 4.5. Articulated blade torsion mode shapes

144 Control settings, deg θ.75 θ 1c θ 1s α s φ s Figure 4.6. Control settings of articulated rotor, µ = Flap Tosrion Lag Blade tip response Azimuthal angle ( ψ ), deg Figure 4.7. Blade tip response of articulated rotor, µ = 0.35

145 Rigid Blade vs. Elastic Blade In order to investigate the vibration level of the rigid blade model, the very large bending and torsion stiffnesses are assumed to simulate a rigid blade. The 4/rev vibratory hub loads are presented in Figure 4.8 and compared to those using the baseline blade without trailing-edge flap. As expected, predicted vibration levels of the rigid blade model are much lower than those of the elastic blade model. This explains snap physics that straightening the blade, which is equivalent to alleviating the curvature of a blade and leads to attenuating the blade inertial loads, results in reducing bending moment and vibration Baseline Active control (dual flap) Rigid blade Fx Fy Fz Figure 4.8. Comparison of vibratory hub loads for active loads control A Single Flap for Moment Reduction n general, a single flap located at R works well for the purpose of vibration reduction of helicopter rotor blade. Vibration and flapwise bending moment are examined to investigate a single flap performance. With typical control inputs 3,4,5/rev and the objective function for vibratory hub loads, it is observed that vibration level is reduced by 88% and flapwise moment is increased by 13%. The

146 119 single flap is also applied to the moment reduction problem. The results show that vibration and flapwise moment are reduced by 54% and 9%, respectively, and the control settings are severely changed. These results indicate that a single flap is not only inadequate to reduce the bending moment but also hard to trim a vehicle when it is used for the purpose of moment reduction. It is observed that reducing the curvature of blade almost always produces lower vibration level, while reducing the vibration level does not necessarily yield lower bending moment Dual Flap Performance Dual flap has been introduced to straighten a rotor blade so that we could reduce the bending moment with a reasonable vibration level. Figure 4.2 shows the dual flap mechanism that generates the additional flapwise bending moment to dynamically straighten the blade. This additional moment is quite effective in the out-board region. Active control authority is, however, small in the in-board region due to geometric location of dual flap and low aerodynamic pressure. This is the reason why we have selected a wider flap at the in-board region. The vibratory hub shear forces are compared to baseline and rigid blade in Figure 4.8. Vibratory hub shear forces are reduced to almost the same level as those of the rigid blade. This is the reason why the dynamically straightened blade yields lower vibration level than baseline blade without the change of control settings for trim. In general, 1/rev control input strongly affects trim since primary control inputs are 1/rev. Dual flap profile presented in Figure 4.9 explains the reason why control settings for trim is barely changed. It is observed that in-board flap deflection is larger than out-board flap, approximately 1.4 times, and there is 180 degree out-ofphase between in-board and out-board flaps. This out-of-phase is reasonable since the bending moment will be reduced by flapwise aerodynamic moment due to dual flap on rotor blade. These out-of-phase and difference between flap deflections

147 120 make the net lift generated by dual flap to be close to zero. Although net lift is almost zero, additional flapwise moment enforces a vehicle with a lateral tilt. Indeed, we can see that trimmed control settings have kept after active control, as shown in Figure Dual flap profile 6 Flap deflections, deg Out-board flap In-board flap Azimuth, deg Figure 4.9. Dual flap profile for moment reduction with 1/rev control Bending moment harmonics along the radial station and maximum bending moment are presented and compared in Figure 4.11 and Table 4.5, respectively. Reduction of maximum flap bending moment and lag bending moment are about 32% and 45%, respectively. Note that our objective function currently includes flapwise curvature harmonics only. There, however, is a cross coupling between flap and lag motion due to blade twist angle. It helps to reduce lagwise curvatures that directly affect lag bending moment. As it is shown in Figure 4.11, maximum of 1 and 2/rev harmonics occur in the out-board region. It is shown that the 1/rev harmonic component is greatly reduced at around 0.6R and maximum is shifted to the in-board region. Flapwise bending moment distributions along the radial station and azimuth are presented in Figures 4.12 and 4.13, before and after control, respectively. It is clearly shown that the maximum moment moved from

148 Baseline Dual flap theta_75 theta_1c theta_1s alpha_s phi_s Figure Control settings of baseline and actively controlled rotors the out-board region to the in-board region after control. These results indicate the superiority of the dual flap configuration over the single flap one for bending moment reduction problem. Table 4.5. Reduction of maximum bending moments maximum moments Maximum bending moments Flap Lag Baseline ( 10 4 ) Dual flap ( 10 4 ) Reduction (%)

149 Baseline 1/rev Baseline 2/rev Retrofit 1/rev Retrofit 2/rev Radial station, r Figure Harmonics of flapwise bending moment along the radial station

150 Z X Retreating side Advancing side Y Figure Flapwise bending moment distribution before control Z X Retreating side Advancing side Y Figure Flapwise bending moment distribution after control

151 Multicyclic Control for Moment and Vibration Reduction In this case, the objective function includes two quadratic functions that are vibratory hub loads and flapwise bending moment harmonics. Multicyclic control inputs, such as 1, 2 and 3/rev, have been considered to improve both bending moment and vibration load reductions. 1 and 2/rev inputs contribute to the reduction of the flapwise bending moments, and 3/rev input helps to reduce the vibration level. Figures 4.14 and 4.15 show the dual flap profiles with multicyclic control inputs. It is observed that there is still 180 degree out-of-phase between in-board and outboard flaps, irrespective of the number of control inputs. These results indicate that in-board flap should deflect to opposite direction of the out-board flap in order to reduce the flapwise bending moment. Multicyclic control yields better results than a single control. As mentioned earlier, 3/rev control input is dominant in reducing vibration level, and tends to increase maximum bending moments. 15 Dual flap profile Flap deflections, deg Out-board TEF In-board TEF Azimuth, deg Figure Dual flap profile with 1 and 2/rev control inputs The reason of this can be deduced from comparisons of vibration index and

152 Dual flap profile Out-board TEF In-board TEF Flap deflections, deg Azimuth, deg Figure Dual flap profile with 1, 2 and 3/rev control inputs maximum flapwise bending moment that are presented in Figure 4.16 and Table 4.6. Control inputs with 1 and 2/rev slightly improve both vibration and moment reductions. The vibratory hub loads are greatly reduced with slightly increasing maximum of bending moment when 3/rev input is added to control input vector. The required flap deflections are, however, increased with adding higher harmonic control inputs. As shown in Figures 4.9, 4.14 and 4.15, the maximum deflections of out-board flap is almost constant with 5 degree, while those of in-board flap is linearly increased from 7 degrees to 14 degrees. As shown in Figure 4.16, a conventional single flap performance for the purpose of vibration reduction is quite good but it normally causes the increase of maximum flapwise bending moment as well as the blade fatigue load. Table 4.6. Reduction of vibration and moment reductions with different control inputs Control inputs Vibration reduction (%) Max. flapwise moment reduction (%) 1/rev and 2/rev , 2 and 3/rev

153 126 Figure Comparison of vibration index and maximum flapwise bending moment with different control inputs Note that dual flap performance with 1/rev control input is not good for vibration reduction since 1/rev does not affect the vibratory hub loads. It, however, does work for vibration reduction when an objective function includes the bending moment harmonics. Dual flap, for the purpose of vibration reduction, is superior over a single flap as reported in Reference [94], but it also yields higher bending moment than a single flap does Active-Passive Hybrid Design The hybrid design can be applied to reduce the required maximum flap deflection. As shown in Figure 4.11, the 1 and 2/rev harmonics are significantly reduced at around 0.7R where the most effective additional aerodynamic moment due to dual flap occurs. Passive design parameters, such as non-structural mass and composite pitch-flap coupling stiffness, maximum flap deflection constraints (4 degrees), and 1/rev control input of active dual flap are considered in the hybrid design, while flap and lag stiffness are not considered. The current dual flap configuration is quite effective on reducing the objective function related to bending moment harmonics. There is no room for further re-

154 127 duction. Thus weighting matrix for bending moment, W K, described in Equation 3.39 has been chosen to avoid being overly minimized at around 0.7R, so that passive parameters help to reduce the required flap deflections, vibration level and the maximum bending moment. In Figure 4.17, vibratory hub loads of hybrid design are presented and compared to those of baseline and retrofit design. Vibration index and maximum flapwise bending moment are compared in Figure 4.18 and Table 4.7. The improvement is not significant when compared to the retrofit design, but the required maximum flap deflection is reduced by 25% (Figure 4.19). This different flap deflection might cause a significant change of control settings, but it turns out that there is no significant change as shown in Figure Table 4.7. Comparisons of maximum moment, vibration index and control efforts Max. moment Vibration Max. TEF Design reduction (%) reduction (%) deflection, deg Flap / Lag Outboard / Inboard Retrofit 32 / / 6.70 Hybrid 34 / / Baseline Retrofit Hybrid Fx Fy Fz Figure Vibratory hub shears comparison for active loads control

155 baseline retrofit hybrid Vib. Index (1e3) Max. Flapw ise moment (2e3) Figure Comparison of vibration index and maximum flapwise moment of hybrid designed rotor 8 6 Flap deflections, deg Out-board flap (Retrofit,Hybrid) In-board flap (Hybrid) -6 In-board flap (Retrofit) Azimuth, deg Figure Dual flap profiles for retrofit and hybrid designs

156 Baseline Retrofit Hybrid theta_75 theta_1c theta_1s alpha_s phi_s Figure Control settings of baseline, retrofit and hybrid designed rotors

157 130 Passive design parameters can take care of the lack of aerodynamic lift at the in-board region. The non-structural mass and composite pitch-flap coupling stiffness, K 25, distributions are shown in Figure The non-structural mass helps to dynamically straighten the blade by increasing the tip mass that will increase the centrifugal forces. The coupling stiffness could help to redistribute the blade pitch in order to compensate the lack of in-board flap lift. Through the fine-tuning of the blade passive parameters, the dual flap works much more efficiently in reducing the bending moment. k25 Mass 1.0E E E E E Figure Blade non-structural mass and pitch-flap composite coupling stiffness, K 25, distribution

158 Summary The purpose of this chapter was to investigate the feasibility of multiple trailingedge flaps for the simultaneous reductions of vibration and blade loads. The concept involved straightening the blade by introducing dual trailing edge flaps in a conventional articulated rotor blade. An active loads control strategy was simulated for the steady-state forward flight condition, µ = Based on the present study, the principal conclusions obtained are summarized as follows: Straightening the rotor blade using dual flap configurations can reduce both vibratory hub loads and bending moments without a significant change of control settings for trim. Only 1/rev control input is required to reduce vibratory hub loads with the present method, which is well suited for resonant actuation systems that will be described in Chapter 6. The proposed active loads control method can reduce the flapwise bending moment by 32% and the vibratory hub loads by 57%, simultaneously, by dynamically straightening the blade with the 1/rev control input. For bending moment reduction, 1/rev control input is dominant, and the 3/rev action is very effective on vibratory hub loads but detrimental on flapwise bending moments. For hybrid design, the maximum flapwise bending moment and vibration are simultaneously reduced by 34% and 62%, respectively. The required maximum flap deflection is reduced by 25% when compared to retrofit design.

159 Chapter 5 Helicopter Vibration Suppression via Multiple Trailing-Edge Flaps with Resonant Actuation Concept The objective of this chapter is to investigate the vibration reduction using the multiple trailing-edge flaps configuration. The concept involves deflecting each individual trailing-edge flap using a compact resonant actuation system (see Chapter 6). Each resonant actuation system could yield high authority, while operating at a single frequency. The rotor blade airloads are calculated using quasi-steady blade element aerodynamics with a free wake model. A compressible unsteady aerodynamics model is employed to accurately predict the incremental trailing edge flap airloads. Both trailing-edge flap finite wing effects and actuator saturation are included in the simulation. A numerical simulation is performed for the steady-state forward flight (µ = ). It is demonstrated that multiple trailing-edge flap configuration with the resonant actuation system can reduce the required trailingedge flap hinge moments. The analysis and parametric study are conducted to explore the finite wing effect of trailing-edge flaps and actuator saturation.

160 Introduction It has been shown that improvements in helicopter vibration reduction can be achieved by smart materials. Piezoelectric actuation system is expected to be compact, light weight, low actuation power, and high bandwidth devices that can be used for multi-functional roles such as to suppress vibration and noise, and increase aeromechanical stability. While piezoelectric materials-based actuators have shown good potential in actuating trailing edge flaps, they can only provide a limited stroke. This limitation can be critical in cases where large trailing-edge flap deflections are required. The efforts to improve the piezoelectric actuator performance have been made by researchers in developing amplification mechanisms of various types (see Section on Page 20). In general, these devices are still limited in their performance. To circumvent this limitation, a resonant actuation system can be used to enhance the effectiveness of piezoelectric actuators, which will be presented in Chapter 6. A multiple piezoelectric actuator configuration has been considered and tested in Eurocopter to adjust the required control power and surface [114]. In this work, a single flap is segmented into three parts, and all actuator is controlled by the same command. On the other hand, multiple trailing-edge flap configurations have been studied to reduce the vibration of helicopter rotor system, in which each actuator operates independently. Myrtle and Friedmann [78] have shown that the dual flap configuration is almost completely unaffected by the change of torsional stiffness of rotor blade. Recently, Cribbs and Friedmann [93] have developed the flap deflection saturation model through an automated approach to reduce the required maximum flap deflection. They have shown that the imposition of saturation of flap deflection could result in the different profile and reduced magnitude of the active flap while maintaining almost the same vibration level as models without actuator saturation. The actuator saturation model has been extended to reduce the vibration due to dynamic stall using a dual flap configuration [94]. They showed that dual flap is superior over a single flap in vibration reduction. In the previous chapter, Chapter 4, it was demonstrated that the dual flap configuration can be also applied to reduce both vibration and blade loads by dynamically straightening the blade.

161 134 In general, a single trailing-edge flap works well for the purpose of vibration reduction of helicopter rotor. With typical control inputs 3,4 and 5/rev, it has been demonstrated via numerical simulations that vibration level can be reduced by about 80%. As mentioned earlier, however, piezoelectric actuators provide a limited stroke. For the same TEF deflections, the actuator design specification in multiple-flap configuration is more flexible than in single-flap configuration, because the required hinge moment is much less due to small control surface area. The goal of this chapter is to develop an active vibration control method using multiple trailing-edge flaps. In this study, it is assumed that each individual trailing-edge flap is operating at a single frequency to utilize a compact resonant actuation system (see Chapter 6). One of the problems associated with a resonant actuation system is the operating bandwidth. This can be resolved using the multiple trailing-edge flaps configuration, in which each flap is designed to operate at a single frequency that is one of the operating frequencies (e.g, 3, 4 and 5/rev for a four-bladed rotor).

162 Description of Analytical Models Aerodynamic loads acting on the blade are calculated using quasi-steady blade element theory. A free wake model, which is extracted from Tauzsig and Gandhi s code [132] (see Section on Page 77), is used to determine the non-uniform inflow distribution over the rotor disk. A compressible unsteady trailing-edge flap aerodynamic model developed by Hariharan and Leishman [73] (Section on Page 98), is formulated and then integrated into the blade model. Multiple partial span trailing-edge flaps, which are plain types, are used to control the rotor vibration. Single-, dual- and multiple-tef configurations (Figure 5.1) are also considered to compare the performance and required trailing-edge flap deflections. Figure 5.1. Various configurations of the rotor with trailing-edge flaps In multiple trailing-edge flaps configuration, a trailing-edge flap is normally smaller than one in a single flap configuration. Thus the aspect ratio could be critical in using multiple-tefs configuration. Based on the approximate lifting surface theory [147], the lifting curve slope of trailing edge flap can be expressed

163 136 as: in which a f a fo = AR AR + 2(AR + 4)/(AR + 2), (5.1) AR = L2 f L f c f, (5.2) where a fo is the nominal lifting curve slope of trailing-edge flap, which is normally 2π. L f and c f represent the flap span and chord, respectively. The unsteady lift of a wing of finite aspect ratio was reported by Jones [148]. Results indicate that the starting lift of finite wing is similar to that of infinite wing, and the change of unsteady lift curve slope with respect to AR is approximately proportional to quasi-steady lift curve slope correction factor. Thus Equation 5.1 is applied to the normal force coefficient C N presented in Equation 3.25 on Page 99, in which C N was calculated by a compressible unsteady trailing edge flap model. A rotor blade is discretized into five or ten finite elements. In order to reduce the computational cost, the finite element equations in terms of physical nodal displacements are transformed into modal space. Four flap, three lag, and two torsion modes are used in this chapter. Eight temporal elements are used, and velocity-continuous shape functions, which are fifth-order polynomials, are used within the temporal elements. A coupled propulsive trim scheme is implemented to simultaneously determine the blade nonlinear steady response, vehicle orientation and control setting. The vibratory hub loads are calculated by integrating the blade and active trailing-edge flap inertial and aerodynamic loads along the blade using the force summation method. The details of aeroelastic analysis were described in Section 2.5 on Page 80. For a four-bladed hingeless rotor, the flap is typically actuated at combinations of 3, 4, and 5/rev to reduce the vibratory hub loads. In this study, the multiple-flap configuration to utilize the resonant actuation system is introduced, so that each flap operates at the specific frequency. Then the control input vector presented in Equation 3.41 is modified as: δ n = δ (1) fc δ (1) fs δ (2) fc δ (2) fs δ (3) fc δ (3) fs T, (5.3)

164 137 where superscript () (i) indicates the i-th trailing-edge flap. Flap deflections d f (ψ) are limited by 2 o 4 o using the actuator saturation algorithm presented in Equation 3.60 on Page 106, which automatically determines the control weight parameter β w. For the purpose of the vibration reduction, the other control weighting parameters, α w and γ w, are set to 0.2 and 0.0, respectively. 5.3 Results and Discussions For numerical studies, a four-bladed hingeless rotor with three plain active flaps is investigated. The baseline blade and trailing-edge flap properties are listed in Table 5.1. Active trailing-edge flaps are set to activate at 3, 4 and 5/rev. Each trailing-edge flap is operated at a single frequency that is one of 3, 4 and 5/rev in the multiple-trailing edge flap configuration to utilize the resonant actuation. Results are obtained at forward flight speeds (µ = 0.15, 0.35). Four cases are computed for comparison among the various trailing-edge flap configurations (see Figure 5.1): 1. Baseline: A generic, uniform, hingeless rotor blade. 2. Single-TEF configuration : Add a single active flap to the generic blade. 3. Dual-TEFs configuration: Add two active flaps to the generic blade. 4. Multiple-TEFs configuration: Add three active flaps, in which each flap operates at a single frequency, to the generic blade. For the purpose of comparison, the vibration index defined in Equation 4.1 on Page 113 is used Baseline Hingeless Rotor Analysis The baseline rotor blades are generic, uniform, hingeless rotor blades. Blade natural frequencies and mode shapes are calculated based on the vehicle trim. In this chapter, four flap modes, three lag modes and two torsion modes are used for the

165 138 Table 5.1. Hingeless rotor and trailing-edge flap properties Main rotor properties Hub type hingeless rotor Number of blades, N b 4 Rotor radius, R 16.2 ft Hover tip speed, ΩR 650 ft/sec Hover tip Mach number, M tip 0.58 Airfoil NACA 0015 Lift coefficients, c o, c 1 0, 5.73 Drag coefficients, d o, d 1, d , 0, 0.2 Blade chord, c/r 0.08 Solidity, σ 0.1 Thrust coefficient over solidity, C T /σ 0.07 Blade linear twist, θ tw 8 o Precone, β p 0 Lock number, γ 6.34 Flap bending stiffness, K22 o Lag bending stiffness, K33 o Torsional stiffness, K55 o Flapwise mass moment of inertia, mkm Lagwise mass moment of inertia, mkm Blade mass/length, m o slugs/in Advance ratio, µ Trailing-edge flap properties Flap chord ratio, c f /c 0.2 Flap mass per unit length, m f /m o Flap chordwise c.g. (after flap hinge), r I /c f Flap radius of gyration about flap hinge, rii 2 /c2 f Offset from blade elastic axis to flap hinge, d/c f 0.55 purpose of modal reduction. The baseline blade natural frequencies are presented in Table 5.2. The blade first four flap mode shapes, the first three lag mode shapes and the first two mode shapes are shown in Figures

166 139 Table 5.2. Natural frequencies of baseline hingeless rotor Mode Frequency, /rev Flap Flap Flap Flap Lag Lag Lag Torsion Torsion Flap Lag Torsion 1st mode Flap Lag Torsion 2nd mode Radial position, x rd mode Radial position, x Flap Lag Torsion 4th mode Flap Lag Torsion Radial position, x Radial position, x Figure 5.2. Hingeless blade coupled flap mode shapes

167 Flap Lag Torsion 1st mode Flap Lag Torsion 2nd mode Radial position, x rd mode Radial position, x Flap Lag Torsion Radial position, x Figure 5.3. Hingeless blade coupled lag mode shapes Flap Lag Torsion 1st mode nd mode Flap Lag Torsion Radial position, x Radial position, x Figure 5.4. Hingeless blade torsion mode shapes

168 141 The rotor thrust coefficient C T is assumed to be 0.007, and quasi-steady blade element aerodynamics with a free-wake model is used to obtain the blade response. The rotor trim control settings are presented in Figures 5.5 and 5.6 for µ = 0.15 and µ = 0.35, respectively. The blade tip responses are shown in Figures 5.7 and 5.8 for low and high speed flight conditions. The wake effect is significant at the low speed flight (µ = 0.15), as shown in Figure 5.7 where showing that the blade tip flapping response contains higher harmonic components Control settings, deg θ 1s φ s 2 θ.75 θ 1c α s 4 Figure 5.5. Control settings of hingeless rotor, µ = Control settings, deg 5 0 θ.75 θ 1c θ 1s α s φ s 5 10 Figure 5.6. Control settings of hingeless rotor, µ = 0.35

169 142 Blade tip response Flap Torsion Lag Azimuthal angle ( ψ ), deg Figure 5.7. Blade tip response of hingeless rotor, µ = Flap Torsion Lag Blade tip response Azimuthal angle ( ψ ), deg Figure 5.8. Blade tip response of hingeless rotor, µ = 0.35

170 Flap Effect to Free-Wake Geometry Multiple trailing-edge flaps could cause discrete lift forces along the rotor spanwise direction. In subsection, the wake geometry, blade tip response and control settings are compared to investigate the flap effect to them. Free-wake vertical geometries are shown in Figures 5.9 and 5.10 for advance ratio of 0.15 and 0.35, respectively. The changes in wake geometry due to the number of beam elements and the flap are not significant, since the tip vortices generated by the flap itself are not considered in this study. Blade tip responses are presented in Figures 5.11 and 5.12 for µ = 0.15 and 0.35, respectively. Torsion tip responses are affected by both the number of beam elements and the flap, especially for low speed flights (Figure 5.11). Trim control settings for these cases are shown in Figure 5.13 for µ = 0.15 and Figure 5.14 for µ = It is observed that changes in control settings due to the number of beam elements and the flap are not significant. In the study of this chapter, five beam elements are used to save the computational time Y wake geomtry for µ=0.15 element 5 element 10 element 10 w/ flap Figure 5.9. Vertical wake geometry with the number of beam elements and the presence of the flap for µ = 0.15

171 Y wake geomtry for µ=0.35 element 5 element 10 element 10 w/ flap Figure Vertical wake geometry with the number of beam elements and the presence of the flap for µ = Blade tip response Flap Torsion Lag 5 beam elements w/ flap 10 beam elements w/ flap Re calculated free wake w/ flap Azimuth, deg Figure Blade tip responses with the number of beam elements and the presence of the flap for µ = 0.15

172 145 Blade tip response Flap Torsion 5 beam elements w/ flap 10 beam elements w/ flap Re calculated free wake w/ flap 0.02 Lag Azimuth, deg Figure Blade tip responses with the number of beam elements and the presence of the flap for µ = 0.35 Figure Control settings with the number of beam elements and the presence of the flap for µ = 0.15

173 Figure Control settings with the number of beam elements and the presence of the flap for µ =

174 Determination of Trailing-Edge Flap Locations To determine the flap locations for multiple-tefs configurations, the parametric study is performed using a small single flap configuration (L f = 0.07R) for lowand high-speed forward flight condition, where the flap motions were limited in amplitude of two degrees to recognize the flap effectiveness in terms of radial locations. Figure 5.15 shows the results showing the effects of flap locations. As expected, the vibration reductions are directly related to the spanwise location of the flap. In low-speed flight (µ = 0.15), 4/rev control input is very effective since the vertical hub shear force (F z ) is dominant. It is noted that the greatest reductions in vibration observed in Figure 5.15 occurs with flap located away from x 0.8R, that is, away from the node of second flapwise bending mode. Apparently the flap benefits by being positioned where it may produce large generalized forces on the second flap bending mode [69, 114]. For high-speed flight (µ = 0.35), 3/rev control input is relatively effective compared to the others. The effect of 5/rev control input is not significant in both low- and high-speed flight conditions. Thus 5/rev is discarded in multiple-tefs configuration to increase the effectiveness of the flaps. Resulting control input sequences for various trailing-edge flap configurations are summarized in Table 5.3. Table 5.3. Control input sequences and flap locations for multiple-flap configurations Flap Single Dual Multiple 1st location R R R control 3,4,5/rev 3,4,5/rev 3/rev 2nd location R R control 3,4,5/rev 4/rev 3rd location R control 4/rev

175 Figure Vibration reduction vs. radial locations of trailing-edge flaps 148

176 Finite Wing Effects Figure 5.16 shows the change of lift curve slope with respect to aspect ratio AR. In single- and multiple-flap configurations, the decreases of lift curve slope due to the finite wing effect are 15% and 38%, respectively. So the lift by multiple-flaps is reduced by 23% compared to a single-flap configuration. Figure Lift curve slope vs. aspect ratio for elliptical lift distribution For multiple-flap configuration with lift flap (the pitching moment of flap is set to zero), vibration reductions are presented in Figure It is observed that performance degradation due to finite wing effect is less than 10% (relative percentage) in both low- and high-speed flight. Recalling that finite wing correction is applied to normal force only, it can be deduced that the kinetic energy due to flap motion contributes to vibration reduction. On the other hand, finite wing effect can be regarded as the weight to control efforts. This is clearly shown in Figure 5.18 showing that 3rd flap deflection increases 1 degree, and 2nd flap phase changes

177 150 slightly for low-speed flight condition (µ = 0.15). Similar trend is also observed in high-speed flight condition (µ = 0.35). This implies that the controller is able to compensate for changes in flap capability by adjusting the flap deflections and phases. This is the reason why trailing-edge flap performance is not reduced by 38% that is estimated by the quasi-steady correction factor presented in Equation 5.1. Figure Vibration reduction by multi-flap configuration with lift flap Figure Flap deflection harmonics of multi-flap configuration with the lift flap, advance ratio: µ = 0.15, actuator saturation: δf sat = 4 o

178 Effectiveness of Multiple-Flap Configuration Single-, dual- and multiple-flap configurations are investigated to explore the multipleflap motion mechanism. For useful insights, the flap motions are limited by 2 degree, and only 4/rev control input is considered for low-speed flight condition. Figure 5.19 shows a polar diagram of trailing-edge flap motion for various singleflap configurations. Each flap span is 20 % of rotor blade length, and location is varied from 0.6R to 0.8R. As previously noted, vibration reduction is less with the flap located at near the node of second flap mode. It is observed that phase difference between inboard and outboard single flap configurations is 110 degree. Similar trends are observed in both dual- and multiple-flap configurations, as presented in Figure 5.20 showing 160 degree phase between inboard and outboard flaps. This clearly indicates that flap motions are nearly out-of-phase to efficiently excite the second flapwise bending mode at inboard and outboard regions from the node (x 0.785R). In all configurations, vibration index is reduced by 56 60% as shown in Figure Flap profile for dual-flap configuration is presented in Figure It is noted that outboard flap deflection (2nd flap) is less than inboard flap one due to the high dynamic pressure. To investigate the effectiveness of multiple-flap configuration, the hinge moments should be compared since both vibration reductions and required maximum flap deflections by single- and multiple-flap configurations are similar. Normalized hinge moments for single- and dual-flap configurations are shown in Figure Maximum hinge moment of single-flap is and occurs at the azimuth angle of 342 degree. Maximum hinge moment of dual-flap is , and occurs at the azimuth angles of 225 and 342 degrees in 2nd and 1st flaps, respectively. Peak-to-peak values of hinge moments are presented in Table 5.4 showing that the peak-to-peak hinge moment at 1st flap is a half of that at single-flap. Total peak-to-peak hinge moment in dual-flap is, however, almost the same as in singleflap.

179 152 Figure Polar diagram of flap motion for single-flap configuration, advance ratio: µ = 0.15, actuator saturation: δf sat = 2 o Table 5.4. Peak-to-peak hinge moments in single- and dual-flap configurations with 4/rev control input, µ = 0.15 Flap configuration Peak-to-peak hinge moments Reduction, % Single flap Dual 1st flap Dual 2nd flap Dual (total)

180 153 Figure Polar diagram of flap motion for dual-flap configuration, advance ratio: µ = 0.15, actuator saturation: δf sat = 2 o

181 154 Figure Flap deflections of dual-flap configuration with 4/rev control input, advance ratio: µ = 0.15, actuator saturation: δf sat = 2 o

182 155 Figure Hinge moments in single- and dual-flap configurations with 4/rev control input, advance ratio: µ = 0.15, actuator saturation: δf sat = 2 o

183 Vibration Reduction with Multicyclic Control The vibratory hub loads for the baseline, single-, dual- and multiple-flap configurations are investigated. Trailing-edge flap deflection limit is set to 4 degree for fair comparison. Control input sequence for multiple-flap configuration is 3/rev, 4/rev and 4/rev from inboard to outboard. Vibratory hub loads for low-speed flight condition are presented in Figure 5.23 showing that all configurations can reduce vibration significantly. Resulting flap profiles for single- and multiple-flap are presented in Figures 5.24 and It is shown that outboard flap deflections are nearly 180 degree out-of-phase with inboard flaps (1st and 2nd flaps) advance ratio = Fx Fy Fz Mx My Mz Figure Comparison of vibratory hub loads, advance ratio: µ = 0.15, actuator saturation: δf sat = 4 o For the high-speed flight, corresponding to an advance ratio of 0.35, the vibratory hub loads are presented in Figure Vibration reductions by single-, dualand multi-flap configurations are 83%, 85% and 79%, respectively. In this case, in-plane hub shear force F x of multi-flap is higher than that of the others, since 5/rev control input is discarded to increase the flap effectiveness in both low- and high-speed flight conditions. Flap profiles of single- and multi-flap are presented in Figure 5.27 showing that both 2nd and 3rd flap deflections are nearly out-of-phase, and their maximum is 1 degree.

184 157 5 Flap profile of single flap, µ= Flap deflections, degrees Azimuth angle, degrees Figure Flap deflections of single-flap configuration, advance ratio: µ = 0.15, actuator saturation: δf sat = 4 o

185 Flap profile of single and multiple flap, µ=0.15 multi 1st TEF multi 2nd TEF multi 3rd TEF Flap deflections, degrees Azimuth angle, degrees Figure Flap deflections of multiple-flap configuration, advance ratio: µ = 0.15, actuator saturation: δf sat = 4 o

186 advance ratio = 0.35 bas eline single (4 deg) dual (4 deg) multi (4 deg) Fx Fy Fz Mx My Mz Figure Comparison of vibratory hub loads, advance ratio: µ = 0.35, actuator saturation: δf sat = 4 o

187 160 Figure Flap deflections of single- and multiple-flap configuration, advance ratio: µ = 0.35, actuator saturation: δf sat = 4 o

188 161 Once again it is observed that all configurations show similar performance in terms of vibration index and flap deflections. To explore differences among various flap configurations, the hinge moments along the azimuth are presented in Figure In multi-flap configuration, maximum hinge moment is and occurs at the azimuth angle of 22 degrees. Although 3rd flap deflections are much less than that of the 1st flap, maximum hinge moment occurs at 3rd flap (outboard flap) because of high dynamic pressure due to high-speed flight. In single-flap configuration, maximum hinge moment is and occurs at the azimuth angle of 170 degree. Peak-to-peak values of hinge moments are presented in Table 5.5. Peak-to-peak hinge moment is reduced by 37.5% to 61.2% in each individual actuator, while total peak-to-peak hinge moment increases by 45.3%. Figure Hinge moments in single- and multiple-flap configuration, advance ratio: µ = 0.35, actuator saturation: δf sat = 4 o

189 162 Table 5.5. µ = 0.35 Peak-to-peak hinge moments in single- and multiple-flap configurations, Flap configuration Peak-to-peak hinge moments Reduction, % Single flap Multiple 1st flap Multiple 2nd flap Multiple 3rd flap Multiple (total) One of the merits of multiple-flap configuration is that the resonant actuation concept can be applied to achieve high action authority, since vibration reduction via single-flap could not be realized due to actuation limitations. This will be discussed in Chapter 7.

190 Summary In this chapter, an active control method for multiple trailing-edge flaps configuration is proposed. The concept involves deflecting the trailing-edge flaps by introducing the resonant actuation, in which each flap operates at a single frequency, in a four-bladed hingeless rotor system. A proposed active vibration control strategy is simulated for the steady-state forward flight condition. Based on the present study, the following conclusions can be made: Blade responses in terms of the number of beam elements are explored. The effect of trailing-edge flaps to the free-wake geometry is also investigated. The number of beam elements lightly affects the free-wake geometry in the high speed flight case, while the flap effect is very small because the tip vortices generated by the flap itself are not considered in the present study. A finite wing effect is not significant in the multiple trailing-edge flap configuration, since the controller is able to compensate for changes in flap capability by adjusting the flap deflections and phases, and kinetic energy also contributes to vibration reduction. For low-speed flight, vibration level is reduced by % with a 4/rev control input. Multiple-flap configuration can reduce the peak-to-peak hinge moments by 49.8% and 50.7% at in- and out-board flaps when compared to the single-flap approach. All flap configurations can reduce the vibration level by 80 85% for both low- and high- speed flight conditions with multi-cyclic control inputs. For high-speed flight, however, peak-to-peak hinge moment is reduced by 37.5% to 61.2% in each individual actuator compared to single-flap configuration.

191 Chapter 6 Piezoelectric Actuation System Synthesis The dissertation to this point has discussed the vibration control and blade loads control of helicopter rotor blades using active trailing-edge flaps. A brief discussion about the resonant actuation concept was given in Chapter 5. With this resonant actuation concept, the helicopter vibration control and blade loads control could be realized. The purpose of the research described in this chapter is to develop such resonant actuation systems and to provide analytical tools. In the first section, the background and objective are presented. Next, a piezoelectric actuation system model is derived for active flap rotors. Utilizing this model, mechanical tuning and electrical tailoring methods are developed, where the optimal tuning parameters for electric networks can be explicitly determined. In the fourth section, an equivalent electric circuit model emulating the physical actuation system is derived and experimentally tested to investigate the initial feasibility of the piezoelectric resonant actuation system. Finally, in the fifth section, summary of this chapter is presented.

192 Introduction Because of their electro-mechanical coupling characteristics, piezoelectric materials have been explored extensively for various engineering applications, where they are often used as sensors or actuators. Some of the advantages of piezoelectric transducers include high bandwidth, high precision, compactness, and easy integration with existing host structures to form smart structures. On the other hand, while piezoelectric material-based actuators have shown good potential, they can only provide limited strain. This limitation can be critical in cases where large actuator strokes are required. The efforts to improve the piezoelectric actuator performance have been made by researchers in developing amplification mechanisms of various types (see Section on Page 20). In general, these devices are still limited in their performance. The goal of this chapter is to develop a new method for actuation authority enhancement to circumvent the aforementioned limitations of piezoelectric actuators. The trailing-edge flap is used as a test bed for illustrating the concept. The approach can be classified into two steps as outlined in the following paragraphs. First, a selected resonant frequency of the actuation system (composed of the piezoelectric actuator, the trailing-edge flap and the amplification mechanism, and under the effect of unsteady aerodynamic loads) is tuned to the desired operating frequency through mechanical tailoring. It is well known that, for harmonic control devices such as the trailing edge flaps, if one can tune the natural frequencies of the actuation system to the actuation frequencies, the actuation authority can be greatly increased due to the mechanical resonance effect. In this case, one can develop much lighter and smaller actuators to activate multiple and smaller trailing-edge flaps, each aiming at different operating frequencies of 3, 4, and 5/rev (for a typical four-bladed rotor). While such a concept is indeed attractive, resonant actuators could be very difficult to control and non-robust, due to its narrow operating bandwidth. This is a critical bottleneck for realizing resonant actuation system in practical applications. To resolve this issue, the second step of this design process is to use electric circuitry tailoring to broaden and flatten the resonant peak, so that one can achieve a

193 166 high authority actuation system with sufficient bandwidth and robustness. In the last decade, piezoelectric materials with electrical networks have been utilized to create shunt damping for structural vibration suppression. It was also recognized that such networks not only can be used for passive damping, they can also be designed to amplify the actuator active authority around the tuned circuit frequency (see see Section on Page 26). Several researchers have proposed the negative capacitance concept to enhance the networks multiple mode and broadband capabilities [129, 130]. The integration of the passive and active approach, often referred to as an active-passive hybrid piezoelectric network, has shown to achieve promising results in vibration control [125]. The electric network tailoring idea proposed in this paper is built upon these previous investigations, but is based on completely different design philosophy and criterion. To demonstrate the proposed concept, the piezoelectric tube actuator developed by Centolenza et. al [116] is selected as a test bed for this study. Coupled piezoelectric actuator, trailing-edge flap and electric network system equations are derived. The proposed method is then analyzed numerically and verified experimentally via an equivalent electric circuit based on the Van Dyke model.

194 Piezoelectric Actuation System Model In this section, a piezoelectric actuation system model is developed for active flap rotors. Fully coupled PZT actuator-flap-circuit system equations are derived via the variational principle. Figure 6.1. A piezoelectric tube actuator configuration The PZT tube shear actuator (Figure 6.1), which uses the shear deformation of piezoelectric materials, is assembled with the piezoelectric ceramic segments of alternating poling signs, and then the accumulation of shear strain around the circumference produces the angle of twist. As opposed to conventional piezoelectric actuators, which are poled in the thickness direction, the PZT tube actuator segments are poled in the length direction [116] Piezoelectric Tube Actuator The potential energy contained in piezoelectric materials is described by a function called the electric enthalpy density function H in the linear piezoelectricity (IEEE 1998) [149] H(S ij, E i ) = 1 2 ce ijkl e kij E k S ij 1 2 εs ije i E j, (6.1) and in the matrix form H(S, E) = 1 2 {S}T [c E ]{S} {S} T [e]{e} 1 2 {E}T [ε S ]{E}. (6.2)

195 168 The constitutive equations for a linear piezoelectricity are given by or T ij = H S ij = c E ijkls kl e ijk E k, (6.3) D i = H E i = e ikl S kl + ε S ike k, (6.4) {T } = [c E ]{S} [e]{e}, (6.5) {D} = [e] T {S} + [ε S ]{E}. (6.6) It can also be expressed in terms of strains and electric displacements. {T } = [c D ]{S} [h]{d}, (6.7) {E} = [h] T {S} + [β S ]{D}. (6.8) where [c D ] = [c E ] + [e][β S ][e] T, (6.9) [h] = [e][β S ], [β S ] = [ε S ] 1. (6.10) Here, nomenclature follows that of ANSI/IEEE standard on piezoelectricity [149]. The governing equations are derived using the variational principle, assuming no body force, as follows: t 0 + V p (ρ p u i δ u i T ij δs ij E i δd i ) dv p dt t 0 S p (p i δu i + φ s δq s ) ds p dt = 0, (6.11) where T ij is the mechanical stress component, S ij the strain component, E i is the electric field component, D i is the electric displacement component, and u i denotes the mechanical displacement component. p i, Q s, and φ s represent the applied traction, charge density, and applied electric potential on the surface, respectively.

196 169 ρ p denotes the piezoelectric material density. For the piezoelectric tube actuator, Equation 6.11 can be written by, in the polar coordinate system, t 0 + v p t 0 ( ρ p u θ δ u θ T zθ δs zθ E θ δ D ) θ dv p dt N s s p ( ) Q p θ δu θ + φ s δ ds p dt = 0, (6.12) N s A s where N s is the number of segments of tube actuator and Q is the electric charge. The angular displacement u θ, shear strain S zθ, and the electric displacement D θ are given by u θ = r p θ, S zθ = r p θ,z, D θ = Q A s, (6.13) in which (),z denotes the partial derivative with respect to z, and A s represents the surface area of the PZT electrode. The equations motion can be discretized by applying the assumed mode method. Then the angular displacement of the piezoelectric tube actuator is given by u θ (r p, z; t) = r p θ(z; t) = r p i=1 Ψ i (z)q i (t), (6.14) where Ψ i (z) is the i-th mode shape function, and q i (t) is the i-th generalized displacement. Substituting Equations 6.7, 6.8, 6.13 and 6.14 into Equation 6.12 yields [ ] { } [ ] { } [ ] { } Mp 0 q Cp 0 q K D + + p K c q = 0 T 0 Q 0 T 0 Q K T c K Q Q where M p = K D p = lp 2π Ro 0 0 lp 2π Ro 0 0 { Fθ V a }, (6.15) R i ( ρp r 2 pψ i Ψ j ) rp dr p dθdz, (6.16) R i ( c D 55 r 2 pψ i,z Ψ j,z ) rp dr p dθdz, (6.17)

197 K c = K Q = F θ = V a = lp 2π Ro 0 0 R i lp 2π Ro 0 0 2π Ro ( h15 r p Ψ i,z 170 ) r p dr p dθdz, (6.18) A s N s ( ) β S 11 r R i A 2 sns 2 p dr p dθdz, (6.19) ( ) h15 r p Ψ i,z r p dr p dθ, (6.20) R i A s N s z=l p ) ( ) φs A s ds = N s = φ s, (6.21) N s A s A s N s 0 ( φs s in which l p is the length of the tube actuator, R o and R i denote the outer radius and inner radius of the tube (see Figure 6.1). The matrix C p represents the structural damping of piezoelectric tube actuator, K Q denotes the inverse of piezoelectric tube actuator capacitance C S p, and V a is the voltage across the segment of piezoelectric tube actuator. For the piezoelectric tube actuator, a single-mode approximation is accurate enough to predict the actuation system response, since the operating frequencies are far below than those of the tube actuator. The tube actuator is modeled as a clamped-free hollow cylinder (see Figure 6.1). clamped torsion bar is given by The first eigenfunction of the ( ) π Ψ(z) = sin z, (6.22) 2l p and then, with assuming no external torque, Equation 6.15 can be rewritten by [ ] } [ ] { } [ ] { } { } Mp 0 { qt Cp 0 qṫ K D + + p K c qt 0 =, (6.23) 0 0 Q 0 R p Q K c K Q Q where q t represents the tip displacement of a piezoelectric tube actuator, which is also referred to as the actuator stroke, and R p represents the electric resistance that includes both the inherent resistance of the tube actuator and the external wire resistance. V a

198 Inertial and Aerodynamic Loads The aerodynamic model for the trailing edge flap used in this study is based on the incompressible thin airfoil theory developed by Theodorsen [143], although the compressible unsteady aerodynamic model, which was presented in Section on Page 98, is available. Since the Theodorsen s theory renders the explicit formulation of aerodynamic loads, it is adequate for the purpose of the present study. The total hinge moment is comprised of the aerodynamic, inertial and centrifugal propeller moments due to the rotation of blade (see Figure 6.2). Assuming that the trailing-edge flap deflection angle, δ f, is small, the inertial and propeller moments can be expressed as [107]: h I = I δ δf, (6.24) h CF = m f Ω 2 dr I sin(δ f ) m f Ω 2 dr I δ f, (6.25) where δ f is a trailing edge flap deflection angle, and Ω is the rotation speed of blade. I δ, m f, d and r I are the flap mass moment of inertia, the flap mass unit per length, the offset from blade elastic axis to flap hinge and the flap chordwise c.g. after flap hinge, respectively. Figure 6.2. Forces and moments acting on the trailing-edge flap The aerodynamic contribution due to trailing edge flap can be obtained from the incompressible aerodynamic model, which was presented in Section on

199 172 Page 97. The hinge moment coefficients in terms of flap deflection angle δ f is given by where coefficients, T 3,4,5 C Hδ = 1 2π T 12T 10 C(k) + 1 2π (T 4T 10 T 5 ), (6.26) C H δ = b T 12 T 11 C(k) + b T 4 T 11, (6.27) 4πU 4πU C H δ = b2 T 2πU 2 3, (6.28) and T 10,11,12, are the geometric parameters defined by Theodorsen [143], U and b are the relative wind velocity at the radial location of the blade and the blade semi-chord, respectively. The Theodorsen s lift deficiency function C(k) is a complex coefficient that can be expressed in terms of Bessel functions. Once all the coefficients are found, aerodynamic and inertial contributions can be expressed in terms of the trailing-edge flap deflection angle δ f : F f (δ f ; t) = M f δf + C f δ f + K f δ f, (6.29) where M f = 1 2 C H δ ρ U 2 c 2 L f + h I, (6.30) C f = 1 2 C H δ ρ U 2 c 2 L f, (6.31) K f = 1 2 C Hδρ U 2 c 2 L f + h CF, (6.32) where c and L f are the blade chord and the trailing-edge flap span, respectively. Underline terms denote inertial contributions from the flap Coupled Actuator-Flap-Circuit System In this section, the coupled actuator-flap-circuit equations are derived to describe the integrated actuation system. For the single-frequency electric networks (see Figure 6.3), a series R-L-C circuit is used and the resulting piezoelectric network

200 173 equation can be written as L Q + R Q + ˆK Q K c q t = V c, (6.33) or L Q + R Q + V a + K a QQ = V c, (6.34) where L is the inductance, R is the resistance, V a and V c denote the voltage across the PZT actuator and the control voltage, respectively. V a is expressed by V a = K Q Q K c q t, (6.35) and ˆK Q is defined by ˆK Q = K Q + K a Q, (6.36) K Q = 1, K a Cp s Q = 1, (6.37) C add in which C s p denotes the capacitance of the PZT tube actuator, and C add represents the added capacitance, which can be either positive or negative. R-L Figure 6.3. Schematic of the PZT tube with R-L circuit and negative capacitance In order to integrate the PZT tube actuator and the flap system, the trailing edge flap deflection angle, δ f, should be expressed in terms of the actuator stroke, q t. This relation can be interpreted as the amplification mechanism, i.e., the linkage

201 174 from the PZT tube actuator to the flap device. The design of this linkage is not a trivial issue, which is beyond the scope of present study. Thus, the simple fulcrum type of amplification mechanism is used to illustrate the concept. It is assumed that the relation between δ f and q t can be expressed in the form (see Figure 6.4) δ f = A M q t. (6.38) Note that this relationship, A M, plays a role of unit conversion. For instance, the trailing-edge flap deflection d f is transformed to the actuator stroke q t of the PZT actuator. On the other hand, using the same relationship, the hinge moments are transformed into either forces or moments, depending on the actuator types. In the tube actuator case, the hinge moment is transformed into the torque. Substituting Equation 6.38 into Equation 6.29 and combining Equation 6.23 and Equation 6.33 yield [ ] } [ ] { } [ ] { } { } M 0 { qt C 0 qṫ K D K c qt =, (6.39) 0 L Q 0 R Q K c ˆKQ Q V c where M = M p + M f A 2 M, C = C p + C f A 2 M, (6.40) K D = K D p + K f A 2 M, where the open-circuit stiffness K D can be also expressed, in terms of the shortcircuit stiffness K E, as: or K E = K D K2 c K Q, (6.41) ˆK E = K D K2 c ˆK Q, (6.42) where K E represents the nominal short-circuit stiffness, which is normally Young s modulus, while ˆK E indicates the situation that the circuit is shorted between the electrode of the PZT and that of the added capacitance.

202 175 PZT tube K f R o R i q t f Flap f A M Figure 6.4. Fulcrum amplification mechanism for the PZT tube actuator q t 6.3 Mechanical Tuning and Electrical Tailoring In this section, mechanical tuning and electrical tailoring are described. A mechanical tuning is needed to tune the resonant frequency of actuation system to the operating frequency. Then electrical tailoring is applied to the actuation system to enhance the actuator operating bandwidth and phase control. After mechanical tuning, the actuation system together with an electric network is referred to as a resonant actuation system (RAS) Mechanical Tuning In the present study, structural resonance of the actuation system is utilized to increase the active authority. The stiffness and/or the mass of the coupled system could be adjusted to tune the resonant frequency of the actuation system to around the desired operating frequency. It is assumed that the stiffness and mass of the PZT tube actuator are fixed, and the modal stiffness and mass of the trailing edge flap and the amplification mechanism can be adjusted. Tuning the mass will be more effective than tuning the stiffness, because the mass of the coupled system mostly comes from the aerodynamic contribution (the flap mass moment of inertia in Equation 6.24). There are two ways to adjust the mass term of the aerodynamic loads. One is to add concentrated mass to the trailing edge flap. This is the simplest method to adjust the mass term, but it may cause aeromechanical instability [31] due to the shift in

203 176 C.G. The other is to design the overhang of the trailing edge flap [84,85]. This will also be effective, but the design detail is beyond the scope of the present study. In this study, the mechanical tuning is achieved by the former method. In the present study, the simple mass tuning is adopted to achieve the mechanical tuning by adding the tuning mass to the trailing-edge flap. This will affect the flap inertial load, mainly the flap mass moment of inertia, I δ, given in Equation Electrical Tailoring It is convenient to express the actuation system equation in the nondimensionalized form for the derivation of electrical tailoring parameters, such as the optimal inductance and resistance tuning ratios, which will be derived based on the transfer functions. From Equation 6.39 and neglecting the added capacitance (i.e., KQ a = 0), the transfer function between actuator stroke and control voltage and that between electric charge and control voltage are, respectively, q t V c = Q V c = and in the nondimesionalized form Q V c = where q t V c = K c, (6.43) ( ω 2 M + jωc + K D )( ω 2 L + jωr + K Q ) Kc 2 ω 2 M + jωc + K D, (6.44) ( ω 2 M + jωc + K D )( ω 2 L + jωr + K Q ) Kc 2 ( ) δ 2 qt, (6.45) (1 + ξ 2 + 2jζ ω ω 2 )(rj ω + δ 2 ω 2 ) δ 2 ξ 2 V a ST δ 2 (1 + ξ 2 + 2ζj ω + ω 2 ) ( 1 (1 + ξ 2 + 2jζ ω ω 2 )(rj ω + δ 2 ω 2 ) δ 2 ξ ξ 2 ω 2 E = KE M, ζ = ω2 c = K Q L, ω = ω ω E, ξ 2 = K2 c K E K Q, C, r = R, 2Mω E Lω E ) ( ) Q V a ST, (6.46) δ = ω c ω E, (6.47)

204 177 and subscripts ST represents the static response that are given by ( qt V a ( ) Q V a ) ST ST = ξ2 K c, (6.48) = 1 + ξ2 K Q. (6.49) Here r and δ are often referred to as the resistance and inductance tuning ratios, and ξ is referred to as the generalized electro-mechanical coupling coefficient, which is also expressed as: ξ 2 = (KE + K 2 c /K Q ) K E K E = KD K E K E = ω2 D ω2 E. (6.50) ωe 2 Now the open-circuit frequency can be expressed, in terms of the generalized electro-mechanical coupling coefficient and the short-circuit frequency, as: ω 2 D = (1 + ξ 2 )ω 2 E. (6.51) It is well known that there are optimal inductance and resistance tuning ratios for the passive shunt circuit configuration proposed by Hagood and Flotow [118]. The optimal values for the shunt inductance and resistance can be derived via a mechanical vibration absorber analogy. These optimal values would be different from those for the best voltage driving responses when the system is used for actuation. In this chapter, the explicit formulae of the optimal tuning ratios for voltage driving responses are derived. Unlike the passive shunt damping system, there is only one invariant point of the voltage driving frequency response curve, which is at the open-circuit frequency. In this case, the optimal inductance tuning strategy is developed by deriving the inductance value such that the transfer function in Equation 6.45 has a stationary value with respect to frequency at this invariant point ω = 1 + ξ 2 that is the open-circuit frequency normalized with the short-circuit frequency. The optimal tuning ratios for the voltage deriving response can be derived based on the

205 178 frequency function approach [125]. The modal damping due to the actuator and trailing-edge flap aerodynamics is neglected in the following derivation. The frequency response function of the piezoelectric actuator with the series R-L-C circuit is given in Equation This can be rewritten, without the mechanical damping, as follows: q t V c = ζ=0 ( ) δ 2 qt. (6.52) (1 + ξ 2 ω 2 )(rj ω + δ 2 ω 2 ) δ 2 ξ 2 V a ST This function has an invariant value at ω 2 = 1 + ξ 2 with respect to the electric resistance tuning ratio r. The optimum inductance-tuning ratio can be found if Equation 6.52 has its stationary value at the invariant point. The derivative of Equation Equation 6.52 with respect to the normalized frequency square ω 2 is given by which yields ω 2 ( q t V c = 0, (6.53) ζ=0,r=0) ω 2 =1+ξ 2 [ (1 + ξ 2 )(δ 2 ω 2 ) (1 + ξ 2 ω 2 ) ] ω 2 =1+ξ 2 = 0. (6.54) From Equation 6.54, the optimal inductance tuning ratio δ can be found by δ = 1 + ξ 2, (6.55) which is the normalized open-circuit frequency and the same as that of passive shunt damping [118]. The voltage driving frequency response function for any given resistance tuning, r, will pass through the invariant point, ω = 1 + ξ 2. The optimal value of r, which will best flatten the frequency response function near the resonant frequency, can be found by equating the magnitude of the frequency response at ω = δ (open-circuit frequency) and that of the resonant frequency, ω = 1 (short-circuit

206 179 frequency), which yields ( qt V c ) ζ=0, ω 2 =1+ξ 2 = ( qt ), (6.56) V c ζ=0, ω 2 =1 r = 2ξ ξ 2. (6.57) This optimal resistance tuning ratio, however, should be calculated by considering the actuation system damping, i.e. ζ 0, because the actuator stroke at the vicinity of the resonant frequency strongly affected by the system damping. An iterative method (e.g., a bisection method) can be used to solve the following nonlinear equation to find the optimal resistance tuning ratio in practice. ( qt V c ) ω 2 =1+ξ 2 = ( qt ). (6.58) V c ω 2 =1 When the additional capacitance is added in series to the piezoelectric material, and with a R-L circuit, which is presented in Equation 6.39, the transfer function between actuator stork and control voltage is ˆq t V c = K c ( ω 2 M + jωc + K D )( ω 2 L + jωr + ˆK Q ) K 2 c, (6.59) which can be written in the dimensionless form using non-dimensionalized parameters defined in Equation 6.47, ˆq t V c = δ 2 ˆξ2 /ξ 2 (1 + ξ 2 + 2jζ ω ω 2 )(rj ω + δ 2 ω 2 ) δ 2 ˆξ 2 ( qt V a ) ST, (6.60) where a hat ˆ() indicates the system response with the added capacitance, the modified coupling coefficient ˆξ is defined by ˆξ 2 = K2 c K E ˆK Q. (6.61)

207 180 The short-circuit frequency, which includes the added capacitance, is given by ˆω 2 E = (1 + ξ 2 ˆξ 2 ) ω 2 E. (6.62) Then, in the same manner described for the system without added capacitance, the optimal inductance and resistance tuning ratios are obtained, respectively, by ˆr = 2ˆξ ˆδ = 1 + ξ 2, (6.63) 1 + ξ 2 0.5ˆξ ξ 2 ˆξ 2. (6.64) Design guidelines and physical insights for the developed resonant actuation system will be discussed in the Section 7.1 of Chapter 7 in detail. 6.4 Equivalent Electric Circuit Model To realize the proposed resonant actuation system and examine its feasibility for the proof-of-concept, an equivalent electric circuit model, which is based on the Van Dyke model, is derived and tested experimentally Van Dyke Model The mechanical and electrical parts of piezoelectric materials are connected through a conversion factor [150]. The electrical property of the piezoelectric actuator can be modeled as a single capacitor, as normally assumed in piezoelectric network analysis. The modal stiffness of the resonant actuation system is governed by that of the PZT tube actuator, although natural frequencies of the PZT tube actuator are relatively high, while the modal mass of the resonant actuation system is governed by trailing edge flap contribution. Therefore, the mechanical stiffness of the PZT tube actuator should be considered to properly emulate the physical system. The PZT tube actuator for trailing edge flap operates at 3/rev, 4/rev, and

208 181 5/rev of the main rotor, which are relatively low frequencies when compared to natural frequencies of the PZT tube actuator. The mechanical part of the actuator and aerodynamic and inertial loads of trailing-edge flap adding together can be considered as a single degree of freedom system. When the negative capacitance, C n, and series inductor-resistor circuit, R and L, are considered, the equivalent electric circuit model of the resonant actuation system can be represented in Figure 6.5. The first and second branch together can be considered as the Van Dyke model that was introduced to model the piezoelectric materials [151]. Figure 6.5. Equivalent electric circuit model of the resonant actuation system The dynamic equations of the equivalent electric circuit model shown in Figure 6.5 can be written as ( 1 L m Q2 + R m Q 2 + C m + Cp e ) Q 2 1 C e p = 0, (6.65) L Q + R Q + 1 Q 1 Q Cp e Cp e 2 = V c, (6.66) where the electric charge Q 2 in the first branch emulates the actuator stroke of the PZT tube actuator. The stiffness, damping, and mass of the mechanical part of the

209 182 PZT actuator and trailing-edge flap contributions are represented by capacitor C m, resistor R m, and inductor L m, respectively. C e p represents the total capacitance, C e p = C nc s p C s p + C n. (6.67) To use this equivalent electric circuit model, the coefficients of the Van Dyke model (C m, R m, L m ) should be determined. It can be resolved by measuring the electrical impedance of the resonant actuation system directly [150]. The equivalent circuit is mainly introduced to experimentally verify the proposed resonant actuation system that includes two active circuit components (synthetic inductor and negative capacitor). These coefficients can be determined through comparing the resonant frequencies between the two systems. There are two types of frequencies in the Van Dyke model, parallel and series frequencies (ω p, ω s ) that are given by ω 2 p = 1 L m C m + C e p C m C e p ωs 2 = 1 = KD Kc 2 / ˆK Q L m C m M = KD M = ω2 D, (6.68) = ˆK E M = ˆω2 E. (6.69) From the equivalence of these two frequencies, the following relationships can be obtained. L m = 1 C e p 1 (ω 2 p ω 2 s), (6.70) R m = L m C M, (6.71) C m = 1, (6.72) L m ωs 2 M ( ) K c = ω 2 Cp e p ωs 2. (6.73) These clearly show that the coupling stiffness K c of the resonant actuation system is proportional to the square root of the difference between the electro-mechanically coupled and de-coupled frequencies, which can be increased by increasing the effective capacitance (using the negative capacitance).

210 Analysis and Experimental Verification In order to evaluate the resonant actuation system for a trailing-edge flap, a Machscaled helicopter rotor blade is considered as an example. The rotor has a blade diameter of 6 feet, a blade chord of 3 inches, and a nominal rotation of 2000 RPM. For comparison purposes, a baseline actuation system is first defined: the length of the baseline PZT tube actuator is 4 inches, outer radius R o is assumed to be inches, the tube wall thickness is 0.1 inches, the flap length is 20% of rotor length, 7.2 inches, and the amplification ratio is assumed to be 13. To demonstrate the improvement of active authority, a smaller 4/rev actuator is considered for the new resonant actuation system. The length of this smaller actuator is 2 inches, which is one-half of the baseline actuator, the trailing edge flap length is 2.4 inches (33% of the baseline), and the other dimensions are the same as the baseline. The piezoelectric material properties used in this study are listed in Table 6.1. Table 6.1. Piezoelectric material properties of PZT-5H for a Mach-scaled rotor Piezoelectric constant Value Charge constant, d m/v or C/N (10 12 ) Voltage constant, g V m/n (10 3 ) Elastic constant (open-circuit), c D N/m 2 (10 10 ) Dielectric constant, ε T F/m (10 8 ) Piezoelectric constant, h V/m or N/C (10 10 ) Impermittivity, β11 S m/f (10 7 ) When appropriate amplification ratio and mechanical tuning frequency (usually, 3, 4 or 5 /rev of the main rotor speed) have been determined, the frequency response between trailing-edge flap deflection and control voltage is evaluated, where the trailing-edge flap deflection is calculated by Equation In the present study, the mechanical resonant tuning is achieved by adjusting the mass moment of inertia of the trailing edge flap. In Figure 6.6, the voltage driving and passive damping frequency responses are plotted, under the optimal resistance-tuning ratio for voltage driving performance. The vertical axis denotes the trailing edge flap deflections. The solid line indicates

211 184 the voltage driving response of the baseline actuation system. The dotted and dashdotted lines denote the voltage driving and passive damping responses, respectively, with both negative capacitor (C n = 2.2Cp) s and shunt circuit. For the voltage driving responses, an electric field of 4 kv/cm is applied. For the passive damping response, the excitation force is assumed to be the same moment as that produced by the 4 kv/cm electric field. The resonant actuation system is tuned to the 4/rev (120 Hz) by modifying the mass moment of inertia of trailing edge flap and amplification ratio. Comparing the baseline and the new resonant actuation systems, once can see that the active authority is significantly increased from 3 degree to 8 degree, as shown in Figure 6.6. The flap deflections around the resonant frequency are almost constant (flat plateau) and the bandwidth reasonably large (approximately 80 Hz). Figure 6.6. Trailing-edge flap deflections of the resonant actuation system for Machscaled rotor An experimental investigation is performed on the resonant actuation system. The equivalent electric circuit model is realized using the synthetic inductance and the negative impdedance converter of capacitance, as shown in Figure 6.7. The mechanical part of the PZT tube actuator and aerodynamic and inertial loads due to the trailing edge flap are emulated by a single degree of freedom series R m -

212 185 L m -C m circuit (left branch in Figure 6.7). This branch and the middle branch (representing the PZT tube capacitance and the negative capacitor) in Figure 6.7 form the Van Dyke model. The coefficients of this model are calculated by Equations , where the resonant frequency of the left branch is tuned to 4/rev (120Hz). The voltage across the capacitor C m is measured, and an HP35665A dynamic signal analyzer is used to extract the frequency response. Figure 6.7. system Realization of the equivalent electric circuit for the resonant actuation To create the negative capacitance, the negative impedance converter of capacitance [128] is used. Since the negative capacitance is non-floating, it should be grounded. In this system, large inductance is required to emulate the trailing-edge flap and actuator dynamics and to create the needed inductance in the R-L circuit. Thus a synthetic inductor [122], which consists of four operational amplifiers, is used. The piezoelectric capacitance is 8 nf, the negative capacitance is nf (- 2.2 Cp) s and the capacitance C m and inductance L m of the left branch in circuit diagram shown in Figure 8 are 6.8 nf and 285 H. The inductance of shunt circuit is 95 H. For this configuration, theoretical optimal resistance for the voltage driving

213 186 performance is estimated to be 83 kilo Ohms. However, only 40 kilo Ohms are added to the circuit since there is significant internal resistance in the actual circuit components. The frequency response of the equivalent electric circuit system with the welltuned parameters is compared to that of analytical prediction, as shown in Figure 6.8, where the experimental results (with 1 volt white noise input) are scaled up to match with analytical results. Clearly, the test results match the theoretical prediction quite closely, implying that the high authority robust resonant actuation system can be realized through the proposed hardware and circuits (synthetic inductance and negative capacitance). Figure 6.8. Comparison of analytical and experimental results of the RAS for a Machscaled rotor

214 Summary In this chapter, a new approach is proposed to enhance the actuation authority of piezoelectric actuation systems. The idea is to first achieve a resonant driver through using mechanical tuning, and then increase the bandwidth and robustness of the resonant actuation system through electrical network tailoring. A Machscaled piezoelectric tube actuator-based trailing-edge flap for helicopter vibration control is used as an example to illustrate the proposed concept. A coupled PZT actuator, trailing edge flap and electrical circuit dynamic model is derived. Utilizing the model, the required electrical circuitry parameters are determined. The inductance is first tuned such that the piezoelectric shunt frequency matches the mechanical resonant frequency. Negative capacitance is then used to broaden the actuation authority bandwidth around the operating frequency. To further enhance the robustness, the optimal resistance tuning ratios are derived such that the actuation authority frequency response near the resonant frequency can be flattened. The negative capacitance and inductance are realized and implemented using operational amplifier-based circuitry. An experiment is set up using equivalent circuit representing the integrated structure-actuator-network system. It is demonstrated that the proposed resonant actuation system can indeed achieve both high active authority and robustness. In this chapter, the electric network with low voltage excitation was realized for the case of a Mach-scaled rotor as the feasibility verification of the resonant actuation system. For full-scale helicopter, however, a high voltage is needed to produce the torque that should overcome strong aerodynamic hinge moments. This issue will be addressed in the following chapter.

215 Chapter 7 Design and Test of Resonant Actuation Systems While investigations on the electro-mechanically tailored piezoelectric resonant actuation system have shown promising results (Chapter 6), there are still research issues to be addressed before such a concept can be realized. One of them is that there is no systematic method for tailoring the electrical parameters of the RAS circuitry network, such that desired actuator authority can be achieved. Thus, in the first section of this chapter, an analytical approach is carried out to derive design guidelines for the RAS circuitry in dimensionless forms. A mechanically tuned resonant actuator is analyzed based on the previous derivation in Chapter 6 and compared to an equivalent mechanical system to provide better physical understanding. In the second section, dynamic characteristics of the RAS will be examined. Vibration reduction performance of various flap configurations is evaluated within the available actuation authority. In the third section, a better method for implementing the electrical circuitry is proposed to realize the actuation system, especially under high voltage operations. In the fourth section, the electric power consumption of the piezoelectric RAS is quantified to evaluate the proposed resonant actuation system. Finally, the efforts and findings of this chapter are summarized.

216 Design Guidelines of the RAS In this section, an analytical approach is carried out to derive design guidelines for the RAS circuitry in dimensionless forms. A mechanically tuned resonant actuator is analyzed based on the previous derivation in Chapter 6 and compared to an equivalent mechanical system to provide better physical understanding Resonant Actuators with R-L elements Given that the electrical property of a piezoelectric actuator is similar to a capacitor, Figure 7.1 illustrates the equivalent dynamics of the piezoelectric material with a series R-L circuitry. The dynamic system equations are presented by Equation 6.45 on Page 176 in Chapter 6. C p R L PZT V a V c V p Figure 7.1. A piezoelectric network with a series R-L circuit For this configuration, the voltage driving frequency responses of resonant piezoelectric actuators with a series R-L circuit are shown in Figure 7.2, where results are plotted with respect to the electro-mechanical coupling coefficient ξ. The frequency responses under the optimal tuning ratios show a flat shape around the resonant frequency. Magnitude at the two points (short- and open-circuit frequencies) under optimal tuning ratios (Equations 6.55 and 6.57 on Page 178) could be used as the performance index of the resonant piezoelectric actuator, that is ( qt V c ) = 1 ω=1, ω= 1+ξ ξ 2 2 ( qt V a ) ST, (7.1)

217 190 where ( qt which was presented in Equation 6.48 on Page 177. V a ) ST = ξ2 K c, (7.2) Frequency response of q t /V c (db) baseline ξ=0.2 ξ=0.4 ξ= Nondimensionalized Frequency Figure 7.2. coefficients Actuator strokes with the optimal tuning ratios and various coupling In general, larger electro-mechanical coupling coefficient, ξ, yields better actuator performance, especially for the static case. There is, however, a tradeoff between actuator authority and bandwidth, as shown in Figure 7.2. From Equation 7.1, one can see that the actuation authority at the operating frequency (near resonant frequency) relative to the static value will decrease with increasing electro-mechanical coupling coefficient. On the other hand, increasing the electromechanical coupling coefficient will increase the bandwidth. Here, the bandwidth is defined by B ω = ω D ω E ω E = 1 + ξ 2 1 = 1 2 ξ2 (1 1 4 ξ2 ). (7.3) Variations of the stroke and bandwidth with different generalized coupling co-

218 191 efficients under the optimal tuning ratios are presented in Figure 7.3, where the stroke and bandwidth are normalized with their maximum values. One can find that there is the generalized coupling coefficient where the two curves intersect. Coupling coefficients near this value could yield the best balanced performance in terms of both stroke and bandwidth. Thus, Equations can serve as design criteria for the resonant actuation systems log(q t /Vc) B ω Coupling coefficient, ξ Figure 7.3. Actuator stroke and bandwidth variations with coupling coefficients Figure 7.4 shows the actuator stroke with different resistance tuning ratios, where the electro-mechanical coupling coefficient is assumed to be 0.4. The frequency response exhibits two resonant peaks when the resistor is below its optimal value. As the resistance tuning ratio is increased, these two peaks coalesce into a single peak which corresponds to the open-circuit frequency ω D. This can be clearly observed considering the equivalent mechanical system (Figure 7.5). The equations of motion for an equivalent mechanical system shown in Figure 7.5 is given by [ } [ ]{ } [ MM 0 ]{ẍe CM 0 xe KM + k m k m m m ÿ e 0 c m y e k m k m ]{ xe y e } { } 0 =, (7.4) f e

219 Frequency response of q t /V c (db) baseline r=0.01 r=r * r= Nondimensionalized Frequency Figure 7.4. Actuator strokes with the optimal inductance tuning and various resistance tuning values, ξ = 0.4 where subscripts M and m represent the structural and electrical properties of the piezoelectric actuation system, respectively. The capital M denotes the mechanical modal mass (actuator and flap inertia). The small m represents the inductor in the circuit. The displacements x e and y e correspond to the actuator stroke q t and electric charge Q, respectively. The small spring k m represents both the electromechanical coupling and the inverse of the piezoelectric actuator capacitance. C M and c m correspond to the mechanical damping and electric resistance, respectively. f e represents the voltage source V c. Note that the damper c m applied on the small mass m m is directly connected to the ground, unlike the classical mechanical vibration absorber case [152]. Here, the open- and short-circuit resonant frequencies can be explained in terms of the electrical damping (i.e., resistor). Assuming m m = 0, the equivalent mechanical system shown in Figure 7.5 becomes a one-degree-of-freedom system. If the damping c m is zero, the displacement x e is not affected by the stiffness k m. The system

220 193 y f(t) mm m k m x M M c m K M C M Figure 7.5. Schematic of an equivalent mechanical system to a piezoelectric actuation system is analogous to the short-circuit situation, and the resonant frequency will become: ω E = KM M M. (7.5) On the other hand, if the damping c m is infinity, the displacement x e is affected by both springs K M and k m, which represents the behavior of piezoelectric actuator under the open-circuit condition, with the resonant frequency ω D = KM + k m M M. (7.6) In the case that the small mass m m is present, the equivalent mechanical system will have two degrees of freedom and two resonant peaks. These two peaks will coalesce into a single resonant peak, as the damping c m is increased. This is the same result as the passive shunt damping case reported in Reference [118] showing that the two resonant frequencies in an inductance- shunt piezoelectric system coalesce into the open-circuit frequency, as the resistance approaches infinity. Electric charges for systems with the optimal tuning ratios are presented in Figure 7.6, where the electro-mechanical coupling coefficient varies from 0.2 to 0.6. It is seen that as the electro-mechanical coupling coefficient increases, electric

221 194 charge near the operating frequency (open-circuit frequency) becomes much lower than that in the static case. This implies that the required electric power could be very low near the open-circuit frequency. This open-circuit frequency can be interpreted as the frequency that yields the minimal displacement of y e in Figure 7.5. This indicates that large displacement x e could be achieved with the smallest excitation of f e at the open-circuit frequency Amplitude of Q (db) ξ=0.2 ξ=0.4 ξ= Nondimensionalized Frequency Figure 7.6. Electric charges with the optimal tuning ratios for various coupling coefficients

222 Resonant Actuation Systems with Additional Capacitance The increase of the electro-mechanical coupling coefficient, as mentioned in the previous section, will increase the static performance. However, there is a trade-off between the stroke amplification (relative to the static stroke) and bandwidth at the operating frequency. For given piezoelectric materials, the nominal generalized electro-mechanical coupling coefficient ξ can be increased by introducing a negative capacitance (or be reduced by adding a positive capacitance) in series with the piezoelectric transducer. The negative capacitance cannot be realized passively, but can be achieved using an operational amplifier to form a negative impedance converter presented in Figure 6.7 on Page 185. When the additional capacitance is added in series to the piezoelectric transducer, and with a R-L circuit (see Figure 7.7), the transfer function between actuator stroke and control voltage is given in Equation 6.59 on Page 179. Now relationship between two generalized coefficients can be obtained by ˆξ 2 = K2 c K E ˆK Q = ξ 2 K Q ˆK Q. (7.7) C p R L PZT V a V c V p C add Figure 7.7. A piezoelectric network with a series R-L circuit and an additional capacitor

223 196 There is a limit on adding capacitance to the piezoelectric transducer. That is, the following criterion has to be satisfied to ensure system stability, K a Q > ξ 2 K Q, or ˆξ 2 < 1 + ξ 2. (7.8) Here it should be noted that the short-circuit condition with the added capacitance refers to the situation that the circuit is shorted between the electrode of the PZT and that of the added capacitance while the PZT and the added capacitance are connected in series. Thus the short-circuit frequency will be changed to, ˆω E = (1 + ξ 2 ˆξ 2 ) ω E, (7.9) and its range is given by 0 < ˆω E < ω D = (1 + ξ 2 ) ω E. (7.10) This means that the new resonant frequency can be assigned to somewhere below the open-circuit frequency ω D. This situation can be explained further by considering an equivalent mechanical system of a piezoelectric actuation system with an additional capacitor (see Figure 7.8). mechanical system are given by [ } [ ]{ } [ MM 0 ]{ẍe CM 0 xe KM + k m k m m m ÿ e 0 c m y e k m k m + k n Dynamic equations of the equivalent ]{ xe y e } { } 0 =, (7.11) where an added spring k n only affects the second equation representing the electrical behavior of actuation systems. Note that here k n could be negative. As shown in Figure 7.8, the spring k n that corresponds to KQ a is directly attached to the ground, so that it only affects the short-circuit frequency. It is clearly seen that if the displacement y e is fixed (i.e., open-circuit condition), the displacement x e is independent of the added spring k n. For a short-circuit condition, the displacement y e is freely moving, the displacement x e is now strongly affected by an added spring. This explains that how the short-circuit frequency can be adjusted. From the system stability point of view, there is a limit on the added spring. This f e

224 197 y f(t) mm m k m x M M c m k n K M C M Figure 7.8. Schematic of an equivalent mechanical system to a piezoelectric actuation system with an additional capacitor is similar to adding capacitance to the electro-mechanical system. The equivalent mechanical system version of Equation 7.8 can be expressed by k n > K Mk m K M + k m. (7.12) This simply indicates that an added spring should be larger than the negative value of the stiffness of two existing springs in series, so that the overall stiffness of the system should be greater than zero. The optimal inductance tuning is the same as that of the electro-mechanical system without an added capacitance, as presented in Equation 6.63, while the optimal resistance tuning given in Equation 6.57 is different, which can be rewritten in terms of the coupling coefficients ξ and ˆξ as: ˆr = 2ˆξ 1 + ξ 2 0.5ˆξ ξ 2 ˆξ 2, (7.13) where one can see that if the coupling coefficient, ˆξ, approaches the nominal coupling coefficient, ξ, the optimal resistance tuning ratio becomes what is shown in Equation 6.57.

225 198 The performance variation due to the added capacitance is presented in Figure 7.9 for various capacitance values, where the nominal coupling coefficient is ξ = 0.5 (the coupling coefficient of the piezoelectric tube actuator). The added capacitance value can be calculated from Equation 7.7 for given coupling coefficients. ^ ^ ^ ^ Figure 7.9. Actuator strokes with additional capacitance for ξ = 0.5 There are two important performance indices, namely the response magnitude at the invariant point ( ω 2 = 1 + ξ 2 ), µ µ qˆt 1 qt = 2, Vc ω = 1+ξ2 ξˆ2, 1+ξ2 ξ Va ST (7.14) and the bandwidth, ˆω B ωd ω ˆE = ωe 1 ˆ2 ξ 1+ 2 p q 1 + ξ 2 ξˆ2, 1 ˆ2 1 2 (ξ ξ ), 4 2 = 1+ ξ2 which can be varied by adjusting the value of the added capacitance. (7.15)

226 199 As noted in Reference [125], the coupling coefficient does not change the magnitude at the invariant point (open-circuit frequency, ω D ), but it does change the bandwidth and the static stroke authority: ( ) ˆqt ˆξ = V c ω=0 2 /ξ ξ 2 ˆξ 2 ( qt V a ) ST. (7.16) The magnitude at the invariant point with respect to the static stroke can be evaluated by ( ) ˆqt V ω= = 1 + ξ2 ˆξ2 c 1+ξ ˆξ 2 2 ( ) ˆqt. (7.17) V c ω=0 As far as the magnitudes at the vicinity of the resonant frequency are concerned, the relative magnitude at the invariant point of Equation 7.17 (i.e., stroke amplification with respect to the static value) is an important design parameter. Variations of relative magnitude and bandwidth due to added capacitance are plotted in Figure 7.10, where the nominal coupling coefficient of the piezoelectric tube actuator, ξ, is 0.5 and the modified coupling coefficient, ˆξ, is assumed to be varying from 0.1 to 0.8. The increased coupling coefficient using negative capacitances tends to increase the bandwidth, while the stroke amplification with respect to the static stroke approaches to unity. This situation is different from that shown in Figure 7.3, where the large coupling coefficients near 0.8 still show stroke amplification Summary of Design Guidelines for the RAS circuitry For electrical tailoring of the resonant actuation system, there are three design parameters, namely the optimal inductance tuning, Equation 6.54, the optimal resistance tuning, Equations 6.57 and 7.13, and the capacitance tuning, Equation 7.15, which are summarized in Table 7.1. The combined inductance-resistance tuning contributes to the bandwidth and flatness of the frequency response function near the operating frequency, while the capacitance tuning contributes to the stroke

227 log(q t /Vc) B ω Modified coupling coefficient, ξ hat Figure Relative actuator stroke and bandwidth variations with modified coupling coefficients amplification with respect to the static stroke as well as the operating frequency bandwidth. Such design parameters are summarized as below: Optimal inductance tuning ratio, δ, to tune the circuit resonant frequency to the open-circuit resonant frequency of system is given by δ = 1 + ξ 2 for with and without added capacitance. Optimal resistance tuning ratio, r, to make the frequency response near the resonant frequency as flat as possible is given by r = 2ξ ξ 2 for without added capacitance, and ˆr = 2ˆξ 1 + ξ 2 0.5ˆξ ξ 2 ˆξ 2

228 201 for with added capacitance. Actuator stroke with respect to the static stroke is expressed by ( qt V c for without added capacitance, and for with added capacitance. ) ( ) qt / ω 2 =1+ξ V 2 a ST = 1 ξ 2 ( ) ( ) ˆqt ˆqt / = 1 + ξ2 ˆξ 2 V c ω 2 =1+ξ V 2 c ω=0 ˆξ 2 Operating bandwidth, B ω, is expressed by B ω = 1 2 ξ2 ( ξ2 ) for without added capacitance, and ˆB ω = 1 [ 2 ˆξ (ˆξ 2 12 )] 4 ξ2 for added capacitance. Best compromised coupling coefficients, ξ and ˆξ, in terms of actuator stroke amplification and operating bandwidth can be found from Figures 7.3 and 7.10 Table 7.1. Design parameters for the RAS circuitry Optimal inductance tuning, δ Equation 6.54 Optimal resistance tuning, r Equations 6.57 and 7.13 Capacitance tuning (stroke) Equations 7.17 Capacitance tuning (bandwidth) Equation 7.15 Capacitance tuning (trade-off) Figure 7.10

229 Dynamic Characteristics of the RAS in Forward Flight In the previous section, it is demonstrated that the high authority actuation system can be achieved near the resonant frequency for general piezoelectric actuation systems. Design guidelines for such systems are provided. In this section, the dynamic characteristics of the RAS will be examined, since the aerodynamic stiffness, K f, presented in Section is not constant. The analysis is performed for piezoelectric resonant actuators in forward flight using a perturbation method. Based on this, the actuation authority is then predicted for the nominal actuation system (i.e., non-resonant actuation system) and the RAS. Vibration reduction performance of various flap configurations is evaluated within the available actuation authority, which can be realized by the actuator saturation algorithm presented in Section on Page A Perturbation Method The aerodynamic contribution of the flap based on Theodorsen s theory with considering the forward flight speed can be written as: F f (δ f ; t) = M f δf + C f [1 + ɛ sin(ωt)] δf + K f [1 + ɛ sin(ωt)] 2 δ f, (7.18) where ɛ = µ r f, (7.19) in which r f denotes the flap location along the rotor spanwise direction. M f, C f and K f were derived in Equation 6.29 on Page 172 in Section With this aerodynamic contribution, coupled actuator-flap-circuit system equations given in Equation 6.39 are rewritten by [ ] } [ ] { } [ ] { } { } M 0 { qt Cµ 0 qṫ K D + + µ K c qt 0 =, (7.20) 0 L Q 0 R Q K c ˆKQ Q V c

230 203 where C µ = C p + C f A 2 M [1 + ɛ sin(ωt)], (7.21) K D µ = K D p + K f A 2 M [1 + ɛ sin(ωt)] 2. (7.22) For mechanical tuning, it is needed to define the nominal stiffness of the actuation system, since its stiffness is a function of time. The resonant frequency of the system in hover could be served as the reference. In this case, the relative wind velocity is constant because the forward speed parameter ɛ becomes zero in Equation The resonant frequency of the system in hover is then obtained as: ˆK ω h = E M, (7.23) where ω h represents the resonant frequency of the RAS in hover, and ˆK E indicates the short-circuit stiffness including the added capacitance. This frequency is tuned to one of operating frequencies that are 3, 4 and 5/rev of the main rotor speed for four-bladed rotor systems by adjusting the flap mass moment of inertia. The resonant frequency in forward flight will be discussed in the next section. It is convenient to explore the simplified version of the electro-mechanical system without circuitry. Equation 7.20, without damping, is then rewritten as follows: M q t + ˆK µ E (t)q t = F p (t), (7.24) where ˆK E µ (t) = ˆK E + K f A 2 M [ 2ɛ sin(ωt) + ɛ 2 sin 2 (Ωt) ], (7.25) F p (t) = K c ˆK Q V c (t), (7.26) V c (t) = V c cos(ωt), (7.27) where V c represents the magnitude of voltage signal, and ω denotes the excitation frequency.

231 204 For the purpose of performing perturbation analysis, Equation 7.24 is expressed in the dimensionless form. A dimensionless quantity is introduced using an azimuth angle ψ. Then q t + [ α 2 m + β 2 m ( 2ɛ sin ψ + ɛ 2 sin 2 ψ )] q t = Γ m (cos γ m ψ), (7.28) where ( ) αm 2 = 1 ˆKE, (7.29) Ω 2 M β 2 m = 1 Ω 2 ( Kf A 2 M M ), (7.30) γ m = ω Ω, (7.31) ( ) Γ m = 1 K c V Ω 2 c. (7.32) M ˆK Q The most elementary version of the perturbation method is to attempt a representation of the solution of Equation 7.28 in the form of a power series in a small parameter ɛ : q t (ψ, ɛ) = q t0 (ψ) + ɛq t1 (ψ) + ɛ 2 q t2 (ψ) +, (7.33) whose coefficients q ti (ψ) are only functions of ψ. To form equations for q ti (ψ), i = 0, 1, 2,, substitute the series Equation 7.33 into Equation 7.28: qt0 + ɛqt1 + ɛ 2 qt2 + + [{ ( αm 2 + βm 2 2ɛ sin ψ + ɛ 2 sin 2 ψ )} ( qt0 + ɛq t1 + ɛ 2 q t2 + )] = Γ m cos(γ m ψ). (7.34) Equating each of the coefficients of ɛ 0, ɛ 1 and ɛ 2 to zero, we have and so on. ɛ 0 : q t0 + α 2 mq t0 = Γ m cos(γ m ψ), (7.35) ɛ 1 : q t1 + α 2 mq t1 = 2β 2 mq t0 sin ψ, (7.36) ɛ 2 : q t2 + α 2 mq t2 = 2β 2 mq t1 sin ψ β 2 mq t0 sin 2 ψ, (7.37)

232 205 As far as the steady-state solutions are concerned, the periodicity of 2π can be applied. That is, q t (ɛ, ψ + 2π) = q t (ɛ, ψ). (7.38) Equations together with the condition of Equation 7.38 are sufficient to provide the required solution. The major term in Equation 7.33 is a periodic solution of the linearized equation (µ = 0, i.e., in hover). It is therefore clear that this process restricts us to finding the solutions of the nonlinear equation which bifurcate from the periodic solutions. The solution of Equation 7.28 is then obtained, which is shown in Equation The method obviously fails if the resonant frequency αm 2 takes one of the values γm, 2 (γ m ± 1) 2, (γ m ± 2) 2,, since certain terms would then be infinite. Such values of correspond to conditions of near-resonance. α m = γ m is so called a primary or main resonance. The other values of α m correspond to nonlinear resonances caused by the harmonics presented in forward flight, which can be regarded as feeding back into the linear equation (hover condition) as forcing terms [153]. Γ m q t (ψ; ɛ) = cos(γ αm 2 γm 2 m ψ) [ ] ao 1 + ɛ αm 2 (1 γ m ) sin(1 γ bo 1 m)ψ + 2 αm 2 (1 + γ m ) sin(1 + γ m)ψ [ 2 + ɛ 2 co 2 ao 2 cos(γ αm 2 γm 2 m ψ) + αm 2 (1 γ m ) sin(2 γ m)ψ ] 2 bo 2 + αm 2 (1 γ m ) sin(2 + γ m)ψ, (7.39) 2 where ao 1 = bo 1 = β2 mγ m, (7.40) αm 2 γm 2 ( ) βm 2 ao 2 = ao 1, (7.41) αm 2 (1 γ m ) ( βm 2 bo 2 = bo 1 αm 2 (1 + γ m ) ), (7.42)

233 co 2 = ao 1 [ β 2 m α 2 m (1 γ m ) βm 2 αm 2 (1 + γ m ) 1 ]. (7.43) Analysis of Time Responses Time responses can be directly calculated using the state-space form of Equation A fourth-order Runge-Kutta method is used to solve the time-varying periodic equation, which is the built-in function ode45 in Matlab. To investigate the time-varying characteristics of the resonant actuation system, the piezoelectric induced-shear tube actuator developed by Centolanza et. al [116] is selected as an example. The rotor has a blade diameter of 34 feet (MD900 class helicopter), a blade chord of 10 inches, and a nominal rotation of 400 RPM. The length of a piezoelectric tube actuator is 8 inches, outer radius R o is 0.35 inches, and inner radius R i is inches. The flap span is 6% of blade length, 12 inches, and the amplification ratio is 5.3. For the voltage driving response, an electric field of 4 kv/cm (1800 V rms ) is applied to the actuation system. The total actuator weight of pounds corresponds to a weight penalty of 1.5 % as MD900 blades weight approximately 55 pounds each [117]. If the resonant actuation system is used, the weight is approximately increased to 1.2 pounds due to the presence of multiple flaps. Time Responses without Circuitry In general, the frequency response functions between the flap deflection and control voltage cannot be directly derived, since there is no fixed stiffness of the actuation system. Peak-to-peak flap deflections are therefore used to obtain the frequency response functions. Flap deflections with several advance ratios are shown in Figure 7.11, where the frequency of the actuation system is tuned to 4/rev (26.6 Hz) by adding mass to the flap. As mentioned earlier, there are several resonant peaks in forward flight conditions (µ =0.15, 0.35) which correspond to 2, 3, 4, 5 and 6/rev frequencies. Among them, the most significant resonant frequencies are the 3, 4 and 5/rev for a resonant actuator tuned to 4/rev frequency, where a

234 207 resonant actuator indicates the actuation system without electric networks. The influence of advance ratios to the major resonance of 4/rev is not significant, while peak-to-peak flap deflections near the static condition are different Flap deflections with various flight speeds hover µ=0.15 µ=0.35 Flap deflections, degrees Excitation frequency, Hz Figure Peak-to-peak flap deflections of a resonant actuator with various flight speeds Figure 7.12 shows the variation of instantaneous frequencies and averaged frequencies. This clearly shows why the influence of advance ratios to the major resonant frequency is not significant. Averaged resonant frequencies are obtained by a constant coefficient approach, and also compared to those by Floquet s theory. Both methods yield almost the same results. Instantaneous stiffness in forward flight is either very stiff or soft. This is the reason why peak-to-peak flap deflections in forward flight are larger than those in hover, as shown in Figure Figure 7.13 shows the time history of flap deflections in forward flight (µ = 0.35) with a 4/rev cosine voltage input. There exists large 4/rev flap motion along with moderate 1/rev content as well as other harmonics, as would be expected by a perturbation analysis. Maximum flap deflection occurs at the fourth quadrant on the retreating blade. Similar trends of flap motions were reported in small-scale rotor experiments by Fulton and Ormiston [30].

235 Variation of instantaneous frequencies hover µ=0.15 µ= Frequency, Hz Averaged frequency Hover : 26.6 Hz µ=0.15 : 26.7 Hz µ=0.35 : 27.3 Hz Azimuth, degrees Figure Variations of instantaneous frequencies along the azimuth 15 Flap time response for a resonant actuator tuned to 4P 10 Flap deflections, degrees ψ=90 ψ=180 ψ= time, sec Figure Time history of flap motions of the actuation system without circuitry with 4/rev voltage excitation, µ = 0.35

236 209 Time Responses with Circuitry To verify the optimal tuning parameters for the circuitry, the frequency responses of the RAS are investigated for forward flight conditions. The optimal tuning parameters are obtained for the hover condition and then applied to the forward flight conditions since the influence of advance ratios to the major resonance of 4/rev was not significant. Peak-to-peak flap deflections with several advance ratios are shown in Figure 7.14, where the resonant frequency is mechanically tuned to the 4/rev frequency. The actuator authority is significantly increased from 1.25 degree to 4.5 degree when compared to the static value (i.e., non-resonant actuator authority). The flap deflections around the operating frequency are almost constant (flat plateau) and the bandwidth reasonably large (approximately 8 Hz). These characteristics of the RAS are also conserved in forward flight conditions (advance ratios of 0.15 and 0.35). This implies that the proposed resonant actuation system can be applied to the forward flights as well as hovering condition. 8 Flap deflections with various flight speeds hover µ=0.15 µ=0.35 Flap deflections, degrees Excitation frequency, Hz Figure Peak-to-peak flap deflections of the RAS with various flight speeds

237 210 Figure 7.15 shows the time history of flap deflections of the RAS in forward flight (µ = 0.35) with a 4/rev cosine voltage input. There exists large 4/rev flap motion. However, there are also large 3/rev and 5/rev flap motions because of the mechanical resonance and aerodynamic excitation. To properly control the resonant actuation system in forward flight, the controller that rejects the side effects (3/rev and 4/rev) is needed, which is also needed for the nominal actuation system (i.e., non-resonant actuation system) due to the time varying aerodynamic excitation, as shown in Figure Maximum flap deflection occurs at the fourth quadrant on the retreating blade, which is the same as the actuation system without circuitry. Figure Time history of flap motions of the RAS with 4/rev voltage excitation, µ = 0.35

238 Figure Time history of flap motions of the nominal actuation system with 4/rev voltage excitation, µ =

239 Vibration Reduction Within Available Actuation Authority In Chapter 5, vibratory hub loads for each flap configuration are estimated based on the assumed actuator authority. In the previous section, it is estimated that actuator authorities of the PZT tube actuator are 1.38 and 4.8 degrees for baseline actuation system (normally, similar to the static performance) and the resonant actuation systems respectively. Thus the saturation angles of the flap deflection are set to 1.38, 1.38 and 4.8 degrees for single-, dual- and multiple-flap configurations, respectively. Flap locations and control input sequences were listed in Section Section 5.3.3, which was determined based on parametric studies, for a four-bladed hingeless helicopter. Vibratory hub loads for low-speed flight are presented in Figure 7.17 showing that vibration reductions are 35%, 34% and 91% by single-, dual- and multipleflap configurations, respectively. Unlike previous cases, it is shown that multipleflap configuration outperforms the other ones when considered available actuator authorities. For high-speed flight case, same trends are observed as shown in Figure 7.18 showing that vibration reductions are 56%, 57% and 79% by single-, dual- and multiple-flap configurations, respectively. This clearly indicates that multiple-flap configuration with the resonant actuation system can significantly reduce the vibration level over the wide range of flight condition as compared to single- and dual- flap configurations.

240 213 Figure Comparison of vibratory hub loads for an advance ratio of 0.15 within the available actuator authority Figure Comparison of vibratory hub loads for an advance ratio of 0.35 within the available actuator authority

241 Experimental Realization of the RAS In Chapter 6, the RAS circuitry was realized via an equivalent electric circuit model to investigate its feasibility. Typical operating voltages of the trailing-edge flap for helicopter vibration controls are normally larger than 500 Volts [112, 114]. To realize the actuation system, especially under high voltage operations, a better method for implementing the electrical circuitry is needed. The previously proposed approach using Op-Amp based circuitry that includes synthetic inductance and negative capacitance may not be realistic under such situations. In addition to this, there is the phase variation near the operating frequency, since the mechanical resonant frequency was utilized to improve the actuation authority. These two issues should be addressed to realize the piezoelectric resonant actuation systems. In this section, a method of implementing the electric network is realized via digital signal processor (DSP), instead of the traditional analogue Op-Amp circuitry. this is more practical approach, especially under high voltage situations. An adaptive feed-forward controller PD controller is designed and implemented to track the phase variation at the vicinity of resonant frequency that is one of the operating frequencies. Through these efforts, performance of the resonant actuation system is validated experimentally Controller Design In order to evaluate the performance of resonant piezoelectric actuators with electric circuitry, the piezoelectric induced shear tube actuator [116] is considered as an example. There are a number of implementation problems associated with realizing the inductor and the negative capacitor. The piezoelectric actuators for certain application, helicopter rotor trailing edge flaps in this case, require high voltage and large inductance values. This, therefore, requires that the synthetic inductor and negative capacitor circuits constructed with high voltage operational amplifiers as discussed in Section 6.4.2, which may induce complexity and cause weight and space penalties. To circumvent this problem, in this investigation, instead of creating analogue circuitry elements, a control voltage signal is directly applied to

242 215 the piezoelectric actuators to emulate the circuitry functions. This signal function can be derived from the frequency response between the actuator stroke and voltage source, which is implemented using a digital signal processor (DSP) unit. Realization of Electric Network From Equations 6.35, 6.39 and 6.47, the voltage across the piezoelectric actuator V a is given as follows: V a V c = K Q ( ) ˆQ V c ( ) ˆqt K c = V c ( ) ˆξ2 ξ 2 ( ) ( ) ˆQ ˆqt ˆK Q K c, (7.44) V c V c which yields ( Va ) V c cty = δ 2 (ˆξ 2 /ξ 2 )(1 + 2jζ ω ω 2 ), (7.45) (1 + ξ 2 + 2jζ ω ω 2 )(rj ω + δ 2 ω 2 ) δ2 ˆξ 2 where subscript cty represents the voltage amplification ratio with a series R-L circuit and the added capacitance. The actuator stroke q t can now be expressed in terms of two transfer functions: one is the transfer function of a piezoelectric resonant actuator without a circuitry, and the other is the transfer function between the voltage across the PZT and the voltage source. q t = ( qt ) ( ) Va V a V c cty V c. (7.46) Here it should be noted that the voltage across the piezoelectric actuator, V a, should not exceed the de-poling limit. This is another constraint for adding capacitances to the piezoelectric RAS (especially for negative capacitance). To incorporate the electric network in high voltage piezoelectric resonant actuation systems, the transfer function of Equation 7.45 is implemented via the DSP system (Matlab and dspace) as a feed-forward controller (Figure 7.19).

243 216 RAS Aerodynamics Piezoelectric Actuator Flap with tuning mass V signal Va/Vc Phaser V ref DSP Figure Controller diagram of the resonant actuation system Phase Controller When a periodic input signal with period To is applied to a resonant actuation system, the output has the same period as the input but is shifted in phase. In this case, one can use a phaser that represents apparent phase shift between input and output [154]. The phaser can be viewed as the counter part of a gain which modifies the magnitude of an input signal but not its phase. The simple phaser based on the Hilbert transform can be rewritten as the PD controller form in the time domain. δ f (t) = cos φ δ δ ref f (t) + sin φ δ ω ref δ f (t). (7.47) A similar strategy was used experimentally in Reference [154]. In this paper, the phase angle φ δf (see Figure 7.20). is adaptively corrected through the feedback of the output signal

244 Figure Diagram of an adaptive phase controller based on Matlab/Simulink 217

245 Bench Top Testing A resonant actuation system using piezoelectric tube actuator developed by Centolanza et. al [116] is evaluated analytically and experimentally. The length of the piezoelectric tube is 8 inches, its diameter is 0.7 inches, and its mass is 0.5 lbs, which is originally designed for the 12 inches flap of MD900 class helicopter rotors. The trailing edge flap mass is 0.15 lbs. A bench top test is conducted to examine the phase controller and the resonant piezoelectric actuation system. Aerodynamic loads, which could be simulated by a physical spring, are not considered in this study. A diagram of the experimental set-up and equipment used in the experiment are shown in Figures 7.21 and The displacement and voltage frequency responses are measured using a Polytec laser vibrometer and a HP dynamic signal analyzer (35665A). The coupled circuit dynamics is realized by a feed-forward type controller of Equation 7.45, which is implemented together with an adaptive phaser controller via a digital signal processor (DSP, dspace ds1102). Figure Experimental set-up for the resonant actuation system

246 219 Figure Equipments used in the experiment The resonant frequency of the actuation system (actuator and flap) is 41Hz. This is tailored to the 4/rev frequency (26.6Hz) by adding a mass of 0.1 lbs to the flap. Frequency responses before and after mechanical tuning are shown in Figure 7.23, where the flap responses are normalized by their static values. Analytical and experimental results are presented in Figures 7.24 and 7.25 with two different modified electro-mechanical coupling coefficients (ˆξ = 0.5 and 0.6). Experimental results show good agreement with analytical predictions near the resonant frequency. It is observed that resonant peaks are reduced due to friction force at the flap hinge. This could be improved by implementing a thrust bearing design [108]. Actuator authority at the tuned frequency (26.6Hz) is increased about times when compared to the static deflection (which would be produced by the original actuation system without resonant tuning). There is a