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

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.

Abstract

The purpose of this thesis is to develop piezoelectric resonant actuation systemsand new active control methods utilizing the multiple trailing-edge flaps’ configu-ration for rotorcraft vibration suppression and blade loads control.

An aeroelastic model is developed for a composite rotor blade with multipletrailing-edge flaps. The rotor blade airloads are calculated using quasi-steady bladeelement aerodynamics with a free wake model for rotor inflow. A compressibleunsteady aerodynamics model is employed to accurately predict the incrementaltrailing edge flap airloads. Both the finite wing effect and actuator saturation fortrailing-edge flaps are also included in an aeroelastic analysis.

For a composite articulated rotor, a new active blade loads control method isdeveloped and tested numerically. The concept involves straightening the bladeby introducing dual trailing edge flaps. The objective function, which includesvibratory hub loads, bending moment harmonics and active flap control inputs,is minimized by an integrated optimal control/optimization process. A numericalsimulation is performed for the steady-state forward flight of an advance ratio of0.35. It is demonstrated that through straightening the rotor blade, which mimicsthe behavior of a rigid blade, both the bending moments and vibratory hub loadscan be significantly reduced by 32% and 57%, respectively.

An active vibration control method is developed and analyzed for a hingelessrotor. The concept involves deflecting each individual trailing-edge flap using acompact resonant actuation system. Each resonant actuation system could yieldhigh authority, while operating at a single frequency. Parametric studies are con-ducted to explore the finite wing effect of trailing-edge flaps and actuator satura-tion. A numerical simulation has been performed for the steady-state forward flight(µ = 0.15 ∼ 0.35). It is demonstrated that multiple trailing-edge flap configura-tion with the resonant actuation system can reduce the required trailing-edge flaphinge moments by 37% to 61% in each individual actuator compared to single-flap

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configuration for high speed flight conditions.A novel resonant actuation concept is developed to efficiently realize the heli-

copter vibration and blade loads control. The resonant actuation system (RAS) isachieved through both mechanical and electrical tailoring. With mechanical tun-ing, the resonant frequencies of the actuation system (includes the piezoelectricactuator and the related mechanical and electrical elements for actuation) can beadjusted to the required actuation frequencies. This obviously will increase theauthority of the actuation system. To further enhance controllability and robust-ness, the actuation resonant peak can be significantly broadened and flattened withelectrical 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 ro-tors. The optimal values of the electrical components are explicitly determined.An equivalent electric circuit model emulating the physical actuation system isderived and experimentally tested to investigate the initial feasibility of the piezo-electric resonant actuation system. It is demonstrated that the proposed resonantactuation system can indeed achieve both high active authority and robustness.It is shown that the actuator authority is significantly increased from 1.25 to 4.5degrees 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 toprovide better physical understanding. Design guidelines of the RAS are derivedin dimensionless forms. Feed-forward controllers are developed to realize the elec-tric network dynamics and to adapt the phase variation. The control strategy isthen implemented via a digital signal processor (DSP) system. Performance of theresonant actuation system is analyzed and verified experimentally on a full-scalepiezoelectric tube actuator for helicopter rotor control. Promising results are illus-trated that the actuator stroke is increased 2 to 3.5 times compared to its staticvalue with bandwidth of 5 to 10 Hz.

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Table of Contents

List of Figures ix

List of Tables xv

List of Symbols xvi

Acknowledgments xxvi

Chapter 1Introduction 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . 21.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Helicopter Vibration Reduction . . . . . . . . . . . . . . . . 71.2.2 Active Trailing-edge Flaps . . . . . . . . . . . . . . . . . . . 141.2.3 Smart Actuation System Development . . . . . . . . . . . . 201.2.4 Piezoelectric Networks . . . . . . . . . . . . . . . . . . . . . 261.2.5 Summary of Literature Review . . . . . . . . . . . . . . . . 30

1.3 Problem Statement and Objectives . . . . . . . . . . . . . . . . . . 321.4 Overview of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 36

Chapter 2Helicopter Model 382.1 Preliminary Background . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1.1 Vehicle Kinematics and Coordinate Systems . . . . . . . . . 392.1.2 Blade Deformed Kinematics and Coordinate Systems . . . . 412.1.3 Nondimensionalization and Ordering Scheme . . . . . . . . . 452.1.4 Variational Formulation . . . . . . . . . . . . . . . . . . . . 47

2.2 Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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2.2.1 Strain Energy of Rotor Blade . . . . . . . . . . . . . . . . . 492.2.2 Kinetic Energy of Rotor Blade . . . . . . . . . . . . . . . . . 57

2.3 Aerodynamic Blade Loads . . . . . . . . . . . . . . . . . . . . . . . 642.3.1 Quasi-steady Airloads . . . . . . . . . . . . . . . . . . . . . 682.3.2 Noncirculatory Airloads . . . . . . . . . . . . . . . . . . . . 722.3.3 Quasi-steady Aerodynamics Implementation . . . . . . . . . 73

2.4 Inflow and Free Wake Model . . . . . . . . . . . . . . . . . . . . . . 762.4.1 Linear Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . 762.4.2 Free Wake Model . . . . . . . . . . . . . . . . . . . . . . . . 77

2.5 Aeroelastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 802.5.1 Aeroelastic Response . . . . . . . . . . . . . . . . . . . . . . 802.5.2 Coupled Propulsive Trim . . . . . . . . . . . . . . . . . . . . 85

Chapter 3Trailing Edge Flap Formulation 913.1 Inertial Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 923.2 Aerodynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.2.1 Incompressible Model . . . . . . . . . . . . . . . . . . . . . . 973.2.2 Compressible Model . . . . . . . . . . . . . . . . . . . . . . 98

3.3 Active Trailing Edge Flap Control Algorithm . . . . . . . . . . . . . 1023.3.1 Feedback Form of Global Controller . . . . . . . . . . . . . . 1023.3.2 Active-Passive Hybrid Design . . . . . . . . . . . . . . . . . 106

Chapter 4Active Loads Control Using a Dual Flap Configuration 1084.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.2 Description of Analytical Models . . . . . . . . . . . . . . . . . . . 1104.3 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . 112

4.3.1 Baseline Articulated Rotor Analysis . . . . . . . . . . . . . . 1134.3.2 Rigid Blade vs. Elastic Blade . . . . . . . . . . . . . . . . . 1184.3.3 A Single Flap for Moment Reduction . . . . . . . . . . . . . 1184.3.4 Dual Flap Performance . . . . . . . . . . . . . . . . . . . . . 1194.3.5 Multicyclic Control for Moment and Vibration Reduction . . 1244.3.6 Active-Passive Hybrid Design . . . . . . . . . . . . . . . . . 126

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Chapter 5Helicopter Vibration Suppression via Multiple Trailing-EdgeFlaps with Resonant Actuation Concept 1325.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

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5.2 Description of Analytical Models . . . . . . . . . . . . . . . . . . . 1355.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 137

5.3.1 Baseline Hingeless Rotor Analysis . . . . . . . . . . . . . . . 1375.3.2 Flap Effect to Free-Wake Geometry . . . . . . . . . . . . . . 1435.3.3 Determination of Trailing-Edge Flap Locations . . . . . . . . 1475.3.4 Finite Wing Effects . . . . . . . . . . . . . . . . . . . . . . . 1495.3.5 Effectiveness of Multiple-Flap Configuration . . . . . . . . . 1515.3.6 Vibration Reduction with Multicyclic Control . . . . . . . . 156

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Chapter 6Piezoelectric Actuation System Synthesis 1646.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1656.2 Piezoelectric Actuation System Model . . . . . . . . . . . . . . . . 167

6.2.1 Piezoelectric Tube Actuator . . . . . . . . . . . . . . . . . . 1676.2.2 Inertial and Aerodynamic Loads . . . . . . . . . . . . . . . . 1716.2.3 Coupled Actuator-Flap-Circuit System . . . . . . . . . . . . 172

6.3 Mechanical Tuning and Electrical Tailoring . . . . . . . . . . . . . . 1756.3.1 Mechanical Tuning . . . . . . . . . . . . . . . . . . . . . . . 1756.3.2 Electrical Tailoring . . . . . . . . . . . . . . . . . . . . . . . 176

6.4 Equivalent Electric Circuit Model . . . . . . . . . . . . . . . . . . . 1806.4.1 Van Dyke Model . . . . . . . . . . . . . . . . . . . . . . . . 1806.4.2 Analysis and Experimental Verification . . . . . . . . . . . . 183

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Chapter 7Design and Test of Resonant Actuation Systems 1887.1 Design Guidelines of the RAS . . . . . . . . . . . . . . . . . . . . . 189

7.1.1 Resonant Actuators with R-L elements . . . . . . . . . . . . 1897.1.2 Resonant Actuation Systems with Additional Capacitance . 1957.1.3 Summary of Design Guidelines for the RAS circuitry . . . . 199

7.2 Dynamic Characteristics of the RAS in Forward Flight . . . . . . . 2027.2.1 A Perturbation Method . . . . . . . . . . . . . . . . . . . . 2027.2.2 Analysis of Time Responses . . . . . . . . . . . . . . . . . . 2067.2.3 Vibration Reduction Within Available Actuation Authority 212

7.3 Experimental Realization of the RAS . . . . . . . . . . . . . . . . . 2147.3.1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 2147.3.2 Bench Top Testing . . . . . . . . . . . . . . . . . . . . . . . 218

7.4 Power Consumption of Piezoelectric Resonant Actuation Systems . 2247.4.1 Piezoelectric resonant actuators without circuitry . . . . . . 224

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7.4.2 Piezoelectric resonant actuators with circuitry . . . . . . . . 2267.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Chapter 8Conclusions and Recommendations 2318.1 Summary of Research Efforts and Achievements . . . . . . . . . . . 2328.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . 235

Appendix ADerivation of Strain Measure 238A.1 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . 240A.2 Foreshortening Term . . . . . . . . . . . . . . . . . . . . . . . . . . 242A.3 Deformed Coordinate System . . . . . . . . . . . . . . . . . . . . . 244

Appendix BRotor System Matrices and Force Vectors 246B.1 Strain Energy of Rotor Blades . . . . . . . . . . . . . . . . . . . . . 247

B.1.1 Stiffness Coefficients of Composite Beam . . . . . . . . . . . 247B.1.2 Stiffness Matrices and Force Vectors . . . . . . . . . . . . . 248

B.2 Kinetic Energy of Rotor Blades . . . . . . . . . . . . . . . . . . . . 249B.2.1 Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 249B.2.2 Mass Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 251B.2.3 Damping Matrix . . . . . . . . . . . . . . . . . . . . . . . . 252B.2.4 Force Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Appendix CFormulations using Mathematica 254C.1 Rotor Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 255C.2 Rotor Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 259C.3 Rotor Quasi-steady Aerodynamic Loads . . . . . . . . . . . . . . . 266C.4 Trailing-Edge Flap’s Inertial Loads . . . . . . . . . . . . . . . . . . 273

Bibliography 279

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List of Figures

1.1 Various sources of a helicopter vibration . . . . . . . . . . . . . . . 21.2 Helicopter vibration variation vs. forward flight speed . . . . . . . . 31.3 Blade vortex interaction [4] . . . . . . . . . . . . . . . . . . . . . . 31.4 Aerodynamic environment in forward flight . . . . . . . . . . . . . . 41.5 Vibratory loads transmitted to fuselage [5] . . . . . . . . . . . . . . 51.6 Schematic of a dynamic vibration observer [5] . . . . . . . . . . . . 81.7 Composite tailoring of helicopter rotor blades [49] . . . . . . . . . . 81.8 Active Control of Structural Response (ACSR) systems: (a) engine

platform (b) cabin [20] . . . . . . . . . . . . . . . . . . . . . . . . . 91.9 Schematic of Higher Harmonic Control (HHC) . . . . . . . . . . . . 101.10 Schematic of Individual Blade Control (IBC) . . . . . . . . . . . . . 111.11 Schematic of Active Trailing-edge Flap (ATF) . . . . . . . . . . . . 121.12 Schematic of Active Twist Rotor (ATR) . . . . . . . . . . . . . . . 131.13 Schematic of semi-active actuators located at blade root region [4] . 131.14 Karman SH-2 Seasprite helicopter with servo-flaps . . . . . . . . . . 151.15 Rotor with on-blade elevons in the NASA Ames Wind Tunnel [30] . 171.16 Elevon motion over one rotor revolution with 4/rev voltage excita-

tion (760 RPM, µ = 0.2) [30] . . . . . . . . . . . . . . . . . . . . . . 171.17 Actuator and flap dynamic system model [83] . . . . . . . . . . . . 181.18 Blade-pitch indexing for the swashplateless rotor configuration [91] . 191.19 Single and dual flap configurations [94] . . . . . . . . . . . . . . . . 201.20 AFC being inserted at active blade assembly [3] . . . . . . . . . . . 221.21 Macro-Fiber Composite (MFC) actuator . . . . . . . . . . . . . . . 221.22 Piezoelectric bender actuators [107] . . . . . . . . . . . . . . . . . . 231.23 MD900 Helicopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.24 Double X-frame actuators for MD900 Helicopter [112] . . . . . . . . 241.25 BK117 Helicopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.26 Flap unit assembly for Eurocopter BK117/EC145 [114] . . . . . . . 25

ix

1.27 Piezoelectric tube actuator for ATF [117] . . . . . . . . . . . . . . . 261.28 Passive piezoelectric vibration absorber [121] . . . . . . . . . . . . . 271.29 Active-Passive Piezoelectric Network (APPN) [121] . . . . . . . . . 271.30 Experimental setup (a) Beam with APPN and negative capacitance,

(b) Circuit diagram of negative capacitance [125] . . . . . . . . . . 281.31 Performance comparison: voltage driving response − ·−: shunt cir-

cuit without negative capacitance; · · · : shunt circuit with negativecapacitance [125] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.32 Scaling of aerodynamic hinge moment and actuator block torque . . 331.33 Conceptual diagram of a resonant actuation system . . . . . . . . . 34

2.1 Vehicle and rotating coordinate systems . . . . . . . . . . . . . . . 392.2 Undeformed blade coordinate systems . . . . . . . . . . . . . . . . . 412.3 Deformed blade coordinate systems . . . . . . . . . . . . . . . . . . 422.4 Cross-section coordinates before and after deformation . . . . . . . 432.5 Deformations in terms of Euler angles . . . . . . . . . . . . . . . . . 442.6 Schematic of the wake, discretized in space and time [131] . . . . . 772.7 Flow chart of an aeroelastic analysis with a free wake . . . . . . . . 792.8 Finite elements for composite rotor blades . . . . . . . . . . . . . . 812.9 Finite element discretization in time . . . . . . . . . . . . . . . . . . 842.10 Vehicle configuration for propulsive trim . . . . . . . . . . . . . . . 89

3.1 Schematic of blade cross-section incorporating a trailing edge flap . 923.2 Nomenclature for a thin airfoil with a flap . . . . . . . . . . . . . . 97

4.1 Dual flap configuration for active loads control . . . . . . . . . . . . 1104.2 Conceptual sketch of dual flap mechanism for active loads control . 1114.3 Articulated blade coupled flap mode shapes . . . . . . . . . . . . . 1154.4 Articulated blade coupled lag mode shapes . . . . . . . . . . . . . . 1164.5 Articulated blade torsion mode shapes . . . . . . . . . . . . . . . . 1164.6 Control settings of articulated rotor, µ = 0.35 . . . . . . . . . . . . 1174.7 Blade tip response of articulated rotor, µ = 0.35 . . . . . . . . . . . 1174.8 Comparison of vibratory hub loads for active loads control . . . . . 1184.9 Dual flap profile for moment reduction with 1/rev control . . . . . . 1204.10 Control settings of baseline and actively controlled rotors . . . . . . 1214.11 Harmonics of flapwise bending moment along the radial station . . 1224.12 Flapwise bending moment distribution before control . . . . . . . . 1234.13 Flapwise bending moment distribution after control . . . . . . . . . 1234.14 Dual flap profile with 1 and 2/rev control inputs . . . . . . . . . . . 1244.15 Dual flap profile with 1, 2 and 3/rev control inputs . . . . . . . . . 125

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4.16 Comparison of vibration index and maximum flapwise bending mo-ment with different control inputs . . . . . . . . . . . . . . . . . . . 126

4.17 Vibratory hub shears comparison for active loads control . . . . . . 1274.18 Comparison of vibration index and maximum flapwise moment of

hybrid designed rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.19 Dual flap profiles for retrofit and hybrid designs . . . . . . . . . . . 1284.20 Control settings of baseline, retrofit and hybrid designed rotors . . . 1294.21 Blade non-structural mass and pitch-flap composite coupling stiff-

ness, K25, distribution . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.1 Various configurations of the rotor with trailing-edge flaps . . . . . 1355.2 Hingeless blade coupled flap mode shapes . . . . . . . . . . . . . . . 1395.3 Hingeless blade coupled lag mode shapes . . . . . . . . . . . . . . . 1405.4 Hingeless blade torsion mode shapes . . . . . . . . . . . . . . . . . 1405.5 Control settings of hingeless rotor, µ = 0.15 . . . . . . . . . . . . . 1415.6 Control settings of hingeless rotor, µ = 0.35 . . . . . . . . . . . . . 1415.7 Blade tip response of hingeless rotor, µ = 0.15 . . . . . . . . . . . . 1425.8 Blade tip response of hingeless rotor, µ = 0.35 . . . . . . . . . . . . 1425.9 Vertical wake geometry with the number of beam elements and the

presence of the flap for µ = 0.15 . . . . . . . . . . . . . . . . . . . . 1435.10 Vertical wake geometry with the number of beam elements and the

presence of the flap for µ = 0.35 . . . . . . . . . . . . . . . . . . . . 1445.11 Blade tip responses with the number of beam elements and the

presence of the flap for µ = 0.15 . . . . . . . . . . . . . . . . . . . . 1445.12 Blade tip responses with the number of beam elements and the

presence of the flap for µ = 0.35 . . . . . . . . . . . . . . . . . . . . 1455.13 Control settings with the number of beam elements and the presence

of the flap for µ = 0.15 . . . . . . . . . . . . . . . . . . . . . . . . . 1455.14 Control settings with the number of beam elements and the presence

of the flap for µ = 0.35 . . . . . . . . . . . . . . . . . . . . . . . . . 1465.15 Vibration reduction vs. radial locations of trailing-edge flaps . . . . 1485.16 Lift curve slope vs. aspect ratio for elliptical lift distribution . . . . 1495.17 Vibration reduction by multi-flap configuration with lift flap . . . . 1505.18 Flap deflection harmonics of multi-flap configuration with the lift

flap, advance ratio: µ = 0.15, actuator saturation: δsatf = 4o . . . . 150

5.19 Polar diagram of flap motion for single-flap configuration, advanceratio: µ = 0.15, actuator saturation: δsat

f = 2o . . . . . . . . . . . . 1525.20 Polar diagram of flap motion for dual-flap configuration, advance

ratio: µ = 0.15, actuator saturation: δsatf = 2o . . . . . . . . . . . . 153

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5.21 Flap deflections of dual-flap configuration with 4/rev control input,advance ratio: µ = 0.15, actuator saturation: δsat

f = 2o . . . . . . . 1545.22 Hinge moments in single- and dual-flap configurations with 4/rev

control input, advance ratio: µ = 0.15, actuator saturation: δsatf =

2o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.23 Comparison of vibratory hub loads, advance ratio: µ = 0.15, actu-

ator saturation: δsatf = 4o . . . . . . . . . . . . . . . . . . . . . . . 156

5.24 Flap deflections of single-flap configuration, advance ratio: µ = 0.15,actuator saturation: δsat

f = 4o . . . . . . . . . . . . . . . . . . . . . 1575.25 Flap deflections of multiple-flap configuration, advance ratio: µ =

0.15, actuator saturation: δsatf = 4o . . . . . . . . . . . . . . . . . . 158

5.26 Comparison of vibratory hub loads, advance ratio: µ = 0.35, actu-ator saturation: δsat

f = 4o . . . . . . . . . . . . . . . . . . . . . . . 1595.27 Flap deflections of single- and multiple-flap configuration, advance

ratio: µ = 0.35, actuator saturation: δsatf = 4o . . . . . . . . . . . . 160

5.28 Hinge moments in single- and multiple-flap configuration, advanceratio: µ = 0.35, actuator saturation: δsat

f = 4o . . . . . . . . . . . . 161

6.1 A piezoelectric tube actuator configuration . . . . . . . . . . . . . . 1676.2 Forces and moments acting on the trailing-edge flap . . . . . . . . . 1716.3 Schematic of the PZT tube with R-L circuit and negative capacitance1736.4 Fulcrum amplification mechanism for the PZT tube actuator . . . . 1756.5 Equivalent electric circuit model of the resonant actuation system . 1816.6 Trailing-edge flap deflections of the resonant actuation system for

Mach-scaled rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.7 Realization of the equivalent electric circuit for the resonant actua-

tion system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.8 Comparison of analytical and experimental results of the RAS for a

Mach-scaled rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7.1 A piezoelectric network with a series R-L circuit . . . . . . . . . . . 1897.2 Actuator strokes with the optimal tuning ratios and various coupling

coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1907.3 Actuator stroke and bandwidth variations with coupling coefficients 1917.4 Actuator strokes with the optimal inductance tuning and various

resistance tuning values, ξ = 0.4 . . . . . . . . . . . . . . . . . . . . 1927.5 Schematic of an equivalent mechanical system to a piezoelectric ac-

tuation system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1937.6 Electric charges with the optimal tuning ratios for various coupling

coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

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7.7 A piezoelectric network with a series R-L circuit and an additionalcapacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.8 Schematic of an equivalent mechanical system to a piezoelectric ac-tuation system with an additional capacitor . . . . . . . . . . . . . 197

7.9 Actuator strokes with additional capacitance for ξ = 0.5 . . . . . . 1987.10 Relative actuator stroke and bandwidth variations with modified

coupling coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007.11 Peak-to-peak flap deflections of a resonant actuator with various

flight speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2077.12 Variations of instantaneous frequencies along the azimuth . . . . . . 2087.13 Time history of flap motions of the actuation system without cir-

cuitry with 4/rev voltage excitation, µ = 0.35 . . . . . . . . . . . . 2087.14 Peak-to-peak flap deflections of the RAS with various flight speeds . 2097.15 Time history of flap motions of the RAS with 4/rev voltage excita-

tion, µ = 0.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2107.16 Time history of flap motions of the nominal actuation system with

4/rev voltage excitation, µ = 0.35 . . . . . . . . . . . . . . . . . . . 2117.17 Comparison of vibratory hub loads for an advance ratio of 0.15

within the available actuator authority . . . . . . . . . . . . . . . . 2137.18 Comparison of vibratory hub loads for an advance ratio of 0.35

within the available actuator authority . . . . . . . . . . . . . . . . 2137.19 Controller diagram of the resonant actuation system . . . . . . . . . 2167.20 Diagram of an adaptive phase controller based on Matlab/Simulink 2177.21 Experimental set-up for the resonant actuation system . . . . . . . 2187.22 Equipments used in the experiment . . . . . . . . . . . . . . . . . . 2197.23 Frequency responses of the PZT tube actuator before and after me-

chanical tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2207.24 Analytical predictions of a resonant actuation system: —– , tuned

system w/o the voltage signal function; − − −, RAS with ξ = 0.5;· · · , RAS with ξ = 0.6 . . . . . . . . . . . . . . . . . . . . . . . . . 221

7.25 Experimental results of a resonant actuation system: —–, tunedsystem w/o the voltage signal function; − − −, RAS with ξ = 0.5;· · · , RAS with ξ = 0.6 . . . . . . . . . . . . . . . . . . . . . . . . . 221

7.26 Time responses of flap deflection signal before/after phase controlwith 24Hz input signal . . . . . . . . . . . . . . . . . . . . . . . . . 222

7.27 Time responses of flap deflection signal before/after phase controlwith 26Hz input signal . . . . . . . . . . . . . . . . . . . . . . . . . 223

7.28 Frequency response of current and phase variation for ξ = 0.5 withdifferent resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

7.29 Frequency response of current and its phase with circuitry for ξ = 0.5226

xiii

7.30 Apparent electric power of a piezoelectric actuator with circuitry forξ = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

7.31 Actuator strokes under the optimal tuning ratios for ξ = 0.5 . . . . 228

A.1 Thin-walled cross section of rotor blade . . . . . . . . . . . . . . . . 239

xiv

List of Tables

2.1 Nondimensionalized parameters . . . . . . . . . . . . . . . . . . . . 462.2 Order of terms used in aeroelastic analysis . . . . . . . . . . . . . . 462.3 Vehicle properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.1 Order of terms for trailing edge flaps . . . . . . . . . . . . . . . . . 94

4.1 Constraints and bounds for design variables . . . . . . . . . . . . . 1124.2 Baseline articulated rotor properties for active loads control . . . . 1134.3 Trailing-edge flap properties for active loads control . . . . . . . . . 1144.4 Natural frequencies of baseline articulated rotor . . . . . . . . . . . 1144.5 Reduction of maximum bending moments maximum moments . . . 1214.6 Reduction of vibration and moment reductions with different control

inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.7 Comparisons of maximum moment, vibration index and control efforts127

5.1 Hingeless rotor and trailing-edge flap properties . . . . . . . . . . . 1385.2 Natural frequencies of baseline hingeless rotor . . . . . . . . . . . . 1395.3 Control input sequences and flap locations for multiple-flap config-

urations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.4 Peak-to-peak hinge moments in single- and dual-flap configurations

with 4/rev control input, µ = 0.15 . . . . . . . . . . . . . . . . . . . 1525.5 Peak-to-peak hinge moments in single- and multiple-flap configura-

tions, µ = 0.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.1 Piezoelectric material properties of PZT-5H for a Mach-scaled rotor 183

7.1 Design parameters for the RAS circuitry . . . . . . . . . . . . . . . 201

xv

List of Symbols

a Lift curve slope

A Cross-section area of blade

a Acceleration vector

Af Cross-section area of the flap

Aij, Bij, Dij Laminate stiffness matrices

AM Amplification ratio in the actuation system

As Surface area of the PZT electrode

b Airfoil semi-chord, b = c/2

c Blade chord

cf The flap chord

co Constant lift coefficient

Cep Total capacitance

Csp Piezoelectric material capacitance

Cp Structural modal damping of the PZT tube actuator

Cadd Added capacitance in the electric network

cD55 Elastic constant (open-circuit)

xvi

c1 Linear lift coefficient

Cl, Cd, Cm Lift, drag, and moment coefficients

Cij Reduced transformed stiffness for laminated composites

C Damping;Modal damping

CT Thrust coefficient

CN , CM , CH Unsteady normal, pitching and hinge moment coefficients of trailing-edge flap

C(k) Theodorsen’s lift deficiency function

d Offset from blade elastic axis to the flap hinge

do Constant drag coefficient

d1 Linear drag coefficient

d2 Quadratic drag coefficient

d15 Charge constant in piezoelectricity

Di Electric displacement component

e Offset between mid-chord and the flap hinge

ed Offset of blade elastic axis and aerodynamic center

eg Offset of blade elastic axis and c.g.

Ei Electric field component

fo Constant moment coefficient

f1 Linear moment coefficient

FA Centrifugal force in Appendix B

F, F Force vector

fe Excitation force in equivalent mechanical system to the piezoelec-tric actuation system

xvii

Fx, Fy, Fz Resultant blade shear forces;Nb/rev vibratory hub shear forces (longitudinal, lateral and verti-cal)

g15 Voltage constant in piezoelectricity

H Electric enthalpy density function

h Rotor height above vehicle c.g.

H1, · · · , H4 Beam element shape functions

Ht1, · · · , Ht6 Temporal element shape functions

h15 Piezoelectric constant

I Identity matrix

Ib Blade flap inertia

If Trailing-edge flap mass moment of inertia

J Objective function

k Reduced frequency;Spring stiffness

K Stiffness;Modal stiffness;Spring stiffness

Kij Composite stiffness coefficient

K22, K33, K55 Flap bending, lag bending, and torsion stiffness

K25 Pitch-flap composite coupling stiffness

Kc Electro-mechanical coupling stiffness

Kn Blade loads vector containing the flapwise curvature harmonics

KQ Inverse of the PZT tube actuator capacitance

KaQ Inverse of the added capacitance

L Inductance

xviii

Lu, Lv, Lw Blade sectional shear forces

LC , DC ,MC Circulatory lift, drag and moment per unit length

LNC ,MNC Noncirculatory lift and moment per unit length

lp Length of the PZT tube actuator

Lel Length of beam element

LLV Vibration index

Lm, Rm, Cm Inductance, resistance and capacitance in Van Dyke circuit

M Mach number;Modal damping

m Blade mass per unit length

mf The flap mass per unit length

MH The flap hinge moment

MM , CM , KM Mass, damping and stiffness of the primary system in equivalentmechanical system to the piezoelectric actuation system

mm, cm, km Mass, damping and stiffness of the secondary system in equivalentmechanical system to the piezoelectric actuation system

mk2m1,mk2

m2 Flapwise and lagwise mass moment of inertia per unit length

mo Blade reference mass per unit length

Mx,My,Mz Resultant blade sectional moments;Nb /rev vibratory hub moments (rolling, pitching and torque)

MLV Blade moment index

N Axial force

Nb Number of blades

Nel Number of beam elements

Ns Number of PZT segments

Nt Number of temporal elements

xix

n, s Contour coordinates in the cross-section of rotor blades

pi Applied traction in the piezoelectric actuator

q Generalized displacement

qR Normal mode displacement

qt Actuator stroke (tip displacement)

Q Electric charge

Qs Applied charge density

r Radial blade station;Resistance tuning ratio

r Position vector

rp Radial coordinate in the polar coordinate system

rI Trailing-edge flap chordwise c.g. from hinge

rII Flap radius of gyration about the flap hinge

R Rotor radius;Resistance

Ri Inner radius of the PZT tube

Ro Outer radius of the PZT tube

S Nondimensional time in unsteady aerodynamics of the flap

s Natural coordinate in the finite element

Sf The flap first sectional moment

Sij Mechanical strain component in piezoelectricity

t Time

T1, · · · , T14 Theodorsen coefficients

THI Rotation matrix between xHi and xI

i

TRH Rotation matrix between xRi and xH

i

xx

TuH Rotation matrix between xui and xH

i

TuR Rotation matrix between xui and xR

i

Tφ St. Venant torsion

Tω Valsov torsion

Tij Mechanical stress component in piezoelectricity

u, v, w Displacements in the undeformed coordinate system

Ux, Uy, Uz Relative wind velocity components in the deformed coordinate sys-tem

~V Total relative wind velocity

~Vb Relative wind velocity due to blade motions

~Vw Relative wind velocity due to vehicle forward speed and rotor inflow

Vx, Vy, Vz Relative wind velocity components in the undeformed coordinatesystem

Va Voltage across the PZT

Vc Control voltage in the actuation system

W Weighting matrices

x, y, z Positions in the undeformed coordinate system

xCG, yCG Longitudinal and lateral distances from vehicle c.g. to rotor hub

xe, ye Degrees of freedom in equivalent mechanical system to the piezo-electric actuation system

xfi , e

fi The flap deformed coordinate system

xFi , eF

i Vehicle coordinate system

xHi , eH

i Hub-fixed coordinate system

xIi , e

Ii Ground-fixed inertial coordinate system

xRi , eR

i Hub-rotating coordinate system

xxi

xui , e

ui Undeformed blade coordinate system

x Vector containing passive design variables

Zn Hub loads vector containing the Nb/rev harmonics

α Angle of attack

αs Longitudinal shaft tilt angle (positive nose down)

αw, βw, γw Scalar weighting parameters used in weighting maxtrices

β Prandtl-Glauert correction factor

βp Precone angle of the undeformed blade

βS11 Impermittivity in piezoelectricity

γ Lock number

γij Engineering shear strain component

δ Variational operator;Inductance tuning ratio

δf Trailing-edge deflection angle

δn Control vector of trailing-edge flap harmonics

δsat Actuator saturation angle

δU First variation of strain energy

δV First variation of kinetic energy

δW First variation of virtual work done by external forces

∆ Incremental quantity

ε Small parameter in perturbation method

ε Small parameter in ordering scheme

εij Normal strain component

εT11 Dielectric constant

xxii

ζ Coordinate in the deformed coordinate system;Nondimensionalized damping coefficient

η Coordinate in the deformed coordinate system

ηt Temporal nodal displacement

θ Coordinate in the polar coordinate system

θo Collective pitch angle (rigid blade pitch)

θtw Linear pretwist angle of a rotor blade

θtr Tail rotor collective pitch

θ1 Total blade pitch angle

θ75 Rigid blade pitch angle at 75% radial location of blade

θ1c Lateral cyclic pitch angle

θ1s Longitudinal cyclic pitch angle

θx, θy, θz Euler angles

Θ Vector of control settings

κij Curvature component

κx, κy Drees’ linear inflow parameters

Λ Axial skew angle

λ Rotor inflow

λi Induced rotor inflow

λT Warping function of blade cross-section

µ Advance ratio (non-dimensionalized forward flight speed)

ξ Coordinate in the deformed coordinate system;Generalized electro-mechanical coupling coefficient

ξ Modified electro-mechanical coupling coefficient

ρ Air density

xxiii

ρp Piezoelectric material density

ρs Material density of rotor blade

σ Rotor solidity

σij Stress component

φs Lateral shaft tilt angle (positive advancing side down)

φ Elastic twist angle of blade

φ Elastic twist angle with respect to undeformed elastic axis

Φ Modal matrix used in normal mode transformation

φs Applied electric potential

ψ Azimuth angle

Ψ Mode shape function of the PZT tube actuator

~ω Angular velocity vector

ω Nondimensionalized frequency, ω/ωE

ωD Open-circuit frequency

ωE Short-circuit frequency

ωE Short-circuit frequency with added capacitance

ωp, ωs Parallel and series frequencies in Van Dyke circuit

Ω 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

()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

() Terms and responses associated with added capacitance

Subscripts

()r Main rotor contributions

()f Trailing-edge flap contributions

()ST Static response

xxv

Acknowledgments

It is a pleasure to thank the many people who made this thesis possible. It isdifficult to overstate my gratitude to my Ph.D. supervisors, Dr. Edward C. Smithand Dr. Kon-Well Wang. With their enthusiasm, their inspiration, and theirgreat efforts to explain things clearly and simply, they helped to make researchfun for me. They provided encouragement, sound advice, good teaching, and lotsof good ideas. I would have been lost without them. I would also like to expressmy 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 engineer-ing: 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 andfun environment in which to learn and grow. I am especially grateful to Dr.Jianhua Zhang, Dr. Joseph Szefi, Dr. Phuriwat Anusonti-Inthra and Jose Palaciosat 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, patientlyteaching me how to use an apparatus.

I wish to thank my korean friends Dr. Dooyong Lee, Youngtae Ahn (especiallyfor providing a ride) and Seongkyu Lee for their friendship. I musk thank all ofthe support staff, specifically Debbie Mottin and Robin Grandy in the aerospacedepartment, and Karen Thal in the mechanical department.

I would like to express my deepest appreciation to my family for their supportand 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 forher love, patience, and the many sacrifices she has made.

xxvi

Dedication

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

xxvii

Chapter 1Introduction

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.

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 in-

ertial 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 compo-

nents 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

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]

4

A typical aerodynamic environment of the helicopter main rotor during for-

ward flight is depicted in Figure 1.4, where helicopter flight velocity adds to the

blade element rotating velocities on the advancing side (0o < ψ < 180o ), and

subtracts from it on the retreating side (180o < ψ < 360o) [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 com-

plicated 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 Nb/rev, where Nb is the number

5

of blades. It has been shown that only the Nb/rev and Nb ± 1/rev rotating frame

loads contribute to the Nb/rev fixed frame loads (see Figure 1.5). More specifically,

the Nb/rev thrust and torque are caused by corresponding Nb/rev rotating frame

loads, whereas the Nb/rev rotor drag- and side forces, and pitching- and rolling

moments are caused by the corresponding Nb ± 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 litera-

ture. 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-

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 ac-

tuation 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 piezo-

electric 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 de-

vice 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 ±3.5o (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 lim-

ited stroke. For the same trailing edge flap deflections, the actuator design specifi-

cation in multiple-flap configuration is more relaxed than that in single-flap config-

uration, because the required hinge moment is less due to the small control surface

area.

7

1.2 Literature Review

This section describes previous studies relevant to the present subject. For conve-

nience, 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 devel-

opment 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.

1.2.1 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 tai-

loring

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 de-

veloped. The most common passive devices are the dynamic vibration absorber

and the isolation mount. A typical dynamic absorber is a single degree of free-

dom 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 dy-

namic 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

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 cou-

plings 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

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 di-

rectly 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 Nb , resulting in blade pitching oscilla-

tions at three frequencies of (Nb−1) , Nb and (Nb+1) of HHC in the rotating frame

(Figure 1.9). These higher harmonic blade pitch motions can generate additional

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

11

Figure 1.10. Schematic of Individual Blade Control (IBC)

to 90 percent. However, both approaches have limitations on their practical im-

plementation. 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 main-

tainability 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 tun-

nel 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

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 1.2.2.

Figure 1.11. 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 trailing-

edge 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 con-

cepts developed to combine the advantages of both purely active and passive con-

cepts. 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 deflec-

tions 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

13

Figure 1.12. Schematic of Active Twist Rotor (ATR)

cyclic variation of the effective flap, lag, and torsion stiffness or damping varia-

tions 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 1.13. 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 methodolo-

gies, 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-

14

ergy density and high bandwidth. The on-blade smart systems open a new domain

for vibration control, aeromechanical stability augmentation, handling qualities en-

hancement and noise reduction. The detailed literature review of smart actuation

system developments will be described in Section 1.2.3.

1.2.2 Active Trailing-edge Flaps

The trailing edge flaps were used to control the 1/rev rotor primary controls (collec-

tive 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 Fig-

ure 1.14) for multicyclic vibration control was conducted,called as Kaman Multi-

cyclic 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

15

Figure 1.14. 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 con-

ducted by McDonnell Douglas Helicopter Systems. A 12-foot diameter Active Flap

Rotor (AFR) model was tested in NASA Langley 14× 22 wind tunnel. The trail-

ing 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 effec-

tiveness of plain trailing edge flaps for vibration suppression. This analysis incor-

porated 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

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 investi-

gated 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 15o are required

17

Figure 1.15. Rotor with on-blade elevons in the NASA Ames Wind Tunnel [30]

Figure 1.16. 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 0.15. 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

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 ac-

tuators. 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 sys-

tem 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 1.17. 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 aerody-

namic performance, the on-blade control concept may also perform the primary

flight control function of the rotor and thereby replace the swashplate, pitch-

links, and hydraulic actuators of the traditional helicopter control system [88].

A few studies of advanced rotor control systems for both primary flight control

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 por-

tions 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 con-

trol 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 1.18. 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 config-

uration (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 satu-

ration 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

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 vibra-

tion 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 1.19. Single and dual flap configurations [94]

1.2.3 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 indi-

vidual blade control, one can potentially obtain advantages in terms of weight and

power consumption when compared to traditional hydraulic systems. These new

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 embed-

ded 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 per-

formance 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 Froude-

scaled rotor model with banks of piezoceramic crystal elements in ±45o 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 ob-

tained 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 1o and 1.5o in the rotating environ-

ment. 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 actu-

ators (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 piezo-

electric fibers are manufactured by dicing from low-cost monolithic piezoceramic

wafers. Thus, it retains most advantageous features of the AFC with a poten-

tially 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

22

Figure 1.20. AFC being inserted at active blade assembly [3]

Figure 1.21. 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 mate-

rial 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

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

[105–108]. One of the piezoelectric bender actuators developed by Koratkar and

Chopra [107] is shown in Figure 1.22.

Figure 1.22. 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 mech-

anism. 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

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 1.23. MD900 Helicopter

Figure 1.24. 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 config-

uration 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

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 1.25. BK117 Helicopter

Figure 1.26. 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 “d15” 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

26

current piezo bender actuators or piezo stack actuators. Moreover, it has a clearer

and simpler amplification mechanism.

Figure 1.27. Piezoelectric tube actuator for ATF [117]

1.2.4 Piezoelectric Networks

Because of their electro-mechanical coupling characteristics, piezoelectric materi-

als 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 cou-

pling 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 config-

uration 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

27

Figure 1.28. 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 [122–125]. 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 1.29. Active-Passive Piezoelectric Network (APPN) [121]

28

Figure 1.30. 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 “neg-

ative capacitor” [125, 128–130]. Tang and Wang performed analysis and experi-

ment 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 signifi-

cantly 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).

29

Figure 1.31. Performance comparison: voltage driving response − · −: shunt circuitwithout negative capacitance; · · · : shunt circuit with negative capacitance [125]

30

1.2.5 Summary of Literature Review

Literature review shows that a great deal of research has been conducted in de-

signing passive and active devices for helicopter vibration reduction. Several top-

ics were addressed including helicopter vibration reduction methodologies, active

trailing-edge flaps, smart actuation system development, and piezoelectric net-

works 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. Vi-

bration 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 con-

trol 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 ap-

proach (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 reduc-

tion, 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 pri-

mary 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).

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 tar-

geted 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 ±3.5o ∼ 5o 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 pas-

sive damping, they can also be designed to amplify the actuator authority around

the tuned circuit frequency. It was demonstrated that the electro-mechanical cou-

pling 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.

32

1.3 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 de-

picted 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 mo-

ment 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 reso-

nant frequencies of the actuation system (including the piezoelectric actuator and

33

Figure 1.32. 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 he-

licopters. 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

34

Figure 1.33. 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 com-

prehensive 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 re-

liable 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

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 electri-

cal 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 sys-

tem 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 op-

erations. A method of implementing the electric networks will be real-

ized 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.

36

1.4 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 sys-

tems 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 aeroe-

lastic 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 si-

multaneous reductions of vibration and blade loads. The concept involves

straightening the blade by introducing dual trailing edge flaps in a conven-

tional articulated rotor blade. Numerical simulation is performed for the

steady-state forward flight of an advance ratio of 0.35.

5. In Chapter 5, vibration reduction using the multiple trailing-edge flap con-

figuration 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 = 0.15 ∼ 0.35) condition.

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 de-

sign 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.

Chapter 2Helicopter 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 quasi-

steady aerodynamic models will be derived. Additional aerodynamic loads gener-

ated 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.

39

2.1 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.

2.1.1 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 iner-

tial 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.

Fx

FyFz

Hz

HxHy

Ix

Iy

Iz

sa

sf

Hx

Hy

Rx

Ry

y¥V

C.G.

Figure 2.1. Vehicle and rotating coordinate systems

40

Position vectors based on each coordinate systems are defined as:

r = xI iI + yI jI + zI kI = xI1e

I1 + xI

2eI2 + xI

3eI3 = xI

i eIi = xH

i eHi = xF

i eFi (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

eH1

eH2

eH3

=

1 0 αs

0 1 −φs

−αs φs 1

eI1

eI2

eI3

, eHi = THI

ij eIj , (2.2)

in which αs is the longitudinal shaft tilt angle (positive nose down), and φs 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 eH3 , with respect to the hub-fixed nonrotating frame (see

Figure 2.1). The transformation between hub-fixed and rotating systems can be

obtained as

eRi = TRH

ij eHj , TRH =

cos ψ sin ψ 0

− sin ψ cos ψ 0

0 0 1

, (2.3)

where ψ is the azimuth angle and equals Ωt.

The undeformed blade coordinate system, superscript u, are attached to the

undeformed blade. The undeformed blade is at a precone angle of βp 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

eui = T uR

ij eRj , TuR =

cos βp 0 sin βp

0 1 0

− sin βp 0 cos βp

. (2.4)

41

Hx

Hy

Hz

Rx

Ry

x

z

pb

y

y

pb

Figure 2.2. Undeformed blade coordinate systems

Letting T uHij = T uR

ik TRHkj , the transformation between the hub-fixed nonrotating

system and the undeformed blade system is written as

eui = T uH

ij eHj , TuH =

cos βp cos ψ cos βp sin ψ sin βp

− sin ψ cos ψ 0

− sin βp cos ψ − sin bp sin ψ cos βp

. (2.5)

2.1.2 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

42

xer

Ver

her

op

p

u v

w

z

y

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, ed2 and ed

3 are aligned with the principal

cross-section axis. A point Po on the undeformed elastic axis undergoes deflections

u, v and w in the eu1 , eu

2 and eu3 directions and moves to a point P as shown in

Figure 2.3. Then the blade cross section containing P undergoes a rotation θ1

about the deformed elastic axis. The total blade pitch, θ1, is defined as

θ1 = θo + φ, (2.6)

where θo 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., θo = θo(r). For a

blade with linear a pretwist, the rigid pitch angle can be expressed by

θo = θ75 + (x− .75)θtw + θ1c cos ψ + θ1s sin ψ. (2.7)

Total blade pitch, θ1, also includes elastic twist φ. The elastic twist is defined

as

φ = φ−∫ x

0

∂w

∂x

∂2v

∂x2dx. (2.8)

The φ is the elastic twist about the undeformed elastic axis, while the φ can be

viewed as the elastic twist about the deformed elastic axis. This is a nonlinear

43

uez

3,r

v

uey

2,r

w

de

2,r

h

de

3,r

z

fqq += o1

^

oqu

ey2

,r

uez

3,r

h

V

Undeformed Cross Section

Deformed Cross Section

V

hA.C. C.G.E.A.

dege

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

edi = T du

ij euj , (2.9)

where

Tdu =

cos θx cos θy cos θy sin θz sin θy

− cos θz sin θx sin θy

− cos θx sin θz

cos θx cos θy

− sin θx sin θy sin θz

cos θy sin θx

− cos θx cos θz sin θy

+ sin θx sin θz

− cos θz sin θx

− cos θx sin θy sin θz

cos θx cos θy

. (2.10)

As shown Figure 2.5, the Euler angles, θx, θy, and θz are defined in terms of blade

44

x

yz

dsdsdxdxdw

dxdv »--= 22 )()(1

ds dsw'

dsv'dxdv

dxdw

xe

heVe

1q

Figure 2.5. Deformations in terms of Euler angles

deformations as

θx = θ1 , − θy ≈ dw

dx= w′ , θz ≈ dv

dx= v′, (2.11)

and the sequence of body fixed rotation (w.r.t. the undeformed axis) is

θz −→ θy −→ θx. (2.12)

Substituting the above relations (Equation 2.11) and simplifying to second

order terms yield the transformation between deformed and undeformed blade

positions

Tdu =

1− v′22− w′2

2v′ w′

−v′ cos θ1 − w′ sin θ1

(1− v′22

) cos θ1

−v′w′ sin θ1

(1− w′22

) sin θ1

v′ sin θ1 − w′ cos θ1

−(1− v′22

) sin θ1

−v′w′ cos θ1

(1− w′22

) cos θ1

. (2.13)

For small elastic twist angle, the trigonometric terms, cos θ1 and sin θ1, can be

45

approximated by

cos θ1 ≈ cos θo − φ sin θo , sin θ1 ≈ sin θo + φ cos θo, (2.14)

By substituting the above equation into Tdu, we can decompose of it as following

constant, linear, and nonlinear matrices

Tdu ≈ (Tdu)o + (Tdu)L + (Tdu)NL : O(ε2), (2.15)

where

(Tdu)o =

1 0 0

0 cos θo sin θo

0 − sin θo cos θo

, (2.16)

(Tdu)L =

0 v′ w′

−v′ cos θo − w′ sin θo −φ sin θo φ cos θo

v′ sin θo − w′ cos θo −φ cos θo −φ sin θo

, (2.17)

(Tdu)NL =

−12(v′2 + w′2) 0 0

φ(v′ sin θo − w′ cos θo)−1

2v′2 cos θo

−v′w′ sin θo

−12w′2 sin θo

φ(v′ cos θo + w′ sin θo)12v′2 sin θo

−v′w′ cos θo

−12w′2 cos θo

. (2.18)

2.1.3 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.

46

Table 2.1. Nondimensionalized parameters

Physical Quantity Reference ParameterLength RTime 1/ΩMass/Length mo

Velocity ΩRAcceleration Ω2RForce moΩ

2R2

Moment moΩ2R3

Energy or Work moΩ2R3

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 ε, such that ε ¿ 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

Cij O(ε−2)x, h, xcg, ycg,m, ∂

∂ψ, ∂

∂xO(1)

µ, cos ψ, sin ψ, θo, θtw, θ75, θ1c, θ1s, c1, d2 O(1)

v, w, φ, βp, η, ζ, n, αs, φs O(ε)λ, co, d1, fo O(ε)u, do, f1, λT O(ε2)ed, eg O(ε3/2)

Special notice is taken to the torsion terms k2m1

and k2m1

and the wall thickness

of blade n. These terms normally would be order ε2, 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, k2m1

, k2m2

and n have

47

been treated as O(ε) and O(ε), respectively.

The definition of the reference mass per unit length, mo, requires special con-

sideration. 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

mo =3Ib

R3=

3∫ R

0mr2dr

R3. (2.19)

Also, time derivatives are slightly different when non-dimensionalized. They are

related by ψ = Ωt, so derivatives are given by

ψ = Ωt, (2.20)

d()

dt=

d()

dψ

dψ

dt= Ω

d()

dψ= Ω()∗, (2.21)

d2()

dt2=

d2()

dψ2

d2ψ

dt2= Ω2d2()

dψ2= Ω()∗∗. (2.22)

2.1.4 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 t1 and final conditions

ate time t2, 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

t1

(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.

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 vari-

ations 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

δU =

Nb∑

b=1

(δUr + δUa) , (2.24)

δT =

Nb∑

b=1

(δTr + δTf + δTa) , (2.25)

δW =

Nb∑

b=1

(δWr + δWf + δWa) , (2.26)

where the subscript r denotes contribution from the rotor, the subscript f de-

notes 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.

49

2.2 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.

2.2.1 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

σ11

σ22

σ33

σ23

σ13

σ12

(k)

=

Q11 Q12 Q13 0 0 0

Q12 Q22 Q23 0 0 0

Q13 Q23 Q33 0 0 0

0 0 0 Q44 0 0

0 0 0 0 Q55 0

0 0 0 0 0 Q66

(k)

ε11

ε22

ε33

γ23

γ13

γ12

(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:

σξξ

σss

σnn

σsn

σξn

σξs

(k)

=

Q11 Q12 Q13 0 0 Q16

Q12 Q22 Q23 0 0 Q26

Q13 Q23 Q33 0 0 Q36

0 0 0 Q44 Q45 0

0 0 0 Q45 Q55 0

Q16 Q26 Q36 0 0 Q66

(k)

εξξ

εss

εnn

γsn

γξn

γξs

(k)

, (2.28)

where Qij denotes the transformed stiffness. These transformation relations can

50

be found in Reference [140].

For a long slender geometry, the uniaxial stress assumption (σnn = θss = θsn =

0) is valid. In addition, it can be assumed that the transverse shear strain θξn 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

σξs

(k)

=

[C11 C16

C16 C66

]

(k)

εξξ

γξs

(k)

, (2.29)

where Cij are defined as

C11 = Q11 +Q2

13Q22 − 2Q12Q13Q23 + Q212Q33

Q223 − Q22Q33

, (2.30)

C16 = Q16 +−Q13Q23Q26 + Q12Q26Q33 + Q13Q22Q36 − Q12Q23Q36

Q223 − Q22Q33

, (2.31)

C66 = Q66 +Q2

26Q33 − 2Q23Q26Q36 + Q22Q236

Q223 − Q22Q33

. (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

Nξξ

Nξs

Mξξ

Mξs

=

A11 A16 B11 B16

A16 A66 B16 B66

B11 B16 D11 D16

B16 B66 D16 D66

εoξξ

γoξs

kξξ

kξs

, (2.33)

where

(Aij, Bij, Dij) =

∫ n/2

n/2

C(k)ij (1, n, n2) dn. (2.34)

51

Then the first variation of strain energy of the blade can be expressed by

δUr =

∫ R

0

∮

s

(Nξξδε

oξξ + Nξsδγ

oξs + Mξξδkξξ + Mξsδkξs

)ds dx, (2.35)

where stress resultant Nij, moment resultant Mij, axial strain εoξξ, and in-plane

shear strain γoξs can be decomposed into linear and nonlinear terms

Nξξ = NLξξ + NNL

ξξ , Mξξ = MLξξ + MNL

ξξ , (2.36)

Nξs = NLξs + NNL

ξs , Mξs = MLξs + MNL

ξs . (2.37)

The strain energy can be also decomposed as

δUr = δULr + δUNL

r , (2.38)

where

δULr =

∫ R

0

∮

s

(NL

ξξδεLξξ + NL

ξsδγLξs + ML

ξξδkξξ + MLξsδkξs

)ds dx, (2.39)

δUNLr =

∫ R

0

∮

s

(NNL

ξξ δεLξξ + NNL

ξs δγLξs + MNL

ξξ δkξξ + MNLξs δkξs

)ds dx

+

∫ R

0

∮

s

(Nξξδε

NLξξ + Nξsδγ

NLξs

)ds dx. (2.40)

The first variation of linear strains and curvatures (Equations A.33 and A.34)

is given by

δεLξξ = δu′e − yo δv′′ − zo δw′′ − λT δφ′′, (2.41)

δγLξs = γt

ξs δφ′, (2.42)

δkξξ = cos θ δw′′ − sin θ δv′′ + q δφ′′, (2.43)

δkξs = 2 δφ′. (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

52

rewritten by

δULr =

∫ R

0

(Nxδu

′e −Myδw

′′ + Mzδv′′ + Tωδφ′′ + Tφδφ

′)

dx, (2.45)

where

Nx =

∮NL

ξξ ds, (2.46)

My = −∮ (−zo NL

ξξ + MLξξ cos θ

)ds, (2.47)

Mz =

∮ (−yo NLξξ −ML

ξξ sin θ)ds, (2.48)

Tω =

∮ (q ML

ξξ − λT NLξξ

)ds, (2.49)

Tφ =

∮ (2 ML

ξs + NLξs γt

ξs

)ds, (2.50)

in which the direction of moments is based on the undeformed coordinate system

eui and the direction of torque and force is based on the deformed coordinate system

edi . Nx denotes the axial force resultant, My is the flapwise bending moment and

Mz represents the lagwise bending moment. Tω is the torque due to warping, which

is so called Valsov torsion, and Tφ is the St. Venant torque.

These force, moment and torque can be expressed in terms of degrees of free-

dom, such as ue, v, w and φ. The constitutive equation of composite beam for

linear strain can be written as follows:

Nx

−My

Mz

Tω

Tφ

=

K11 K12 K13 K14 K15

K12 K22 K23 K24 K25

K13 K23 K33 K34 K35

K14 K24 K34 K44 K45

K15 K25 K35 K45 K55

u′ew′′

v′′

φ′′

φ′

, (2.51)

where Kij are defined as

K12 = K12 cos θo + K13 sin θo, (2.52)

K13 = K13 cos θo −K12 sin θo, (2.53)

53

K22 = K22 cos2 θo + K33 sin2 θo + K23 sin 2θo, (2.54)

K23 = (K33 −K22) sin θo cos θo + K23 cos 2θo, (2.55)

K24 = K24 cos θo + K34 sin θo, (2.56)

K25 = K25 cos θo + K35 cos θo, (2.57)

K33 = K33 cos2 θo + K22 sin2 θo −K23 sin 2θo, (2.58)

K34 = K34 cos θo −K24 sin θo, (2.59)

K35 = K35 cos θo −K25 sin θo. (2.60)

The other Kij terms are defined to equal to Kij which denotes Kij of which the

rigid pitch angle θo is set to zero, i.e.,

Kij = Kij|θo→0. (2.61)

The Kij are defined in Appendix B.1.1. The stiffness coefficients K11, K22, K33,

K44, and K55 are related to in-plane stiffness EA, flap bending rigidity EIy, lag

bending rigidity EIz, warping rigidity EIωω, and torsional rigidity GJ in the

isotropic beam, respectively. The coupling terms K25 and K35 denote the flap-

torsion coupling and the lag-torsion coupling stiffness, respectively. The coupling

term K13 is also related to EAeAof the isotropic beam. Definite of EA, GJ , and

EAeAetc. can be found in Reference [76].

It is assumed that the other coupling terms except K13, K15 K25, and K35

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 flap-

torsion, or the lag-torsion coupling. The terms related to warping,∮

ζ A11ds and∮ζη A11ds, are therefore discarded. For such a configuration, the stiffness matrices

can be rewritten, in terms of shape function, as follows:

[Kuu] =

∫K11 HT

u

′Hu

′ dx, (2.62)

[Kuw] =

∫K13 sin θo HT

u

′Hw

′′ dx, (2.63)

[Kuv] =

∫K13 cos θo HT

u

′Hv

′′ dx, (2.64)

54

[Kup] =

∫K15 HT

u

′Hp

′ dx, (2.65)

[Kww] =

∫(K22 cos2 θo + K33 sin2 θo) HT

w

′′Hw

′′ dx, (2.66)

[Kwv] =

∫(K33 −K22) cos θo sin θo HT

w

′′Hv

′′ dx, (2.67)

[Kwp] =

∫(K25 cos θo + K35 sin θo) HT

w

′′Hp

′ dx, (2.68)

[Kvv] =

∫(K33 cos2 θo + K22 sin2 θo) HT

v

′′Hv

′′ dx, (2.69)

[Kvp] =

∫(K35 cos θo −K25 sin θo) HT

v

′′Hp

′ dx, (2.70)

[Kpp] =

∫(K44 HT

p

′′Hp

′′ + K55 HTp

′Hp

′) dx. (2.71)

The first variations of nonlinear strains (Equations A.33 and A.34) are given

by

δεNLξξ = (η2 + ζ2)θ′o

(δφ′ + w′δv′′ + v′′δw′

)

+ (η2 + ζ2)(φ′δφ′ + w′v′′δφ′ + w′φ′δv′′ + v′′φ′δw′

)

+ (zov′′ − yow

′′)δφ′ + φ′(zoδv′′ − yoδw

′′), (2.72)

δγNLξs = γt

ξsv′′δw′ + γt

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

δUNLr =

∫ R

0

(NNL

x δu′e −MNLy δw′′ + MNL

z δv′′ + TNLω δφ′′ + TNL

φ δφ′

+ NNLw′ δw′ + NNL

φδφ

)dx, (2.74)

where

NNLx =

∮NNL

ξξ ds, (2.75)

55

MNLy = −

∮ (−zo NNL

ξξ + MNLξξ cos θ − yo Nξξ φ

)ds, (2.76)

MNLz =

∮ −yo NNLξξ −MNL

ξξ sin θ

+ Nξξ

(zoφ + (ζ2 + η2)(w′θ′o + w′φ′ + w′2v′′)

)+ Nξsγ

tξsw

′

ds, (2.77)

TNLω =

∮ (q MNL

ξξ − λT NNLξξ

)ds, (2.78)

TNLφ =

∮ 2 MNL

ξs + NNLξs γt

ξs + Nξξ(η2 + ζ2)(φ′ + θ′o + w′v′′)

ds, (2.79)

NNLw′ =

∮ Nξξ(η

2 + ζ2)(φ′v′′ + θ′ov′′ + w′v′′2) + Nξsγ

tξsv

′′

ds, (2.80)

NNLφ =

∮Nξξ (yow

′′ + zov′′) ds. (2.81)

Note that some linear terms are appeared in Equations 2.75 – 2.81, 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(ε2) based on Table 2.2 is applied to reduce the complexity. The laminate

stiffness Bij and Dij 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.

NNLx = Ak2

11(1

2φ′2 + θ′ov

′′w′) + Ak211φ

′θ′o

+ K13(w′′φ cos θo − v′′φ sin θo) + K15v

′′w′, (2.82)

−MNLy = (K33 − K22)(v

′′φ cos 2θo + w′′φ sin 2θo) + K13 cos θou′eφ

+ K25(v′′w′ cos θo − φφ′ sin θo) + K35(v

′′w′ sin θo + φφ′ cos θo)

+ (AB1 sin θo + AB2 cos θo)φ′θ′o, (2.83)

MNLz = (K33 − K22)(w

′′φ cos 2θo − v′′φ sin 2θo)− K13 sin θou′eφ

+ K55w′φ′ + Ak2

11θ′ou′ew

′ + K15u′ew

′

+ K25

(w′′w′ − φφ′) cos θo − 2w′v′′ sin θo

+ K35

(w′′w′ − φφ′) sin θo + 2w′v′′ cos θo

+ (AB1 cos θo − AB2 sin θo)φ′θ′o, (2.84)

TNLω = K44w

′v′′, (2.85)

56

TNLφ = K35(w

′′φ cos θo − v′′φ sin θo)− K25(w′′φ sin θo + v′′φ cos θo)

+ K55w′v′′ + Ak2

11u′eφ′ + Ak2

11u′eθ′o, (2.86)

NNLw′ = K25(v

′′w′′ cos θo − v′′2 sin θo) + K35(v′′w′′ sin θo + v′′2 cos θo)

+ K55v′′φ′ + Ak2

11θ′ou′ev′′ + K15u

′ev′′, (2.87)

NNLφ = (K33 − K22)

v′′w′′ cos 2θo + (w′′2 − v′′2) sin θo cos θo

+ K13(w′′u′e cos θo − v′′u′e sin θo)

+ K35(w′′φ′ cos θo − v′′φ′ sin θo)− K25(w

′′φ′ sin θo + v′′φ′ cos θo),(2.88)

where

Ak211 =

∮A11 (η2 + ζ2) ds, (2.89)

AB1 = −∮

A11 η(η2 + ζ2) ds, (2.90)

AB2 = −∮

A11 ζ(η2 + ζ2) ds, (2.91)

the terms Kij are defined as Kij of which the stretching bending coupling Bij and

bending rigidity Dij of thin wall are set to zero, i.e.,

Kij = Kij|Bij ,Dij→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

[Kup] =

∫Ak2

11 HTu

′H ′

pdx, (2.93)

[Kwp] =

∫(AB1 sin θo + AB2 cos θo)θ

′o HT

w

′′H ′

p dx, (2.94)

[Kvp] =

∫(AB1 cos θo − AB2 sin θo)θ

′o HT

v

′′H ′

p dx, (2.95)

and the nonlinear forces in terms of shape functions are given by

Fu =−∫

NNLx HT

u

′dx, (2.96)

57

Fw =−∫ (

−MNLy HT

w

′′+ NNL

w′ HTw

′)dx, (2.97)

Fv =−∫

MNLz HT

v

′′dx, (2.98)

Fp =−∫ (

TNLω HT

p

′′+ TNL

φ HTp

′+ NN

φHT

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.1.2.

2.2.2 Kinetic Energy of Rotor Blade

The kinetic energy of the rotor blade, δTr, 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 = ~Vb + ~Vf , (2.100)

where ~Vb is the velocity of the blade relative to the hub and ~Vf 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 + ue − λT φ′)ed1 + ηed

2 + ζed3,

= (x + u− λT φ′)eu1 + veu

2 + weu3 + ηed

2 + ζed3, (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

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 + due),

≈

1− 1

2(v′2 + w′2)

(ds + due), (2.102)

and

du ≈ due − 1

2(v′2 + w′2)dx, (2.103)

u = ue − 1

2

∫ x

0

(v′2 + w′2)dx, (2.104)

where ue 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

sed1 =

x− 1

2

∫ x

0

(v′2 + w′2)dx

eu

1 + veu2 + weu

3 . (2.105)

Referring to Equation 2.14, the relationship between the deformed and undeformed

coordinate system is given by

ed2 ≈ −(v′ cos θ1 + w′ sin θ1) eu

1 + cos θ1eu2 + sin θ1e

u3 , (2.106)

ed3 ≈ (v′ sin θ1 − w′ cos θ1) eu

1 − sin θ1eu2 + cos θ1e

u3 . (2.107)

By substituting Equations 2.106 and 2.107 into Equation 2.101, the deformed

position vector can be rewritten, in the undeformed coordinate system, as follows:

r = x1eu1 + y1e

u2 + z2e

u3 , (2.108)

where

x1 = x + ue − 1

2

∫ x

0

(v′2 + w′2)dx− λT φ′ − v′y − w′z, (2.109)

y1 = v + y, z1 = w + z, (2.110)

59

y = η cos θ1 − ζ sin θ1 , z = η sin θ1 + ζ cos θ1. (2.111)

Then the velocity of an arbitrary point on the blade can be calculated as

~Vb =∂r

∂t+ ~ω × r , ~ω = Ω eH

3 ≈ Ω(βpeu1 + eu

3), (2.112)

and the velocity components are given by

~Vb = Vbxeu1 + Vbye

u2 + Vbze

u3 , (2.113)

where

Vbx = x1 − Ωy1, (2.114)

Vby = y1 + Ωx1 − Ωz1βp, (2.115)

Vbz = z1 + Ωy1βp. (2.116)

The derivatives of the arbitrary point on the blade with respect to time are

given as

x1 = ue −∫ x

0

(v′v′ + w′w′)dx− λT φ′

− (v′ + w′θ1)y − (w′ − v′θ1)z, (2.117)

y1 = v − θ1z, (2.118)

z1 = w + θ1y, (2.119)

and the kinetic energy variation can be expressed by

δTr =

∫ R

0

∫

A

ρ~V δ~V dAdx. (2.120)

Substituting Equations 2.114 to 2.116 into Equation 2.120 and integrating by parts

yield

δTr =δTr

moΩ2R3=

∫ 1

0

∫

A

ρs (Tx1δx1 + Ty1δy1 + Tz1δz1) dAdx, (2.121)

60

where ρs is the material density of structure and

Tx1 =−x∗∗1 + 2y∗1 + x1 − z1βp, (2.122)

Ty1 =−y∗∗1 − 2x∗1 + y1 + 2z∗1βp, (2.123)

Tz1 =−z∗∗1 − 2y∗1βp − x1βp + z1β2p , (2.124)

in which

x∗∗1 = u∗∗e −∫ x

0

(v∗′v∗′ + v′v∗∗′ + w∗′w∗′ + w′w∗∗′) dx− λT φ∗∗′

− (v∗∗′ + w′θ∗∗1 + 2θ∗1w

∗′) y − (w∗∗′ − v′θ∗∗1 − 2θ∗1v

∗′) z, (2.125)

y∗∗1 = v∗∗ − θ∗∗1 z − θ∗12y, (2.126)

z∗∗1 = w∗∗ + θ∗∗1 y − θ∗12z. (2.127)

The variations of the deformed position vector components, x1, y1 and z1, are

given by

δx1 = δue − λT δφ′ − yδv′ − v′δy − zδw′ − w′δz

−∫ x

0

(v′δv′ + w′δw′)dx, (2.128)

δy1 = δv + δy, (2.129)

δz1 = δw + δz, (2.130)

where

δy = −z δφ , δz = y δφ. (2.131)

Now the variation of kinetic energy (Equation 2.121) can be rewritten by

δTr =

∫ 1

0

(V I

x δu + V Iy δv + V I

z δw + M Iz δv′ −M I

y δw′ + M Ixδφ

)dx, (2.132)

where

V Ix = m

[x + ue − 1

2

∫ x

0

(v′2 + w′2)dx− u∗∗e

61

+

∫ x

0

(v∗′v∗′ + v′v∗∗′ + w∗′w∗′ + w′w∗∗′) dx

+ 2v∗ − wβp − eg cos θo

(v′ − v∗∗′ + 2θ∗oφ + θ∗o

2v′ − θ∗∗o w′ − 2θ∗ow∗′)

− eg sin θo

(2θ∗1 + βp + w′ − w∗∗′ + θ∗∗o v′ + θ∗o

2w′ + 2θ∗ov∗′)] , (2.133)

V Iy = m

[v − v∗∗ + 2w∗βp − 2u∗e + 2

∫ x

0

(v′v∗′ + w′w∗′)dx

+ eg cos θo

(1 + 2v∗′ + 2φ∗θ∗o + θ∗o

2 + θ∗∗o φ + 2θ∗oβp + 2θ∗ow′)

+ eg sin θo

(θ∗∗1 − φ + 2w∗′ − θ∗o

2φ− 2θ∗ov′)]

, (2.134)

V Iz = m

[−w∗∗ − (x + 2v∗)βp − eg cos θo(θ

∗∗1 − θ∗o

2φ)

+ eg sin θo

(2θ∗oφ

∗ + θ∗o2 + θ∗∗o φ + 2θ∗oβp

)], (2.135)

M Iz = m

[eg

(−x− 2v∗) cos θo + xφ sin θo

+ (k2m1 sin2 θo + k2

m2 cos2 θo)(2θ∗oφ + v′ − v∗∗′ + θ∗o

2v′ − θ∗∗o w′ − 2θ∗ow∗′)

− (k2m2 sin2 θo + k2

m1 cos2 θo)2θ∗oφ (2.136)

+ cos θo sin θo(k2m2 − k2

m1)(w′ − w∗∗′ + 2θ∗1 + βp + θ∗∗o v′ + θ∗o

2w′ + 2θ∗ov∗′)] ,

−M Iy = m

[eg

(−x− 2v∗) sin θo − xφ cos θo

+ (cos2 θok2m1 + sin2 θok

2m2)βp

+ sin θo cos θo(k2m2 − k2

m1)(v′ − v∗∗′ + 4θ∗oφ + θ∗o

2v′ − θ∗∗o w′ − 2θ∗ow∗′)

+ (k2m1 cos2 θo + k2

m2 sin2 θo) (2.137)(w′ + 2θ∗1 − w∗∗′ + θ∗∗o v′ + θ∗o

2w′ + 2θ∗ov∗′)] ,

M Ix = m

[eg sin θo (−v + v∗∗ + xv′)− eg cos θo

(w∗∗ + xβp + xw′+2v∗βp

)

+ sin θo cos θo(k2m1 − k2

m2)(1 + 2v∗′+2θ∗1w′) (2.138)

− k2mθ∗∗1 + φ(k2

m1 − k2m2) cos 2θo − 2w∗′(k2

m1 cos2 θo + k2m2 sin2 θo)

],

where mk2m1 and mk2

m2 represent the flapwise and lagwise mass moments of inertia

per unit length, respectively, and are given by

m(eg, k2m1, k

2m2) =

∫

A

ρs(η, ζ2, η2)dA, (2.139)

62

and

mk2m = mk2

m1 + mk2m2. (2.140)

By substituting Equation 2.104 into Equation 2.132, the variation of kinetic

energy can be also expressed by

δTr =

∫ 1

0

(V I

x δue + V Iy δv + V I

z δw + M Iz δv′ −M I

y δw′ + M Ixδφ

)dx, (2.141)

where

V Ix = V I

x + m1

2

∫ x

0

(v′2 + w′2)dξ ≈ V Ix , (2.142)

V Iy = V I

y −m(x + 2v∗)v′ + v′′∫ 1

x

m(ξ + 2v∗)dξ (2.143)

' V Iy −mv − 2

∫ x

0

m v′v∗′dξ, (2.144)

V Iz = V I

z −m(x + 2v∗)w′ + w′′∫ 1

x

m(ξ + 2v∗)dξ (2.145)

' V Iz −mw − 2

∫ x

0

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 ue.

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

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

(v′δv′ + w′δw′)dξ =

[∫ 1

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.

64

2.3 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 de-

formed 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 = −~Vw + ~Vb, (2.148)

where ~Vw is the wind velocity with contributions from the vehicle forward speed

and the rotor inflow and ~Vb 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

~Vw = µΩR eH1 − λΩR eH

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 αs + λi ≈ µαs + λi. (2.150)

These velocities, ~Vw and ~Vb, must be transformed through the blade precone,

βp, 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 pre-

cone. From Equation 2.5, the transformation matrix between the hub coordinates,

eHi , and the blade coordinates that are the undeformed coordinates, eu

i , is given

by

eui = T uH

ij eHj . (2.151)

The wind velocity on the undeformed coordinate systems with a small angle

65

assumption for precone, βp, is expressed as

~Vw = Vwieu

i

= (µΩR cos ψ − λΩRβp) eu1 − µΩR sin ψ eu

2

− (µΩR cos ψβp + λΩR) eu3 . (2.152)

The blade velocity, ~Vb, with respect to the rotating undeformed frame can be

written as

~Vb =∂r

∂t+ ~ω × r = Vbi eu

i . (2.153)

From Equations 2.112-2.119, the blade velocity components are given by

~Vb1 = u− λT φ′ − (v′ + w′θ1)y − (w′ − v′θ1)z − Ω(v + y), (2.154)

~Vb2 = v − θ1z + Ω(x + u− λT φ′ − v′y − w′z)− Ω(w + z)βp, (2.155)

~Vb3 = w + θ1y + Ω(v + y)βp, (2.156)

where

u = ue − 1

2

∫ x

0

(v′2 + w′2)dx , u = ue −∫ x

0

(v′v′ + w′w′)dx, (2.157)

φ′ = φ′ + w′v′′ , θ1 = θo + φ, (2.158)

y = η cos θ1 − ζ sin θ1 , z = η sin θ1 + ζ cos θ1. (2.159)

For quasi-steady aerodynamics, the rotor blade aerodynamic loads are calcu-

lated using a blade section strip analysis based on the angle of attack at the three-

quarter chord location. In the cross section, this point is at η = ηr and ζ = 0,

simplifying the blade velocity expressions. Then the total velocity, ~V = ~Vb − ~Vw,

can be written as

~V = Vi eui , (2.160)

66

in which

Vx = u− λT˙φ′ − Ω(v + ηr cos θ1) + λΩRβp − µΩR cos ψ

− ηr(θ1w′ + v′) cos θ1 − ηr(w

′ − θ1v′) sin θ1, (2.161)

Vy = v + Ω(u + x) + µΩ sin ψ − ηrθ1 sin θ1 − βpΩ(w + ηr sin θ1)

− Ωηr(v′ cos θ1 + w′ sin θ1), (2.162)

Vz = w + λΩR + µΩRβp cos ψ + Ω(v + ηr cos θ1)βp + ηrθ1 cos θ1. (2.163)

The blade section loads are calculated using the resultant velocity and aerody-

namic 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 2.13.

edi = T du

ij euj , ~V = Vi eu

i = ViTduij

−1ed

j = Uj edj . (2.164)

Thus the velocity components, Uj, using the transformation matrix property,

T duij−1

= T duji , are given by

Vi T duji = T du

ji Vi = Uj. (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 transfor-

mation, θ1 is replaced with θo + φ to modify cos θ1 and sin θ1, and assuming φ is a

small angle.

sin θ1 = sin(θo + φ) ≈ sin θo + φ cos θo, (2.166)

cos θ1 = cos(θo + φ) ≈ cos θo − φ sin θo. (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

67

Ux

ΩR= u∗ − v + λ(βp + w′) + µv′ sin ψ + +xv′ + v∗v′ + w∗w′

+ µ cos ψ

−1 + βpw

′ +1

2(v′2 + w′2)

− ηr(1 + v∗′) cos θo + ηr(φ− w∗′) sin θo, (2.168)

Uy

ΩR= cos θo

u + x + v∗ + (λ + w∗)φ− wβp + vv′ − λβpv

′ − 1

2xv′2

+ µ sin ψ(1− v′2

2) + µ cos ψ(βpφ + w′φ + v′)

+ sin θo

λ(1− βpw

′ − 1

2w′2) + w∗ − x(φ + v′w′)− v∗φ + vβp + vw′

+ µ cos ψ(βp − φv′ + w′)− µ sin ψ(φ + v′w′)

, (2.169)

Uz

ΩR= ηr(φ

∗ + θ∗o + βp + w′)

+ sin θo

−u− x− v∗ + λ(βpv

′ − φ)− w∗φ + wβp − vv′ +1

2xv′2

+ µ sin ψ(−1 +v′2

2)− µ cos ψ(βpφ + w′φ + v′)

+ cos θo

λ + w∗ − xφ− v∗φ + vβp + vw′ − λ(βpw

′ +w′2

2)− xv′w′

+ µ cos ψ(βp − v′φ + w′)− µ sin ψ(φ− v′w′)

, (2.170)

where Ux, Uy, and Uz denote the radial, tangential, and perpendicular velocities,

respectively. These velocities are also referred to as UR, UT , and UP . The displace-

ments, 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.

68

2.3.1 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,

edi , can be written as

LC =1

2ρU2cCl, (2.171)

DC =1

2ρU2cCd, (2.172)

MC =1

2ρU2c2Cm, (2.173)

where U is the incident velocity and Cl, Cd, Cm 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

Cl = co + clα, (2.174)

Cd = do + d1|α|+ d2α2, (2.175)

Cm = fo + f1α = cmac + f1α, (2.176)

in which co is the lift coefficient when the airfoil is at a zero angle of attack, c1 is the

lift curve slope, often given by Clα or Cnα , and do is the profile drag of the blades

due to viscous effects and the nonlinear term, d2, causes a drag rise with large

angles of attack. fo or cmac is the zero angle pitching moment coefficient about

aerodynamic center and f1 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, cl, as

cl =clM=∞

β, β =

√1−M2, (2.177)

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 UT 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

LCu =−DC sin Λ, (2.178)

LCv = LC sin α−DC cos α, (2.179)

LCw = LC cos α + DC sin α, (2.180)

MCφ

= MC − ed LCw , (2.181)

where LCi are the external loads along the deformed frame, ed

i , MCφ

is the pitching

moment about the deformed elastic axis, Λ is the axial skew angle due to the radial

component of incident velocity, UR, acting on the blade, and ed is the chordwise

offset of the aerodynamic center behind the elastic axis.

The equations are non-dimensionalized by the Lock number, γ, the blade iner-

tia, Ib, and the standard non-dimensionalization factor Ω and R. The Lock number

and uniform blade inertia are given by

γ =ρacR4

Ib

, Ib =moR

3

3, (2.182)

where a is the lift curve slope, Clα . All forces, moments, and velocities are non-

dimensionalized using the scheme described in Table 2.1. The section lift and drag

forces, LC and DC , are non-dimensionalized as follows:

1

2ρcU2 =

1

2

(ρR2

mo

) ( c

R

) (U2

Ω2R2

)=

γ

6a

(U

ΩR

)2

=γ

6aU

2. (2.183)

70

The airloads, Equations 2.178-2.181, are nondimensionalized by Equation 2.183.

Then,

LC

u =γU

2

6a(−Cd sin Λ) , (2.184)

LC

v =γU

2

6a(Cl sin α− Cd cos α) , (2.185)

LC

w =γU

2

6a(Cl cos α + Cd sin α) , (2.186)

MC

φ =γU

2

6a

( c

RCm − edL

C

w

). (2.187)

Substituting the expression for Cl,Cd, and Cm from Equations 2.174–2.176, and

using the following approximations

sin α ≈ α , cos α ≈ 1 , U ≈ UT , α ≈ −UP

UT

, sin Λ ≈ UR

UT

, (2.188)

yield

LC

u =γ

6a

doURUT

, (2.189)

LC

v =γ

6a

−doU

2

T − (coUP − d1|UP |)UT + (c1 − d2)U2

P

, (2.190)

LC

w =γ

6a

coU

2

T − (c1 + do)UT UP + d1|UP |UP

, (2.191)

MC

φ =γ

6a

c

R

(fo(U

2

T + U2

P )− f1UP UT

)− edL

C

w

. (2.192)

The nondimensionalized aerodynamic forces and pitching moment acting on a

blade section can be expressed by the following vector form.

LC = LC

i edi , MC = M

C

φ ed1. (2.193)

It is consistent with the velocity in the deformed frame, ~V = Ui edi . The aerody-

namic forces in the undeformed frame, eui , are obtained by using the orthogonal

coordinate transformation. The transformation matrix between the deformed and

undeformed coordinate systems, edi = T du

ij euj , has already developed in Equation

71

2.13 on Page 44. The aerodynamics forces in the undeformed frame are given by

LC = LC

i edi = L

C

i T duij eu

j = LCi eu

i , MC = MC

φ ed1 ≈ MC

φeu

1 , (2.194)

LCi = L

C

j T duji = T du

ij LC

j , MCφ≈ M

C

φ , (2.195)

where the pitching moment, MC

φ , is assumed small, so that higher order terms can

be neglected as follows:

MC

φ ed1 = M

C

φ (1− v′2

2− w′2

2)eu

1 + MC

φ v′ eu2 + M

C

φ w′ eu3 ,

≈ MC

φ eu1 . (2.196)

The virtual work done by aerodynamic forces can be written by

δWCaero =

∫ 1

0

(LC

u δu + LCv δv + LC

wδw + MCφ

δφ)

dx. (2.197)

This variational form should be expressed in terms of the elastic axis displacement,

ue, 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.

δWCaero =

∫ 1

0

(LC

u δue + LCv δv + LC

wδw + MCφ

δφ)

dx, (2.198)

where

LCu ≈ LC

u , (2.199)

LCv = LC

v − LCu v′ + v′′

∫ 1

x

LCu dξ, (2.200)

LCw = LC

w − LCu w′ + w′′

∫ 1

x

LCu dξ, (2.201)

and another form is available for finite element formulation, which is called the

72

weak form. The weak form of Equation 2.198 is given by

δWCaero =

∫ 1

0

LC

u δue + LCv δv + LC

wδw + MCφ

δφ

+ (v′δv′ + w′δw′)∫ 1

x

LCu dξ

dx, (2.202)

where the underline term could be neglected, because it is the higher order term,

LCu× the foreshortening term. Thus Equation 2.202 can be simplified as:

δWCaero ≈

∫ 1

0

(LC

u δue + LCv δv + LC

wδw + MCφ

δφ)

dx. (2.203)

2.3.2 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,

LNCw = L2 + L3 = ρπb2(h− ahbα) + ρπb2Uα), (2.204)

MNCφ

= ahbL2 − (1

2− ah)bL3 + Ma , Ma = −ρπb4

8α, (2.205)

in which L2 is a lift force with center of pressure at the mid-chord, of amount

equal to the apparent mass, ρπb2, times the vertical acceleration at the mid-chord

point. L3 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, ρπb2,

times Uα. Ma is a nose-down moment equal to the apparent moment of inertia,

73

ρπb2(b2/8), times the angular acceleration, α.

U = ΩR(x + µ sin ψ) , ahb = −(ed +c

4) , h = −w (2.206)

α = θ1 = θo + φ , b =c

2(2.207)

where U denotes the free stream tangential velocity, ahb is the distance from mid-

chord to the elastic axis (positive aft), h is the plunge acceleration (positive down),

α is the pitch angle (positive nose up) and b is the airfoil semi-chord. The noncir-

culatory 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

δWNCaero =

∫ 1

0

(LNCw δw + MNC

φδφ)dx, (2.208)

where

LNCw =

γπc

12a

−w∗∗ + (c/4 + ed)(θ

∗∗o + φ∗∗) + (x + µ sin ψ)(θ∗o + φ∗)

, (2.209)

MNCφ

=γπc

12a

(c/4 + ed)w

∗∗ − (c/4 + ed)2(θ∗∗o + φ∗∗)

− (c/2 + ed)(x + µ sin ψ)(θ∗o + φ∗)− c2

32(θ∗∗o + φ∗∗)

. (2.210)

2.3.3 Quasi-steady Aerodynamics Implementation

The virtual work by the total airloads, which includes both the circulatory and

noncirculatory airloads contributions, is given by

δWaero = δWCaero + δWNC

aero,

=

∫ 1

0

LC

u δue + LCv δv + (LC

w + LNCw )δw

+ (MCφ

+ MNCφ

)δφ

dx. (2.211)

74

This equation is discretized using the finite element method. Because many terms

in Equation 2.211 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 2.168–2.170) are divided into constant, linear, and non-linear

groups as follows:

UR = URC + URL + URNL, (2.212)

UT = UTC + UTL + UTNL, (2.213)

UP = UPC + UPL + UPNL, (2.214)

where

URC = λβp − µ cos ψ − ηr cos θo, (2.215)

URL = u∗ − v + xv′ + λw′ + µ (v′ sin ψ + βpw′ cos ψ)

− v∗′ηr cos θo + ηr

(φ− w∗′

)sin θo, (2.216)

URNL = v∗v′ + w∗w′ +1

2µ

(v′2 + w′2

)cos ψ, (2.217)

UTC = (x + µ sin ψ) cos θo + (λ + µβp cos ψ) sin θo, (2.218)

UTL =(u + v∗ + λφ− wβp + µβpφ cos ψ + µv′ cos ψ − λβpv

′)

cos θo

+(w∗ − xφ− µφ sin ψ + vβp + w′µ cos ψ − λβpw

′)

sin θo, (2.219)

UTNL =

w∗φ + vv′ − 1

2v′2 (x + µ sin ψ) + µw′φ cos ψ

cos θo

+

(−v∗φ + vw′ − xv′w′ − µv′φ cos ψ − µv′w′ sin ψ − 1

2λw′2

)sin θo, (2.220)

UPC = (θ∗o + βp) ηr + (λ + µβp cos ψ) cos θo − (x + µ sin ψ) sin θo, (2.221)

UPL =(φ∗ + w′

)ηr −

u + v∗ + λ(φ− βpv

′)− βpw + µ cos ψ(βpφ + v′)

sin θo

+

w∗ − xφ + vβp − λβpw′ + w′µ cos ψ − φµ sin ψ

cos θo, (2.222)

UPNL =

(−w∗φ− vv′ +

1

2xv′2 +

1

2v′2µ sin ψ − w′φµ cos ψ

)sin θo

+

(−v∗φ + vw′ − 1

2λw′2 − xv′w′ − v′φµ cos ψ + v′w′µ sin ψ

)cos θo. (2.223)

75

By substituting Equations 2.212 – 2.214 into Equations 2.189 – 2.192 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.

76

2.4 Inflow and Free Wake Model

There are several types of inflow models, such as uniform or linear inflow , pre-

scribed wake, free wake, and finite state wake models. Accurate modeling of the

induced inflow is important for analysis of flying qualities, detailed power calcu-

lations, 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.

2.4.1 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 + λi (1 + kx x cos ψ + ky x sin ψ) , (2.224)

where λi is the induced inflow ratio, r represents the radial location along the rotor

blade, and

λi =CT

2√

µ2 + λ2, (2.225)

kx =4

3

(1− 1.8µ2)

√1−

(λ

µ

)2

− λ

µ

, (2.226)

ky = −2µ, (2.227)

which is the Dree’s model.

77

2.4.2 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 as-

suming that for each point in the flow the velocity is convected at the local velocity,

~Vloc. Considering a single element of a trailed vortex filament, the fundamental

equation describing the transport of the filament is

d~r

dt= ~Vloc(~r, t), (2.228)

78

where ~r = ~r(ψw, ψb) is the position vector of a point on the filament at a time or

wake age ψw that was trailed from the blade when it was at an azimuth angle ψb.

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

∂ψw

+∂~r

∂ψb

=~V∞Ω

+1

Ω~Vind[~r(ψb, ψw)], (2.229)

where the summation is carried out over the total number of trailed vortex fil-

aments, which is the number of azimuthal discretization, that contribute to the

induced velocity field at any given point. The induced velocity ~Vind, can be deter-

mined using the Biot-Savart law.

The numerical solution of Equation 2.228 can be accomplished by discretiz-

ing the wake and using finite difference to approximate the derivatives. Land-

grebe [145] developed the explicit time finite difference schemes to solve the Equa-

tion. 2.228. 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 ne-

cessitating the use of artificial numerical damping. Bagai and Leishman [75] devel-

oped 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 ψb and space ψw. 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 (L2 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.

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.

80

2.5 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 trailing-

edge flaps are treated as an additional load vector for the blade response. This

will be discussed in Chapter 3 on Page 91.

2.5.1 Aeroelastic Response

The blade finite element discretization is shown in Figure 2.8. The blade is dis-

cretized 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

wi for the i-th beam element is interpolated by

wi = b H1 H2 H3 H4 c

w1

w′1

w2

w′2

i

, (2.230)

where

H1 =1

4(s + 2) (s− 1)2, (2.231)

81

H2 =L

(i)el

8(1− s2) (1− s), (2.232)

H3 =1

4(2− s) (s + 1)2, (2.233)

H4 =L

(i)el

8(s2 − 1) (s + 1), (2.234)

in which L(i)el is the length of the i-th beam element.

1 2

Trailing edge flaps

S-1 +1

Figure 2.8. Finite elements for composite rotor blades

The extended Hamilton’s principle presented in Section 2.1.4 can be rewritten

in the discretized form as

∫ 2π

0

[Nel∑i=1

δqTi ([M ]iq∗∗i + [C]iq∗i + [K]iqi − F (q)i)

]dψ = 0, (2.235)

in which subscript ()i indicates the i-th beam element, Nel is the number of element,

and the elemental nodal displacement vector, qi, is given by

qTi = b v1 v′1 w1 w′

1 φ′1 v2 v′2 w2 w′2 φ′2 ci, (2.236)

where w and v represent flap and lag displacements, respectively. φ′ denotes the

elastic twist of a blade.

Derivatives with respect to the azimuth ψ are placed in the damping and mass

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] qR, (2.239)

where [Φ] is the matrix of eigenvectors or madal matrix, and qR 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]qR = f(qR), (2.240)

where [m], [c], [k], and f are the modal mass, damping, and stiffness matrices

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 2.240 can be

numerically solved using the finite element method. This equation can be rewritten

as:

∫ 2π

0

δqRT ([m]q∗∗R + [c]q∗R+ [k]qR − f(qR)) 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, qRi, for the i-th temporal element can be interpolated by

qRi = b Ht1 Ht2 Ht3 Ht4 Ht5 Ht6 c

ηt1

η∗t1ηt2

η∗t2ηt3

η∗t3

i

(2.246)

where

Ht1 = s2 − 5

4s3 − 1

2s4 +

3

4s5, (2.247)

Ht2 =∆ψi

8

(s2 − s3 − s4 + s5

), (2.248)

Ht3 = s2 +5

4s3 − 1

2s4 − 3

4s5, (2.249)

Ht4 =∆ψi

8

(−s2 − s3 + s4 + s5), (2.250)

84

Ht5 = 1− 2s2 + s4, (2.251)

Ht6 =∆ψi

2

(s− 2s3 + s5

), (2.252)

which are the velocity continuous shape functions based on Hermitian polynomials,

where ∆ψi represents the time-interval of the i-th temporal element, and ηti 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.

1

2

3

45

6

7

8

1 2

S-1 +1

3

Figure 2.9. Finite element discretization in time

Substituting Equation 2.246 into Equation 2.245 yields

Nt∑i=1

∫ ψi+1

ψi

δηtTi ( [Kt(ψ)]iηti − Ft(ηt, ψ)i ) dψ = 0, (2.253)

85

where

[Kt(ψ)]i = −H∗t T

i [m(ψ)]H∗t i

+ HtTi [c(ψ)]H∗

t i + HtTi [k(ψ)]Hti, (2.254)

Ft(ηt, ψ)i = HtTi f(ηt, ψ), (2.255)

and assembling the temporal elements, Equation 2.253 takes the form

[Kt]ηt = Ft(ηt). (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 2.256 about the

current condition, ηt(i), yields

∆ηt(i) =

([Kt]− ∂Ft

∂ηt

∣∣∣∣ηt=η

(i)t

)−1 ([Kt]ηt(i) − Ft(η

(i)t )

), (2.257)

where

∆ηt(i) = ηt(i+1) − ηt(i). (2.258)

The initial guess of the response ηt(0) is solved by

ηt(0) = [Kt]−1Ft(0). (2.259)

When the error vector given in Equation 2.258, ∆ηt(i), is sufficiently close to

zero, the response is considered converged.

2.5.2 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

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 2.141 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 2.211 on Page

73 in Section 2.3.3. Then the resulting blade section loads are expressed by

Lu = LAu + V I

x + Lfu,

Lv = LAv + V I

y + Lfv , (2.260)

Lw = LAw + V I

z + Lfw,

Mu = MAφ

+ M Ix + M f

u ,

Mv = v′MAφ

+ M Iy + M f

v , (2.261)

Mw = w′MAφ

+ M Iz + M f

w,

where superscripts ()f 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 2.260 and 2.261 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,

Fx

Fy

Fz

=

∫ 1

0

Lu

Lv

Lw

dx, (2.262)

87

Mx

My

Mz

=

∫ 1

0

−Lv w + Lw v + Mu

Lu w − Lw x + Mv

−Lu v + Lv x + Mw

dx. (2.263)

The hub loads are obtained by summing the blade root loads between the Nb

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

FHX (ψ) =

Nb∑i=1

(F i

x cos ψi − F iy sin ψi − βpF

iz cos ψi

), (2.264)

FHY (ψ) =

Nb∑i=1

(F i

x sin ψi + F iy cos ψi − βpF

iz sin ψi

), (2.265)

FHZ (ψ) =

Nb∑i=1

(F i

z + βpFix

), (2.266)

MHX (ψ) =

Nb∑i=1

(M i

x cos ψi −M iy sin ψi − βpM

iz cos ψi

), (2.267)

MHY (ψ) =

Nb∑i=1

(M i

x sin ψi + M iy cos ψi − βpM

iz sin ψi

), (2.268)

MHZ (ψ) =

Nb∑i=1

(M i

z + βpMix

), (2.269)

where F ix, F i

y, and F iz are the blade root shear forces from Equation 2.262 due to

the i-th blade, and M ix, M i

y, and M iz are the blade root moments due to the i-th

blade from Equation 2.263.

The time histories of the hub forces and moments presented in Equations 2.264

– 2.269 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 Nb/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

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 vehi-

cle controls and orientation. The controls and vehicle orientation states, such as

collective pitch θo, cyclic pitch θ1c and θ1s, tail rotor collective pitch θtr, and the

shaft tilt angles αs and φs, 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) = b F Vx F V

y F Vz MV

x MVy MV

z cT , (2.271)

Θ = b θ75 θ1c θ1s αs φs θtr cT (2.272)

in which F Vx , F V

y , and F Vx are the vehicle force residuals, and MV

x , MVy , and MV

z

are the moments. The detailed formulations can be found in References [76, 133].

The nonlinear vehicle equilibrium equations given in Equation 2.270 can be

solved to find the control settings Θ via a first order Taylor’s series expansion with

respect to the control settings.

∆Θi = −(

∂F V ∂Θ

∣∣∣∣Θ=Θi

)−1

F V (Θi), (2.273)

where

∆Θi = Θi+1 −Θi, (2.274)

the Jacobian matrix is approximated by a forward finite difference,

∂F V ∂Θ

∣∣∣∣Θ=Θi

≈ F V (Θi+1) − F V (Θi)∆Θi

, (2.275)

and the initial guess Θ0 is estimated by a rigid blade trim analysis based on blade

flap dynamics.

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 2.10. Vehicle configuration for propulsive trim

90

Table 2.3. Vehicle properties

Total vehicle weight 5800 lbsVehicle longitudinal c.g. offset, xCG/R 0Vehicle lateral c.g. offset, yCG/R 0Hub location above vehicle c.g., h/R 0.2Vehicle flat plate area, f/πR2 0.01Tail rotor radius 3.24 ftTail rotor solidity, σtr 0.15Tail rotor location, xtr/R 1.2Tail rotor location above vehicle c.g., htr/R 0.2Tail rotor lift coefficient, c0tr, c1tr 0, 6Horizontal tail location, xht/R 0.95Horizontal tail planform area, Sht/πR2 0.011Horizontal tail lift coefficient, c0ht, c1ht 0, 6

Chapter 3Trailing 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 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, 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.

92

3.1 Inertial Contribution

The trailing-edge flap coordinate system is presented in Figure 3.1 showing that

flap coordinates xfi are attached on the trailing edge flap and move along with the

flap by the deflection angle δf .

f

d

y

z

d

x fy f

z f

w

v

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 efi is given by

rf = b 0 xf2 xf

3 c ef (3.1)

This position vector can be transformed into the blade deformed coordinate system

edi presented in Section 2.1.2.

rf = xfdi ed

i , (3.2)

93

in which

xfd1 = 0, xfd

2 = xf2 − δfx

f3 − d, xfd

3 = xf3 + δfx

f2 , (3.3)

where δf 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:

rfu = rf + b (x + u) v w ceu, (3.4)

where

rf = xfui eu

i , (3.5)

and

xfui = xfd

j T duji . (3.6)

The position vector rfu presented in Equation 3.4 is used in the formulation of

flap inertia forces. The position vector rf 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:

af = R∗∗ + Ω∗ × rfu + Ω× (Ω×Ω) + rf

u

∗∗+ 2Ω× rf

u

∗, (3.7)

where R represents the hub position vector with respect to an inertial frame, and

Ω ≈ βp eu1 + eu

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

94

constant.

The inertial contribution to the blade sectional loads associated with a trailing-

edge flap mass element dmf can be written as:

LfI = −∫

Af

afdmf , (3.9)

where Af 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 eu, are calculated by

MfI = −∫

Af

rf × afdmf . (3.10)

Table 3.1. Order of terms for trailing edge flaps

Term List Order

δf O(ε1/2)

δf , δf , d, xf O(ε)yf O(ε3/2)zf O(ε5/2)

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 δf , δf and δf , therefore needs to be considered as the flap

inertial contributions. By applying the small angle assumptions and the ordering

scheme up to O(ε3) (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

LfIu = 0, (3.11)

LfIv = −Sf

(δf sin θo − 2δ∗fθ

∗o cos θo

+ δfθ∗2o sin θo − δ∗∗f sin θo − δfθ

∗∗o cos θo

), (3.12)

95

LfIw = −Sf

(−2δ∗fθ∗o sin θo − δfθ

∗2o cos to

+ δ∗∗f cos θo − δfθ∗∗o sin θo

), (3.13)

, and the inertial sectional moments are

M fIu = −Sf

(δfv cos θo − d δf cos2 θo − xδfβp sin θo

− δfv∗∗ cos θo − δfw

∗∗ sin θo − d δ∗∗f)− Ifδ

∗∗f , (3.14)

M fIv = Sfxδf cos θo, (3.15)

M fIw = Sfxδf sin θo, (3.16)

where Sf and If represent the flap first sectional moment and flap hinge moment

of inertia, respectively, and they are defined by

Sf =

∫

Af

xf2 dmf , (3.17)

If =

∫

Af

(xf2)

2 dmf . (3.18)

The inertial trailing-edge flap hinge moment is calculated by

M IH = −

∫

Af

(rfh × af ) · eh

1 dmf , (3.19)

where

rfh = b 0 (xf

2 − δfxf3) (xf

3 + δfxf2) c eh, (3.20)

in which eh 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 af

is expressed in the deformed coordinate system ed by transforming Equation 3.7

using Equation 2.9. Applying the ordering scheme up to O(ε3.5), one can obtain

the inertial contribution to the trailing-edge flap hinge moment.

M IH = −Sf

−1

2d sin 2θo + (v − x v′ − v∗∗) sin θo

+ (xβp + xw′ + w∗∗) cos θo − d θ∗∗o

96

− If

1

2sin 2θo + δ∗∗f + θ∗∗o

, (3.21)

where the symmetry of trailing-edge flap about xf1 − xf

2 plane is assumed,

∫

Af

xf3 dmf = 0, (3.22)

∫

Af

xf2 xf

3 dmf = 0. (3.23)

Note that the inertial hinge moment, Equation 3.21, includes the blade motion as

well as the trailing edge flap motion.

97

3.2 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.

3.2.1 Incompressible Model

An unsteady airfoil theory with an oscillating trailing edge flap has been devel-

oped by Theodorsen [143]. The most important assumptions behind this approach

are that thin airfoil theory holds and flow is incompressible. Although the incom-

pressible 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 δf (t) relative to the chord line.

adf

ab eb

c

-b +b

U

Elastic axis Flap hinge

h

Figure 3.2. Nomenclature for a thin airfoil with a flap

98

The unsteady lift coefficient, CN , of a thin airfoil with a flap undergoing os-

cillatory motion resulting from trailing-edge flap deflection δf and rate δf is given

by

CN =b

U2

(−UT4δf − bT1δf

)+ 2πC(k)

(T10

πδf +

bT11

2πUδf

), (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 T4, T1, T10, and T11 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 assump-

tion 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, CM , and hinge moment, CH , generated by trailing-edge flap

motions. The detailed derivation can be found in References [72, 135,143].

3.2.2 Compressible Model

In the subsonic compressible flow, both the magnitude and phase of the aerody-

namic loads are affected by compressibility effects, thus the simple correction by

the Prandtl-Glauert factor cannot address the phase issue. Unlike the incompress-

ible 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

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 CN in terms of coefficient

due to flap deflection δf and coefficient due to the flap rate δf . Then,

CN(S) = CNδ(S) + CNδ

(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=

2Uψ

c, (3.26)

where () represents the non-dimensional quantity, and U ≈ x + µ sin ψ.

Considering an arbitrary flap motion, δf , the lift coefficient due to δf is given

by

CNδ(S) =

2T10

βδefff (S) +

2(1− e)

MT ′

Nδ

(Kn

Nδ−K ′n

Nδ

), (3.27)

where β is the Prandtl-Glauert compressibility factor, β =√

1−M2, and δefff is

the effective flap deflection and is given by

δefff (S) = δn

f −Xn1 − Y n

1 , (3.28)

in which δnf is the geometry flap deflection at the given instant in time step n.

KnNδ

, K ′nNδ

, Xn1 , and Y n

1 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 ′Nδ

is the non-circulatory time

constant.

100

For an arbitrary flap rate δf , the lift coefficient is given by

CNδ(S) =

T11

2β

(δfc

U

)eff

(S) +(1− e)2

2MT ′

Nδ

(Kn

Nδ−K ′n

Nδ

), (3.29)

where the first term represents the circulatory term, and the second term indicates

the non-circulatory term.(

δf c

U

)eff

(S) is the effective flap rate, which is given by,

(δfc

U

)eff

(S) =

(δfc

U

)n

−Xn2 − Y n

2 , (3.30)

in which(

δf c

U

)n

is the geometry flap deflection at the given instant in time step n,

and KnNδ

, K ′nNδ

, Xn2 and Y n

2 are the deficiency functions. T ′Nδ

is the non-circulatory

time constant.

Similar expressions can be derived for the pitching moment, CM(S), and hinge

moment, CH(S), due to the flap deflection δf and flap rate δf . 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 2.183 on Page 69,

LfAu = 0, (3.31)

LfAv =

γU2

6aCN sin α, (3.32)

LfAw =

γU2

6aCN cos α, (3.33)

M fA

φ=

γU2

6ac CM , (3.34)

and the flap hinge moment in the blade deformed coordinate system is given by

MAH =

γU2

6ac CH . (3.35)

101

These forces and moment should be transformed into the blade undeformed coor-

dinate system to be consistent with the formulation of rotor blade, except the flap

hinge moment. Then, via the transformation matrix T duij presented in Equation

2.9,

LfAi = T du

ij

TLfA

j , M fA

φ≈ M fA

φ. (3.36)

The trailing-edge flap’s inertial and aerodynamic loads are treated as an ad-

ditional loads for the aeroelastic analysis. The inertial contribution of a flap was

presented in Equations 3.11–3.16. Now the total force contribution of trailing-edge

flaps to the blade sectional loads can be written as:

Lfu = LfA

u + LfIu ,

Lfv = LfA

u + LfIv , (3.37)

Lfw = LfA

u + LfIw ,

and the moment contribution

M fu = M fA

φ+ M fI

u ,

M fv = v′M fA

φ+ M fI

v , (3.38)

M fw = w′M fA

φ+ M fI

w .

These forces and moments are added into the blade sectional forces and moments,

which are presented in Equations 2.260 and 2.261 on Page 86 in Section 2.5.2.

102

3.3 Active Trailing Edge Flap Control Algorithm

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.

3.3.1 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 = ZTn WZ Zn + KT

n WK Kn + δTn Wδ δn, (3.39)

where Zn is a hub loads vector containing the Nb/rev cos and sin harmonic com-

ponents of the fixed hub loads at time step n (three hub shear forces and three hub

moments). Kn is a blade loads vector containing the flapwise curvature harmonics,

Kn = b κ(i)1c κ

(i)1s κ

(i)2c κ

(i)2s cT , i = 1, 2, · · · , nK , (3.40)

in which κ(i) represents the flapwise curvature 1/rev and 2/rev harmonics at the

i-th radial station. nK is the number of curvature sensing locations along the rotor

spanwise direction. δn represents the harmonics of the control inputs. For a typical

103

four-bladed rotor,

δn = b δf1c δf1s · · · δf5c δf5s cT , (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

δf (ψ) =5∑

i=1

[ δfic cos(iψ) + δfis sin(iψ) ] . (3.42)

The matrices W contain penalty weights for the harmonics of the curvature WK ,

the vibration WZ and the control inputs Wδ, they are given by

WK = (1− βw) γw I, (3.43)

WZ = (1− βw)

[αwI 0

0 (1− αw)I

], (3.44)

Wδ = βw I, (3.45)

where I and 0 represent identity and null matrices, respectively. αw, βw and γw

are scalar weighting parameters. By changing αw, 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 Zn and Kn with respect to a current control

input vector δn−1 yield

Zn = Zn−1 + TZ (δn − δn−1), (3.46)

Kn = Kn−1 + TK (δn − δn−1), (3.47)

where the sensitivity matrices T relate the linearized system response to multi-

cyclic 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,

TZ =∆Zn−1

∆δn−1

, TK =∆Kn−1

∆δn−1

, (3.48)

104

Substituting Equations 3.46 and 3.47 into Equation 3.39 yields

J = [Zn−1 + TZ(δn − δn−1)]T WZ [Zn−1 + TZ(δn − δn−1)]

+ [Kn−1 + TK(δn − δn−1)]T WK [Kn−1 + TK(δn − δn−1)]

+ δTn Wδ δn. (3.49)

Then minimizing J by solving∂J

∂δn

= 0, (3.50)

yields

δn = D[TT

ZWZTZ + TTKWKTK

]δn−1

− D[TT

ZWZZn−1 + TTKWKKn−1

], (3.51)

where

D ≡ [TT

ZWZTZ + TTKWKTK + Wδ

]−1. (3.52)

This can be further simplified by defining

CZ ≡ −D TTZ WZ , (3.53)

CK ≡ −D TTK WK . (3.54)

Then, Equation 3.51 can be rewritten as:

δn = CZ Zn−1 + CK Kn−1 − (CZTZ + CKTK) δn−1. (3.55)

The update law of the optimal controller that minimizes the objective function

J is expressed by

∆δn = δn − δn−1,

= CZZn−1 + CKKn−1 − (CZTZ + CKTK + I) δn−1. (3.56)

105

From the definitions presented in Equations 3.52–3.54, one can find that

DWδ = CZTZ + CKTK + I, (3.57)

≡ Cδ. (3.58)

Then, the update law of the controller given in Equation 3.56 can be concisely

expressed as:

∆δn = CZZn−1 + CKKn−1 −Cδ δn−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 satu-

ration. Saturation can be due to limitations associated with piezoelectric actuation

which can provide flap deflections of 4o or less. Alternatively, when larger flap de-

flections 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

106

find an optimal weighting value of βw. This can be expresses as:

minimize[‖ δn(βw) ‖2 −δsat

]2

subject to 0 ≤ βw < 1 (3.60)

where δn represents the optimal control gain vector, and δsat 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 Wδ is appropriately modified to relax or tighten the flap

deflection constraint by adjusting βw. Then the new weighting matrix is input

into the optimal control calculation routine and the process repeats until the flap

is properly constrained.

3.3.2 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 pa-

rameters and active control actions are optimized.

For an active-passive approach, the constrained optimization problem can be

formulated as:

minimize minδn

[f(x, δn)]

subject to gi(x) i = 1, 2, · · · , nc (3.61)

xL ≤ x ≤ xU

in which x represent the vector containing passive design variables, such as the

blade mass and stiffness distributions. nc is the number of constraints. f(x, δf ) is

107

the objective function given in Equation 3.39. The design variable vector is given

by

x = b m K22 K33 K55 K25 K35 cT(i), i = 1, 2, · · · , Nel (3.62)

where K22, K33 and K55 represent the flap bending, lag bending and torsional

bending stiffness, respectively. K25 and K35 denote the pitch-flap and pitch-lag

composite coupling stiffness. The constraints gi(x) are given by

gj = 1− ωj

ωLj

, j = 1, 2, 3 (3.63)

gk =ωk

ωUk

− 1, k = 4, 5, 6 (3.64)

where ωj 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 vari-

ables 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 K25 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 Equa-

tion 3.61. 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].

Chapter 4Active 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 con-

ventional articulated rotor blade. A classical incompressible theory presented in

Section 3.2.1 is employed to predict the incremental trailing edge flap airloads.

The objective function, which includes vibratory hub loads, bending moment har-

monics and active flap control inputs, is minimized by an integrated optimal con-

trol/optimization process described in Section 3.3. Numerical simulation has been

performed for the steady-state forward flight of an advance ratio of 0.35. It is

demonstrated that through straightening the rotor blade, which mimics the be-

havior of a rigid blade, both the bending moments and vibratory hub loads can be

significantly reduced.

109

4.1 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 care-

fully 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 vi-

bration 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.

110

4.2 Description of Analytical Models

The trailing-edge flap model, which is based on the incompressible theory de-

veloped 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 flap0.4~0.6R

Out-board flap0.9~1.0R

Figure 4.1. Dual flap configuration for active loads control

In the present analysis, motion of the fuselage is not considered. The aerody-

namics 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 incom-

pressible 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-

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 re-

tains 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 harmon-

ics is dominant, and its maximum occurs at around mid-span. For this reason,

1/rev control input is considered. The control weighting parameters αw, βw and

γw 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 3.3.2 is imple-

mented by combining an optimal control law with nonlinear optimization program-

ming for a composite rotor blade model. The passive optimization is solved with

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 limitω1 / ωo

1 (flap) 0.95 1.05ω2 / ωo

2 (lag) 0.9 1.1ω3 / ωo

3 (torsion) 0.8 1.2

Design variables Lower bound Upper boundm/mo 0.90 1.10Kii / Ko

ii 0.90 1.10K25 −1× 10−4 1× 10−4

4.3 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 0.35. 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 cou-

pling stiffness K25 and active control inputs.

113

Table 4.2. Baseline articulated rotor properties for active loads control

Hub type articulated rotorNumber of blades, Nb 4Rotor radius, R 16.2 ftHover tip speed, ΩR 650 ft/secHover tip Mach number, Mtip 0.58Airfoil NACA 0015Lift coefficients, co, c1 0, 5.73Drag coefficients, do, d1, d2 0.0095, 0, 0.2Blade chord, c/R 0.08Solidity, σ 0.1Thrust coefficient over solidity, CT /σ 0.07Blade linear twist, θtw −8o

Precone, βp 0Lock number, γ 6.34Flap bending stiffness, Ko

22 0.002429Lag bending stiffness, Ko

33 0.022825Torsional stiffness, Ko

55 0.002195Flapwise mass moment of inertia, mk2

m1 0.00002114Lagwise mass moment of inertia, mk2

m2 0.000369Blade mass/length, mo 0.011947 slugs/inAdvance ratio, µ 0.35

For the purpose of comparison, the vibration and blade moment indices are

defined:

VLV =√

F 2x + F 2

y + F 2z + M2

x + M2z , (4.1)

MLV = max [Mflap(r, ψ)] , (4.2)

where Fx, Fy and Fz are 4/rev vibratory hub shear forces, and Mx, My and Mz

are 4/rev vibratory hub moments. Mflap represents the flapwise bending moment

in the rotating frame.

4.3.1 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

114

Table 4.3. Trailing-edge flap properties for active loads control

In-board flap location 0.4 ∼ 0.6Out-board flap location 0.9 ∼ 1.0Flap chord ratio, cf/c 0.2Flap mass per unit length, mf/mo 0.0844Flap chordwise c.g. (after flap hinge), rI/cf 0.149Flap radius of gyration about flap hinge, r2

II/c2f 0.109

Offset from blade elastic axis to flap hinge, d/cf 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 4.3 – 4.5.

Table 4.4. Natural frequencies of baseline articulated rotor

Mode Frequency, /revFlap 1 1.03072Flap 2 2.68559Flap 3 5.03369Flap 4 8.24382Lag 1 0.24988Lag 2 3.47887Lag 3 9.24582Torsion 1 3.99194Torsion 2 11.6810

The rotor thrust coefficient CT is assumed to be 0.007, and quasi-steady blade

element aerodynamics with a linear inflow model is used to obtain the blade re-

sponse. 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.

115

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.351st mode

Radial position, x

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.1

−0.05

0

0.05

0.1

0.152nd mode

Radial position, x

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.13rd mode

Radial position, x

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.064th mode

Radial position, x

FlapLagTorsion

Figure 4.3. Articulated blade coupled flap mode shapes

116

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.351st mode

Radial position, x

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.1

−0.05

0

0.05

0.1

0.152nd mode

Radial position, x

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.13rd mode

Radial position, x

FlapLagTorsion

Figure 4.4. Articulated blade coupled lag mode shapes

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Radial position, x

1st mode

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.152nd mode

Radial position, x

FlapLagTorsion

Figure 4.5. Articulated blade torsion mode shapes

117

−10

−8

−6

−4

−2

0

2

4

6

8

10

Con

trol

set

tings

, deg

θ.75

θ1c

θ1s

αs

φs

Figure 4.6. Control settings of articulated rotor, µ = 0.35

0 90 180 270 360−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Azimuthal angle ( ψ ), deg

Bla

de ti

p re

spon

se

FlapTosrionLag

Figure 4.7. Blade tip response of articulated rotor, µ = 0.35

118

4.3.2 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.

0

0.001

0.002

0.003

0.004

0.005

Fx Fy Fz

Baseline

Active control (dual flap)

Rigid blade

Figure 4.8. Comparison of vibratory hub loads for active loads control

4.3.3 A Single Flap for Moment Reduction

n general, a single flap located at 0.6 ∼ 0.8R 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

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.

4.3.4 Dual Flap Performance

Dual flap has been introduced to straighten a rotor blade so that we could re-

duce 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-of-

phase 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

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 4.10.

0 90 180 270 360-8

-6

-4

-2

0

2

4

6

8

Out-board flapIn-board flap

Fla

p d

efl

ecti

on

s, d

eg

Azimuth, deg

Dual flap profile

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

121

-10

-8

-6

-4

-2

0

2

4

6

8

10

theta_75

theta_1c

theta_1s

alpha_s

phi_s

Baseline

Dual flap

Figure 4.10. 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 LagBaseline (×104) 2.57 2.50Dual flap (×104) 1.75 1.36Reduction (%) 31.9 45.4

122

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Radial station, r

Baseline 1/rev

Baseline 2/rev

Retrofit 1/rev

Retrofit 2/rev

Figure 4.11. Harmonics of flapwise bending moment along the radial station

123

-1 -0.5 0 0.5 1Y

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

XZ

0.01210.01040.00860.00690.00510.00340.0017

-0.0001-0.0018-0.0036-0.0053-0.0071-0.0088-0.0106-0.0123

Advancing sideRetreating side

Figure 4.12. Flapwise bending moment distribution before control

-1 -0.5 0 0.5 1Y

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

X

Z

0.01210.01040.00860.00690.00510.00340.0017

-0.0001-0.0018-0.0036-0.0053-0.0071-0.0088-0.0106-0.0123

Advancing sideRetreating side

Figure 4.13. Flapwise bending moment distribution after control

124

4.3.5 Multicyclic Control for Moment and Vibration Re-

duction

In this case, the objective function includes two quadratic functions that are vi-

bratory hub loads and flapwise bending moment harmonics. Multicyclic control

inputs, such as 1, 2 and 3/rev, have been considered to improve both bending mo-

ment 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 out-

board 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.

0 90 180 270 360-10

-5

0

5

10

15

Out-board TEF

In-board TEF

Dual flap profile

Fla

p d

efl

ecti

on

s, d

eg

Azimuth, deg

Figure 4.14. Dual flap profile with 1 and 2/rev control inputs

The reason of this can be deduced from comparisons of vibration index and

125

0 90 180 270 360-10

-5

0

5

10

15

Out-board TEF

In-board TEF

Dual flap profile

Fla

p d

efl

ecti

on

s, d

eg

Azimuth, deg

Figure 4.15. 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 momentreduction (%)

1/rev 56.68 31.961 and 2/rev 60.09 34.60

1, 2 and 3/rev 96.01 24.42

126

Figure 4.16. Comparison of vibration index and maximum flapwise bending momentwith different control inputs

Note that dual flap performance with 1/rev control input is not good for vi-

bration 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.

4.3.6 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-

127

duction. Thus weighting matrix for bending moment, WK , described in Equation

3.39 has been chosen to avoid being overly minimized at around 0.7R, so that pas-

sive 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 de-

sign, 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 4.20.

Table 4.7. Comparisons of maximum moment, vibration index and control efforts

Max. moment Vibration Max. TEFDesign reduction (%) reduction (%) deflection, deg

Flap / Lag Outboard / InboardRetrofit 32 / 45 56.7 4.88 / 6.70Hybrid 34 / 44 61. 8 5.09 / 4.90

0

0.001

0.002

0.003

0.004

0.005

Fx Fy Fz

Baseline

Retrofit

Hybrid

Figure 4.17. Vibratory hub shears comparison for active loads control

128

0

1

2

3

4

5

6

7

8

Vib. Index (1e3) Max. Flapw ise moment (2e3)

baseline

retrofit

hybrid

Figure 4.18. Comparison of vibration index and maximum flapwise moment of hybriddesigned rotor

0 90 180 270 360-8

-6

-4

-2

0

2

4

6

8

Out-board flap(Retrofit,Hybrid)

In-board flap(Retrofit)

In-board flap(Hybrid)

Fla

p d

efl

ecti

on

s, d

eg

Azimuth, deg

Figure 4.19. Dual flap profiles for retrofit and hybrid designs

129

-10

-8

-6

-4

-2

0

2

4

6

8

10

theta_75

theta_1c

theta_1s

alpha_s

phi_s

BaselineRetrofitHybrid

Figure 4.20. Control settings of baseline, retrofit and hybrid designed rotors

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, K25, distributions are shown in Figure 4.21. 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

-1.0E-04

-5.0E-05

0.0E+00

5.0E-05

1.0E-04

1 2 3 4 5

Mass

-0.1

-0.05

0

0.05

0.1

1 2 3 4 5

Figure 4.21. Blade non-structural mass and pitch-flap composite coupling stiffness,K25, distribution

131

4.4 Summary

The purpose of this chapter was to investigate the feasibility of multiple trailing-

edge flaps for the simultaneous reductions of vibration and blade loads. The con-

cept involved straightening the blade by introducing dual trailing edge flaps in a

conventional articulated rotor blade. An active loads control strategy was simu-

lated for the steady-state forward flight condition, µ = 0.35.

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 flap-

wise 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.

Chapter 5Helicopter 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 in-

dividual 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 aerody-

namics model is employed to accurately predict the incremental trailing edge flap

airloads. Both trailing-edge flap finite wing effects and actuator saturation are in-

cluded in the simulation. A numerical simulation is performed for the steady-state

forward flight (µ = 0.15 ∼ 0.35). It is demonstrated that multiple trailing-edge flap

configuration with the resonant actuation system can reduce the required trailing-

edge flap hinge moments. The analysis and parametric study are conducted to

explore the finite wing effect of trailing-edge flaps and actuator saturation.

133

5.1 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 per-

formance have been made by researchers in developing amplification mechanisms

of various types (see Section 1.2.3 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.

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 us-

ing 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 mul-

tiple 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).

135

5.2 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 2.4.2 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 3.2.2 on

Page 98), is formulated and then integrated into the blade model. Multiple par-

tial 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

136

as:af

afo

=AR

AR + 2(AR + 4)/(AR + 2), (5.1)

in which

AR =L2

f

Lfcf

, (5.2)

where afo is the nominal lifting curve slope of trailing-edge flap, which is normally

2π. Lf and cf 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 CN presented in Equation 3.25 on Page 99,

in which CN 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:

δn = b δ(1)fc δ

(1)fs δ

(2)fc δ

(2)fs δ

(3)fc δ

(3)fs cT , (5.3)

137

where superscript ()(i) indicates the i-th trailing-edge flap. Flap deflections df (ψ)

are limited by 2o ∼ 4o using the actuator saturation algorithm presented in Equa-

tion 3.60 on Page 106, which automatically determines the control weight param-

eter βw. For the purpose of the vibration reduction, the other control weighting

parameters, αw and γw, 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 op-

erates 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.

5.3.1 Baseline Hingeless Rotor Analysis

The baseline rotor blades are generic, uniform, hingeless rotor blades. Blade nat-

ural 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

138

Table 5.1. Hingeless rotor and trailing-edge flap properties

Main rotor propertiesHub type hingeless rotorNumber of blades, Nb 4Rotor radius, R 16.2 ftHover tip speed, ΩR 650 ft/secHover tip Mach number, Mtip 0.58Airfoil NACA 0015Lift coefficients, co, c1 0, 5.73Drag coefficients, do, d1, d2 0.0095, 0, 0.2Blade chord, c/R 0.08Solidity, σ 0.1Thrust coefficient over solidity, CT /σ 0.07Blade linear twist, θtw −8o

Precone, βp 0Lock number, γ 6.34Flap bending stiffness, Ko

22 0.008345Lag bending stiffness, Ko

33 0.023198Torsional stiffness, Ko

55 0.00225Flapwise mass moment of inertia, mk2

m1 0.0001Lagwise mass moment of inertia, mk2

m2 0.0004Blade mass/length, mo 0.011947 slugs/inAdvance ratio, µ 0.15 ∼ 0.35

Trailing-edge flap propertiesFlap chord ratio, cf/c 0.2Flap mass per unit length, mf/mo 0.0844Flap chordwise c.g. (after flap hinge), rI/cf 0.149Flap radius of gyration about flap hinge, r2

II/c2f 0.109

Offset from blade elastic axis to flap hinge, d/cf 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 5.2 – 5.4.

139

Table 5.2. Natural frequencies of baseline hingeless rotor

Mode Frequency, /revFlap 1 1.1469Flap 2 3.4041Flap 3 7.4938Flap 4 13.4995Lag 1 0.7499Lag 2 4.3682Lag 3 11.0249Torsion 1 3.5546Torsion 2 10.4414

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.31st mode

Radial position, x

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.122nd mode

Radial position, x

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.083rd mode

Radial position, x

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.054th mode

Radial position, x

FlapLagTorsion

Figure 5.2. Hingeless blade coupled flap mode shapes

140

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.31st mode

Radial position, x

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.122nd mode

Radial position, x

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.083rd mode

Radial position, x

FlapLagTorsion

Figure 5.3. Hingeless blade coupled lag mode shapes

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

Radial position, x

1st mode

FlapLagTorsion

0 0.2 0.4 0.6 0.8 1−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.12nd mode

Radial position, x

FlapLagTorsion

Figure 5.4. Hingeless blade torsion mode shapes

141

The rotor thrust coefficient CT 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.

−4

−2

0

2

4

6

8

10

Con

trol

set

tings

, deg

θ.75

θ1c

θ1s

αs

φs

Figure 5.5. Control settings of hingeless rotor, µ = 0.15

−10

−5

0

5

10

15

Con

trol

set

tings

, deg

θ.75

θ1c

θ1s

αs

φs

Figure 5.6. Control settings of hingeless rotor, µ = 0.35

142

0 90 180 270 360−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Azimuthal angle ( ψ ), deg

Bla

de ti

p re

spon

se

FlapTorsionLag

Figure 5.7. Blade tip response of hingeless rotor, µ = 0.15

0 90 180 270 360−0.04

−0.02

0

0.02

0.04

0.06

0.08

Azimuthal angle ( ψ ), deg

Bla

de ti

p re

spon

se

FlapTorsionLag

Figure 5.8. Blade tip response of hingeless rotor, µ = 0.35

143

5.3.2 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 = 0.35. 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.

−1 0 1 2 3 4 5−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2Y wake geomtry for µ=0.15

element 5element 10element 10 w/ flap

Figure 5.9. Vertical wake geometry with the number of beam elements and the presenceof the flap for µ = 0.15

144

−2 0 2 4 6 8 10−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0Y wake geomtry for µ=0.35

element 5element 10element 10 w/ flap

Figure 5.10. Vertical wake geometry with the number of beam elements and the pres-ence of the flap for µ = 0.35

0 90 180 270 360−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Bla

de ti

p re

spon

se

Azimuth, deg

5 beam elements w/ flap10 beam elements w/ flapRe−calculated free−wake w/ flap

Flap

Torsion

Lag

Figure 5.11. Blade tip responses with the number of beam elements and the presenceof the flap for µ = 0.15

145

0 90 180 270 360−0.04

−0.02

0

0.02

0.04

0.06

0.08

Bla

de ti

p re

spon

se

Azimuth, deg

5 beam elements w/ flap10 beam elements w/ flapRe−calculated free−wake w/ flap

Flap

Torsion

Lag

Figure 5.12. Blade tip responses with the number of beam elements and the presenceof the flap for µ = 0.35

Figure 5.13. Control settings with the number of beam elements and the presence ofthe flap for µ = 0.15

146

Figure 5.14. Control settings with the number of beam elements and the presence ofthe flap for µ = 0.35

147

5.3.3 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 (Lf = 0.07R) for low-

and 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 (Fz) 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 Multiple1st location 0.6 ∼ 0.8R 0.6 ∼ 0.74R 0.6 ∼ 0.67R

control 3,4,5/rev 3,4,5/rev 3/rev2nd location 0.9 ∼ 0.97R 0.7 ∼ 0.77R

control 3,4,5/rev 4/rev3rd location 0.9 ∼ 0.97R

control 4/rev

148

Figure 5.15. Vibration reduction vs. radial locations of trailing-edge flaps

149

5.3.4 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 5.16. 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 5.17. 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

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 5.17. Vibration reduction by multi-flap configuration with lift flap

Figure 5.18. Flap deflection harmonics of multi-flap configuration with the lift flap,advance ratio: µ = 0.15, actuator saturation: δsat

f = 4o

151

5.3.5 Effectiveness of Multiple-Flap Configuration

Single-, dual- and multiple-flap configurations are investigated to explore the multiple-

flap 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 single-

flap 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 pre-

sented 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 5.20.

Flap profile for dual-flap configuration is presented in Figure 5.21. 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 configura-

tion, 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 5.22. Maximum hinge moment of single-flap is 6.8 × 10−6 and

occurs at the azimuth angle of 342 degree. Maximum hinge moment of dual-flap

is 4.5× 10−6, 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 single-

flap.

152

Figure 5.19. Polar diagram of flap motion for single-flap configuration, advance ratio:µ = 0.15, actuator saturation: δsat

f = 2o

Table 5.4. Peak-to-peak hinge moments in single- and dual-flap configurations with4/rev control input, µ = 0.15

Flap configuration Peak-to-peak hinge moments Reduction, %Single flap 8.04× 10−6 -

Dual 1st flap 4.04× 10−6 49.8Dual 2nd flap 3.96× 10−6 50.7Dual (total) 8.00× 10−6 0.47

153

Figure 5.20. Polar diagram of flap motion for dual-flap configuration, advance ratio:µ = 0.15, actuator saturation: δsat

f = 2o

154

Figure 5.21. Flap deflections of dual-flap configuration with 4/rev control input, ad-vance ratio: µ = 0.15, actuator saturation: δsat

f = 2o

155

Figure 5.22. Hinge moments in single- and dual-flap configurations with 4/rev controlinput, advance ratio: µ = 0.15, actuator saturation: δsat

f = 2o

156

5.3.6 Vibration Reduction with Multicyclic Control

The vibratory hub loads for the baseline, single-, dual- and multiple-flap config-

urations 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 5.25. It is shown that outboard flap deflections

are nearly 180 degree out-of-phase with inboard flaps (1st and 2nd flaps).

advance ratio = 0.15

0.000

0.002

0.004

0.006

0.008

Fx Fy Fz Mx My Mz

Figure 5.23. Comparison of vibratory hub loads, advance ratio: µ = 0.15, actuatorsaturation: δsat

f = 4o

For the high-speed flight, corresponding to an advance ratio of 0.35, the vibra-

tory hub loads are presented in Figure 5.26. Vibration reductions by single-, dual-

and multi-flap configurations are 83%, 85% and 79%, respectively. In this case,

in-plane hub shear force Fx 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.

157

0 90 180 270 360−5

−4

−3

−2

−1

0

1

2

3

4

5

Fla

p de

flect

ions

, deg

rees

Azimuth angle, degrees

Flap profile of single−flap, µ=0.15

Figure 5.24. Flap deflections of single-flap configuration, advance ratio: µ = 0.15,actuator saturation: δsat

f = 4o

158

0 90 180 270 360−5

−4

−3

−2

−1

0

1

2

3

4

5

Fla

p de

flect

ions

, deg

rees

Azimuth angle, degrees

Flap profile of single− and multiple−flap, µ=0.15

multi 1st TEFmulti 2nd TEFmulti 3rd TEF

Figure 5.25. Flap deflections of multiple-flap configuration, advance ratio: µ = 0.15,actuator saturation: δsat

f = 4o

159

advance ratio = 0.35

0.000

0.001

0.001

0.002

0.002

0.003

0.003

Fx Fy Fz Mx My Mz

baseline

s ingle (4 deg)

dual (4 deg)

mult i (4 deg)

Figure 5.26. Comparison of vibratory hub loads, advance ratio: µ = 0.35, actuatorsaturation: δsat

f = 4o

160

Figure 5.27. Flap deflections of single- and multiple-flap configuration, advance ratio:µ = 0.35, actuator saturation: δsat

f = 4o

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

5.28. In multi-flap configuration, maximum hinge moment is 5.2×10−6 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 5.2 × 10−6 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 5.28. Hinge moments in single- and multiple-flap configuration, advance ratio:µ = 0.35, actuator saturation: δsat

f = 4o

162

Table 5.5. Peak-to-peak hinge moments in single- and multiple-flap configurations,µ = 0.35

Flap configuration Peak-to-peak hinge moments Reduction, %Single flap 1.06× 10−5 -

Multiple 1st flap 4.10× 10−6 61.2Multiple 2nd flap 4.65× 10−6 56.0Multiple 3rd flap 6.60× 10−6 37.5Multiple (total) 1.53× 10−5 -45.3

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.

163

5.4 Summary

In this chapter, an active control method for multiple trailing-edge flaps configura-

tion is proposed. The concept involves deflecting the trailing-edge flaps by intro-

ducing 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 con-

figuration, since the controller is able to compensate for changes in flap ca-

pability by adjusting the flap deflections and phases, and kinetic energy also

contributes to vibration reduction.

• For low-speed flight, vibration level is reduced by 56 ∼ 60 % 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.

Chapter 6Piezoelectric 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 param-

eters 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.

165

6.1 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 trans-

ducers 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 1.2.3 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 actu-

ators. 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 para-

graphs. 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 oper-

ating 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 frequen-

cies 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, reso-

nant 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 cir-

cuitry tailoring to broaden and flatten the resonant peak, so that one can achieve a

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 de-

signed to amplify the actuator active authority around the tuned circuit frequency

(see see Section 1.2.4 on Page 26). Several researchers have proposed the “nega-

tive 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 devel-

oped 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 experi-

mentally via an equivalent electric circuit based on the Van Dyke model.

167

6.2 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 piezoelec-

tric actuators, which are poled in the thickness direction, the PZT tube actuator

segments are poled in the length direction [116].

6.2.1 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(Sij, Ei) =1

2cEijkl − ekijEkSij − 1

2εS

ijEiEj, (6.1)

and in the matrix form

H(S,E) =1

2ST [cE]S − ST [e]E − 1

2ET [εS]E. (6.2)

168

The constitutive equations for a linear piezoelectricity are given by

Tij =∂H

∂Sij

= cEijklSkl − eijkEk, (6.3)

Di = − ∂H

∂Ei

= eiklSkl + εSikEk, (6.4)

or

T = [cE]S − [e]E, (6.5)

D = [e]TS+ [εS]E. (6.6)

It can also be expressed in terms of strains and electric displacements.

T = [cD]S − [h]D, (6.7)

E = −[h]TS+ [βS]D. (6.8)

where

[cD] = [cE] + [e][βS][e]T , (6.9)

[h] = [e][βS], [βS] = [εS]−1. (6.10)

Here, nomenclature follows that of ANSI/IEEE standard 176-1987 on piezoelec-

tricity [149].

The governing equations are derived using the variational principle, assuming

no body force, as follows:

∫ t

0

∫

Vp

(ρpuiδui − TijδSij − EiδDi) dVpdt

+

∫ t

0

∫

Sp

(piδui + φsδQs) dSpdt = 0, (6.11)

where Tij is the mechanical stress component, Sij the strain component, Ei is the

electric field component, Di is the electric displacement component, and ui denotes

the mechanical displacement component. pi, Qs, and φs represent the applied

traction, charge density, and applied electric potential on the surface, respectively.

169

ρp denotes the piezoelectric material density.

For the piezoelectric tube actuator, Equation 6.11 can be written by, in the

polar coordinate system,

∫ t

0

∫

vp

(ρpuθδuθ − TzθδSzθ − Eθδ

Dθ

Ns

)dvpdt

+

∫ t

0

∫

sp

(pθδuθ + φsδ

Q

NsAs

)dspdt = 0, (6.12)

where Ns is the number of segments of tube actuator and Q is the electric charge.

The angular displacement uθ, shear strain Szθ, and the electric displacement Dθ

are given by

uθ = rpθ, Szθ = rpθ,z, Dθ =Q

As

, (6.13)

in which (),z denotes the partial derivative with respect to z, and As 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

uθ(rp, z; t) = rpθ(z; t) = rp

∞∑i=1

Ψi(z)qi(t), (6.14)

where Ψi(z) is the i-th mode shape function, and qi(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

0T 0

]q

Q

+

[Cp 0

0T 0

] q

Q

+

[KD

p −Kc

−KTc KQ

]q

Q

=

Fθ

Va

,

(6.15)

where

Mp =

∫ lp

0

∫ 2π

0

∫ Ro

Ri

(ρpr

2pΨiΨj

)rpdrpdθdz, (6.16)

KDp =

∫ lp

0

∫ 2π

0

∫ Ro

Ri

(cD55r

2pΨi,zΨj,z

)rpdrpdθdz, (6.17)

170

Kc =

∫ lp

0

∫ 2π

0

∫ Ro

Ri

(h15rpΨi,z

AsNs

)rpdrpdθdz, (6.18)

KQ =

∫ lp

0

∫ 2π

0

∫ Ro

Ri

(βS

11

A2sN

2s

)rpdrpdθdz, (6.19)

Fθ =

∫ 2π

0

∫ Ro

Ri

(h15rpΨi,z

AsNs

)rpdrpdθ

∣∣∣∣z=lp

, (6.20)

Va =

∫

s

(φs

NsAs

)ds = Ns

(φsAs

AsNs

)= φs, (6.21)

in which lp is the length of the tube actuator, Ro and Ri denote the outer radius and

inner radius of the tube (see Figure 6.1). The matrix Cp represents the structural

damping of piezoelectric tube actuator, KQ denotes the inverse of piezoelectric tube

actuator capacitance CSp , and Va 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). The first eigenfunction of the

clamped torsion bar is given by

Ψ(z) = sin

(π

2lpz

), (6.22)

and then, with assuming no external torque, Equation 6.15 can be rewritten by

[Mp 0

0 0

]qt

Q

+

[Cp 0

0 Rp

] qt

Q

+

[KD

p −Kc

−Kc KQ

]qt

Q

=

0

Va

, (6.23)

where qt represents the tip displacement of a piezoelectric tube actuator, which is

also referred to as the ‘actuator stroke’, and Rp represents the electric resistance

that includes both the inherent resistance of the tube actuator and the external

wire resistance.

171

6.2.2 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

3.2.2 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 cen-

trifugal propeller moments due to the rotation of blade (see Figure 6.2). Assuming

that the trailing-edge flap deflection angle, δf , is small, the inertial and propeller

moments can be expressed as [107]:

hI = Iδ δf , (6.24)

hCF = mfΩ2drI sin(δf ) ≈ mfΩ

2drIδf , (6.25)

where δf is a trailing edge flap deflection angle, and Ω is the rotation speed of

blade. Iδ, mf , d and rI 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 3.2.1 on

172

Page 97. The hinge moment coefficients in terms of flap deflection angle δf is given

by

CHδ = − 1

2πT12T10C(k) +

1

2π(T4T10 − T5) , (6.26)

CHδ = − b

4πU∞T12T11C(k) +

b

4πU∞T4T11, (6.27)

CHδ =b2

2πU2∞T3, (6.28)

where coefficients, T3,4,5 and T10,11,12, 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 δf :

Ff (δf ; t) = Mf δf + Cf δf + Kfδf , (6.29)

where

Mf =1

2CHδρ∞U2

∞c2Lf + hI , (6.30)

Cf =1

2CHδρ∞U2

∞c2Lf , (6.31)

Kf =1

2CHδρ∞U2

∞c2Lf + hCF , (6.32)

where c and Lf are the blade chord and the trailing-edge flap span, respectively.

Underline terms denote inertial contributions from the flap.

6.2.3 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

173

equation can be written as

LQ + RQ + KQ −Kcqt = Vc, (6.33)

or

LQ + RQ + Va + KaQQ = Vc, (6.34)

where L is the inductance, R is the resistance, Va and Vc denote the voltage across

the PZT actuator and the control voltage, respectively. Va is expressed by

Va = KQQ−Kcqt, (6.35)

and KQ is defined by

KQ = KQ + KaQ, (6.36)

KQ =1

Csp

, KaQ =

1

Cadd

, (6.37)

in which Csp denotes the capacitance of the PZT tube actuator, and Cadd 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, δf , should be expressed in terms of the actuator stroke,

qt. This relation can be interpreted as the amplification mechanism, i.e., the linkage

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 δf and qt can be expressed in the form (see Figure 6.4)

δf = AMqt. (6.38)

Note that this relationship, AM , plays a role of unit conversion. For instance, the

trailing-edge flap deflection df is transformed to the actuator stroke qt 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

0 L

]qt

Q

+

[C 0

0 R

]qt

Q

+

[KD −Kc

−Kc KQ

]qt

Q

=

0

Vc

, (6.39)

where

M = Mp + MfA2M ,

C = Cp + CfA2M , (6.40)

KD = KDp + KfA

2M ,

where the open-circuit stiffness KD can be also expressed, in terms of the short-

circuit stiffness KE, as:

KE = KD − K2c

KQ

, (6.41)

or

KE = KD − K2c

KQ

, (6.42)

where KE represents the nominal short-circuit stiffness, which is normally Young’s

modulus, while KE indicates the situation that the circuit is shorted between the

electrode of the PZT and that of the added capacitance.

175

PZT tube

K f

Flap

d =f tqAM

q t df

Ro

R i

Figure 6.4. Fulcrum amplification mechanism for the PZT tube actuator

6.3 Mechanical Tuning and Electrical Tailoring

In this section, mechanical tuning and electrical tailoring are described. A mechan-

ical 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).

6.3.1 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 aero-

dynamic 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

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 mechan-

ical 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, Iδ, given in Equation

6.24.

6.3.2 Electrical Tailoring

It is convenient to express the actuation system equation in the nondimensional-

ized 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., KaQ = 0), the

transfer function between actuator stroke and control voltage and that between

electric charge and control voltage are, respectively,

qt

Vc

=Kc

(−ω2M + jωC + KD)(−ω2L + jωR + KQ)−K2c

, (6.43)

Q

Vc

=−ω2M + jωC + KD

(−ω2M + jωC + KD)(−ω2L + jωR + KQ)−K2c

, (6.44)

and in the nondimesionalized form

qt

Vc

=δ2

(1 + ξ2 + 2jζω − ω2)(rjω + δ2 − ω2)− δ2ξ2

(qt

Va

)

ST

, (6.45)

Q

Vc

=δ2(1 + ξ2 + 2ζjω + ω2)

(1 + ξ2 + 2jζω − ω2)(rjω + δ2 − ω2)− δ2ξ2

(1

1 + ξ2

)(Q

Va

)

ST

, (6.46)

where

ω2E =

KE

M, ω2

c =KQ

L, ω =

ω

ωE

, ξ2 =K2

c

KEKQ

,

ζ =C

2MωE

, r =R

LωE

, δ =ωc

ωE

, (6.47)

177

and subscripts ST represents the static response that are given by

(qt

Va

)

ST

=ξ2

Kc

, (6.48)

(Q

Va

)

ST

=1 + ξ2

KQ

. (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:

ξ2 =(KE + K2

c /KQ)−KE

KE=

KD −KE

KE=

ω2D − ω2

E

ω2E

. (6.50)

Now the open-circuit frequency can be expressed, in terms of the generalized

electro-mechanical coupling coefficient and the short-circuit frequency, as:

ω2D = (1 + ξ2)ω2

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 station-

ary value with respect to frequency at this invariant point ω =√

1 + ξ2 that is

the open-circuit frequency normalized with the short-circuit frequency. The opti-

mal tuning ratios for the voltage deriving response can be derived based on the

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 6.45. This can be rewritten, without the me-

chanical damping, as follows:

qt

Vc

∣∣∣∣ζ=0

=δ2

(1 + ξ2 − ω2)(rjω + δ2 − ω2)− δ2ξ2

(qt

Va

)

ST

. (6.52)

This function has an invariant value at ω2 = 1 + ξ2 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 ω2 is

given by

∂

∂ω2

(qt

Vc

∣∣∣∣ζ=0,r=0

)∣∣∣∣∣ω2=1+ξ2

= 0, (6.53)

which yields

[−(1 + ξ2)(δ2 − ω2)− (1 + ξ2 − ω2)]∣∣

ω2=1+ξ2 = 0. (6.54)

From Equation 6.54, the optimal inductance tuning ratio δ∗ can be found by

δ∗ =√

1 + ξ2, (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 + ξ2. 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

179

frequency), (qt

Vc

)∣∣∣∣ζ=0,ω2=1+ξ2

=

(qt

Vc

)∣∣∣∣ζ=0,ω2=1

, (6.56)

which yields

r∗ =√

2ξ√

1 + 0.5ξ2. (6.57)

This optimal resistance tuning ratio, however, should be calculated by consid-

ering the actuation system damping, i.e. ζ 6= 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

Vc

)∣∣∣∣ω2=1+ξ2

=

(qt

Vc

)∣∣∣∣ω2=1

. (6.58)

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

qt

Vc

=Kc

(−ω2M + jωC + KD)(−ω2L + jωR + KQ)−K2c

, (6.59)

which can be written in the dimensionless form using non-dimensionalized param-

eters defined in Equation 6.47,

qt

Vc

=δ2ξ2/ξ2

(1 + ξ2 + 2jζω − ω2)(rjω + δ2 − ω2)− δ2ξ2

(qt

Va

)

ST

, (6.60)

where a hat () indicates the system response with the added capacitance, the

modified coupling coefficient ξ is defined by

ξ2 =K2

c

KEKQ

. (6.61)

180

The short-circuit frequency, which includes the added capacitance, is given by

ω2E = (1 + ξ2 − ξ2) ω2

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

δ∗ =√

1 + ξ2, (6.63)

r∗ =√

2ξ

√1 + ξ2 − 0.5ξ2

1 + ξ2 − ξ2. (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.

6.4.1 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 gov-

erned 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

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,

Cn, 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

LmQ2 + RmQ2 +

(1

Cm + Cep

)Q2 − 1

Cep

= 0, (6.65)

LQ + RQ +1

Cep

Q− 1

Cep

Q2 = Vc, (6.66)

where the electric charge Q2 in the first branch emulates the actuator stroke of the

PZT tube actuator. The stiffness, damping, and mass of the mechanical part of the

182

PZT actuator and trailing-edge flap contributions are represented by capacitor Cm,

resistor Rm, and inductor Lm, respectively. Cep represents the total capacitance,

Cep =

CnCsp

Csp + Cn

. (6.67)

To use this equivalent electric circuit model, the coefficients of the Van Dyke

model (Cm, Rm, Lm) should be determined. It can be resolved by measuring the

electrical impedance of the resonant actuation system directly [150]. The equiv-

alent 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 frequen-

cies in the Van Dyke model, parallel and series frequencies (ωp, ωs) that are given

by

ω2p =

1

Lm

Cm + Cep

CmCep

=KD

M= ω2

D, (6.68)

ω2s =

1

LmCm

=KD −K2

c /KQ

M=

KE

M= ω2

E. (6.69)

From the equivalence of these two frequencies, the following relationships can be

obtained.

Lm =1

Cep

1

(ω2p − ω2

s), (6.70)

Rm = LmC

M, (6.71)

Cm =1

Lmω2s

, (6.72)

Kc =

√M

Cep

(ω2

p − ω2s

). (6.73)

These clearly show that the coupling stiffness Kc 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 ef-

fective capacitance (using the negative capacitance).

183

6.4.2 Analysis and Experimental Verification

In order to evaluate the resonant actuation system for a trailing-edge flap, a Mach-

scaled 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 Ro is assumed to be

0.175 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 ValueCharge constant, d15 741 m/V or C/N (10−12)Voltage constant, g15 26.8 V m/N (10−3)Elastic constant (open-circuit), cD

55 2.3 N/m2 (1010)Dielectric constant, εT

11 2.765 F/m (10−8)Piezoelectric constant, h15 6.164 V/m or N/C (1010)Impermittivity, βS

11 5.865 m/F (107)

When appropriate amplification ratio and mechanical tuning frequency (usu-

ally, 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 6.38. 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

184

the voltage driving response of the baseline actuation system. The dotted and dash-

dotted lines denote the voltage driving and passive damping responses, respectively,

with both negative capacitor (Cn = −2.2Csp) 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 Mach-scaled 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 Rm-

185

Lm-Cm 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 6.70–6.72, where the resonant frequency of the left branch is tuned to

4/rev (120Hz). The voltage across the capacitor Cm is measured, and an HP35665A

dynamic signal analyzer is used to extract the frequency response.

Figure 6.7. Realization of the equivalent electric circuit for the resonant actuationsystem

To create the negative capacitance, the negative impedance converter of capac-

itance [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 -18.4 nF (-

2.2 Csp) and the capacitance Cm and inductance Lm 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

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 well-

tuned 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 Mach-scaled rotor

187

6.5 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 Mach-

scaled 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 de-

termined. 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 us-

ing operational amplifier-based circuitry. An experiment is set up using equivalent

circuit representing the integrated structure-actuator-network system. It is demon-

strated 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.

Chapter 7Design 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.

189

7.1 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.

7.1.1 Resonant Actuators with R-L elements

Given that the electrical property of a piezoelectric actuator is similar to a capaci-

tor, 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.

R C

p

V p

V c

L

PZT V

a

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 fre-

quencies) 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

Vc

)

ω=1,ω=√

1+ξ2

= − 1

ξ2

(qt

Va

)

ST

, (7.1)

190

where (qt

Va

)

ST

=ξ2

Kc

, (7.2)

which was presented in Equation 6.48 on Page 177.

0 0.4 0.8 1.2 1.6 2−20

−10

0

10

20

30

40

50Frequency response of q

t/V

c (dB)

Nondimensionalized Frequency

baselineξ=0.2ξ=0.4ξ=0.6

Figure 7.2. Actuator strokes with the optimal tuning ratios and various couplingcoefficients

In general, larger electro-mechanical coupling coefficient, ξ, yields better ac-

tuator performance, especially for the static case. There is, however, a trade-

off 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 electro-

mechanical coupling coefficient will increase the bandwidth. Here, the bandwidth

is defined by

Bω =ωD − ωE

ωE

=√

1 + ξ2 − 1 =1

2ξ2(1− 1

4ξ2). (7.3)

Variations of the stroke and bandwidth with different generalized coupling co-

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 7.1–7.3 can serve as design

criteria for the resonant actuation systems.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Coupling coefficient, ξ

log(qt/Vc)

Bω

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 fre-

quency 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 ωD. 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

0 mm

]xe

ye

+

[CM 0

0 cm

]xe

ye

+

[KM + km −km

−km km

]xe

ye

=

0

fe

, (7.4)

192

0 0.4 0.8 1.2 1.6 2−20

−10

0

10

20

30

40

50Frequency response of q

t/V

c (dB)

Nondimensionalized Frequency

baseliner=0.01r=r*

r=1

Figure 7.4. Actuator strokes with the optimal inductance tuning and various resistancetuning 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 xe and ye correspond to the actuator stroke qt and

electric charge Q, respectively. The small spring km represents both the electro-

mechanical coupling and the inverse of the piezoelectric actuator capacitance. CM

and cm correspond to the mechanical damping and electric resistance, respectively.

fe represents the voltage source Vc.

Note that the damper cm applied on the small mass mm 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 mm = 0, the equivalent mechanical

system shown in Figure 7.5 becomes a one-degree-of-freedom system. If the damp-

ing cm is zero, the displacement xe is not affected by the stiffness km. The system

193

MM

m

x

y

KM

km

CM

cm

f(t)

mm

Figure 7.5. Schematic of an equivalent mechanical system to a piezoelectric actuationsystem

is analogous to the short-circuit situation, and the resonant frequency will become:

ωE =

√KM

MM

. (7.5)

On the other hand, if the damping cm is infinity, the displacement xe is affected by

both springs KM and km, which represents the behavior of piezoelectric actuator

under the open-circuit condition, with the resonant frequency

ωD =

√KM + km

MM

. (7.6)

In the case that the small mass mm 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 cm 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

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 ye in Figure

7.5. This indicates that large displacement xe could be achieved with the smallest

excitation of fe at the open-circuit frequency.

0 0.4 0.8 1.2 1.6 2−40

−30

−20

−10

0

10

20

Nondimensionalized Frequency

Amplitude of Q (dB)

ξ=0.2ξ=0.4ξ=0.6

Figure 7.6. Electric charges with the optimal tuning ratios for various coupling coeffi-cients

195

7.1.2 Resonant Actuation Systems with Additional Capac-

itance

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 trans-

ducer, and with a R-L circuit (see Figure 7.7), the transfer function between ac-

tuator stroke and control voltage is given in Equation 6.59 on Page 179. Now

relationship between two generalized coefficients can be obtained by

ξ2 =K2

c

KEKQ

= ξ2KQ

KQ

. (7.7)

R C

p

V p

V c

L

PZT

C add

V a

Figure 7.7. A piezoelectric network with a series R-L circuit and an additional capacitor

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,

KaQ > − 1

1 + ξ2KQ, or ξ2 < 1 + ξ2. (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,

ωE =

√(1 + ξ2 − ξ2) ωE, (7.9)

and its range is given by

0 < ωE < ωD =√

(1 + ξ2) ωE. (7.10)

This means that the new resonant frequency can be assigned to somewhere below

the open-circuit frequency ωD. This situation can be explained further by con-

sidering an equivalent mechanical system of a piezoelectric actuation system with

an additional capacitor (see Figure 7.8). Dynamic equations of the equivalent

mechanical system are given by

[MM 0

0 mm

]xe

ye

+

[CM 0

0 cm

]xe

ye

+

[KM + km −km

−km km + kn

]xe

ye

=

0

fe

, (7.11)

where an added spring kn only affects the second equation representing the elec-

trical behavior of actuation systems. Note that here kn could be negative.

As shown in Figure 7.8, the spring kn that corresponds to KaQ is directly at-

tached to the ground, so that it only affects the short-circuit frequency. It is clearly

seen that if the displacement ye is fixed (i.e., open-circuit condition), the displace-

ment xe is independent of the added spring kn. For a short-circuit condition, the

displacement ye is freely moving, the displacement xe 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

197

MM

m

x

y

KM

km

CM

cm

f(t)

mm

kn

Figure 7.8. Schematic of an equivalent mechanical system to a piezoelectric actuationsystem 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

kn > − KMkm

KM + km

. (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 ξ as:

r∗ =√

2ξ

√1 + ξ2 − 0.5ξ2

1 + ξ2 − ξ2, (7.13)

where one can see that if the coupling coefficient, ξ, approaches the nominal cou-

pling coefficient, ξ, the optimal resistance tuning ratio becomes what is shown in

Equation 6.57.

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 (ω2 = 1 + ξ2),

(qt

Vc

)∣∣∣∣ω=√

1+ξ2−ξ2,√

1+ξ2

= − 1

ξ2

(qt

Va

)

ST

, (7.14)

and the bandwidth,

Bω =ωD − ωE

ωE

=√

1 + ξ2 −√

1 + ξ2 − ξ2,

≈ 1

2ξ2

[1 +

1

4(ξ2 − 1

2ξ2)

], (7.15)

which can be varied by adjusting the value of the added capacitance.

199

As noted in Reference [125], the coupling coefficient does not change the mag-

nitude at the invariant point (open-circuit frequency, ωD), but it does change the

bandwidth and the static stroke authority:

(qt

Vc

)∣∣∣∣ω=0

=ξ2/ξ2

1 + ξ2 − ξ2

(qt

Va

)

ST

. (7.16)

The magnitude at the invariant point with respect to the static stroke can be

evaluated by (qt

Vc

)∣∣∣∣ω=√

1+ξ2

=1 + ξ2 − ξ2

ξ2

(qt

Vc

)∣∣∣∣ω=0

. (7.17)

As far as the magnitudes at the vicinity of the resonant frequency are con-

cerned, 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 parame-

ter. Variations of relative magnitude and bandwidth due to added capacitance

are plotted in Figure 7.10, where the nominal coupling coefficient of the piezoelec-

tric tube actuator, ξ, is 0.5 and the modified coupling coefficient, ξ, 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.

7.1.3 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

200

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Modified coupling coefficient, ξhat

log(qt/Vc)

Bω

Figure 7.10. Relative actuator stroke and bandwidth variations with modified couplingcoefficients

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 + ξ2

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√

1 + 0.5ξ2

for without added capacitance, and

r∗ =√

2ξ

√1 + ξ2 − 0.5ξ2

1 + ξ2 − ξ2

201

for with added capacitance.

• Actuator stroke with respect to the static stroke is expressed by

(qt

Vc

)

ω2=1+ξ2

/

(qt

Va

)

ST

=1

ξ2

for without added capacitance, and

(qt

Vc

)

ω2=1+ξ2

/

(qt

Vc

)

ω=0

=1 + ξ2 − ξ2

ξ2

for with added capacitance.

• Operating bandwidth, Bω, is expressed by

Bω =1

2ξ2

(1− 1

4ξ2

)

for without added capacitance, and

Bω =1

2ξ2

[1 +

1

4

(ξ2 − 1

2ξ2

)]

for added capacitance.

• Best compromised coupling coefficients, ξ and ξ, 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 6.54Optimal resistance tuning, r∗ Equations 6.57 and 7.13Capacitance tuning (stroke) Equations 7.17Capacitance tuning (bandwidth) Equation 7.15Capacitance tuning (trade-off) Figure 7.10

202

7.2 Dynamic Characteristics of the RAS in For-

ward 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 sys-

tems. Design guidelines for such systems are provided. In this section, the dynamic

characteristics of the RAS will be examined, since the aerodynamic stiffness, Kf ,

presented in Section 6.2.2 is not constant. The analysis is performed for piezoelec-

tric 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 perfor-

mance of various flap configurations is evaluated within the available actuation

authority, which can be realized by the actuator saturation algorithm presented in

Section 3.3.1 on Page 106.

7.2.1 A Perturbation Method

The aerodynamic contribution of the flap based on Theodorsen’s theory with con-

sidering the forward flight speed can be written as:

Ff (δf ; t) = Mf δf + Cf [1 + ε sin(Ωt)] δf + Kf [1 + ε sin(Ωt)]2 δf , (7.18)

where

ε =µ

rf

, (7.19)

in which rf denotes the flap location along the rotor spanwise direction. Mf , Cf

and Kf were derived in Equation 6.29 on Page 172 in Section 6.2.2.

With this aerodynamic contribution, coupled actuator-flap-circuit system equa-

tions given in Equation 6.39 are rewritten by

[M 0

0 L

]qt

Q

+

[Cµ 0

0 R

] qt

Q

+

[KD

µ −Kc

−Kc KQ

]qt

Q

=

0

Vc

, (7.20)

203

where

Cµ = Cp + CfA2M [1 + ε sin(Ωt)] , (7.21)

KDµ = KD

p + KfA2M [1 + ε sin(Ωt)]2 . (7.22)

For mechanical tuning, it is needed to define the nominal stiffness of the ac-

tuation 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 ε becomes zero in

Equation 7.20. The resonant frequency of the system in hover is then obtained as:

ωh =

√KE

M, (7.23)

where ωh represents the resonant frequency of the RAS in hover, and KE 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 sys-

tem without circuitry. Equation 7.20, without damping, is then rewritten as fol-

lows:

Mqt + KEµ (t)qt = Fp(t), (7.24)

where

KEµ (t) = KE + KfA

2M

[2ε sin(Ωt) + ε2 sin2(Ωt)

], (7.25)

Fp(t) =Kc

KQ

Vc(t), (7.26)

Vc(t) = Vc cos(ωt), (7.27)

where Vc represents the magnitude of voltage signal, and ω denotes the excitation

frequency.

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 +[α2

m + β2m

(2ε sin ψ + ε2 sin2 ψ

)]qt = Γm(cos γmψ), (7.28)

where

α2m =

1

Ω2

(KE

M

), (7.29)

β2m =

1

Ω2

(KfA

2M

M

), (7.30)

γm =ω

Ω, (7.31)

Γm =1

Ω2M

(Kc

KQ

)Vc. (7.32)

The most elementary version of the perturbation method is to attempt a rep-

resentation of the solution of Equation 7.28 in the form of a power series in a small

parameter ε :

qt(ψ, ε) = qt0(ψ) + εqt1(ψ) + ε2qt2(ψ) + · · · , (7.33)

whose coefficients qti(ψ) are only functions of ψ. To form equations for qti(ψ),

i = 0, 1, 2, · · · , substitute the series Equation 7.33 into Equation 7.28:

q∗∗t0 + εq∗∗t1 + ε2q∗∗t2 + · · ·+ [α2

m + β2m

(2ε sin ψ + ε2 sin2 ψ

)

(qt0 + εqt1 + ε2qt2 + · · · )] = Γm cos(γmψ). (7.34)

Equating each of the coefficients of ε0, ε1 and ε2 to zero, we have

ε0 : q∗∗t0 + α2mqt0 = Γm cos(γmψ), (7.35)

ε1 : q∗∗t1 + α2mqt1 = −2β2

mqt0 sin ψ, (7.36)

ε2 : q∗∗t2 + α2mqt2 = −2β2

mqt1 sin ψ − β2mqt0 sin2 ψ, (7.37)

and so on.

205

As far as the steady-state solutions are concerned, the periodicity of 2π can be

applied. That is,

qt(ε, ψ + 2π) = qt(ε, ψ). (7.38)

Equations 7.35 – 7.35 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

7.39. The method obviously fails if the resonant frequency α2m takes one of the

values γ2m, (γm ± 1)2, (γm ± 2)2, · · · , since certain terms would then be infinite.

Such values of correspond to conditions of near-resonance. αm = γm is so called

a primary or main resonance. The other values of αm 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].

qt(ψ; ε) =Γm

α2m − γ2

m

cos(γmψ)

+ ε

[ao1

α2m − (1− γm)2

sin(1− γm)ψ +bo1

α2m − (1 + γm)2

sin(1 + γm)ψ

]

+ ε2

[co2

α2m − γ2

m

cos(γmψ) +ao2

α2m − (1− γm)2

sin(2− γm)ψ

+bo2

α2m − (1− γm)2

sin(2 + γm)ψ

], (7.39)

where

ao1 = bo1 = − β2mΓm

α2m − γ2

m

, (7.40)

ao2 = ao1

(β2

m

α2m − (1− γm)2

− 1

4

), (7.41)

bo2 = bo1

(β2

m

α2m − (1 + γm)2

− 1

4

), (7.42)

206

co2 = −ao1

[β2

m

α2m − (1− γm)2

+β2

m

α2m − (1 + γm)2

− 1

2

]. (7.43)

7.2.2 Analysis of Time Responses

Time responses can be directly calculated using the state-space form of Equa-

tion 7.20. 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 Ro is 0.35 inches,

and inner radius Ri is 0.225 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 Vrms) is applied to the actuation system. The total actuator

weight of 0.775 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

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.

0 10 20 30 40 500

2

4

6

8

10

12

14Flap deflections with various flight speeds

Fla

p de

flect

ions

, deg

rees

Excitation frequency, Hz

hoverµ=0.15µ=0.35

Figure 7.11. Peak-to-peak flap deflections of a resonant actuator with various flightspeeds

Figure 7.12 shows the variation of instantaneous frequencies and averaged fre-

quencies. 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 the-

ory. 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 7.11.

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].

208

0 90 180 270 3600

5

10

15

20

25

30

35

40Variation of instantaneous frequencies

Fre

quen

cy, H

z

Azimuth, degrees

hoverµ=0.15µ=0.35

Averaged frequencyHover : 26.6 Hzµ=0.15 : 26.7 Hzµ=0.35 : 27.3 Hz

Figure 7.12. Variations of instantaneous frequencies along the azimuth

1.05 1.0875 1.125 1.1625 1.2−15

−10

−5

0

5

10

15Flap time response for a resonant actuator tuned to 4P

time, sec

Fla

p de

flect

ions

, deg

rees

ψ=90 ψ=180 ψ=270

Figure 7.13. Time history of flap motions of the actuation system without circuitrywith 4/rev voltage excitation, µ = 0.35

209

Time Responses with Circuitry

To verify the optimal tuning parameters for the circuitry, the frequency re-

sponses 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 reso-

nance 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.

0 10 20 30 40 500

2

4

6

8Flap deflections with various flight speeds

Fla

p de

flect

ions

, deg

rees

Excitation frequency, Hz

hoverµ=0.15µ=0.35

Figure 7.14. Peak-to-peak flap deflections of the RAS with various flight speeds

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 7.16. Maximum flap deflection occurs

at the fourth quadrant on the retreating blade, which is the same as the actuation

system without circuitry.

Figure 7.15. Time history of flap motions of the RAS with 4/rev voltage excitation,µ = 0.35

211

Figure 7.16. Time history of flap motions of the nominal actuation system with 4/revvoltage excitation, µ = 0.35

212

7.2.3 Vibration Reduction Within Available Actuation Au-

thority

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 multiple-

flap configurations, respectively. Unlike previous cases, it is shown that multiple-

flap 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.

213

Figure 7.17. Comparison of vibratory hub loads for an advance ratio of 0.15 withinthe available actuator authority

Figure 7.18. Comparison of vibratory hub loads for an advance ratio of 0.35 withinthe available actuator authority

214

7.3 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 pro-

posed 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 mechani-

cal 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 cir-

cuitry. 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 op-

erating frequencies. Through these efforts, performance of the resonant actuation

system is validated experimentally.

7.3.1 Controller Design

In order to evaluate the performance of resonant piezoelectric actuators with elec-

tric circuitry, the piezoelectric induced shear tube actuator [116] is considered as an

example. There are a number of implementation problems associated with realiz-

ing 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 ampli-

fiers 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

215

the piezoelectric actuators to emulate the circuitry functions. This signal func-

tion 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 actu-

ator Va is given as follows:

Va

Vc

= KQ

(Q

Vc

)−Kc

(qt

Vc

)=

(ξ2

ξ2

)KQ

(Q

Vc

)−Kc

(qt

Vc

), (7.44)

which yields

(Va

Vc

)

cty

=δ2(ξ2/ξ2)(1 + 2jζω − ω2)

(1 + ξ2 + 2jζω − ω2)(rjω + δ2 − ω2)− δ2ξ2, (7.45)

where subscript ‘cty’ represents the voltage amplification ratio with a series R-L

circuit and the added capacitance.

The actuator stroke qt 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.

qt =

(qt

Va

) (Va

Vc

)

cty

Vc. (7.46)

Here it should be noted that the voltage across the piezoelectric actuator, Va,

should not exceed the de-poling limit. This is another constraint for adding ca-

pacitances to the piezoelectric RAS (especially for negative capacitance).

To incorporate the electric network in high voltage piezoelectric resonant actu-

ation 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).

216

Piezoelectric

Actuator

Flap with

tuning mass

Aerodynamics

Va/Vc Phaser V ref

V signal

DSP

RAS

Figure 7.19. 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.

δf (t) = cos φδ δreff (t) +

sin φδ

ωδf

ref(t). (7.47)

A similar strategy was used experimentally in Reference [154]. In this paper, the

phase angle φδfis adaptively corrected through the feedback of the output signal

(see Figure 7.20).

217

Figure 7.20. Diagram of an adaptive phase controller based on Matlab/Simulink

218

7.3.2 Bench Top Testing

A resonant actuation system using piezoelectric tube actuator developed by Cen-

tolanza 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 7.22.

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 7.21. Experimental set-up for the resonant actuation system

219

Figure 7.22. 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 (ξ = 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 2.1 ∼ 3.5 times when compared to the static deflection (which would be

produced by the original actuation system without resonant tuning). There is a

220

0 10 20 30 40 50 600

1

2

3

4

5

6

Frequency, Hz

Fla

p de

flect

ion

ratio

s

Mechanical tuning 4/rev (26.6Hz)

Baseline (Experiment)Baseline (Analysis)Mechanically tuned system (Experiment)Mechanically tuned system (Analysis)

Figure 7.23. Frequency responses of the PZT tube actuator before and after mechanicaltuning

trade off between bandwidth and stroke as illustrated in the figures.

Phase variations of a RAS (with the voltage signal function) shows more grad-

ual change than those of pure mechanically tuned system around the resonant

frequency, as is shown in Figures 7.24 and 7.25. This can help to track the phase

variation at the vicinity of resonant frequency. Required inductance values in these

cases are more than 1000 H, which is hard to implement using physical circuitry

components due to the high voltage across the inductor.

221

0 10 20 30 40 500

1

2

3

4

5

Fla

p de

flect

ion

ratio

s

Analytical predictions

0 10 20 30 40 50−180

−90

0

90

180

Frequency, Hz

Pha

se, d

eg

Figure 7.24. Analytical predictions of a resonant actuation system: —– , tuned systemw/o the voltage signal function; −−−, RAS with ξ = 0.5; · · · , RAS with ξ = 0.6

0 10 20 30 40 500

1

2

3

4

5

Fla

p de

flect

ion

ratio

s

Expreimental results

0 10 20 30 40 50−180

−90

0

90

180

Frequency, Hz

Pha

se, d

eg

Figure 7.25. Experimental results of a resonant actuation system: —–, tuned systemw/o the voltage signal function; −−−, RAS with ξ = 0.5; · · · , RAS with ξ = 0.6

222

While the voltage signal function, which emulating the electrical network, can

help to broaden and flatten the resonant peak of actuation system, the phase near

a resonant frequency varies with the control input frequencies. To track the phase

variation, an adaptive ‘phaser’ is implemented and tested experimentally. Signal

errors and flap signals before/after implementing the controller with a 24Hz input

are shown in Figure 7.26, where the solid and dash-dotted lines represent the

flap response and reference input, respectively. It is shown that the implemented

adaptive controller is able to accurately follow the reference input.

0 5 10 15 20−2

0

2

Sig

nal e

rror

s

24 Hz input signal

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−2

0

2

Bef

ore

cont

rol

10 10.05 10.1 10.15 10.2 10.25 10.3 10.35 10.4−2

0

2

Afte

r co

ntro

l

time, sec

referenceoutput

Figure 7.26. Time responses of flap deflection signal before/after phase control with24Hz input signal

Reference inputs of 26Hz and 29Hz are also tested in the experiment. Time

responses with 26Hz input signal are shown in Figure 7.27. It is illustrated that

the controller works well, implying that the it can adapt for the phase variation

near the operating frequency. Such a controller together with the voltage signal

function yields a robust and high performance flap actuation system.

223

0 5 10 15 20−2

0

2

Sig

nal e

rror

s

26 Hz input signal

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−2

0

2

Bef

ore

cont

rol

10 10.05 10.1 10.15 10.2 10.25 10.3 10.35 10.4−2

0

2

Afte

r co

ntro

l

time, sec

referenceoutput

Figure 7.27. Time responses of flap deflection signal before/after phase control with26Hz input signal

224

7.4 Power Consumption of Piezoelectric Reso-

nant Actuation Systems

In previous sections, it is demonstrated, analytically and experimentally, that the

high actuation authority with a wide bandwidth can be achieved near the operating

frequency for piezoelectric resonant actuation systems. In this section, the electric

power consumption of the present RAS is evaluated and compared to that of

piezoelectric resonant actuators without circuitry.

7.4.1 Piezoelectric resonant actuators without circuitry

The electrical property of a piezoelectric actuator is similar to a capacitor. The

electrical power consumption for the case of the non-ideal capacitor can be esti-

mated as [155,156]:

P = V × I × cos φc (7.48)

where V and I represent the absolute values of voltage and current respectively,

φc is the phase angle between voltage V and current I, and cos φc is known as the

power factor.

The power factor of a piezoelectric actuator is approximately zero when the

operating frequency is far away from the resonance frequency. In order to increase

the efficiency, it is desirable to increase the power factor and make it as close

to unity as possible [155]. In our case, this is achieved through the mechanical

resonance tailoring. That is, by tuning the actuation system natural frequency

to the operating frequency, the power factor becomes unity at two points in the

frequency response function, which are the short- and open-circuit frequencies.

From Equation 6.23 on Page 170, the frequency response between the current

and voltage source can be evaluated as:

I

Va

= ω

(Q

Va

), ∠

(I

Va

)= ∠

(Q

Va

)+ 90o. (7.49)

225

The frequency response and phase plot of a piezoelectric resonant actuator for

ξ = 0.5 with different resistances are shown in Figure 7.28, where rp = ωERp/KQ.

The resonant and anti-resonant frequencies correspond to the short- and open-

circuit frequencies, respectively. It is seen that the apparent electric power is

minimal at the open-circuit frequency. This is obvious since the open-circuit con-

dition is to have constant electric displacement which includes zero electric charge.

The optimal operating frequency would then be the open-circuit frequency ωD be-

cause the electric power to the actuators is minimal. As far as the power factor is

concerned, short- and open-circuit frequencies are the best operating frequencies

for electric power efficiency, in which the power factor is unity, (Since the phase,

φc, between current and voltage is zero at these frequencies, as shown in the phase

plot in Figure 7.28). The electric resistance can help to vary the phase gradu-

ally around the resonant frequency, but not the phase around the anti-resonant

frequency.

Figure 7.28. Frequency response of current and phase variation for ξ = 0.5 withdifferent resistances

226

7.4.2 Piezoelectric resonant actuators with circuitry

When a series R-L circuit is connected to the PZT, the frequency response between

the current and voltage source is given by, from Equation 6.46 on Page 176,

I

Vc

= ω

(Q

Vc

), ∠

(I

Vc

)= ∠

(Q

Vc

)+ 90o. (7.50)

The frequency response and phase plot of a piezoelectric resonant actuator with

a series R-L circuit for ξ = 0.5 are shown in Figure 7.29, where results are plotted

with the optimal inductance tuning. There are now three points where the power

factor is unity (i.e., the phase angle, φc, is zero), as shown in the second plot

of Figure 7.29. One is the open-circuit frequency, and the others are at the two

resonant frequencies. The voltage across the PZT should be considered to evaluate

the power consumption since it is not constant in this case.

Figure 7.29. Frequency response of current and its phase with circuitry for ξ = 0.5

227

The voltage across the piezoelectric actuator, which is similar to Equation 7.45,

can be evaluated as:

(Va

Vc

)=

δ2(1 + 2jζω − ω2)

(1 + ξ2 + 2jζω − ω2)(rjω + δ2 − ω2)− δ2ξ2, (7.51)

and the apparent electric power is then evaluated,

(P

Vc

)= ω

(Va

Vc

)(Q

Vc

)Vc, (7.52)

which has two poles at the short- and open-circuit frequencies. This is clearly

illustrated in Figure 7.30 showing the apparent electric power of a piezoelectric

resonant actuator with circuitry for ξ = 0.5. Results are obtained under the

optimal inductance tuning and compared to the baseline (without circuitry).

Figure 7.30. Apparent electric power of a piezoelectric actuator with circuitry forξ = 0.5

There is now the banded frequency range between the short- and open-circuit

frequencies. This bandwidth can be evaluated by Equation 7.3 on Page 190 that

is closely related to the electro-mechanical coupling coefficient ξ. Corresponding

228

actuator strokes are presented in Figure 7.31. From these figures, one can conclude

that a piezoelectric resonant actuation system with proper tuning parameters not

only yields high authority with broad bandwidth but also is electrically efficient

when compared to that without circuitry.

Figure 7.31. Actuator strokes under the optimal tuning ratios for ξ = 0.5

229

7.5 Summary

The investigation of this chapter addresses research issues related to the synthesis

and realization of a piezoelectric resonant actuation system described in Chapter

6. The efforts and findings can be summarized as follows:

• Design guidelines for the RAS circuitry (inductance, resistance and addi-

tional capacitance) are provided based on the nondimensionalized formulae.

The RAS system is also compared to an equivalent mechanical system to

provide better physical understanding.

• Time varying characteristics of the RAS in forward flights are investigated.

The analysis shows that the actuator authority is significantly increase from

1.25 to 4.5 degrees when compared to the static value. The main characteris-

tics of the RAS are conserved in forward flight conditions with the mechanical

tuning and electrical tailoring for hover condition.

• Within the available actuation authority, multiple-flap configuration with

resonant actuation systems can significantly reduce the vibration level over

the wide range of flight condition as compared to single- and dual- flap config-

urations. The multiple-flap configuration reduces the vibration by 91% and

79%, in low- and high-speed flight conditions, while single-flap configuration

does it by 35% and 56%.

• A method is proposed to emulate the circuitry functions by directly applying

a voltage signal function to the actuator. This will avoid the difficulties

of implementing high voltage analogue electrical components. The voltage

signal function is derived and implemented using a digital signal processor

(DSP) system, forming a feed-forward type controller. An adaptive ‘phaser’

is also implemented to track the phase variation near the resonant frequency.

• The concepts presented are applied to the piezoelectric induced-shear tube

actuator for MD900 class helicopters, and verified experimentally. The the-

oretical predictions match well with the experimental findings. Promising

results are illustrated in this study. The actuator stroke is increased 2 ∼ 3.5

times compared to its static value with wide bandwidth of 5 to 10 Hz.

230

• In addition to the actuator authority and bandwidth, the electric power

consumption is also investigated to truly evaluate the performance of the

RAS. It is found that the electric power consumption of the RAS is low

between the short- and open-circuit frequencies, where the required voltage

is very low at the short-circuit frequency and the required current is very low

at the open-circuit frequency. Based on this investigation, it is concluded that

the proposed piezoelectric resonant actuation system (RAS) with properly

tuned circuitry parameters not only yields a high actuation authority with a

wide bandwidth, but also is electrically very efficient.

Chapter 8Conclusions and Recommendations

In this thesis, active control methods utilizing the multiple trailing-edge flaps’

configuration and the piezoelectric resonant actuation system are proposed for ro-

torcraft vibration suppression and blade loads control, and such resonant actuation

systems are developed and tested. The general conclusions of this dissertation are

summarized, and recommendations for future work are presented in this chapter.

232

8.1 Summary of Research Efforts and Achieve-

ments

In Chapter 2, an aeroelastic analysis has been formulated to examine the per-

formance and loads of rotorcraft with elastically coupled composite blades. The

analysis includes the free-wake model developed by Tauzsig and Gandhi [132] to

accurately predict the blades loads. A nonlinear composite beam theory, which

includes the cross-section model based on Vlasov theory, is employed to obtain the

blade structural model. The derivation of strain measure used in this formulation

is presented in Appendix A. The explicit forms of strain and kinetic energy con-

tributions used in the finite elements implementation are provided in Appendix B.

Blade loads calculation and vehicle propulsive trim procedure are also provided.

In Chapter 3, the trailing-edge flap model and optimal controller for both

vibration and blade loads have been developed. The inertial loads of trailing-

edge flaps are derived based on the previous work [69]. The incompressible and

compressible aerodynamic models were presented to predict the aerodynamic loads

generated by flap motions. The optimal controller with the actuator saturation

algorithm were developed to reduce the vibration and blade loads. The hybrid

design approach developed by Zhang [57] was briefly reviewed.

Based on the formulations presented in Chapter 2 and Chapter 3, the feasibility

of multiple trailing-edge flaps for the simultaneous reductions of vibration and

blade loads was investigated in Chapter 4. The concept involved straightening the

blade by introducing dual trailing edge flaps in a conventional articulated rotor

blade. An active loads control strategy was numerically tested for the steady-state

forward flight condition, µ = 0.35. It was shown that straightening the rotor blade

using dual flap configurations can reduces both vibratory hub loads and bending

moments without a significant change of control settings for trim. Only 1/rev

control input was required to reduce vibratory hub loads with the present method,

which is well suited for resonant actuation systems. The proposed active loads

control method was shown that the flapwise bending moment and the vibratory

hub loads can be reduced by 32% and 57%, respectively, through dynamically

233

straightening the blade with the 1/rev control input. The required maximum

flap deflection was reduced by 25% when compared to retrofit design using the

composite tailoring of pitch-flap coupling stiffness via a hybrid design approach.

In Chapter 5, an active control method for multiple trailing-edge flaps configu-

ration was developed. The concept involved deflecting the trailing-edge flaps by in-

troducing the resonant actuation, in which each flap operates at a single frequency,

in a four-bladed hingeless rotor system. A proposed active vibration control strat-

egy was simulated for the steady-state forward flight conditions (µ = 0.15, 0.35).

The finite wing effect of the flap was not significant in the multiple trailing-edge

flap configuration, since the controller was able to compensate for changes in flap

capability by adjusting the flap deflections and phases. For the low-speed flight,

the vibration level was reduced by 56 ∼ 60 % with a 4/rev control input. It was

shown that 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. It was also shown that 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, the peak-to-peak hinge moments

were reduced by 37.5% to 61.2% in each individual actuator compared to single-flap

configuration.

In Chapter 6, a new approach was developed to enhance the actuation author-

ity of piezoelectric actuation systems. The idea was to first achieve a resonant

driver through using mechanical tuning, and then increase the bandwidth and ro-

bustness of the resonant actuation system through electrical network tailoring. A

coupled PZT actuator, trailing edge flap and electrical circuit dynamic model was

derived. Utilizing the developed model, the required electrical circuitry parameters

were systematically determined. A Mach-scaled piezoelectric tube actuator-based

trailing-edge flap for helicopter vibration control was used as an example to illus-

trate the proposed concept. The negative capacitance and inductance were real-

ized and implemented using operational amplifier-based circuitry. An experiment

were set up using equivalent circuit representing the integrated structure-actuator-

network system. It was demonstrated that the proposed resonant actuation system

234

can indeed achieve both high active authority and robustness.

While investigations on the electro-mechanically tailored piezoelectric resonant

actuation system have shown promising results in Chapter 6, there were still re-

search issues to be addressed before such a concept can be realized. In Chapter

7, such issues related to the synthesis and realization of a piezoelectric resonant

actuation system were addressed.

First, design guidelines for the RAS circuitry (inductance, resistance and addi-

tional capacitance) were provided based on the nondimensionalized formulae. The

RAS system were also compared to an equivalent mechanical system to provide

better physical understanding.

Second, time varying characteristics of the RAS in forward flights were inves-

tigated. It was shown that the actuator authority is significantly increase from

1.25 to 4.5 degrees when compared to the static value. The main characteristics of

the RAS were conserved in forward flight conditions with the mechanical tuning

and electrical tailoring for hover condition. It was also demonstrated that within

the available actuation authority prediced, the multiple-flap configuration with

resonant actuation systems can significantly reduce the vibration level over the

wide range of flight condition. The multiple-flap configuration reduces the vibra-

tion by 91% and 79%, in low- and high-speed flight conditions, while single-flap

configuration does it by 35% and 56%.

Third, a method was developed to emulate the circuitry functions by directly

applying a voltage signal function to the actuator. The voltage signal function was

derived and implemented using a digital signal processor (DSP) system, forming

a feed-forward type controller. An adaptive ‘phaser’ was also implemented to

track the phase variation near the resonant frequency. The concept presented

were applied to the piezoelectric induced-shear tube actuator for MD900 class

helicopters, and verified experimentally. The theoretical predictions matched well

with the experimental findings. Promising results were illustrated in this study.

The actuator stroke was increased 2 ∼ 3.5 times compared to its static value with

wide bandwidth of 5 to 10 Hz.

Fourth, in addition to the actuator authority and bandwidth, the electric power

consumption is also investigated to truly evaluate the performance of the RAS.

235

Based on this investigation, it is concluded that the proposed piezoelectric resonant

actuation system (RAS) with properly tuned circuitry parameters not only yields

a high actuation authority with a wide bandwidth, but also is electrically very

efficient.

8.2 Recommendations for Future Work

The work described in this dissertation is to reduce the helicopter vibration and

blade loads using multiple trailing-edge flaps together with resonant actuation

systems and to develop such resonant actuation systems. Although the present

research has demonstrated that the multiple-trailing edge flap configuration with

resonant actuation systems can achieve the vibration reduction and blade loads

reduction within the available actuation authority, there are still room to improve

and extend the present research. Several suggestions for future work are described

below.

1. Trailing-Edge Flap Dynamics in Comprehensive Rotor Analysis:

The first recommendation for future study is that it is needed to consider the

trailing-edge flap dynamic model in an aeroelastic comprehensive rotor anal-

ysis. The trailing-edge flap response is strongly affected by the time-varying

aerodynamic loads as shown in Section 7.2.2. Thus implementing accurate

dynamic model of the actuation system is important for more realistic anal-

ysis, especially for the case of using the resonant actuation concept.

2. Active Loads Control for Heavy Lift Class Helicopter:

A heavy lift class helicopter suffers from the excessive blade loads. Dynami-

cally straightening the blade could yield low blade loads as well as low vibra-

tions using dual flap configuration. In this research, the blade loads control

was explored only for the high speed flight. To consider the low speed flight,

the wake-model should be used in this case. Multiple flaps configuration can

be also applied to reduce the blade loads via the resonant actuation concept.

It is also needed to reduce the pitch-link loads and lag bending moment as

well as the flap bending moment.

236

3. Realistic Adaptive Controller for Helicopter Vibration and Blade Loads Con-

trols:

It is needed to design the realistic controller that performs the actuator dy-

namic compensation as well as system variations (e.g. air turbulence)

4. Further Investigation of Composite Tailoring:

Composite tailoring was considered in active loads control using dual flap

configuration in Section 4. To reduce both vibration and blade loads, com-

posite tailoring can be used in multiple-trailing edge flap configuration. This

can be further extended to reduce the pitch-link loads together with dynamic

shock absorber located at the pitch-link place.

5. Swashplateless Helicopter Control Using Multiple Flaps:

As shown in Chapter 4 and Chapter 5, the resonant actuation concept is well

suited for the multiple flap configuration. Using only 1/rev control input can

reduce both blade loads and vibration. This can be further extended for

swashplateless helicopter control, since it requires 1/rev control as the pri-

mary vehicle control. Multiple flaps together with 1/rev resonant actuation

could improve the swashplateless helicopter control.

6. Nonlinear Modeling of Piezoelectric Resonant Actuators:

The present model of piezoelectric resonant actuation system was based on

the linear piezoelectricity. There are nonlinearities in piezoelectric materials

due to the high voltage excitation, high dynamic stress, and temperature. In

order to consider these nonlinearities, nonlinear modeling is needed. Thus

one can achieve better physical understandings and appropriate controller

design for the piezoelectric resonant actuation system.

7. Controller for Hysteresis and Nonlinear Resonances in Resonant Actuation

Systems:

There are two types of nonlinearities in piezoelectric materials, such as hys-

teresis and material nonlinearity. These two nonlinearities are important in

designing the controller of piezoelectric actuators, especially for the resonant

type actuators. In the case of the resonant actuation systems for active flap

rotors, they are under both high voltage and high stroke driving operations.

237

8. Wind Tunnel Test of Resonant Actuation Systems:

The resonant actuation system developed was verified experimentally only

for the hovering condition. It is needed to test it for forward flight conditions

for the investigation of aerodynamic damping and structural damping that

are important parameters in trailing-flap actuation performance. This can

be achieved through the wind tunnel test.

Appendix ADerivation of Strain Measure

The strain-displacement relations reflect the nonlinearity due to moderately large

rotation of the flexible beam. Additional terms are also visible due to blade pitch,

θo. This pitch is generally due to combinations of elastic torsion, pretwist, and

pitch control inputs.

The strain measure with small strains but moderately large rotation was de-

veloped by Hodges and Dowell [134]. They developed the strain measure based

on Eulerian formulation. The cross-section of beam, however, was treated as a

solid section. In this section, the strain measure is re-derived for a thin-walled

cross-section. The strain measure is given by

εxx = u′ +v′2

2+

w′2

2− λT φ′′ + (η2 + ζ2)

(θo′ + φ′ +

φ′2

2

)− yv′′ − zw′′, (A.1)

2εxη = γxη = −(ζ + λT,η)φ′, (A.2)

2εxζ = γxζ = (η − λT,ζ)φ′, (A.3)

where λT is a warping function of beam cross-section, (),x and ()′ are equivalent

to ∂()∂x

,

y = η cos θ1 − ζ sin θ1 , z = η sin θ1 + ζ cos θ1 , (A.4)

and γxη and γxζ are engineering shear strains.

Here, note that the coordinate system eci is based on the undeformed coordinate

system eui but rotated by pitch input angle θo due to rigid input pitch θ75 and linear

239

V

h

V

h

s

n

cq

cP

P

)cos(

)sin(

)cos()sin(),(),(

),(

32

32

c

c

c

c

c

cc

cc

c

n

n

enenPP

eeP

qzV

qhh

qqzhVh

zhzh

-=

+=

-+=

+=rr

r

rr

r

ce2

r

ce3

r

h

z

Figure A.1. Thin-walled cross section of rotor blade

twist angles θtw. This relation can be obtained as:

eci = T cu

ij euj , Tcu =

1 0 0

0 cos θo sin θo

0 − sin θo cos θo

≡ (Tdu)o . (A.5)

Figure A.1 shows the coordinate systems of the thin-walled cross section of rotor

blade. The contour line of the cross section is represented by the local coordinate

system, s and n. The distance from torsion center to contour line can be expressed

by η and ζ, and relations between (η, ζ) and (η, ζ) are given by

η = η + n sin θc , ζ = ζ − n cos θc, (A.6)

λT = λT − n (η cos θc + ζ sin θc), (A.7)

where the overbar () represents quantities on the contour line. These relationships

can be found in Reference [137].

Note that θc and θ1 are different angles. The former indicates the relative

orientation of the two coordinate systems that are the local coordinate system

(ξ, η, ζ) and the undeformed coordinate system eci after rotated by rigid pitch

input and linear twist angle θo. The latter represents the total blade pitch angles.

240

A.1 Coordinate Transformation

For convenience, the strains based on (x, η, ζ) coordinates are transformed into

the strains based on (x, s, n) coordinates. It is assumed that these two coordinate

systems are on the same plane, which is consistent with Euler-Bernoulli hypothesis.

Then shear strains are given as follows:

γxs = γxη cos θc + γxζ sin θc, (A.8)

γxn = γxη sin θc − γxζ cos θc, (A.9)

in which γxs represents the shear strain due to torsion, while γxn represents the

shear strain due to flexure. The shear strain γxn may be cancelled out because

it was developed under Euler-Bernoulli hypothesis, which means that the cross

section remains rigid and perpendicular to the elastic axis.

The shear strains, γxs and γxn, can be expressed in the coordinate system (η,

ζ, n) as follows:

γxs = (r − λT,s)φ′ + n 2φ′ = γt

xs + n kxs, (A.10)

γxn = −qφ′ − (λT,η sin θc − λT,ζcos θc)φ

′ ≡ 0, (A.11)

where

r = η sin θc − ζ cos θc , q = η cos θc + ζ sin θc. (A.12)

As shown in Equation A.11, the shear strains due to flexure γxn turns out to

be zero because

∮q ds = 0 , cos θc =

dη

ds, sin θc =

dζ

ds. (A.13)

Thus q term can be neglected because it does not affect the resultant force. This

is consistent with Reference [137]. Then the warping function of contour line λT

is simplified by

λT =

∮ (r − γt

xs

)ds −→ λT,s = r − γt

xs , γtxs = γt

xsφ′, (A.14)

in which γtxs denotes the shear strains due to torsion in the beam with the closed

241

cross section. In case of the open cross section, this strain may be vanished, which

is Vlasov’s assumption. The various definition of γtxs can be found in References

[136, 138]. Note that the sign of warping function is different from References

because of a different sign convention. Finally, the shear strains can be obtained

as follows:

γxs = γtxs + n 2φ′. (A.15)

Now normal strain (Equation A.1) and shear strain (Equation A.15) are ex-

pressed in the coordinate system (x, s, n). It can be achieved by substituting the

total pitch input (Equation 2.6) and elastic torsion relations (Equation 2.8) into

the strain components (Equations A.1 and A.15). Then the following linear and

nonlinear strain components are obtained.

εxx = εLxx + εNL

xx + n kLxx = εo

xx + n kxx, (A.16)

γxs = γLxs + γNL

xs + n kLxs = γo

xs + n kxs, (A.17)

in which

εLxx = u′ − yov

′′ − zow′′ − λT φ′′, (A.18)

εNLxx =

1

2(v′2 + w′2)

+ (η2 + ζ2)

θ′o(φ

′ + w′v′′) +1

2φ′2 + φ′w′v′′

+ φ(zov

′′ − yow′′), (A.19)

kLxx = w′′ cos θ − v′′ sin θ + q φ′′, (A.20)

γLxs = γt

xs φ′ , γNLxs = γt

xsw′v′′ , kL

xs = 2 φ′, (A.21)

where

yo = η cos θo − ζ sin θo , zo = η sin θo + ζ cos θo , θ = θo + θc . (A.22)

Superscripts L and NL denote linear and nonlinear terms, respectively. Nonlinear

terms have been neglected in the curvature equations, because the nonlinear cur-

vature is physically very small and higher order O(ε5). Some unusual linear terms

242

have been treated as nonlinear term. The linear strain components are the same

as those in Reference [137] except the terms related to the pitch angle θo.

A.2 Foreshortening Term

The axial deflection u is represented by two separate components; an elastic ax-

ial deflection, ue, and a kinematic axial deflection, uF , due to foreshortening or

moderately large rotation. That is,

u = ue − uF , uF =1

2

∫ x

0

(v′2 + w′2)dx, (A.23)

As mentioned in section 2.1.2, the base vector of displacements u, v, w is based

on the undeformed coordinate system, eui , which is the rectangular cartesian co-

ordinate system. In this section, some explanation for replacing u with ue will be

given.

For simplicity, let us consider the two-dimensional beam deflection problem

with moderately large rotation (see Figure A.1). In developing the strain mea-

sure, the undeformed coordinate system was used in Equation A.1. The deformed

position in terms of displacement, in the undeformed coordinate system, can be

expressed as:

r1 = (x + u)eu1 + weu

3 , ro = xeu1 , (A.24)

and the infinitesimal increment of position vector can be obtained as,

dr1 = (1 + u′)dxeu1 + w′dxeu

3 , dro = dxeu1 , (A.25)

2εij(dx)2 = dr1dr1 − drodro ≈ (2u′ + w′2)(dx)2. (A.26)

This strain measure is well known as von Karman partial nonlinearity, which

causes a foreshortening of the beam. In moderately large rotation case, the axial

displacement u is no longer just an elastic axial displacement. It includes a fore-

shortening term, and its magnitude is greater than the elastic axial displacement.

This makes the system stiffer, as the rotation increases. Eventually, a lateral de-

flection w will be restricted to certain small value. The added foreshortening term

243

w′2 reduces the strain, u′, compared to the strain in an undeformed blade. The

strain energy of the system is therefore smaller, and the system is more flexible,

which allows larger deflections.

It is possible to define the deformed position vector based on the orthogonal

curvilinear coordinate system. Then the deformed and undeformed position vector

can be written as

r1 = (r + ue) ed1 , ro = xeu

1 , (A.27)

Note that the deformed covariant base vector ed1 already includes all information

related to a deformation, and the displacement is measured along the deformed

elastic axis ed1. Another difference is that ed

1 is not a constant but a function of x.

The increment of position vector for this case can be expressed by

dr1 = (dr + due) ed1 + (r + ue) ded

1 , dro = dx eu1 , (A.28)

where the underline term represents the curvature effect of the deformed elastic

axis. The strain measure based on Equation A.28 is given by

2εij(dx)2 = dr1dr1 − drodro = (1 + u+e )2(dr)2 − (dx)2 ≈ 2u+

e (dx)2, (A.29)

where superscript + denotes ∂()∂r

. In second order approximation, ()+ becomes

equivalent to ()′.

Assuming the system undergoes moderately large deformations, the two strain

terms presented by Equations A.26 and A.29 should be equivalent and independent

of the coordinate systems, for a second order approximation. Then, the following

relationship can be obtained as

u′e = u′ +1

2w′2, (A.30)

In case of three-dimensional problem, Equation A.30 can be rewritten by

u′e = u′ +1

2(v′2 + w′2) , or u′ = u′e −

1

2(v′2 + w′2), (A.31)

As mentioned in the above, we can see that u′ already includes a foreshortening

term. Now the deformed position vector based on the undeformed coordinate

244

system can be rewritten by

r =

(x + ue − 1

2

∫ x

0

(v′2 + w′2)dx

)eu

1 + weu3 . (A.32)

Then we obtain the same strain energy as the linear strain energy, while we keep

the foreshortening term in the position vector (Equation A.32). This explicit form

of position vector does not produce an additional energy due to the foreshortening

term, which comes from a moderately large blade deformation, but affects the

kinetic energy.

A.3 Deformed Coordinate System

In this section, the strain measure is expressed in the mixed coordinate system.

The deformed coordinate ξ in edi and the undeformed coordinates y and z in eu

i

are combined together to get the simple expression.

Among the strain measure, the only underline terms of the axial strain com-

ponent εoxx is altered (see Equation A.19). Referring to the previous sections, the

strain measure in the mixed coordinate system is given by

εξξ = εoξξ + n kξξ , εo

ξξ = εLξξ + εNL

ξξ (A.33)

γξs = γoξs + n kξs , γo

ξs = γLξs + γNL

ξs (A.34)

where

εLξξ = u′e − yov

′′ − zow′′ − λT φ′′ (A.35)

εNLξξ = (η2 + ζ2)

θ′o(φ

′ + w′v′′) +1

2φ′2 + φ′w′v′′

+ φ(zov

′′ − yow′′) (A.36)

kLξξ = w′′ cos θ − v′′ sin θ + q φ′′ (A.37)

γLξs = γt

ξs φ′ (A.38)

γNLξs = γt

ξsw′v′′ (A.39)

kLξs = 2 φ′ (A.40)

245

where some approximation has been made using ordering scheme of O(ε2). Note

that the axial displacement ue is measured in the direction of ed1, while the lateral

displacements v and w are measured in the directions of eu2 and eu

3 , respectively.

In Chapter 2, the strain measure or strains represent Equations A.33-A.40,

where overbar () in η and ζ will be dropped.

Appendix BRotor System Matrices and Force

Vectors

Total system matrices and force vectors are defined as follows:

[M ] = [M ]k + [M ]a (B.1)

[C] = [C]k + [C]a (B.2)

[K] = [K]s + [K]k + [K]a (B.3)

F = Fs + Fk + Fa + Ff (B.4)

where the superscript s and k represent the contributions from strain energy and

kinetic energy of rotor blades, respectively. The superscript a denotes the con-

tribution from aerodynamic loads on the rotor. The superscript f indicates the

contribution from inertial and aerodynamic loads on trailing-edge flaps.

In this study, the aerodynamic and inertial contributions of trailing-edge flaps

are treated as an additional load vector. This is discussed in Chapter 3. In this

Appendix B, among others, the system matrices and force vectors by strain energy

and kinetic energy of rotor blades will be given . The underline terms indicate

terms kept in Reference [76], throughout this Appendix B.

247

B.1 Strain Energy of Rotor Blades

The system matrices and force vectors due to the strain energy of rotor blades are

defined in this section.

B.1.1 Stiffness Coefficients of Composite Beam

The stiffness coefficients of composite beam are defined as

K11 =

∮A11ds (B.5)

K12 =

∮(−ζA11 + B11 cos θc) ds (B.6)

K13 =

∮(−ηA11 −B11 sin θc) ds (B.7)

K14 =

∮(−λT A11 + qB11) ds (B.8)

K15 =

∮ (γt

ξsA16 + 2B16

)ds (B.9)

K22 =

∮ (ζ2A11 − 2ζB11 cos θc + D11 cos2 θc

)ds (B.10)

K23 =

∮(ζηA11 − ηB11 cos θc + ζB11 sin θc −D11 cos θc sin θc) ds (B.11)

K24 =

∮(ζλT A11 − qζB11 − λT B11 cos θc + qD11 cos θc) ds (B.12)

K25 =

∮ (−ζγtξsA16 − 2ζB16 + γt

ξsB16 cos θc + 2D16 cos θc

)ds (B.13)

K33 =

∮ (η2A11 + 2ηB11 sin θc + D11 sin2 θc

)ds (B.14)

K34 =

∮(ηλT A11 − qηB11 + λT B11 sin θc − qD11 sin θc) ds (B.15)

K35 =

∮ (−ηγtξsA16 − 2ηB16 − γt

ξsB16 sin θc − 2D16 sin θc

)ds (B.16)

K44 =

∮ (−λT γtξsA16 + qγt

ξsB16 − 2λT B16 + 2qD16

)ds (B.17)

K45 =

∮ ((γt

ξs)2A66 + 4γt

ξsB66 + 4D66

)ds (B.18)

K55 =

∮ (qγt

ξsA66 + γtξsλT A16 + 2qB66 + 2λT B16

)ds (B.19)

248

B.1.2 Stiffness Matrices and Force Vectors

The linear stiffness matrices, [K]s, are given by

[Kuu] =

∫K11 HT

u

′Hu

′ dx (B.20)

[Kuw] =

∫K13 sin θo HT

u

′Hw

′′ dx (B.21)

[Kuv] =

∫K13 cos θo HT

u

′Hv

′′ dx (B.22)

[Kup] =

∫ (K15 HT

u

′Hp

′ + Ak211 HT

u

′H ′

p

)dx (B.23)

[Kww] =

∫(K22 cos2 θo + K33 sin2 θo) HT

w

′′Hw

′′ dx (B.24)

[Kwv] =

∫(K33 −K22) cos θo sin θo HT

w

′′Hv

′′ dx (B.25)

[Kwp] =

∫ (K25 cos θo + K35 sin θo)

+ (AB1 sin θo + AB2 cos θo) θ′o HTw

′′H ′

p dx (B.26)

[Kvv] =

∫(K33 cos2 θo + K22 sin2 θo) HT

v

′′Hv

′′ dx (B.27)

[Kvp] =

∫ (K35 cos θo −K25 sin θo)

+ (AB1 cos θo − AB2 sin θo) θ′o HTv

′′H ′

p dx (B.28)

[Kpp] =

∫(K44 HT

p

′′Hp

′′ + K55 HTp

′Hp

′) dx (B.29)

and the nonlinear forces, Fs, are given by

FuNL = −∫ [

Ak211(

1

2φ′2 + θ′ov

′′w′)

+ K13(w′′φ cos θo − v′′φ sin θo) + K15v

′′w′]HT

u

′dx (B.30)

FwNL = −∫ [

(K33 − K22)(v′′φ cos 2θo + w′′φ sin 2θo) + K13 cos θou

′eφ

+ K25(v′′w′ cos θo − φφ′ sin θo) + K35(v

′′w′ sin θo + φφ′ cos θo)]HT

w

′′dx

−∫ [

K25(v′′w′′ cos θo − v′′2 sin θo) + K35(v

′′w′′ sin θo + v′′2 cos θo)

+ K55v′′φ′ + Ak2

11θ′ou′ev′′ + K15u

′ev′′]HT

w

′dx (B.31)

249

FvNL = −∫ [

(K33 − K22)(w′′φ cos 2θo − v′′φ sin 2θo)− K13 sin θou

′e φ

+ K55w′φ′ + Ak2

11θ′ou′ew

′ + K15u′ew

′

+ K25

(w′′w′ − φφ′) cos θo − 2w′v′′ sin θo

+ K35

(w′′w′ − φφ′) sin θo + 2w′v′′ cos θo

]HT

v

′′dx (B.32)

FpNL = −∫ [

K44w′v′′HT

p

′′+

K35(w

′′φ cos θo − v′′φ sin θo)

− K25(w′′φ sin θo + v′′φ cos θo) + K55w

′v′′ + Ak211u

′eφ′

HTp

′

+

(K33 − K22)(v′′w′′ cos 2θo + (w′′2 − v′′2) sin θo cos θo

)

+ K13(w′′u′e cos θo − v′′u′e sin θo) + K35(w

′′φ′ cos θo − v′′φ′ sin θo)

− K25(w′′φ′ sin θo + v′′φ′ cos θo)

HT

p

]dx (B.33)

B.2 Kinetic Energy of Rotor Blades

The system matrices and force vectors due to the kinetic energy of rotor blades

are defined in this section. Note that, derivatives with respect to the azimuth are

represented by (), instead of using ()∗, in this section.

B.2.1 Stiffness Matrix

The linear stiffness matrices, [K]k, are given by

[Kuu] =

∫−mHT

u Hu dx (B.34)

[Kuw] =

∫ (meg sin θo HT

u Hw′ + mβp HT

u Hw

)dx

+

∫meg

((θ2

o + 2θo) sin θo − θo cos θo

)HT

u Hw′ dx (B.35)

[Kuv] =

∫meg cos θo HT

u Hv′ dx

+

∫meg(θ

2o cos θo + θo sin θo) Hu

T Hv′ dx (B.36)

[Kup] =

∫2megθo cos θoH

Tu Hp dx (B.37)

[Kwu] = 0 (B.38)

250

[Kww] =

∫FA(x) HT

w

′Hw

′dx

−∫

m(k2m1 cos2 θo + k2

m2 sin2 θo) (1 + θ2o)H

Tw

′Hw

′ dx

+

∫m cos θo sin θo(k

2m2 − k2

m1) θoHTw

′Hw

′ dx (B.39)

[Kwv] = −∫

m(k2m1 cos2 θo + k2

m2 sin2 θo) θoHTw

′Hv

′ dx

+

∫m cos θo sin θo(k

2m1 − k2

m2) (1 + θ2o)H

Tw

′Hv

′ dx (B.40)

[Kwp] = −∫

meg

(θ2

o + θo

)HT

wHp dx

+

∫m

(egx cos θo + 4θo sin θo cos θo(k

2m1 − k2

m2))

HTw

′Hp dx (B.41)

[Kvu] = 0 (B.42)

[Kvw] = −∫

2megθo cos θoHTv H ′

w dx

+

∫(k2

m1 sin2 θo + k2m2 cos2 θo)θoH

Tv

′H ′

w dx

+

∫cos θo sin θo(k

2m1 − k2

m2)(1 + θ2o)H

Tv

′H ′

w dx (B.43)

[Kvv] =

∫FA(x) HT

v

′Hv

′dx

−∫ [

mHTv Hv + 2megθo sin θoH

Tv Hv

′]dx

−∫

(k2m1 sin2 θo + k2

m2 cos2 θo)(1 + θ2o)H

Tv

′H ′

v dx

+

∫cos θo sin θo(k

2m1 − k2

m2)θoHTv

′H ′

v dx (B.44)

[Kvp] =

∫meg

(sin θo + θ2

o sin θo − θo cos θo

)HT

v Hp dx

+

∫meg

(2θo(k

2m1 − k2

m2) cos 2θo−x sin θo

)HT

v

′Hp dx (B.45)

[Kpu] = 0 (B.46)

[Kpw] =

∫m

(xeg cos θo + sin 2θo(k

2m2 − k2

m1)θo

)HT

p Hw′ dx (B.47)

[Kpv] =

∫meg

(sin θoH

Tp Hv − x sin θoH

Tp Hv

′)

dx (B.48)

[Kpp] =

∫m(k2

m2 − k2m1) cos 2θoH

Tp Hp dx (B.49)

251

where FA(x) denotes the centrifugal force and has been defined as

FA(x) =

∫ 1

x

mξdx (B.50)

B.2.2 Mass Matrix

The linear mass matrices, [M ]k, are given by

[Muu] =

∫mLT

u Ludx (B.51)

[Muw] = −∫

meg sin θoLT

uLw′dx (B.52)

[Muv] = −∫

meg cos θoLT

uLv′dx (B.53)

[Mup] = 0 (B.54)

[Mwu] = 0 (B.55)

[Mww] =

∫ (mLT

wLw + m(k2m1 cos2 θo + k2

m2 sin2 θo)HTw

′Hw

′)

dx (B.56)

[Mwv] =

∫m(k2

m2 − k2m1) sin θo cos θoH

Tw

′Hv

′dx (B.57)

[Mwp] =

∫meg cos θoH

TwHpdx (B.58)

[Mvu] = 0 (B.59)

[Mvw] =

∫m(k2

m2 − k2m1) sin θo cos θoH

Tv

′Hw

′dx (B.60)

[Mvv] =

∫ (mHT

v Hv + m(k2m1 sin2 θo + k2

m2 cos2 θo)HTv

′Hv

′)

dx (B.61)

[Mvp] =

∫−meg sin θoH

Tv Hpdx (B.62)

[Mpu] = 0 (B.63)

[Mpw] =

∫meg cos θoH

Tp Hwdx (B.64)

[Mpv] = −∫

meg sin θoHTp Hvdx (B.65)

[Mpp] =

∫mk2

mHTp Hpdx (B.66)

252

B.2.3 Damping Matrix

The linear mass matrices, [C]k, are given by

[Cuu] = 0 (B.67)

[Cuw] = −∫

2megθo cos θoHTu Hw

′dx (B.68)

[Cuv] =

∫ (−2mHT

u Hv + 2megθo sin θoHTu Hv

′)

dx (B.69)

[Cup] =

∫2meg sin θoH

Tu Hpdx (B.70)

[Cwu] = 0 (B.71)

[Cww] =

∫mθo(k

2m2 − k2

m1) sin 2θoHTw

′Hw

′dx (B.72)

[Cwv] =

∫ 2mβpH

TwHv + 2meg sin θoH

Tw

′Hv

− 2mθo(k2m1 cos2 θo + k2

m2 sin2 θo)HTw

′Hv

′

dx (B.73)

[Cwp] =

∫ −2megθo sin θoH

TwHp

+ 2m(k2m1 cos2 θo + k2

m2 sin2 θo)HTw

′Hv

′

dx (B.74)

[Cvu] =

∫2mHT

v Hudx (B.75)

[Cvw] =

∫ −2mβpH

Tv Hw − 2meg sin θoH

Tv Hw

′

+ 2mθo(k2m1 sin2 θo + k2

m2 cos2 θo)HTv

′Hw

′

dx (B.76)

[Cvv] =

∫ −2meg cos θoH

Tv Hv

′ + 2meg cos θoHTv

′Hv

+ 2mθo(k2m1 − k2

m2) sin θo cos θoHTv

′Hv

′

dx (B.77)

[Cvp] =

∫ −2megθo cos θoH

Tv Hp

+ m sin 2θo(k2m1 − k2

m2)HTv

′Hp

dx (B.78)

[Cpu] = 0 (B.79)

[Cpw] =

∫2m(k2

m1 cos2 θo + k2m2 sin2 θo)H

Tp Hw

′dx (B.80)

[Cpv] =

∫ 2megβp cos θoH

Tp Hv

253

− 2m(k2m1 − k2

m2) sin θo cos θoHTp Hv

′ dx (B.81)

[Cpp] = 0 (B.82)

B.2.4 Force Vectors

The constant force vectors, Fk, are defined as

Fk = Fko + Fk

NL (B.83)

where the subscript o denotes the constant terms and NL denotes the nonlinear

terms, and

Fuo =

∫m

x− eg(βp + 2θ) sin θo

HT

u dx (B.84)

Fwo =

∫m

−xβp − egθo cos θo + eg sin θo(θ

2o + 2θoβp)

HT

wdx

+

∫m

−x sin θo

+ (k2m1 cosθ

o +k2m2 sin2 θo)(βp + 2θo)

HT

w

′dx (B.85)

Fvo =

∫m

eg cos θo(1 + 2θoβp + θ2

o) + egθo sin θo

HT

v dx

+

∫m

−egx cos θo + cos θo sin θo(βp + 2θo)

HT

v

′dx (B.86)

Fpo =

∫m

(k2

m1 − k2m2) sin θo cos θo

− k2mθo − egxβp cos θo

HT

p dx (B.87)

and the nonlinear force vector terms are given as follows:

FuNL =

∫m

(∫ x

0

(v′v′ + v′v′ + w′w′ + w′w′) dξ

)HT

u dx (B.88)

FwNL = −∫ (

2w′∫ 1

x

mvdξ

)HT

w

′dx (B.89)

FvNL =

∫ (2m

∫ x

0

(v′v′ + w′w′)dξ

)HT

v dx

−∫ (

2v′∫ 1

x

mvdξ

)HT

v

′dx (B.90)

Appendix CFormulations using Mathematica

In this Appendix, the Mathematica 4.0 programs used to derive the rotor strain

energy, rotor kinetic energy, rotor quasi-aerodynamic loads, and trailing-edge flaps’

inertial loads are presented. Some functions, which are modified based on the func-

tions developed by Milgram [69], are provided to implement the ordering scheme.

The first three sections describe the strain energy, kinetic energy, quasi-steady

aerodynamic contributions of rotor blade based on the ordering scheme presented

in Table 2.2. In the fourth section, the trailing-edge flaps’ inertial contributions

are derived using the ordering schemes presented in Tables 2.2 and 3.1.

255

C.1 Rotor Strain Energy

256

257

258

259

C.2 Rotor Kinetic Energy

260

261

262

263

264

265

266

C.3 Rotor Quasi-steady Aerodynamic Loads

267

268

269

270

271

272

273

C.4 Trailing-Edge Flap’s Inertial Loads

274

275

276

277

278

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Vita

Jun-Sik Kim

EducationThe Pennsylvania State University University Park, PA 2000–2005

Ph.D. in Aerospace EngineeringArea of Specialization: Rotorcraft Dynamics and Smart Structures

Inha University Inchon, Korea 1994–1996M.S. in Aerospace EngineeringArea of Specialization: Solid Mechanics and Finite Element Method

Inha University Inchon, Korea 1990–1994

B.S. in Aerospace Engineering

Awards and HonorsPSU Rotorcraft Center Fellowship 2000–2001

Research ExperienceResearch Assistant The Pennsylvania State University 2001–2005

Research Project sponsored by NRTC

Research Assistant Inha University 1996, 1999

Teaching ExperiencePart-time Instructor Inha Technical College, Inchon, Korea 1999–2000

“The strength of materials” in undergraduate level.

Selected Publications• J.-S. Kim, K. W. Wang, and E. C. Smith, “Design and Analysis of Piezoelec-

tric Transducer Based Resonant Actuation Systems,” Proceedings of the ASMEIMECE, Orlando, FL, Nov 5–11, 2005.

• J.-S. Kim, K. W. Wang, and E. C. Smith, “High Authority PiezoelectricSystem Synthesis through Mechanical Resonance and Electrical Tailoring,” Journalof Intelligent Material Systems and Structures, Vol. 16, No. 1, 2005, pp. 21–32.

• J.-S. Kim, E. C. Smith, and K. W. Wang, “Helicopter Vibration Suppressionvia Multiple Trailing Edge Flaps Controlled by Resonance Actuation System,”Proceedings of the 60th AHS Forum, Bltimore, Maryland, June 8–10, 2004.

• J.-S. Kim, “Reconstruction of First-order Shear Deformation Theory for Lam-inated and Sandwich Shells,” AIAA Journal, Vol. 41, No. 5, 2004, pp. 1685–1697.