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AEROELASTIC RESPONSE AND
STABILITY OF TILTROTORS WITH
ELAS TICALLY- C O UP LED
COMPOSITE ROTOR BLADES
Mark W. Nixon
Dissertation submitted to the Faculty of the Graduate School
of The University of Maryland in partial fulf_llme,t
of the requirements for the degree of
Doctor of Philosophy
1993
Advisory Committee:
Professor Inderjit Chopra, Advisor
Associate Professor Roberto Celi
Associate Professor Anthony J. Vizzini
Associate Professor J. Gordon Leishman
Professor David Schelling (Dean's Representative)
(NA SA- T_- IL)']7_ _ ) AEROELASTIC N94-24953
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TILTR_IT_!_S WITH _LASTICALLY-COUPLEC)
_- )MPJS_T_: ;,OTn_ FJLAD_S Ph.D. Thesis Unclas
('!_rfl,nd Univ.) G4a p
G3/O5 0201587
Acknowledgements
I am forever indebted to my advisor, Dr. Inderjit Chopra, not only for his
guidance through my doctoral studies and research, but for his investment in me as
an undergraduate and our fellowship since that time. I thank the other members of
my dissertation committee, Dr. Roberto Celi, Dr. Anthony Vizzini, Dr. J. Gordon
Leishman, and Dr. David Schelling for their generous support of this research, and
I thank the rest of the faculty and staff of the University of Maryland who helped
make my experience memorable, especially Professor Alfred Gessow and Dr. Gunjit
Bir. I have fond recollections of the graduate students of the Rotorcraft Center
with whom I shared this experience- Carl, Dave, Wes, Ranjan, Swami, Steve, Joe,
Wes, Gil, Fred, Anita, and especially Ed Smith and Anne Marie Spence; thanks
for the help when I needed it.
I also appreciate the support of my supervisors and colleagues from Langley
Research Center. Foremost, I acknowledge and thank my NASA research advisor,
Dr. Raymond Kvatnernik, whose expert guidance made this research possible. I
have many supervisors to thank, all of whom steadfastly encouraged my doctoral
pursuits- Dr. Wolf Elber, Dan Hoad, Dr. Irving Abel, Dr. Thomas Noll, Boyd Perry,
Dr. Jaroslaw Sobieski, Dr. Howard Adelman, Robert Doggett, Rodney Ricketts,
and especially Dr. Felton Bartlett. The support and understanding of my coworkers
is gratefully acknowledged, especially my allies-in-reorganization, Howard Hinnant
and Renee Lake, whose help enabled me to complete this research in a timely
manner.
Lastly, I offer thanks to my family whose love and encouragement made this
experience worthwhile- my parents, Nancy and Bill; my parents-in-law, Paul and
Maria; my sisters, Valerie and Leanne, and their families; and most of all my wife,
Sandra, and our children, Amanda, Caitlin, and Eric.
Abstract
Title of Dissertation: AEROELASTIC RESPONSE AND
STABILITY OF TILTROTORS WITH
ELASTICALLY-COUPLED COMPOSITE
ROTOR BLADES
Mark W. Nixon, Doctor of Philosophy, 1993
Dissertation directed by: Dr. Inderjit Chopra, Professor
Department of Aerospace Engineering
There is a potential for improving the performance and aeroelastic stability
of tiltrotors through the use of elastically-coupled composite rotor blades. To
study the characteristics of tiltrotors with these types of rotor blades it is nec-
essary to formulate a new analysis which has the capabilities of modeling both a
tiltrotor configuration and an anisotropic rotor blade. Background for these formu-
lations is established in two preliminary investigations. In the first, the influence
of several system design parameters on tiltrotor aeroelastic stability is examined
for the high-speed axial flight mode using a newly-developed rigid-blade analysis
with an elastic wing finite element model. The second preliminary investigation
addresses the accuracy of using a one-dimensional beam analysis to predict fre-
quencies of elastically-coupled highly-twisted rotor blades. Important aspects of
the new aeroelastic formulations are the inclusion of a large steady pylon angle
which controls tilt of the rotor system with respect to the airflow, the inclusion of
elastic pitch-lag coupling terms related to rotor precone,the inclusion of hub-
related degrees of freedom which enable modeling of a gimballed rotor system and
engine drive-train dynamics, and additional elastic coupling terms which enable
modeling of the anisotropic features for both the rotor blades and the tiltrotor
wing. Accuracy of the new tiltrotor analysis is demonstrated by a comparison of
the results produced for a baseline case with analytical and experimental results
reported in the open literature. Two investigations of elastically tailored blades
on a baseline tiltrotor are then conducted. One investigation shows that elastic
bending-twist coupling of the rotor blade is a very effective means for increasing
the flutter velocity of a tiltrotor, and the magnitude of coupling required does not
have an adverse effect on performance or blade loads. The second investigation
shows that passive blade twist control via elastic extension-twist coupling of the
rotor blade has the capability of significantly improving tiltrotor aerodynamic per-
formance. This concept, however, is shown to have, in general, a negative impact
on stability characteristics.
Contents
1 Introduction
1.1 Problem Statement ...........................
1.2 Background and Motivation ......................
1.2.1 Advantages of the Tiltrotor Configuration ..........
1.2.2 Important Considerations in Tiltrotor Design ........
1.2.3 History of Tiltrotor Development ............... 10
1.2.4 Elastic Tailoring of Composite Rotor Blades ......... 14
Survey of Tiltrotor Aeroelastic Research ............... 20
Survey of Anisotropic Blade Modeling ................. 29
1.4.1 Important Considerations in Rotor Blade Analysis ...... 29
1.4.2 General Anisotropic Beams .................. 32
1.4.3 Anisotropic Beam Modeling for Rotor Blades ........ 33
1.5 Scope of the Present Research ..................... 36
1.5.1 Fundamental Study of Tiltrotor Stability ........... 37
1.5.2 Dynamic Analysis of Elastically-Coupled Blades ....... 38
1.5.3 Tiltrotor Aeroelastic Theory Development .......... 39
1.5.4 Comparison Studies ....................... 40
2 Fundamental Study of Tiltrotor Whirl Flutter 43
2.1 Description of the Math Model .................... 44
2.2 Frequency and Damping Characteristics of the Baseline System . . 45
111
2.3 Rotor Frequency ............................ 46
2.4 Wing Frequency ............................. 52
2.5 Forward Swept Wing .......................... 54
2.6 Pitch-Flap Coupling .......................... 56
2.7 Summary ................................ 58
3 Dynamic Analysis of Pretwisted Elastically-Coupled Rotor Blades 80
3.1 Energy Formulations .......................... 82
3.1.1 Geometry and Coordinates ................... 83
3.1.2 Strain Energy Derivation .................... 84
3.1.3 Kinetic Energy Derivation ................... 96
3.2 Implementation ............................. 100
3.2.1 Finite Element Discretization ................. 101
3.3 Analysis Application .......................... 105
3.3.1 Analysis Verification ...................... 106
3.3.2 Warping Influences on the Anti-Symmetric Beam ...... 10S
3.3.3 Warping Influences on the Symmetric Beam ......... 111
3.3.4 Influence of Large Pretwist on Nonclassical Effects ...... 112
3.3.5 Convergence Study ....................... 113
3.4 Summary ................................ ll4
4 Structural Modeling of a Tiltrotor with Anisotropic Blades 130
4.1 The Tiltrotor Model .......................... 131
4.2 Frames of Reference ........................... 132
4.3 Nondimensionalization and Ordering Scheme ............. 138
4.4 Formulation Using Hamilton's Principle ................ 139
4.4.1 Formulation of Elastic Strain Energy ............. 140
4.4.2 Formulation of Kinetic Energy ................. 146
iv
4.5 Structural Contributions to Mass, Damping, and Stiffness ...... 161
4.5.1 Blade Matrices ......................... 163
4.5.2 Hub Matrices .......................... 168
4.5.3 Wing Matrices ......................... 172
5 Aerodynamic Modeling 180
5.1 Derivation of Local Rotor Blade Velocities .............. 181
5.2 Quasi-Steady Airloads ......................... 191
5.2.1 Reverse Flow .......................... 193
5.2.2 Mach Number Perturbations .................. 193
5.2.3 Noncirculatory Airloads .................... 195
5.3 Finite Element Discretization of Work ................ 196
5.3.1 Derivation of the TEL Matrix ................. 198
5.3.2 Discretization of the Blade Equations ............. 204
5.3.3 Discretization of the Hub Equations .............. 207
5.3.4 Nonlinear Force Contributions ................. 209
5.4 Wing Aerodynamics .......................... 212
6 Vehicle Trim and Blade Response Analysis 216
6.1 Vehicle Trim Equations ......................... 217
6.1.1 Free-Flight Trim ........................ 217
6.1.2 Wind Tunnel Trim ....................... 220
6.1.3 Axisymmetric Trim ....................... 221
6.2 Blade Response Equations ....................... 222
6.2.1 Finite Element in Time ..................... 223
6.3 Coupled Trim Procedure ........................ 226
6.3.1 Initial Controls Estimate .................... 227
6.3.2 Blade Steady Periodic Response ................ 228
V
6.3.3 Computation of Blade and Hub Loads ............ 228
6.3.4 Inflow Update .......................... 230
6.3.5 Computation of Jacobian and Controls Update ....... 231
6.3.6 Converged Blade Response and Vehicle Trim ......... 233
6.3.7 Large Twist Deformations ................... 233
7 Stability Analysis 241
7.1 Assembly of the System Equations .................. 242
7.1.1 Element Integration ....................... 246
7.1.2 Assembly of the Element Matrices ............... 247
7.1.3 Normal Mode Transformation ................. 251
7.,1.4 Addition of the Wing Equations ................ 253
7.1.5 Engine Drive Train Dynamics ................. 255
7.2 Stability Analysis Procedure ...................... 256
7.2.1 Fixed Coordinate Transformation . .............. 257
7.2.2 Floquet Theory ......................... 261
7.2.3 Constant Coefficient Approximation .............. 263
8 Results and Discussion 265
8.1 Baseline Design ............................. 266
8.2 Validation ................................ 266
8.2.1 Blade Frequencies ........................ 267
8.2.2 Stability in High-Speed Axial Flight .............. 269
8.2.3 Stability in Helicopter, Conversion, and Airplane Modes . . 271
8.2.4 Performance ........................... 272
8.2.5 Free-Flight Trim and Blade Loads ............... 273
8.2.6 Summary of Validation Results ................ 275
8.3 Bending-Twist-Coupled Rotor Blade ................. 275
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